Calculus Illustrated. Volume 3: Integral Calculus [3, 1 ed.] 9798657180145

This is the third volume of the series Calculus Illustrated, a textbook for undergraduate students.Mathematical thinking

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Table of contents :
Preface
Integration
What we can do with integral calculus
Several variables
Series
Exercises
Index
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Calculus Illustrated. Volume 3: Integral Calculus [3, 1 ed.] 9798657180145

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✷

❚♦ t❤❡ st✉❞❡♥t✳✳✳

❚♦ t❤❡ st✉❞❡♥t ▼❛t❤❡♠❛t✐❝s ✐s ❛ s❝✐❡♥❝❡✳ ❏✉st ❛s t❤❡ r❡st ♦❢ t❤❡ s❝✐❡♥t✐sts✱ ♠❛t❤❡♠❛t✐❝✐❛♥s ❛r❡ tr②✐♥❣ t♦ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤❡ ❯♥✐✈❡rs❡ ♦♣❡r❛t❡s ❛♥❞ ❞✐s❝♦✈❡r ✐ts ❧❛✇s✳

❲❤❡♥ s✉❝❝❡ss❢✉❧✱ t❤❡② ✇r✐t❡ t❤❡s❡ ❧❛✇s ❛s s❤♦rt st❛t❡♠❡♥ts

❝❛❧❧❡❞ ✏t❤❡♦r❡♠s✑✳ ■♥ ♦r❞❡r t♦ ♣r❡s❡♥t t❤❡s❡ ❧❛✇s ❝♦♥❝❧✉s✐✈❡❧② ❛♥❞ ♣r❡❝✐s❡❧②✱ ❛ ❞✐❝t✐♦♥❛r② ♦❢ t❤❡ ♥❡✇ ❝♦♥❝❡♣ts ✐s ❛❧s♦ ❞❡✈❡❧♦♣❡❞❀ ✐ts ❡♥tr✐❡s ❛r❡ ❝❛❧❧❡❞ ✏❞❡✜♥✐t✐♦♥s✑✳ ❚❤❡s❡ t✇♦ ♠❛❦❡ ✉♣ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ♣❛rt ♦❢ ❛♥② ♠❛t❤❡♠❛t✐❝s ❜♦♦❦✳ ❚❤✐s ✐s ❤♦✇ ❞❡✜♥✐t✐♦♥s✱ t❤❡♦r❡♠s✱ ❛♥❞ s♦♠❡ ♦t❤❡r ✐t❡♠s ❛r❡ ✉s❡❞ ❛s ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ♦❢ t❤❡ s❝✐❡♥t✐✜❝ t❤❡♦r② ✇❡ ♣r❡s❡♥t ✐♥ t❤✐s t❡①t✳ ❊✈❡r② ♥❡✇ ❝♦♥❝❡♣t ✐s ✐♥tr♦❞✉❝❡❞ ✇✐t❤ ✉t♠♦st s♣❡❝✐✜❝✐t②✳

❉❡✜♥✐t✐♦♥ ✵✳✵✳✶✿ sq✉❛r❡ r♦♦t ❙✉♣♣♦s❡

x✱

a

✐s ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ❚❤❡♥ t❤❡ sq✉❛r❡ r♦♦t ♦❢ x2 = a✳

a

✐s ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r

s✉❝❤ t❤❛t

❚❤❡ t❡r♠ ❜❡✐♥❣ ✐♥tr♦❞✉❝❡❞ ✐s ❣✐✈❡♥ ✐♥ ✐t❛❧✐❝s✳ ❚❤❡ ❞❡✜♥✐t✐♦♥s ❛r❡ t❤❡♥ ❝♦♥st❛♥t❧② r❡❢❡rr❡❞ t♦ t❤r♦✉❣❤♦✉t t❤❡ t❡①t✳ ◆❡✇ s②♠❜♦❧✐s♠ ♠❛② ❛❧s♦ ❜❡ ✐♥tr♦❞✉❝❡❞✳

❙q✉❛r❡ r♦♦t √

a

❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ♥♦t❛t✐♦♥ ✐s ❢r❡❡❧② ✉s❡❞ t❤r♦✉❣❤♦✉t t❤❡ t❡①t✳ ❲❡ ♠❛② ❝♦♥s✐❞❡r ❛ s♣❡❝✐✜❝ ✐♥st❛♥❝❡ ♦❢ ❛ ♥❡✇ ❝♦♥❝❡♣t ❡✐t❤❡r ❜❡❢♦r❡ ♦r ❛❢t❡r ✐t ✐s ❡①♣❧✐❝✐t❧② ❞❡✜♥❡❞✳

❊①❛♠♣❧❡ ✵✳✵✳✷✿ ❧❡♥❣t❤ ♦❢ ❞✐❛❣♦♥❛❧ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ ❛

1 × 1 sq✉❛r❡❄ ❚❤❡ sq✉❛r❡ ✐s ♠❛❞❡ ♦❢ t✇♦ r✐❣❤t tr✐❛♥❣❧❡s ❛♥❞ t❤❡ a✳ ❚❤❡♥✱ ❜② t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ✱ t❤❡ sq✉❛r❡ ♦❢

❞✐❛❣♦♥❛❧ ✐s t❤❡✐r s❤❛r❡❞ ❤②♣♦t❡♥✉s❡✳ ▲❡t✬s ❝❛❧❧ ✐t a ✐s 12 + 12 = 2✳ ❈♦♥s❡q✉❡♥t❧②✱ ✇❡ ❤❛✈❡✿

a2 = 2 . ❲❡ ✐♠♠❡❞✐❛t❡❧② s❡❡ t❤❡ ♥❡❡❞ ❢♦r t❤❡ sq✉❛r❡ r♦♦t✦ ❚❤❡ ❧❡♥❣t❤ ✐s✱ t❤❡r❡❢♦r❡✱

a=

√

2✳

❨♦✉ ❝❛♥ s❦✐♣ s♦♠❡ ♦❢ t❤❡ ❡①❛♠♣❧❡s ✇✐t❤♦✉t ✈✐♦❧❛t✐♥❣ t❤❡ ✢♦✇ ♦❢ ✐❞❡❛s✱ ❛t ②♦✉r ♦✇♥ r✐s❦✳ ❆❧❧ ♥❡✇ ♠❛t❡r✐❛❧ ✐s ❢♦❧❧♦✇❡❞ ❜② ❛ ❢❡✇ ❧✐tt❧❡ t❛s❦s✱ ♦r q✉❡st✐♦♥s✱ ❧✐❦❡ t❤✐s✳

❊①❡r❝✐s❡ ✵✳✵✳✸ ❋✐♥❞ t❤❡ ❤❡✐❣❤t ♦❢ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s✐❞❡ ♦❢ ✇❤✐❝❤ ✐s

1✳

❚❤❡ ❡①❡r❝✐s❡s ❛r❡ t♦ ❜❡ ❛tt❡♠♣t❡❞ ✭♦r ❛t ❧❡❛st ❝♦♥s✐❞❡r❡❞✮ ✐♠♠❡❞✐❛t❡❧②✳ ▼♦st ♦❢ t❤❡ ✐♥✲t❡①t ❡①❡r❝✐s❡s ❛r❡ ♥♦t ❡❧❛❜♦r❛t❡✳

❚❤❡② ❛r❡♥✬t✱ ❤♦✇❡✈❡r✱ ❡♥t✐r❡❧② r♦✉t✐♥❡ ❛s t❤❡② r❡q✉✐r❡

✉♥❞❡rst❛♥❞✐♥❣ ♦❢✱ ❛t ❧❡❛st✱ t❤❡ ❝♦♥❝❡♣ts t❤❛t ❤❛✈❡ ❥✉st ❜❡❡♥ ✐♥tr♦❞✉❝❡❞✳ ❆❞❞✐t✐♦♥❛❧ ❡①❡r❝✐s❡ s❡ts ❛r❡ ♣❧❛❝❡❞ ✐♥ t❤❡ ❛♣♣❡♥❞✐① ❛s ✇❡❧❧ ❛s ❛t t❤❡ ❜♦♦❦✬s ✇❡❜s✐t❡✿ ❝❛❧❝✉❧✉s✶✷✸✳❝♦♠✳ ❉♦ ♥♦t st❛rt ②♦✉r st✉❞② ✇✐t❤ t❤❡ ❡①❡r❝✐s❡s✦ ❑❡❡♣ ✐♥ ♠✐♥❞ t❤❛t t❤❡ ❡①❡r❝✐s❡s ❛r❡ ♠❡❛♥t t♦ t❡st ✕ ✐♥❞✐r❡❝t❧② ❛♥❞ ✐♠♣❡r❢❡❝t❧② ✕ ❤♦✇ ✇❡❧❧ t❤❡ ❝♦♥❝❡♣ts ❤❛✈❡ ❜❡❡♥ ❧❡❛r♥❡❞✳ ❚❤❡r❡ ❛r❡ s♦♠❡t✐♠❡s ✇♦r❞s ♦❢ ❝❛✉t✐♦♥ ❛❜♦✉t ❝♦♠♠♦♥ ♠✐st❛❦❡s ♠❛❞❡ ❜② t❤❡ st✉❞❡♥ts✳

❚♦ t❤❡ st✉❞❡♥t✳✳✳

✸

❲❛r♥✐♥❣✦ 2 √ (−1) = 1✱ 1✱ 1 = 1✳

■♥ s♣✐t❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ♦♥❡ sq✉❛r❡ r♦♦t ♦❢

t❤❡r❡ ✐s ♦♥❧②

❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❢❛❝ts ❛❜♦✉t t❤❡ ♥❡✇ ❝♦♥❝❡♣ts ❛r❡ ♣✉t ❢♦r✇❛r❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛♥♥❡r✳

❚❤❡♦r❡♠ ✵✳✵✳✹✿ Pr♦❞✉❝t ♦❢ ❘♦♦ts ❋♦r ❛♥② t✇♦ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs

a

b✱

❛♥❞

√

a·

√

✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t②✿

b=

√

a·b

❚❤❡ t❤❡♦r❡♠s ❛r❡ ❝♦♥st❛♥t❧② r❡❢❡rr❡❞ t♦ t❤r♦✉❣❤♦✉t t❤❡ t❡①t✳ ❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡♦r❡♠s ♠❛② ❝♦♥t❛✐♥ ❢♦r♠✉❧❛s❀ ❛ t❤❡♦r❡♠ s✉♣♣❧✐❡s ❧✐♠✐t❛t✐♦♥s ♦♥ t❤❡ ❛♣♣❧✐❝❛❜✐❧✐t② ♦❢ t❤❡ ❢♦r♠✉❧❛ ✐t ❝♦♥t❛✐♥s✳

❋✉rt❤❡r♠♦r❡✱ ❡✈❡r② ❢♦r♠✉❧❛ ✐s ❛ ♣❛rt ♦❢ ❛ t❤❡♦r❡♠✱ ❛♥❞ ✉s✐♥❣ t❤❡ ❢♦r♠❡r ✇✐t❤♦✉t

❦♥♦✇✐♥❣ t❤❡ ❧❛tt❡r ✐s ♣❡r✐❧♦✉s✳ ❚❤❡r❡ ✐s ♥♦ ♥❡❡❞ t♦ ♠❡♠♦r✐③❡ ❞❡✜♥✐t✐♦♥s ♦r t❤❡♦r❡♠s ✭❛♥❞ ❢♦r♠✉❧❛s✮✱ ✐♥✐t✐❛❧❧②✳ ❲✐t❤ ❡♥♦✉❣❤ t✐♠❡ s♣❡♥t ✇✐t❤ t❤❡ ♠❛t❡r✐❛❧✱ t❤❡ ♠❛✐♥ ♦♥❡s ✇✐❧❧ ❡✈❡♥t✉❛❧❧② ❜❡❝♦♠❡ ❢❛♠✐❧✐❛r ❛s t❤❡② ❝♦♥t✐♥✉❡ t♦ r❡❛♣♣❡❛r ✐♥ t❤❡ t❡①t✳ ❲❛t❝❤ ❢♦r ✇♦r❞s ✏✐♠♣♦rt❛♥t✑✱ ✏❝r✉❝✐❛❧✑✱ ❡t❝✳ ❚❤♦s❡ ♥❡✇ ❝♦♥❝❡♣ts t❤❛t ❞♦ ♥♦t r❡❛♣♣❡❛r ✐♥ t❤✐s t❡①t ❛r❡ ❧✐❦❡❧② t♦ ❜❡ s❡❡♥ ✐♥ t❤❡ ♥❡①t ♠❛t❤❡♠❛t✐❝s ❜♦♦❦ t❤❛t ②♦✉ r❡❛❞✳ ❨♦✉ ♥❡❡❞ t♦✱ ❤♦✇❡✈❡r✱ ❜❡ ❛✇❛r❡ ♦❢ ❛❧❧ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥s ❛♥❞ t❤❡♦r❡♠s ❛♥❞ ❜❡ ❛❜❧❡ t♦ ✜♥❞ t❤❡ r✐❣❤t ♦♥❡ ✇❤❡♥ ♥❡❝❡ss❛r②✳ ❖❢t❡♥✱ ❜✉t ♥♦t ❛❧✇❛②s✱ ❛ t❤❡♦r❡♠ ✐s ❢♦❧❧♦✇❡❞ ❜② ❛ t❤♦r♦✉❣❤ ❛r❣✉♠❡♥t ❛s ❛ ❥✉st✐✜❝❛t✐♦♥✳

Pr♦♦❢✳ ❙✉♣♣♦s❡

A=

√

a

❛♥❞

B=

√

b✳

❚❤❡♥✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

a = A2

❛♥❞

b = B2 .

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

❍❡♥❝❡✱

√

a · b = A2 · B 2 = A · A · B · B = (A · B) · (A · B) = (AB)2 . ab = A · B ✱

❛❣❛✐♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥✳

❙♦♠❡ ♣r♦♦❢s ❝❛♥ ❜❡ s❦✐♣♣❡❞ ❛t ✜rst r❡❛❞✐♥❣✳ ■ts ❤✐❣❤❧② ❞❡t❛✐❧❡❞ ❡①♣♦s✐t✐♦♥ ♠❛❦❡s t❤❡ ❜♦♦❦ ❛ ❣♦♦❞ ❝❤♦✐❝❡ ❢♦r s❡❧❢✲st✉❞②✳ ■❢ t❤✐s ✐s ②♦✉r ❝❛s❡✱ t❤❡s❡ ❛r❡ ♠② s✉❣❣❡st✐♦♥s✳ ❲❤✐❧❡ r❡❛❞✐♥❣ t❤❡ ❜♦♦❦✱ tr② t♦ ♠❛❦❡ s✉r❡ t❤❛t ②♦✉ ✉♥❞❡rst❛♥❞ ♥❡✇ ❝♦♥❝❡♣ts ❛♥❞ ✐❞❡❛s✳ ❤♦✇❡✈❡r✱ t❤❛t s♦♠❡ ❛r❡ ♠♦r❡ ✐♠♣♦rt❛♥t t❤❛t ♦t❤❡rs❀ t❤❡② ❛r❡ ♠❛r❦❡❞ ❛❝❝♦r❞✐♥❣❧②✳

❑❡❡♣ ✐♥ ♠✐♥❞✱

❈♦♠❡ ❜❛❝❦ ✭♦r ❥✉♠♣

❢♦r✇❛r❞✮ ❛s ♥❡❡❞❡❞✳ ❈♦♥t❡♠♣❧❛t❡✳ ❋✐♥❞ ♦t❤❡r s♦✉r❝❡s ✐❢ ♥❡❝❡ss❛r②✳ ❨♦✉ s❤♦✉❧❞ ♥♦t t✉r♥ t♦ t❤❡ ❡①❡r❝✐s❡ s❡ts ✉♥t✐❧ ②♦✉ ❤❛✈❡ ❜❡❝♦♠❡ ❝♦♠❢♦rt❛❜❧❡ ✇✐t❤ t❤❡ ♠❛t❡r✐❛❧✳ ❲❤❛t t♦ ❞♦ ❛❜♦✉t ❡①❡r❝✐s❡s ✇❤❡♥ s♦❧✉t✐♦♥s ❛r❡♥✬t ♣r♦✈✐❞❡❞❄ ❋✐rst✱ ✉s❡ t❤❡ ❡①❛♠♣❧❡s✳ ▼❛♥② ♦❢ t❤❡♠ ❝♦♥t❛✐♥ ❛ ♣r♦❜❧❡♠ ✕ ✇✐t❤ ❛ s♦❧✉t✐♦♥✳ ❚r② t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ✕ ❜❡❢♦r❡ ♦r ❛❢t❡r r❡❛❞✐♥❣ t❤❡ s♦❧✉t✐♦♥✳ ❨♦✉ ❝❛♥ ❛❧s♦ ✜♥❞ ❡①❡r❝✐s❡s ♦♥❧✐♥❡ ♦r ♠❛❦❡ ✉♣ ②♦✉r ♦✇♥ ♣r♦❜❧❡♠s ❛♥❞ s♦❧✈❡ t❤❡♠✦ ■ str♦♥❣❧② s✉❣❣❡st t❤❛t ②♦✉r s♦❧✉t✐♦♥ s❤♦✉❧❞ ❜❡ t❤♦r♦✉❣❤❧② ✇r✐tt❡♥✳ ❨♦✉ s❤♦✉❧❞ ✇r✐t❡ ✐♥ ❝♦♠♣❧❡t❡ s❡♥t❡♥❝❡s✱ ✐♥❝❧✉❞✐♥❣ ❛❧❧ t❤❡ ❛❧❣❡❜r❛✳ ❋♦r ❡①❛♠♣❧❡✱ ②♦✉ s❤♦✉❧❞ ❛♣♣r❡❝✐❛t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦✿

❲r♦♥❣✿

1+1 2

❘✐❣❤t✿

1+1 =2

✹

❚♦ t❤❡ st✉❞❡♥t✳✳✳

❚❤❡ ❧❛tt❡r r❡❛❞s ✏♦♥❡ ❛❞❞❡❞ t♦ ♦♥❡ ✐s t✇♦✑✱ ✇❤✐❧❡ t❤❡ ❢♦r♠❡r ❝❛♥♥♦t ❜❡ r❡❛❞✳ ❨♦✉ s❤♦✉❧❞ ❛❧s♦ ❥✉st✐❢② ❛❧❧ ②♦✉r st❡♣s ❛♥❞ ❝♦♥❝❧✉s✐♦♥s✱ ✐♥❝❧✉❞✐♥❣ ❛❧❧ t❤❡ ❛❧❣❡❜r❛✳ ❋♦r ❡①❛♠♣❧❡✱ ②♦✉ s❤♦✉❧❞ ❛♣♣r❡❝✐❛t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦✿ ❲r♦♥❣✿

2x = 4 x=2

❘✐❣❤t✿

2x = 4 ; x = 2.

t❤❡r❡❢♦r❡✱

❚❤❡ st❛♥❞❛r❞s ♦❢ t❤♦r♦✉❣❤♥❡ss ❛r❡ ♣r♦✈✐❞❡❞ ❜② t❤❡ ❡①❛♠♣❧❡s ✐♥ t❤❡ ❜♦♦❦✳ ◆❡①t✱ ②♦✉r s♦❧✉t✐♦♥ s❤♦✉❧❞ ❜❡ t❤♦r♦✉❣❤❧② r❡❛❞✳ ❚❤✐s ✐s t❤❡ t✐♠❡ ❢♦r s❡❧❢✲❝r✐t✐❝✐s♠✿ ▲♦♦❦ ❢♦r ❡rr♦rs ❛♥❞ ✇❡❛❦ s♣♦ts✳ ■t s❤♦✉❧❞ ❜❡ r❡✲r❡❛❞ ❛♥❞ t❤❡♥ r❡✇r✐tt❡♥✳ ❖♥❝❡ ②♦✉ ❛r❡ ❝♦♥✈✐♥❝❡❞ t❤❛t t❤❡ s♦❧✉t✐♦♥ ✐s ❝♦rr❡❝t ❛♥❞ t❤❡ ♣r❡s❡♥t❛t✐♦♥ ✐s s♦❧✐❞✱ ②♦✉ ♠❛② s❤♦✇ ✐t t♦ ❛ ❦♥♦✇❧❡❞❣❡❛❜❧❡ ♣❡rs♦♥ ❢♦r ❛ ♦♥❝❡✲♦✈❡r✳ ◆❡①t✱ ②♦✉ ♠❛② t✉r♥ t♦ ♠♦❞❡❧✐♥❣ ♣r♦❥❡❝ts✳ ❙♣r❡❛❞s❤❡❡ts ✭▼✐❝r♦s♦❢t ❊①❝❡❧ ♦r s✐♠✐❧❛r✮ ❛r❡ ❝❤♦s❡♥ t♦ ❜❡ ✉s❡❞ ❢♦r ❣r❛♣❤✐♥❣ ❛♥❞ ♠♦❞❡❧✐♥❣✳ ❖♥❡ ❝❛♥ ❛❝❤✐❡✈❡ ❛s ❣♦♦❞ r❡s✉❧ts ✇✐t❤ ♣❛❝❦❛❣❡s s♣❡❝✐✜❝❛❧❧② ❞❡s✐❣♥❡❞ ❢♦r t❤❡s❡ ♣✉r♣♦s❡s✱ ❜✉t s♣r❡❛❞s❤❡❡ts ♣r♦✈✐❞❡ ❛ t♦♦❧ ✇✐t❤ ❛ ✇✐❞❡r s❝♦♣❡ ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✳ ♦♣t✐♦♥✳ ●♦♦❞ ❧✉❝❦✦ ❏✉♥❡ ✷✸✱ ✷✵✷✵

Pr♦❣r❛♠♠✐♥❣ ✐s ❛♥♦t❤❡r

❚♦ t❤❡ t❡❛❝❤❡r

✺

❚♦ t❤❡ t❡❛❝❤❡r ❚❤❡ ❜✉❧❦ ♦❢ t❤❡ ♠❛t❡r✐❛❧ ✐♥ t❤❡ ❜♦♦❦ ❝♦♠❡s ❢r♦♠ ♠② ❧❡❝t✉r❡ ♥♦t❡s✳ ❚❤❡r❡ ✐s ❧✐tt❧❡ ❡♠♣❤❛s✐s ♦♥ ❝❧♦s❡❞✲❢♦r♠ ❝♦♠♣✉t❛t✐♦♥s ❛♥❞ ❛❧❣❡❜r❛✐❝ ♠❛♥✐♣✉❧❛t✐♦♥s✳ ■ ❞♦ t❤✐♥❦ t❤❛t ❛ ♣❡rs♦♥ ✇❤♦ ❤❛s ♥❡✈❡r ✐♥t❡❣r❛t❡❞ ❜② ❤❛♥❞ ✭♦r ❞✐✛❡r❡♥t✐❛t❡❞✱ ♦r ❛♣♣❧✐❡❞ t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠✉❧❛✱ ❡t❝✳✮ ❝❛♥♥♦t ♣♦ss✐❜❧② ✉♥❞❡rst❛♥❞ ✐♥t❡❣r❛t✐♦♥ ✭♦r ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ♦r q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s✱ ❡t❝✳✮✳ ❍♦✇❡✈❡r✱ ❛ ❧❛r❣❡ ♣r♦♣♦rt✐♦♥ ♦❢ t✐♠❡ ❛♥❞ ❡✛♦rt ❝❛♥ ❛♥❞ s❤♦✉❧❞ ❜❡ ❞✐r❡❝t❡❞ t♦✇❛r❞✿

•

✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❝♦♥❝❡♣ts ❛♥❞

•

♠♦❞❡❧✐♥❣ ✐♥ r❡❛❧✐st✐❝ s❡tt✐♥❣s✳

❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ t❤✐s ❛♣♣r♦❛❝❤ ✐s t❤❛t ✐t r❡q✉✐r❡s ♠♦r❡ ❛❜str❛❝t✐♦♥ r❛t❤❡r t❤❛♥ ❧❡ss✳ ❱✐s✉❛❧✐③❛t✐♦♥ ✐s t❤❡ ♠❛✐♥ t♦♦❧ ✉s❡❞ t♦ ❞❡❛❧ ✇✐t❤ t❤✐s ❝❤❛❧❧❡♥❣❡✳ ■❧❧✉str❛t✐♦♥s ❛r❡ ♣r♦✈✐❞❡❞ ❢♦r ❡✈❡r② ❝♦♥❝❡♣t✱ ❜✐❣ ♦r s♠❛❧❧✳ ❚❤❡ ♣✐❝t✉r❡s t❤❛t ❝♦♠❡ ♦✉t ❛r❡ s♦♠❡t✐♠❡s ✈❡r② ♣r❡❝✐s❡ ❜✉t s♦♠❡t✐♠❡s s❡r✈❡ ❛s ♠❡r❡ ♠❡t❛♣❤♦rs ❢♦r t❤❡ ❝♦♥❝❡♣ts t❤❡② ✐❧❧✉str❛t❡✳ ❚❤❡ ❤♦♣❡ ✐s t❤❛t t❤❡② ✇✐❧❧ s❡r✈❡ ❛s ✈✐s✉❛❧ ✏❛♥❝❤♦rs✑ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ✇♦r❞s ❛♥❞ ❢♦r♠✉❧❛s✳ ■t ✐s ✉♥❧✐❦❡❧② t❤❛t ❛ ♣❡rs♦♥ ✇❤♦ ❤❛s ♥❡✈❡r ♣❧♦tt❡❞ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❜② ❤❛♥❞ ❝❛♥ ✉♥❞❡rst❛♥❞ ❣r❛♣❤s ♦r ❢✉♥❝t✐♦♥s✳ ❍♦✇❡✈❡r✱ ✇❤❛t ✐❢ ✇❡ ✇❛♥t t♦ ♣❧♦t ♠♦r❡ t❤❛♥ ❥✉st ❛ ❢❡✇ ♣♦✐♥ts ✐♥ ♦r❞❡r t♦ ✈✐s✉❛❧✐③❡ ❝✉r✈❡s✱ s✉r❢❛❝❡s✱ ✈❡❝t♦r ✜❡❧❞s✱ ❡t❝✳❄

❙♣r❡❛❞s❤❡❡ts ✇❡r❡ ❝❤♦s❡♥ ♦✈❡r ❣r❛♣❤✐❝ ❝❛❧❝✉❧❛t♦rs ❢♦r ✈✐s✉❛❧✐③❛t✐♦♥ ♣✉r♣♦s❡s

❜❡❝❛✉s❡ t❤❡② r❡♣r❡s❡♥t t❤❡ s❤♦rt❡st st❡♣ ❛✇❛② ❢r♦♠ ♣❡♥ ❛♥❞ ♣❛♣❡r✳

■♥❞❡❡❞✱ t❤❡ ❞❛t❛ ✐s ♣❧♦tt❡❞ ✐♥ t❤❡

s✐♠♣❧❡st ♠❛♥♥❡r ♣♦ss✐❜❧❡✿ ♦♥❡ ❝❡❧❧ ✲ ♦♥❡ ♥✉♠❜❡r ✲ ♦♥❡ ♣♦✐♥t ♦♥ t❤❡ ❣r❛♣❤✳ ❋♦r ♠♦r❡ ❛❞✈❛♥❝❡❞ t❛s❦s s✉❝❤ ❛s ♠♦❞❡❧✐♥❣✱ s♣r❡❛❞s❤❡❡ts ✇❡r❡ ❝❤♦s❡♥ ♦✈❡r ♦t❤❡r s♦❢t✇❛r❡ ❛♥❞ ♣r♦❣r❛♠♠✐♥❣ ♦♣t✐♦♥s ❢♦r t❤❡✐r ✇✐❞❡ ❛✈❛✐❧❛❜✐❧✐t② ❛♥❞✱ ❛❜♦✈❡ ❛❧❧✱ t❤❡✐r s✐♠♣❧✐❝✐t②✳ ◆✐♥❡ ♦✉t ♦❢ t❡♥✱ t❤❡ s♣r❡❛❞s❤❡❡t s❤♦✇♥ ✇❛s ✐♥✐t✐❛❧❧② ❝r❡❛t❡❞ ❢r♦♠ s❝r❛t❝❤ ✐♥ ❢r♦♥t ♦❢ t❤❡ st✉❞❡♥ts ✇❤♦ ✇❡r❡ ❧❛t❡r ❛❜❧❡ t♦ ❢♦❧❧♦✇ ♠② ❢♦♦tst❡♣s ❛♥❞ ❝r❡❛t❡ t❤❡✐r ♦✇♥✳ ❆❜♦✉t t❤❡ t❡sts✳ ❚❤❡ ❜♦♦❦ ✐s♥✬t ❞❡s✐❣♥❡❞ t♦ ♣r❡♣❛r❡ t❤❡ st✉❞❡♥t ❢♦r s♦♠❡ ♣r❡❡①✐st✐♥❣ ❡①❛♠❀ ♦♥ t❤❡ ❝♦♥tr❛r②✱ ❛ss✐❣♥♠❡♥ts s❤♦✉❧❞ ❜❡ ❜❛s❡❞ ♦♥ ✇❤❛t ❤❛s ❜❡❡♥ ❧❡❛r♥❡❞✳ ❚❤❡ st✉❞❡♥ts✬ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❝♦♥❝❡♣ts ♥❡❡❞s t♦ ❜❡ t❡st❡❞ ❜✉t✱ ♠♦st ♦❢ t❤❡ t✐♠❡✱ t❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ♦♥❧② ✐♥❞✐r❡❝t❧②✳ ❚❤❡r❡❢♦r❡✱ ❛ ❝❡rt❛✐♥ s❤❛r❡ ♦❢ r♦✉t✐♥❡✱ ♠❡❝❤❛♥✐❝❛❧ ♣r♦❜❧❡♠s ✐s ✐♥❡✈✐t❛❜❧❡✳ ◆♦♥❡t❤❡❧❡ss✱ ♥♦ t♦♣✐❝ ❞❡s❡r✈❡s ♠♦r❡ ❛tt❡♥t✐♦♥ ❥✉st ❜❡❝❛✉s❡ ✐t✬s ❧✐❦❡❧② t♦ ❜❡ ♦♥ t❤❡ t❡st✳ ■❢ ❛t ❛❧❧ ♣♦ss✐❜❧❡✱ ❞♦♥✬t ♠❛❦❡ t❤❡ st✉❞❡♥ts ♠❡♠♦r✐③❡ ❢♦r♠✉❧❛s✳ ■♥ t❤❡ ♦r❞❡r ♦❢ t♦♣✐❝s✱ t❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ❢r♦♠ ❛ t②♣✐❝❛❧ ❝❛❧❝✉❧✉s t❡①t❜♦♦❦ ✐s t❤❛t s❡q✉❡♥❝❡s ❝♦♠❡ ❜❡❢♦r❡ ❡✈❡r②t❤✐♥❣ ❡❧s❡✳ ❚❤❡ r❡❛s♦♥s ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

•

❙❡q✉❡♥❝❡s ❛r❡ t❤❡ s✐♠♣❧❡st ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥s✳

•

▲✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ❛r❡ s✐♠♣❧❡r t❤❛♥ ❧✐♠✐ts ♦❢ ❣❡♥❡r❛❧ ❢✉♥❝t✐♦♥s ✭✐♥❝❧✉❞✐♥❣ t❤❡ ♦♥❡s ❛t ✐♥✜♥✐t②✮✳

•

❚❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✱ ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♠❛❦❡ ♠♦r❡ s❡♥s❡ t♦ ❛ st✉❞❡♥t ✇✐t❤

•

❆ q✉✐❝❦ tr❛♥s✐t✐♦♥ ❢r♦♠ s❡q✉❡♥❝❡s t♦ s❡r✐❡s ♦❢t❡♥ ❧❡❛❞s t♦ ❝♦♥❢✉s✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦✳

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❙❡q✉❡♥❝❡s ❛r❡ ♥❡❡❞❡❞ ❢♦r ♠♦❞❡❧✐♥❣✱ ✇❤✐❝❤ s❤♦✉❧❞ st❛rt ❛s ❡❛r❧② ❛s ♣♦ss✐❜❧❡✳

❛ s♦❧✐❞ ❜❛❝❦❣r♦✉♥❞ ✐♥ s❡q✉❡♥❝❡s✳

❋r♦♠ t❤❡ ❞✐s❝r❡t❡ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s

✻

❋r♦♠ t❤❡ ❞✐s❝r❡t❡ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ■t✬s ♥♦ s❡❝r❡t t❤❛t ❛ ✈❛st ♠❛❥♦r✐t② ♦❢ ❝❛❧❝✉❧✉s st✉❞❡♥ts ✇✐❧❧ ♥❡✈❡r ✉s❡ ✇❤❛t t❤❡② ❤❛✈❡ ❧❡❛r♥❡❞✳ P♦♦r ❝❛r❡❡r ❝❤♦✐❝❡s ❛s✐❞❡✱ ❛ ❢♦r♠❡r ❝❛❧❝✉❧✉s st✉❞❡♥t ✐s ♦❢t❡♥ ✉♥❛❜❧❡ t♦ r❡❝♦❣♥✐③❡ t❤❡ ♠❛t❤❡♠❛t✐❝s t❤❛t ✐s s✉♣♣♦s❡❞ t♦ s✉rr♦✉♥❞ ❤✐♠✳ ❲❤② ❞♦❡s t❤✐s ❤❛♣♣❡♥❄ ❈❛❧❝✉❧✉s ✐s t❤❡ s❝✐❡♥❝❡ ♦❢ ❝❤❛♥❣❡✳ ❋r♦♠ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣✱ ✐ts ♣❡❝✉❧✐❛r ❝❤❛❧❧❡♥❣❡ ❤❛s ❜❡❡♥ t♦ st✉❞② ❛♥❞

❝♦♥t✐♥✉♦✉s ❝❤❛♥❣❡✿ ❝✉r✈❡s ❛♥❞ ♠♦t✐♦♥ ❛❧♦♥❣ ❝✉r✈❡s✳ ❢♦r♠✉❧❛s✳ ❙❦✐❧❧❢✉❧ ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ t❤♦s❡ ❢♦r♠✉❧❛s ✐s ✇❤❛t

♠❡❛s✉r❡

❚❤❡s❡ ❝✉r✈❡s ❛♥❞ t❤✐s ♠♦t✐♦♥ ❛r❡ r❡♣r❡s❡♥t❡❞

❜②

s♦❧✈❡s ❝❛❧❝✉❧✉s ♣r♦❜❧❡♠s✳ ❋♦r ♦✈❡r ✸✵✵ ②❡❛rs✱

t❤✐s ❛♣♣r♦❛❝❤ ❤❛s ❜❡❡♥ ❡①tr❡♠❡❧② s✉❝❝❡ss❢✉❧ ✐♥ s❝✐❡♥❝❡s ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✳

❚❤❡ s✉❝❝❡ss❡s ❛r❡ ✇❡❧❧✲❦♥♦✇♥✿

♣r♦❥❡❝t✐❧❡ ♠♦t✐♦♥✱ ♣❧❛♥❡t❛r② ♠♦t✐♦♥✱ ✢♦✇ ♦❢ ❧✐q✉✐❞s✱ ❤❡❛t tr❛♥s❢❡r✱ ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥✱ ❡t❝✳ ❚❡❛❝❤✐♥❣ ❝❛❧❝✉❧✉s ❢♦❧❧♦✇s t❤✐s ❛♣♣r♦❛❝❤✿ ❆♥ ♦✈❡r✇❤❡❧♠✐♥❣ ♠❛❥♦r✐t② ♦❢ ✇❤❛t t❤❡ st✉❞❡♥t ❞♦❡s ✐s ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ ❢♦r♠✉❧❛s ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✳ ❇✉t t❤✐s ♠❡❛♥s t❤❛t ❛❧❧ t❤❡ ♣r♦❜❧❡♠s t❤❡ st✉❞❡♥t ❢❛❝❡s ✇❡r❡ ✭♦r ❝♦✉❧❞ ❤❛✈❡ ❜❡❡♥✮ s♦❧✈❡❞ ✐♥ t❤❡ ✶✽t❤ ♦r ✶✾t❤ ❝❡♥t✉r✐❡s✦ ❚❤✐s ✐s♥✬t ❣♦♦❞ ❡♥♦✉❣❤ ❛♥②♠♦r❡✳ ❲❤❛t ❤❛s ❝❤❛♥❣❡❞ s✐♥❝❡ t❤❡♥❄ ❚❤❡ ❝♦♠♣✉t❡rs ❤❛✈❡ ❛♣♣❡❛r❡❞✱ ♦❢ ❝♦✉rs❡✱ ❛♥❞ ❝♦♠♣✉t❡rs ❞♦♥✬t ♠❛♥✐♣✉❧❛t❡ ❢♦r♠✉❧❛s✳

❚❤❡② ❞♦♥✬t ❤❡❧♣ ✇✐t❤ s♦❧✈✐♥❣ ✕ ✐♥ t❤❡ tr❛❞✐t✐♦♥❛❧ s❡♥s❡ ♦❢

t❤❡ ✇♦r❞ ✕ t❤♦s❡ ♣r♦❜❧❡♠s ❢r♦♠ t❤❡ ♣❛st ❝❡♥t✉r✐❡s✳

✐♥❝r❡♠❡♥t❛❧

■♥st❡❛❞ ♦❢

❝♦♥t✐♥✉♦✉s✱

❝♦♠♣✉t❡rs ❡①❝❡❧ ❛t ❤❛♥❞❧✐♥❣

♣r♦❝❡ss❡s✱ ❛♥❞ ✐♥st❡❛❞ ♦❢ ❢♦r♠✉❧❛s t❤❡② ❛r❡ ❣r❡❛t ❛t ♠❛♥❛❣✐♥❣ ❞✐s❝r❡t❡ ✭❞✐❣✐t❛❧✮ ❞❛t❛✳ ❚♦ ✉t✐❧✐③❡

t❤❡s❡ ❛❞✈❛♥t❛❣❡s✱ s❝✐❡♥t✐sts ✏❞✐s❝r❡t✐③❡✑ t❤❡ r❡s✉❧ts ♦❢ ❝❛❧❝✉❧✉s ❛♥❞ ❝r❡❛t❡ ❛❧❣♦r✐t❤♠s t❤❛t ♠❛♥✐♣✉❧❛t❡ t❤❡ ❞✐❣✐t❛❧ ❞❛t❛✳

❚❤❡ s♦❧✉t✐♦♥s ❛r❡ ❛♣♣r♦①✐♠❛t❡ ❜✉t t❤❡ ❛♣♣❧✐❝❛❜✐❧✐t② ✐s ✉♥❧✐♠✐t❡❞✳

❙✐♥❝❡ t❤❡ ✷✵t❤ ❝❡♥t✉r②✱

t❤✐s ❛♣♣r♦❛❝❤ ❤❛s ❜❡❡♥ ❡①tr❡♠❡❧② s✉❝❝❡ss❢✉❧ ✐♥ s❝✐❡♥❝❡s ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✿ ❛❡r♦❞②♥❛♠✐❝s ✭❛✐r♣❧❛♥❡ ❛♥❞ ❝❛r ❞❡s✐❣♥✮✱ s♦✉♥❞ ❛♥❞ ✐♠❛❣❡ ♣r♦❝❡ss✐♥❣✱ s♣❛❝❡ ❡①♣❧♦r❛t✐♦♥✱ str✉❝t✉r❡ ♦❢ t❤❡ ❛t♦♠ ❛♥❞ t❤❡ ✉♥✐✈❡rs❡✱ ❡t❝✳ ❚❤❡ ❛♣♣r♦❛❝❤ ✐s ❛❧s♦ ❝✐r❝✉✐t♦✉s✿ ❊✈❡r② ❝♦♥❝❡♣t ✐♥ ❝❛❧❝✉❧✉s

st❛rts

✕ ♦❢t❡♥ ✐♠♣❧✐❝✐t❧② ✕ ❛s ❛ ❞✐s❝r❡t❡ ❛♣♣r♦①✐♠❛t✐♦♥

♦❢ ❛ ❝♦♥t✐♥✉♦✉s ♣❤❡♥♦♠❡♥♦♥✦

❈❛❧❝✉❧✉s ✐s t❤❡ s❝✐❡♥❝❡ ♦❢ ❝❤❛♥❣❡✱

❜♦t❤

✐♥❝r❡♠❡♥t❛❧ ❛♥❞ ❝♦♥t✐♥✉♦✉s✳ ❚❤❡ ❢♦r♠❡r ♣❛rt ✕ t❤❡ s♦✲❝❛❧❧❡❞ ❞✐s❝r❡t❡

❝❛❧❝✉❧✉s ✕ ♠❛② ❜❡ s❡❡♥ ❛s t❤❡ st✉❞② ♦❢ ✐♥❝r❡♠❡♥t❛❧ ♣❤❡♥♦♠❡♥❛ ❛♥❞ t❤❡ q✉❛♥t✐t✐❡s

✐♥❞✐✈✐s✐❜❧❡

❜② t❤❡✐r

✈❡r② ♥❛t✉r❡✿ ♣❡♦♣❧❡✱ ❛♥✐♠❛❧s✱ ❛♥❞ ♦t❤❡r ♦r❣❛♥✐s♠s✱ ♠♦♠❡♥ts ♦❢ t✐♠❡✱ ❧♦❝❛t✐♦♥s ♦❢ s♣❛❝❡✱ ♣❛rt✐❝❧❡s✱ s♦♠❡ ❝♦♠♠♦❞✐t✐❡s✱ ❞✐❣✐t❛❧ ✐♠❛❣❡s ❛♥❞ ♦t❤❡r ♠❛♥✲♠❛❞❡ ❞❛t❛✱ ❡t❝✳ ❲✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ ❝❛❧❝✉❧✉s ♠❛❝❤✐♥❡r② ❝❛❧❧❡❞ ✏❧✐♠✐ts✑✱ ✇❡ ✐♥✈❛r✐❛❜❧② ❝❤♦♦s❡ t♦ tr❛♥s✐t✐♦♥ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ♣❛rt ♦❢ ❝❛❧❝✉❧✉s✱ ❡s♣❡❝✐❛❧❧② ✇❤❡♥ ✇❡ ❢❛❝❡ ❝♦♥t✐♥✉♦✉s ♣❤❡♥♦♠❡♥❛ ❛♥❞ t❤❡ q✉❛♥t✐t✐❡s

✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡

❡✐t❤❡r ❜② t❤❡✐r ♥❛t✉r❡ ♦r ❜② ❛ss✉♠♣t✐♦♥✿ t✐♠❡✱

s♣❛❝❡✱ ♠❛ss✱ t❡♠♣❡r❛t✉r❡✱ ♠♦♥❡②✱ s♦♠❡ ❝♦♠♠♦❞✐t✐❡s✱ ❡t❝✳ ❈❛❧❝✉❧✉s ♣r♦❞✉❝❡s ❞❡✜♥✐t✐✈❡ r❡s✉❧ts ❛♥❞ ❛❜s♦❧✉t❡ ❛❝❝✉r❛❝② ✕ ❜✉t ♦♥❧② ❢♦r ♣r♦❜❧❡♠s ❛♠❡♥❛❜❧❡ t♦ ✐ts ♠❡t❤♦❞s✦ ■♥ t❤❡ ❝❧❛ssr♦♦♠✱ t❤❡ ♣r♦❜❧❡♠s ❛r❡ s✐♠♣❧✐✜❡❞ ✉♥t✐❧ t❤❡② ❜❡❝♦♠❡ ♠❛♥❛❣❡❛❜❧❡❀ ♦t❤❡r✇✐s❡✱ ✇❡ ❝✐r❝❧❡ ❜❛❝❦ t♦ t❤❡ ❞✐s❝r❡t❡ ♠❡t❤♦❞s ✐♥ s❡❛r❝❤ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❲✐t❤✐♥ ❛ t②♣✐❝❛❧ ❝❛❧❝✉❧✉s ❝♦✉rs❡✱ t❤❡ st✉❞❡♥t s✐♠♣❧② ♥❡✈❡r ❣❡ts t♦ ❝♦♠♣❧❡t❡ t❤❡ ✏❝✐r❝❧❡✑✦

▲❛t❡r ♦♥✱ t❤❡

❣r❛❞✉❛t❡ ✐s ❧✐❦❡❧② t♦ t❤✐♥❦ ♦❢ ❝❛❧❝✉❧✉s ♦♥❧② ✇❤❡♥ ❤❡ s❡❡s ❢♦r♠✉❧❛s ❛♥❞ r❛r❡❧② ✇❤❡♥ ❤❡ s❡❡s ♥✉♠❡r✐❝❛❧ ❞❛t❛✳ ■♥ t❤✐s ❜♦♦❦✱ ❡✈❡r② ❝♦♥❝❡♣t ♦❢ ❝❛❧❝✉❧✉s ✐s ✜rst ✐♥tr♦❞✉❝❡❞ ✐♥ ✐ts ❞✐s❝r❡t❡✱ ✏♣r❡✲❧✐♠✐t✑✱ ✐♥❝❛r♥❛t✐♦♥ ✕ ❡❧s❡✇❤❡r❡ t②♣✐❝❛❧❧② ❤✐❞❞❡♥ ✐♥s✐❞❡ ♣r♦♦❢s ✕ ❛♥❞ t❤❡♥ ✉s❡❞ ❢♦r ♠♦❞❡❧✐♥❣ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ✇❡❧❧ ❜❡❢♦r❡ ✐ts ❝♦♥t✐♥✉♦✉s ❝♦✉♥t❡r♣❛rt ❡♠❡r❣❡s✳ ❚❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❢♦r♠❡r ❛r❡ ❞✐s❝♦✈❡r❡❞ ✜rst ❛♥❞ t❤❡♥ t❤❡ ♠❛t❝❤✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❧❛tt❡r ❛r❡ ❢♦✉♥❞ ❜② ♠❛❦✐♥❣ t❤❡ ✐♥❝r❡♠❡♥t s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r✱ ❛t t❤❡ ❞✐s❝r❡t❡ ❝❛❧❝✉❧✉s

∆x→0

−−−−−−−−−−→

❧✐♠✐t ✿

❝♦♥t✐♥✉♦✉s ❝❛❧❝✉❧✉s

❚❤❡ ✈♦❧✉♠❡ ❛♥❞ ❝❤❛♣t❡r r❡❢❡r❡♥❝❡s ❢♦r ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞

✼

❚❤❡ ✈♦❧✉♠❡ ❛♥❞ ❝❤❛♣t❡r r❡❢❡r❡♥❝❡s ❢♦r ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞ ❚❤✐s ❜♦♦❦ ✐s ❛ ♣❛rt ♦❢ t❤❡ s❡r✐❡s ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❚❤❡ s❡r✐❡s ❝♦✈❡rs t❤❡ st❛♥❞❛r❞ ♠❛t❡r✐❛❧ ♦❢ t❤❡ ✉♥❞❡r✲ ❣r❛❞✉❛t❡ ❝❛❧❝✉❧✉s ✇✐t❤ ❛ s✉❜st❛♥t✐❛❧ r❡✈✐❡✇ ♦❢ ♣r❡❝❛❧❝✉❧✉s ❛♥❞ ❛ ♣r❡✈✐❡✇ ♦❢ ❡❧❡♠❡♥t❛r② ♦r❞✐♥❛r② ❛♥❞ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❇❡❧♦✇ ✐s t❤❡ ❧✐st ♦❢ t❤❡ ❜♦♦❦s ♦❢ t❤❡ s❡r✐❡s✱ t❤❡✐r ❝❤❛♣t❡rs✱ ❛♥❞ t❤❡ ✇❛② t❤❡ ♣r❡s❡♥t ❜♦♦❦ ✭♣❛r❡♥t❤❡t✐❝❛❧❧②✮ r❡❢❡r❡♥❝❡s t❤❡♠✳ 

✶ P❈✲✶ ✶ P❈✲✷ ✶ P❈✲✸ ✶ P❈✲✹ ✶ P❈✲✺



✷ ❉❈✲✶ ✷ ❉❈✲✷ ✷ ❉❈✲✸ ✷ ❉❈✲✹ ✷ ❉❈✲✺ ✷ ❉❈✲✻

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✹✿ ❈❛❧❝✉❧✉s ✐♥ ❍✐❣❤❡r ❉✐♠❡♥s✐♦♥s

❋✉♥❝t✐♦♥s ✐♥ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s P❛r❛♠❡tr✐❝ ❝✉r✈❡s ❋✉♥❝t✐♦♥s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❚❤❡ ❣r❛❞✐❡♥t ❚❤❡ ✐♥t❡❣r❛❧ ❱❡❝t♦r ✜❡❧❞s 

✺ ❉❊✲✶ ✺ ❉❊✲✷ ✺ ❉❊✲✸ ✺ ❉❊✲✹ ✺ ❉❊✲✺

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✸✿ ■♥t❡❣r❛❧ ❈❛❧❝✉❧✉s

❚❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ■♥t❡❣r❛t✐♦♥ ❲❤❛t ✇❡ ❝❛♥ s♦ ✇✐t❤ ✐♥t❡❣r❛❧ ❝❛❧❝✉❧✉s ❙❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ❙❡r✐❡s 

✹ ❍❉✲✶ ✹ ❍❉✲✷ ✹ ❍❉✲✸ ✹ ❍❉✲✹ ✹ ❍❉✲✺ ✹ ❍❉✲✻

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✷✿ ❉✐✛❡r❡♥t✐❛❧ ❈❛❧❝✉❧✉s

▲✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ❚❤❡ ❞❡r✐✈❛t✐✈❡ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡ ♠❛✐♥ t❤❡♦r❡♠s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s ❲❤❛t ✇❡ ❝❛♥ ❞♦ ✇✐t❤ ❝❛❧❝✉❧✉s 

✸ ■❈✲✶ ✸ ■❈✲✷ ✸ ■❈✲✸ ✸ ■❈✲✹ ✸ ■❈✲✺

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✶✿ Pr❡❝❛❧❝✉❧✉s

❈❛❧❝✉❧✉s ♦❢ s❡q✉❡♥❝❡s ❙❡ts ❛♥❞ ❢✉♥❝t✐♦♥s ❈♦♠♣♦s✐t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s ❈❧❛ss❡s ♦❢ ❢✉♥❝t✐♦♥s ❆❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr②

❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞✳ ❱♦❧✉♠❡ ✺✿ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

❖r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❱❡❝t♦r ❛♥❞ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡s ❙②st❡♠s ♦❢ ❖❉❊s ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❖❉❊s P❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

❊❛❝❤ ✈♦❧✉♠❡ ❝❛♥ ❜❡ r❡❛❞ ✐♥❞❡♣❡♥❞❡♥t❧②✳

❚❤❡ ✈♦❧✉♠❡ ❛♥❞ ❝❤❛♣t❡r r❡❢❡r❡♥❝❡s ❢♦r ❈❛❧❝✉❧✉s ■❧❧✉str❛t❡❞

✽

❆ ♣♦ss✐❜❧❡ s❡q✉❡♥❝❡ ♦❢ ❝❤❛♣t❡rs ✐s ♣r❡s❡♥t❡❞ ❜❡❧♦✇✳ ❆♥ ❛rr♦✇ ❢r♦♠ ❆ t♦ ❇ ♠❡❛♥s t❤❛t ❝❤❛♣t❡r ❇ s❤♦✉❧❞♥✬t ❜❡ r❡❛❞ ❜❡❢♦r❡ ❝❤❛♣t❡r ❆✳

❆❜♦✉t t❤❡ ❛✉t❤♦r

❆❜♦✉t t❤❡ ❛✉t❤♦r P❡t❡r ❙❛✈❡❧✐❡✈ ✐s ❛ ♣r♦❢❡ss♦r ♦❢ ♠❛t❤❡♠❛t✐❝s ❛t ▼❛rs❤❛❧❧ ❯♥✐✈❡rs✐t②✱ ❍✉♥t✲ ✐♥❣t♦♥✱ ❲❡st ❱✐r❣✐♥✐❛✱ ❯❙❆✳ ❆❢t❡r ❛ P❤✳❉✳ ❢r♦♠ t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ■❧❧✐♥♦✐s ❛t ❯r❜❛♥❛✲❈❤❛♠♣❛✐❣♥✱ ❤❡ ❞❡✈♦t❡❞ t❤❡ ♥❡①t ✷✵ ②❡❛rs t♦ t❡❛❝❤✐♥❣ ♠❛t❤❡♠❛t✐❝s✳ P❡t❡r ✐s t❤❡ ❛✉t❤♦r ♦❢ ❛ ❣r❛❞✉❛t❡ t❡①t❜♦♦❦ ❚♦♣♦❧♦❣② ■❧❧✉str❛t❡❞ ♣✉❜❧✐s❤❡❞ ✐♥ ✷✵✶✻✳ ❍❡ ❤❛s ❛❧s♦ ❜❡❡♥ ✐♥✈♦❧✈❡❞ ✐♥ r❡s❡❛r❝❤ ✐♥ ❛❧❣❡❜r❛✐❝ t♦♣♦❧♦❣② ❛♥❞ s❡✈❡r❛❧ ♦t❤❡r ✜❡❧❞s✳ ❍✐s ♥♦♥✲❛❝❛❞❡♠✐❝ ♣r♦❥❡❝ts ❤❛✈❡ ❜❡❡♥✿ ❞✐❣✐t❛❧ ✐♠❛❣❡ ❛♥❛❧②s✐s✱ ❛✉t♦♠❛t❡❞ ✜♥❣❡r♣r✐♥t ✐❞❡♥t✐✜❝❛t✐♦♥✱ ❛♥❞ ✐♠❛❣❡ ♠❛t❝❤✐♥❣ ❢♦r ♠✐s✲ s✐❧❡ ♥❛✈✐❣❛t✐♦♥✴❣✉✐❞❛♥❝❡✳

✾

❈♦♥t❡♥ts Pr❡❢❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷

 ❈❤❛♣t❡r ✶✿ ❚❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✶✳✶ ❚❤❡ ❆r❡❛ Pr♦❜❧❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❚❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✿ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❍♦✇ t♦ ❛♣♣r♦①✐♠❛t❡ ❛r❡❛s ❛♥❞ ❞✐s♣❧❛❝❡♠❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✻ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✳ ✳ ✳ ✶✳✼ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s ✳ ✶✳✽ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❝♦♥t✐♥✉❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✾ ❇❛s✐❝ ✐♥t❡❣r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✵ ❋r❡❡ ❢❛❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✶✷ ✶✽ ✸✸ ✹✸ ✹✾ ✺✺ ✻✺ ✼✹ ✽✹ ✾✵

 ❈❤❛♣t❡r ✷✿ ■♥t❡❣r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻

✷✳✶ Pr♦♣❡rt✐❡s ♦❢ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ■♥t❡❣r❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ❛♥❞ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡✿ ❧✐♥❡❛r✐t② ✷✳✸ ■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ✳ ✳ ✳ ✳ ✷✳✹ ▲✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺ ■♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥✿ ❝♦♠♣♦s✐t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽ ❚r✐❣♦♥♦♠❡tr✐❝ s✉❜st✐t✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✾ ■♥t❡❣r❛t✐♦♥ ❜② ♣❛rts✿ ♣r♦❞✉❝ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✵ ❆♣♣r♦❛❝❤❡s t♦ ✐♥t❡❣r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✶ ❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✷✳✶✷ Pr♦♣❡rt✐❡s ♦❢ ♣r♦♣❡r ❛♥❞ ✐♠♣r♦♣❡r ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s ✳ ✳ ✷✳✶✸ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✳ ✶✽✵ ✳ ✶✽✵ ✳ ✶✽✽ ✳ ✶✾✷ ✳ ✷✵✶ ✳ ✷✵✹ ✳ ✷✵✽ ✳ ✷✶✺ ✳ ✷✶✾ ✳ ✷✸✵ ✳ ✷✸✻ ✳ ✷✹✹ ✳ ✷✺✶

 ❈❤❛♣t❡r ✸✿ ❲❤❛t ✇❡ ❝❛♥ ❞♦ ✇✐t❤ ✐♥t❡❣r❛❧ ❝❛❧❝✉❧✉s

✸✳✶ ❚❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t✇♦ ❣r❛♣❤s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ❚❤❡ ❧✐♥❡❛r ❞❡♥s✐t② ❛♥❞ t❤❡ ♠❛ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹ ❚❤❡ r❛❞✐❛❧ ❞❡♥s✐t② ❛♥❞ t❤❡ ♠❛ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✺ ❚❤❡ ✢♦✇ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ✢✉① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✻ ❚❤❡ ❢♦r❝❡ ❛♥❞ t❤❡ ✇♦r❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✼ ❚❤❡ t♦t❛❧ ❛♥❞ t❤❡ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✸✳✽ ◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✾ ▲❡♥❣t❤s ♦❢ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✵ ❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ t❤r❡❡ ✳ ✸✳✶✶ ❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ✈✐❛ t❤❡✐r ❝r♦ss✲s❡❝t✐♦♥s ✳ ✳ ✸✳✶✷ ❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ♦❢ r❡✈♦❧✉t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✾✻ ✶✵✶ ✶✵✾ ✶✷✶ ✶✷✽ ✶✸✸ ✶✸✽ ✶✹✷ ✶✹✼ ✶✺✶ ✶✺✸ ✶✻✺ ✶✼✵

 ❈❤❛♣t❡r ✹✿ ❙❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺✽

✹✳✶ ❆ ❜❛❧❧ ✐s t❤r♦✇♥✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺✽ ✹✳✷ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥ t❤❡ ♣❧❛♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✹ ✹✳✸ ❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼✹

❈♦♥t❡♥ts

✶✶

✹✳✹ ■♥tr♦❞✉❝t✐♦♥ t♦ ❝❛❧❝✉❧✉s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✹✳✺ ❚❤❡ ❝❡♥tr♦✐❞ ♦❢ ❛ ✢❛t ♦❜❥❡❝t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✻ ❈♦♦r❞✐♥❛t❡ s②st❡♠s❀ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✼ ❱❡❝t♦rs ♦♥ t❤❡ ♣❧❛♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✽ ❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✾ ❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2 ✳ ✳ ✳ ✹✳✶✵ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿ C ✐s♥✬t ❥✉st R2 ✹✳✶✶ ❉✐s❝r❡t❡ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✶✷ ❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✷✽✺ ✷✾✸ ✷✾✼ ✸✵✼ ✸✶✷ ✸✶✾ ✸✷✸ ✸✷✽ ✸✸✼

 ❈❤❛♣t❡r ✺✿ ❙❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹✹

✺✳✶ ❋r♦♠ ❧✐♥❡❛r t♦ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✺✳✷ ❚❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✸ ❙❡q✉❡♥❝❡s ♦❢ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✹ ■♥✜♥✐t❡ s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✺ ▼❛✐♥ ❝❧❛ss❡s ♦❢ s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✻ ❋r♦♠ ✜♥✐t❡ s✉♠s ✈✐❛ ❧✐♠✐ts t♦ s❡r✐❡s ✳ ✳ ✳ ✺✳✼ ❉✐✈❡r❣❡♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✽ ❙❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s ✳ ✳ ✳ ✳ ✳ ✺✳✾ ❈♦♠♣❛r✐s♦♥ ♦❢ s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✶✵ ❆❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✶✶ ❚❤❡ ❘❛t✐♦ ❚❡st ❛♥❞ t❤❡ ❘♦♦t ❚❡st ✳ ✳ ✳ ✺✳✶✷ P♦✇❡r s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✶✸ ❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✸✹✹ ✸✺✷ ✸✻✷ ✸✻✽ ✸✼✺ ✸✽✹ ✸✾✷ ✸✾✺ ✹✵✷ ✹✵✽ ✹✶✺ ✹✷✶ ✹✷✽

❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❊①❡r❝✐s❡s✿ ❇❛❝❦❣r♦✉♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❊①❡r❝✐s❡s✿ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❊①❡r❝✐s❡s✿ ❉❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ❊①❡r❝✐s❡s✿ ■♥❢♦r♠❛t✐♦♥ ❢r♦♠ t❤❡ ❞❡r✐✈❛t✐✈❡s ✳ ✻ ❊①❡r❝✐s❡s✿ ❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✼ ❊①❡r❝✐s❡s✿ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ❊①❡r❝✐s❡s✿ ■♥t❡❣r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ❊①❡r❝✐s❡s✿ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✶✵ ❊①❡r❝✐s❡s✿ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ r❡❧❛t❡❞ ✳ ✶✶ ❊①❡r❝✐s❡s✿ ❙❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ❊①❡r❝✐s❡s✿ ❙❡q✉❡♥❝❡s ❛♥❞ s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ❊①❡r❝✐s❡s✿ P♦✇❡r s❡r✐❡s ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s

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✳ ✹✹✶ ✳ ✹✹✶ ✳ ✹✹✻ ✳ ✹✹✽ ✳ ✹✺✶ ✳ ✹✺✸ ✳ ✹✺✹ ✳ ✹✺✺ ✳ ✹✺✼ ✳ ✹✺✾ ✳ ✹✻✶ ✳ ✹✻✹ ✳ ✹✻✻ ✳ ✹✻✽

■♥❞❡① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼✷

❈❤❛♣t❡r ✶✿ ❚❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

❈♦♥t❡♥ts

✶✳✶ ❚❤❡ ❆r❡❛ Pr♦❜❧❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❚❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✿ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❍♦✇ t♦ ❛♣♣r♦①✐♠❛t❡ ❛r❡❛s ❛♥❞ ❞✐s♣❧❛❝❡♠❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✻ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✳ ✳ ✳ ✶✳✼ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s ✶✳✽ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❝♦♥t✐♥✉❡❞ ✳ ✳ ✳ ✳ ✳ ✶✳✾ ❇❛s✐❝ ✐♥t❡❣r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✵ ❋r❡❡ ❢❛❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷ ✶✽ ✸✸ ✹✸ ✹✾ ✺✺ ✻✺ ✼✹ ✽✹ ✾✵

✶✳✶✳ ❚❤❡ ❆r❡❛ Pr♦❜❧❡♠

❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ✇❛②s t♦ ❡♥t❡r ❝❛❧❝✉❧✉s✿ • ❙t✉❞② ♦❢ ♠♦t✐♦♥✿ ❜❛❝❦ ❛♥❞ ❢♦rt❤ ❜❡t✇❡❡♥ ♣♦s✐t✐♦♥s ❛♥❞ ✈❡❧♦❝✐t✐❡s✳

• ❙t✉❞② ♦❢ ❝✉r✈❡❞ s❤❛♣❡s✿ ❜❛❝❦ ❛♥❞ ❢♦rt❤ ❜❡t✇❡❡♥ t❛♥❣❡♥ts ❛♥❞✳✳✳ ✇❤❛t❄ ❚❤❡ ❛r❡❛s✳

❲❡ ❦♥♦✇ t❤❛t t❤❡ ❛r❡❛ ♦❢ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s r ✐s s✉♣♣♦s❡❞ t♦ ❜❡ A = πr2 ✳ ❊①❛♠♣❧❡ ✶✳✶✳✶✿ ❛r❡❛ ♦❢ ❝✐r❝❧❡

▲❡t✬s r❡✈✐❡✇ ❤♦✇ ✇❡ ❝❛♥ ❝♦♥✜r♠ t❤❡ ❢♦r♠✉❧❛ ✇✐t❤ ♥♦t❤✐♥❣ ❜✉t ❛ s♣r❡❛❞s❤❡❡t✳ ❋✐rst✱ ✇❡ ♣❧♦t t❤❡ ❣r❛♣❤ y=

√

1 − x2 ,

❜② ❧❡tt✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ x r✉♥ ❢r♦♠ −1 t♦ 1 ❡✈❡r② h = .1 ❛♥❞ ✜♥❞✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ y ✇✐t❤ t❤❡ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛✿ ❂❙◗❘❚✭✶✲❘❈❬✲✷❪✂ ✷✮

❲❡ ♣❧♦t t❤❡s❡ 20 ♣♦✐♥ts❀ t❤❡ r❡s✉❧t ✐s ❛ ❤❛❧❢✲❝✐r❝❧❡✿

✶✳✶✳

❚❤❡ ❆r❡❛ Pr♦❜❧❡♠

✶✸

❲❡ ♥❡①t ❝♦✈❡r✱ ❛s ❜❡st ✇❡ ❝❛♥✱ t❤✐s ❤❛❧❢✲❝✐r❝❧❡ ✇✐t❤ ✈❡rt✐❝❛❧ ❜❛rs t❤❛t st❛♥❞ ♦♥ t❤❡ ✐♥t❡r✈❛❧

[−1, 1]✳

❲❡

r❡✲✉s❡ t❤❡ ❞❛t❛✿

• •

❚❤❡ ❜❛s❡s ♦❢ t❤❡ ❜❛rs ❛r❡ ♦✉r ✐♥t❡r✈❛❧s ✐♥ t❤❡ ❚❤❡ ❤❡✐❣❤ts ❛r❡ t❤❡ ✈❛❧✉❡s ♦❢

x✲❛①✐s✳

y✳

❚♦ s❡❡ t❤❡ ❜❛rs✱ ✇❡ s✐♠♣❧② ❝❤❛♥❣❡ t❤❡ t②♣❡ ♦❢ t❤❡ ❝❤❛rt ♣❧♦tt❡❞ ❜② t❤❡ s♣r❡❛❞s❤❡❡t✿

◆❡①t✱ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ❞♦ t❤❡ ✇♦r❦ ❢♦r ✉s✿

• •

s✉♠ ♦❢ t❤❡ ❛r❡❛s ♦❢ t❤❡ ❜❛rs✳ ❚♦ ❧❡t t❤❡ s♣r❡❛❞s❤❡❡t

▼✉❧t✐♣❧② t❤❡ ❤❡✐❣❤ts ❜② t❤❡ ✭❝♦♥st❛♥t✮ ✇✐❞t❤s ✐♥ t❤❡ ♥❡①t ❝♦❧✉♠♥✳ ❆❞❞ t❤❡♠ ✉♣ ❛t t❤❡ t♦♣ ❝❡❧❧ ✭②❡❧❧♦✇✮✳

❲❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛s ❢♦r

❛r❡❛s ✿

❂❘❈❬✲✶❪✯✭❘❈❬✲✷❪✲❘❬✲✶❪❈❬✲✷❪✮ ❚❤❡ r❡s✉❧t ♣r♦❞✉❝❡❞ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❆♣♣r♦①✐♠❛t❡ ❛r❡❛ ♦❢ t❤❡ s❡♠✐❝✐r❝❧❡

= 1.552 .

■t ✐s ❝❧♦s❡ t♦ t❤❡ t❤❡♦r❡t✐❝❛❧ r❡s✉❧t ♦❜t❛✐♥❡❞ ❧❛t❡r ✐♥ t❤✐s ❝❤❛♣t❡r✿ ❊①❛❝t ❛r❡❛ ♦❢ t❤❡ s❡♠✐❝✐r❝❧❡

= π/2 ≈ 1.571 .

A ✐s t❤❡ s✉♠ ♦❢ t❤❡ s❡q✉❡♥❝❡✿ p an = 1 − x2n · 0.1, ✇❤❡r❡ xn = −1.0 , −0.9 , −0.8 , ... , 0.8 , 0.9 , 1.0 .

■♥ s✉♠♠❛r②✱ t❤❡ ❛r❡❛

■♥ ♦t❤❡r ✇♦r❞s✿

A=

20 X n=1

0.1 ·

p 1 − x2n .

✶✳✶✳

❚❤❡ ❆r❡❛ Pr♦❜❧❡♠

✶✹

❚❤❡r❡ ✐s ♠♦r❡✦ ❚❤❡ ❛♣♣r♦❛❝❤ ✇❡ ❤❛✈❡ ✉s❡❞ ♠❛♥② t✐♠❡s ✐♥ ❝❛❧❝✉❧✉s ✭❈❤❛♣t❡r ✷❉❈✲✸✮ ✐s t♦ ✐♠♣r♦✈❡ ♦✉r ❛♣♣r♦①✐♠❛t✐♦♥ ❜② ♠❛❦✐♥❣ t❤❡ ✐♥t❡r✈❛❧s s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r✳ ❘❡❞♦✐♥❣ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ✇✐t❤ 40 ✐♥t❡r✈❛❧s ❣✐✈❡s ❛ ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥✱ 1.564✿

❚❤❡ q✉❛❧✐t② ♦❢ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s s❡❡♥ ❛s t❤❡ s✐③❡ ♦❢ t❤❡ ♣❛rts ♦❢ t❤❡ ❜❛rs st✐❝❦✐♥❣ ♦✉t ♦❢ t❤❡ ❝✐r❝❧❡ ❛s ✇❡❧❧ ❛s t❤❡ ♣❛rts ♦❢ t❤❡ ✐♥s✐❞❡ ♦❢ t❤❡ ❝✐r❝❧❡ ♥♦t ❝♦✈❡r❡❞ ❜② t❤❡ ❜❛rs✿

◆♦t❤✐♥❣ st♦♣s ✉s ❢r♦♠ ✐♠♣r♦✈✐♥❣ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ❢✉rt❤❡r ❛♥❞ ❢✉rt❤❡r ✇✐t❤ ❧❛r❣❡r ❛♥❞ ❧❛r❣❡r ✈❛❧✉❡s ♦❢ n✳ ❊①❡r❝✐s❡ ✶✳✶✳✷

❆♣♣r♦①✐♠❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s 1 ✇✐t❤✐♥ 0.0001✳ ❲❡ ❤❛✈❡ s❤♦✇❡❞ t❤❛t ✐♥❞❡❡❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ ✐s ❝❧♦s❡ t♦ ✇❤❛t✬s ❡①♣❡❝t❡❞✳ ❇✉t t❤❡ r❡❛❧ q✉❡st✐♦♥ ✐s✿ ◮ ❲❤❛t

✐s t❤❡ ❛r❡❛❄

❉♦ ✇❡ ❡✈❡♥ ✉♥❞❡rst❛♥❞ ✇❤❛t ✐t ✐s❄ ❲❛r♥✐♥❣✦ ❚❤❡ ❛r❡❛ ✐s ❛ ♥✉♠❜❡r✳

❚❤❡ ✐♥t✉✐t✐♦♥ ✐s t♦ s♣❡❛❦ ♦❢ t❤❡ ❛♠♦✉♥t ♦❢ ♠❛t❡r✐❛❧ ❤❡❧❞ ❜② t❤❡ ❝✉r✈❡✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ✜❧❧ ❛ s❡♠✐❝✐r❝✉❧❛r ❜✉❝❦❡t ✇✐t❤ ✇❛t❡r ❛♥❞ t❤❡ ♣♦✉r ✐t ✐♥t♦ ❛ r❡❝t❛♥❣✉❧❛r ♦♥❡ s♦ t❤❛t ✇❡ ❝❛♥ ♠❡❛s✉r❡ t❤❡ ❝♦♥t❡♥ts✿

✶✳✶✳

❚❤❡ ❆r❡❛ Pr♦❜❧❡♠

✶✺

❊①❡r❝✐s❡ ✶✳✶✳✸

❈♦♥✜r♠ t❤❛t t❤❡ s❡♠✐❝✐r❝✉❧❛r ❜✉❝❦❡t ❛♥❞ t❤❡ r❡❝t❛♥❣✉❧❛r ♦♥❡ ❝♦♥t❛✐♥ t❤❡ s❛♠❡ ❛♠♦✉♥t ♦❢ ✇❛t❡r✳

❍♦✇ ❞♦ ✇❡ ♠❛❦❡ ♠❛t❤❡♠❛t✐❝❛❧ s❡♥s❡ ♦❢ t❤✐s❄ ❖♥❡ t❤✐♥❣ ✇❡ ❞♦ ❦♥♦✇✳ ❚❤❡ ❛r❡❛ ♦❢ ❛ r❡❝t❛♥❣❧❡

a×b

✐s

ab✳

❋✉rt❤❡r♠♦r❡✱ ❛♥② r✐❣❤t tr✐❛♥❣❧❡ ✐s s✐♠♣❧② ❛ ❤❛❧❢ ♦❢ ❛ ❞✐❛❣♦♥❛❧❧② ❝✉t r❡❝t❛♥❣❧❡✿

❲❡ ❝❛♥ ❛❧s♦ ❝✉t ❛♥② tr✐❛♥❣❧❡ ✐♥t♦ ❛ ♣❛✐r ♦❢ r✐❣❤t tr✐❛♥❣❧❡s✿

❋✐♥❛❧❧②✱ ❛♥② ♣♦❧②❣♦♥ ❝❛♥ ❜❡ ❝✉t ✐♥t♦ tr✐❛♥❣❧❡s✿

❙♦✱ ✇❡ ❝❛♥ ✜♥❞ ✕ ❛♥❞ ✇❡ ✉♥❞❡rst❛♥❞ ✕ t❤❡ ❛r❡❛s ♦❢ ❛❧❧ ♣♦❧②❣♦♥s✳ ❚❤❡② ❛r❡ ❣❡♦♠❡tr✐❝ ♦❜❥❡❝ts ✇✐t❤

str❛✐❣❤t

❡❞❣❡s✳ ❇✉t ✇❤❛t ❛r❡ t❤❡

❛r❡❛s ♦❢ ❝✉r✈❡❞ ♦❜❥❡❝ts❄

❊①❛♠♣❧❡ ✶✳✶✳✹✿ ❝✐r❝❧❡

▲❡t✬s r❡✈✐❡✇ t❤❡ ❛♥❝✐❡♥t ●r❡❡❦s✬ ❛♣♣r♦❛❝❤ t♦ ✉♥❞❡rst❛♥❞✐♥❣ ❛♥❞ ❝♦♠♣✉t✐♥❣ t❤❡ ❛r❡❛ ♦❢ ❛ ❝✐r❝❧❡✳ ❚❤❡② ❛♣♣r♦①✐♠❛t❡❞ t❤❡ ❝✐r❝❧❡ ✇✐t❤ r❡❣✉❧❛r ♣♦❧②❣♦♥s✿ ❡q✉❛❧ s✐❞❡s ❛♥❞ ❛♥❣❧❡s✳ ❲❡ ♣✉t s✉❝❤ ♣♦❧②❣♦♥s ❛r♦✉♥❞ t❤❡ ❝✐r❝❧❡ s♦ t❤❛t ✐t t♦✉❝❤❡s t❤❡♠ ❢r♦♠ t❤❡ ✐♥s✐❞❡ ✭✏❝✐r❝✉♠s❝r✐❜✐♥❣✑ ♣♦❧②❣♦♥s✮✿

❚❤❡ ❆r❡❛ Pr♦❜❧❡♠

✶✳✶✳

❋♦r ❡❛❝❤

✶✻

n = 3, 4, 5, 6, ...✱

t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❝❛rr✐❡❞ ♦✉t✳ ❲❡ s♣❧✐t ❡❛❝❤ s✉❝❤ ♣♦❧②❣♦♥ ✇✐t❤

n

s✐❞❡s ✐♥t♦

2n

r✐❣❤t tr✐❛♥❣❧❡s ❛♥❞ ❝♦♠♣✉t❡ t❤❡✐r ❞✐♠❡♥s✐♦♥s✿

❲❡ t❤✐♥❦ ♦❢ ✐s

2π ✱

π

❤❡r❡ ❛s t❤❡ ❛♥❣❧❡ ♠❡❛s✉r❡ ✭♣♦ss✐❜❧② ✉♥❦♥♦✇♥✮ ♦❢ ❛ ❤❛❧❢ ♦❢ t❤❡ ❢✉❧❧ t✉r♥✳ ❚❤❡ ❢✉❧❧ t✉r♥

❛♥❞ ✐t ✐s ❝✉t ✐♥t♦

2n

❛♥❣❧❡s✿

αn =

2π π = . 2n n

❚❤❡ s✐❞❡ t❤❛t t♦✉❝❤❡s t❤❡ ❝✐r❝❧❡ ✐s ✐ts r❛❞✐✉s ❛♥❞ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡ ✐s

r tan πn ✳

❚❤❡r❡❢♦r❡✱ t❤❡ ❛r❡❛ ♦❢

t❤❡ tr✐❛♥❣❧❡ ✐s

r2 1 π an = · r · r tan n = tan πn . 2 2 ❲❡ ❦♥♦✇ ♥♦✇ t❤❡ ❛r❡❛ ♦❢ t❤❡ ✇❤♦❧❡ ♣♦❧②❣♦♥✿

An = an · 2n =

r2 tan πn · 2n . 2

❲❡ ❝❛♥ ❡①❛♠✐♥❡ t❤❡ ❞❛t❛✿

n 3 4 5 6 7 8 9 10 11 12 13 14 15 ... An 5.196 4.000 3.633 3.464 3.371 3.314 3.276 3.249 3.230 3.215 3.204 3.195 3.188 ... ❚❤❡ ♥✉♠❜❡rs s❡❡♠ t♦

■♥❞❡❡❞✱ t❤❡ s❡q✉❡♥❝❡

♠❡❛♥✐♥❣

❝♦♥✈❡r❣❡ An

✦

✐s ❜♦t❤ ♠♦♥♦t♦♥❡ ❛♥❞ ❜♦✉♥❞❡❞ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❝♦♥✈❡r❣❡♥t✳

♦❢ t❤❡ ❛r❡❛ ✭❛♥❞ ♦❢

π ✮✳

■ts ❧✐♠✐t ✐s t❤❡

✶✳✶✳ ❚❤❡ ❆r❡❛ Pr♦❜❧❡♠

✶✼

▲❡t✬s ❝♦♠♣✉t❡ t❤❡ ❧✐♠✐t✿

r2 tan πn · 2n 2 sin π 1 = πr2 · π n · cos πn n

An =

|| πr2

↓

↓

·1

❲❡ r❡❛rr❛♥❣❡ t❤❡ t❡r♠s ✐♥ ♦r❞❡r t♦ ✉s❡✳✳✳

❛s

n→∞

♦♥❡ ♦❢ t❤❡ ❢❛♠♦✉s tr✐❣ ❧✐♠✐ts ✭❈❤❛♣t❡r ✷❉❈✲✶).

·1

= πr2 . ■♥ ♦r❞❡r t♦ ❢✉❧❧② ❥✉st✐❢② t❤❡ r❡s✉❧t✱ t❤❡ ●r❡❡❦s ❛❧s♦ ♣✉t s✉❝❤ ♣♦❧②❣♦♥s ❛r♦✉♥❞ t❤❡ ❝✐r❝❧❡ s♦ t❤❛t ✐t t♦✉❝❤❡s t❤❡♠ ❢r♦♠ t❤❡ ♦✉ts✐❞❡ ✭✏✐♥s❝r✐❜✐♥❣✑ ♣♦❧②❣♦♥s✮✳ ❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ♠❡t❤♦❞s ❝❛♥♥♦t ❜❡ ❡❛s✐❧② ❛♣♣❧✐❡❞ t♦✱ s❛②✱ ♣❛r❛❜♦❧❛s✳ ❚❤❛t ✐s ✇❤② ✇❡ ✇✐❧❧ s❡❡❦ ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❛♣♣r♦❛❝❤✳

❊①❡r❝✐s❡ ✶✳✶✳✺

❊①♣❧❛✐♥ ✐♥ ❞❡t❛✐❧ ✇❤②

An

❝♦♥✈❡r❣❡s✳

❊①❡r❝✐s❡ ✶✳✶✳✻

❊①♣❧❛✐♥ t❤❡ ♦t❤❡r ❧✐♠✐t ✐♥ ♦✉r ❝♦♠♣✉t❛t✐♦♥✳

❊①❡r❝✐s❡ ✶✳✶✳✼

❈❛rr② ♦✉t t❤✐s ❝♦♥str✉❝t✐♦♥ ❢♦r t❤❡ ✐♥s❝r✐❜❡❞ ♣♦❧②❣♦♥s✳

▲❡t✬s ❝♦♠♣❛r❡ t❤❡s❡ t✇♦ s❡❡♠✐♥❣❧② ✉♥r❡❧❛t❡❞ ♣r♦❜❧❡♠s ❛♥❞ ❤♦✇ t❤❡② ❛r❡ s♦❧✈❡❞✿ ❚❤❡ ❚❛♥❣❡♥t Pr♦❜❧❡♠

❚❤❡ ❆r❡❛ Pr♦❜❧❡♠

●❡♦♠❡tr②✿ ❋♦r ❛ ❣✐✈❡♥ ❝✉r✈❡✱ ✜♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣✿

t❤❡ ❧✐♥❡ t♦✉❝❤✐♥❣ t❤❡ ❝✉r✈❡ ❛t ❛ ♣♦✐♥t

t❤❡ ❛r❡❛ ❡♥❝❧♦s❡❞ ❜② t❤❡ ❝✉r✈❡

❚❤❡ ♣r♦❜❧❡♠s ❛r❡ ❡❛s② ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ❛❝t✉❛❧ ❝✉r✈✐♥❣✿ str❛✐❣❤t ❧✐♥❡s✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡ s♦❧✈❡❞ t❤❡s❡ ♣r♦❜❧❡♠s ❢♦r ❛ s♣❡❝✐✜❝ ❝✉r✈❡✿ t❤❡ ❝✐r❝❧❡✳ ❚❤❡ ❢✉rt❤❡r ♣r♦❣r❡ss ❞❡♣❡♥❞s ♦♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛❧❣❡❜r❛✱ t❤❡ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡ s②st❡♠✱ ❛♥❞ t❤❡ ✐❞❡❛ ♦❢ ❢✉♥❝t✐♦♥ ✭❈❤❛♣t❡rs ✶P❈✲✷✱ ✶P❈✲✸✱ ❛♥❞ ✶P❈✲✹✮✳

✶✳✷✳ ❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s

✶✽

▼♦t✐♦♥✿ ❚❤❡ t✇♦ ♣r♦❜❧❡♠s ❛r❡ ✐♥t❡r♣r❡t❡❞ ❛s ❢♦❧❧♦✇s✿ ❚❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❚❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✈❡❧♦❝✐t② ❢✉♥❝t✐♦♥ t♦ t❤❡ ♣♦s✐t✐♦♥ ❢✉♥❝t✐♦♥ ✐s t❤❡ ✈❡❧♦❝✐t② ❛t t❤❛t ♠♦♠❡♥t✳ ✐s t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✳ ❚❤❡ ❢✉rt❤❡r ❞❡✈❡❧♦♣♠❡♥t ✉t✐❧✐③❡s ❞✐✈✐❞✐♥❣ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐♥t♦ s♠❛❧❧❡r ♣✐❡❝❡s✱ ∆x ❧♦♥❣✱ s❛♠♣❧✐♥❣ t❤❡ ❢✉♥❝t✐♦♥✱ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ ❜② ♠❡❛♥s ♦❢ str❛✐❣❤t ❧✐♥❡s✳

❋♦r ♠❛♥② ♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s✱ t❤✐s st❛❣❡ ✐s s✉✣❝✐❡♥t✳ ■♥ ✐❞❡❛❧✐③❡❞ s✐t✉❛t✐♦♥s✱ ✇❡ ❝❛♥ ❞♦ ♠♦r❡✳

❈❛❧❝✉❧✉s✿ ❚❤❡ ❧✐♠✐ts ♦❢ t❤❡s❡ ❛♣♣r♦①✐♠❛t✐♦♥s✱ ❛s ∆x → 0✱ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ t❤❡ ❞❡r✐✈❛t✐✈❡

t❤❡ ✐♥t❡❣r❛❧

■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐❧❧ ♣✉rs✉❡ t❤❡ ♣❧❛♥ ❢♦r t❤❡ ❧❛tt❡r ♣r♦❜❧❡♠ ❛s ♦✉t❧✐♥❡❞ ✐♥ t❤❡ r✐❣❤t ❝♦❧✉♠♥✳

✶✳✷✳ ❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s ❲❡ t✉r♥ t♦ ♠♦t✐♦♥ ♥♦✇✳ ❖✉r st❛rt✐♥❣ ♣♦✐♥t ✐s t❤❡ ❢♦r♠✉❧❛✿ ❞✐st❛♥❝❡ = s♣❡❡❞ × t✐♠❡ ❘❡❝❛❧❧ t✇♦ ❢❛♠✐❧✐❛r ♣r♦❜❧❡♠s ✭❈❤❛♣t❡r ✷❉❈✲✶✮✳

Pr♦❜❧❡♠✿ ■♠❛❣✐♥❡ t❤❛t ♦✉r s♣❡❡❞♦♠❡t❡r ✐s ❜r♦❦❡♥ ❛♥❞ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛ ✇❛② t♦ ❡st✐♠❛t❡ ❤♦✇ ❢❛st ✇❡ ❛r❡ ❞r✐✈✐♥❣✳

✶✳✷✳

❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s

❲❡ ❧♦♦❦ ❛t t❤❡ ♦❞♦♠❡t❡r

✶✾

s❡✈❡r❛❧ t✐♠❡s ❞✉r✐♥❣ t❤❡ tr✐♣ ❛♥❞ r❡❝♦r❞ t❤❡ ♠✐❧❡❛❣❡ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✿

✶✳ ✐♥✐t✐❛❧ r❡❛❞✐♥❣✿ 10, 000 ♠✐❧❡s ✷✳ ❛❢t❡r t❤❡ ✜rst ❤♦✉r✿ 10, 055 ♠✐❧❡s ✸✳ ❛❢t❡r t❤❡ s❡❝♦♥❞ ❤♦✉r✿ 10, 095 ♠✐❧❡s ✹✳ ❛❢t❡r t❤❡ t❤✐r❞ ❤♦✉r✿ 10, 155 ♠✐❧❡s ✺✳ ❡t❝✳ ❚❤❛t✬s ❛ s❡q✉❡♥❝❡✳ ❲❡ ♥♦✇ ✉s❡ t❤❡

❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ s❡q✉❡♥❝❡ t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠✿

✶✳ ❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❞✉r✐♥❣ t❤❡ ✜rst ❤♦✉r✿ 10, 055 − 10, 000 = 55 ♠✐❧❡s

✷✳ ❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❞✉r✐♥❣ t❤❡ s❡❝♦♥❞ ❤♦✉r✿ 10, 095 − 10, 055 = 40 ♠✐❧❡s ✸✳ ❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❞✉r✐♥❣ t❤❡ t❤✐r❞ ❤♦✉r✿ 10, 155 − 10, 095 = 60 ♠✐❧❡s

✹✳ ❡t❝✳

❚❤❛t✬s ❛♥♦t❤❡r s❡q✉❡♥❝❡✳ ❲❡ s❡❡ ❜❡❧♦✇ ❤♦✇ t❤❡s❡ ♥❡✇ ♥✉♠❜❡rs ❛♣♣❡❛r ❛s t❤❡ ❤❡✐❣❤ts ♦❢ t❤❡ st❡♣s ♦❢ ♦✉r ❧❛st ♣❧♦t ✭t♦♣✮✿

❆s ②♦✉ ❝❛♥ s❡❡✱ ✇❡ ✐❧❧✉str❛t❡ t❤❡ ♥❡✇ ❞❛t❛ ✐♥ s✉❝❤ ❛ ✇❛② ❛s t♦ s✉❣❣❡st t❤❛t t❤❡ s♣❡❡❞ r❡♠❛✐♥s ❞✉r✐♥❣ ❡❛❝❤ ♦❢ t❤❡s❡ ❤♦✉r✲❧♦♥❣ ♣❡r✐♦❞s✳

❝♦♥st❛♥t

■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ✢✐♣ s✐❞❡ ♦❢ t❤❡ ❧❛st ♣r♦❜❧❡♠✳

Pr♦❜❧❡♠✿ ■♠❛❣✐♥❡ t❤❛t ✐t ✐s t❤❡ ♦❞♦♠❡t❡r t❤❛t ✐s ❜r♦❦❡♥ ❛♥❞ ✜♥❞ ❛ ✇❛② t♦ ❡st✐♠❛t❡ ❤♦✇ ❢❛r ✇❡ ✇✐❧❧ ❤❛✈❡ ❣♦♥❡✳

❖❢ ❝♦✉rs❡✱ ✇❡ ❧♦♦❦ ❛t t❤❡ s♣❡❡❞♦♠❡t❡r r❡❛❞✐♥❣s ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✿

s❡✈❡r❛❧ t✐♠❡s ✕ s❛②✱ ❡✈❡r② ❤♦✉r ✕ ❞✉r✐♥❣ t❤❡ tr✐♣ ❛♥❞ r❡❝♦r❞ ✐ts

✶✳ ❞✉r✐♥❣ t❤❡ ✜rst ❤♦✉r✿ 35 ♠✐❧❡s ❛♥ ❤♦✉r ✷✳ ❞✉r✐♥❣ t❤❡ s❡❝♦♥❞ ❤♦✉r✿ 65 ♠✐❧❡s ❛♥ ❤♦✉r ✸✳ ❞✉r✐♥❣ t❤❡ t❤✐r❞ ❤♦✉r✿ 50 ♠✐❧❡s ❛♥ ❤♦✉r ✹✳ ❡t❝✳ ❚❤❛t✬s ❛ s❡q✉❡♥❝❡✳ ❲❤❛t ❞♦❡s t❤✐s t❡❧❧ ✉s ❛❜♦✉t ♦✉r ❧♦❝❛t✐♦♥❄ ◆♦t❤✐♥❣✱ ✇✐t❤♦✉t ❛❧❣❡❜r❛✦ ❋♦rt✉♥❛t❡❧②✱ ✇❡ ❝❛♥ ❥✉st ✉s❡ t❤❡ s❛♠❡ ❢♦r♠✉❧❛ ❛s ❜❡❢♦r❡✳ ■♥ ❝♦♥tr❛st t♦ t❤❡ ❢♦r♠❡r ♣r♦❜❧❡♠✱ ✇❡ ♥❡❡❞ ❛♥♦t❤❡r ❜✐t ♦❢ ✐♥❢♦r♠❛t✐♦♥✳ ❲❡ ♠✉st ❦♥♦✇ t❤❡ st❛rt✐♥❣ ♣♦✐♥t ♦❢ ♦✉r tr✐♣✱ s❛②✱ t❤❡ 100✲♠✐❧❡ ♠❛r❦✳ ❚❤❡ t✐♠❡ ✐♥t❡r✈❛❧ ✇❛s ❝❤♦s❡♥ t♦ ❜❡ 1 ❤♦✉r s♦ t❤❛t ✇❡ ♥❡❡❞ ♦♥❧② t♦ ❛❞❞✱ ❛♥❞ ❦❡❡♣ ❛❞❞✐♥❣✱ t❤❡ s♣❡❡❞ ❛t ✇❤✐❝❤ ✕ ✇❡ ❛ss✉♠❡ ✕ ✇❡ ❞r♦✈❡ ❞✉r✐♥❣ ❡❛❝❤ ♦❢ t❤❡s❡ ♦♥❡✲❤♦✉r ♣❡r✐♦❞s✿ ✶✳ t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✿ 100✲♠✐❧❡ ♠❛r❦ ✷✳ t❤❡ ❧♦❝❛t✐♦♥ ❛❢t❡r t❤❡ ✜rst ❤♦✉r✿ 100 + 35 = 135✲♠✐❧❡ ♠❛r❦ ✸✳ t❤❡ ❧♦❝❛t✐♦♥ ❛❢t❡r t❤❡ s❡❝♦♥❞ ❤♦✉r✿ 135 + 65 = 200✲♠✐❧❡ ♠❛r❦ ✹✳ t❤❡ ❧♦❝❛t✐♦♥ ❛❢t❡r t❤❡ t❤✐r❞ ❤♦✉r✿ 200 + 50 = 250✲♠✐❧❡ ♠❛r❦

✶✳✷✳ ❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s

✷✵

✺✳ ❡t❝✳ ❚❤❛t✬s ❛♥♦t❤❡r s❡q✉❡♥❝❡✳ ■t ✐s ❝❛❧❧❡❞ t❤❡ s✉♠ ♦❢ t❤❡ s❡q✉❡♥❝❡✳ ■♥ ♦r❞❡r t♦ ✐❧❧✉str❛t❡ t❤✐s ❛❧❣❡❜r❛✱ ✇❡ ✉s❡ t❤❡ s♣❡❡❞s ❛s t❤❡ ❤❡✐❣❤ts ♦❢ t❤❡ ❝♦♥s❡❝✉t✐✈❡ st❡♣s ♦❢ t❤❡ st❛✐r❝❛s❡✿

❚❤❡♥ t❤❡ ♥❡✇ ♥✉♠❜❡rs s❤♦✇ ❤♦✇ ❤✐❣❤ ✇❡ ❤❛✈❡ t♦ ❝❧✐♠❜ ✐♥ ♦✉r ❧❛st ♣❧♦t✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞✦ ❲❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ t❤❛t ✇❡ ❤❛✈❡ ♣r♦❣r❡ss❡❞ t❤r♦✉❣❤ t❤❡ r♦✉❣❤❧② 135✲✱ 200✲✱ ❛♥❞ 250✲♠✐❧❡ ♠❛r❦s ❞✉r✐♥❣ t❤✐s t✐♠❡✳ ■♥ s✉♠♠❛r②✱ ✇❤❡♥ t❤❡ ✐♥t❡r✈❛❧s ♦❢ t✐♠❡ ❛r❡ ✉♥✐ts✱ ✇❡ ❝❛♥ ❣♦ ❢r♦♠ ❧♦❝❛t✐♦♥s t♦ ✈❡❧♦❝✐t✐❡s ❛♥❞ ❜❛❝❦ ✇✐t❤ t❤❡ t✇♦ s✐♠♣❧❡ ♦♣❡r❛t✐♦♥s✿

❚❤✐s ✐s t❤❡ s✉♠♠❛r②✿ ✶✳ ■❢ ❡❛❝❤ t❡r♠ ♦❢ ❛ s❡q✉❡♥❝❡ r❡♣r❡s❡♥ts ❛ ❧♦❝❛t✐♦♥✱ t❤❡ ♣❛✐r✇✐s❡ ❞✐✛❡r❡♥❝❡s ✇✐❧❧ ❣✐✈❡ ②♦✉ t❤❡ ✈❡❧♦❝✐t✐❡s✳ ❆ ❜r♦❦❡♥ s♣❡❡❞♦♠❡t❡r ✐s s✉❜st✐t✉t❡❞ ✇✐t❤ ❛♥ ♦❞♦♠❡t❡r ❛♥❞ ❛ ✇❛t❝❤✳ ❚❤✐s st✉❞② ✐s ❛♣♣❡❛r❡❞ ✐♥ ❱♦❧✉♠❡ ✷✱ ❈❤❛♣t❡r ✷❉❈✲✸✳ ✷✳ ■❢ ❡❛❝❤ t❡r♠ ♦❢ ❛ s❡q✉❡♥❝❡ r❡♣r❡s❡♥ts ❛ ✈❡❧♦❝✐t②✱ t❤❡✐r s✉♠ ✉♣ t♦ t❤❛t ♣♦✐♥t ✇✐❧❧ ❣✐✈❡ ②♦✉ t❤❡ ❧♦❝❛t✐♦♥✳ ❆ ❜r♦❦❡♥ ♦❞♦♠❡t❡r ✐s s✉❜st✐t✉t❡❞ ✇✐t❤ ❛ s♣❡❡❞♦♠❡t❡r ❛♥❞ ❛ ✇❛t❝❤✳ ❚❤✐s st✉❞② ✇✐❧❧ ❛♣♣❡❛r ✐♥ t❤✐s ❝❤❛♣t❡r✳ ■♥ t❤❡ ❛❜str❛❝t✱ t❤❡ ♣❛✐r✇✐s❡ ❞✐✛❡r❡♥❝❡s r❡♣r❡s❡♥t t❤❡ ❝❤❛♥❣❡ ✇✐t❤✐♥ t❤❡ s❡q✉❡♥❝❡✱ ❢r♦♠ ❡❛❝❤ ♦❢ ✐ts t❡r♠s t♦ t❤❡ ♥❡①t✿

❉❡✜♥✐t✐♦♥ ✶✳✷✳✶✿ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ❋♦r ❛ s❡q✉❡♥❝❡ an ✱ ✐ts s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s✱ ♦r s✐♠♣❧② t❤❡ ❞✐✛❡r❡♥❝❡✱ ✐s ❛ ♥❡✇ s❡q✉❡♥❝❡✱ s❛② dn ✱ ❞❡✜♥❡❞ ❢♦r ❡❛❝❤ n ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ dn = an+1 − an .

■t ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ ∆an = an+1 − an

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ st❛rt ♦♥ t❤❡ ♣❛t❤ ♦❢ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛♥ ✐❞❡❛ t❤❛t ❝✉❧♠✐♥❛t❡s ✇✐t❤ t❤❡ s❡❝♦♥❞ ❢✉♥❞❛♠❡♥t❛❧ ❝♦♥❝❡♣t ♦❢ ❝❛❧❝✉❧✉s t♦ ❜❡ s❡❡♥ ❧❛t❡r ✐♥ t❤✐s ❝❤❛♣t❡r✳

✶✳✷✳

❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s

✷✶

❚❤❡ s✉♠ r❡♣r❡s❡♥ts t❤❡ t♦t❛❧✐t② ♦❢ t❤❡ ✏❜❡❣✐♥♥✐♥❣✑ ♦❢ ❛ s❡q✉❡♥❝❡✱ ❢♦✉♥❞ ❜② ❛❞❞✐♥❣ ❡❛❝❤ ♦❢ ✐ts t❡r♠s t♦ t❤❡ ♥❡①t✱ ✉♣ t♦ t❤❛t ♣♦✐♥t✳

❊①❛♠♣❧❡ ✶✳✷✳✷✿ s❡q✉❡♥❝❡s ❣✐✈❡♥ ❜② ❧✐sts ❲❡ ❥✉st ❛❞❞ t❤❡ ❝✉rr❡♥t t❡r♠ t♦ ✇❤❛t ✇❡ ❤❛✈❡ ❛❝❝✉♠✉❧❛t❡❞ s♦ ❢❛r✿ s❡q✉❡♥❝❡✿ s✉♠s✿

♥❡✇ s❡q✉❡♥❝❡✿

2 4 7 1 −1 ↓ ↓ ↓ ↓ ↓ 2 2 + 4 = 6 6 + 7 = 13 13 + 1 = 14 14 + (−1) = 13 ↓ ↓ ↓ ↓ ↓ 2 6 13 14 13

... ...

... ...

❲❡ ❤❛✈❡ ❛ ♥❡✇ ❧✐st✦

❊①❛♠♣❧❡ ✶✳✷✳✸✿ s❡q✉❡♥❝❡s ❣✐✈❡♥ ❜② ❣r❛♣❤s ❲❡ tr❡❛t t❤❡ ❣r❛♣❤ ♦❢ ❛ s❡q✉❡♥❝❡ ❛s ✐❢ ♠❛❞❡ ♦❢ ❜❛rs ❛♥❞ t❤❡♥ ❥✉st st❛❝❦ ✉♣ t❤❡s❡ ❜❛rs ♦♥ t♦♣ ♦❢ ❡❛❝❤ ♦t❤❡r ♦♥❡ ❜② ♦♥❡✿

❚❤❡s❡ st❛❝❦❡❞ ❜❛rs ✕ ♦r r❛t❤❡r t❤❡ ♣r♦❝❡ss ♦❢ st❛❝❦✐♥❣ ✕ ♠❛❦❡ ❛ ♥❡✇ s❡q✉❡♥❝❡✳

❉❡✜♥✐t✐♦♥ ✶✳✷✳✹✿ s❡q✉❡♥❝❡ ♦❢ s✉♠s an ✱ ✐ts s❡q✉❡♥❝❡ ♦❢ s✉♠s✱ ♦r s✐♠♣❧② t❤❡ s✉♠✱ ✐s ❛ ♥❡✇ s❡q✉❡♥❝❡ sn ❞❡♥♦t❡❞ ❢♦r ❡❛❝❤ n ≥ m ✇✐t❤✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ an ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

❋♦r ❛ s❡q✉❡♥❝❡ ❞❡✜♥❡❞ ❛♥❞

✭r❡❝✉rs✐✈❡✮ ❢♦r♠✉❧❛✿

sm = 0,

sn+1 = sn + an+1

■♥ ♦t❤❡r ✇♦r❞s✱

sn = am + am+1 + ... + an

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥ ❛❜❜r❡✈✐❛t❡❞ ✇❛② t♦ ♣r❡s❡♥t s✉♠♠❛t✐♦♥ ♦❢ ❛ s❡q✉❡♥❝❡✳

❙✐❣♠❛ ♥♦t❛t✐♦♥ ❢♦r s✉♠♠❛t✐♦♥ sn = am + am+1 + ... + an =

n X

ak

k=m

❲❛r♥✐♥❣✦ ❯s✐♥❣ ❡✐t❤❡r ✏✳✳✳✑ ❛♥❞ ✏

X

✑ ♠✐❣❤t ♦❜s❝✉r❡ t❤❡ r❡✲

❝✉rs✐✈❡ ♥❛t✉r❡ ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥✳

✶✳✷✳

❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s

✷✷

▲❡t✬s t❛❦❡ ❛ ❝❧♦s❡r ❧♦♦❦ ❛t t❤❡ ♥❡✇ ♥♦t❛t✐♦♥✳ ❚❤❡ ✜rst ❝❤♦✐❝❡ ♦❢ ❤♦✇ t♦ r❡♣r❡s❡♥t t❤❡ s✉♠ ♦❢ ❛ s❡❣♠❡♥t ✕ ❢r♦♠

m

t♦

n

✕ ♦❢ ❛ s❡q✉❡♥❝❡

an

✐s t❤✐s✿

+am+1 +... | {z }

am |{z}

st❡♣ 1

+a +... |{z}k

st❡♣ k

st❡♣ 2

+a |{z}n

st❡♣ n−m

❚❤✐s ♥♦t❛t✐♦♥ r❡✢❡❝ts t❤❡ r❡❝✉rs✐✈❡ ♥❛t✉r❡ ♦❢ t❤❡ ♣r♦❝❡ss ❜✉t ✐t ❝❛♥ ❛❧s♦ ❜❡ r❡♣❡t✐t✐✈❡ ❛♥❞ ❝✉♠❜❡rs♦♠❡✳ ❚❤❡ ♥❡✇ ♥♦t❛t✐♦♥ ✐s ♠❡❛♥t t♦ ♠❛❦❡ ✐t ♠♦r❡ ❝♦♠♣❛❝t✳ ❚❤❡ ✐❞❡❛ ✐s t♦ ✐♥tr♦❞✉❝❡ ❛♥ ✏✐♥t❡r♥❛❧ ✈❛r✐❛❜❧❡✑

k

❛s ❢♦❧❧♦✇s✿

❙✐❣♠❛ ♥♦t❛t✐♦♥ ❜❡❣✐♥♥✐♥❣

3 X k=0

 k 2 + k = 20

❛♥❞ ❡♥❞ ✈❛❧✉❡s ❢♦r

↓ 3 X

−→

k2 + k

k=0

↑

k



= 20

❛ s♣❡❝✐✜❝ s❡q✉❡♥❝❡ ♦❢

k

↑

❛ s♣❡❝✐✜❝ ♥✉♠❜❡r

❲❛r♥✐♥❣✦ ■t ✇♦✉❧❞ ❛❧s♦ ♠❛❦❡ s❡♥s❡ t♦ ❤❛✈❡ ✏ k

= 3✑

❛❜♦✈❡ t❤❡

s✐❣♠❛✿

k=3 X k=0

❍❡r❡ t❤❡ ●r❡❡❦ ❧❡tt❡r

Σ

 k2 + k .

st❛♥❞s ❢♦r t❤❡ ❧❡tt❡r ❙ ♠❡❛♥✐♥❣ ✏s✉♠✑✳

❊①❛♠♣❧❡ ✶✳✷✳✺✿ ❡①♣❛♥❞✐♥❣ ❢r♦♠ s✐❣♠❛ ♥♦t❛t✐♦♥ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ❛❜♦✈❡ ✐s ❡①♣❛♥❞❡❞ ❤❡r❡✿

k k2 + k 3 X

2



k +k =

k=0

0 02 + 0 = 0

+

1 12 + 1 = 2

+

2

2 2 +2 =6

+

3 32 + 3 = 12 = 20

❊①❡r❝✐s❡ ✶✳✷✳✻✿ ❝♦♥tr❛❝t✐♥❣ t♦ s✐❣♠❛ ♥♦t❛t✐♦♥ ❍♦✇ ✇✐❧❧ t❤❡ s✉♠ ❝❤❛♥❣❡ ✐❢ ✇❡ r❡♣❧❛❝❡ ✇✐t❤

k=0

✇✐t❤

k = 1✱

♦r

4❄

k = −1❄

❲❤❛t ✐❢ ✇❡ r❡♣❧❛❝❡

3

❛t t❤❡ t♦♣

❊①❛♠♣❧❡ ✶✳✷✳✼✿ ❝♦♥tr❛❝t✐♥❣ s✉♠♠❛t✐♦♥ ❚❤✐s ✐s ❤♦✇ ✇❡

❝♦♥tr❛❝t

t❤❡ s✉♠♠❛t✐♦♥✿

12 + 22 + 32 + ... + 172 =

17 X

k2 .

k=1

❚❤✐s ✐s ♦♥❧② ♣♦ss✐❜❧❡ ✐❢ ✇❡ ✜♥❞ t❤❡ ❆♥❞ t❤✐s ✐s ❤♦✇ ✇❡

❡①♣❛♥❞

nt❤✲t❡r♠

❢♦r♠✉❧❛ ❢♦r t❤❡ s❡q✉❡♥❝❡❀ ✐♥ t❤✐s ❝❛s❡✱

ak = k 2 ✳

❜❛❝❦ ❢r♦♠ t❤✐s ❝♦♠♣❛❝t ♥♦t❛t✐♦♥✱ ❜② ♣❧✉❣❣✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢

k = 1, 2, ..., 17

✶✳✷✳

❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s

✷✸

✐♥t♦ t❤❡ ❢♦r♠✉❧❛✿

17 X k=1

172 . 32 +... + |{z} 22 + |{z} k 2 = |{z} 12 + |{z} k=2

k=1

k=17

k=3

❙✐♠✐❧❛r❧②✱ ✇❡ ❤❛✈❡✿

10

X 1 1 1 1 1 . 1 + + 2 + 3 + ... + 10 = k 2 2 2 2 2 k=0 ❊①❡r❝✐s❡ ✶✳✷✳✽

❈♦♥✜r♠ t❤❛t ✇❡ ❝❛♥ st❛rt ❛t ❛♥② ♦t❤❡r ✐♥✐t✐❛❧ ✐♥❞❡① ✐❢ ✇❡ ❥✉st ♠♦❞✐❢② t❤❡ ❢♦r♠✉❧❛✿

?

1+

?

X 1 X 1 1 1 1 1 = = ... + 2 + 3 + ... + 10 = k−1 k−2 2 2 2 2 2 2 k=? k=?

❊①❡r❝✐s❡ ✶✳✷✳✾

❈♦♥tr❛❝t t❤✐s s✉♠♠❛t✐♦♥✿

1+

1 1 1 + + =? 3 9 27

❊①❡r❝✐s❡ ✶✳✷✳✶✵

❊①♣❛♥❞ t❤✐s s✉♠♠❛t✐♦♥✿

4 X

(k/2) = ?

k=0

❊①❡r❝✐s❡ ✶✳✷✳✶✶

❘❡✇r✐t❡ ✉s✐♥❣ t❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✿ ✶✳ ✷✳ ✸✳ ✹✳ ✺✳ ✻✳ ✼✳

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 0.9 + 0.99 + 0.999 + 0.9999 1/2 − 1/4 + 1/8 − 1/16 1 + 1/2 + 1/3 + 1/4 + ... + 1/n 1 + 1/2 + 1/4 + 1/8 2 + 3 + 5 + 7 + 11 + 13 + 17 1 − 4 + 9 − 16 + 25

❚❤❡ ♥♦t❛t✐♦♥ ❛♣♣❧✐❡s t♦ ❛❧❧ s❡q✉❡♥❝❡s✱ ❜♦t❤ ✜♥✐t❡ ❛♥❞ ✐♥✜♥✐t❡✳ ❋♦r ✐♥✜♥✐t❡ s❡q✉❡♥❝❡s✱ r❡❝♦❣♥✐③❡❞ ❜② ✏✳✳✳✑ ❛t t❤❡ ❡♥❞✱ t❤❡ s✉♠ s❡q✉❡♥❝❡ ✐s ❛❧s♦ ❝❛❧❧❡❞ ✏♣❛rt✐❛❧ s✉♠s✑ ❛s ✇❡❧❧ ❛s ✏s❡r✐❡s✑ ✭t♦ ❜❡ ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✹✮✳ ❚❤✐s ✐s t❤❡ r❡❝✉rs✐✈❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s r❡✲✇r✐tt❡♥ ✇✐t❤ t❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✿

✶✳✷✳

❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s

✷✹

❙❡q✉❡♥❝❡ ♦❢ s✉♠s

❛ s❡q✉❡♥❝❡✿

✐ts s✉♠s✿

t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s✿

t❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✿

a1 ↓ a1 a1

a2 ↓

a3 ↓

+ a2 =

↓ s1 || 1 X ak

s2 s2

+ a3 =

↓ s2 || 2 X ak

k=1

a4 ↓ s3 s3 ↓ s3 || 3 X ak

k=1

k=1

+ a4 =

... ...

s4 s4 ↓ s4 || 4 X ak

... ... ... ... ... ...

k=1

❊①❛♠♣❧❡ ✶✳✷✳✶✷✿ s✉♠s ❛r❡ ❞✐s♣❧❛❝❡♠❡♥ts ❲❡ ❝❛♥ ✉s❡ ❝♦♠♣✉t❡rs t♦ s♣❡❡❞ ✉♣ t❤❡s❡ ❝♦♠♣✉t❛t✐♦♥s✳

❋♦r ❡①❛♠♣❧❡✱ ♦♥❡ ♠❛② ❤❛✈❡ ❜❡❡♥ r❡❝♦r❞✐♥❣

♦♥❡✬s ✈❡❧♦❝✐t✐❡s ❛♥❞ ♥♦✇ ❧♦♦❦✐♥❣ ❢♦r t❤❡ ❧♦❝❛t✐♦♥✳ ❚❤✐s ✐s ❛ ❢♦r♠✉❧❛ ❢♦r ❛ s♣r❡❛❞s❤❡❡t ✭t❤❡ ❧♦❝❛t✐♦♥s✮✿

❂❘❬✲✶❪❈✰❘❈❬✲✶❪ ❲❤❡t❤❡r t❤❡ s❡q✉❡♥❝❡ ❝♦♠❡s ❢r♦♠ ❛ ❢♦r♠✉❧❛ ♦r ✐t✬s ❥✉st ❛ ❧✐st ♦❢ ♥✉♠❜❡rs✱ t❤❡ ❢♦r♠✉❧❛ ❛♣♣❧✐❡s✿

❆s ❛ r❡s✉❧t✱ ❛ ❝✉r✈❡ ❤❛s ♣r♦❞✉❝❡❞ ❛ ♥❡✇ ❝✉r✈❡✿

❊①❛♠♣❧❡ ✶✳✷✳✶✸✿ t❤r❡❡ r✉♥♥❡rs✱ ❝♦♥t✐♥✉❡❞ ❚❤❡ ❣r❛♣❤ s❤♦✇s t❤❡ ✈❡❧♦❝✐t✐❡s ♦❢ t❤r❡❡ r✉♥♥❡rs ✐♥ t❡r♠s ♦❢ t✐♠❡✱

n✿

✶✳✷✳

❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s

■t✬s ❡❛s② t♦ ❞❡s❝r✐❜❡

• A • B • C ❇✉t

❤♦✇

✷✺

t❤❡② ❛r❡ ♠♦✈✐♥❣✿

st❛rts ❢❛st ❛♥❞ t❤❡♥ s❧♦✇s ❞♦✇♥✳ ♠❛✐♥t❛✐♥s t❤❡ s❛♠❡ s♣❡❡❞✳ st❛rts ❧❛t❡ ❛♥❞ t❤❡♥ r✉♥s ❢❛st✳

✇❤❡r❡

❛r❡ t❤❡②✱ ❛t ❡✈❡r② ♠♦♠❡♥t❄

B ❛♥❞ A ✇♦✉❧❞ r❡q✉✐r❡ ♠♦r❡ s✉❜t❧❡ ❛♥❛❧②s✐s✳

❆ s✐♠♣❧❡ ❡①❛♠✐♥❛t✐♦♥ ♦❢ t❤❡ ✜rst ❣r❛♣❤ s❤♦✇s t❤❛t

C ✇✐❧❧ ❛rr✐✈❡ ❛t t❤❡ ✜♥✐s❤ ❧✐♥❡ ❛t t❤❡ s❛♠❡ t✐♠❡✳

❚♦ s❛② t❤❛t ❛❜♦✉t

❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ♣♦ss✐❜❧❡ ❛♥s✇❡rs✿

❲❤✐❝❤ ♦♥❡ ✐s t❤❡ r✐❣❤t ♦♥❡ ❞❡♣❡♥❞s ♦♥ t❤❡ st❛rt✐♥❣ ♣♦✐♥t✳ ❋✉rt❤❡r♠♦r❡✱ ✐❢ t❤❡ r❡q✉✐r❡♠❡♥t t❤❛t t❤❡② ❛❧❧ st❛rt ❛t t❤❡ s❛♠❡ ❧♦❝❛t✐♦♥ ✐s ❧✐❢t❡❞✱ t❤❡ r❡s✉❧t ✇✐❧❧ ❜❡ ❞✐✛❡r❡♥t✱ ❢♦r ❡①❛♠♣❧❡✿

❊①❡r❝✐s❡ ✶✳✷✳✶✹

❙✉❣❣❡st ♦t❤❡r ❣r❛♣❤s t❤❛t ♠❛t❝❤ t❤❡ ❞❡s❝r✐♣t✐♦♥ ❛❜♦✈❡✳

❊①❡r❝✐s❡ ✶✳✷✳✶✺

P❧♦t t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ tr✐♣✿ ✏■ ❞r♦✈❡ s❧♦✇❧②✱ ❣r❛❞✉❛❧❧② s♣❡❞ ✉♣✱ st♦♣♣❡❞ ❢♦r ❛ ✈❡r② s❤♦rt ♠♦♠❡♥t❀ st❛rt❡❞ ❛❣❛✐♥ ❜✉t ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✱ q✉✐❝❦❧② ❛❝❝❡❧❡r❛t❡❞✱ ❛♥❞ ❢r♦♠ t❤❛t ♣♦✐♥t ♠❛✐♥t❛✐♥❡❞ t❤❡ s♣❡❡❞✳✑ ▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ st♦r② ❛♥❞ r❡♣❡❛t t❤❡ t❛s❦✳

❊①❡r❝✐s❡ ✶✳✷✳✶✻

❉r❛✇ ❛ ❝✉r✈❡ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r✱ ✐♠❛❣✐♥❡ t❤❛t ✐t r❡♣r❡s❡♥ts ②♦✉r ✈❡❧♦❝✐t②✱ ❛♥❞ t❤❡♥ s❦❡t❝❤ ✇❤❛t ②♦✉r ❧♦❝❛t✐♦♥s ✇♦✉❧❞ ❧♦♦❦ ❧✐❦❡✳ ❘❡♣❡❛t✳

❍♦✇ ❞♦ ✇❡ ❞❡❛❧ ✇✐t❤ ♠♦t✐♦♥ ✇❤❡♥ t❤❡ t✐♠❡ ♠♦♠❡♥ts ❛r❡♥✬t ✐♥t❡❣❡rs❄ ❲❤❛t ✐s t❤❡ ❙✉♣♣♦s❡

• xn

✐s t❤❡ s❡q✉❡♥❝❡ ♦❢ ♠♦♠❡♥ts ♦❢ t✐♠❡✱ ❛♥❞

• vn

✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✈❡❧♦❝✐t✐❡s✳

❚❤❡② ❛r❡ ✐❧❧✉str❛t❡❞ ❛s ❢♦❧❧♦✇s✿

❞✐s♣❧❛❝❡♠❡♥t

t❤❡♥❄

✶✳✷✳ ❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s

✷✻

❚❤❡ ✈❡❧♦❝✐t✐❡s ❛r❡ t❤❡ ❤❡✐❣❤ts ♦❢ t❤❡ r❡❝t❛♥❣❧❡s✳ ❚❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ✐♥t❡r✈❛❧s ❛r❡ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ❜❛s❡s ♦❢ t❤❡ r❡❝t❛♥❣❧❡s✿ ∆xn = xn − xn−1 .

❆s ✇❡ ❦♥♦✇✱ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❞✉r✐♥❣ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧ [xn−1 , xn ] ✐s t❤❡ ♣r♦❞✉❝t ♦❢ vn ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ xn ✿ ❞✐s♣❧❛❝❡♠❡♥t = vn · ∆xn . ❚❤❡♥ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ❢♦r♠ ❛ s❡q✉❡♥❝❡✿ ❞✐s♣❧❛❝❡♠❡♥ts✿ v1 · ∆x1 , v2 · ∆x2 , ..., vn · ∆xn , ...

❚❤❡♥ t❤❡ t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t ❞✉r✐♥❣ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧ [x0 , xn ] ✐s t❤❡ s✉♠ ♦❢ t❤✐s s❡q✉❡♥❝❡✿ t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t = v1 · ∆x1 + v2 · ∆x2 + ... + vn · ∆xn =

n X k=1

vk · ∆xk .

❲✐t❤ t❤✐s ✐❞❡❛✱ ✇❡ st❛rt t♦ ❞❡✈❡❧♦♣ t❤❡ ♠❛t❤❡♠❛t✐❝s ♦❢ t❤❡ ❜❛❝❦✲❛♥❞✲❢♦rt❤ ✐♥t❡r❛❝t✐♦♥ ✇✐t❤✐♥ t❤❡s❡ t✇♦ ♣❛✐rs✿ • ♣♦s✐t✐♦♥s ❛♥❞ ✈❡❧♦❝✐t✐❡s • t❛♥❣❡♥ts ❛♥❞ ❛r❡❛s

❚❤❡ ❣❡♥❡r❛❧ s❡t✉♣ ❢♦❧❧♦✇s✳ ❚❤❡r❡ ✇✐❧❧ ❜❡ ♥♦ r❡str✐❝t✐♦♥s ✇❤❛ts♦❡✈❡r ♦♥ t❤❡ s❡❣♠❡♥ts ♦r t❤❡ s❛♠♣❧❡ ♣♦✐♥ts ✐♥ t❤✐s ❝♦♥str✉❝t✐♦♥✳ ❋✐rst✱ ✇❡ ♥❡❡❞ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b]✳ ❲❡ ❝❤♦♦s❡ ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r n ❛♥❞ t❤❡♥ ♣❧❛❝❡ n + 1 ♣♦✐♥ts ♦♥ t❤❡ ✐♥t❡r✈❛❧✿ ■t✬s ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ xi ✿

a = x0 ≤ x1 ≤ x2 ≤ ... ≤ xn−1 ≤ xn = b .

❆s ❛ r❡s✉❧t✱ t❤❡ ✐♥t❡r✈❛❧ ✐s ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ n s♠❛❧❧❡r ✐♥t❡r✈❛❧s ♦❢ ♣♦ss✐❜❧② ❞✐✛❡r❡♥t ❧❡♥❣t❤s✿ [x0 , x1 ], [x1 , x2 ], ..., [xn−1 , xn ] .

❉❡✜♥✐t✐♦♥ ✶✳✷✳✶✼✿ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ ❆ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b] ✐s ✐ts r❡♣r❡s❡♥t❛t✐♦♥ ❛s t❤❡ ✉♥✐♦♥ ♦❢ ✐♥t❡r✈❛❧s t❤❛t ✐♥t❡rs❡❝t ♦♥❧② ❛t t❤❡ ❡♥❞✲♣♦✐♥ts✿ [a, b] = [x0 , x1 ] ∪ [x1 , x2 ] ∪ ... ∪ [xn−1 , xn ] .

❚❤❡s❡ ❡♥❞✲♣♦✐♥ts ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤✐s s❡q✉❡♥❝❡ ❣✐✈❡s ✉s t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡s❡ ✐♥t❡r✈❛❧s✳

✶✳✷✳ ❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s

✷✼

❉❡✜♥✐t✐♦♥ ✶✳✷✳✶✽✿ ✐♥❝r❡♠❡♥ts ♦❢ ♣❛rt✐t✐♦♥ ❚❤❡ ✐♥❝r❡♠❡♥ts ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛r❡ ❣✐✈❡♥ ❜②✿ ∆xi = xi − xi−1 , i = 1, 2, ..., n .

■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ♥♦❞❡s✱ t❤❡ ♣r✐♠❛r② ♥♦❞❡s✱ ✇❡ ♠❛② ❛❧s♦ ❜❡ ❣✐✈❡♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ✐♥ ❡❛❝❤ ✐♥t❡r✈❛❧ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✿ c1 ✐♥ [x0 , x1 ], c2 ✐♥ [x1 , x2 ], ..., cn ✐♥ [xn−1 , xn ] .

■♥ s✉♠♠❛r②✿

❉❡✜♥✐t✐♦♥ ✶✳✷✳✶✾✿ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ ❆♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥✱ ♦r s✐♠♣❧② ❛ ♣❛rt✐t✐♦♥✱ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b] ❝♦♥s✐sts ♦❢ t✇♦ s❡q✉❡♥❝❡s✿ ✶✳ ♣r✐♠❛r② ♥♦❞❡s a = x0 , x1 , x2 , ..., xn−1 , xn = b ❀ ✷✳ s❡❝♦♥❞❛r② ♥♦❞❡s c1 , c2 , c3 , ..., cn−1 , cn ❀ t❤❛t s❛t✐s❢② t❤❡s❡ ✐♥❡q✉❛❧✐t✐❡s✿ x0 ≤ c1 ≤ x1 ≤ c2 ≤ x2 ≤ ... ≤ xn−1 ≤ cn ≤ xn .

❲❛r♥✐♥❣✦ ❲❡ ❝❛♥ ❝❤♦s❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❢r♦♠ t❤❡ ❧✐st ♦❢ ♣r✐✲ ♠❛r② ♥♦❞❡s ❜❡❝❛✉s❡ t❤❡ ✐♥❡q✉❛❧✐t✐❡s ❛r❡ ♥♦♥✲str✐❝t✳

❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ✭✐✳❡✳✱ t❤❡ r✐s❡s✮ ✐s ♦✉t❧✐♥❡❞ ❜❡❧♦✇✿

✶✳✷✳ ❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s

✷✽

❚❤❡s❡ ❛r❡ t❤❡ st❛❣❡s t❤❛t ✇❡ s❡❡ ❤❡r❡✿ ✶✳ ❛ ❢✉♥❝t✐♦♥ ✷✳ ❛ ♣❛rt✐t✐♦♥ ❛♥❞ ✐ts ♣r✐♠❛r② ♥♦❞❡s ✸✳ s❛♠♣❧✐♥❣ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ✹✳ r❡♠♦✈✐♥❣ t❤❡ r❡st ♦❢ t❤❡ ❣r❛♣❤ ✺✳ ♣❧♦tt✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡s ✭r✐s❡s✮ ✻✳ ♣❧❛❝✐♥❣ t❤❡s❡ ❞✐✛❡r❡♥❝❡s ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❚❤❡ r❡s✉❧t ✐s ❛ ♥❡✇ ❢✉♥❝t✐♦♥✳ ■t ✐s ❞❡✜♥❡❞ ❛❧❣❡❜r❛✐❝❛❧❧② ❛s ❢♦❧❧♦✇s✿

❉❡✜♥✐t✐♦♥ ✶✳✷✳✷✵✿ ❞✐✛❡r❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥ ❙✉♣♣♦s❡ y = f (x) ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s xk , k = 0, 1, 2, ..., n✱ ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❚❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ f ✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ ❡❛❝❤ ✐♥t❡r✈❛❧ ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛♥❞✱ ❝♦rr❡s♣♦♥❞✐♥❣❧②✱ ❛t ❡✈❡r② s❡❝♦♥❞❛r② ♥♦❞❡ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ❛♥❞ ❞❡♥♦t❡❞✱ ❛s ❢♦❧❧♦✇s✿ ∆ f [x

k−1 ,xk ]

= ∆f (ck ) = f (xk+1 ) − f (xk )

▲❡t✬s r❡♠❡♠❜❡r t❤❛t ❤❡r❡ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ f ✐s ❡✈❛❧✉❛t❡❞ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧✱ [xk−1 , xk ]✱ ✇❤✐❧❡ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ∆f ✐s ❞❡✜♥❡❞ ❛t ck ✿ ∆ f [x

k−1 ,xk ]

= ∆f (ck ) .

❲❡ ❝❛♥ ❛❧s♦ ✉s❡ ♣❛r❡♥t❤❡s❡s t♦ s❡♣❛r❛t❡ ❢✉♥❝t✐♦♥s ❢r♦♠ t❤❡✐r ✐♥♣✉ts ✇❤❡♥ ✐t✬s ♥♦t ❝✉♠❜❡rs♦♠❡✿  ∆ f [x

k−1 ,xk ]

❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s ❝♦♠♣✉t❡❞ ♦♥❡ s❡❣♠❡♥t ❛t ❛ t✐♠❡✿



= (∆f )(ck ) .

✶✳✷✳ ❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s

✷✾

◆♦✇ ✇❡ ❣♦ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✿ ❢r♦♠ ✈❡❧♦❝✐t✐❡s t♦ ♣♦s✐t✐♦♥s ❛♥❞ ❢r♦♠ ❛r❡❛s t♦ t❛♥❣❡♥ts✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ✭✐✳❡✳✱ t❤❡ r✐s❡s✮ ✐s ♦✉t❧✐♥❡❞ ❜❡❧♦✇✿

❚❤❡s❡ ❛r❡ t❤❡ st❛❣❡s t❤❛t ✇❡ s❡❡ ❤❡r❡✿ ✶✳ ❛ ❢✉♥❝t✐♦♥ ✷✳ ❛ ♣❛rt✐t✐♦♥ ❛♥❞ ✐ts s❡❝♦♥❞❛r② ♥♦❞❡s ✸✳ s❛♠♣❧✐♥❣ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ✹✳ r❡♠♦✈✐♥❣ t❤❡ r❡st ♦❢ t❤❡ ❣r❛♣❤ ✺✳ ♣✉tt✐♥❣ t❤❡s❡ ✈❛❧✉❡s ♦♥ t♦♣ ♦❢ ❡❛❝❤ ♦t❤❡r ✻✳ ♣❧❛❝✐♥❣ t❤❡s❡ s✉♠s ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ❚❤❡ r❡s✉❧t ✐s ❛ ♥❡✇ ❢✉♥❝t✐♦♥✳ ■t ✐s ❞❡✜♥❡❞ ❛❧❣❡❜r❛✐❝❛❧❧② ❛s ❢♦❧❧♦✇s✿

❉❡✜♥✐t✐♦♥ ✶✳✷✳✷✶✿ s✉♠ ♦❢ ❢✉♥❝t✐♦♥ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ g ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ P ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b]✳ ❚❤❡ s✉♠ ♦❢ g ✐s t❤❡ ❢✉♥❝t✐♦♥ h ❞❡✜♥❡❞ ♦♥ t❤❡ ♣r✐♠❛r② ♥♦❞❡s ❛s t❤❡ s✉♠ ♦❢ t❤❡ s❡q✉❡♥❝❡✿ g(c1 ), g(c2 ), ..., g(ck ), ...

■♥ ♦t❤❡r ✇♦r❞s✱ ✐t ✐s ❞❡✜♥❡❞ ✭r❡❝✉rs✐✈❡❧②✮✿ h(x0 ) = 0,

h(xm ) = h(xm−1 ) + g(cm ), m > 0

✶✳✷✳

❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s

✸✵

■t ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

Σ g [a,c ] = Σg (xk ) k

❲❡ ❛❧s♦ ❤❛✈❡✿

h(xm ) = g(c1 ) + g(c2 ) + ... + g(cm ) . ■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s ❛♥ ❛❜❜r❡✈✐❛t✐♦♥ ♦❢ t❤❡

s✐❣♠❛ ♥♦t❛t✐♦♥

✭✇❤✐❝❤ ✐s ❛❧s♦ ❛♥ ❛❜❜r❡✈✐❛t✐♦♥✮✿

g(c1 ) + g(c2 ) + ... + g(cm ) =

m X

g(ci ) .

i=1

❍❡r❡ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ ❛t

g

✐s ❡✈❛❧✉❛t❡❞ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧✱

xk ✿

[a, ck ]✱

✇❤✐❧❡ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥

h = Σg

✐s ❞❡✜♥❡❞

Σ g [a,c ] = Σg (xk ) . k

❲❡ ❝❛♥ ❛❧s♦ ✉s❡ ♣❛r❡♥t❤❡s❡s t♦ s❡♣❛r❛t❡ ❢✉♥❝t✐♦♥s ❢r♦♠ t❤❡✐r ✐♥♣✉ts ✇❤❡♥ ✐t✬s ♥♦t ❝✉♠❜❡rs♦♠❡✿

  Σ g [a,c ] = (Σg)(xk ) . k

❚❤❡ s✉♠ ✐s ❝♦♠♣✉t❡❞ r❡❝✉rs✐✈❡❧② ♦♥❡ s❡❣♠❡♥t ❛t ❛ t✐♠❡✿

■♥ t❤✐s ❝♦♥t❡①t✱ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡ ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡

s❛♠♣❧❡ ♣♦✐♥ts✳

■♥ t❤❡ ❡①❛♠♣❧❡s✱ ✇❡ ❝❤♦s❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ✐♥ ❛ ❝♦♥s✐st❡♥t ✇❛②✳ ❋✐rst ✇❡ ❝❤♦♦s❡ t♦ ❤❛✈❡ ❡q✉❛❧ ✐♥❝r❡♠❡♥ts✿

h = ∆x =

b−a . n

❋✉rt❤❡r♠♦r❡✱ t❤❡r❡ ❛r❡ t❤r❡❡ ♠❛✐♥ ✏s❝❤❡♠❡s✑ ❢♦r ❝❤♦♦s✐♥❣ s❡❝♦♥❞❛r② ♥♦❞❡s✳ ❖♥❡ ✐s s❡❡♥ ❛❜♦✈❡✿ ❚❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡ ♣❧❛❝❡❞ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ✐♥t❡r✈❛❧✳ ■t ✐s ❝❛❧❧❡❞ t❤❡

•

Pr✐♠❛r② ♥♦❞❡s✿

•

❙❡❝♦♥❞❛r② ♥♦❞❡s✿

❚❤✐s ✐s t❤❡

x = a, a + h, a + 2h, ... c = a + h, a + 2h, ...

❧❡❢t✲❡♥❞ s❝❤❡♠❡ ✿

•

Pr✐♠❛r② ♥♦❞❡s✿

•

❙❡❝♦♥❞❛r② ♥♦❞❡s✿

x = a, a + h, a + 2h, ... c = a, a + h, ...

❆♥♦t❤❡r ❝♦♥✈❡♥✐❡♥t ❝❤♦✐❝❡ ✐s t❤❡

•

Pr✐♠❛r② ♥♦❞❡s✿

•

❙❡❝♦♥❞❛r② ♥♦❞❡s✿

♠✐❞✲♣♦✐♥t s❝❤❡♠❡ ✿

x = a, a + h, a + 2h, ... c = a + h/2, a + 3h/2, ...

❚❤❡② ❛r❡ ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

r✐❣❤t✲❡♥❞ s❝❤❡♠❡ ✿

✶✳✷✳ ❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s

✸✶

❇❡❧♦✇ ✐s t❤❡ s✉♠♠❛r② ♦❢ ♦✉r ❜❛❝❦✲❛♥❞✲❢♦rt❤ ❝♦♥str✉❝t✐♦♥✿

◆♦✇✱ ❡✈❡♥ ✐❢ t❤❡ ♣❡rs♦♥ ❞✐❞♥✬t s♣❡♥❞ ❛♥② t✐♠❡ ❞r✐✈✐♥❣✱ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t st✐❧❧ ♠❛❦❡s s❡♥s❡✳ ■t✬s ③❡r♦✦ ❲❡ ❡①t❡♥❞ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ t♦ ✐♥❝❧✉❞❡ ③❡r♦ ✐♥❝r❡♠❡♥ts✿

❉❡✜♥✐t✐♦♥ ✶✳✷✳✷✷✿ ❞✐✛❡r❡♥❝❡ ♦✈❡r ③❡r♦✲❧❡♥❣t❤ ✐♥t❡r✈❛❧ ❲❤❡♥ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ x ✐s ③❡r♦✿ ∆xk = xk − xk−1 = 0,

t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ g ♦✈❡r t❤✐s s❡❣♠❡♥t ✐s ❞❡✜♥❡❞ t♦ ❜❡ ③❡r♦✿  ∆ g [x

k ,xk ]



=0

❲❡ ❡①t❡♥❞ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ s✉♠ t♦ ✐♥❝❧✉❞❡ ✐♥t❡r✈❛❧s ♦❢ ③❡r♦ ❧❡♥❣t❤✳

❉❡✜♥✐t✐♦♥ ✶✳✷✳✷✸✿ s✉♠ ♦✈❡r ③❡r♦✲❧❡♥❣t❤ ✐♥t❡r✈❛❧ ❚❤❡ s✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥ g ♦✈❡r ❛♥ ✐♥t❡r✈❛❧ [a, a] ✐s ❞❡✜♥❡❞ t♦ ❜❡ ③❡r♦✿   Σ g [a,a] = 0

❚♦ ❝❛♣✐t❛❧✐③❡ ♦♥ t❤✐s ✐❞❡❛ ❡✈❡♥ ❢✉rt❤❡r✱ ✇❡ ❡①t❡♥❞ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ t❤❡ s✉♠ t♦ ❛❧❧ ♦r✐❡♥t❡❞ s❡❣♠❡♥ts✿

✶✳✷✳ ❉✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s • a • b

✸✷

b ❚❤❡ ✐♥t❡r✈❛❧ [a, b] ✐s ♣♦s✐t✐✈❡❧② ♦r✐❡♥t❡❞ ✇❤❡♥ a < b✳ a ❚❤❡ ✐♥t❡r✈❛❧ [a, b] ✐s ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞ ✇❤❡♥ a > b✳

■♥ t❤❡ ❧❛tt❡r ❝❛s❡✱ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✐♥t❡r✈❛❧ ✐s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ a t♦ b✱ ✐✳❡✳✱ b − a✱ ✇❤✐❝❤ ✐s ♥❡❣❛t✐✈❡✦

❉❡✜♥✐t✐♦♥ ✶✳✷✳✷✹✿ ❞✐✛❡r❡♥❝❡ ♦✈❡r ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞ ✐♥t❡r✈❛❧

❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ♦✈❡r ❛ ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞ ✐♥t❡r✈❛❧ [xk , xk−1 ], xk−1 < xk ✱ ♦❢ ❛ ♣❛rt✐t✐♦♥ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ❞✐❢✲ ❢❡r❡♥❝❡ ♦✈❡r [xk−1 , xk ]✿  ∆ f [x

k ,xk−1 ]



 = −∆ f [x

k−1 ,xk ]



❉❡✜♥✐t✐♦♥ ✶✳✷✳✷✺✿ s✉♠ ♦✈❡r ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞ ✐♥t❡r✈❛❧ ❚❤❡ s✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥ g ♦✈❡r ❛ ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞ ✐♥t❡r✈❛❧ [b, a], a < b✱ ♦✈❡r ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ P ♦❢ [a, b]✱ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ s✉♠ ♦✈❡r P ✿     Σ g [b,a] = −Σ g [a,b]

❲❛r♥✐♥❣✦ ■♥ ❛♥t✐❝✐♣❛t✐♦♥ ♦❢ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥❛❧ ❝❛❧❝✉❧✉s✱ ❛♥ ♦r✐❡♥t❡❞ s❡❣♠❡♥t ✐s

♥♦t

❛ ✈❡❝t♦r✳

❲❡ ❝❛♥ ❛❧s♦ t❤✐♥❦ ♦❢ ✢✐♣♣✐♥❣ ❛♥ ✐♥t❡r✈❛❧ ❛s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② −1✳ ■❢ I st❛♥❞s ❢♦r s✉❝❤ ❛♥ ✐♥t❡r✈❛❧✱ ✇❡ ❤❛✈❡✿ X −I

f =−

X

f.

I

❲❡ ♥❡①t ❝♦♥s✐❞❡r t❤❡ s✐♠♣❧❡st ❝❛s❡✳

❚❤❡♦r❡♠ ✶✳✷✳✷✻✿ ❙✉♠ ♦❢ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥ g ✐s ❝♦♥st❛♥t ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ [a, b]✱ ✐✳❡✳✱ g(ci ) = p i = 1, 2, ..., n ❛♥❞ s♦♠❡ r❡❛❧ ♥✉♠❜❡r p✳ ❚❤❡♥

❙✉♣♣♦s❡ ❢♦r ❛❧❧

Σg [a,b] = p · n

❚❤✐♥❣s ❛r❡ ♣❛✐r❡❞ ✉♣✿

❉✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s

■♥t❡❣r❛❧ ❝❛❧❝✉❧✉s

f ❞❡✜♥❡❞ ♦♥ t❤❡ ♣r✐♠❛r② ♥♦❞❡s

g ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s

✶✳ ❞✐✛❡r❡♥❝❡✱ ∆f

✶✳ s✉♠✱ Σg

❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❞❡✜♥❡❞ ♦♥ t❤❡ ♣r✐♠❛r② ♥♦❞❡s ∆f (ck ) = f (xk ) − f (xk−1 )

Σg (xk ) = g(c1 ) + ... + g(ck )

✷✳

✷✳

✶✳✸✳ ❚❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✿ t❤❡ ❘✐❡♠❛♥♥ s✉♠

✸✸

❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤✐s t❛❜❧❡ ❜❡❧♦✇✳

✶✳✸✳ ❚❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✿ t❤❡ ❘✐❡♠❛♥♥ s✉♠

▲❡t✬s ❣❡♥❡r❛❧✐③❡ t❤❡ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❊①❛♠♣❧❡ ✶✳✸✳✶✿ s❛♠♣❧✐♥❣ ❧♦❝❛t✐♦♥

❋✐rst✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ tr✐♣ ✇✐t❤ ❛ ❜r♦❦❡♥ s♣❡❡❞♦♠❡t❡r✳ ❲❡ ❤❛✈❡ ❛ t✐♠❡ ✐♥t❡r✈❛❧

[a, b]✳

■♥ ♦r❞❡r t♦ ❡st✐♠❛t❡ ♦✉r s♣❡❡❞✱ ✇❡ ❞❡❝✐❞❡ t♦ ❧♦♦❦ ❛t t❤❡ ♦❞♦♠❡t❡r s❡✈❡r❛❧

t✐♠❡s ❞✉r✐♥❣ t❤❡ tr✐♣✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣ ❛♥❞ t❤❡ ✈❡r② ❡♥❞ ♦❢ ✐t✳ ❖t❤❡r✇✐s❡✱ t❤❡ ♠♦♠❡♥ts ♦❢ t✐♠❡ ♠❛② ❜❡ ❛r❜✐tr❛r②✿

a = x0 , x1 , x2 , ..., xn−1 , xn = b . ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐s ❛ ❢✉♥❝t✐♦♥

p

[a, b]✱

❛♥❞ t❤❡ ♣❧❛♥ ✐s t♦ s❛♠♣❧❡ t❤❡ ❧♦❝❛t✐♦♥✳

❚❤❡ ❧♦❝❛t✐♦♥

❞❡✜♥❡❞ ♦♥ t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✱ ❜✉t ♥♦✇ ♦♥❧② ✐ts ✈❛❧✉❡s ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡

♣❛rt✐t✐♦♥ ❛r❡ r❡❝♦r❞❡❞✳

❲❡ ❝♦♠♣✉t❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤✐s s❡q✉❡♥❝❡ ♣r♦❞✉❝✐♥❣ t❤❡ ❛✈❡r❛❣❡

✈❡❧♦❝✐t②✳ ❊✈❡♥ t❤♦✉❣❤ t❤❡ ✈❡❧♦❝✐t② ✐s ❛ ❢✉♥❝t✐♦♥

v

❞❡✜♥❡❞ ♦♥ t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✱ ✇❡ ❤❛✈❡ ♥♦✇ ♦♥❧② ✐ts

❛♣♣r♦①✐♠❛t✐♦♥s ❛ss✐❣♥❡❞ t♦ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳

❊①❛♠♣❧❡ ✶✳✸✳✷✿ s❛♠♣❧✐♥❣ ✈❡❧♦❝✐t②

❖♥ t❤❡ ✢✐♣ s✐❞❡✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ tr✐♣ ✇✐t❤ ❛ ❜r♦❦❡♥ ♦❞♦♠❡t❡r✳ ❲❡ st✐❧❧ ❤❛✈❡ ❛ t✐♠❡ ✐♥t❡r✈❛❧

[a, b]✳

❲❡ s♣❧✐t

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ ❛ ♣❛rt✐t✐♦♥ ♦❢

[a, b]

[a, b]✱

✐♥t♦ s♠❛❧❧❡r t✐♠❡ ✐♥t❡r✈❛❧s ✐♥ ❛♥ ❛r❜✐tr❛r② ♠❛♥♥❡r✳

❛♥❞ t❤❡ ♣❧❛♥ ✐s t♦ s❛♠♣❧❡ t❤❡ ✈❡❧♦❝✐t②✳ ❉✉r✐♥❣ ❡❛❝❤ ♦❢

t❤❡♠ ✇❡ ❧♦♦❦❡❞ ❛t t❤❡ s♣❡❡❞♦♠❡t❡r ✇✐t❤ t❤❡ ❡①❛❝t ♠♦♠❡♥ts✿

c1 , c2 , c3 , ..., cn−1 , cn . ❚❤❡② ❛r❡ ❥✉st ❛ ♠❛tt❡r ♦❢ ❜♦♦❦❦❡❡♣✐♥❣✳ ❚❤❡ ✈❡❧♦❝✐t② ✐s ❛ ❢✉♥❝t✐♦♥

v

❞❡✜♥❡❞ ♦♥ t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✱

❜✉t ♥♦✇ ♦♥❧② ✐ts ✈❛❧✉❡s ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛r❡ r❡❝♦r❞❡❞✳

❲❡ ♠✉❧t✐♣❧② t❤❡

✈❡❧♦❝✐t② ❜② t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝✉rr❡♥t t✐♠❡ ✐♥t❡r✈❛❧ ♣r♦❞✉❝✐♥❣ t❤❡ ❛♣♣r♦①✐♠❛t❡ ❞✐s♣❧❛❝❡♠❡♥t✳ ❚❤❡♥ ✇❡ ❛❞❞ t❤❡♠ t♦❣❡t❤❡r ✭t❤❡ ✏❘✐❡♠❛♥♥ s✉♠✑✮✳ ❊✈❡♥ t❤♦✉❣❤ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✐s ❛ ❢✉♥❝t✐♦♥

p

❞❡✜♥❡❞ ♦♥

t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✱ ✇❡ ❤❛✈❡ ♥♦✇ ♦♥❧② ✐ts ❛♣♣r♦①✐♠❛t✐♦♥s ❛ss✐❣♥❡❞ t♦ t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❚❤✐s ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢ s✉❝❤ ❛ ❝♦♠♣✉t❛t✐♦♥✿

t✐♠❡ ✐♥t❡r✈❛❧s ✭❤♦✉rs✮✿ t✐♠❡ ✭❤♦✉rs✮✿ ✈❡❧♦❝✐t② ✭♠✐❧❡s✴❤♦✉r✮✿ ❞✐s♣❧❛❝❡♠❡♥t ✭♠✐❧❡s✮✿

t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t ✭♠✐❧❡s✮✿

[0, 2]

[2, 4]

[4, 6]

[6, 8]

c1 = 1

c2 = 4

c3 = 5

c4 = 6

60

100

−80

−80

60 · 2

100 · 2

−80 · 2

−80 · 2

= 120

= 200

= −160

= −160

120 120 + 200 340 − 160 180 − 160 = 120

= 340

= 180

= 20

✶✳✸✳ ❚❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✿ t❤❡ ❘✐❡♠❛♥♥ s✉♠

✸✹

❲❡ ❤❛✈❡ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❛s ❛ ❢✉♥❝t✐♦♥ p ♦❢ t✐♠❡ ❜✉t ✇❡ ❝❛♥ ❛❧s♦ ❛ss✐❣♥ t❤❡s❡ ♥✉♠❜❡rs t♦ t❤❡ ♣r✐♠❛r② ♥♦❞❡s✿ t✐♠❡ ✭❤♦✉rs✮✿ x0 = 0 x1 = 2 x2 = 4 x3 = 6 x4 = 8 t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t ✭♠✐❧❡s✮✿

0

120

340

180

20

❖❢ ❝♦✉rs❡✱ ✐❢ ✇❡ ♥❡❡❞ t❤❡ ❧♦❝❛t✐♦♥✱ ✇❡ ♥❡❡❞ ♦✉r ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ ✜rst✱ ✐✳❡✳✱ t❤❡ ✈❛❧✉❡ ♦❢ p(a)✱ ✐♥ ♦r❞❡r t♦ st❛rt t❤❡ ❝♦♠♣✉t❛t✐♦♥❀ ✇❡ ❛ss✉♠❡❞ ✐t t♦ ❜❡ 0 ❛❜♦✈❡✳ ■❢ ✐t✬s 10✱ ✇❡ ❤❛✈❡✿ t✐♠❡ ✭❤♦✉rs✮✿

x0 = 0 x1 = 2 x2 = 4 x3 = 6 x4 = 8

t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t ✭♠✐❧❡s✮✿

10

130

350

190

30

❊①❡r❝✐s❡ ✶✳✸✳✸ ❲❤❛t ✐s t❤❡ ❛♣♣r♦①✐♠❛t❡ ❞✐s♣❧❛❝❡♠❡♥t ✐❢ t❤❡ ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥ ✇❛s −50❄ ❲❤❛t ✐s t❤❡ ❛♣♣r♦①✐♠❛t❡ ❞✐s♣❧❛❝❡♠❡♥t ✐❢ t❤❡ ♣♦s✐t✐♦♥ ❛t t✐♠❡ 2 ✇❛s 20❄

❊①❡r❝✐s❡ ✶✳✸✳✹ ❈♦♠♣✉t❡ t❤❡ ❛♣♣r♦①✐♠❛t❡ ❞✐s♣❧❛❝❡♠❡♥t ❢♦r t❤❡ s❛♠❡ s❛♠♣❧✐♥❣ ❞❛t❛ ❛s ✐♥ t❤❡ ❡①❛♠♣❧❡ ❜✉t ❢♦r ❛ ❞✐✛❡r❡♥t ✭✉♥❡q✉❛❧✮ ♣❛rt✐t✐♦♥✿ t✐♠❡ ✐♥t❡r✈❛❧s ✭❤♦✉rs✮✿

[0, 1] [1, 4] [4, 5] [5, 8]

❙♦✱ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ❧❛st s❡❝t✐♦♥✱ t❤❡ ♥❡✇ st❡♣ ✐s t❤❡ ❞✐✈✐s✐♦♥ ♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥t ∆x✿

▲❡t✬s ♣r♦✈✐❞❡ ♣r❡❝✐s❡ ❞❡✜♥✐t✐♦♥s✳ ❏✉st ❛s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ t❤❡r❡ ✇✐❧❧ ❜❡ ♥♦ r❡str✐❝t✐♦♥s ✇❤❛ts♦❡✈❡r ♦♥ t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✶✳✸✳✺✿ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ ❢✉♥❝t✐♦♥ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s xk , k = 0, 1, 2, ..., n✱ ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❚❤❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡

✶✳✸✳

✸✺

❚❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✿ t❤❡ ❘✐❡♠❛♥♥ s✉♠

s❡❝♦♥❞❛r② ♥♦❞❡s ck , k = 1, 2, ..., n ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢r❛❝t✐♦♥✿

∆f f (xk+1 ) − f (xk ) f (xk + ∆xk ) − f (xk ) (ck ) = = ∆x xk+1 − xk ∆xk ■t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ f ❞✐✈✐❞❡❞ ❜② ∆xk ✳ ■t ✐s t❤❡ r❡❧❛t✐✈❡ ❝❤❛♥❣❡ ✕ t❤❡

r❛t❡

♦❢ ❝❤❛♥❣❡ ✕ ♦❢ t❤❡ t✇♦ s❡q✉❡♥❝❡s✳

▲❡t✬s r❡♠❡♠❜❡r t❤❛t ❤❡r❡ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ f ✐s ❡✈❛❧✉❛t❡❞ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧✱ [xk−1 , xk ]✱ ✇❤✐❧❡ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ∆f ✐s ❞❡✜♥❡❞ ❛t ck ✿ ∆ ∆f f [x ,x ] = (ck ) . k−1 k ∆x ∆x ❲❡ ❝❛♥ ❛❧s♦ ✉s❡ ♣❛r❡♥t❤❡s❡s t♦ s❡♣❛r❛t❡ ❢✉♥❝t✐♦♥s ❢r♦♠ t❤❡✐r ✐♥♣✉ts ✇❤❡♥ ✐t✬s ♥♦t ❝✉♠❜❡rs♦♠❡✿   ∆f  ∆  f [x ,x ] = (ck ) . k−1 k ∆x ∆x

❲❤❡r❡ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ✇❡ ❧♦♦❦❡❞ ❢♦r t❤❡ r✐s❡s✱ ✇❡ ✜♥❞ t❤❡ s❧♦♣❡s ♥♦✇✳ ❖♥ t❤❡ ❣r❛♣❤✱ ✇❡ ❝❛♥ s❡❡ ❤♦✇ ❡❛❝❤ ❝♦♥s❡❝✉t✐✈❡ ♣❛✐r ♦❢ ♣♦✐♥ts ♣r♦❞✉❝❡s ❛ ❧✐♥❡ s❡❣♠❡♥t ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡ ❝♦♥♥❡❝t✐♥❣ t❤❡♠✿

◆♦✇ ✐♥ r❡✈❡rs❡✳ ❚❤❡ ❡①❛♠♣❧❡ ♦❢ ❞✐s♣❧❛❝❡♠❡♥t ✐s ❛♥ ❡s♣❡❝✐❛❧❧② ✐♠♣♦rt❛♥t ❝❛s❡ ♦❢ t❤❡ s✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❝♦♠♣✉t❡❞ ❢r♦♠ ❛♥♦t❤❡r ♦♥❡ ❛s ❢♦❧❧♦✇s✿ g(ck ) = f (ck )∆xk . ❚❤❡ ❝♦♥str✉❝t✐♦♥ ✐s t❤❡ s❛♠❡ ❛s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ❜✉t t❤❡ ✐♥❝r❡♠❡♥ts ♦❢ x ♠❛❦❡ t❤❡✐r ❛♣♣❡❛r❛♥❝❡✿

❚❤❡s❡ ❛r❡ ♦✉r t✇♦ ✐♥t❡r♣r❡t❛t✐♦♥s ♦❢ t❤❡ s❛♠❡ ❝♦♥str✉❝t✐♦♥✿ ♠♦t✐♦♥

❣❡♦♠❡tr②

∆xk

t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ k t❤ t✐♠❡ ✐♥t❡r✈❛❧

t❤❡ ✇✐❞t❤ ♦❢ t❤❡ k t❤ r❡❝t❛♥❣❧❡

f (ck )

t❤❡ ✈❡❧♦❝✐t② ❞✉r✐♥❣ t❤❡ k t❤ t✐♠❡ ✐♥t❡r✈❛❧

t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ k t❤ r❡❝t❛♥❣❧❡

g(ck ) = f (ck )∆xk

t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❞✉r✐♥❣ t❤❡ k t❤ t✐♠❡ ✐♥t❡r✈❛❧

t❤❡ ❛r❡❛ ♦❢ t❤❡ k t❤ r❡❝t❛♥❣❧❡

✶✳✸✳

✸✻

❚❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✿ t❤❡ ❘✐❡♠❛♥♥ s✉♠

❲❛r♥✐♥❣✦ ❊✈❡♥ t❤♦✉❣❤ ✇❡ ♦❢t❡♥ s❛② t❤❛t ❡❛❝❤ s✉❝❤ t❡r♠ ✐s t❤❡ ✏❛r❡❛✑ ♦❢ t❤❡ r❡❝t❛♥❣❧❡✱ t❤✐s q✉❛♥t✐t② ✐s tr✉❧② t❤❡ ❛r❡❛ ♦♥❧② ✐❢ t❤❡ ✉♥✐t ♦❢ ❜♦t❤ t❤❡

y ✲❛①✐s

x✲

❛♥❞ t❤❡

✐s t❤❛t ♦❢ ❧❡♥❣t❤✳

❲❡ ❞❡✜♥❡ ❛ s♣❡❝✐❛❧ ❦✐♥❞ ♦❢ s✉♠✳

❉❡✜♥✐t✐♦♥ ✶✳✸✳✻✿ ❘✐❡♠❛♥♥ s✉♠ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ P ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b]✳ ❚❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ f ✐s t❤❡ ❢✉♥❝t✐♦♥ h ❞❡✜♥❡❞ ♦♥ t❤❡ ♣r✐♠❛r② ♥♦❞❡s ❛s t❤❡ s✉♠ ♦❢ t❤❡ s❡q✉❡♥❝❡✿ f (c1 )∆x1 , f (c2 )∆x2 , ..., f (cn )∆xn .

■♥ ♦t❤❡r ✇♦r❞s✱ ✐t ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡ ❞❡✜♥❡❞ r❡❝✉rs✐✈❡❧②✿ h(x0 ) = 0,

h(xk ) = h(xk−1 ) + f (ck )∆xk , k = 1, 2, ..., n .

■t ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ Σ f · ∆x [a,c ] = Σf · ∆x (xk ) = f (c1 )∆x1 + f (c2 )∆x2 + ... + f (ck )∆xk k

■♥ s✐❣♠❛ ♥♦t❛t✐♦♥✱ ✐t ✐s✿

k X

f (ci )∆xi .

i=1

❍❡r❡ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ f ✐s ❡✈❛❧✉❛t❡❞ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧✱ [a, ck ]✱ ✇❤✐❧❡ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ h = Σf · ∆x ✐s ❞❡✜♥❡❞ ❛t xk ✿ Σ f · ∆x [a,c ] = Σf · ∆x (xk ) . k

❲❡ ❝❛♥ ❛❧s♦ ✉s❡ ♣❛r❡♥t❤❡s❡s t♦ s❡♣❛r❛t❡ ❢✉♥❝t✐♦♥s ❢r♦♠ t❤❡✐r ✐♥♣✉ts ✇❤❡♥ ✐t✬s ♥♦t ❝✉♠❜❡rs♦♠❡✿   Σ f · ∆x [a,c ] = (Σf · ∆x)(xk ) . k

❆t ✜rst ✇❡ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦✈❡r t❤❡ ✇❤♦❧❡

✳

✐♥t❡r✈❛❧

▲❡t✬s ❝♦♥s✐❞❡r s♦♠❡ s♣❡❝✐✜❝ ❡①❛♠♣❧❡s ♦❢ ❘✐❡♠❛♥♥ s✉♠s✳ ❲❡ ❛ss✉♠❡ ❛❣❛✐♥ t❤❛t t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ✐♥t❡r✈❛❧s ❛r❡ ❡q✉❛❧✿ ∆xi = ∆x = (b − a)/n .

❊①❛♠♣❧❡ ✶✳✸✳✼✿ x2 ▲❡t f (x) = x2 ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ [0, 1] ✇✐t❤ n = 4✳ ❚❤❡♥ ∆x = 1/4 ❛♥❞ t❤❡ ✐♥t❡r✈❛❧ ✐s s✉❜❞✐✈✐❞❡❞ ❛s ❜❡❢♦r❡ ❛♥❞ ✇❡ ❤❛✈❡ ✜✈❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t♦

✶✳✸✳

✸✼

❚❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✿ t❤❡ ❘✐❡♠❛♥♥ s✉♠

✇♦r❦ ✇✐t❤✿ •

−−

|

−−

|

−−

|

−−

•

♣❛rt✐t✐♦♥✿ x =

0

1/4

1/2

3/4

1

✈❛❧✉❡s✿ x2 =

0

1/16

1/4

9/16

1

❧❡❢t✲❡♥❞ ❘❙ = 0 · 1/4

+

r✐❣❤t✲❡♥❞ ❘❙ =

1/16 · 1/4

+

1/4 · 1/4

+

9/16 · 1/4

1/16 · 1/4

+

1/4 · 1/4

+

9/16 · 1/4

≈ 0.22 +

1 · 1/4 ≈ 0.47

❚❤❡ s✐♠♣❧❡st ❝❤♦✐❝❡s ❢♦r t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❛r❡ t❤❡ ❧❡❢t✲❡♥❞ ♦r t❤❡ r✐❣❤t✲❡♥❞ ♣♦✐♥ts ♦❢ t❤❡ ✐♥t❡r✈❛❧s✳ ❚❤✐s ❢✉❧❧② ❛❧❣❡❜r❛✐❝ ❝♦♥str✉❝t✐♦♥ ❝❛♥ ❜❡ ✈✐s✉❛❧✐③❡❞✿

❊①❡r❝✐s❡ ✶✳✸✳✽

❲❤❛t ✐s ②♦✉r ❜❡st ❡st✐♠❛t❡ ♦❢ t❤❡ ❝✉r✈❡❞ ❛r❡❛ ❜❛s❡❞ ♦♥ t❤❡ ❞❛t❛ ♣r♦✈✐❞❡❞❄ ❊①❡r❝✐s❡ ✶✳✸✳✾

❘❡♣❡❛t t❤❡ ❝♦♠♣✉t❛t✐♦♥s ❢♦r ✭❛✮ ♥❂✽✱ ✭❜✮ [a, b] = [−1, 0]✱ ✭❝✮ f (x) = x3 ✳ ❚❤✐s ✐s t❤❡ ❧❡❢t✲❡♥❞ ♣❛rt✐t✐♦♥✿ a = x0

x1

x2 ... xn−1

a = c1

c2

c3

...

xn = b ◦

cn

♣r✐♠❛r② ♥♦❞❡s s❡❝♦♥❞❛r② ♥♦❞❡s

❚❤❡ t❛❜❧❡ s❤♦✇s t❤❡ nt❤ ❧❡❢t✲❡♥❞ ❘✐❡♠❛♥♥ s✉♠ ❛♥❞ t❤❡ nt❤ r✐❣❤t✲❡♥❞ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ♦♥ [a, b]✿ i 0 [xi , xi+1 ] |

−−

xi a f (x) f (x0 )

❧❡❢t✲❡♥❞ ❘❙ = f (x0 ) · ∆x

+

r✐❣❤t✲❡♥❞ ❘❙ =

1

...

n−1

|

− ... −

|

a + ∆x

...

a + (n − 1) ∆x

f (x1 )

...

f (xn−1 )

f (x1 ) · ∆x + ... +

f (xn−1 ) · ∆x

f (x1 ) · ∆x + ... +

f (xn−1 ) · ∆x

❚❤❡ ❢♦r♠✉❧❛ ❢♦r s✉❝❤ ❛ s❡q✉❡♥❝❡ ♦❢ s❡❝♦♥❞❛r② ♥♦❞❡s ✐s t❤❡ s❛♠❡ ❢♦r ❜♦t❤✿ ci = a + i ∆x ,

❜✉t t❤❡ ✐♥❞✐❝❡s r✉♥ ♦✈❡r ❞✐✛❡r❡♥t s❡ts✿

n −−

| b f (xn )

+

f (xn ) · ∆x

✶✳✸✳

✸✽

❚❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✿ t❤❡ ❘✐❡♠❛♥♥ s✉♠

• i = 0, 1, 2, ..., n − 1 ❢♦r t❤❡ ❧❡❢t✲❡♥❞ ♣♦✐♥ts❀

• i = 1, 2, ..., n ❢♦r t❤❡ r✐❣❤t✲❡♥❞ ♣♦✐♥ts✳

❚❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❝❛♥ ❛❧s♦ ❜❡ ❝❤♦s❡♥ t♦ ❜❡ t❤❡ ♠✐❞✲♣♦✐♥ts ♦❢ t❤❡ ✐♥t❡r✈❛❧s✿ i 0 [xi , xi+1 ] | ci f (x)

♠✐❞✲♣♦✐♥t ❘❙ =

1 −·−

|

−·−

2

...

|

− ...

a + 21 ∆x

a + 32 ∆x

...

f (c0 )

f (c1 )

...

f (c1 ) · ∆x + f (c2 ) · ∆x +

n−1 −

|

n −·− a+

2n−1 2

| ∆x

f (cn )

...

+

f (cn ) · ∆x

❚❤❡ ✐❧❧✉str❛t✐♦♥ s❤♦✇s t❤❡ nt❤ ♠✐❞✲♣♦✐♥t ❘✐❡♠❛♥♥ s✉♠ ♦❢ f ♦♥ [a, b]✿

❚❤❡ s✐♥❣❧❡ ❢♦r♠✉❧❛ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ♠✐❞✲♣♦✐♥ts ✐s ci = a + i +

1 2



∆x, i = 0, 1, 2, ..., n − 1 .

❊①❛♠♣❧❡ ✶✳✸✳✶✵✿ ♥❡❣❛t✐✈❡ ❛r❡❛

❲❤❡♥ t❤❡ r❡❣✐♦♥ ❧✐❡s ✇✐t❤✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✱ t❤❡ it❤ t❡r♠ ✐♥ ❛❧❧ t❤r❡❡ ❘✐❡♠❛♥♥ s✉♠s ✐s✿ ✇✐❞t❤ ♦❢ r❡❝t❛♥❣❧❡

t❤❡ ❛r❡❛ ♦❢ it❤ r❡❝t❛♥❣❧❡ =

f (ci ) | {z }

❤❡✐❣❤t ♦❢ r❡❝t❛♥❣❧❡

×

z}|{ ∆xi

❚❤❡ ❘✐❡♠❛♥♥ s✉♠ ✐s t❤❡ t♦t❛❧ ❛r❡❛ ♦❢ t❤❡ r❡❝t❛♥❣❧❡s✳ ❇✉t ✇❤❛t ✐❢ t❤❡ ✈❛❧✉❡s ♦❢ f ❛r❡ ♥❡❣❛t✐✈❡✱ f (ci ) < 0?

◆♦t❤✐♥❣ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♣r❡✈❡♥ts t❤❛t✿

✶✳✸✳ ❚❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✿ t❤❡ ❘✐❡♠❛♥♥ s✉♠

❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ ❡❛❝❤ t❡r♠

f (ci ) ∆x

f (ci ) ∆x

✸✾

t❤❡♥❄ ❏✉st ✐♠❛❣✐♥❡ ❛❣❛✐♥ t❤❛t

f

✐s t❤❡ ✈❡❧♦❝✐t②✳ ❚❤❡♥

✐s st✐❧❧ ❞✐st❛♥❝❡ ✏❝♦✈❡r❡❞✑ ❜✉t ✇✐t❤ t❤❡ ♥❡❣❛t✐✈❡ s♣❡❡❞✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t ②♦✉ ❛r❡ ♠♦✈✐♥❣ ✐♥

t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✦ ❚❤❡♥ ②♦✉r ❞✐s♣❧❛❝❡♠❡♥t ✐s ♥❡❣❛t✐✈❡✳ ❆s ❢♦r t❤❡ ❛r❡❛ ♠❡t❛♣❤♦r✱ s✐♥❝❡ t❤❡ ♠❡❛♥✐♥❣ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♠❡❛♥✐♥❣ ♦❢

∆x > 0✱

f (ci )✿

❲❡ ❢♦❧❧♦✇ ✉♣ t♦ t❤❡ ❧❛st s❡❝t✐♦♥ ❛♥❞ ❡①t❡♥❞ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠ t♦ ✐♥❝❧✉❞❡ ✐♥t❡r✈❛❧s ♦❢ ③❡r♦ ❧❡♥❣t❤✳

❉❡✜♥✐t✐♦♥ ✶✳✸✳✶✶✿ ❘✐❡♠❛♥♥ s✉♠ ♦✈❡r ③❡r♦✲❧❡♥❣t❤ ✐♥t❡r✈❛❧ ❚❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥

f

♦✈❡r ❛♥ ✐♥t❡r✈❛❧

[a, a]

✐s ❞❡✜♥❡❞ t♦ ❜❡ ③❡r♦✿

  Σ f · ∆x [a,a] = 0

❉❡✜♥✐t✐♦♥ ✶✳✸✳✶✷✿ ❘✐❡♠❛♥♥ s✉♠ ♦✈❡r ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞ ✐♥t❡r✈❛❧ f ♦✈❡r ❛ ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞ ✐♥t❡r✈❛❧ [b, a], a ≤ b✱ P ♦❢ [a, b]✱ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡

❚❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦✈❡r ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ❘✐❡♠❛♥♥ s✉♠ ♦✈❡r

P✿     Σ f · ∆x [b,a] = −Σ f · ∆x [a,b]

❊①❛♠♣❧❡ ✶✳✸✳✶✸✿ ♥❡❣❛t✐✈❡ ✇✐❞t❤ ❲❤❡♥ t❤❡ r❡❣✐♦♥ ❧✐❡s ✇✐t❤✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✱ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❡❝t❛♥❣❧❡ ✐♥ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ✐s ❛❧s♦ ♥❡❣❛t✐✈❡✱ ❛♥❞ s♦ ✐s ✐ts ❛r❡❛ ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♣♦s✐t✐✈❡✿

✶✳✸✳

✹✵

❚❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✿ t❤❡ ❘✐❡♠❛♥♥ s✉♠

❚❤❡ ♥♦t✐♦♥ ♦❢ ❛ ♥❡❣❛t✐✈❡ ❛r❡❛ ✐s ❥✉st✐✜❡❞ ❜② t❤❡ ❢❛❝t t❤❛t t❤❡ r❡❝t❛♥❣❧❡ ✐s ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞✳✳✳ ❚❤❡ ♠♦t✐♦♥ ♠❡t❛♣❤♦r ♠✐❣❤t ❡①♣❧❛✐♥ ✐t ❜❡tt❡r t❤♦✉❣❤✳ ■t ✐s ❛s ✐❢ t✐♠❡ ✐s r❡✈❡rs❡❞ ✭♦r ❛ ✈✐❞❡♦t❛♣❡ ❣♦❡s ❜❛❝❦✇❛r❞✮✱ s♦ t❤❛t t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ♠♦t✐♦♥ ✐s ♦♣♣♦s✐t❡✱ ❛♥❞ ❛❧❧ ❣❛✐♥s ❛r❡ r❡✈❡rs❡❞✿

❲❡ ♥❡①t ❝♦♥s✐❞❡r t❤❡ s✐♠♣❧❡st ❝❛s❡ ♦❢ ❛ ❘✐❡♠❛♥♥ s✉♠✳ ❚❤❡♦r❡♠ ✶✳✸✳✶✹✿ ❘✐❡♠❛♥♥ ❙✉♠ ♦❢ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥

f ✐s ❝♦♥st❛♥t ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ [a, b]✱ ✐✳❡✳✱ f (ci ) = p i = 1, 2, ..., n ❛♥❞ s♦♠❡ r❡❛❧ ♥✉♠❜❡r p✳ ❚❤❡♥

❙✉♣♣♦s❡ ❢♦r ❛❧❧



 Σ f · ∆x [a,b] = p(b − a) Pr♦♦❢✳

❙✐♥❝❡ f (ci ) = p ❢♦r ❛❧❧ i✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ✐s ❡q✉❛❧ t♦✿ Σf · ∆x = f (c1 ) ∆x1 + f (c2 ) ∆x2 + ... + f (cn ) ∆xn = p ∆x1 + p ∆x2 + ... + p ∆xn   = p ∆x1 + ∆x2 + ... + ∆xn   = p (x1 − x0 ) + (x2 − x1 ) + (x3 − x2 ) + ... + (xn − xn−1 ) = p(b − a) .

❖♥❝❡ ✇❡ ③♦♦♠❡❞ ♦✉t✱ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t t❤❡ ❘✐❡♠❛♥♥ s✉♠ r❡♣r❡s❡♥ts t❤❡ r❡❝t❛♥❣❧❡ ✇✐t❤ ✇✐❞t❤ b − a ❛♥❞ ❤❡✐❣❤t p✿

✶✳✸✳ ❚❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✿ t❤❡ ❘✐❡♠❛♥♥ s✉♠

❊①❝❡♣t✱ ✐t ✐s ♥❡❣❛t✐✈❡ ✇❤❡♥

p

✹✶

✐s ♥❡❣❛t✐✈❡✳

❊①❛♠♣❧❡ ✶✳✸✳✶✺✿ ❘✐❡♠❛♥♥ s✉♠s ❛r❡ ❢✉♥❝t✐♦♥s ▲❡t✬s ❝♦♥s✐❞❡r ❛❧❧ ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ♦❢ ♦❢

❲❡ ❝❤♦♦s❡ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥t❡r✈❛❧s t♦ ❜❡

f (x) = x2

n=4

♦✈❡r

[0, 1]✿

✇✐t❤ ❡q✉❛❧ ✐♥t❡r✈❛❧s ♦❢ ❧❡♥❣t❤

h = 1/4✱

❛♥❞ ✇❡ ❝❤♦♦s❡✱

❛s t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✱ t❤❡ ❧❡❢t✲❡♥❞ ♦r t❤❡ r✐❣❤t✲❡♥❞ ♦❢ ❡❛❝❤ ✐♥t❡r✈❛❧✿

❆t t❤♦s❡ ♣♦✐♥ts✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❡✈❛❧✉❛t❡❞✳ ❚❤✐s ✐s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❧❡❢t✲❡♥❞ ❘✐❡♠❛♥♥ s✉♠

•

−−

|

−−

|

−− |

−− •

x

0

1/4

1/2

3/4

1

x2

0

1/16

1/4

9/16

1

L4 0 · 1/4

+

1/16 · 1/4

+

1/4 · 1/4 +

9/16 · 1/4

Σ [0,0] 0 · 1/4

❲❡✱

≈ 0.22 =0

Σ [0,1/4] 0 · 1/4

+

1/16 · 1/4

Σ [0,1/2] 0 · 1/4

+

1/16 · 1/4

+

1/4 · 1/4

Σ [0,3/4] 0 · 1/4

+

1/16 · 1/4

+

1/4 · 1/4 +

❢✉rt❤❡r♠♦r❡✱

L4 ✿

≈ 0.04 ≈ 0.10 9/16 · 1/4

r❡❛❧✐③❡ t❤❛t ✇❡ ❛r❡ ❝♦♠♣✉t✐♥❣ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❢✉♥❝t✐♦♥

♣❛rt✐t✐♦♥✳ ■ts ❢♦✉r ✈❛❧✉❡s ❛r❡ s❤♦✇♥ ✐♥ ❜♦tt♦♠ ♦❢ t❤❡ t❛❜❧❡✳

❲❡ ❝❛♥ ❛❧s♦ ❝❤♦♦s❡ t❤❡ ♠✐❞✲♣♦✐♥ts ❛s t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✿

≈ 0.22 ❢♦r t❤✐s ❛✉❣♠❡♥t❡❞

✶✳✸✳

✹✷

❚❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✿ t❤❡ ❘✐❡♠❛♥♥ s✉♠

❚❤✐s ✐s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ♠✐❞✲♣♦✐♥t ❘✐❡♠❛♥♥ s✉♠ M4 ❢♦r t❤❡ s❛♠❡ ✐♥t❡❣r❛❧✿ • x f (x) = x2 M4

−−

|

−−

|

−−

1/8

3/8

5/8

(1/8)2

(3/8)2

(5/8)2

|

−−

•

7/8 (7/8)2

(1/8)2 · 1/4 + (3/8)2 · 1/4 + (5/8)2 · 1/4 + (7/8)2 · 1/4 = 0.328125

Σ [0,1/8]

(1/8)2 · 1/4

≈ 0.004

Σ [0,3/8]

(1/8)2 · 1/4 + (3/8)2 · 1/4

≈ 0.040

Σ [0,5/8]

(1/8)2 · 1/4 + (3/8)2 · 1/4 + (5/8)2 · 1/4

≈ 0.230

Σ [0,7/8]

(1/8)2 · 1/4 + (3/8)2 · 1/4 + (5/8)2 · 1/4 + (7/8)2 · 1/4 ≈ 0.328

■t ✐s ♠✉❝❤ ❝❧♦s❡r t❤❛♥ L4 t♦ t❤❡ tr✉❡ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧✱ 1/3✳ ❍❡r❡ ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ✉♣ t♦ t❤✐s ♣♦✐♥t✿

❉✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s

■♥t❡❣r❛❧ ❝❛❧❝✉❧✉s

f ❞❡✜♥❡❞ ♦♥ t❤❡ ♣r✐♠❛r② ♥♦❞❡s

g ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s

✶✳ ❞✐✛❡r❡♥❝❡✱ ∆f

✶✳ s✉♠✱ Σg

❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❞❡✜♥❡❞ ♦♥ t❤❡ ♣r✐♠❛r② ♥♦❞❡s ∆f (ck ) = f (xk ) − f (xk−1 )

Σg (xk ) = g(c1 ) + ... + g(ck )

❞✐✈✐❞❡❞ ❜② ∆x

∆x ✐s ❢❛❝t♦r❡❞ ✐♥ ❡❛❝❤ t❡r♠

✷✳ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱

∆f ∆x

✷✳ ❘✐❡♠❛♥♥ s✉♠✱ Σf · ∆x

❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❞❡✜♥❡❞ ♦♥ t❤❡ ♣r✐♠❛r② ♥♦❞❡s ∆f f (xk ) − f (xk−1 ) (ck ) = ∆x ∆x

Σf · ∆x (xk ) = f (c1 )∆x + ... + f (ck )∆x

✸✳

✸✳

❚❤❡ t✇♦ ❝♦❧✉♠♥s ❛r❡ ❝♦♥str✉❝t❡❞ ✐♥❞❡♣❡♥❞❡♥t❧② ❜✉t ❝❛♥ ❡❛s✐❧② ❜❡ ❧✐♥❦❡❞ t♦❣❡t❤❡r✳

✶✳✹✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

✹✸

✶✳✹✳ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

❆s ✇❡ ❦♥♦✇✱ ❛❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥ ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✳ ❘❡♣❡❛t❡❞ ❛❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥ ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✱ t♦♦✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ s✉♠ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ✕ ❛s ♦♣❡r❛t✐♦♥s ♦♥ s❡q✉❡♥❝❡s ✕ ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✳ ❲❡ ✐❧❧✉str❛t❡ t❤✐s ✐❞❡❛ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡ s✉♠ st❛❝❦s ✉♣ t❤❡ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦♥ t♦♣ ♦❢ ❡❛❝❤ ♦t❤❡r ✇❤✐❧❡ t❤❡ ❞✐✛❡r❡♥❝❡ t❛❦❡s t❤✐s ❜❛❝❦ ❞♦✇♥✳ ❲❤❛t ❞♦ t❤❡ s✉♠ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ❤❛✈❡ t♦ ❞♦ ✇✐t❤ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts❄ ❏✉st ♣❧✉❣ ✐♥ ∆xk = 1 ❛♥❞ ②♦✉ ❣❡t t❤❡ ❢♦r♠❡r ❢r♦♠ t❤❡ ❧❛tt❡r✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ✐s ❢✉♥❞❛♠❡♥t❛❧✳ ❊①❛♠♣❧❡ ✶✳✹✳✶✿ ❝❛♥❝❡❧❧❛t✐♦♥

❲❡ ❦♥♦✇ ❤♦✇ t♦ ❣❡t t❤❡ ✈❡❧♦❝✐t② ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥ ✕ ❛♥❞ t❤❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✳ ❖❢ ❝♦✉rs❡✱ ❡①❡❝✉t✐♥❣ t❤❡s❡ t✇♦ ♦♣❡r❛t✐♦♥s ❝♦♥s❡❝✉t✐✈❡❧② s❤♦✉❧❞ ❜r✐♥❣ ✉s ❜❛❝❦ ✇❤❡r❡ ✇❡ st❛rt❡❞✳ ❲❡ ♥♦✇ t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t t❤❡ s♣❡❡❞♦♠❡t❡r ✕ ♣r❡s❡♥t❡❞ ❡❛r❧✐❡r✳

t✇♦ ❝♦♠♣✉t❛t✐♦♥s ❛❜♦✉t ♠♦t✐♦♥ ✕ ❛ ❜r♦❦❡♥ ♦❞♦♠❡t❡r ❛♥❞ ❛ ❜r♦❦❡♥

❋✐rst✱ ❜❡❧♦✇ ✇❡ s❡❡ ❤♦✇ t❤❡ ✈❡❧♦❝✐t✐❡s ❛r❡ ✉s❡❞ t♦ ❛❝q✉✐r❡ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ✈✐❛ t❤❡ s✉♠s✱ ❜✉t ✇❡ ❛❧s♦ ❞✐s❝♦✈❡r t❤❛t ✇❡ ❝❛♥ ❣❡t t❤❡ ❢♦r♠❡r ❢r♦♠ t❤❡ ❧❛tt❡r ✈✐❛ t❤❡ ❞✐✛❡r❡♥❝❡s✿

✶✳✹✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

✹✹

❙❡❝♦♥❞✱ ❜❡❧♦✇ ✇❡ s❡❡ ❤♦✇ t❤❡ ♣♦s✐t✐♦♥s ❛r❡ ✉s❡❞ t♦ ❛❝q✉✐r❡ t❤❡ ✈❡❧♦❝✐t✐❡s ✈✐❛ t❤❡ ❞✐✛❡r❡♥❝❡s✱ ❜✉t ✇❡ ❛❧s♦ ❞✐s❝♦✈❡r t❤❛t ✇❡ ❝❛♥ ❣❡t t❤❡ ❢♦r♠❡r ❢r♦♠ t❤❡ ❧❛tt❡r ✈✐❛ t❤❡ s✉♠s✿

❲❡ ❦♥❡✇ t❤✐s ✇♦✉❧❞ ❤❛♣♣❡♥✿ ❆❢t❡r ❛❧❧✱ t❤❡ ♣♦s✐t✐♦♥ ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ❛r❡♥✬t r❡❛❧❧② ❞❡r✐✈❡❞ ❢r♦♠ ❡❛❝❤ ♦t❤❡r ❜✉t ❝♦✲❡①✐sts ❛s t✇♦ ❛ttr✐❜✉t❡s ♦❢ t❤❡ s❛♠❡ ♠♦t✐♦♥✳ ❙✉♣♣♦s❡ ♦♥❝❡ ❛❣❛✐♥ t❤❛t ✇❡ ❤❛✈❡ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ P ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b] ✇✐t❤ n ✐♥t❡r✈❛❧s ❛♥❞ • t❤❡ ♣r✐♠❛r② ♥♦❞❡s✿ xk , k = 0, 1, ..., n

• t❤❡ ✐♥❝r❡♠❡♥ts✿ ∆xk = xk − xk−1 , i = 1, 2, ..., n

• t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✿ ck , i = 1, 2, ..., n ✐♥ [xk−1 , xk ]

❚❤❡r❡ ❛r❡ t✇♦ ♣❛rts✿ ✶✳

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ g ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ck ✳ ❲❡ ❝♦♠♣✉t❡ ✐ts s✉♠ ✇✐t❤ ❛ ✈❛r✐❛❜❧❡ r✐❣❤t ❡♥❞✱ xk ✳ ❲❡ ❛ss✐❣♥ t❤❡s❡ ✈❛❧✉❡s t♦ t❤❡s❡ ♣r✐♠❛r② ♥♦❞❡s✳ ❚❤✐s ❞❡✜♥❡s ❛ ♥❡✇ ❢✉♥❝t✐♦♥✱

❙✉♠✿

✶✳✹✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

✹✺

G✱ ❞❡✜♥❡❞ ♦♥ t❤❡ ♣r✐♠❛r② ♥♦❞❡s✳ ❚❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❡①♣❧✐❝✐t❧②✿ G(xk ) = Σg (xk ) = g(c1 ) + g(c2 ) + ... + g(ck ) , ♦r ❝♦♠♣✉t❡❞ r❡❝✉rs✐✈❡❧②✿

G(xk ) = G(xk−1 ) + g(ck ) . ✷✳

❉✐✛❡r❡♥❝❡✿

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ F ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❏✉st ❛s ❛❧✇❛②s✱ ✐ts ❞✐✛❡r❡♥❝❡ ✐s ❝♦♠♣✉t❡❞ ♦✈❡r ❡❛❝❤ ✐♥t❡r✈❛❧ [xk−1 , xk ] ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ❛s ❢♦❧❧♦✇s✿

F (xk ) − F (xk−1 ), k = 1, 2..., n . ❲❡ t❤❡♥ ❤❛✈❡ ❛ ♥✉♠❜❡r ✕ r❡♣r❡s❡♥t✐♥❣ t❤❡ r✐s❡ ✕ ❢♦r ❡❛❝❤ k ✳ ❚❤✐s ❞❡✜♥❡s ❛ ♥❡✇ ❢✉♥❝t✐♦♥✱ f ✱ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✳ ❚❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ✐s ❝♦♠♣✉t❡❞ ❜② ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛✿

f (ck ) = F (xk ) − F (xk−1 ) . ❚❤❡ ✜rst q✉❡st✐♦♥ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❛♥s✇❡r ✐s✿

◮

❲❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ s✉♠ ❄

❲❡ ❤❛✈❡ g ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛♥❞ ✐ts s✉♠ G ❞❡✜♥❡❞ r❡❝✉rs✐✈❡❧② ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s✿ G(xk ) = G(xk−1 ) + g(ck ) . ❆❧s♦✱ t❤❡ ❞✐✛❡r❡♥❝❡ f ♦❢ G ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❜②✿

f (ck ) = G(xk ) − G(xk−1 ) . ❲❡

s✉❜st✐t✉t❡ t❤❡ ❧❛tt❡r ✐♥t♦ t❤❡ ❢♦r♠❡r✿ f (ck ) = G(xk ) − G(xk−1 ) = g(ck ) .

❙♦✱ t❤❡ ❛♥s✇❡r ✐s✱

t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥✳

❚❤❡ r❡s✉❧t t❛❦❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣❛❝t ❢♦r♠✿

❚❤❡♦r❡♠ ✶✳✹✳✷✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ■ ✶✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ s✉♠ ♦❢ g ✐s g ✿ ∆ (Σg) = g

✷✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ g ✐s g ✿ ∆ (Σg · ∆x) =g ∆x ❚❤❡ t✇♦ ♦♣❡r❛t✐♦♥s

❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✦

Pr♦♦❢✳ ❋♦r t❤❡ s❡❝♦♥❞ ♣❛rt✱ ✇❡ ♥❡❡❞ ❛ s❧✐❣❤t ♠♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡ ❛r❣✉♠❡♥t✳ ❲❡ ❤❛✈❡ g ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛♥❞ ✐ts ✭✈❛r✐❛❜❧❡✲❡♥❞✮ ❘✐❡♠❛♥♥ s✉♠ G ❞❡✜♥❡❞ r❡❝✉rs✐✈❡❧② ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s✿ G(xk ) = G(xk−1 ) + g(ck ) ∆xk .

✶✳✹✳ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

✹✻

❆❧s♦✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t f ♦❢ G ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❜②✿ f (ck ) =

G(xk ) − G(xk−1 ) . ∆xk

❲❡ s✉❜st✐t✉t❡ t❤❡ ❧❛tt❡r ✐♥t♦ t❤❡ ❢♦r♠❡r✿ G(xk ) − G(xk−1 ) ∆xk g(ck ) ∆xk = ∆xk

f (ck ) =

= g(ck ) .

❚❤❡ s❡❝♦♥❞ q✉❡st✐♦♥ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❛♥s✇❡r ✐s✿ ◮

❲❤❛t ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ❄

❲❡ ❤❛✈❡ F ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛♥❞ ✐ts ❞✐✛❡r❡♥❝❡ f ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✿ f (ck ) = F (xk ) − F (xk−1 ) .

❆❧s♦✱ t❤❡ s✉♠ G ♦❢ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ❜②✿ G(xk ) = G(xk−1 ) + f (ck ) .

❲❡ s✉❜st✐t✉t❡ t❤❡ ❢♦r♠❡r ✐♥t♦ t❤❡ ❧❛tt❡r✿ G(xk ) − G(xk−1 ) = f (ck ) = F (xk ) − F (xk−1 ) .

❋✉rt❤❡r♠♦r❡✱       G(xk ) − G(x0 ) = G(xk ) − G(xk−1 ) + G(xk−1 ) − G(xk−2 ) + ... + G(x1 ) − G(x0 )       = F (xk ) − F (xk−1 ) + F (xk−1 ) − F (xk−2 ) + ... + F (x1 ) − F (x0 ) = F (xk ) − F (x0 ) .

❙♦✱ t❤❡ ❛♥s✇❡r ✐s✱ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ ♣❧✉s ❛ ❝♦♥st❛♥t✳ ❚❤❡♦r❡♠ ✶✳✹✳✸✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ■■

✶✳ ❚❤❡ s✉♠ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ F ✐s F + C ✱ ✇❤❡r❡ C ✐s ❛ ❝♦♥st❛♥t✿ Σ (∆F ) = F + C

✷✳ ❚❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ F ✐s F + C ✱ ✇❤❡r❡ C ✐s ❛ ❝♦♥st❛♥t✿ Σ



∆F ∆x



∆x = F + C

Pr♦♦❢✳

❋♦r t❤❡ s❡❝♦♥❞ ♣❛rt✱ ✇❡ ♥❡❡❞ ❛ s❧✐❣❤t ♠♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡ ❛r❣✉♠❡♥t✳ ❲❡ ❤❛✈❡ F ❞❡✜♥❡❞ ❛t t❤❡

✶✳✹✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

✹✼

♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛♥❞ ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t f ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✿

f (ck ) =

F (xk ) − F (xk−1 ) . ∆xk

❆❧s♦✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠ G ♦❢ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ❜②✿

G(xk ) = G(xk−1 ) + f (ck ) ∆xk . ❲❡ s✉❜st✐t✉t❡ t❤❡ ❢♦r♠❡r ✐♥t♦ t❤❡ ❧❛tt❡r✿

G(xk ) − G(xk−1 ) = f (ck ) ∆xk =

F (xk ) − F (xk−1 ) ∆xk ∆xk

= F (xk ) − F (xk−1 ) . ❋✉rt❤❡r♠♦r❡✱

      G(xk ) − G(x0 ) = G(xk ) − G(xk−1 ) + G(xk−1 ) − G(xk−2 ) + ... + G(x1 ) − G(x0 )       = F (xk ) − F (xk−1 ) + F (xk−1 ) − F (xk−2 ) + ... + F (x1 ) − F (x0 ) = F (xk ) − F (x0 ) .

❚❤❡ t✇♦ ♦♣❡r❛t✐♦♥s ✕ ❛❧♠♦st ✕ ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✱ ❛❣❛✐♥✦ ❚❤❡② ❞♦♥✬t ❝❛♥❝❡❧ ❢✉❧❧② ❜❡❝❛✉s❡ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✐s♥✬t ❛ ♦♥❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥✳ ■♥ s✉♠♠❛r②✱ t❤❡s❡ ❛r❡ t❤❡ ♦♣❡r❛t✐♦♥s ✐♥✈♦❧✈❡❞✿ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱

∆f : → ∆x

s✉❜tr❛❝t✐♦♥

→

❞✐✈✐s✐♦♥

↓ ❘✐❡♠❛♥♥ s✉♠s✱ Σg · ∆x : ←

❛❞❞✐t✐♦♥

←

♠✉❧t✐♣❧✐❝❛t✐♦♥

❲❡ ❝❛rr② ♦✉t t❤❡s❡ ❢♦✉r ♦♣❡r❛t✐♦♥s ❝♦♥s❡❝✉t✐✈❡❧②✳ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ❜② ∆x ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r ✜rst✿ ❞✐✛❡r❡♥❝❡✱ ∆f : →

s✉❜tr❛❝t✐♦♥

↓ ❚❤❛t✬s ❛♥♦t❤❡r ❝❛♥❝❡❧❧❛t✐♦♥✦

s✉♠✱ Σg : ←

❛❞❞✐t✐♦♥

❊①❛♠♣❧❡ ✶✳✹✳✹✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ✇✐t❤ s♣r❡❛❞s❤❡❡t

❋♦r ❝♦♠♣❧❡① ❞❛t❛✱ ✇❡ ✉s❡ ❛ s♣r❡❛❞s❤❡❡t ✇✐t❤ t❤❡ ❢♦r♠✉❧❛s ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡rs ✷❉❈✲✷ ❛♥❞ ✷❉❈✲✸✳ ❋r♦♠ ❛ ❢✉♥❝t✐♦♥ t♦ ✐ts ❘✐❡♠❛♥♥ s✉♠✿

❂❘❬✲✶❪❈✰❘❈❬✲✶❪✯❘✶❈✷ ❋r♦♠ ❛ ❢✉♥❝t✐♦♥ t♦ ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿

❂✭❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪✮✴❘✶❈✷ ❲❤❛t ✐❢ ✇❡ ❡①❡❝✉t❡ t❤❡ t✇♦ ❝♦♠♣✉t❛t✐♦♥s ❝♦♥s❡❝✉t✐✈❡❧②❄

✶✳✹✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s

■♥ t❤✐s ♦r❞❡r ✜rst✱ ❢r♦♠ ❛ ❢✉♥❝t✐♦♥ t♦ ✐ts ❘✐❡♠❛♥♥ s✉♠ t♦ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❧❛tt❡r✿

■t✬s t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥✦ ◆♦✇ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ♦r❞❡r✱ ❢r♦♠ ❛ ❢✉♥❝t✐♦♥ t♦ ✐ts ❘✐❡♠❛♥♥ s✉♠ t♦ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❧❛tt❡r✿

■t✬s t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ s❤✐❢t❡❞ ❞♦✇♥✦

✹✽

✶✳✺✳

❍♦✇ t♦ ❛♣♣r♦①✐♠❛t❡ ❛r❡❛s ❛♥❞ ❞✐s♣❧❛❝❡♠❡♥ts

✹✾

✶✳✺✳ ❍♦✇ t♦ ❛♣♣r♦①✐♠❛t❡ ❛r❡❛s ❛♥❞ ❞✐s♣❧❛❝❡♠❡♥ts

❊①❛♠♣❧❡ ✶✳✺✳✶✿ ❡st✐♠❛t❡ ❛r❡❛s

▲❡t✬s ❡st✐♠❛t❡ ✕ ✐♥ s❡✈❡r❛❧ ✇❛②s ✕ t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ ✐♥t❡r✈❛❧

y = f (x) = x2

t❤❛t ❧✐❡s ❛❜♦✈❡ t❤❡

[0, 1]✳

❆s ❜❡❢♦r❡✱ ✇❡ ✇✐❧❧ ▲❡t✬s st❛rt ✇✐t❤

f (1) = 12 = 1✱

s❛♠♣❧❡ ♦✉r ❢✉♥❝t✐♦♥ ❛t s❡✈❡r❛❧ ✈❛❧✉❡s ♦❢ x✱ ❛ t♦t❛❧ ♦❢ n ♦❢ t✐♠❡s✳

n = 1

❛♥❞ ♣✐❝❦ t❤❡

t❤❡ ❛r❡❛ ✐s

1✱

r✐❣❤t ❡♥❞

♦❢ t❤❡ ✐♥t❡r✈❛❧ ❛s t❤❡ ♦♥❧② s❡❝♦♥❞❛r② ♥♦❞❡✳

❍❡r❡✱

t❤❡ ✇❤♦❧❡ sq✉❛r❡✳ ❲❡ r❡❝♦r❞ t❤✐s r❡s✉❧t ❛s ❢♦❧❧♦✇s✿

R1 = 1 , ✇❤❡r❡

R

st❛♥❞s ❢♦r ✏r✐❣❤t✑✳

■♥ t❤❡ ♠❡❛♥t✐♠❡✱ ✐❢ ✇❡ ❝❤♦♦s❡ t❤❡

❧❡❢t

❡♥❞✱ ✇❡ ❤❛✈❡

f (1) = 02 = 0✱

s♦ t❤❡ ❛r❡❛ ✐s

r❡s✉❧t ❛s ❢♦❧❧♦✇s✿

L1 = 0 , ✇❤❡r❡

L

◆❡①t✱

n = 2✳

st❛♥❞s ❢♦r ✏❧❡❢t✑✳ ❚❤❡♥

∆x = 1/2

❛♥❞ t❤❡ ✐♥t❡r✈❛❧ ✐s s✉❜❞✐✈✐❞❡❞ ❛s ❢♦❧❧♦✇s✿

•

−−−

|

x

0

1/2

f (x) = x2

0

1/4

L2 0 · 1/2 R2

+

−−−

• 1

1/4 · 1/2 1/4 · 1/2

= 1/8 +

1 · 1/2

= 5/8

0✳

❲❡ r❡❝♦r❞ t❤✐s

✶✳✺✳

❍♦✇ t♦ ❛♣♣r♦①✐♠❛t❡ ❛r❡❛s ❛♥❞ ❞✐s♣❧❛❝❡♠❡♥ts

✺✵

◆♦t❡ t❤❛t t❤❡s❡ t✇♦ q✉❛♥t✐t✐❡s ✉♥❞❡r❡st✐♠❛t❡ ❛♥❞ ♦✈❡r❡st✐♠❛t❡ t❤❡ ❛r❡❛✱ r❡s♣❡❝t✐✈❡❧②✳ ◆♦✇✱ n = 4✳ ❚❤❡♥ ∆x = 1/4 ❛♥❞ t❤❡ ✐♥t❡r✈❛❧ ✐s s✉❜❞✐✈✐❞❡❞ ❛s ❢♦❧❧♦✇s✿ •

−−

|

−−

|

−−

|

−−

•

x

0

1/4

1/2

3/4

1

x2

0

1/16

1/4

9/16

1

L4 0 · 1/4 R4

+

1/16 · 1/4

+

1/4 · 1/4

+

9/16 · 1/4

1/16 · 1/4

+

1/4 · 1/4

+

9/16 · 1/4

≈ 0.22 +

1 · 1/4 ≈ 0.47

◆♦t❡ t❤❛t t❤❡s❡ t✇♦ q✉❛♥t✐t✐❡s ✉♥❞❡r❡st✐♠❛t❡ ❛♥❞ ♦✈❡r❡st✐♠❛t❡ t❤❡ ❛r❡❛✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ❛❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐♥ t❤❡ t❛❜❧❡✿ 1 1 1 1 1 9 1 · + · + · + ·1 4 16 4 4 4 16  4 4 9 16 1 1 + + + = 4 16 16 16 16 1 30 = 4 16

R4 =

≈ 0.47 .

❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ ❧❛r❣❡r ❛♥❞ ❧❛r❣❡r ✈❛❧✉❡s ♦❢ n✳ ❲❡ ❡♥❞ ✉♣ ✇✐t❤ ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs✳ ❚❤❡♥ t❤❡ ❧✐♠✐t ♦❢ t❤✐s s❡q✉❡♥❝❡ ✐s ♠❡❛♥t t♦ ♣r♦❞✉❝❡ t❤❡ ❛r❡❛ ♦❢ t❤✐s ❝✉r✈❡❞ ✜❣✉r❡✳ ❚❤❡s❡ ❡①♣r❡ss✐♦♥s ❛r❡ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❚❤✐s ❤♦✇ ✐t ✇♦r❦s✿

❛♣♣r♦①✐♠❛t❡ t❤❡ ❡①❛❝t ❛r❡❛✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ✐s t❤❡ ❡①❛❝t ❛r❡❛✳

• ❚❤❡ ❘✐❡♠❛♥♥ s✉♠s •

✶✳✺✳

❍♦✇ t♦ ❛♣♣r♦①✐♠❛t❡ ❛r❡❛s ❛♥❞ ❞✐s♣❧❛❝❡♠❡♥ts

✺✶

❊①❛♠♣❧❡ ✶✳✺✳✷✿ ♠✐❞✲♣♦✐♥ts ❚❤❡r❡ ❛r❡ ♦t❤❡r ❘✐❡♠❛♥♥ s✉♠s ✐❢ ✇❡ ❝❤♦♦s❡ ♦t❤❡r s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ✐♥t❡r✈❛❧s✱ s✉❝❤ ❛s ♠✐❞✲♣♦✐♥ts❀ 2 ✐t ✐s ❞❡♥♦t❡❞ ❜② Mn ✳ ▲❡t✬s ❛❣❛✐♥ ❡st✐♠❛t❡ t❤❡ ❛r❡❛ ✉♥❞❡r y = x ✇✐t❤ n = 4✳ ❙✐♥❝❡ L4 ❛♥❞ R4 ✉♥✲ ❞❡r❡st✐♠❛t❡ ♦r ♦✈❡r❡st✐♠❛t❡ t❤❡ ❛r❡❛ r❡s♣❡❝t✐✈❡❧②✱ ♦♥❡ ♠✐❣❤t ❡①♣❡❝t t❤❛t ❚❤❡ ✈❛❧✉❡ ♦❢

∆x

✐s st✐❧❧

• x f (x) = x2 M4

M4

✇✐❧❧ ❜❡ ❝❧♦s❡r t♦ t❤❡ tr✉t❤✳

1/4✿ −−

|

−−

|

−−

|

−−

1/8

3/8

5/8

7/8

(1/8)2

(3/8)2

(5/8)2

(7/8)2

•

(1/8)2 · 1/4 + (3/8)2 · 1/4 + (5/8)2 · 1/4 + (7/8)2 · 1/4 ≈ 0.328

❊①❡r❝✐s❡ ✶✳✺✳✸ ❋✐♥✐s❤ t❤❡ s❡♥t❡♥❝❡ ✏ Ln ❛♥❞

Rn

✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ✉♥❞❡r❡st✐♠❛t❡ ❛♥❞ ♦✈❡r❡st✐♠❛t❡ t❤❡ ❛r❡❛ ♥♦ ♠❛tt❡r

❤♦✇ ♠❛♥② ✐♥t❡r✈❛❧s ✇❡ ❤❛✈❡ ❜❡❝❛✉s❡

f

✐s✳✳✳✑

❊①❡r❝✐s❡ ✶✳✺✳✹ ❋✐♥✐s❤ t❤❡ s❡♥t❡♥❝❡ ✏ Mn ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ✉♥❞❡r❡st✐♠❛t❡ t❤❡ ❛r❡❛ ♥♦ ♠❛tt❡r ❤♦✇ ♠❛♥② ✐♥t❡r✈❛❧s ✇❡ ❤❛✈❡ ❜❡❝❛✉s❡

f

✐s✳✳✳✑

❊①❛♠♣❧❡ ✶✳✺✳✺✿ ❛r❡❛ ♦❢ tr✐❛♥❣❧❡ ▲❡t✬s t❡st t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❛♣♣r♦❛❝❤ t♦ ❝♦♠♣✉t✐♥❣ ❛r❡❛s t♦ ❛♥♦t❤❡r ❢❛♠✐❧✐❛r r❡❣✐♦♥✱ ❛ tr✐❛♥❣❧❡✳ ❙✉♣✲ ♣♦s❡

f (x) = x . ❲❤❛t ✐s t❤❡ ❛r❡❛ ✉♥❞❡r ✐ts ❣r❛♣❤ ❢r♦♠

x=0

t♦

x = 1❄

❲❡ st❛rt ✇✐t❤ ❛ ❧❡❢t✲❡♥❞ ♣❛rt✐t✐♦♥ ❛♥❞ t❤❡s❡ s❡❝♦♥❞❛r② ♥♦❞❡s✿

0

1 n

2 n

0 = c1

c2

c3 ...

❲❡ ♥❡①t ♣❧♦t t❤✐s ❝❤❛rt ❢♦r

n = 2 , 4 , 8 , 16 , 80

...

n−1 n

1

cn

◦

t♦ ✈✐s✉❛❧✐③❡ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿

✶✳✺✳ ❍♦✇ t♦ ❛♣♣r♦①✐♠❛t❡ ❛r❡❛s ❛♥❞ ❞✐s♣❧❛❝❡♠❡♥ts

✺✷

❚❤❡ ❧❛st ♦♥❡ ❤❛s ✈✐rt✉❛❧❧② ♥♦ ❣❛♣s✳ ❲❡ ❧❡t t❤❡ s♣r❡❛❞s❤❡❡t t♦ ✜♥❞ t❤❡ t♦t❛❧ ❛r❡❛s ✭❝♦r♥❡rs✮✿ n

2

∆x 1/2

4

8

16

80

... n

1/4

1/8

1/16

1/80

... 1/n

Dn 0.250 0.375 0.438 0.469 0.494 ... ?

❚❤❡ ♥✉♠❜❡rs s❡❡♠ t♦ ❛♣♣r♦❛❝❤ 0.5 ❛s ❡①♣❡❝t❡❞✳ ❚♦ ✜♥❞ t❤❡ ❢✉❧❧ tr✉t❤✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t❤❡ ❧✐♠✐t ♦❢ t❤✐s s❡q✉❡♥❝❡✦ ❋♦r ❛♥ ❛r❜✐tr❛r② n✱ t❤❡ t♦t❛❧ ❛r❡❛ ✐s ♦♥❧② ✇r✐tt❡♥ r❡❝✉rs✐✈❡❧②✳ ▲❡t✬s tr② t♦ s✐♠♣❧✐❢② ✭f (x) = x✮✿ Dn

= =  =

❙✉❜st✐t✉t❡✳

f (c1 ) · ∆x + f (c2 ) · ∆x + ... + f (cn ) · ∆x 1 1 · n n

+

1

+

2 1 · n n

+ ... +

2

+ ... +

n−1 1 · n n (n − 1)

❋❛❝t♦r ✐t✳ 

·

1 . n2

❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤✐s ❧✐♠✐t❄ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ s✉♠ ✐♥ ♣❛r❡♥t❤❡s❡s ✐s ✐♥✜♥✐t❡ ❛♥❞ s♦ ✐s t❤❛t ♦❢ n2 ✳ ❲❡ ❤❛✈❡ ❛♥ ✐♥❞❡t❡r♠✐♥❛t❡ ❡①♣r❡ss✐♦♥✿ ∞ · 0✳ ❏✉st ❛s ❜❡❢♦r❡✱ t❤❡ ♦♥❧② ✇❛② t♦ r❡s♦❧✈❡ ✐t ✐s ❛❧❣❡❜r❛✳ ❲❡ ♥❡❡❞ t♦ ✜♥❞ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r t❤❡ s✉♠✳ ❋♦rt✉♥❛t❡❧②✱ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❡①♣r❡ss✐♦♥ ✐♥ ♣❛r❡♥t❤❡s❡s ✐s ❦♥♦✇♥ ✭❈❤❛♣t❡r ✶P❈✲✶✮ ❛s t❤❡ s✉♠ ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✿ n(n − 1) . 2 ❚❤❡r❡❢♦r❡✱ t❤❡ t♦t❛❧ ❛r❡❛ ♦❢ t❤❡ ❜❛rs ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t ❛s n → ∞✿ 1 + 2 + ... + (n − 1) =

Dn =

n(n − 1) 1 1 n2 − n 1 = → , 2 2 2 n 2 n 2

❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠ ▲✐♠✐ts ♦❢ ❘❛t✐♦♥❛❧ ❋✉♥❝t✐♦♥s ✭❈❤❛♣t❡r ✷❉❈✲✷✮✳ ❚❤❡ r❡s✉❧t ♠❛t❝❤❡s ✇❤❛t ✇❡ ❦♥♦✇ ❢r♦♠ ❣❡♦♠❡tr②✳ ■t s❤♦✉❧❞ ❜❡ ❝❧❡❛r t❤❛t ✇❡ ✇♦✉❧❞♥✬t ❜❡ ❛❜❧❡ t♦ ❛♣♣❧② t❤❡ r✉❧❡s ❛♥❞ ♠❡t❤♦❞s ♦❢ ❝♦♠♣✉t✐♥❣ ❧✐♠✐ts ✇✐t❤♦✉t t❤✐s s✐♠♣❧✐✜❝❛t✐♦♥ st❡♣✳ ❯♥❧✐❦❡ ♠♦st s❡q✉❡♥❝❡s ✇❡ s❛✇ ✐♥ ❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s ✭❱♦❧✉♠❡ ✷✱ ❈❤❛♣t❡r ✷❉❈✲✶✮✱ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❤❛s n t❡r♠s✳ ❚❤✐s ♠❡❛♥s t❤❛t ✇❡ ❞♦ ♥♦t ❤❛✈❡ ❛ ❞✐r❡❝t✱ ♦r ❡①♣❧✐❝✐t✱ ❢♦r♠✉❧❛ ❢♦r t❤❡ nt❤ t❡r♠ ♦❢ t❤✐s s❡q✉❡♥❝❡✳ ❈♦♥✈❡rt✐♥❣ t❤❡ ❢♦r♠❡r ✐♥t♦ t❤❡ ❧❛tt❡r r❡q✉✐r❡s s♦♠❡ ❝❤❛❧❧❡♥❣✐♥❣ ❛❧❣❡❜r❛✳ ❆ ❢❡✇ s✉❝❤ ❢♦r♠✉❧❛s ❛r❡ ❦♥♦✇♥✳ ❚❤❡♦r❡♠ ✶✳✺✳✻✿ ❋♦r♠✉❧❛s ❢♦r ❋✐♥✐t❡ ❙✉♠s ❚❤❡ s✉♠s ♦❢

m

❝♦♥s❡❝✉t✐✈❡ ♥✉♠❜❡rs✱ t❤❡✐r sq✉❛r❡s✱ ❛♥❞ t❤❡✐r ❝✉❜❡s ❛r❡ t❤❡

✶✳✺✳

✺✸

❍♦✇ t♦ ❛♣♣r♦①✐♠❛t❡ ❛r❡❛s ❛♥❞ ❞✐s♣❧❛❝❡♠❡♥ts

❢♦❧❧♦✇✐♥❣✿

m X

k

=

k=1

m X

m(m + 1)(2m + 1) m3 m2 m = + + 6 3 2 6 2  3 4 m m2 m m(m + 1) + + = = 2 4 2 4

k2 =

k=1

m X

m(m + 1) 2

k3

k=1

Pr♦♦❢✳

❚❤❡ ✜rst ♦♥❡ ✐s ♣r♦✈❡♥ ✐♥ ❈❤❛♣t❡r ✶P❈✲✶✳ ❊①❛♠♣❧❡ ✶✳✺✳✼✿ ❛r❡❛ ✉♥❞❡r ♣❛r❛❜♦❧❛

❙✉♣♣♦s❡ ✇❡ ❛r❡ ❢♦❧❧♦✇✐♥❣ ❛ ❧❛♥❞✐♥❣ ♠♦❞✉❧❡ ♦♥ ✐ts tr✐♣ t♦ t❤❡ ♠♦♦♥✳ ❚❤❡ ✈❡r② ❧❛st ♣❛rt ♦❢ t❤❡ tr✐♣ ✐s ✈❡r② ❝❧♦s❡ t♦ t❤❡ s✉r❢❛❝❡ ❛♥❞ ✐t ✐s s✉♣♣♦s❡❞ t♦ ❝♦✈❡r 2/3 ♦❢ ❛ ♠✐❧❡ ✐♥ ♦♥❡ ♠✐♥✉t❡✳ ❈♦♥s✐❞✲ ❡r✐♥❣ t❤❛t t❤❡r❡ ✐s ♥♦ ✇❛② t♦ ♠❡❛s✉r❡ ❛❧t✐t✉❞❡ ❛❜♦✈❡ t❤❡ s✉r❢❛❝❡✱ ❤♦✇ ✇♦✉❧❞ ✇❡ ❦♥♦✇ t❤❛t ✐t ❤❛s ❧❛♥❞❡❞❄ Pr♦❜❧❡♠✿

❚❤✐s ✐s ✇❤❛t ✇❡ ❞♦ ❦♥♦✇✿

❚❤❡ ✈❡❧♦❝✐t②✱ ❛t ❛❧❧ t✐♠❡s✿

❲❡ ❛♣♣r♦❛❝❤ t❤❡ ♣r♦❜❧❡♠ ❜② ♣r❡❞✐❝t✐♥❣ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❡✈❡r②✿ • 1 ♠✐♥✉t❡✱ ♦r • 1/2 ♠✐♥✉t❡✱ ♦r • 1/4 ♠✐♥✉t❡✱ ♦r • ❡t❝✳✱ ❜② ✉s✐♥❣ t❤❡ s♣❡❡❞ r❡❝♦r❞❡❞ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❡❛❝❤ ♦❢ t❤❡s❡ ✐♥t❡r✈❛❧s✿

❙♦❧✉t✐♦♥✿

♣r♦❥❡❝t❡❞ ❞✐s♣❧❛❝❡♠❡♥t = ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② · t✐♠❡ ♣❛ss❡❞. ❲❡ ❤❛✈❡ ❛❧r❡❛❞② ♣❧♦tt❡❞ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ ♥♦✇ ✇❡ ❛r❡ ❛❧s♦ t♦ ✈✐s✉❛❧✐③❡ t❤❡ ♣r♦❥❡❝t❡❞ ❞✐s♣❧❛❝❡♠❡♥t✳ ■t ✐s ❛ ♣❧♦t ♦❢ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ ❜✉t ♣r❡s❡♥t❡❞ ❛s ❛ ❞✐✛❡r❡♥t ❦✐♥❞ ♦❢ ♣❧♦t✱ ❛ ❜❛r ❝❤❛rt✿

❲❡ ♣❧♦t t❤✐s ❝❤❛rt ❢♦r ❞✐✛❡r❡♥t ♥✉♠❜❡r ♦❢ t✐♠❡ ✐♥t❡r✈❛❧s✱ n✱ ❛♥❞ ❞✐✛❡r❡♥t ❝❤♦✐❝❡s ♦❢ t❤❡✐r ❧❡♥❣t❤s✱ ∆x = 1/n✳ ❲❡ ❛❧s♦ ❧❡t t❤❡ s♣r❡❛❞s❤❡❡t ❛❞❞ t❤❡s❡ ♥✉♠❜❡rs t♦ ♣r♦❞✉❝❡ t❤❡ t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t✱ Dn ✱

✶✳✺✳

❍♦✇ t♦ ❛♣♣r♦①✐♠❛t❡ ❛r❡❛s ❛♥❞ ❞✐s♣❧❛❝❡♠❡♥ts

✺✹

♦✈❡r t❤❡ ✇❤♦❧❡ ♦♥❡✲♠✐♥✉t❡ ✐♥t❡r✈❛❧✳ ❚❤✐s ✐s t❤❡ ❞❛t❛✿ n

2

∆x 1/2

4

8

16

80

1/4

1/8

1/16

1/80

Dn 0.869 0.778 0.725 0.697 0.673

❚❤❡ ❞❛t❛ s✉❣❣❡sts t❤❛t t❤❡ ❝r❛❢t ❤❛s ❧❛♥❞❡❞ ❛s t❤❡ ❡st✐♠❛t❡❞ ❞✐s♣❧❛❝❡♠❡♥t s❡❡♠s t♦ ❜❡ ❝❧♦s❡ t♦ 2/3 ♠✐❧❡s✳ ❍♦✇ ❝❧♦s❡ ❛r❡ ✇❡ t♦ t❤✐s ❞✐st❛♥❝❡❄ ❆s t❤✐s q✉❡st✐♦♥ ♠❡❛♥s ❞✐✛❡r❡♥t t❤✐♥❣s t♦ ❞✐✛❡r❡♥t ♣❡♦♣❧❡✱ ❧❡t✬s tr② t♦ ✜♥❞ ❛ r✉❧❡ ❢♦r ❛♥s✇❡r✐♥❣ ✐t✿ ❋✐♥❞ t❤❡ ♣r♦❥❡❝t❡❞ ❞✐s♣❧❛❝❡♠❡♥t Dn ❛s ✐t ❞❡♣❡♥❞s ♦♥ n✳ ❋✐rst✱ t❤❡ ❧❡♥❣t❤ ♦❢ ❡❛❝❤ t✐♠❡ ✐♥t❡r✈❛❧ ✐s✿ ∆x =

❛♥❞ t❤❡ ♠♦♠❡♥ts ♦❢ t✐♠❡ ❢♦r s❛♠♣❧✐♥❣ ❛r❡✿ x1 = 0, x2 =

1 , n

3 n−1 2 , x3 = , ..., xn = . n n n

❋✉rt❤❡r♠♦r❡✱ ✇❡ ♥❡❡❞ ❡①❛❝t✱ ❝♦♠♣❧❡t❡ ❞❛t❛ ❛❜♦✉t t❤❡ ✈❡❧♦❝✐t②✳ ❙✉♣♣♦s❡ t❤❡ ✈❡❧♦❝✐t② y ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✱ x✱ ✐s ❣✐✈❡♥ ❜② t❤✐s✱ ❡①❛❝t ❢♦r♠✉❧❛✿ y = f (x) = 1 − x2 .

❚❤❡♥✱ t❤❡ t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t ✐s Dn = = = 1− = 1−





f (x1 ) · ∆x +  2 ! 1 1 + · 1− n n 12 + n2 12

f (x2 ) · ∆x + ... +  2 ! 2 1 1− + ... + · n n 22 + ... + n2 22

+

+ ... +

f (xn ) · ∆x  2 ! n−1 1 1− · n n  1 (n − 1)2 · 2 n  n 1 · 3. (n − 1)2 n

❲❡ ❛r❡ ✐♥ t❤❡ s❛♠❡ ♣❧❛❝❡ ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✿ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ✐♠♣r♦✈✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥s ✐s t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s ♦❢ ❛ ❝❡rt❛✐♥ s❡q✉❡♥❝❡✳ ❆♥❞✱ ♦♥❝❡ ❛❣❛✐♥✱ t❤❡ r❡❝✉rs✐✈❡ ❡①♣r❡ss✐♦♥ ♠✉st ❜❡ ❝♦♥✈❡rt❡❞ t♦ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ✐♥ ♦r❞❡r t♦ ❛♣♣❧② t❤❡ ♠❡t❤♦❞s ♦❢ ❝♦♠♣✉t✐♥❣ ❧✐♠✐ts t❤❛t ✇❡ ❦♥♦✇✳ ▲❡t✬s s✐♠♣❧✐❢② ✐t ✉s✐♥❣ t❤❡ ❧❛st t❤❡♦r❡♠✿ 12 + 22 + ... + (n − 1)2 =

❚❤❡r❡❢♦r❡✱ Dn



(n − 1)3 (n − 1)2 n − 1 + + . 3 2 6

(n − 1)3 (n − 1)2 n − 1 =1− + + 3 2 6 2 3 (n − 1) n−1 (n − 1) − − . =1− 3 3 3n 2n 6n3



1 n3

❲✐t❤ t❤✐s ❢♦r♠✉❧❛✱ ✇❡ ❝❛♥ ✐♠♣r♦✈❡ t❤❡ ❛❝❝✉r❛❝② ♦❢ ♦✉r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t t♦ ❛♥② ❞❡❣r❡❡ ✇❡ ❞❡s✐r❡ ❜② ❝❤♦♦s✐♥❣ ❧❛r❣❡r ❛♥❞ ❧❛r❣❡r ✈❛❧✉❡s ♦❢ n✳

✶✳✻✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

✺✺

◆❡①t✱ ✇❤❛t ✐s t❤❡ ❡①❛❝t ❞✐s♣❧❛❝❡♠❡♥t D❄ ■t ✐s t❤❡ ❧✐♠✐t ♦❢ t❤❡ s❡q✉❡♥❝❡ Dn ✳ ❲❡ ❛♣♣❧② t❤❡ t❤❡♦r❡♠ ▲✐♠✐ts ♦❢ ❘❛t✐♦♥❛❧ ❋✉♥❝t✐♦♥s ✭❈❤❛♣t❡r ✷❉❈✲✶✮✮ ❛♥❞ ❝♦♠♣❛r❡ t❤❡ ❧❡❛❞✐♥❣ t❡r♠s ♦❢ t❤❡ t✇♦ ❢r❛❝t✐♦♥s✿ Dn → 1 −

2 1 −0−0= . 3 3

❊①❡r❝✐s❡ ✶✳✺✳✽

❘❡❞♦ t❤❡ ❡①❛♠♣❧❡ ❢♦r f (x) = x2 ✳ ❊①❡r❝✐s❡ ✶✳✺✳✾

❯s❡ t❤❡ ❧❛st ❢♦r♠✉❧❛ ✐♥ t❤❡ t❤❡♦r❡♠ t♦ ✜♥❞ t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ y = x3 ✳ ■♥ ♦r❞❡r t♦ ❡s❝❛♣❡ t❤❡ ♥❡❡❞ ❢♦r t❤❡♦r❡♠s ❧✐❦❡ t❤❡ ♦♥❡ ❛❜♦✈❡✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t♦ ❞❡✈❡❧♦♣ ❛ ✇♦r❦✲❛r♦✉♥❞✳ ❆♥ ✐♥❞✐r❡❝t ❛♣♣r♦❛❝❤ ✇✐❧❧ ♣r♦✈❡ ♠♦r❡ ❡✛❡❝t✐✈❡✳

✶✳✻✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

❲❡ ✐♠♣r♦✈❡ ♦✉r ❛♣♣r♦①✐♠❛t✐♦♥s t♦ ♣r♦❞✉❝❡ ❜❡tt❡r ❛♥❞ ❡st✐♠❛t❡s ♦❢ t❤❡ ❛r❡❛s ♦r ❞✐s♣❧❛❝❡♠❡♥ts ❜② ♠❛❦✐♥❣ t❤❡ ♣❛rt✐t✐♦♥ ✜♥❡r ❛♥❞ ✜♥❡r✳ ❲❡ t❤❡♥ ❝♦♥s✐❞❡r t❤❡ ❧✐♠✐t ♦❢ t❤✐s ♣r♦❝❡ss✳ ❲❡ ❛r❡✱ ✐♥ ❢❛❝t✱ ❢♦❧❧♦✇✐♥❣ t❤❡ r♦✉t❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿ s❛♠♣❧✐♥❣

s❛♠♣❧✐♥❣

s❡q✉❡♥❝❡ ←−−−−−−−− ❢✉♥❝t✐♦♥ ♦♥ [a, b]

s❡q✉❡♥❝❡ ←−−−−−−−− ❢✉♥❝t✐♦♥ ♦♥ [a, b]

s❡q✉❡♥❝❡ −−−∆x→0 −−−−−→ ❢✉♥❝t✐♦♥ ♦♥ [a, b]

s❡q✉❡♥❝❡ −−−∆x→0 −−−−−→ ❢✉♥❝t✐♦♥ ♦♥ [a, b]

  ❞❡r✐✈❛t✐✈❡ y

  ❉◗ y

  ✐♥t❡❣r❛❧ y

  ❘❙ y

▲❡t✬s r❡✈✐❡✇ t❤❡ s❡t✉♣✳ ❙✉♣♣♦s❡ t❤✐s t✐♠❡ t❤❛t ❛ ❢✉♥❝t✐♦♥ f ✐s ❞❡✜♥❡❞ ❛t ❛❧❧ ♣♦✐♥ts ♦❢ ✐♥t❡r✈❛❧ [a, b]✳ ❲❡ ❛r❡ ❛❧s♦ ❝♦♥s✐❞❡r✐♥❣ ❛❧❧ ♣♦ss✐❜❧❡ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥s P ♦❢ [a, b]✿ a = x0 ≤ c1 ≤ x1 ≤ c2 ≤ x2 ≤ . . . < xn−1 ≤ c1 ≤ xn = b .

❚❤❡♥ ❡❛❝❤ s✉❝❤ ♣❛rt✐t✐♦♥ ✇✐❧❧ ♣r♦❞✉❝❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿ f (ck+1 ) − f (ck ) ∆f = , ∆x ∆xk

❛♥❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿ Σf · ∆x = f (c1 ) ∆x1 + f (c2 ) ∆x2 + ... + f (cn ) ∆xn =

✇❤❡r❡

n X

f (ci ) ∆xi ,

i=1

∆xi = xi − xi−1 .

◆♦✇✱ ✐♥ ♦r❞❡r t♦ ✐♠♣r♦✈❡ t❤❡s❡ ❛♣♣r♦①✐♠❛t✐♦♥s✱ ✇❡ r❡✜♥❡ t❤❡ ♣❛rt✐t✐♦♥✿ ❚❤❡r❡ ✇✐❧❧ ❜❡ ♠♦r❡ ✐♥t❡r✈❛❧s ❛♥❞ t❤❡② ❛r❡ s♠❛❧❧❡r✳ ❲❡ ❦❡❡♣ r❡✜♥✐♥❣✳ ❚❤❡ r❡s✉❧t ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐t✐♦♥s✱ Pn ✳ ❍❡r❡ ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ t②♣✐❝❛❧ s❡q✉❡♥❝❡ ♦❢ r❡✜♥✐♥❣ ♣❛rt✐t✐♦♥s✳ ❲❡ s✐♠♣❧② ❝✉t ❡✈❡r② ✐♥t❡r✈❛❧ ✐♥ ❤❛❧❢ ❡✈❡r② t✐♠❡ ✭❧❡❢t✮ ❛❣❛✐♥ ❛♥❞ ❛❣❛✐♥✿

✶✳✻✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

✺✻

❚❤❡ ❧❡❢t ❡♥❞s ❛r❡ ❝❤♦s❡♥ ❛s s❡❝♦♥❞❛r② ♥♦❞❡s t♦ s❛♠♣❧❡ t❤❡ ❢✉♥❝t✐♦♥ ✭r✐❣❤t✮✳ ❚❤❡ r❡s✉❧t ✐s ✇❤❛t ✇❡ ❝❛♥ t❤✐♥❦ ♦❢ ❛s ❛ ♥❡✇ ❢✉♥❝t✐♦♥ fn ✳ ❊①❛♠♣❧❡ ✶✳✻✳✶✿

sin x

▲❡t✬s ❝♦♥s✐❞❡r f (x) = sin x .

❋♦r ❡❛❝❤ n = 2, 3, 4, ...✱ t❤❡ ✐♥❝r❡♠❡♥t ✐s ❢♦✉♥❞✱ ∆x = (b − a)/n✱ ❛♥❞ ✇❡ ❤❛✈❡ n s❡❣♠❡♥ts ✐♥ ♦✉r ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡r✈❛❧✳ ❖♥ ❡❛❝❤ ♦❢ ✐ts s❡❣♠❡♥ts✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐s ❝♦♠♣✉t❡❞✱ t❤❡ ✈❛❧✉❡ ✐s r❡❝♦r❞❡❞ ❛s t❤❡ ✈❛❧✉❡ ♦❢ ❛ ♥❡✇ ❢✉♥❝t✐♦♥✱ ❛♥❞ t❤❡ r❡s✉❧t ✐s ♣❧♦tt❡❞ ✐♥ t❤❡ ❜♦tt♦♠ r♦✇✿

❲❡ ❢❛❝❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s✳ ■ts ❧✐♠✐t ✐s t❤❡ ❞❡r✐✈❛t✐✈❡✳ ❙✐♠✐❧❛r❧② t❤❡ ❘✐❡♠❛♥♥ s✉♠ ✐s ❝♦♠♣✉t❡❞✱ t❤❡ ✈❛❧✉❡ ✐s r❡❝♦r❞❡❞ ❛s t❤❡ ✈❛❧✉❡ ♦❢ ❛ ♥❡✇ ❢✉♥❝t✐♦♥✱ ❛♥❞ t❤❡ r❡s✉❧t ✐s ♣❧♦tt❡❞ ✐♥ t❤❡ ❜♦tt♦♠ r♦✇✿

❲❡ ❢❛❝❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ❛❣❛✐♥✳ ❲❤❛t ✐s ✐ts ❧✐♠✐t❄ ❖✉r s❛♠♣❧✐♥❣ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❣❡tt✐♥❣ ❞❡♥s❡r ❛♥❞ ❞❡♥s❡r✳ ■♥ t❤❡ ♠❡❛♥t✐♠❡✱ t❤❡ ♣♦✐♥ts t❤❛t ♠❛❦❡ ✉♣ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❛♥❞ t❤♦s❡ t❤❛t ♠❛❦❡ ✉♣ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛r❡ ❣❡tt✐♥❣ ❝❧♦s❡r ❛♥❞ ❝❧♦s❡r t♦❣❡t❤❡r✳ ❲❤❛t ✐s ❛t t❤❡ ❡♥❞ ♦❢ t❤✐s ♣r♦❝❡ss❄ ❆ ♥❡✇ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✦ ❋♦r s✐♠♣❧✐❝✐t②✱ ❜❡❧♦✇ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❘✐❡♠❛♥♥ s✉♠s ♦♥ ❛ ✜①❡❞ ✐♥t❡r✈❛❧✳ ❲❡ ✇✐❧❧ ❜❡ ✉s✐♥❣ [a, b] ✐♥st❡❛❞ ♦❢ ❛❧❧

✶✳✻✳

❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

♦❢ t❤❡s❡✿

✺✼

[a, xk ]✳

❚❤❡ ❘✐❡♠❛♥♥ s✉♠s ♦✈❡r

[a, b]

♦❢ t❤❡ ❢✉♥❝t✐♦♥s

fn

❛❝q✉✐r❡❞ ❢r♦♠ s❛♠♣❧✐♥❣✱

Sn = Σfn ∆x , ❢♦r♠ ❛

♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡✱ Sn ✳ n → ∞✿

❆s ✇❡ ❤❛✈❡ s❡❡♥✱ t❤✐s s❡q✉❡♥❝❡ ♠❛② ❝♦♥✈❡r❣❡ ❛s

❚❤❡ ❧✐♠✐t ♦❢ t❤✐s s❡q✉❡♥❝❡✱

lim Sn = S ✱

n→∞

✐s ✇❤❛t ✇❡ ❛r❡ ❛❢t❡r✳

❍♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ ❧❡❢t✲❡♥❞✱ r✐❣❤t✲❡♥❞✱ ♠✐❞✲♣♦✐♥t ♣❛rt✐t✐♦♥s✱ ♣❛rt✐t✐♦♥s ✇✐t❤ ✉♥❡q✉❛❧ ✐♥❝r❡♠❡♥ts✱ ❛♥❞ ✐♥✲ ✜♥✐t❡❧② ♠❛♥② ♦t❤❡r s❡q✉❡♥❝❡s ♦❢ ♣❛rt✐t✐♦♥s✳ ❏✉st ❛s ♦♥❡ ❝❛♥ ❛♣♣r♦①✐♠❛t❡ ❛ ❝✐r❝❧❡ ✇✐t❤ ♥♦♥✲r❡❣✉❧❛r ♣♦❧②❣♦♥s✦ ❲❡✱ t❤❡r❡❢♦r❡✱ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r

❛❧❧

♦❢ t❤❡♠ ❛♥❞ r❡q✉✐r❡ t❤❛t

Ln

Mn

♦❢ t❤❡♠ ♣r♦❞✉❝❡ t❤❡ s❛♠❡ ♥✉♠❜❡r✿

Rn

ց

↓

ւ

Sn →

S

←

ր

↑

տ

?

❛❧❧

?

?

?

❚❤❡ ♦♥❧② r❡str✐❝t✐♦♥ ✐s t❤❛t ❡❛❝❤ ♦❢ t❤❡s❡ s❡q✉❡♥❝❡s ♦❢ ♣❛rt✐t✐♦♥s ❤❛✈❡ t♦ ❜❡ ❝❤♦s❡♥ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t

∆xi → 0

❛s

n → ∞.

❚♦ ♠❛❦❡ s❡♥s❡ ♦❢ t❤✐s✱ ✇❡ ❞❡✜♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❉❡✜♥✐t✐♦♥ ✶✳✻✳✷✿ ♠❡s❤ ♦❢ ♣❛rt✐t✐♦♥ ❚❤❡

♠❡s❤ ♦❢ ❛ ♣❛rt✐t✐♦♥ P

✐s t❤❡ ♠❛①✐♠❛❧ ✈❛❧✉❡ ♦❢ ✐ts ✐♥❝r❡♠❡♥t✿

|P | = max ∆xi . i

■t ✐s ❛ ♠❡❛s✉r❡ ♦❢ t❤❡ ❞❡❣r❡❡ ♦❢ ✏r❡✜♥❡♠❡♥t✑ ♦❢

P✳

❲❤❡♥❡✈❡r ✐t ❣♦❡s t♦

0✱

❛❧❧ ✐♥❝r❡♠❡♥ts ❣♦ t♦

0

t♦♦✳ ❲❡

✉s❡ ✐t ❛s ❢♦❧❧♦✇s✿

❉❡✜♥✐t✐♦♥ ✶✳✻✳✸✿ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ❚❤❡

❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✱ ♦r t❤❡ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧✱ ♦❢ ❛ ❢✉♥❝t✐♦♥ f

✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❧✐♠✐t ♣❛rt✐t✐♦♥s

Pn

I

♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ✐ts ❘✐❡♠❛♥♥ s✉♠s

✇✐t❤ t❤❡✐r ♠❡s❤ ❛♣♣r♦❛❝❤✐♥❣

Sn → I,

0

♣r♦✈✐❞❡❞

❛s

n → ∞❀

♦✈❡r ✐♥t❡r✈❛❧

Sn

[a, b]

♦✈❡r ❛✉❣♠❡♥t❡❞

✐✳❡✳✱

|Pn | → 0 ,

✇❤❡♥ ❛❧❧ t❤❡s❡ ❧✐♠✐ts ❛r❡ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✳ ❲❤❡♥ t❤✐s ❧✐♠✐t ✐s ❛ ♥✉♠❜❡r✱

✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ ♦✈❡r [a, b]✳ ❲❤❡♥ ✭♦r −∞✮✱ ✇❡ s❛② t❤❛t t❤❡ ✐♥t❡❣r❛❧ ✐s ✐♥✜♥✐t❡✳

❝❛❧❧❡❞ ❛♥

❚❤❡ ♥♦t❛t✐♦♥ ✐s s✐♠✐❧❛r t♦ t❤❛t ❢♦r ❛♥t✐✲❞❡r✐✈❛t✐✈❡s✿

❛❧❧ t❤❡s❡ ❧✐♠✐ts ❛r❡ ❡q✉❛❧ t♦

f ✐s +∞

✶✳✻✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

✺✽

■♥t❡❣r❛❧

Z

b

f dx a

■t r❡❛❞s ✏t❤❡ ✐♥t❡❣r❛❧ ♦❢

a

f

❢r♦♠

b✑✳

t♦

❆❜❜r❡✈✐❛t❡❞✱ t❤❡ ❞❡✜♥✐t✐♦♥ ✐s ✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿

Z ✇❤❡r❡

fn

✐s

f

s❛♠♣❧❡❞ ♦✈❡r t❤❡ ♣❛rt✐t✐♦♥

b

f dx = lim Σfn ∆x , a

n→∞

Pn ✳

■♥ t❤❡ ✐♥✜♥✐t❡ ❝❛s❡✱ ✇❡ s✐♠♣❧② ✇r✐t❡ ✭❥✉st ❛s ✇✐t❤ ♦t❤❡r ❧✐♠✐ts✮✿

Z

b a

f dx = +∞ (♦r − ∞) .

❙♦✱ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✐s t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✱ ✐♥ t❤✐s s♣❡❝✐❛❧ s❡♥s❡✳ ❇♦t❤ ❝❛♥ ❜❡ s❡❡♥ ❛s t❤❡ ❛r❡❛s ✉♥❞❡r ❝❡rt❛✐♥ ❣r❛♣❤s✿

❚❤❡ s②♠❜♦❧ ✏

Z

✭❛s ❞♦❡s ❧❡tt❡r

✑ ✐s ❝❛❧❧❡❞ t❤❡ ✐♥t❡❣r❛❧ s✐❣♥✳ ■t ❧♦♦❦s ❧✐❦❡ ❛ str❡t❝❤❡❞ ❧❡tt❡r ❙✱ ✇❤✐❝❤ st❛♥❞s ❢♦r ✏s✉♠♠❛t✐♦♥✑

Σ✮✿

❚❤❡ ♥♦t❛t✐♦♥ ✐s s✐♠✐❧❛r✱ ❜❡❝❛✉s❡ ✐t ✐s r❡❧❛t❡❞✱ t♦ t❤❛t ❢♦r ❘✐❡♠❛♥♥ s✉♠s ✭✐t ✐s ❛❧s♦ s✐♠✐❧❛r✱ ❜❡❝❛✉s❡ ✐t ✐s r❡❧❛t❡❞✱ t♦ t❤❛t ❢♦r ❛♥t✐❞❡r✐✈❛t✐✈❡s✮✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ♥♦t❛t✐♦♥ ✐s ❞❡❝♦♥str✉❝t❡❞✿

✶✳✻✳

❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

✺✾

❘✐❡♠❛♥♥ s✉♠ ❧❡❢t

❛♥❞ r✐❣❤t ❜♦✉♥❞s ❢♦r

x

↓ 1

"

Z 

❞♦♠❛✐♥

−1

3x

3

+ sin x



dx = 0

↑

↑

❧❡❢t ❛♥❞ ❲❡ r❡❢❡r t♦

a

❛♥❞

b

↑

r✐❣❤t ❜r❛❝❦❡ts

❛ s♣❡❝✐✜❝ ♥✉♠❜❡r

❛s t❤❡ ✏❧♦✇❡r ❜♦✉♥❞✑ ❛♥❞ t❤❡ ✏✉♣♣❡r ❜♦✉♥❞✑ ♦❢ t❤❡ ✐♥t❡❣r❛❧✱ ✇❤✐❧❡

✏✐♥t❡❣r❛♥❞✑✳ ❚❤❡ ✐♥t❡r✈❛❧

[a, b]

f

✐s t❤❡

✐s r❡❢❡rr❡❞ t♦ ❛s t❤❡ ✏❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥✑✳

❈♦♠♣❛r❡ t♦ t❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✿

❧❡❢t

❛♥❞ r✐❣❤t ❜♦✉♥❞s ❢♦r

k

↓ "

10

❞♦♠❛✐♥

k=0

X

3k

3

+ sin k

↑



= 9076.411188

↑

❧❡❢t ❛♥❞

↑

r✐❣❤t ❜r❛❝❦❡ts

❛ s♣❡❝✐✜❝ ♥✉♠❜❡r

❊✐t❤❡r t❤❡ ✐♥t❡❣r❛❧ s✐❣♥ ♦r t❤❡ s✐❣♠❛ s✐❣♥ ❞❡s✐❣♥❛t❡s ❛ ❝❡rt❛✐♥ ❢✉♥❝t✐♦♥ t❤❛t t❛❦❡s ❛ ❢✉♥❝t✐♦♥ ✕ ♦❢

x

♦r

k

✕

❛s ✐ts ✐♥♣✉t✳ ❚❤❡ ♦✉t♣✉t ✐s ❛ ♥✉♠❜❡r✳

❲❛r♥✐♥❣✦

a ❛♥❞ b ❛r❡♥✬t ❥✉st ❧♦✇❡r ❛♥❞ ✉♣♣❡r ✐♥t❡r✈❛❧ [a, b] ❜✉t ✐ts ♠✐♥✐♠✉♠ ❛♥❞

❚❤❡s❡ ✏❜♦✉♥❞s✑ ❜♦✉♥❞s ♦❢ t❤❡

♠❛①✐♠✉♠✳ ■t ✐s ❛❧s♦ ✈❡r② ❝♦♠♠♦♥ t♦ ✉s❡ ✏❧✐♠✐ts✑ ✐♥st❡❛❞ ♦❢ ✏❜♦✉♥❞s✑✳

❊①❡r❝✐s❡ ✶✳✻✳✹ ❲❤❛t ❞♦ t❤♦s❡ ✏❜♦✉♥❞s✑ ❤❛✈❡ t♦ ❞♦ ✇✐t❤ t❤❡ ✇♦r❞ ✏❜♦✉♥❞❛r②✑❄

❲❤✐❧❡

dx

s❡❡♠s t♦ ❜❡ ♥♦t❤✐♥❣ ❜✉t ❛ ✏❜♦♦❦❡♥❞✑ ✐♥ t❤❡ ❛❜♦✈❡ ♥♦t❛t✐♦♥✱ ❧❡t✬s ♥♦t ❢♦r❣❡t t❤❛t t❤✐s ✐s t❤❡

❞✐✛❡r❡♥t✐❛❧ ✭❈❤❛♣t❡r ✷❉❈✲✸✮✳ ❆♥ ❛❧t❡r♥❛t✐✈❡ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ f (x) · dx ✭❈❤❛♣t❡r ✹✮✿

♥♦t❛t✐♦♥ r❡✢❡❝ts t❤❡ ❢❛❝t t❤❛t t❤❡ ✐♥t❡❣r❛❧ ✐s ❞❡t❡r♠✐♥❡❞ ❜②

❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

Z

f dx [a,b]

❙♦✱ t❤✐s ✐s ❛ ❢✉♥❝t✐♦♥ ❝r❡❛t❡❞ ❜② t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ❛♥❞ ✐ts ✐♥♣✉ts ❛r❡ ✐♥t❡r✈❛❧s✳ ❚♦ ❡♠♣❤❛s✐③❡ t❤❡ ❧❛tt❡r ♣♦✐♥t✱ ✇❡ ♠♦✈❡ t❤❡

❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥ [a, b] t♦ t❤❡ s✉❜s❝r✐♣t✳

✶✳✻✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

✻✵

❚❤✐s ✐s ❤♦✇ t❤❡ ♥♦t❛t✐♦♥ ✐s ❞❡❝♦♥str✉❝t❡❞✿ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

✏❡✈❛❧✉❛t❡✑

✐♥♣✉t

↓ Z

→

[−1, 1]

 3x3 + sin x dx = 0 | {z } ↑

↑

❞✐✛❡r❡♥t✐❛❧ ❢♦r♠

♦✉t♣✉t

❚❤✐s tr❛♥s✐t✐♦♥ ❜❡❝♦♠❡s ✐♥❡✈✐t❛❜❧❡ ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s ✭❱♦❧✉♠❡ ✹✮✳ ❲❤❡♥ ✇❡ s♣❡❛❦ ♦❢ t❤❡ ❛r❡❛✱ ✇❡ ❤❛✈❡ ✭✇✐t❤

Z

n

b

f dx | {z } a

t❤❡ ❡①❛❝t ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤ ❊①❛♠♣❧❡ ✶✳✻✳✺✿ ♥❡❣❛t✐✈❡ ❛r❡❛❄

k ✮✿ n X f (ci ) ∆xi . = lim

❞❡♣❡♥❞❡♥t ♦♥

❏✉st ❛s ✇✐t❤ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✱ ✇❤❡♥ t❤❡ ✈❛❧✉❡s ♦❢

k→∞

i=1

|

{z

}

❛r❡❛s ♦❢ t❤❡ ❜❛rs

f

❛r❡ ♥❡❣❛t✐✈❡✱ s♦ ✐s t❤❡ ✏❛r❡❛✑ ✉♥❞❡r ✐ts ❣r❛♣❤✿

❚❤✐s ❞❡♠♦♥str❛t❡s ❛ ❞r❛✇❜❛❝❦ ♦❢ t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ❛s t❤❡ ❛r❡❛ ✐♥ ❝♦♥tr❛st t♦ t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ❛s t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✳

▲❡t✬s ✈❡r✐❢② t❤❛t t❤❡ ❞❡✜♥✐t✐♦♥ ♠❛❦❡s s❡♥s❡ ❢♦r t❤❡ s✐♠♣❧❡st ❢✉♥❝t✐♦♥✳ ❚❤❡♦r❡♠ ✶✳✻✳✻✿ ■♥t❡❣r❛❧ ♦❢ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥ ❙✉♣♣♦s❡ ♥✉♠❜❡r

f ✐s ❝♦♥st❛♥t ♦♥ [a, b]✱ ✐✳❡✳✱ f (x) = c ❢♦r ❛❧❧ x c✳ ❚❤❡♥ f ✐s ✐♥t❡❣r❛❜❧❡ ♦♥ [a, b] ❛♥❞ ✇❡ ❤❛✈❡✿ Z

b a

f dx = c(b − a)

Pr♦♦❢✳

❋r♦♠ t❤❡ ❈♦♥st❛♥t ❋✉♥❝t✐♦♥ ❘✉❧❡ ❢♦r ❘✐❡♠❛♥♥ ❙✉♠s✱ ✇❡ ❦♥♦✇✿

Σ[a,b] fk ∆x = c(b − a) . ❙✐♥❝❡ t❤✐s ❡①♣r❡ss✐♦♥ ❞♦❡s♥✬t ❞❡♣❡♥❞ ♦♥

k✱

t❤❡ ❝♦♥❝❧✉s✐♦♥ ❢♦❧❧♦✇s✳

✐♥

[a, b]

❛♥❞ s♦♠❡ r❡❛❧

✶✳✻✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

✻✶

❖❢ ❝♦✉rs❡✱ ✇❡ ❤❛✈❡ r❡❝♦✈❡r❡❞ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛r❡❛ ♦❢ ❛ r❡❝t❛♥❣❧❡✿ ❛r❡❛ ❊①❝❡♣t✱ t❤✐s ❛r❡❛ ✐s ♥❡❣❛t✐✈❡ ✇❤❡♥

c

=

❤❡✐❣❤t

·

✇✐❞t❤ .

✐s ♥❡❣❛t✐✈❡✳

❊✈❡♥ t❤♦✉❣❤ t❤❡ ❛♥s✇❡r t♦ t❤❡ q✉❡st✐♦♥ ✏❍♦✇ ❞♦ ✇❡ ❞♦ ✐t❄✑ ✐s st✐❧❧ t♦ ❝♦♠❡✱ ✇❡ ❛s❦ ✏■s ✐t ❛❧✇❛②s ♣♦ss✐❜❧❡❄✑ ❙♦♠❡ ❧✐♠✐ts ❞♦♥✬t ❡①✐st✳ ❚❤❡♥✱ ❛s ❛ ❧✐♠✐t✱ t❤❡ ✐♥t❡❣r❛❧ ♠✐❣❤t ♥♦t ❡①✐st ❡✐t❤❡r✳ ■♥ ❝♦♥tr❛st t♦ ❞✐✛❡r❡♥t✐❛❜✐❧✐t②✱ ✇❡ ❝❛♥✬t t❡❧❧ ❜② ❥✉st ❧♦♦❦✐♥❣ ❛t t❤❡ ❣r❛♣❤✳

❊①❛♠♣❧❡ ✶✳✻✳✼✿ ✐♥✜♥✐t❡ ❛r❡❛ ❍❡r❡ ✐s ❛ s✐♠♣❧❡ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♥♦♥✲✐♥t❡❣r❛❜❧❡ ♦✈❡r

 1 f (x) = x2 0

■t s✉✣❝❡s t♦ ❧♦♦❦ ❛t t❤❡ ✜rst t❡r♠ ♦❢ t❤❡

nt❤

[0, 1]✿

✐❢

x > 0,

✐❢

x = 0.

r✐❣❤t✲❡♥❞ ❘✐❡♠❛♥♥ s✉♠s✿

❚❤✐s ❜❛r ✐s ❣❡tt✐♥❣ t❛❧❧❡r ❛♥❞ t❤✐♥♥❡r ❛t t❤❡ s❛♠❡ t✐♠❡✳ ❲❤❛t ❛❜♦✉t ✐ts ❛r❡❛❄ ❲❡ ❝♦♠♣✉t❡✿

  1 1 1 1 · =n→∞ = f (x1 ) ∆x = f 2 n n 1/n n

❛s

n → ∞.

■t t✉r♥s ♦✉t t❤✐s ❜❛r ✐s ❣❡tt✐♥❣ t❛❧❧❡r ❢❛st❡r t❤❛♥ ✐t ✐s ❣❡tt✐♥❣ t❤✐♥♥❡r✦ ❚❤❡♦r❡♠ ❢♦r ❉✐✈❡r❣❡♥❝❡ ✭❈❤❛♣t❡r ✷❉❈✲✶✮✱ ✐t ❢♦❧❧♦✇s t❤❛t

Rn → ∞ ❚❤❡r❡❢♦r❡✱

Z

❊①❡r❝✐s❡ ✶✳✻✳✽ ❈♦♥s✐❞❡r t❤✐s ❝♦♥str✉❝t✐♦♥ ❢♦r

1 1 ❛♥❞ 3 ✳ x x

❛s

n → ∞.

1

f dx = +∞ . 0

❚❤❡♥✱ ❢r♦♠ t❤❡ P✉s❤ ❖✉t

✶✳✻✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

✻✷

❊①❛♠♣❧❡ ✶✳✻✳✾✿ ❉✐r✐❝❤❧❡t ❢✉♥❝t✐♦♥

❚❤❡ ❉✐r✐❝❤❧❡t ❢✉♥❝t✐♦♥ ✐s ❛❧s♦ ♥♦♥✲✐♥t❡❣r❛❜❧❡✿

IQ (x) =

(

1 0

✐❢ ✐❢

x x

✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r, ✐s ❛♥ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r.

❚♦ ♣r♦✈❡ t❤✐s✱ ✇❡ ❝♦♥s✐❞❡r t✇♦ ❞✐✛❡r❡♥t s❡q✉❡♥❝❡s ♦❢ ♣❛rt✐t✐♦♥s✿

• •

■❢ ✇❡ ❝❤♦♦s❡ ❛❧❧ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ♣❛rt✐t✐♦♥s s✉♠s ✇✐❧❧ ❜❡ ❡q✉❛❧ t♦

1❀

Pn

0❀

t❤❡r❡❢♦r❡✱ t❤❡ ❧✐♠✐t ✐s

b − a✳

Qn t♦ ❜❡ ✐rr❛t✐♦♥❛❧✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❘✐❡♠❛♥♥ ❛❧s♦ ❡q✉❛❧ t♦ 0✳

■❢ ✇❡ ❝❤♦♦s❡ ❛❧❧ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ♣❛rt✐t✐♦♥s s✉♠s ✇✐❧❧ ❜❡ ❡q✉❛❧ t♦

t♦ ❜❡ r❛t✐♦♥❛❧✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❘✐❡♠❛♥♥

t❤❡r❡❢♦r❡✱ t❤❡ ❧✐♠✐t ✇✐❧❧ ❜❡ ❡q✉❛❧ t♦

❚❤❡ ♠✐s♠❛t❝❤ ❜❡t✇❡❡♥ t❤❡ ❧✐♠✐ts ♦❢ ❘✐❡♠❛♥♥ s✉♠s ♣r♦✈❡s t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ♥♦t ✐♥t❡❣r❛❜❧❡✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ♣r♦✈❡s t❤❛t ♦✉r ❞❡✜♥✐t✐♦♥ ♠❛❦❡s s❡♥s❡ ❢♦r ❛ ❧❛r❣❡ ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s✳ ❚❤❡♦r❡♠ ✶✳✻✳✶✵✿ ■♥t❡❣r❛❜✐❧✐t② ♦❢ ❈♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥s ❆❧❧ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥

[a, b]

❛r❡ ✐♥t❡❣r❛❜❧❡ ♦♥

[a, b]✳

❲❡ ❛❝❝❡♣t ✐t ✇✐t❤♦✉t ♣r♦♦❢✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐s♥✬t tr✉❡ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❊①❛♠♣❧❡ ✶✳✻✳✶✶✿ s✐❣♥ ❢✉♥❝t✐♦♥

❚❤❡ s✐❣♥ ❢✉♥❝t✐♦♥✱

f (x) = sign(x)✱

❤❛s ❛ ✈❡r② s✐♠♣❧❡ ❣r❛♣❤ ❛♥❞✱ ✐t ❛♣♣❡❛rs✱ t❤❡ ❛r❡❛ ✉♥❞❡r ✐t ✇♦✉❧❞

✏♠❛❦❡ s❡♥s❡✑✿

■♥❞❡❡❞✱ t❤✐s ❢✉♥❝t✐♦♥ ✐s ✐♥t❡❣r❛❜❧❡ ♦✈❡r ❛♥② ✐♥t❡r✈❛❧ ❚❤❡r❡ ❛r❡ t✇♦ ❝❛s❡s✳ ❲❤❡♥

b≤0

♦r

a ≥ 0✱

[a, b]✳

t❤❡ ❢✉♥❝t✐♦♥ ✐s s✐♠♣❧② ❝♦♥st❛♥t ♦♥ t❤✐s ✐♥t❡r✈❛❧✳ ❲❤❡♥

a < 0 < b✱ ❛❧❧ t❡r♠s ♦❢ ❛❧❧ ❘✐❡♠❛♥♥ s✉♠s ❛r❡ −1 · ∆xi ✱ ♦r 1 · ∆xi ✱ ♦r 0 ✭❛t ♠♦st t✇♦✮✳ ▲❡t✬s s✉♣♣♦s❡ 0 ✐s♥✬t ❛ ♥♦❞❡ ♦❢ ❛♥② ♦❢ t❤❡ ♣❛rt✐t✐♦♥s Pk ❛♥❞✱ ✐♥ ❢❛❝t✱ ✐t ✐s ♦♥❡ ♦❢ ✐ts s❡❝♦♥❞❛r② ♥♦❞❡s✱ 0 = cm ✳

t❤❛t

✶✳✻✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

❚❤❡♥

Σf · ∆x =

m−1 X

(−1) · ∆xi +0 · ∆xm

i=1 m−1 X

=−

✻✸

∆xi

+

i=1

∆xi

+(b − xm+1 ) −xm − xm+1 .

=a+b |Pk | → 0✱

i=m+1 n X

1 · ∆xi

i=m+1

= −(xm − a)

❚❤❡♥✱ ❛s

n X

+

✇❡ ❤❛✈❡✿

xm → 0

❚❤❡r❡❢♦r❡✱ t❤❡ ❡q✉❛t✐♦♥✬s ❧✐♠✐t ✐s✿

Z

❛♥❞

xm+1 → 0 .

b

f dx = a + b . a

❊①❡r❝✐s❡ ✶✳✻✳✶✷ ❈♦♥s✐❞❡r t❤❡ ♠✐ss✐♥❣ ❝❛s❡s ✐♥ t❤❡ ❛❜♦✈❡ ♣r♦♦❢✳

❚❤❡s❡ ❛r❡ t❤❡ ♠❛✐♥ ❝❧❛ss❡s ♦❢ ❢✉♥❝t✐♦♥ ✇❡ ❤❛✈❡ s❡❡♥ ❛♥❞ t❤❡✐r r❡❧❛t✐♦♥s✿

❚❤❡② ❛r❡ s✉❜s❡ts ♦❢ ❡❛❝❤ ♦t❤❡r✦ ❲❡ ❛❝❝❡♣t t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ✇✐t❤♦✉t ♣r♦♦❢✳

❚❤❡♦r❡♠ ✶✳✻✳✶✸✿ ■♥t❡❣r❛❜✐❧✐t② ♦❢ ❘❡str✐❝t✐♦♥

f ✐s ✐♥t❡❣r❛❜❧❡ A < B ≤ b✳ ■❢

♦✈❡r

[a, b]✱

t❤❡♥ ✐t ✐s ❛❧s♦ ✐♥t❡❣r❛❜❧❡ ♦✈❡r ❛♥②

[A, B]

✇✐t❤

a≤

❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ ♦❜s❡r✈❡ t❤❛t ❡✈❡♥ ✐❢ t❤❡ ♣❡rs♦♥ ❞✐❞♥✬t s♣❡♥❞ ❛♥② t✐♠❡ ❞r✐✈✐♥❣✱ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t st✐❧❧ ♠❛❦❡s s❡♥s❡❀ ✐t✬s ③❡r♦✳

❚❤❡♦r❡♠ ✶✳✻✳✶✹✿ ■♥t❡❣r❛❧ ❖✈❡r ❩❡r♦✲❧❡♥❣t❤ ■♥t❡r✈❛❧ ❚❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♦❢ ❛ ❢✉♥❝t✐♦♥

f

♦✈❡r ❛ ✏③❡r♦✲❧❡♥❣t❤✑ ✐♥t❡r✈❛❧

t♦ ③❡r♦✿

Z ❆♥❞ t❤❡ ❛r❡❛ ♦❢ ❛ r❡❣✐♦♥ ♦♥❡✲♣♦✐♥t t❤✐❝❦ ✐s ③❡r♦✱ t♦♦✳

a

f dx = 0 a

[a, a]✱ ✐s ❡q✉❛❧

✶✳✻✳ ❚❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

✻✹

❊①❡r❝✐s❡ ✶✳✻✳✶✺

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

■♥ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♥♦t❛t✐♦♥✱ t❤✐s ✐♥t❡❣r❛❧ ✇✐t❤ ❡q✉❛❧ ❜♦✉♥❞s ❤❛s ❛ ❝❧❡❛r❡r ♠❡❛♥✐♥❣✿

Z

f dx = 0 {a}

❚❤❡ ♦r✐❡♥t❡❞ ✐♥t❡r✈❛❧s ❛r❡ ❛❧s♦ ✐♥❝❧✉❞❡❞✳

❲❡ ♦♥❝❡ ❛❣❛✐♥ ✉t✐❧✐③❡ t❤❡ ✐❞❡❛ ♦❢ ♦r✐❡♥t❡❞ ✐♥t❡r✈❛❧s ❛♥❞ ♦r✐❡♥t❡❞ r❡❝t❛♥❣❧❡s✳ ❚❤❡♦r❡♠ ✶✳✻✳✶✻✿ ■♥t❡❣r❛❧ ❖✈❡r ◆❡❣❛t✐✈❡❧② ❖r✐❡♥t❡❞ ■♥t❡r✈❛❧ ❚❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♦❢ ❛ ❢✉♥❝t✐♦♥

a

f

♦✈❡r ❛ ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞ ✐♥t❡r✈❛❧

✐s ❡q✉❛❧ t♦ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ♦✈❡r

Z

a b

f dx = −

Z

[b, a], b >

[a, b]✿

b

f dx a

❊①❡r❝✐s❡ ✶✳✻✳✶✼

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

■♥ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♥♦t❛t✐♦♥✱ t❤✐s ✏✢✐♣♣✐♥❣✑ ♦❢ t❤❡ ❜♦✉♥❞s ♦❢ ✐♥t❡❣r❛❧ ❤❛s ❛ ♠♦r❡ ♣r❡❝✐s❡ ♠❡❛♥✐♥❣✳ ❲❡ t❤✐♥❦ ♦❢ t❤❡ ♥❡❣❛t✐✈❡❧② ♦r✐❡♥t❡❞ ✐♥t❡r✈❛❧ ❛s t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❡ ♦r✐❣✐♥❛❧✿

[b, a] = −[a, b] . ❚❤❡♥ t❤❡ ❢♦r♠✉❧❛ t❛❦❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿

Z

−[a,b]

f dx = −

Z

f dx [a,b]

❊①❛♠♣❧❡ ✶✳✻✳✶✽✿ ❛r❡❛ ✈s✳ ✐♥t❡❣r❛❧

❚❤✉s✱ ✇❡ ❤❛✈❡ ❡①♣❧❛✐♥❡❞ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❛r❡❛ ♦❢ ❛ ❝✉r✈❡❞ r❡❣✐♦♥✿ ■t ✐s ❛♥ ✐♥t❡❣r❛❧✳ ❈♦♥✈❡rs❡❧②✱ ✐s t❤❡ ✐♥t❡❣r❛❧ ❛♥ ❛r❡❛❄ ❨❡s✱ ✐♥ ❛ s❡♥s❡✳ ■t ❞❡♣❡♥❞s ♦♥ t❤❡ ✉♥✐ts ♦❢ t❤❡ ✈❛r✐❛❜❧❡s✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❤❡♥ ❜♦t❤

x

❛♥❞

y

❛r❡ ♠❡❛s✉r❡❞ ✐♥ ❢❡❡t✱ t❤❡ ✐♥t❡❣r❛❧ ✐s ✐♥❞❡❡❞ t❤❡ ❛r❡❛ ✐♥ t❤❡ ✉s✉❛❧ s❡♥s❡❀ ✐t ✐s ♠❡❛s✉r❡❞ ✐♥

sq✉❛r❡ ❢❡❡t✳ ❍♦✇❡✈❡r✱ ✇❤❛t ✐❢ ❛♥❞

y

x

❛♥❞

y

❛r❡ s♦♠❡t❤✐♥❣ ❡❧s❡❄ ❲❡ ❤❛✈❡ ❛❧r❡❛❞② s❡❡♥ t❤❛t

x

♠❛② ❜❡ t✐♠❡

t❤❡ ✈❡❧♦❝✐t②❀ t❤❡② ❛r❡ ♠❡❛s✉r❡❞ ✐♥ s❡❝♦♥❞s ❛♥❞ ❢❡❡t ♣❡r s❡❝♦♥❞ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♥ t❤❡ ✐♥t❡❣r❛❧

✐s ♠❡❛s✉r❡❞ ✐♥ s❡❝♦♥❞s

·

❢❡❡t ♣❡r s❡❝♦♥❞✱ ✐✳❡✳✱ ❢❡❡t✳ ❚❤❛t ❝❛♥✬t ❜❡ ❛r❡❛✳✳✳ ■♥ ❢❛❝t✱ ❜♦t❤

q✉❛♥t✐t✐❡s ♦❢ ❛r❜✐tr❛r② ♥❛t✉r❡❀ t❤❡♥ t❤❡ ✉♥✐ts ♦❢ t❤❡ ✐♥t❡❣r❛❧ ♠✐❣❤t ❜❡✿ ♣♦✉♥❞ ❚❤❡♥ t❤❡ ✐♥t❡❣r❛❧ ✐s ❞❡s❝r✐❜❡❞ ❛s ✏t❤❡ t♦t❛❧ ✈❛❧✉❡✑ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳

❍❡r❡ ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❝❛❧❝✉❧✉s ✉♣ t♦ t❤✐s ♣♦✐♥t✿

·

x

❛♥❞

y

♠❛② ❜❡

❞❡❣r❡❡✱ ♠❛♥✲❤♦✉r✱ ❡t❝✳

✶✳✼✳

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s

✻✺

❉✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s

■♥t❡❣r❛❧ ❝❛❧❝✉❧✉s

f ❞❡✜♥❡❞ ♦♥ t❤❡ ♣r✐♠❛r② ♥♦❞❡s

g ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s

✶✳ ❞✐✛❡r❡♥❝❡✱ ∆f

✶✳ s✉♠✱ Σg

❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s

❞❡✜♥❡❞ ♦♥ t❤❡ ♣r✐♠❛r② ♥♦❞❡s

∆f (ck ) = f (xk ) − f (xk−1 )

Σg (xk ) = g(c1 ) + ... + g(ck )

❞✐✈✐❞❡ ❜② ∆x

∆x ✐s ❢❛❝t♦r❡❞ ✐♥ ❡❛❝❤ t❡r♠

✷✳ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱

∆f ∆x

✷✳ ❘✐❡♠❛♥♥ s✉♠✱ Σg · ∆x

❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s

❞❡✜♥❡❞ ♦♥ t❤❡ ♣r✐♠❛r② ♥♦❞❡s

∆f f (xk − f (xk−1 ) (ck ) = ∆x ∆x

Σg · ∆x (xk ) = g(c1 )∆x + ... + g(ck )∆x

∆x → 0

∆x → 0

df ✸✳ ❞❡r✐✈❛t✐✈❡✱ dx ❞❡✜♥❡❞ ♦♥ t❤❡ ✐♥t❡r✈❛❧

✸✳ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✱

Z

g dx

❞❡✜♥❡❞ ♦♥ t❤❡ ✐♥t❡r✈❛❧

❚❤❡ t✇♦ ❝♦❧✉♠♥s ❛r❡ ❝♦♥str✉❝t❡❞ ✐♥❞❡♣❡♥❞❡♥t❧②✳ ❘♦✇s ✶ ❛♥❞ ✷ ❛r❡ ❧✐♥❦❡❞ t♦❣❡t❤❡r ❜② t❤❡ ❞✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✳ ❘♦✇ ✸ ✇✐❧❧ ❜❡ ❧✐♥❦❡❞ t♦❣❡t❤❡r ❜② ✐ts ✜♥❛❧ ✈❡rs✐♦♥✳

✶✳✼✳ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡✲ ❣r❛❧s

❲❡ ❣♦ ❜❛❝❦ t♦ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥ ♦r❞❡r t♦ ❝♦♥✜r♠ t❤❛t ♦✉r t❤❡♦r② ♠❛❦❡s s❡♥s❡ ❜② ♠❛t❝❤✐♥❣ t❤❡ ♣❡r❝❡✐✈❡❞ ✐❞❡❛s ♦❢ ❤♦✇ t❤❡s❡ ❝♦♥❝❡♣ts ❛r❡ s✉♣♣♦s❡❞ t♦ ♦♣❡r❛t❡ ✐♥ r❡❛❧ ✕ t❤♦✉❣❤ ✐❞❡❛❧✐③❡❞ ✕ ❧✐❢❡✳ ❚❤❡ ♠❛✐♥ ❛r❡❛s ❛r❡✱ ❛s ❜❡❢♦r❡✱ ♠♦t✐♦♥ ❛♥❞ ❣❡♦♠❡tr②✳ ❚❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❝♦♠❡ ❢r♦♠ ♣✉r❡ ❛❧❣❡❜r❛✳ ❲❤❛t✬s ❧❡❢t✱ t❤❡♥✱ ✐s t♦ ♠❛❦❡ s✉r❡ t❤❛t t❤♦s❡ r❡❧❛t✐♦♥s ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r t❤❡ ❧✐♠✐t t♦ ♣r♦❞✉❝❡ t❤❡ ♠❛t❝❤✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s✳ ❚❤❡ ✐♥t❡❣r❛❧ ❢♦❧❧♦✇s t❤❡ ❘✐❡♠❛♥♥ s✉♠✱ ❡✈❡r② t✐♠❡✿

❚❤❡ ❢♦r♠✉❧❛ ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ✐s✿

Σf · ∆x = f (c1 ) ∆x1 + f (c2 ) ∆x2 + ... + f (cn ) ∆xn ,

✶✳✼✳

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s

✻✻

✇❤❡r❡ t❤❡ ♣♦✐♥ts

a = x0 ≤ c1 ≤ x1 ≤ c2 ≤ x2 ≤ ... ≤ cn ≤ xn = b ♠❛❦❡ ✉♣ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢ ❚❤❡ ❘✐❡♠❛♥♥ s✉♠ r❡♠❛✐♥s ❥✉st ❛

[a, b]✳

s✉♠✳ ❲❡✱ t❤❡r❡❢♦r❡✱ ❝❛♥ ✉s❡ s♦♠❡ ♦❢ t❤❡ ✈❡r② ❡❧❡♠❡♥t❛r② ❛❧❣❡❜r❛✐❝ ❢❛❝ts✳

❲❤✐❧❡ ❛❞❞✐♥❣✱ ✇❡ ❝❛♥ r❡✲❣r♦✉♣ t❤❡ t❡r♠s ❢r❡❡❧②❀ ✐♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❝❛♥ r❡♠♦✈❡ t❤❡ ♣❛r❡♥t❤❡s❡s✿

(a + b) + c = a + (b + c) = a + b + c . ❋♦r s✉♠s✱ ✇❡ ❤❛✈❡✿

(u1 + u2 + ... + un ) + (v1 + v2 + ... + vm ) = u1 + u2 + ... + un + v1 + v2 + ... + vm . ❚❤❡ st❛t❡♠❡♥t ✐s ❛❜♦✉t t❤❡ ❢❛❝t t❤❛t ✇❤❡♥ ❛❞❞✐♥❣✱ ✇❡ ❝❛♥ ❝❤❛♥❣❡ t❤❡ ♦r❞❡r ♦❢ t❡r♠s ❢r❡❡❧②❀ t❤✐s ✐s ❝❛❧❧❡❞ t❤❡

❆ss♦❝✐❛t✐✈✐t② Pr♦♣❡rt② ♦❢ ❛❞❞✐t✐♦♥✳ ❚❤✐s ✐s ❛❧s♦ t❤❡ ❆❞❞✐t✐✈✐t② ❘✉❧❡ ❢♦r ❙✉♠s ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r

✶P❈✲✶✮✳ ◆♦t ♠✉❝❤ t♦ ❝❤❛♥❣❡ ✇❤❡♥ ✇❡ ❝♦♥s✐❞❡r t❤❡ s✉♠s ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ ❛ ♣❛rt✐t✐♦♥✳ ■❢ ✇❡ ❤❛✈❡ ♣❛rt✐t✐♦♥s ♦❢ t✇♦ ❛❞❥❛❝❡♥t ✐♥t❡r✈❛❧s✱ ✇❡ ❝❛♥ ❥✉st ❝♦♥t✐♥✉❡ t♦ ❛❞❞ t❡r♠s✱ t❤✉s ❝r❡❛t✐♥❣ ❛ ✏❧♦♥❣❡r✑ s✉♠✿

❖r ✇❡ t❤✐♥❦ ♦❢ t❤❡s❡ ❛s ❛r❡❛s✳ ❚❤❡ ❛❧❣❡❜r❛ ✐s ❛s ❢♦❧❧♦✇s✿ ❚❤❡♦r❡♠ ✶✳✼✳✶✿ ❆❞❞✐t✐✈✐t② ♦❢ ❙✉♠s ❚❤❡ s✉♠ ♦❢ t❤❡ s✉♠s ♦✈❡r t✇♦ ❝♦♥s❡❝✉t✐✈❡ ♣❛rts ♦❢ ❛♥ ✐♥t❡r✈❛❧ ✐s t❤❡ s✉♠ ♦✈❡r t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✳

■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ ♦❢ ✐♥t❡r✈❛❧s

[a, b]

❛♥❞

[b, c]✱

f

❛♥❞ ❢♦r ❛♥② ❢✉♥❝t✐♦♥

f

❛♥❞ ❢♦r ❛♥② ♣❛rt✐t✐♦♥s

✇❡ ❤❛✈❡✿

Σg [a,b] + Σg [b,c] = Σg [a,c]

❲❤❛t ✐❢ t❤❡s❡ s✉♠s ❛r❡

❘✐❡♠❛♥♥ s✉♠s❄ ◆♦t ♠✉❝❤ ❝❤❛♥❣❡s✿

❚❤❡♦r❡♠ ✶✳✼✳✷✿ ❆❞❞✐t✐✈✐t② ♦❢ ❘✐❡♠❛♥♥ ❙✉♠s ❚❤❡ s✉♠ ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ♦✈❡r t✇♦ ❝♦♥s❡❝✉t✐✈❡ ♣❛rts ♦❢ ❛♥ ✐♥t❡r✈❛❧ ✐s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦✈❡r t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✳

■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥

[b, c]✱

f

❛♥❞ ❢♦r ❛♥② ♣❛rt✐t✐♦♥s ♦❢ ✐♥t❡r✈❛❧s

✇❡ ❤❛✈❡✿

Σf · ∆x [a,b] + Σf · ∆x [b,c] = Σf ∆x [a,c]

[a, b]

❛♥❞

✶✳✼✳

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s

✻✼

Pr♦♦❢✳

❙✉♣♣♦s❡ t❤❡ t✇♦ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥s✱ P ❛♥❞ Q✱ ❛r❡ ❣✐✈❡♥ ❜②✿ P :

a = x0 ≤ c1 ≤ x1 ≤ c2 ≤ x2 ≤ ... ≤ cn ≤ xn = b

Q:

b = y0 ≤ d1 ≤ y1 ≤ d2 ≤ y2 ≤ ... ≤ dm ≤ xm = c

❲❡ r❡♥❛♠❡ t❤❡ ✐t❡♠s ♦♥ t❤❡ ❧❛tt❡r ❧✐st ❛♥❞ ❢♦r♠ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢ [a, c]✿ P ∪ Q : a = x0 ≤ c1 ≤ x1 ≤ c2 ≤ ... ≤ cn ≤ xn ≤ cn+1 ≤ xn+1 ≤ cn+2 ≤ xn+2 ≤ ... ≤ cn+m ≤ xn+m = c

❆♣♣❧②✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ t♦ t❤❡ ✐♥t❡r✈❛❧s [a, b] ❛♥❞ [b, c] ❛♥❞ t❤❡♥ t♦ [a, c]✱ ✇❡ ❤❛✈❡✿ Σf · ∆x [a,b] + Σf · ∆x [b,c] = (f (c1 ) + f (c2 ) + ... + f (cn ))

+ (f (cn+1 ) + f (cn+2 ) + ... + f (cn+m )) = Σf ∆x [a,c] .

❚♦ ✉s❡ t❤❡ ❛r❡❛ ♠❡t❛♣❤♦r✱ ✐♠❛❣✐♥❡ t❤❛t ✇❡ ❤❛✈❡ ③♦♦♠❡❞ ♦✉t ♦❢ t❤❡ ♣✐❝t✉r❡ ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿

❚❤❡ r❡s✉❧t ✐s ❡q✉❛❧❧② ❛♣♣❧✐❝❛❜❧❡ t♦ t❤❡ ✐♥t❡❣r❛❧s❀ t❤❡ ❢♦r♠✉❧❛ ✐♥ t❤❡ t❤❡♦r❡♠ ❛❞❞s t❤❡ ❛r❡❛s ♦❢ t❤❡s❡ t✇♦ ❛❞❥❛❝❡♥t r❡❣✐♦♥s✿ ❚❤❡♦r❡♠ ✶✳✼✳✸✿ ❆❞❞✐t✐✈✐t② ♦❢ ■♥t❡❣r❛❧ ❚❤❡ s✉♠ ♦❢ t❤❡ ✐♥t❡❣r❛❧s ♦✈❡r t✇♦ ❝♦♥s❡❝✉t✐✈❡ ♣❛rts ♦❢ ❛♥ ✐♥t❡r✈❛❧ ✐s t❤❡ ✐♥t❡❣r❛❧ ♦✈❡r t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✳

■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ ✐♥t❡❣r❛❜❧❡

[a, c]✱

f

✐♥t❡❣r❛❜❧❡ ♦✈❡r

[a, b]

❛♥❞ ♦✈❡r

[b, c]

✐s ❛❧s♦

❛♥❞ ✇❡ ❤❛✈❡✿

Z

b

f dx + a

Z

c

f dx = b

Z

c

f dx a

Pr♦♦❢✳

❚❤❡ ✐♥t❡❣r❛❜✐❧✐t② ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❧❛st t❤❡♦r❡♠ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❚♦ ♣r♦✈❡ t❤❡ ❢♦r♠✉❧❛✱ ✇❡ st❛rt ✇✐t❤ t❤✐s ❢❛❝t ❛❜♦✉t t❤❡ ♠❡s❤ ♦❢ ♣❛rt✐t✐♦♥s✿ |P ∪ Q| = max{|P |, |Q|} .

✶✳✼✳

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s

✻✽

❖♥❝❡ ✇❡ ♠♦✈❡ t♦ s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐t✐♦♥s✱ t❤✐s ❢❛❝t ✐♠♣❧✐❡s t❤❡ ❢♦❧❧♦✇✐♥❣✿

|Pk ∪ Qk | → 0 ⇐⇒ |Pk | → 0

❛♥❞

|Qk | → 0 .

◆❡①t✱ ✇❡ t❛❦❡ t❤❡ ❢♦r♠✉❧❛ ✐♥ ♣❛rt ✭❆✮✱ t❤❡ ❆❞❞✐t✐✈✐t② ♦❢ ❘✐❡♠❛♥♥ s✉♠s✱ t❛❦❡ t❤❡ ❧✐♠✐t ✇✐t❤ ❛♥❞ ✉s❡ t❤❡

❙✉♠ ❘✉❧❡ ❢♦r ▲✐♠✐ts ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✶✳

k→∞

❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡ ✇♦r❞ ✏❛❞❞✐t✐✈✐t②✑ ✐♥ t❤❡ ♥❛♠❡ ♦❢ t❤❡ t❤❡♦r❡♠ ❞♦❡s♥✬t r❡❢❡r t♦ ❛❞❞✐♥❣ t❤❡ t❡r♠s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❜✉t t♦

❛❞❞✐♥❣ t❤❡ ❞♦♠❛✐♥s ♦❢ ✐♥t❡❣r❛t✐♦♥✱ ✐✳❡✳✱ t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ t✇♦ ✐♥t❡r✈❛❧s✳

❚❤❡ ✐❞❡❛

❜❡❝♦♠❡s ❡s♣❡❝✐❛❧❧② ✈✐✈✐❞ ✇❤❡♥ t❤❡ ❢♦r♠✉❧❛ ✐s ✇r✐tt❡♥ ✐♥ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♥♦t❛t✐♦♥✿

Z

f dx + [a,b]

+

♦r❛♥❣❡

Z

f dx = [b,c]

=

❣r❡❡♥

Z

f dx [a,b]∪[b,c]

❜❧✉❡

❚❤❡ ✐♥t❡r✈❛❧ ❜❡❝♦♠❡s t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡✱ s✉❜❥❡❝t t♦ s♦♠❡ ❛❧❣❡❜r❛✳ ❊①❡r❝✐s❡ ✶✳✼✳✹

❋✐♥✐s❤ t❤❡ ❢♦r♠✉❧❛✿

❋♦r t❤❡

Z

f dx + [a,b]

Z

f dx = ... [c,d]

♠♦t✐♦♥ ♠❡t❛♣❤♦r✱ ✇❡ ❤❛✈❡✿

❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❞✉r✐♥❣ t❤❡ ✶st ❤♦✉r

+

❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❞✉r✐♥❣ t❤❡ ✷♥❞ ❤♦✉r

=

❞✐st❛♥❝❡ ❞✉r✐♥❣ t❤❡ t✇♦ ❤♦✉rs

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥ ✐♠♣♦rt❛♥t ❝♦r♦❧❧❛r②✳ ❲❡ ❛❧s♦ ❛❝❝❡♣t ✐t ✇✐t❤♦✉t ♣r♦♦❢ t❤❡ ❝♦r♦❧❧❛r② ❜❡❧♦✇✳ ❚❤❡♦r❡♠ ✶✳✼✳✺✿ ■♥t❡❣r❛❜✐❧✐t② ♦❢ P✐❡❝❡✇✐s❡ ❈♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥s

❆❧❧ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✭✐✳❡✳✱ ❝♦♥t✐♥✉♦✉s ♦♥ ❛❧❧ ❜✉t ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ✇✐t❤ ♦♥❧② ❥✉♠♣ ❞✐s❝♦♥t✐♥✉✐t✐❡s✮ ♦♥ [a, b] ❛r❡ ✐♥t❡❣r❛❜❧❡ ♦♥ [a, b]✳ Pr♦♦❢✳

■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ✐♥t❡❣r❛❜✐❧✐t② ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✭❧❛st s❡❝t✐♦♥✮ ❛♥❞ t❤❡

❆❞❞✐t✐✈✐t② ❘✉❧❡✳

■♥ ♣❛rt✐❝✉❧❛r✱ ❛❧❧ st❡♣✲❢✉♥❝t✐♦♥s ❛r❡ ✐♥t❡❣r❛❜❧❡✳ ❆❢t❡r ❛❧❧✱ t❤❡② ❡✈❡♥ ❧♦♦❦ ❧✐❦❡ ❘✐❡♠❛♥♥ s✉♠s✿

✶✳✼✳

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s

✻✾

❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ✕ ✈✐❛ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r✉❧❡s ♦❢ ❧✐♠✐ts✳ ❚❤❡ ❣r❛♣❤✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥s ♦❢ t❤❡ t✇♦ ♣r♦♣❡rt✐❡s ❛r❡ ❛❧s♦ t❤❡ s❛♠❡❀ ✇❡ ❥✉st ③♦♦♠ ♦✉t✳

n

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤❡ r✐❣❤t✲❡♥❞ s✉♠❀ t❤❡r❡ ❛r❡

✐♥t❡r✈❛❧s ♦❢ ❡q✉❛❧ ❧❡♥❣t❤✱

∆xi = h = ❜❡t✇❡❡♥

a

❛♥❞

b

b−a , n

❛♥❞ t❤❡s❡ ❛r❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✿

ci = a, a + h, ... , b − h . ❆ ♣r♦♦❢ ❢♦r ❧❡❢t✲❡♥❞ s✉♠ ❛♥❞ ♠✐❞✲♣♦✐♥ts s✉♠s ✇♦✉❧❞ ❜❡ ✈✐rt✉❛❧❧② ✐❞❡♥t✐❝❛❧✳ ❇❡❧♦✇✱ ✇❡ ✇✐❧❧ ❜❡ r❡✈✐❡✇✐♥❣ s♦♠❡ ❢❛❝ts ❛❜♦✉t

s✉♠s ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✶P❈✲✶ ❛♥❞ t❤❡♥ ❛♣♣❧②✐♥❣ t❤❡♠ t♦

t❤❡ ❘✐❡♠❛♥♥ s✉♠s ♣r♦❞✉❝✐♥❣✱ ✈✐❛ ❧✐♠✐ts✱ r❡s✉❧ts ❛❜♦✉t t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s✳ ❈❛♥ ✇❡ ❝♦♠♣❛r❡ t❤❡ ✈❛❧✉❡s ♦❢ t✇♦ ❘✐❡♠❛♥♥ s✉♠s❄ ❈♦♥s✐❞❡r t❤✐s s✐♠♣❧❡ ❛❧❣❡❜r❛✿

≤ U

u

≤ V

v

u+v ≤ U +V ❲❡ ❝❛♥ ❦❡❡♣ ❛❞❞✐♥❣ t❡r♠s✿

up ≤ Up , up+1 ≤ Up+1 ✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

uq ≤ Uq up + ... + uq ≤ Up + ... + Uq q q X X un ≤ Un n=p

❚❤❛t✬s t❤❡

n=p

❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ❙✉♠s ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✶✮✳

❋♦r ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦✈❡r ♣❛rt✐t✐♦♥s✱ ✐❢ ♦♥❡ ❢✉♥❝t✐♦♥ ✏❞♦♠✐♥❛t❡s✑ ❛♥♦t❤❡r✱ t❤❡♥ s♦ ❞♦❡s ✐ts s✉♠✱ ❘✐❡♠❛♥♥ s✉♠✱ ❛♥❞ ❧❛t❡r t❤❡ ✐♥t❡❣r❛❧✿

✶✳✼✳

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s

✼✵

❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✶✳✼✳✻✿ ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ❙✉♠s ❚❤❡ s✉♠ ♦❢ ❛ s♠❛❧❧❡r ❢✉♥❝t✐♦♥ ✐s s♠❛❧❧❡r✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢

[a, b]✱

f

❛♥❞

g

❛r❡ ❢✉♥❝t✐♦♥s✱ t❤❡♥ ❢♦r ❛♥② ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧

✇❡ ❤❛✈❡✿

f (x) ≥ g(x)

♦♥

[a, b] =⇒ Σf ≥ Σg

✇❤❡r❡ t❤❡ s✉♠♠❛t✐♦♥s ❛r❡ ♦✈❡r ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢

[a, b]✳

❲❤❛t ✐❢ t❤❡s❡ s✉♠s ❛r❡ ❘✐❡♠❛♥♥ s✉♠s❄ ❚❤❡♦r❡♠ ✶✳✼✳✼✿ ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ❘✐❡♠❛♥♥ ❙✉♠s ❚❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ s♠❛❧❧❡r ❢✉♥❝t✐♦♥ ✐s s♠❛❧❧❡r✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢

[a, b]✱

f

❛♥❞

g

❛r❡ ❢✉♥❝t✐♦♥s✱ t❤❡♥ ❢♦r ❛♥② ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧

✇❡ ❤❛✈❡✿

f (x) ≥ g(x)

♦♥

[a, b] =⇒ Σf · ∆x ≥ Σg · ∆x

✇❤❡r❡ t❤❡ s✉♠♠❛t✐♦♥s ❛r❡ ♦✈❡r ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢

[a, b]✳

Pr♦♦❢✳

❆♣♣❧②✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠ t♦ t❤❡ ❢✉♥❝t✐♦♥ f ❛♥❞ t❤❡♥ t♦ g ✱ ✇❡ ❤❛✈❡✿ Σf · ∆x (b) = f (a) + f (a + h) + f (a + 2h) + ... + f (b − h) ≥ g(a) + g(a + h) + g(a + 2h) + ... + g(b − h)

= Σg · ∆x (b) .

■❢ ✇❡ ③♦♦♠ ♦✉t✱ ✇❡ s❡❡ t❤❛t t❤❡ ❧❛r❣❡r ❢✉♥❝t✐♦♥ ❛❧✇❛②s ❝♦♥t❛✐♥s ❛ ❧❛r❣❡r ❛r❡❛ ✉♥❞❡r ✐ts ❣r❛♣❤✿

✶✳✼✳

✼✶

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s

❙♦✱ ✇❡ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ t❤✐s ✐♥❡q✉❛❧✐t② ❛s ∆x → 0✿ ❚❤❡♦r❡♠ ✶✳✼✳✽✿ ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ❘✐❡♠❛♥♥ ■♥t❡❣r❛❧s ❚❤❡ ✐♥t❡❣r❛❧ ♦❢ ❛ s♠❛❧❧❡r ❢✉♥❝t✐♦♥ ✐s s♠❛❧❧❡r✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢

f

❛♥❞

g

❛r❡ ❢✉♥❝t✐♦♥s✱ t❤❡♥ ❢♦r ❛♥②

f (x) ≥ g(x) ♣r♦✈✐❞❡❞

f

❛♥❞

g

♦♥

[a, b] =⇒

❛r❡ ✐♥t❡❣r❛❜❧❡ ♦✈❡r

Z

a, b

✇✐t❤

Z

g dx

b a

f dx ≥

a < b✱

✇❡ ❤❛✈❡✿

b a

[a, b]✳

Pr♦♦❢✳

◆♦✇ t❛❦❡ t❤❡ ❧✐♠✐t ✇✐t❤ n → ∞ ❛♥❞ ✉s❡ t❤❡ ❈♦♠♣❛r✐s♦♥

❘✉❧❡ ❢♦r ▲✐♠✐ts

❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✶✳

❊①❛♠♣❧❡ ✶✳✼✳✾✿ ❝♦♠♣❛r✐s♦♥ ♦❢ ♥❡❣❛t✐✈❡ ✐♥t❡❣r❛❧s

❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ❛r❡❛ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✇✐❧❧ ♠❛t❝❤ ♦✉r ✐♥t✉✐t✐♦♥ ♦♥❧② ❢♦r ❛s ❧♦♥❣ ❛s t❤❡ ❛r❡❛s ❛r❡ ♣♦s✐t✐✈❡✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t ✇❡ ♠❡❛s✉r❡ t❤❡ ❛r❡❛s ❢r♦♠ t❤❡ x✲❛①✐s t♦ t❤❡ ❣r❛♣❤✳ ❚❤❛t ✐s ✇❤② ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥s ❛r❡ ♥❡❣❛t✐✈❡✱ t❤❡ ❛r❡❛s ❛r❡ ♥❡❣❛t✐✈❡ t♦♦✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥ t❤❡♥ ❛♣♣❡❛rs t♦ ❜❡ ✇r♦♥❣✿

❇✉t t❤❡ ❝♦♠♣❛r✐s♦♥ ❤❛s♥✬t ❜❡❡♥ ✢✐♣♣❡❞✦ ❚❤❡r❡ ♠❛② ❜❡ ❛ s✐♠✐❧❛r ❝♦♥✢✐❝t ❢♦r t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤✐s r❡s✉❧t ✐♥ t❡r♠s ♦❢ ♠♦t✐♦♥✳ ❚❤✐s s✐♠♣❧❡ ✐♥t❡r♣r❡t❛t✐♦♥ ❢❛✐❧s ❜❡❝❛✉s❡ ✐t ❞♦❡s♥✬t t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❞✐r❡❝t✐♦♥s✿ ❚❤❡ ❢❛st❡r ❝♦✈❡rs t❤❡ ❧♦♥❣❡r ❞✐st❛♥❝❡✳ ❇❡tt❡r st✐❧❧✿ ❚❤❡ ❢❛st❡r ②♦✉ ❣♦ ✐♥ ❛ ♣❛rt✐❝✉❧❛r ❞✐r❡❝t✐♦♥✱ t❤❡ ❢❛rt❤❡r ②♦✉ ♣r♦❣r❡ss✳ ❊①❡r❝✐s❡ ✶✳✼✳✶✵

Pr♦✈❡ t❤❡ r❡st ♦❢ t❤❡ t❤❡♦r❡♠✳

✶✳✼✳

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s

✼✷

❊①❡r❝✐s❡ ✶✳✼✳✶✶

▼♦❞✐❢② t❤❡ ♣r♦♦❢ ❢♦r✿ ✭❛✮ ❧❡❢t✲❡♥❞✱ ✭❜✮ ♠✐❞✲♣♦✐♥t✱ ❛♥❞ ✭❝✮ ❣❡♥❡r❛❧ ❘✐❡♠❛♥♥ s✉♠s✳ ❊①❡r❝✐s❡ ✶✳✼✳✶✷

❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ a > b❄ ❘❡❧❛t❡❞ r❡s✉❧ts ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✶✳✼✳✶✸✿ ❙tr✐❝t ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ❙✉♠s ❚❤❡ s✉♠ ♦❢ ❛ str✐❝t❧② s♠❛❧❧❡r ❢✉♥❝t✐♦♥ ✐s str✐❝t❧② s♠❛❧❧❡r✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢

[a, b]

✇✐t❤

a < b✱

f

❛♥❞

g

❛r❡ ❢✉♥❝t✐♦♥s✱ t❤❡♥ ❢♦r ❛♥② ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧

✇❡ ❤❛✈❡✿

f (x) < g(x)

♦♥

[a, b] =⇒ Σf < Σg

✇❤❡r❡ t❤❡ s✉♠♠❛t✐♦♥s ❛r❡ ♦✈❡r ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢

[a, b]✳

❚❤❡♦r❡♠ ✶✳✼✳✶✹✿ ❙tr✐❝t ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ❘✐❡♠❛♥♥ ❙✉♠s ❚❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ str✐❝t❧② s♠❛❧❧❡r ❢✉♥❝t✐♦♥ ✐s str✐❝t❧② s♠❛❧❧❡r✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢

[a, b]

✇✐t❤

a < b✱

f

❛♥❞

g

❛r❡ ❢✉♥❝t✐♦♥s✱ t❤❡♥ ❢♦r ❛♥② ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧

✇❡ ❤❛✈❡✿

f (x) < g(x)

♦♥

[a, b] =⇒ Σf · ∆x < Σg · ∆x

✇❤❡r❡ t❤❡ s✉♠♠❛t✐♦♥s ❛r❡ ♦✈❡r ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢

[a, b]✳

❊①❡r❝✐s❡ ✶✳✼✳✶✺

❙t❛t❡ ❛♥❞ ♣r♦✈❡ t❤❡ ❙tr✐❝t ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ✐♥t❡❣r❛❧s✳ ❍✐♥t✿ ❚❤❡r❡ ✐s ♥♦ ❙tr✐❝t ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ❧✐♠✐ts✳ ❲❤❛t ✐❢ ✇❡ ❦♥♦✇ ♦♥❧② ❛ ♣r✐♦r✐ ❜♦✉♥❞s ♦❢ t❤❡ ❢✉♥❝t✐♦♥❄ ❙✉♣♣♦s❡ t❤❡ r❛♥❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ❧✐❡s ✇✐t❤✐♥ t❤❡ ✐♥t❡r✈❛❧ [m, M ]✳ ❚❤❡♥ ✐ts ❣r❛♣❤ ❧✐❡s ❜❡t✇❡❡♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❧✐♥❡s y = m ❛♥❞ y = M ✱ ✇✐t❤ m < M ✳

❚❤❡♥✱ ✇❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ f ❄ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✜♥❞ ❛♥ ❡st✐♠❛t❡ ❢♦r t❤❡ ❛r❡❛ ♦❢ t❤❡ ♦r❛♥❣❡ r❡❣✐♦♥ ✐♥ t❡r♠s ♦❢ a✱ b✱ m✱ M ✳ ❇❡❧♦✇✱ t❤❡ ②❡❧❧♦✇ r❡❣✐♦♥ ♦♥ t❤❡ ❧❡❢t ✐s ❧❡ss t❤❛♥ t❤❡ ♦r❛♥❣❡ ❛r❡❛✳ ❖♥ t❤❡ r✐❣❤t✱ t❤❡ ❣r❡❡♥ ❛r❡❛ ✐s ❧❛r❣❡r✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡s❡ t✇♦ r❡❣✐♦♥s ❛r❡ r❡❝t❛♥❣❧❡s ❛♥❞ t❤❡✐r ❛r❡❛s ❛r❡ ❡❛s② t♦ ❝♦♠♣✉t❡✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❝❧✉s✐♦♥s✿ t❤❡ s♠❛❧❧❡r r❡❝t❛♥❣❧❡ ⊂ t❤❡ r❡❣✐♦♥ ✉♥❞❡r t❤❡ ❣r❛♣❤ ⊂ t❤❡ ❧❛r❣❡r r❡❝t❛♥❣❧❡ .

✶✳✼✳

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s

✼✸

❚❤❡② ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❡q✉❛❧✐t✐❡s✿ t❤❡ ❛r❡❛ ♦❢ t❤❡ s♠❛❧❧❡r r❡❝t❛♥❣❧❡

≤

t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤

≤

t❤❡ ❛r❡❛ ♦❢ t❤❡ ❧❛r❣❡r r❡❝t❛♥❣❧❡ .

❋♦r t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✶✳✼✳✶✻✿ ❇♦✉♥❞s ❢♦r ❘✐❡♠❛♥♥ ❙✉♠s ❙✉♣♣♦s❡

f

✐s ❛ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❛t ❢♦r ❛♥② ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧

[a, b]✱

✇❡ ❤❛✈❡✿

m ≤ f (x) ≤ M , ❢♦r ❛❧❧

x

✇✐t❤

a ≤ x ≤ b✳

❚❤❡♥✱ ✇❡ ❤❛✈❡✿

m(b − a) ≤ Σf · ∆x ≤ M (b − a) ✇❤❡r❡ t❤❡ s✉♠♠❛t✐♦♥ ✐s ♦✈❡r ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢

[a, b]✳

Pr♦♦❢✳

❋♦r ❡✐t❤❡r t❤✐s ✐♥❡q✉❛❧✐t✐❡s ❛♥❞ t❤❡ ♦♥❡ ✐♥ t❤❡ ♥❡①t t❤❡♦r❡♠✱ ✇❡ ❛♣♣❧② t❤❡ ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥ ❛♥❞ t❤❡ t❤❡

❈♦♠♣❛r✐s♦♥ ❚❤❡♦r❡♠

❇♦✉♥❞s ❘✉❧❡ ❢♦r ▲✐♠✐ts ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✶✳

❈♦♥st❛♥t ❋✉♥❝t✐♦♥ ❘✉❧❡

❛❜♦✈❡✳ ◆♦✇ t❛❦❡ t❤❡ ❧✐♠✐t ✇✐t❤

k→∞

❛♥❞ ✉s❡

◆♦✇ ✇❡ t❛❦❡ t❤❡ ❧✐♠✐t✳ ❚❤❡♦r❡♠ ✶✳✼✳✶✼✿ ❇♦✉♥❞s ❢♦r ■♥t❡❣r❛❧s ❙✉♣♣♦s❡

f

✐s ❛ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❛t ❢♦r ❛♥②

a, b

✇✐t❤

a < b✱

✇❡ ❤❛✈❡✿

m ≤ f (x) ≤ M , ❢♦r ❛❧❧

x

✇✐t❤

a ≤ x ≤ b✳

■❢

f

✐s ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ ♦✈❡r

m(b − a) ≤

Z

[a, b]✱

t❤❡♥ ✇❡ ❤❛✈❡✿

b a

f dx ≤ M (b − a)

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❜♦✉♥❞s ❢♦r ❛ ❢✉♥❝t✐♦♥ ❝r❡❛t❡ ❜♦✉♥❞s ❢♦r t❤❡ ✐♥t❡❣r❛❧✳ ❊①❡r❝✐s❡ ✶✳✼✳✶✽

❙t❛t❡ ❛♥❞ ♣r♦✈❡ t❤❡ ❙tr✐❝t ❇♦✉♥❞s ❢♦r ■♥t❡❣r❛❧s✳ ❊①❛♠♣❧❡ ✶✳✼✳✶✾✿ ❛ ♣r✐♦r✐ ❡st✐♠❛t❡s

❲❡ ❦♥♦✇✱

❛ ♣r✐♦r✐✱ t❤❛t t❤❡ ✈❛❧✉❡s ♦❢ s✉❝❤ ❛ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥ ❛s sin ✭❛♥❞ cos✮ ❧✐❡ ✇✐t❤✐♥ [−1, 1]✳

❚❤❡♥✱ ✇❡ ❤❛✈❡✿

−1 ≤ −1(b − a) ≤

sin x Z

b a

≤ 1

❢♦r ❛❧❧

x

=⇒

sin x dx ≤ 1(b − a)

❙♦✱ ❡✈❡♥ t❤♦✉❣❤ ❛❧❧ ✇❡ ❦♥♦✇ ❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥ ✐s ❥✉st t❤❡s❡ t✇♦✱ ✈❡r② ❝r✉❞❡✱ ❡st✐♠❛t❡s✱ ✇❡ t❤❛t t❤❡ ✐♥t❡❣r❛❧ ✇✐❧❧ ❧✐❡ ✇✐t❤✐♥ t❤❡ ✐♥t❡r✈❛❧

[a − b, b − a]✳

❣✉❛r❛♥t❡❡

✶✳✽✳

✼✹

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❝♦♥t✐♥✉❡❞

✶✳✽✳ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❝♦♥t✐♥✉❡❞

❙♦ ❢❛r✱ ✇❡ ❤❛✈❡ ❧❡❛r♥❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ◮ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ✉♥❞♦ t❤❡ ❡✛❡❝t ♦❢ ❡❛❝❤ ♦t❤❡r✳

❚❤❡ s✐♠♣❧❡ r❡❛s♦♥ ✐s t❤❛t t❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥ t❤❡② ❛r❡ ❝r❡❛t❡❞ ❜② ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r ♦♥❡ ❜② ♦♥❡✳ ◆♦✇✱ s✐♥❝❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ❛r❡ t❤❡ r❡❛s♦♥ t❤❛t t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ r❡❧❛t✐♦♥✿

❧✐♠✐ts

♦❢ t❤❡s❡ t✇♦ r❡s♣❡❝t✐✈❡❧②✱ ✐t st❛♥❞s t♦

◮ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✉♥❞♦ t❤❡ ❡✛❡❝t ♦❢ ❡❛❝❤ ♦t❤❡r✳

❇❡❧♦✇✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s s❛♠♣❧❡❞ t♦ ♣r♦❞✉❝❡ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✱ ✇❤✐❝❤ ✉♥❞❡r t❤❡ ❧✐♠✐t ♣r♦❞✉❝❡ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ t♦ ❜❡ ❞✐✛❡r❡♥t✐❛t❡❞✿

❲✐❧❧ ✇❡ ♠❛❦❡ t❤❡ ❢✉❧❧ ❝✐r❝❧❡❄ ❙♦✱ ✇❤❛t ❞♦❡s t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ❤❛✈❡ t♦ ❞♦ ✇✐t❤ ❛♥t✐❞❡r✐✈❛t✐✈❡s❄ ❙❛♠❡ ❛s ✇❤❛t t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❞♦❡s✦ ❊①❛♠♣❧❡ ✶✳✽✳✶✿ ♣♦s✐t✐♦♥s ❛♥❞ ✈❡❧♦❝✐t✐❡s

❲❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❡ ❛♥s✇❡r✱ ❢♦r t❤❡ ♠♦t✐♦♥ ♠❡t❛♣❤♦r✳ ■❢ y = f (x) ✐s t❤❡ ✈❡❧♦❝✐t② ❛♥❞ x ✐s t❤❡ t✐♠❡✱ t❤❡♥ ✇❡ ❤❛✈❡✿ • ❖♥❡ ♦❢ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ✐s t❤❡ ♣♦s✐t✐♦♥ ❢✉♥❝t✐♦♥✱ F (x)✳ • ❚❤❡ ✐♥t❡❣r❛❧ ✐s t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❞✉r✐♥❣ [a, b]✱

Zb

f dx✳

a

■❢ ✇❡ ❦♥♦✇ t❤❡ ♣♦s✐t✐♦♥ ❛t ❛❧❧ t✐♠❡s✱ ✇❡ ❝❛♥ ❝❡rt❛✐♥❧② ❝♦♠♣✉t❡ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❛t ❛♥② ♠♦♠❡♥t✿ F (b) |{z}

❝✉rr❡♥t ♣♦s✐t✐♦♥

−

F (a) | {z }

✐♥✐t✐❛❧ ♣♦s✐t✐♦♥

❈♦♥✈❡rs❡❧②✱ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✐s ❢♦✉♥❞ ❢♦r ❛♥② x > a ❛s

Zx

f dx✳ ❆❢t❡r ❛❧❧✱ t❤❡ t✇♦ ❛r❡ ❛ttr✐❜✉t❡s ♦❢ t❤❡

a

s❛♠❡ ♠♦t✐♦♥✱ ♥♦ ♠❛tt❡r ❤♦✇ t❤❡② ❛r❡ ❝♦♠♣✉t❡❞❀ t❤❡② ❝♦✲❡①✐st✳ ❊①❛♠♣❧❡ ✶✳✽✳✷✿ ❛r❡❛s ❛♥❞ t❛♥❣❡♥ts

◆♦✇✱ ✇❤❛t ❛❜♦✉t t❤❡ ❛r❡❛ ♠❡t❛♣❤♦r❄ ■t✬s ♥♦t tr❛♥s♣❛r❡♥t✳

✶✳✽✳

✼✺

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❝♦♥t✐♥✉❡❞

❆♥❛❧♦❣♦✉s t♦ ✇❤❛t ✇❡ ❞✐❞ ❛❜♦✈❡ ❢♦r t❤❡ ❘✐❡♠❛♥♥ s✉♠✱ ✇❡ ❞❡✜♥❡ t❤❡ ❛r❡❛

A(x) =

❢✉♥❝t✐♦♥

♦❢ f t♦ ❜❡✿

✈❛r✐❡s z}|{ Zx

f (t) dt = ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦✈❡r [a, x] .

a

❚❤✐s ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ❤❛s ❛ ✈❛r✐❛❜❧❡ ❜❡②♦♥❞✳

✳ ■t✬s ❛❧s♦ ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿ x r✉♥s ❢r♦♠ a t♦ b ❛♥❞

✉♣♣❡r ❜♦✉♥❞

❲❡ ❝❛♥ s❡❡ ❤♦✇✱ ✇❤❡r❡✈❡r f ❤❛s ♣♦s✐t✐✈❡ ✈❛❧✉❡s✱ ❡✈❡r② ✐♥❝r❡❛s❡ ♦❢ x > a ❛❞❞s ❛ s❧✐❝❡ t♦ t❤❡ ❛r❡❛ ✭❧❡ss ✐♥ t❤❡ ♠✐❞❞❧❡✮ ❝❛✉s✐♥❣ A t♦ ✐♥❝r❡❛s❡ ✭s❧♦✇❡r ✐♥ t❤❡ ♠✐❞❞❧❡✮✳ ▼♦r❡♦✈❡r✱ ✇❤❡r❡✈❡r f ❤❛s ♥❡❣❛t✐✈❡ ✈❛❧✉❡s✱ t❤✐s s❧✐❝❡ ✇✐❧❧ ❤❛✈❡ ❛ ♥❡❣❛t✐✈❡ ❛r❡❛✱ t❤❡r❡❜② ❝❛✉s✐♥❣ A t♦ ❞❡❝r❡❛s❡✳ ❚❤❛t✬s ❛ ❜❡❤❛✈✐♦r ♦❢ ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ f ✦ ▲❡t✬s ❝♦♥s✐❞❡r ✐ts ❞❡r✐✈❛t✐✈❡✱ A′ ✳ ❋✐rst✱ ✐t✬s t❤❡ s❧♦♣❡✿

❲❡ ❣♦ ❛❧❧ t❤❡ ✇❛② ❜❛❝❦ t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✿ A(x + h) − A(x) h→0 h Z x+h  Z x 1 = lim f dt f dt − h→0 h a a Z 1 x+h f dx . ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❆❞❞✐t✐✈✐t② ❘✉❧❡✳ = lim h→0 h x

A′ (x) = lim

❚❤❡ ❧❛st ✈❛❧✉❡ ✐s ✐❧❧✉str❛t❡❞ ♦♥ t❤❡ r✐❣❤t✿

✶✳✽✳ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❝♦♥t✐♥✉❡❞

✼✻

❲❡ ❛r❡ ❧❡❢t ✇✐t❤ ❛ s❧✐❝❡ ♦❢ t❤❡ ❛r❡❛ ❛❜♦✈❡ t❤❡ s❡❣♠❡♥t

[x, x + h]❀

✐t ❧♦♦❦s ❧✐❦❡ ❛ tr❛♣❡③♦✐❞ ✇❤❡♥

h

✐s

s♠❛❧❧✳ ❲❤❛t ❤❛♣♣❡♥s t♦ t❤✐s ❧✐♠✐t❄ ▲❡t✬s ♠❛❦❡

h

s♠❛❧❧✳ ❚❤❡ ✏tr❛♣❡③♦✐❞✑ ✇✐❧❧ ❜❡ t❤✐♥♥❡r ❛♥❞ t❤✐♥♥❡r ❛♥❞ ✐ts t♦♣ ❡❞❣❡ ✇✐❧❧ ❧♦♦❦ ♠♦r❡ ❛♥❞

♠♦r❡ str❛✐❣❤t✱ ❛ss✉♠✐♥❣

f

✐s ❝♦♥t✐♥✉♦✉s✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡ ❛r❡❛ ♦❢ ❛ tr❛♣❡③♦✐❞ ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡

♠✐❞✲❧✐♥❡ t✐♠❡s t❤❡ ❤❡✐❣❤t✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ ❛r❡❛ ♦❢ tr❛♣❡③♦✐❞ ✇✐❞t❤

=

❤❡✐❣❤t ✐♥ t❤❡ ♠✐❞❞❧❡ .

■t ❢♦❧❧♦✇s t❤❛t ✐ts ❤❡✐❣❤t ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜②

R x+h x

❚❤✐s ♦✉t❝♦♠❡ s✉❣❣❡sts t❤❛t

A′ = f ✳

f dx . h

❍♦✇❡✈❡r✱ ♣r♦✈✐♥❣ t❤❛t t❤❡ ❧✐♠✐t ❡①✐sts ✇♦✉❧❞ r❡q✉✐r❡ ❛ s✉❜t❧❡r

❛r❣✉♠❡♥t✳

❊①❡r❝✐s❡ ✶✳✽✳✸

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❛r❡❛ ❢✉♥❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥

f

♣❧♦tt❡❞ ❜❡❧♦✇ ❢♦r

a = 1✿

❊①❡r❝✐s❡ ✶✳✽✳✹

❙✉♣♣♦s❡ t❤❡ ❢✉♥❝t✐♦♥ ♣❧♦tt❡❞ ❛❜♦✈❡ ✐s t❤❡ ❛r❡❛ ❢✉♥❝t✐♦♥ ♦❢ s♦♠❡ ❢✉♥❝t✐♦♥✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❛t ❢✉♥❝t✐♦♥✳ ❊①❡r❝✐s❡ ✶✳✽✳✺

Pr♦✈❡ t❤❛t t❤❡ ❛r❡❛ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s✳

✶✳✽✳

✼✼

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❝♦♥t✐♥✉❡❞

❊①❡r❝✐s❡ ✶✳✽✳✻

❋✐♥✐s❤ t❤❡ ♣r♦♦❢ ❜② ✉s✐♥❣ t❤❡ ❙q✉❡❡③❡

❚❤❡♦r❡♠

f (x) ≤

❢♦r✿

A(x + h) − A(x) ≤ f (x + h) . h

❊①❛♠♣❧❡ ✶✳✽✳✼✿ tr✐❛♥❣❧❡

▲❡t✬s ❝♦♥✜r♠ t❤❡ ✐❞❡❛ ✇✐t❤ ❛ ❢❛♠✐❧✐❛r s❤❛♣❡✳ ❈♦♥s✐❞❡r f (x) = 2x .

▲❡t✬s ❝♦♠♣✉t❡ t❤❡ ❛r❡❛ ❢✉♥❝t✐♦♥✳

❲❡ ❤❛✈❡✿ A(x) =

Z

x

2x dx 0

= ❛r❡❛ ♦❢ tr✐❛♥❣❧❡ ✇✐t❤ ❜❛s❡ [0, x] 1 ✇✐❞t❤ · ❤❡✐❣❤t 2 1 = x · 2x 2 =

= x2 .

❲❤❛t✬s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ f (x) = 2x ❛♥❞ A(x) = x2 ❄ ❲❡ ❦♥♦✇ t❤❡ ❛♥s✇❡r✿ x2

s♦ f ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ f ✦

′

= 2x ,

❚❤✐s ✐❞❡❛ ❛♣♣❧✐❡s t♦ ❛❧❧ ❢✉♥❝t✐♦♥s✳ ❚❤❡♦r❡♠ ✶✳✽✳✽✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ■

●✐✈❡♥ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ f ♦♥ [a, b]✱ t❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜② F (x) =

Z

x

f dx a

✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ f ♦♥ (a, b)❀ ✐✳❡✱ F′ = f .

✶✳✽✳

✼✽

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❝♦♥t✐♥✉❡❞

❚❤✉s✱ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❝❛♥❝❡❧s t❤❡ ❡✛❡❝t ♦❢ t❤❡ ✭✈❛r✐❛❜❧❡✲❡♥❞✮ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✿ F ′ = f ✳ ❊①❡r❝✐s❡ ✶✳✽✳✾

❋✐♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ✇✐t❤ F (a) = 1✳ ❊①❡r❝✐s❡ ✶✳✽✳✶✵

❋✐♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢

Z

b

f dx✳

x

❇✉t t❤✐s ✐s ♦♥❧② ❛ ❤❛❧❢✦ ◆❡①t✱ ✇❡ s❡❡ ❤♦✇ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ❝❛♥❝❡❧s ✕ ✉♣ t♦ ❛ ❝♦♥st❛♥t ✕ t❤❡ ❡✛❡❝t ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❚❤❡♦r❡♠ ✶✳✽✳✶✶✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ■■ ✭◆❡✇t♦♥✲ ▲❡✐❜♥✐③ ❋♦r♠✉❧❛✮ ❋♦r ❛♥② ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥

f Z

♦♥

[a, b]

❛♥❞ ❛♥② ♦❢ ✐ts ❛♥t✐❞❡r✐✈❛t✐✈❡s

F✱

✇❡ ❤❛✈❡

b a

f dx = F (b) − F (a)

Pr♦♦❢✳

❲❡ st❛rt ✇✐t❤ ❛ ♣❛rt✐t✐♦♥ P ♦❢ ✐♥t❡r✈❛❧ [a, b] ✇✐t❤ n ✐♥t❡r✈❛❧s✳ ❚❤❡ ♥♦❞❡s xi , i = 0, 1, ..., n, ❝❛♥ ❜❡ ❛r❜✐✲ tr❛r②❀ ✇❡ ❝❛♥ ❡✈❡♥ ❝❤♦♦s❡ ❡q✉❛❧ ❧❡♥❣t❤s ❢♦r t❤❡ ✐♥t❡r✈❛❧s✿ ∆xi = xi − xi−1 = (b − a)/n, i = 1, 2, ..., n✳ ❚❤❡r❡ ❛r❡ ♥♦ s❡❝♦♥❞❛r② ♥♦❞❡s ②❡t❀ t❤❡✐r ❝❤♦✐❝❡ ✇✐❧❧ ❜❡ ❞✐❝t❛t❡❞ ❜② F ✳ ❚❤✐s ✐s ❤♦✇ ❡❛❝❤ ♦❢ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ci , i = 1, 2, ..., n, ✐s ❝❤♦s❡♥✿

▲❡t✬s t❛❦❡ ♦♥❡ ✐♥t❡r✈❛❧ [xi−1 , xi ] ❛♥❞ ✇❡ ❛♣♣❧② t❤❡ ▼❡❛♥ ❱❛❧✉❡ ✐♥t❡r✈❛❧ s✉❝❤ t❤❛t F (xi ) − F (xi−1 ) = F ′ (ci ) . xi − xi−1

❚❤❡♦r❡♠

t♦ F ✿ ❚❤❡r❡ ✐s ❛ ci ✐♥ t❤❡

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ✜♥❞ ❛ ♣♦✐♥t ✇✐t❤✐♥ ❡❛❝❤ ✐♥t❡r✈❛❧ t❤❛t ❤❛s t❤❡ s❧♦♣❡ ♦❢ ✐ts t❛♥❣❡♥t ❧✐♥❡ ❡q✉❛❧ t♦ t❤❡ s❧♦♣❡ ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡ ♦✈❡r t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧ [xi−1 , xi ]✿

✶✳✽✳

✼✾

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❝♦♥t✐♥✉❡❞

❲❡ ♠♦❞✐❢② t❤❡ ❢♦r♠✉❧❛✿

F (xi ) − F (xi−1 ) = f (ci ) ∆x . ❍❡r❡ ✇❡ ❤❛✈❡✿

• •

❚❤❡

[xi−1 , xi ]✳ ❚❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s t❤❡ ❡❧❡♠❡♥t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ f ♦✈❡r [xi−1 , xi ]✳ ♥❡①t st❡♣ ✐s t♦ ❡①♣r❡ss t❤❡ t♦t❛❧ ♥❡t ❝❤❛♥❣❡ ♦❢ F ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ [a, b] ❛s t❤❡ ❚❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ✐s t❤❡

♥❡t ❝❤❛♥❣❡

✭t❤❡ r✐s❡✮ ♦❢

F

♦✈❡r t❤❡ ✐♥t❡r✈❛❧

s✉♠ ♦❢ t❤❡ ♥❡t

❝❤❛♥❣❡s ♦✈❡r t❤❡ ✐♥t❡r✈❛❧s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✿

❲❡ ♥♦✇ ❝♦♥✈❡rt t❤♦s❡ ♥❡t ❝❤❛♥❣❡s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛✿

F (b) − F (a) = F (xn )

=

−F (x0 ) ❚❤❡ ❧❛st ❡①♣r❡ss✐♦♥ ✐s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢

f✱

F (xn ) − F (xn−1 )

=

f (cn ) ∆x

+F (xn−1 ) − F (xn−2 )

+f (cn−1 ) ∆x

+...

+...

+F (xi+1 ) − F (xi )

+f (ci ) ∆x

+...

+...

+F (x1 ) − F (x0 )

+f (c1 ) ∆x .

❛♥❞ s✐♥❝❡

∆x → 0✳ ❊①❡r❝✐s❡ ✶✳✽✳✶✷

❯s❡ t❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥ t♦ r❡✲✇r✐t❡ t❤❡ ❧❛st ❝♦♠♣✉t❛t✐♦♥✳

f

✐s ✐♥t❡❣r❛❜❧❡✱ ✐t ❝♦♥✈❡r❣❡s t♦

Z

b

f dx a

❛s

✶✳✽✳

✽✵

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❝♦♥t✐♥✉❡❞

❊①❡r❝✐s❡ ✶✳✽✳✶✸

Pr♦✈✐❞❡ t❤❡ ❞❡t❛✐❧s ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ■ ❢r♦♠ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ■■ ✭◆▲❋✮✿ d dx

Z

x

f dx a



=

d (F (x) − F (a)) = F ′ (x) = f (x) . dx ❲❛r♥✐♥❣✦ ❙✐♥❝❡ t❤❡ t❤❡♦r❡♠ ❛♣♣❧✐❡s t♦

❛❧❧

❛♥t✐❞❡r✐✈❛t✐✈❡s✱ ✇❡

❝❛♥ ♦♠✐t ✏ +C ✑✳

❊①❛♠♣❧❡ ✶✳✽✳✶✹✿ ♣❛r❛❜♦❧❛

▲❡t✬s ❝♦♠♣✉t❡ t❤❡ ❛r❡❛ ✉♥❞❡r y = x2 ❢r♦♠ 0 t♦ 1✿

❲❡ ❞✐❞ ❛ s✐♠✐❧❛r ♦♥❡ t❤❡ ❤❛r❞ ✇❛②✱ ✈✐❛ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✱ ✇❤✐❝❤ r❡q✉✐r❡❞ ❛ ❢♦r♠✉❧❛ ❢♦r 1 + 2 + ... + n✳ ▲❡t✬s t❡st t❤❡ ❢♦r♠✉❧❛✳ ❲❡ ♥❡❡❞ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = x2 ✳ ❇② ❝♦♠♣✉t❡❞ ❜❡❢♦r❡✿

r❡❝❛❧❧✐♥❣

t❤❡ ❞❡r✐✈❛t✐✈❡ ✇❡

(x3 )′ = 3x2 .

✭♦r ✉s✐♥❣ t❤❡ P♦✇❡r

❋♦r♠✉❧❛

❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✺✮✱ ✇❡ ✜♥❞ ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡✿ F (x) =

❚❤❡♥✱ ❜② ◆▲❋✱ ✇❡ ❤❛✈❡✿ ❚❤❡ r❡s✉❧t ✐s ♣❧❛✉s✐❜❧❡✿

Z

x3 . 3

1 0

x2 dx = F (1) − F (0) =

13 03 1 − = . 3 3 3

❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ❤❛s ❛❧s♦ s❤♦✇♥ t❤❛t ✐❢ ✇❡ ✐♥❝r❡❛s❡ t❤❡ s♣❡❡❞ ❢r♦♠ 0 t♦ 1 q✉❛❞r❛t✐❝❛❧❧② ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ ♦❢ t✐♠❡ [0, 1]✱ t❤❡♥ t❤❡ ❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ✐s 1/3✳

✶✳✽✳

✽✶

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❝♦♥t✐♥✉❡❞

❚❤❡ P❛rt ■ s✉♣♣❧✐❡s ✉s ✇✐t❤ ❛ s♣❡❝✐❛❧ ❝❤♦✐❝❡ ♦❢ ❛♥t✐❞❡r✐✈❛t✐✈❡ ✕ t❤❡ ♦♥❡ t❤❛t s❛t✐s✜❡s F (a) = 0✳ ❚❤❡ r❡st✱ ❛❝❝♦r❞✐♥❣ t♦ P❛rt ■■✱ ❛r❡ ❛❝q✉✐r❡❞ ❜② ✈❡rt✐❝❛❧ s❤✐❢ts✿ G = F + C ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✷✱ ❈❤❛♣t❡r ✷❉❈✲✹✮✳ ❚❤❡ ✐❞❡❛ ✐s t❤❛t ✐❢ t❤❡ ❝❡✐❧✐♥❣ ❛♥❞ t❤❡ ✢♦♦r ♦❢ ❛ t✉♥♥❡❧ ❛r❡ ❡q✉❛❧ ❛t ❡✈❡r② ♣♦✐♥t✱ ✐ts ❤❡✐❣❤t ✐s ❝♦♥st❛♥t✿

❘❡❝❛❧❧ t❤❡

s✉❜st✐t✉t✐♦♥ ♥♦t❛t✐♦♥

✿

F (x)

= F (a) . x=a

■♥ ✐ts s♣✐r✐t✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇ ♥♦t❛t✐♦♥ ❢♦r t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❢♦r♠✉❧❛✳ ■t ✐s s✐♠♣❧② t✇♦ s✉❜st✐t✉t✐♦♥s s✉❜tr❛❝t❡❞✿

❙✉❜st✐t✉t✐♦♥ ♥♦t❛t✐♦♥ ❢♦r ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ Z

b a

x=b b = F (b) − F (a) f dx = F (x) = F (x) x=a

a

❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡ ❜♦✉♥❞s ♦❢ ✐♥t❡❣r❛t✐♦♥ ❥✉st ❥✉♠♣ ♦✈❡r ❛♥❞ ❛r❡ ❦❡♣t ❢♦r r❡❢❡r❡♥❝❡✳ ❚❤✐s ✐s ❤♦✇ ✇❡ ❝❛♥ r❡❝♦r❞ t❤❡ ❛❜♦✈❡ ❝♦♠♣✉t❛t✐♦♥✿ 1 Z 1 3 x x2 dx = . 3 0 0

❈♦♠♣✉t❛t✐♦♥s ♦❢ ❛r❡❛s ❜❡❝♦♠❡ ❡❛s② ✕ ❛s ❧♦♥❣ ❛s ✇❡ ❤❛✈❡ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐♥✈♦❧✈❡❞✳

❊①❛♠♣❧❡ ✶✳✽✳✶✺✿ ❡①♣♦♥❡♥t ❈♦♠♣✉t❡ t❤❡ ❛r❡❛ ✉♥❞❡r y = ex ❢r♦♠ 0 t♦ 1✿

❇②

✱ ✇❡ ❤❛✈❡✿

◆▲❋

Z

1 0

1 x x e dx = e = e1 − e0 = e − 1 . 0

✶✳✽✳

✽✷

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❝♦♥t✐♥✉❡❞

❊①❛♠♣❧❡ ✶✳✽✳✶✻✿ ❝♦s✐♥❡

❈♦♠♣✉t❡ t❤❡ ❛r❡❛ ✉♥❞❡r y = cos x ❢r♦♠ 0 t♦ π/2✿

❇②

✱ ✇❡ ❤❛✈❡✿

◆▲❋

Z ❊①❛♠♣❧❡ ✶✳✽✳✶✼✿ ❝✐r❝❧❡

π 2

0

π 2 π cos x dx = sin x = sin − sin 0 = 1 − 0 = 1 . 2 0

❲❤❛t ✐s t❤❡ ❛r❡❛ ♦❢ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s R❄ ❊✈❡♥ t❤♦✉❣❤ ❡✈❡r②♦♥❡ ❦♥♦✇s t❤❡ ❛♥s✇❡r✱ t❤✐s ✐s t❤❡ t✐♠❡ t♦ t❤❡ ❢♦r♠✉❧❛✳ ❲❡ ❛❧s♦ ❣❛✈❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ ❛♥s✇❡r ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤✐s ❝❤❛♣t❡r✳

♣r♦✈❡

❲❡ ✜rst ♣✉t t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝✐r❝❧❡ ❛t t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ xy ✲♣❧❛♥❡✿

■♥ ♦r❞❡r t♦ ✜♥❞ ✐ts ❛r❡❛✱ ✇❡ r❡♣r❡s❡♥t t❤❡

f (x) =

✉♣♣❡r ❤❛❧❢

√

♦❢ t❤❡ ❝✐r❝❧❡ ❛s t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢

R2 − x2 , −R ≤ x ≤ R .

❚❤❡♥✱

1 ❆r❡❛ = 2

Z

R −R

√

R2 − x2 dx .

❲❤❛t✬s t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄ ❯♥❢♦rt✉♥❛t❡❧②✱ ✇❡ ❞♦♥✬t ❦♥♦✇❀ ✐t✬s ♥♦t ♦♥ ♦✉r ✕ ✈❡r② s❤♦rt ✕ ❧✐st✳ ❲❤❡♥ t❤✐s ❤❛♣♣❡♥s✱ ✇❡ ❛❧✇❛②s ❞♦ t❤❡ s❛♠❡ t❤✐♥❣✿ ❲❡ ❣♦ t♦ t❤❡ ❜❛❝❦ ♦❢ t❤❡ ❜♦♦❦ t♦ ✜♥❞ ❛ ❧♦♥❣❡r ❧✐st✳ ❚❤✐s t❛❜❧❡ ❝♦♥t❛✐♥s t❤❡ r❡❧❡✈❛♥t ❢♦r♠✉❧❛✿ Z √ u√ 2 a2 u a2 − u2 dx = a − u2 + sin−1 + C . 2 2 a ◆♦✇✱ ❡✈❡♥ t❤♦✉❣❤ ✇❡ ❣❡t t❤❡ ✐♥t❡❣r❛t✐♦♥ ❢♦r♠✉❧❛ ❢r♦♠ ❡❧s❡✇❤❡r❡✱ ♦♥❝❡ ✇❡ ❤❛✈❡ ✐t✱ ✇❡ ❝❛♥ ❝❡rt❛✐♥❧② ♣r♦✈❡ ✐t ✕ ❜② ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❲❡ ❦♥♦✇ t❤❡s❡ t✇♦ ❞❡r✐✈❛t✐✈❡s ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✹✿

√

a2 − u2

′

= −√

u ❛♥❞ a2 − u2

sin−1 y

′

1 =p . 1 − y2

✶✳✽✳ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❝♦♥t✐♥✉❡❞

✽✸

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿



a2 u√ 2 u a − u2 + sin−1 2 2 a

′

1√ 2 a − u2 − 2 1√ 2 a − u2 + = 2 √ = a2 − u 2 . =

u a2 u 1 √ + √ 2 2 2 2 a −u 2 a − u2 1 a2 − u2 √ 2 a2 − u2

❈♦♥✜r♠❡❞✦ ❲❡ r❡♣❧❛❝❡

a

✇✐t❤

1 ❆r❡❛ 2

R

=

❛♥❞

Z

R −R

u

√

✇✐t❤

x

✐♥ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛ ❛♥❞ ❛♣♣❧② t❤❡ ◆❡✇t♦♥✲▲❡✐❜♥✐③ ❢♦r♠✉❧❛ ✿

R2 − x2 dx

R 2 x x√ 2 R sin−1 = R − x2 + 2 2 R −R     2 2 √ √ R R R R −R −R −1 −1 R2 − R2 + R2 − R2 + = − sin sin 2 2 R 2 2 R 2  R sin−1 (1) + 0 − 0 − sin−1 (−1) = 2 R2 = (π/2 − (−π/2)) 2 R2 . =π 2

❲❡ ❤❛✈❡ ❝♦♥✜r♠❡❞ t❤❡ ❢♦r♠✉❧❛✦ ❈♦♥✈❡rs❡❧②✱ ✇❡ ❝❛♥ ✉s❡ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❛r❡❛ ♦❢ t❤❡ ✉♥✐t ❝✐r❝❧❡ ❛s ❛ ✇❛② t♦ ✜♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ t❤❡ ✈❛❧✉❡ ♦❢

π✳

❲❤❛t ✐❢ ✇❡ ❝❛♥✬t ✜♥❞ ❛♥ ✐♥t❡❣r❛t✐♦♥ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ✐♥ q✉❡st✐♦♥❄ ❚❤❡r❡ ❛r❡ ❛❧✇❛②s ❧❛r❣❡r ❧✐sts✱ ❜✉t ✐t✬s ♣♦ss✐❜❧❡ t❤❛t ♥♦ ❜♦♦❦ ❝♦♥t❛✐♥s t❤❡ ❢✉♥❝t✐♦♥ ②♦✉ ♥❡❡❞✳ ■♥ ❢❛❝t✱ ✇❡ ♠❛② ❤❛✈❡ t♦ ✐♥tr♦❞✉❝❡ ♥❡✇ ❢✉♥❝t✐♦♥s s✉❝❤ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ●❛✉ss ❡rr♦r ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛s t❤❡ ✐♥t❡❣r❛❧✿

2 erf(x) = √ π

Z

2

e−x dx .

❚❤✐s ❢✉♥❝t✐♦♥ ✐s ✐♠♣♦rt❛♥t ❛s ✐t r❡♣r❡s❡♥ts ❛ ❞✐str✐❜✉t✐♦♥ ♦❢ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ s♦♠❡ ❝♦♠♠♦♥ q✉❛♥t✐t✐❡s ✭❤❡✐❣❤ts✱ ✇❡✐❣❤ts✱ ■◗s✱ ❡t❝✳✮ ❛r♦✉♥❞ t❤❡ ❛✈❡r❛❣❡✿

❊✈❡♥ ✐❢ ✇❡ ❦❡❡♣ ❛❞❞✐♥❣ t❤❡s❡ ♥❡✇ ❢✉♥❝t✐♦♥s t♦ t❤❡ ❧✐st ♦❢ ✏❢❛♠✐❧✐❛r✑ ❢✉♥❝t✐♦♥s✱ t❤❡r❡ ✇✐❧❧ ❛❧✇❛②s r❡♠❛✐♥ ✐♥t❡❣r❛❧s ♥♦t ♦♥ t❤❡ ❧✐st✳ ■♥ ❝♦♥tr❛st t♦ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ✇✐❧❧ ♦❢t❡♥ t❛❦❡ ✉s ♦✉ts✐❞❡ ♦❢ t❤❡ r❡❛❧♠ ♦❢ ❡❧❡♠❡♥t❛r② ❢✉♥❝t✐♦♥s ✭r✐❣❤t✮✿

✶✳✾✳

❇❛s✐❝ ✐♥t❡❣r❛t✐♦♥

✽✹

✶✳✾✳ ❇❛s✐❝ ✐♥t❡❣r❛t✐♦♥

❘❡❝❛❧❧ t❤❡ ♣r♦❝❡❞✉r❡ ❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ ❞❡r✐✈❛t✐✈❡✿ s❛♠♣❧✐♥❣

s❡q✉❡♥❝❡ ←−−−−−−−− ❢✉♥❝t✐♦♥ ♦♥ [a, b]    ❉◗ y

 ❞❡r✐✈❛t✐✈❡ y

s❡q✉❡♥❝❡ −−−∆x→0 −−−−−→ ❢✉♥❝t✐♦♥ ♦♥ [a, b]

❚❤❡ ♦♥❡ ❢♦r t❤❡ ✐♥t❡❣r❛❧ ❤❛s ❛ ✈❡r② s✐♠✐❧❛r r❡♣r❡s❡♥t❛t✐♦♥✿ s❛♠♣❧✐♥❣

s❡q✉❡♥❝❡ ←−−−−−−−− ❢✉♥❝t✐♦♥ ♦♥ [a, b]    ❘❙ y

 ✐♥t❡❣r❛❧ y

s❡q✉❡♥❝❡ −−−∆x→0 −−−−−→ ❢✉♥❝t✐♦♥ ♦♥ [a, b]

❇❡❝❛✉s❡ ♦❢ t❤❡ r❡❝✉rs✐✈❡ ♥❛t✉r❡ ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✱ ❤♦✇❡✈❡r✱ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ❜② ❢❛r ♠♦r❡ ❝♦♠♣❧❡① ♦r ❡✈❡♥ ✐♠♣♦ss✐❜❧❡✳ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ✇✐❧❧ ❣✐✈❡ t❤❡ ❛♥s✇❡r ✐❢ ✇❡ ❝❛♥ ❡①❡❝✉t❡ t❤❡ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ r❡q✉✐r❡❞✳ ❊①❛♠♣❧❡ ✶✳✾✳✶✿

x

❋✐♥❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢ f (x) = x .

❲❡ r✉♥ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ✜rst✿

❚❤❡ ❛♥s✇❡r ❛♣♣❡❛r t♦ ❜❡ q✉❛❞r❛t✐❝✳ ❇✉t (ax2 + bx + c)′ = 2ax + b .

✶✳✾✳

❇❛s✐❝ ✐♥t❡❣r❛t✐♦♥

■t ✇♦r❦s ✐❢ ✇❡ ❝❤♦♦s❡✿

❊①❛♠♣❧❡ ✶✳✾✳✷✿

✽✺

a = 1/2

❛♥❞

b = 0✳ ❲❡ ❝♦♥❝❧✉❞❡✿ Z 1 x dx = x2 + C . 2

x2

❋✐♥❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢

f (x) = x2 . ❲❡ r✉♥ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ✜rst✿

▼✐❣❤t ✐t ❜❡ ❝✉❜✐❝❄ ▲❡t✬s tr②✿

(ax3 + bx2 + cx + d)′ = 3ax2 + 2bx + c . ■t ✇♦r❦s ✐❢ ✇❡ ❝❤♦♦s❡✿

a = 1/3✱ b = 0✱

❛♥❞

Z

c = 0✳

❲❡ ❝♦♥❝❧✉❞❡✿

1 x2 dx = x3 + C . 3

❖❢ ❝♦✉rs❡✱ ✇❡ ❝❛♥ ❥✉st ✏r❡✈❡rs❡✑ t❤❡ P♦✇❡r ❋♦r♠✉❧❛✿

Z ❊①❛♠♣❧❡ ✶✳✾✳✸✿

sin x

❛♥❞

xn dx =

1 xn+1 + C n+1

cos x

❋✐♥❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢

f (x) = sin x . ❲❡ r✉♥ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ✜rst✿

■t ❧♦♦❦s ❧✐❦❡

cos x + 1✳

■♥❞❡❡❞✿

(cos x)′ = sin x .

✶✳✾✳

❇❛s✐❝ ✐♥t❡❣r❛t✐♦♥

✽✻

❋✐♥❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢

f (x) = cos x . ❲❡ r✉♥ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ✜rst✿

■t ❧♦♦❦s ❧✐❦❡ t❤❡ ♥❡❣❛t✐✈❡ s✐♥❡✳ ■♥❞❡❡❞✿

(sin x)′ = cos x . ❖❢ ❝♦✉rs❡✱ ✇❡ ❝❛♥ ❥✉st ✏r❡✈❡rs❡✑ ❛ ❝♦✉♣❧❡ ♦❢ ❢❛♠✐❧✐❛r ❢♦r♠✉❧❛s✿

Z ❊①❛♠♣❧❡ ✶✳✾✳✹✿

sin x dx = cos x + C

Z

cos x dx = − sin x + C

ex

❋✐♥❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢

f (x) = ex . ❲❡ r✉♥ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ✜rst✿

■t ❛❧s♦ ❧♦♦❦s ❡①♣♦♥❡♥t✐❛❧✳ ❲❡ ❝♦♥❝❧✉❞❡✿

Z

ex dx = ex + C .

❲❡ ✏r❡✈❡rs❡✑ ❛ ❢❛♠✐❧✐❛r ❢♦r♠✉❧❛ ❢♦r ❞✐✛❡r❡♥t✐❛t✐♦♥✿

Z

ex dx = ex + C

▲❡t✬s r❡✈✐❡✇ t❤❡ ♠❛✐♥ ✐❞❡❛s ♦❢ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✳

✶✳✾✳

❇❛s✐❝ ✐♥t❡❣r❛t✐♦♥

✽✼

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ y = g(x) ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✱ c✱ ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❍♦✇ ❞♦ ✇❡ ✜♥❞ ❛ ❢✉♥❝t✐♦♥ y = f (x) ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s✱ x✱ ♦❢ t❤❡ ♣❛rt✐t✐♦♥ s♦ t❤❛t g ✐s ✐ts ❞✐✛❡r❡♥❝❡ ✿

∆f (c) = g(c)? ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❢❛❝❡ ❛♥ ❡q✉❛t✐♦♥✿ ❙♦❧✈❡ ❢♦r f : ∆f = g ❙♦❧✈✐♥❣ t❤✐s ❡q✉❛t✐♦♥ ✐s♥✬t ❤❛r❞✳ ❙✉♣♣♦s❡ t❤✐s ❢✉♥❝t✐♦♥ g ✐s ❦♥♦✇♥ ❜✉t f ✐s♥✬t✱ ❡①❝❡♣t ❢♦r ♦♥❡ ✭✐♥✐t✐❛❧✮ ✈❛❧✉❡✿ y0 = f (a) = f (x0 )✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿

f (xk+1 ) = f (xk ) + g(ck ) ❚❤✐s ❢♦r♠✉❧❛ ✐s

r❡❝✉rs✐✈❡ ✿ ❲❡ ♥❡❡❞ t♦ ❦♥♦✇ t❤❡ ❧❛st ✈❛❧✉❡ ♦❢ f ✐♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ♥❡①t✳

◆♦✇✱ t❤❡ ❞✐✛❡r❡♥❝❡ s✉❜tr❛❝t✐♦♥✿

q✉♦t✐❡♥t✳ ■♥ ❝♦♠♣❛r✐s♦♥✱ t❤❡r❡ ✐s ❛♥♦t❤❡r ♦♣❡r❛t✐♦♥ ✕ ❞✐✈✐s✐♦♥ ✭❜② ∆x✮ ✕ ❢♦❧❧♦✇✐♥❣ t❤❡

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ y = v(x) ✭✇❤✐❝❤ ❝♦✉❧❞ ❜❡ t❤❡ ✈❡❧♦❝✐t②✮ ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✱ c✱ ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❍♦✇ ❞♦ ✇❡ ✜♥❞ ❛ ❢✉♥❝t✐♦♥ y = p(x) ✭✇❤✐❝❤ ❝♦✉❧❞ ❜❡ t❤❡ ♣♦s✐t✐♦♥✮ ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s✱ x✱ ♦❢ t❤❡ ♣❛rt✐t✐♦♥ s♦ t❤❛t v ✐s ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✿

∆p (c) = v(c)? ∆x ■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s t❤❡ ❡q✉❛t✐♦♥ ✇❡ ❢❛❝❡✿ ❙♦❧✈❡ ❢♦r p :

∆p =v ∆x

❲❡ ❥✉st ❢♦❧❧♦✇ ❡①❛❝t❧② t❤❡ ♣r♦❝❡ss ❛❜♦✈❡✳ ❙✉♣♣♦s❡ t❤✐s ❢✉♥❝t✐♦♥ v ✐s ❦♥♦✇♥ ❜✉t p ✐s♥✬t✱ ❡①❝❡♣t ❢♦r ♦♥❡ ✭✐♥✐t✐❛❧✮ ✈❛❧✉❡✿ y0 = p(a) = p(x0 )✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿

p(xk+1 ) = p(xk ) + v(ck )∆xk ◆♦✇✱ t❤❡

❞❡r✐✈❛t✐✈❡✳ ❲❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ✐ts ❞❡r✐✈❛t✐✈❡✿

✶✳✾✳

❇❛s✐❝ ✐♥t❡❣r❛t✐♦♥

✽✽

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ ✜♥❞ ❛ ❢✉♥❝t✐♦♥ ❡❛❝❤

y = p(x)

y = v(x)

✭✇❤✐❝❤ ❝♦✉❧❞ ❜❡ t❤❡ ✈❡❧♦❝✐t②✮ ❞❡✜♥❡❞ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧✳ ❍♦✇ ❞♦ ✇❡

✭✇❤✐❝❤ ❝♦✉❧❞ ❜❡ t❤❡ ♣♦s✐t✐♦♥✮ ❞❡✜♥❡❞ ♦♥ t❤✐s ✐♥t❡r✈❛❧ s♦ t❤❛t

x✿

v

✐s ✐ts ❞❡r✐✈❛t✐✈❡ ❢♦r

dp (x) = v(x)? dx

■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ✐s t❤❡ ❡q✉❛t✐♦♥ ✇❡ ❢❛❝❡✿

❙♦❧✈❡ ❢♦r

dp =v dx

p:

❚❤✐s ❡q✉❛t✐♦♥ ✐s♥✬t ❛s ❡❛s② t♦ s♦❧✈❡ ❛s t❤❡ ❧❛st✳ ❚❤❡ s♦❧✉t✐♦♥ ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✇❛s ❜✉✐❧t ❜② ❧✐♥❦✐♥❣

n

♣✐❡❝❡s t♦❣❡t❤❡r✳ ❇❡❝❛✉s❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ❛ ❧✐♠✐t✱ ✐t✬s ❛s ✐❢ ✇❡ ✇❡r❡ t♦ ❧✐♥❦ t♦❣❡t❤❡r ✐♥✜♥✐t❡❧② ♠❛♥② ✐♥✜♥✐t❡❧② s♠❛❧❧ ♣✐❡❝❡s✦ ❚❤❡r❡ ✐s ♥♦ s✐♠♣❧❡✱ ❡✈❡♥ r❡❝✉rs✐✈❡✱ ❢♦r♠✉❧❛✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ❝❛♥♥♦t ❜❡ ✏♠❛❞❡✑ ❢r♦♠ t❤❡ ❢✉♥❝t✐♦♥s ✇❡ ❛r❡ ❢❛♠✐❧✐❛r ✇✐t❤✿

dp 2 = ex . dx ❚❤❡s❡ t❤r❡❡ ♣r♦❜❧❡♠s ❛r❡ s✐♠✐❧❛r t♦ t❤❛t ♦❢ ✜♥❞✐♥❣ t❤❡

✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✳

❛♣♣❡❛r ✐♥ ❛❧❣❡❜r❛❀ t❤❡② ❝♦♠❡ ❢r♦♠ s♦❧✈✐♥❣ ❡q✉❛t✐♦♥s✱ ❢♦r

❚❤✐s ✐s ❤♦✇ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥s

x✿

x2

= 4 =⇒ x = 2

✈✐❛

√

2x

= 8 =⇒ x = 3

✈✐❛

log2 ( )

sin x = 0 =⇒ x = 0

✈✐❛

sin−1 ( )

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❛t ❞♦ ✇❡ ❞♦ ✐❢ ✇❡ ❦♥♦✇ t❤❡ ♦✉t♣✉t ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ ✇❛♥t t♦ ❦♥♦✇ t❤❡ ✐♥♣✉t❄ ■♥✐t✐❛❧❧②✱ ✇❡ ❝❛♥ ♦♥❧②

r❡❝♦❧❧❡❝t

❛ ♣❛st ❡①♣❡r✐❡♥❝❡ ✇✐t❤ ❛ ❢✉♥❝t✐♦♥✦ ❋♦r ❛ r❡♣❡❛t❡❞ ✉s❡✱ ✇❡ ❞❡✈❡❧♦♣ t❤❡

✐♥✈❡rs❡

♦❢ t❤❡

❢✉♥❝t✐♦♥✳ ❙✐♠✐❧❛r❧②✱ ✇❤❛t ❞♦ ✇❡ ❞♦ ✐❢ ✇❡ ❦♥♦✇ t❤❡ r❡s✉❧t ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ✇❛♥t t♦ ❦♥♦✇ ✇❤❡r❡ ✐t ❝❛♠❡ ❢r♦♠❄ ■♥✐t✐❛❧❧②✱ ✇❡ ❝❛♥ ♦♥❧②

r❡❝♦❧❧❡❝t

❛ ♣❛st ❡①♣❡r✐❡♥❝❡ ✇✐t❤ ❞✐✛❡r❡♥t✐❛t✐♦♥✿

f ′ = 2x

=⇒ f = x2

f ′ = cos x =⇒ f = sin x f ′ = ex ❋♦r ❛ r❡♣❡❛t❡❞ ✉s❡✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t♦ ❞❡✈❡❧♦♣ t❤❡

=⇒ f = ex

✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳

❲❡ ✐❧❧✉str❛t❡ t❤❡ ✐❞❡❛ ♦❢ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ✇✐t❤ ❛ ❞✐❛❣r❛♠✿

x2 → 2x →

d dx  −1 d dx

→ 2x → x2

✶✳✾✳

❇❛s✐❝ ✐♥t❡❣r❛t✐♦♥

✽✾

❖❢ ❝♦✉rs❡✱ t❤❡r❡ ❛r❡ ♠♦r❡ s♦❧✉t✐♦♥s✿

x2 + 1 ր ...

2x → x2 ց ...

x2 − 1

❆s ❛ ❢✉♥❝t✐♦♥✱ ❛ ❢✉♥❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s✱

d ✐s♥✬t ♦♥❡✲t♦✲♦♥❡✳ ❲❡ ✉s❡ ✏❛♥✑ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ♠❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡s dx

❢♦r ❡❛❝❤ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ✐s t❤❡ ❝✉❧♠✐♥❛t✐♦♥ ♦❢ ♦✉r st✉❞② ♦❢ ❝❛❧❝✉❧✉s✱ s♦ ❢❛r✳ ❚❤❡ ♠✐❧❡st♦♥❡s ♦❢ t❤✐s st✉❞② ✉♣ t♦ t❤✐s ♣♦✐♥t ❛r❡ ♦✉t❧✐♥❡❞ ❜❡❧♦✇✿ ❚❤❡ ❚❛♥❣❡♥t Pr♦❜❧❡♠

❚❤❡ ❆r❡❛ Pr♦❜❧❡♠

❆♣♣r♦①✐♠❛t✐♦♥s✿ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

∆F ∆x

t❤❡ ❘✐❡♠❛♥♥ s✉♠

Σf · ∆x

t❤❡ s❧♦♣❡s ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡s ♦♥ t❤❡ ❣r❛♣❤ ♦❢

y = F (x)

t❤❡ t♦t❛❧ ❛r❡❛ ♦❢ t❤❡ r❡❝t❛♥❣❧❡s ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢

❲✐t❤ ✈❛r✐❛❜❧❡ ❧♦❝❛t✐♦♥s✱ t❤❡ ❧✐♠✐ts ♦❢ t❤❡s❡ ❛♣♣r♦①✐♠❛t✐♦♥s✱ ❛s

t❤❡ ❞❡r✐✈❛t✐✈❡ ❢✉♥❝t✐♦♥

∆x → 0✱

y = f (x)

❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

t❤❡ ✐♥❞❡✜♥✐t❡ ✐♥t❡❣r❛❧

✶✳✶✵✳ ❋r❡❡ ❢❛❧❧

✾✵

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠s✱ t❤❡ ♦♣❡r❛t✐♦♥s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ✐♥t❡❣r❛t✐♦♥ ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✿ ❋❚❈■✿ f → ❋❚❈■■✿ F →

Z

 dx d  dx

d  dx

→ F → → f

→

Z

 dx

→

f

→ F +C

❲❛r♥✐♥❣✦ ■♥t❡❣r❛t✐♦♥ ✐s ❛ tr✉❡ ❢✉♥❝t✐♦♥ ♦♥❧② ✇❤❡♥ ❛♥ ❡①tr❛ ❝♦♥❞✐t✐♦♥✱ s✉❝❤ ❛s

F (a) = 0✱

✐s ✐♠♣♦s❡❞✳

✶✳✶✵✳ ❋r❡❡ ❢❛❧❧

❊①❛♠♣❧❡ ✶✳✶✵✳✶✿ ♠♦✈✐♥❣ ❜❛❧❧

▲❡t✬s r❡✈✐❡✇ ❛♥ ❡①❛♠♣❧❡ ❢r♦♠ ❱♦❧✉♠❡ ✷✳ ❚❤❡ ✈❡❧♦❝✐t② ♦❢ ❛ ❜❛❧❧ t❤r♦✇♥ ✉♣ ✐♥ t❤❡ ❛✐r ✐s ❝♦♥st❛♥t❧② ❝❤❛♥❣❡❞ ❜② t❤❡ ❣r❛✈✐t②✳ ■♠❛❣✐♥❡ t❤❛t ✇❡ ❤❛✈❡ t❤✐s ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛ ♦❢ t❤❡ ❤❡✐❣❤ts ♦❢ ❛ ♣✐♥❣✲♣♦♥❣ ❜❛❧❧ ❢❛❧❧✐♥❣ ❞♦✇♥ r❡❝♦r❞❡❞ ❛❜♦✉t ❡✈❡r② 0.1 s❡❝♦♥❞✱ ♠❡❛s✉r❡❞ ✐♥ ✐♥❝❤❡s✿

❲❡ ♣❧♦t t❤❡ ❧♦❝❛t✐♦♥ s❡q✉❡♥❝❡✱ pn ✭r❡❞✮✳ ❲❡ t❤❡♥ ❝♦♠♣✉t❡ t❤❡ t❤❡ ✈❡❧♦❝✐t②✱ vn ✭❣r❡❡♥✮✿

❲❡ ❝♦♠♣✉t❡ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ an ✭❜❧✉❡✮✱ t♦♦✳ ■t ❛♣♣❡❛rs ❝♦♥st❛♥t✳ ▲❡t✬s ❛❝❝❡♣t t❤❡ ♣r❡♠✐s❡ ✇❡✬✈❡ ♣✉t ❢♦r✇❛r❞✿

✶✳✶✵✳

✾✶

❋r❡❡ ❢❛❧❧

◮

✳

❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ ❢r❡❡ ❢❛❧❧ ✐s ❝♦♥st❛♥t

❚❤❡♥ ✇❡ ❝❛♥ tr② t♦ ♣r❡❞✐❝t t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❛♥ ♦❜❥❡❝t t❤r♦✇♥ ✐♥ t❤❡ ❛✐r ✕ ❢r♦♠ ❛♥② ✐♥✐t✐❛❧ ❤❡✐❣❤t ❛♥❞ ✇✐t❤ ❛♥② ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✿

◮ ❲❡ ✉s❡ ♦✉r ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ t♦ ❞❡r✐✈❡ t❤❡ ✈❡❧♦❝✐t②✱ ❛♥❞ t❤❡♥ ❞❡r✐✈❡ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ♦❜❥❡❝t ✐♥ t✐♠❡✳ ❲❡ ♣❧♦t t❤❡ ♣♦s✐t✐♦♥ ❛❣❛✐♥st t✐♠❡✿

❲❡ ✉s❡❞ t❤❡s❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢♦r♠✉❧❛s t♦ ✜♥❞ t❤❡ ✈❡❧♦❝✐t② ❢r♦♠ t❤❡ ♣♦s✐t✐♦♥ ❛♥❞ t❤❡♥ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✿

∆v pn+1 − pn vn+1 − vn ∆p ❉◗ −−−−−→ an = = = ∆t h ∆t h ❍❡r❡ h ✐s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦♥ t❤❡ ♣♦s✐t✐♦♥ ❛♥❞ ♦❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦♥ t❤❡ ✈❡❧♦❝✐t② ✐s✱ ♦❢ ❝♦✉rs❡✱ ✐❞❡♥t✐❝❛❧✳

❉◗

pn −−−−−→ vn =

❚♦ ❝r❡❛t❡ ❢♦r♠✉❧❛s ❢♦r ❛ s✐♠✉❧❛t✐♦♥ ♦❢ ❢r❡❡ ❢❛❧❧✱ t❤❡ ❞❡r✐✈❛t✐♦♥ ❣♦❡s ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✿

• t❤❡ ✈❡❧♦❝✐t② ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ ❛♥❞ t❤❡♥ • t❤❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳

❚❤❡ t✇♦ ❢♦r♠✉❧❛s ❛❜♦✈❡ ❛r❡ s♦❧✈❡❞ ❛s ❡q✉❛t✐♦♥s ❢♦r pn+1 ❛♥❞ vn+1 r❡s♣❡❝t✐✈❡❧②✿

pn+1 − pn =⇒ pn+1 = pn + hvn h vn+1 − vn an = =⇒ vn+1 = vn + han h ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦♥ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❛♥❞ ♦❢ t❤❡ ♣♦s✐t✐♦♥ ♦♥ t❤❡ ✈❡❧♦❝✐t② ✐s✱ ♦❢ ❝♦✉rs❡✱ ✐❞❡♥t✐❝❛❧✳ vn =

❲❛r♥✐♥❣✦ ❯♥❧✐❦❡ t❤❡ ❢♦r♠❡r✱ t❤❡s❡ ❛r❡

r❡❝✉rs✐✈❡

s❡q✉❡♥❝❡s✳

❊①❛♠♣❧❡ ✶✳✶✵✳✷✿ ❢r❡❡ ❢❛❧❧

◮ Pr♦❜❧❡♠✿ ❋r♦♠ ❛ 100✲❢♦♦t ❜✉✐❧❞✐♥❣✱ ❛ ❜❛❧❧ ✐s t❤r♦✇♥ ✉♣ ❛t 50 ❢❡❡t ♣❡r s❡❝♦♥❞ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ✐t ❢❛❧❧s ♦♥ t❤❡ ❣r♦✉♥❞✳ ❍♦✇ ❤✐❣❤ ✇✐❧❧ t❤❡ ❜❛❧❧ ❣♦❄ ❲❡ ✉s❡ t❤❡ s❛♠❡ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛ ❢♦r t❤❡ ✈❡❧♦❝✐t② ❛♥❞ ♣♦s✐t✐♦♥✿

❂❘❬✲✶❪❈✰❘❬✲✶❪❈❬✲✶❪✯❘✷❈✶ ◆♦✇ ✐♥ t❤❡ s♣❡❝✐✜❝ ❝❛s❡ ♦❢ ❢r❡❡ ❢❛❧❧✱ t❤❡r❡ ✐s ❥✉st ♦♥❡ ❢♦r❝❡✱ t❤❡ ❣r❛✈✐t②✱ ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❦♥♦✇♥ ✭❢r♦♠ ❛ ♣❤②s✐❝s t❡①t❜♦♦❦✮ t♦ ❜❡ a = −g ✱ ✇❤❡r❡ g ✐s t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❝♦♥st❛♥t✿

g = 32 ❢t/s❡❝2 .

✶✳✶✵✳ ❋r❡❡ ❢❛❧❧

✾✷

◆❡①t✱ ✇❡ ❛❝q✉✐r❡ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ • ❚❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿ p0 = 100✳ • ❚❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② ✐s ❣✐✈❡♥ ❜②✿ v0 = 50✳ ❲❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛s t♦ ❡✈❛❧✉❛t❡ t❤❡ ❧♦❝❛t✐♦♥ ❡✈❡r② h = 0.20 s❡❝♦♥❞✳ ❚❤✐s ✐s ✇❤❛t t❤❡ ❣r❛♣❤s ❧♦♦❦ ❧✐❦❡✿

❇② s✐♠♣❧② ❡①❛♠✐♥✐♥❣ t❤❡ ❞❛t❛✱ ✇❡ ❝❛♥ s♦❧✈❡ ✈❛r✐♦✉s ♣r♦❜❧❡♠s ❛❜♦✉t t❤✐s ❡①♣❡r✐♠❡♥t✿ ✶✳ ❚♦ ✜♥❞ t❤❡ ❤✐❣❤❡st ❡❧❡✈❛t✐♦♥✱ ✇❡ ❧♦♦❦ ❛t t❤❡ r♦✇ ✇✐t❤ p✳ ❚❤❡ ❧❛r❣❡st ✈❛❧✉❡ s❡❡♠s t♦ ❜❡ ❝❧♦s❡ t♦ y = 144 ❢❡❡t✳ ✷✳ ❚♦ ✜♥❞ ✇❤❡♥ t❤❡ ❜❛❧❧ ❤✐ts t❤❡ ❣r♦✉♥❞✱ ✇❡ s❝r♦❧❧ ❞♦✇♥ t♦ ✜♥❞ t❤❡ r♦✇ ✇✐t❤ p ❝❧♦s❡ t♦ 0✳ ■t ❤❛♣♣❡♥s s♦♠❡t✐♠❡ ❝❧♦s❡ t♦ t = 4.7 s❡❝♦♥❞s✳ ✸✳ ❚♦ ✜♥❞ ❤♦✇ ❢❛st t❤❡ ❜❛❧❧ ❤✐ts t❤❡ ❣r♦✉♥❞✱ ✇❡ s❝r♦❧❧ ❞♦✇♥ ❛❣❛✐♥ t♦ ✜♥❞ t❤❡ r♦✇ ✇✐t❤ p ❝❧♦s❡ t♦ 0 ❛♥❞ ❧♦♦❦ ✉♣ t❤❡ ✈❛❧✉❡ ♦❢ v ✳ ■t ✐s ❝❧♦s❡ t♦ v = 200 ❢❡❡t ♣❡r s❡❝♦♥❞✳ ❖❢ ❝♦✉rs❡✱ ✇❡ ❝❛♥ ❞❡❝r❡❛s❡ t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥t h = ∆x ❛♥❞ ❣❡t ♠♦r❡ ❛❝❝✉r❛t❡ ❛♥s✇❡rs✳ ❲✐t❤ ♦✉r ❢r❡❡ ❢❛❧❧ s♣r❡❛❞s❤❡❡t✱ ✇❡ ❝❛♥ ❛s❦ ❛♥❞ ❛♥s✇❡r ❛ ✈❛r✐❡t② ♦❢ ♦t❤❡r q✉❡st✐♦♥s ❛❜♦✉t s✉❝❤ ♠♦t✐♦♥ ✭❤♦✇ ❤❛r❞ ✐t ❤✐ts t❤❡ ❣r♦✉♥❞✱ ❡t❝✳✮✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ ♦♥❧② ❞♦ ♦♥❡ ❡①❛♠♣❧❡ ❛t ❛ t✐♠❡✦ ❚❤❡ ❝♦♥❝❧✉s✐♦♥s ✇❡ ❞r❛✇ ❛r❡ s♣❡❝✐✜❝ t♦ t❤❡s❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✳ ❚❤❡② ❛r❡ ❛❧s♦ s♣❡❝✐✜❝ t♦ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❣r❛✈✐t②✱ ❛♥❞ ✇♦✉❧❞ ❜❡ ❞✐✛❡r❡♥t ♦♥ ▼❛rs✳ ❆♥❞ s♦ ♦♥✳ ❚❤❡ r❡s✉❧ts ❛r❡ ❛❧s♦ ❞❡♣❡♥❞❡♥t ♦♥ ♦✉r ❝❤♦✐❝❡ ♦❢ t❤❡ ✐♥❝r❡♠❡♥t h = ∆x✳ ❚❤✐s ✐s ✇❤② ✇❡ ♥♦✇ ♣r♦❝❡❡❞ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡ ❛♥❞ t❛❦❡ t❤❡ ❧✐♠✐t✿ h = ∆x → 0 .

❚❤✐s t✐♠❡✱ ✐♥st❡❛❞ ♦❢ s❡q✉❡♥❝❡s✱ ✇❡ ❤❛✈❡ t❤❡s❡ ❢✉♥❝t✐♦♥s ♦❢ t✐♠❡ ✿ ✶✳ p ✐s t❤❡ ❤❡✐❣❤t✱ t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥✳ ✷✳ v ✐s t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t②✳ ✸✳ a ✐s t❤❡ ✈❡rt✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥✳ ❲❡ ❤❛✈❡ ✜rst✿

v = p′ , a = v ′ ,

✶✳✶✵✳

✾✸

❋r❡❡ ❢❛❧❧

❛♥❞✱ ❛❝❝♦r❞✐♥❣❧②✱ p=

Z

v dx, v =

Z

a dx .

◆♦✇ t❤❡ s♣❡❝✐✜❝ ❝❛s❡ ♦❢ ❢r❡❡ ❢❛❧❧✿ a = −g .

❲❡ ❦♥♦✇ t❤❛t✿ ✶✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ❧✐♥❡❛r✳ ✷✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ✐s ❝♦♥st❛♥t✳ ❆♥❞✱ ❝♦♥✈❡rs❡❧②✿ ✶✳ ❚❤❡ ♦♥❧② ❢✉♥❝t✐♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤✐❝❤ ✐s ❧✐♥❡❛r ✐s ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✳ ✷✳ ❚❤❡ ♦♥❧② ❢✉♥❝t✐♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤✐❝❤ ✐s ❝♦♥st❛♥t ✐s ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧✳ ❋r♦♠ t❤❡ ❧❛tt❡r t✇♦✱ ✇❡ ❞❡r✐✈❡✿ a = a(t) ✐s ❝♦♥st❛♥t =⇒ v = v(t) ✐s ❧✐♥❡❛r =⇒ p = p(t) ✐s q✉❛❞r❛t✐❝✳

▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ ❤❛✈❡✿ = At2 + Bt + C

1. p(t)

2. v(t) = p′ (t) = 2At + B 3. a(t) = v ′ (t) = 2A

❲❤❛t ♠❛❦❡s t❤❡s❡ ❝♦❡✣❝✐❡♥ts✱ A✱ B ✱ ❛♥❞ C ✱ s♣❡❝✐✜❝ ❛r❡ t❤❡ ✐♥✐t✐❛❧

❝♦♥❞✐t✐♦♥s

❛♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✿

✶✳ p0 ✐s t❤❡ ✐♥✐t✐❛❧ ❤❡✐❣❤t✱ p0 = p(0)✳ ✷✳ v0 ✐s t❤❡ ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ✈❡❧♦❝✐t②✱ v0 = v(0)✳ ✸✳ −g ✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ −g = a(t)✳

❲❡ ✉s❡ t❤❡ t❤✐r❞ ✐t❡♠✿

❲❡ ✉s❡ t❤❡ s❡❝♦♥❞ ✐t❡♠✿ ❲❡ ✉s❡ t❤❡ ✜rst ✐t❡♠✿

g 2A = −g =⇒ A = − . 2 2At + B

t=0

At2 + Bt + C

= B = v0 .

t=0

= C = p0 .

❚❤❡♥✱ ♦✉r ♠♦❞❡❧ ♦❢ ♠♦t✐♦♥ t❛❦❡s ✐ts ✜♥❛❧ ❢♦r♠✿

1 1. p(t) = p0 +v0 t − gt2 2 2. v(t) = v0 −gt 3. a(t) =

−g

❚❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ✇✐t❤ ❛❜s♦❧✉t❡ ❛❝❝✉r❛❝②✦ ❊①❛♠♣❧❡ ✶✳✶✵✳✸✿ ❢r❡❡ ❢❛❧❧

❋r♦♠ ❛ 100✲❢♦♦t ❜✉✐❧❞✐♥❣✱ ❛ ❜❛❧❧ ✐s t❤r♦✇♥ ✉♣ ❛t 50 ❢❡❡t ♣❡r s❡❝♦♥❞ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ✐t ❢❛❧❧s ♦♥ t❤❡ ❣r♦✉♥❞✳ ❍♦✇ ❤✐❣❤ ✇✐❧❧ t❤❡ ❜❛❧❧ ❣♦❄

◮

Pr♦❜❧❡♠✿

✶✳✶✵✳ ❋r❡❡ ❢❛❧❧

✾✹

❲❡ ❤❛✈❡✿ p0 = 100, v0 = 50 .

❖✉r ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s✿ p = 100 +50t −16t2 .

■♥ ❝♦♥tr❛st t♦ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✱ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ♣♦s✐t✐♦♥ ✐s♥✬t r❡❝✉rs✐✈❡ ❜✉t ❞✐r❡❝t ❛♥❞ ❡①♣❧✐❝✐t✦ ❇❡❢♦r❡ ✇❡ ✉t✐❧✐③❡ t❤❡ ❡①♣❧✐❝✐t ❛❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥✱ ✇❡ ✈✐s✉❛❧✐③❡ t❤❡ r❡s✉❧ts ❜② ♣❧♦tt✐♥❣ t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ♥❡①t t♦ t❤❡ ♦♥❡ ♦❜t❛✐♥❡❞ r❡❝✉rs✐✈❡❧②✿

❚❤❡ ❧❛tt❡r ✐s ❛ s❛♠♣❧✐♥❣ ♦❢ t❤❡ ❡①❛❝t s♦❧✉t✐♦♥✳ ❏✉st ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ✉s❡ t❤✐s ♣❧♦t t♦ s♦❧✈❡ ❛ ✈❛r✐❡t② ♦❢ ♣r♦❜❧❡♠s ❛❜♦✉t t❤✐s ♠♦t✐♦♥✳ ▲❡t✬s r❡✈✐s✐t t❤❡ t✇♦ ♣r♦❜❧❡♠s ❛❜♦✉t t❤✐s s♣❡❝✐✜❝ t❤r♦✇ ✇❡ s♦❧✈❡❞ ♥✉♠❡r✐❝❛❧❧②✳ ❚❤❡② ❛r❡ s♦❧✈❡❞ t❤❡ s❛♠❡ ✇❛②✿ ✶✳ ❚♦ ✜♥❞ t❤❡ ❤✐❣❤❡st ❡❧❡✈❛t✐♦♥✱ ✇❡ ❧♦♦❦ ❛t t❤❡ r♦✇ ✇✐t❤ p✳ ❚❤❡ ❧❛r❣❡st ✈❛❧✉❡ s❡❡♠s t♦ ❜❡ ❝❧♦s❡ t♦ y = 139 ❢❡❡t✳ ✷✳ ❚♦ ✜♥❞ ✇❤❡♥ t❤❡ ❜❛❧❧ ❤✐ts t❤❡ ❣r♦✉♥❞✱ ✇❡ s❝r♦❧❧ ❞♦✇♥ t♦ ✜♥❞ t❤❡ r♦✇ ✇✐t❤ p ❝❧♦s❡ t♦ 0✳ ■t ❤❛♣♣❡♥s s♦♠❡t✐♠❡ ❝❧♦s❡ t♦ t = 4.5 s❡❝♦♥❞s✳ ❚❤❡s❡ ❛r❡ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s ❛s t❤❡② ❝♦♠❡ ❢r♦♠ ❛ s❛♠♣❧❡❞ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❢♦r♠✉❧❛✱ ❤♦✇❡✈❡r✱ ❣✐✈❡s ✉s ❛ ✇❛② t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥s ✇✐t❤ ❛❜s♦❧✉t❡ ❛❝❝✉r❛❝②✳ ❲❡ ❝❛♥ ❡✈❡♥ ❛✈♦✐❞ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❋♦r t❤❡ ✜rst ♣r♦❜❧❡♠✱ ✇❡ r❡❛❧✐③❡ t❤❛t p = −16t2 + 50t + 100 ✐s ❛ ♣❛r❛❜♦❧❛✦ ❆♥❞ t❤❡ ✈❡rt❡① ♦❢ y = ax2 + bx + c ✐s ❛t x = −b/a ✭❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❤✐❣❤❡st ♣♦✐♥t ✐s r❡❛❝❤❡❞ ❛t t✐♠❡ t = −50/(−2 · 16) = 1.5625 .

❚❤❡♥✱ t❤❡ ❤✐❣❤❡st ❡❧❡✈❛t✐♦♥ ✐s

p = 100 + 50 · 1.5625 − 16 · 1.56252 = 139.0625 .

❚❤❡ r❡s✉❧t ♠❛t❝❤❡s ♦✉r ❡st✐♠❛t❡✦ ❋♦r t❤❡ s❡❝♦♥❞ ♣r♦❜❧❡♠✱ t❤❡ ❛❧t✐t✉❞❡ ❛t t❤❡ ❡♥❞ ✐s 0✱ s♦ t♦ ✜♥❞ ✇❤❡♥ ✐t ❤❛♣♣❡♥❡❞✱ ✇❡ s❡t p = 0✱ ♦r −16t2 + 50t + 100 = 0 ,

❛♥❞ s♦❧✈❡ ❢♦r t✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ ✭s❡❡♥ ✐♥ ❱♦❧✉♠❡ ✶✱ ❈❤❛♣t❡r ✶P❈✲✹✮✱ ✇❡ ❤❛✈❡✿ t=

❚❤❡ r❡s✉❧t ♠❛t❝❤❡s ♦✉r ❡st✐♠❛t❡✦

−50 −

p

502 − 4(−16)100 ≈ 4.5106 . 2 · (−16)

✶✳✶✵✳ ❋r❡❡ ❢❛❧❧

✾✺

❊①❡r❝✐s❡ ✶✳✶✵✳✹

❲❤❛t ❤❛♣♣❡♥❡❞ t♦ ±❄ ❊①❡r❝✐s❡ ✶✳✶✵✳✺

❍♦✇ ❤✐❣❤ ❞♦❡s t❤❡ ♣r♦❥❡❝t✐❧❡ ❣♦ ✐♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡❄ ❊①❡r❝✐s❡ ✶✳✶✵✳✻

❯s✐♥❣ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✱ ❤♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ ❢♦r t❤❡ ♣r♦❥❡❝t✐❧❡ t♦ r❡❛❝❤ t❤❡ ❣r♦✉♥❞ ✐❢ ✜r❡❞ ❞♦✇♥ ❄ ❊①❡r❝✐s❡ ✶✳✶✵✳✼

❯s❡ t❤❡ ❛❜♦✈❡ ♠♦❞❡❧ t♦ ❞❡t❡r♠✐♥❡ ❤♦✇ ❧♦♥❣ ✐t ✇✐❧❧ t❛❦❡ ❢♦r ❛♥ ♦❜❥❡❝t t♦ r❡❛❝❤ t❤❡ ❣r♦✉♥❞ ✐❢ ✐t ✐s ❞r♦♣♣❡❞✳ ▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ q✉❡st✐♦♥s ❛❜♦✉t t❤❡ s✐t✉❛t✐♦♥ ❛♥❞ ❛♥s✇❡r t❤❡♠✳ ❘❡♣❡❛t✳ ❲✐t❤ ♦✉r s✐♠♣❧❡ ♠♦❞❡❧ ♦❢ ♠♦t✐♦♥✱ ❛❧❧ ♣♦ss✐❜❧❡ s❝❡♥❛r✐♦s ❤❛✈❡ ❜❡❡♥ ❢♦✉♥❞ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛✦ ❲❡ ❝❛♥✬t ❡①♣❡❝t t♦ ❛✈♦✐❞ ❛♣♣r♦①✐♠❛t✐♦♥s t❤♦✉❣❤✿ ✶✳ ❚❤❡r❡ ♠❛② ❜❡ ♥♦ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r t❤❡ ✈❡❧♦❝✐t②✱ ❛♥❞ t❤❡ ♣♦s✐t✐♦♥✱ ✇❤❡♥ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛❝❝❡❧❡r✲ ❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✐s ❝♦♠♣❧❡① ❡♥♦✉❣❤✳ ❲❡ ❤❛✈❡ t♦ ❣♦ ❜❛❝❦ t♦ ♦✉r ❞✐s❝r❡t❡ ♠♦❞❡❧✳ ✷✳ ❊✈❡♥ ✇❤❡♥ t❤❡r❡ ❛r❡ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s ❢♦r t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ♣♦s✐t✐♦♥✱ t❤❡ ❡q✉❛t✐♦♥ ♠❛② ❤❛✈❡ ♥♦ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r t❤❡ s♦❧✉t✐♦♥ ✭t❤❡ t✐♠❡ ✐♥ ✢✐❣❤t✮ ✇❤❡♥ t❤❡ ❢♦r♠❡r ✐s t♦♦ ❝♦♠♣❧❡①✳ ❲❡ ❤❛✈❡ t♦ s❡❡❦ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥✳ ✸✳ ❊✈❡♥ ✐♥ ♦✉r ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r t❤❡ t✐♠❡ ✐♥ ✢✐❣❤t ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ t❤❡ sq✉❛r❡ r♦♦t ✇✐❧❧ st✐❧❧ ❤❛✈❡ t♦ ❜❡ ❛♣♣r♦①✐♠❛t❡❞✳

❈❤❛♣t❡r ✷✿ ■♥t❡❣r❛t✐♦♥

❈♦♥t❡♥ts

✷✳✶ Pr♦♣❡rt✐❡s ♦❢ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ■♥t❡❣r❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ❛♥❞ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡✿ ❧✐♥❡❛r✐t② ✷✳✸ ■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ▲✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺ ■♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥✿ ❝♦♠♣♦s✐t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽ ❚r✐❣♦♥♦♠❡tr✐❝ s✉❜st✐t✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✾ ■♥t❡❣r❛t✐♦♥ ❜② ♣❛rts✿ ♣r♦❞✉❝ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✵ ❆♣♣r♦❛❝❤❡s t♦ ✐♥t❡❣r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✶ ❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✷✳✶✷ Pr♦♣❡rt✐❡s ♦❢ ♣r♦♣❡r ❛♥❞ ✐♠♣r♦♣❡r ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✷✳✶✸ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✻ ✶✵✶ ✶✵✾ ✶✷✶ ✶✷✽ ✶✸✸ ✶✸✽ ✶✹✷ ✶✹✼ ✶✺✶ ✶✺✸ ✶✻✺ ✶✼✵

✷✳✶✳ Pr♦♣❡rt✐❡s ♦❢ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s

❈♦♥s✐❞❡r t❤✐s

◮

♦❜✈✐♦✉s

st❛t❡♠❡♥t ❛❜♦✉t ♠♦t✐♦♥✿

✏■❢ ♠② s♣❡❡❞ ✐s ③❡r♦✱ ■ ❛♠ st❛♥❞✐♥❣ st✐❧❧ ✭❛♥❞ ✈✐❝❡ ✈❡rs❛✮✳✑

Pr♦✈✐♥❣ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ✈❡rs✐♦♥ ♦❢ t❤✐s st❛t❡♠❡♥t ✇✐❧❧ ❝♦♥✜r♠ t❤❛t ♦✉r t❤❡♦r② ♠❛t❝❤❡s t❤❡ r❡❛❧✐t② ❛♥❞ t❤❡ ❝♦♠♠♦♥ s❡♥s❡✳ ❲❡ ❞✐❞ t❤✐s ✐♥ ❱♦❧✉♠❡ ✷✳ ❆s ❛ r❡✈✐❡✇✱ ✇❡ ✇✐❧❧ ❣♦ t❤r♦✉❣❤ t❤❡s❡ t❤r❡❡ st❛❣❡s ❛❣❛✐♥✿ ✶✳ t❤❡ ❞✐✛❡r❡♥❝❡ ✷✳ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✸✳ t❤❡ ❞❡r✐✈❛t✐✈❡ ❚❤✐s t✐♠❡✱ ✇❡ ✇✐❧❧ ❛❝❝♦♠♣❛♥② ❡❛❝❤ ♦❢ t❤♦s❡ st❛t❡♠❡♥ts ✇✐t❤ ✐ts ❡q✉✐✈❛❧❡♥t ✐♥ t❡r♠s ♦❢✱ r❡s♣❡❝t✐✈❡❧②✿ ✶✳ t❤❡ s✉♠ ✷✳ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ✸✳ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

❚❤❡♦r❡♠ ✷✳✶✳✶✿ ❈♦♥st❛♥❝② ✈s✳ ❩❡r♦ ❉✐✛❡r❡♥❝❡ ❆ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ ❤❛s ❛ ③❡r♦ ❞✐✛❡r❡♥❝❡ ✭❢♦r ❛❧❧ s❡❝♦♥❞❛r② ♥♦❞❡s ✐♥ t❤❡ ♣❛rt✐t✐♦♥✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t✳

✷✳✶✳

Pr♦♣❡rt✐❡s ♦❢ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s

✾✼

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ∆F = 0 ⇐⇒ F = ❝♦♥st❛♥t.

❈♦r♦❧❧❛r② ✷✳✶✳✷✿ ❩❡r♦ ❋✉♥❝t✐♦♥ ✈s✳ ❈♦♥st❛♥t ❙✉♠ ❆ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ ❤❛s ❛ ❝♦♥st❛♥t s✉♠ ✭♦✈❡r ❛❧❧ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s ③❡r♦✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ k = 0 ⇐⇒ Σk = ❝♦♥st❛♥t. ❲❡ ❝❛♥ t❤✐♥❦ ♦❢

F

❛s t❤❡ ✐♥❝r❡♠❡♥t❛❧ ♣♦s✐t✐♦♥ ❛♥❞

k

❛s t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✳

❊①❡r❝✐s❡ ✷✳✶✳✸ ❉❡r✐✈❡ t❤❡ ❧❛tt❡r r❡s✉❧t ❢r♦♠ t❤❡ ❢♦r♠❡r✳ ◆❡①t✱ ✇❡ ❞✐✈✐❞❡

∆x

♦r ❢❛❝t♦r ♦✉t

∆x

✭✐✳❡✳✱

k = f ∆x✮✿

❚❤❡♦r❡♠ ✷✳✶✳✹✿ ❈♦♥st❛♥❝② ✈s✳ ❩❡r♦ ❉✐✛❡r❡♥❝❡ ◗✉♦t✐❡♥t ❆ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ ❤❛s ❛ ③❡r♦ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✭❢♦r ❛❧❧ s❡❝♦♥❞❛r② ♥♦❞❡s ✐♥ t❤❡ ♣❛rt✐t✐♦♥✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ∆F = 0 ⇐⇒ F = ❝♦♥st❛♥t . ∆x

❈♦r♦❧❧❛r② ✷✳✶✳✺✿ ❩❡r♦ ❋✉♥❝t✐♦♥ ✈s✳ ❈♦♥st❛♥t ❘✐❡♠❛♥♥ ❙✉♠ ❆ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ ❤❛s ❛ ❝♦♥st❛♥t ❘✐❡♠❛♥♥ s✉♠ ✭♦✈❡r ❛❧❧ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s ③❡r♦✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f = 0 ⇐⇒ Σf · ∆x = ❝♦♥st❛♥t. ❲❡ ❝❛♥ t❤✐♥❦ ♦❢

F

❛s t❤❡ ✐♥❝r❡♠❡♥t❛❧ ♣♦s✐t✐♦♥ ❛♥❞

f

❛s t❤❡ ✈❡❧♦❝✐t②✳

❊①❡r❝✐s❡ ✷✳✶✳✻ ❉❡r✐✈❡ t❤❡ ❧❛tt❡r r❡s✉❧t ❢r♦♠ t❤❡ ❢♦r♠❡r✳ ◆❡①t✱ ✇❡ ♠❛❦❡

∆x → 0✿

❚❤❡♦r❡♠ ✷✳✶✳✼✿ ❈♦♥st❛♥❝② ✈s✳ ❩❡r♦ ❉❡r✐✈❛t✐✈❡ ❆ ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ✐♥t❡r✈❛❧ ❢✉♥❝t✐♦♥ ❤❛s ❛ ③❡r♦ ❞❡r✐✈❛t✐✈❡ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t✳

✷✳✶✳

Pr♦♣❡rt✐❡s ♦❢ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s

✾✽

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ dF = 0 ⇐⇒ F = ❝♦♥st❛♥t. dx ❈♦r♦❧❧❛r② ✷✳✶✳✽✿ ❩❡r♦ ❋✉♥❝t✐♦♥ ✈s✳ ❈♦♥st❛♥t ❘✐❡♠❛♥♥ ■♥t❡❣r❛❧

❆♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ ❛♥ ✐♥t❡r✈❛❧ ❤❛s ❛ ❝♦♥st❛♥t ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s ③❡r♦✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f = 0 ⇐⇒ ❲❡ ❝❛♥ t❤✐♥❦ ♦❢

F

Z

x

f dx = ❝♦♥st❛♥t.

a

❛s t❤❡ ❝♦♥t✐♥✉♦✉s❧② ❝❤❛♥❣✐♥❣ ♣♦s✐t✐♦♥ ❛♥❞

f

❛s t❤❡ ✐♥st❛♥t❛♥❡♦✉s ✈❡❧♦❝✐t②✳

❊①❡r❝✐s❡ ✷✳✶✳✾

❉❡r✐✈❡ t❤❡ ❧❛tt❡r r❡s✉❧t ❢r♦♠ t❤❡ ❢♦r♠❡r✳

❙✉♣♣♦s❡ ♥♦✇ t❤❛t t❤❡r❡ ❛r❡

t✇♦ r✉♥♥❡rs r✉♥♥✐♥❣ ✇✐t❤ t❤❡ s❛♠❡ s♣❡❡❞❀ ✇❤❛t ❝❛♥ ✇❡ s❛② ❛❜♦✉t t❤❡✐r ♠✉t✉❛❧

❧♦❝❛t✐♦♥s❄ ❚❤❡② ❛r❡ ♥♦t✱ ♦❢ ❝♦✉rs❡✱ st❛♥❞✐♥❣ st✐❧❧✱ ❜✉t t❤❡②

❛r❡

st✐❧❧ r❡❧❛t✐✈❡ t♦ ❡❛❝❤ ♦t❤❡r✦

❲❡ ❤❛✈❡ ❛ s❧✐❣❤t❧② ❧❡ss

♦❜✈✐♦✉s ❢❛❝t ❛❜♦✉t ♠♦t✐♦♥✿

◮

✏■❢ t✇♦ r✉♥♥❡rs r✉♥ ✇✐t❤ t❤❡ s❛♠❡ s♣❡❡❞✱ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡♠ ❞♦❡s♥✬t ❝❤❛♥❣❡ ✭❛♥❞ ✈✐❝❡

✈❡rs❛✮✑✳ ■t✬s ❛s ✐❢ t❤❡② ❛r❡ ❤♦❧❞✐♥❣ t❤❡ t✇♦ ❡♥❞s ♦❢ ❛ ♣♦❧❡ ✇✐t❤♦✉t ♣✉❧❧✐♥❣ ♦r ♣✉s❤✐♥❣✿

❚❤❡ ❢❛❝t r❡♠❛✐♥s ✈❛❧✐❞ ❡✈❡♥ ✐❢ t❤❡② s♣❡❡❞ ✉♣ ❛♥❞ s❧♦✇ ❞♦✇♥ ❛❧❧ t❤❡ t✐♠❡✳ ❚❤❡② ♠♦✈❡ ❛s ✐❢ ❛ s✐♥❣❧❡ ❜♦❞②✿

❖♥❝❡ ❛❣❛✐♥✱ ❢♦r ❢✉♥❝t✐♦♥s

y = F (x)

❛♥❞

y = G(x)

r❡♣r❡s❡♥t✐♥❣ t❤❡✐r ♣♦s✐t✐♦♥s✱ ✇❡ ❝❛♥ r❡st❛t❡ t❤✐s ✐❞❡❛

♠❛t❤❡♠❛t✐❝❛❧❧② ✐♥ ♦r❞❡r t♦ ❝♦♥✜r♠ t❤❛t ♦✉r t❤❡♦r② ♠❛❦❡s s❡♥s❡✳ ❲❡ ❢♦❧❧♦✇ t❤❡ s❛♠❡ t❤r❡❡ st❛❣❡s st❛rt✐♥❣ ✇✐t❤ t❤❡ ❞✐✛❡r❡♥❝❡s✿

✷✳✶✳

Pr♦♣❡rt✐❡s ♦❢ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s

✾✾

❚❤❡♦r❡♠ ✷✳✶✳✶✵✿ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❢♦r ❉✐✛❡r❡♥❝❡s ❚✇♦ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ ❤❛✈❡ t❤❡ s❛♠❡ ❞✐✛❡r❡♥❝❡s ✭❢♦r ❛❧❧ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ t✇♦ ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ∆F = ∆G ⇐⇒ F − G = ❝♦♥st❛♥t.

❈♦r♦❧❧❛r② ✷✳✶✳✶✶✿ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❢♦r ❙✉♠s ❚✇♦ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ ❛r❡ ❡q✉❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s✉♠s ♦❢ t❤❡ t✇♦ ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t ✭♦♥ ❛❧❧ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✮✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f = g ⇐⇒ Σf − Σg = ❝♦♥st❛♥t. ❲❡ ❝❛♥ t❤✐♥❦ ♦❢

F

❛♥❞

G

❛s t❤❡ ✐♥❝r❡♠❡♥t❛❧ ♣♦s✐t✐♦♥s ❛♥❞

f

❛♥❞

g

❛s t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts✳

❊①❡r❝✐s❡ ✷✳✶✳✶✷ ❉❡r✐✈❡ t❤❡ ❧❛tt❡r r❡s✉❧t ❢r♦♠ t❤❡ ❢♦r♠❡r✳

◆❡①t✱ ✇❡ ❞✐✈✐❞❡

∆x

♦r ❢❛❝t♦r ♦✉t

∆x✿

❚❤❡♦r❡♠ ✷✳✶✳✶✸✿ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❢♦r ❉✐✛❡r❡♥t ◗✉♦t✐❡♥ts ❚✇♦ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ ❤❛✈❡ t❤❡ s❛♠❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ t✇♦ ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ∆G ∆F = ⇐⇒ F − G = ❝♦♥st❛♥t. ∆x ∆x

❈♦r♦❧❧❛r② ✷✳✶✳✶✹✿ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❢♦r ❘✐❡♠❛♥♥ ❙✉♠s ❚✇♦ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ ❛r❡ ❡q✉❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ♦❢ t❤❡ t✇♦ ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t ✭♦♥ ❛❧❧ ♣r✐♠❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✮✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f = g ⇐⇒ Σf · ∆x − Σg · ∆x = ❝♦♥st❛♥t. ❲❡ ❝❛♥ t❤✐♥❦ ♦❢

F

❛♥❞

G

❛s t❤❡ ✐♥❝r❡♠❡♥t❛❧ ♣♦s✐t✐♦♥s ❛♥❞

❊①❡r❝✐s❡ ✷✳✶✳✶✺ ❉❡r✐✈❡ t❤❡ ❧❛tt❡r r❡s✉❧t ❢r♦♠ t❤❡ ❢♦r♠❡r✳

◆❡①t✱ ✇❡ ♠❛❦❡

∆x → 0✿

f

❛♥❞

g

❛s t❤❡ ✈❡❧♦❝✐t✐❡s✳

✷✳✶✳

Pr♦♣❡rt✐❡s ♦❢ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s

✶✵✵

❚❤❡♦r❡♠ ✷✳✶✳✶✻✿ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❢♦r ❉❡r✐✈❛t✐✈❡s ❚✇♦ ❢✉♥❝t✐♦♥s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ❛♥ ✐♥t❡r✈❛❧ ❤❛✈❡ t❤❡ s❛♠❡ ❞❡r✐✈❛t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ t✇♦ ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ d d F = G ⇐⇒ F − G = ❝♦♥st❛♥t. dx dx

❈♦r♦❧❧❛r② ✷✳✶✳✶✼✿ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❢♦r ■♥t❡❣r❛❧s ❚✇♦ ❢✉♥❝t✐♦♥s ✐♥t❡❣r❛❜❧❡ ♦♥ ❛♥ ✐♥t❡r✈❛❧ ❤❛✈❡ t❤❡ s❛♠❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s ♦❢ t❤❡ t✇♦ ❞✐✛❡r ❜② ❛ ❝♦♥st❛♥t✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f = g ⇐⇒

Z

x a

f dx −

Z

x

g dx = ❝♦♥st❛♥t.

a

❲❡ ❝❛♥ t❤✐♥❦ ♦❢ F ❛♥❞ G ❛s t❤❡ ❝♦♥t✐♥✉♦✉s❧② ❝❤❛♥❣✐♥❣ ♣♦s✐t✐♦♥s ❛♥❞ f ❛♥❞ g ❛s t❤❡ ✐♥st❛♥t❛♥❡♦✉s ✈❡❧♦❝✐t✐❡s✳

❊①❡r❝✐s❡ ✷✳✶✳✶✽ ❉❡r✐✈❡ t❤❡ ❧❛tt❡r r❡s✉❧t ❢r♦♠ t❤❡ ❢♦r♠❡r✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ♠♦t✐♦♥ ✐♥t❡r♣r❡t❛t✐♦♥✱ t❤❡r❡ ✐s ❛❧s♦ ♦♥❡ ✐♥ t❡r♠s ♦❢ ❣❡♦♠❡tr②✳ ❚❤❡ ❧❛st t❤❡♦r❡♠ s❛②s✿ ◮ ■❢ t❤❡ ❣r❛♣❤s ♦❢ y = F (x) ❛♥❞ y = G(x) ❤❛✈❡ ♣❛r❛❧❧❡❧ t❛♥❣❡♥t ❧✐♥❡s ❢♦r ❡✈❡r② ✈❛❧✉❡ ♦❢ x✱ t❤❡♥ t❤❡ ❣r❛♣❤ ♦❢ F ✐s ❛ ✈❡rt✐❝❛❧ s❤✐❢t ♦❢ t❤❡ ❣r❛♣❤ ♦❢ G ✭❛♥❞ ✈✐❝❡ ✈❡rs❛✮✳

❲❡ ❝❛♥ ✉♥❞❡rst❛♥❞ t❤✐s ✐❞❡❛ ✐❢ ✇❡ ✐♠❛❣✐♥❡ ❛ t✉♥♥❡❧ ❛♥❞ ❛ ♣❡rs♦♥ ✇❤♦s❡ ❤❡❛❞ ✐s t♦✉❝❤✐♥❣ t❤❡ ❝❡✐❧✐♥❣✳ ■❢ t❤❡ ❝❡✐❧✐♥❣ ✐s s❧♦♣❡❞ ❞♦✇♥✱ s❤♦✉❧❞ ❤❡ ❜❡ ❝♦♥❝❡r♥❡❞ ❛❜♦✉t ❤✐tt✐♥❣ ❤✐s ❤❡❛❞❄ ◆♦t ✐❢ t❤❡ ✢♦♦r ✐s s❧♦♣❡❞ ❞♦✇♥ ❛s ♠✉❝❤✿

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ t❤❡ s❧♦♣❡ ♦❢ t❤❡ t✉♥♥❡❧✬s t♦♣ ✐s ❡q✉❛❧ t♦ t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❜♦tt♦♠ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥✱ t❤❡♥ t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ t✉♥♥❡❧ r❡♠❛✐♥s t❤❡ s❛♠❡ t❤r♦✉❣❤♦✉t ✐ts ❧❡♥❣t❤✳

✷✳✷✳ ■♥t❡❣r❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ❛♥❞ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡✿ ❧✐♥❡❛r✐t②

✶✵✶

❇❛s❡❞ ♦♥ t❤❡ t❤❡♦r❡♠✱ ✇❡ ❝❛♥ ♥♦✇ ✉♣❞❛t❡ t❤✐s ❧✐st ♦❢ s✐♠♣❧❡ ❜✉t ✐♠♣♦rt❛♥t ❢❛❝ts✿ ✐♥❢♦ ❛❜♦✉t F

✐♥❢♦ ❛❜♦✉t F

′

F ✐s ❝♦♥st❛♥t✳

⇐⇒ F ✐s ③❡r♦✳

F ✐s ❧✐♥❡❛r✳

⇐⇒ F ✐s ❝♦♥st❛♥t✳

′ ′

F ✐s q✉❛❞r❛t✐❝✳ ⇐⇒ F ✐s ❧✐♥❡❛r✳ ′

✐♥❢♦ ❛❜♦✉t

Z

✐♥❢♦ ❛❜♦✉t f

f dx

Z

f dx ✐s ❝♦♥st❛♥t✳

⇐⇒ f ✐s ③❡r♦✳

f dx ✐s ❧✐♥❡❛r✳

⇐⇒ f ✐s ❝♦♥st❛♥t✳

Z

f dx ✐s q✉❛❞r❛t✐❝✳ ⇐⇒ f ✐s ❧✐♥❡❛r✳

Z

❲❡ ✉s❡ t❤❡ ❧❛st t✇♦ ❢❛❝ts t♦ ❥✉st✐❢② ♦✉r ❛♥❛❧②s✐s ♦❢ ❢r❡❡ ✢✐❣❤t ✿ ❋✉♥❝t✐♦♥s ♦❢ t✐♠❡ ❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❝♦♥st❛♥t✳ =⇒ ❚❤❡ ✈❡❧♦❝✐t② ✐s ❧✐♥❡❛r✳ =⇒ ❚❤❡ ❧♦❝❛t✐♦♥ ✐s q✉❛❞r❛t✐❝✳ ❆♥♦t❤❡r ✐♠♣♦rt❛♥t ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡ s❛♠❡ ❞❡r✐✈❛t✐✈❡ F ′ = f ✳ ❚❤❡✐r ❣r❛♣❤s ❛r❡ ❡❛s② t♦ ♣❧♦t ❜❡❝❛✉s❡ t❤❡② ❞✐✛❡r ❜② ❛ ✈❡rt✐❝❛❧ s❤✐❢t✿

❙♦✱ ❡✈❡♥ ✐❢ ✇❡ ❝❛♥ r❡❝♦✈❡r t❤❡ ❢✉♥❝t✐♦♥ F ❢r♦♠ ✐ts ❞❡r✐✈❛t✐✈❡ F ′ ✱ t❤❡r❡ ❛r❡ ♠❛♥② ♦t❤❡rs ✇✐t❤ t❤❡ s❛♠❡ ❞❡r✐✈❛t✐✈❡✱ s✉❝❤ ❛s G = F + C ❢♦r ❛♥② ❝♦♥st❛♥t r❡❛❧ ♥✉♠❜❡r C ✳ ❆r❡ t❤❡r❡ ♦t❤❡rs❄ ◆♦t ❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠✳ ❲❛r♥✐♥❣✦ ■t✬s ♦♥❧② tr✉❡ ✇❤❡♥ t❤❡ ❞♦♠❛✐♥ ✐s ❛♥ ✐♥t❡r✈❛❧✳

✷✳✷✳ ■♥t❡❣r❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ❛♥❞ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡✿ ❧✐♥❡❛r✲ ✐t②

■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐❧❧ ❜❡ t❛❦✐♥❣ ❛ ❜r♦❛❞❡r ❧♦♦❦ ❛t ❤♦✇ ✇❡ ❝♦♠♣✉t❡ t❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✳ ❋♦r t❤❡ ♣r♦❝❡❞✉r❡ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ✭❧❡❢t✮✱ ✇❡ ♥♦✇ ❤❛✈❡ s❡✈❡r❛❧ s❤♦rt❝✉ts ✭r✐❣❤t✮✿ s❛♠♣❧✐♥❣

s❡q✉❡♥❝❡ ←−−−−−−−− ❢✉♥❝t✐♦♥ ♦♥ ✐♥t❡r✈❛❧    ❘❙ y

R  y

s❡q✉❡♥❝❡ −−−∆x→0 −−−−−→ ❢✉♥❝t✐♦♥ ♦♥ ✐♥t❡r✈❛❧

xn  R  y

1 xn+1 n+1

ex  R  y

ex

sin x  R  y

− cos x

cos x  R  y sin x

✷✳✷✳

■♥t❡❣r❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ❛♥❞ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡✿ ❧✐♥❡❛r✐t②

✶✵✷

▲❡t✬s r❡✈✐❡✇ t❤❡ ❞❡t❛✐❧s✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ [a, b]✳ ❚❤❡s❡ ❛r❡ t❤❡ ♣r✐♠❛r② ♥♦❞❡s✿ a = x0 , ..., xn = b .

❚❤❡s❡ ❛r❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✿ c1 , ..., cn .

❚❤❡② s❛t✐s❢②✿ xk ≤ ck ≤ xk+1 .

❲❡ st❛rt ✇✐t❤ ❛ ❢✉♥❝t✐♦♥ g ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✳ ■t ✐s s✐♠♣❧② ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs✳ ❍❡r❡ ✐s ✐ts s✉♠ ✿ m Σg(xk ) =

X

g(ck ) .

k=1

■t ✐s ❛❧s♦ ❞❡✜♥❡❞ ♦♥ t❤❡ ♣❛rt✐t✐♦♥✳ ❖♥ t❤❡ ♦♥❡ ❤❛♥❞✱ ✐t ✐s s✐♠♣❧② ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs✳ ❖♥ t❤❡ ♦t❤❡r✱ ✐t✬s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ ♣r✐♠❛r② ♥♦❞❡s✳ ❲❤❛t t❤✐s ♠❡❛♥s ✐s t❤❛t t❤✐s ♣r♦❝❡❞✉r❡ ✐s ❛ s♣❡❝✐❛❧ ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥✱ ❛ ❢✉♥❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s✿ ❢✉♥❝t✐♦♥ →

→ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥✱ s❡q✉❡♥❝❡

Σ

◆♦✇ ✇❡ ❤❛✈❡ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✳ ❚♦ ✜♥❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠ Σ ♦❢ f ✱ ✇❡ ❥✉st ❛♣♣❧② t❤❡ s✉♠ ❝♦♥str✉❝t✐♦♥ t♦ f ✇✐t❤ ✐ts ✈❛❧✉❡s ♠✉❧t✐♣❧✐❡❞ ❜② ∆x = xk+1 − xk ✳ ❲❡ ❤❛✈❡ ❝r❡❛t❡❞ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s✿ ❢✉♥❝t✐♦♥ →

→ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥✱ s❡q✉❡♥❝❡

Σ · ∆x

◆❡①t✱ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✐s ❞❡✜♥❡❞ ❛s ❛ ❧✐♠✐t✳ ■t ✐s t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ♦❢ f ❢ ♦✈❡r ❛❧❧ ♣❛rt✐t✐♦♥s ♦❢ t❤❡ ✐♥t❡r✈❛❧✳ ❚❤✐s ♣r♦❝❡ss ❝r❡❛t❡s ❛ s♣❡❝✐❛❧ ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥ t♦♦✱ ❛❧s♦ ❛ ❢✉♥❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s✿ ❢✉♥❝t✐♦♥ →

Z

 dx

→ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥

❆s ❛ r❡♠✐♥❞❡r✱ t❤❡ ♠♦t✐♦♥ ❛♥❛❧♦❣② ❢♦r t❤❡s❡ t❤r❡❡ ♦♣❡r❛t✐♦♥s ✐s ❛s ❢♦❧❧♦✇s✿ ✈❡❧♦❝✐t② 

 s❛♠♣❧✐♥❣ y

s❛♠♣❧❡❞ ✈❡❧♦❝✐t②  ·∆ y

s❛♠♣❧❡❞ ❞✐s♣❧❛❝❡♠❡♥ts  P  y

s❛♠♣❧❡❞ t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t  ∆x→0 y

t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t

❲❡ ♥❡❡❞ t♦ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤❡s❡ t❤r❡❡ ❢✉♥❝t✐♦♥s ♦♣❡r❛t❡✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ s✉♠s✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✱ ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s✳ ■♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ✇❡ ❝❛♥ ♣r❡❞✐❝t t❤❛t t❤❡② ✇✐❧❧ ♠❛t❝❤ t❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✱ ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡s✳

✷✳✷✳

■♥t❡❣r❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ❛♥❞ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡✿ ❧✐♥❡❛r✐t②

✶✵✸

❋✉rt❤❡r♠♦r❡✱ t❤❡ ❧❛st ✐t❡♠ ♦♥ ❡✐t❤❡r ❧✐st ❝♦♠❡s ❢r♦♠ t❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❧✐♠✐ts✳ ■♥ ❢❛❝t✱ t❤❡ ✐❞❡❛s ♦❢ ✇❤❛t s❤♦rt❝✉ts t♦ ❧♦♦❦ ❢♦r ❝♦♠❡ ❢r♦♠ t❤♦s❡ ❢♦r ❧✐♠✐ts ✿ t❤❡ ❙✉♠ ❘✉❧❡✱ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✱ ❡t❝✳ ❚❤❡ q✉❡st✐♦♥ ✇❡ ✇✐❧❧ ❜❡ ❛s❦✐♥❣ ✐s✿ ◮ ❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ♦✉t♣✉t ❢✉♥❝t✐♦♥ ♦❢ ✐♥t❡❣r❛t✐♦♥ ❛s ✇❡ ♣❡r❢♦r♠ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✇✐t❤

t❤❡ ✐♥♣✉t ❢✉♥❝t✐♦♥s❄

❚❤❡r❡ ❛r❡ ❛ ❢❡✇ s❤♦rt❝✉t ♣r♦♣❡rt✐❡s✳ ▲❡t✬s t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t t❤✐s ❡❧❡♠❡♥t❛r② st❛t❡♠❡♥t ❛❜♦✉t ♠♦t✐♦♥ ✿ ◮ ■❋ t✇♦ r✉♥♥❡rs ❛r❡ r✉♥♥✐♥❣ ❛✇❛② ❢r♦♠ ❛ ♣♦st✱ ❚❍❊◆ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡♠ ✐s t❤❡ s❛♠❡

❛s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ♣♦st ♦❢ ❛ t❤✐r❞ ♣❡rs♦♥ r✉♥♥✐♥❣ ❢♦r t❤❡ ❜♦t❤ ♦❢ t❤❡♠ ✇✐t❤ t❤❡ ❝♦♠❜✐♥❡❞ s♣❡❡❞✳

❲❤❡♥ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❛❞❞❡❞✱ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡✐r s✉♠s❄ ❚❤✐s s✐♠♣❧❡ ❛❧❣❡❜r❛✱ t❤❡ ❆ss♦❝✐❛t✐✈❡

✱ t❡❧❧s t❤❡ ✇❤♦❧❡ st♦r②✿

Pr♦♣❡rt②

u

+ U

v

+ V

+ = (u + v) + (U + V )

❚❤❡ r✉❧❡ ❛♣♣❧✐❡s ❡✈❡♥ ✐❢ ✇❡ ❤❛✈❡ ♠♦r❡ t❤❛♥ ❥✉st t✇♦ t❡r♠s❀ ✐t✬s ❥✉st r❡✲❛rr❛♥❣✐♥❣ t❤❡ t❡r♠s✿ up + Up up+1 + Up+1

✳✳ ✳

✳✳ ✳

✳✳ ✳

uq + Uq up + ... + uq + Up + ... + Uq = (up + Up )+ ... +(uq + Uq ) q X (un + Un ) = n=p

❚❤❛t✬s t❤❡ ♣❛rt✐t✐♦♥s✿

❙✉♠ ❘✉❧❡ ❢♦r ❙✉♠s ♦❢ ❙❡q✉❡♥❝❡s

✭❈❤❛♣t❡r ✶P❈✲✶✮✳ ❲❡ r❡st❛t❡ ✐t ❢♦r ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥

❚❤❡♦r❡♠ ✷✳✷✳✶✿ ❙✉♠ ❘✉❧❡ ❢♦r ❙✉♠s ❋♦r ❛♥② ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧✱ t❤❡ s✉♠ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ t❤❡ s✉♠s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

Σ (f + g) = Σf + Σg

❋♦r t❤❡ ❘✐❡♠❛♥♥ s✉♠s✱ t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ✐s ❛❧s♦ s✐♠♣❧❡❀ t❤❡ ♣✐❝t✉r❡ ❜❡❧♦✇ ✐❧❧✉str❛t❡s ❤♦✇ ❛❞❞✐♥❣ t✇♦ ❢✉♥❝t✐♦♥s ❝❛✉s❡s ❛❞❞✐♥❣ t❤❡ ❛r❡❛s ✉♥❞❡r t❤❡✐r ❣r❛♣❤s✿

✷✳✷✳

■♥t❡❣r❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ❛♥❞ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡✿ ❧✐♥❡❛r✐t②

✶✵✹

❚❤❡♦r❡♠ ✷✳✷✳✷✿ ❙✉♠ ❘✉❧❡ ❢♦r ❘✐❡♠❛♥♥ ❙✉♠s ❋♦r ❛♥② ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

Σ (f + g) · ∆x = Σf · ∆x + Σg · ∆x Pr♦♦❢✳

❆♣♣❧②✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ t♦ t❤❡ ❢✉♥❝t✐♦♥ f + g ✱ ✇❡ ❤❛✈❡✿    Σ f + g · ∆x = f (a) + g(a) + f (a + h) + g(a + h) + ...  + f (x − h) + g(x − h) · ∆x   = f (a) + f (a + h) + f (a + 2h) + ... + f (x − h) + f (x) · ∆x   + g(a) + g(a + h) + g(a + 2h) + ... + g(x − h) + g(x) · ∆x = Σf · ∆x (b) + Σg · ∆x .

■t ✐s ❛s ✐❢ ♦♥❡ ♠❛❦❡s t✇♦ ❝♦♥s❡❝✉t✐✈❡ ❞r♦♣s ✐♥ ❛ ❣❛♠❡ ♦❢ ❚❡tr✐s✿

◆♦✇ ∆x → 0 ✐♥ t❤❡s❡ ❘✐❡♠❛♥♥ s✉♠s✿

✷✳✷✳ ■♥t❡❣r❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ❛♥❞ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡✿ ❧✐♥❡❛r✐t②

✶✵✺

❚❤❡♦r❡♠ ✷✳✷✳✸✿ ❙✉♠ ❘✉❧❡ ❢♦r ■♥t❡❣r❛❧s ❚❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ t❤❡ ✐♥t❡❣r❛❧s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢

f

❛♥❞

g

❛r❡ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥s ♦✈❡r

[a, b]✱

t❤❡♥ s♦ ✐s

f +g

❛♥❞ ✇❡ ❤❛✈❡✿

Z

x

(f + g) dx = a

Z

x

f dx + a

Z

x

g dx a

Pr♦♦❢✳ ❲❡ ✉s❡ t❤❡ ❙✉♠ ❘✉❧❡ ❢♦r ▲✐♠✐ts ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✶✳

❙♦✱

∆✬s

❜❡❝♦♠❡

d✬s✦

❚❤❡ ♣✐❝t✉r❡ ❜❡❧♦✇ ✐❧❧✉str❛t❡s ✇❤❛t ❤❛♣♣❡♥s ✇❤❡♥ t❤❡ ❜♦tt♦♠ ❞r♦♣s ❢r♦♠ ❛ ❜✉❝❦❡t ♦❢ s❛♥❞ ❛♥❞ ✐t ❢❛❧❧s ♦♥ ❛ ✉♥❡✈❡♥ s✉r❢❛❝❡✿

❚❤❡ ❧❛st t✇♦ t❤❡♦r❡♠s ❞❡♠♦♥str❛t❡ t❤❛t t❤✐s ✐s tr✉❡ ✇❤❡t❤❡r t❤❡ s✉r❢❛❝❡ ✐s st❛✐r❝❛s❡✲❧✐❦❡ ♦r ❝✉r✈❡❞✳

❊①❡r❝✐s❡ ✷✳✷✳✹ ▼♦❞✐❢② t❤❡ ♣r♦♦❢ ❢♦r✿ ✭❛✮ ❧❡❢t✲❡♥❞✱ ✭❜✮ ♠✐❞✲♣♦✐♥t✱ ❛♥❞ ✭❝✮ ❣❡♥❡r❛❧ ❘✐❡♠❛♥♥ s✉♠s✳

❈♦♠♣✉t❛t✐♦♥❛❧❧②✱ ✇❡ ❤❛✈❡ t❤❡ s❛♠❡ ✐❞❡❛ ❛s t❤❡ ♦♥❡ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡s✿ ❚❤❡ ✐♥t❡❣r❛❧ ✐s s♣❧✐t ✐♥ ❤❛❧❢✳ ❊✐t❤❡r ♦❢ t❤❡ ❧❛st t✇♦ t❤❡♦r❡♠s ❝❛♥ ❛❧s♦ ❜❡ ❞❡♠♦♥str❛t❡❞ ✇✐t❤ t❤❡ s♣r❡❛❞s❤❡❡t✿

❚❤❡ s❛♠❡ ♣r♦♦❢ ❛♣♣❧✐❡s t♦ s✉❜tr❛❝t✐♦♥ ♦❢ t❤❡ s✉♠s✳

✷✳✷✳

■♥t❡❣r❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ❛♥❞ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡✿ ❧✐♥❡❛r✐t②

✶✵✻

❊①❡r❝✐s❡ ✷✳✷✳✺

❙t❛t❡ t❤❡ ❉✐✛❡r❡♥❝❡ ❘✉❧❡✳ ◆❡①t✱ ✇❤❡♥ ❛ ❢✉♥❝t✐♦♥ ✐s ♠✉❧t✐♣❧✐❡❞ ❜② ❛ ❝♦♥st❛♥t✱ ✇❤❛t ❤❛♣♣❡♥s t♦ ✐ts s✉♠s❄ ❚❤✐s s✐♠♣❧❡ ❛❧❣❡❜r❛✱ t❤❡ ❉✐str✐❜✉t✐✈❡

✱ t❡❧❧s t❤❡ ✇❤♦❧❡ st♦r②✿

Pr♦♣❡rt②

c·( u

=

+ U)

cu + cU

❚❤❡ r✉❧❡ ❛♣♣❧✐❡s ❡✈❡♥ ✐❢ ✇❡ ❤❛✈❡ ♠♦r❡ t❤❛♥ ❥✉st t✇♦ t❡r♠s❀ ✐t✬s ❥✉st ❢❛❝t♦r✐♥❣ ✿ c · up

c · up+1

✳✳ ✳✳ ✳✳ ✳ ✳ ✳

c · uq

c · (up + ... + uq ) q X =c · un n=p

❚❤❛t✬s t❤❡ ❈♦♥st❛♥t ♦♥ ♣❛rt✐t✐♦♥s✿

▼✉❧t✐♣❧❡ ❢♦r ❙✉♠s

❢♦r s❡q✉❡♥❝❡s ✭❈❤❛♣t❡r ✶P❈✲✶✮✳ ❲❡ r❡st❛t❡ ✐t ❢♦r ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞

❚❤❡♦r❡♠ ✷✳✷✳✻✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❙✉♠s ❋♦r ❛♥② ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧✱ t❤❡ s✉♠ ♦❢ ❛ ♠✉❧t✐♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s t❤❡ ♠✉❧t✐♣❧❡ ♦❢ ✐ts s✉♠✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

Σ(cf ) = c (Σf )

❋♦r t❤❡ ❘✐❡♠❛♥♥ s✉♠s✱ t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ✐s ❛❧s♦ s✐♠♣❧❡❀ t❤❡ ♣✐❝t✉r❡ ❜❡❧♦✇ ✐❧❧✉str❛t❡s t❤❡ ✐❞❡❛ ♦❢ ♠✉❧t✐♣❧✐✲ ❝❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱ ✈✐③✳ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❛r❡❛ ✉♥❞❡r ✐ts ❣r❛♣❤✿

✷✳✷✳

✶✵✼

■♥t❡❣r❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ❛♥❞ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡✿ ❧✐♥❡❛r✐t②

❚❤❡♦r❡♠ ✷✳✷✳✼✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❘✐❡♠❛♥♥ ❙✉♠s ❚❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ ♠✉❧t✐♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s t❤❡ ♠✉❧t✐♣❧❡ ♦❢ ✐ts ❘✐❡♠❛♥♥ s✉♠✳

■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛♥② r❡❛❧

c

✇❡ ❤❛✈❡✿

Σ(cf ) · ∆x = c (Σf · ∆x) Pr♦♦❢✳

❆♣♣❧②✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ t♦ t❤❡ ❢✉♥❝t✐♦♥ f ✱ ✇❡ ❤❛✈❡✿ Σ(cf )∆x = cf (a) + cf (a + h) + cf (a + 2h) + ... + cf (x − h) = c (f (a) + f (a + h) + f (a + 2h) + ... + f (x − h))

= c Σf · ∆x .

■t ✐s ❛s ✐❢ ♦♥❡ ✐s ♠❛❦❡s s❡✈❡r❛❧ ✐❞❡♥t✐❝❛❧ ❞r♦♣s ✐♥ ❛ ❣❛♠❡ ♦❢ ❚❡tr✐s✿

◆♦✇ ∆x → 0 ✐♥ t❤❡s❡ ❘✐❡♠❛♥♥ s✉♠s✿ ❚❤❡♦r❡♠ ✷✳✷✳✽✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ■♥t❡❣r❛❧s ❚❤❡ ✐♥t❡❣r❛❧ ♦❢ ❛ ♠✉❧t✐♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s t❤❡ ♠✉❧t✐♣❧❡ ♦❢ ✐ts ✐♥t❡❣r❛❧✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ r❡❛❧

c✱

f

✐s ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ ♦✈❡r

[a, b]✱

❛♥❞ ✇❡ ❤❛✈❡✿

Z

x

(cf ) dx = c a

Z

t❤❡♥ s♦ ✐s

c·f

❢♦r ❛♥②

x

f dx a

Pr♦♦❢✳

❲❡ ✉s❡ t❤❡

❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ▲✐♠✐ts

❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✶✳

❆♥❞ ∆✬s ❜❡❝♦♠❡ d✬s ❛❣❛✐♥✳ ❈♦♠♣✉t❛t✐♦♥❛❧❧②✱ ✇❡ ❤❛✈❡ t❤❡ s❛♠❡ ✐❞❡❛ ❛s t❤❡ ♦♥❡ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡s✿ ♦✉t ♦❢ t❤❡ ✐♥t❡❣r❛❧✳

❚❤❡ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ✐s ❢❛❝t♦r❡❞

❚❤❡ ♣✐❝t✉r❡ ❜❡❧♦✇ ✐❧❧✉str❛t❡s t❤❡ ✐❞❡❛ t❤❛t tr✐♣❧✐♥❣ t❤❡ ❤❡✐❣❤t ♦❢ ❛ r♦❛❞ ✇✐❧❧ ♥❡❡❞ tr✐♣❧✐♥❣ t❤❡ ❛♠♦✉♥t ♦❢ s♦✐❧ ✉♥❞❡r ✐t✿

✷✳✷✳ ■♥t❡❣r❛t✐♦♥ ♦✈❡r ❛❞❞✐t✐♦♥ ❛♥❞ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡✿ ❧✐♥❡❛r✐t②

✶✵✽

❚❤❡ t✇♦ ❧❛st t❤❡♦r❡♠s ❞❡♠♦♥str❛t❡ t❤❛t t❤✐s ✐s tr✉❡ ✇❤❡t❤❡r t❤❡ s✉r❢❛❝❡ ✐s st❛✐r❝❛s❡✲❧✐❦❡ ♦r ❝✉r✈❡❞✳ ❋♦r t❤❡ ♠♦t✐♦♥ ♠❡t❛♣❤♦r✱ ✐❢ ②♦✉r ✈❡❧♦❝✐t② ✐s tr✐♣❧❡❞✱ t❤❡♥ s♦ ✐s t❤❡ ❞✐st❛♥❝❡ ②♦✉ ❤❛✈❡ ❝♦✈❡r❡❞ ♦✈❡r t❤❡ s❛♠❡ ♣❡r✐♦❞ ♦❢ t✐♠❡✳ ❊✐t❤❡r ♦❢ t❤❡ t✇♦ ❧❛st t❤❡♦r❡♠s ❝❛♥ ❜❡ ❞❡♠♦♥str❛t❡❞ ✇✐t❤ t❤❡ s♣r❡❛❞s❤❡❡t✿

❊①❡r❝✐s❡ ✷✳✷✳✾

▼♦❞✐❢② t❤❡ ♣r♦♦❢ ❢♦r✿ ✭❛✮ ❧❡❢t✲❡♥❞✱ ✭❜✮ ♠✐❞✲♣♦✐♥t✱ ❛♥❞ ✭❝✮ ❣❡♥❡r❛❧ ❘✐❡♠❛♥♥ s✉♠s✳

❚❤❡s❡ t✇♦ ♦♣❡r❛t✐♦♥s ❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ ✐♥t♦ ♦♥❡ ♣r♦❞✉❝✐♥❣ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ✿

αx + βy , ✇❤❡r❡

α, β

❛r❡ t✇♦ ❝♦♥st❛♥t ♥✉♠❜❡rs✳ ❚❤❡ ✐❞❡❛ ❛♣♣❧✐❡s t♦ ❢✉♥❝t✐♦♥s❀ ❢♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s t❤❡ ❛✈❡r❛❣❡ ♦❢ t✇♦

❢✉♥❝t✐♦♥s ✭❧❡❢t✮✿

❲❡ ❛❧s♦ ♥♦t✐❝❡ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡✐r ✐♥t❡❣r❛❧s ✭r✐❣❤t✮✿

◮

❚❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤❡ ❛✈❡r❛❣❡ ✐s t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧s✳

❚❤❡ q✉❡st✐♦♥ ❜❡❝♦♠❡s✿ ❲❤❛t ❤❛♣♣❡♥s t♦ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s ✉♥❞❡r ✐♥t❡❣r❛t✐♦♥❄ ❘❡❝❛❧❧ t❤❛t ❛ ❢✉♥❝t✐♦♥

F

✐s ❧✐♥❡❛r ✐❢ ✐t ✏♣r❡s❡r✈❡s✑ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s✿

αx + βy →

F

→ αF (x) + βF (y)

❲✐t❤ t❤✐s ✐❞❡❛✱ t❤❡s❡ t✇♦ ❢♦r♠✉❧❛s ❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ ✐♥t♦ ♦♥❡✳ ❚❤❡ s✉♠✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠✱ ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ❛r❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s✳ ❆ ♣r❡❝✐s❡ ✈❡rs✐♦♥ ✐s ❜❡❧♦✇✿

✷✳✸✳ ■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠

✶✵✾

❚❤❡♦r❡♠ ✷✳✷✳✶✵✿ ▲✐♥❡❛r✐t② ♦❢ ■♥t❡❣r❛t✐♦♥ ❚❤❡ s✉♠✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠✱ ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♦❢ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡✐r s✉♠s✱ ❘✐❡♠❛♥♥ s✉♠s✱ ❛♥❞ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s✱ r❡s♣❡❝t✐✈❡❧②✱ ✇❤❡♥❡✈❡r t❤❡② ❡①✐st✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

Σ(αf + βg)

= αΣf

+βΣg

Σ(αf + βg) · ∆x = αΣf · ∆x +βΣg · ∆x Z x Z x Z x g dx f dx +β (αf + βg) dx = α a

a

a

❚❤❡ ❧❛st ❢♦r♠✉❧❛ ✐s ✐❧❧✉str❛t❡❞ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠✿ αf + βg →

Z

→ α

Z

f dx + β

Z

g dx

❆s ✇❡ s❡❡✱ t❤❡ ✐♥t❡❣r❛❧ ❢♦❧❧♦✇s t❤❡ ❘✐❡♠❛♥♥ s✉♠✱ ❡✈❡r② t✐♠❡✿

■♥ s✉♠♠❛r②✱ ✇❡ t❛❦❡ t❤❡ s❤♦rt❝✉t ✐♥ ♦✉r ❞✐❛❣r❛♠ ♦♥ ♠❛♥② ♦❝❝❛s✐♦♥s ❛♥❞ ✐❣♥♦r❡ t❤❡ r❡st✿ s❛♠♣❧✐♥❣

s❡q✉❡♥❝❡ ←−−−−−−−− ❢✉♥❝t✐♦♥ ♦♥ [a, b]    ❘❙ y

 ✐♥t❡❣r❛❧ y

s❡q✉❡♥❝❡ −−−−−−−−→ ❢✉♥❝t✐♦♥ ♦♥ [a, b] ∆x→0

−→

Z

f  R  y

f dx

❲❛r♥✐♥❣✦ ❚❤❡s❡ r✉❧❡s ❝❛♥ ❜❡ ❛❝q✉✐r❡❞ ✈✐❛ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✳

✷✳✸✳ ■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠

▲❡t✬s r❡✈✐❡✇ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✳

✷✳✸✳

■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠

✶✶✵

❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✿ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦♥ ♣❛rt✐t✐♦♥ f

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ ✐♥t❡r✈❛❧

I✳

✐s ❞❡✜♥❡❞ ♦♥ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥

❚❤❡♥ ❛ ❢✉♥❝t✐♦♥

F

❞❡✜♥❡❞ ♦♥ t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ t❤❛t s❛t✐s✜❡s

t❤❡ ❡q✉❛t✐♦♥✿

∆F (c) = f (c) ∆x ❢♦r ❡✈❡r② s❡❝♦♥❞❛r② ♥♦❞❡

c✱

✐s ❝❛❧❧❡❞ ❛♥

❛♥t✐❞❡r✐✈❛t✐✈❡

♦❢

f✳

■♥ ❱♦❧✉♠❡ ✶✱ ✇❡ ❢♦✉♥❞ ❛ r❡❝✉rs✐✈❡ s♦❧✉t✐♦♥ ♦❢ t❤✐s ❡q✉❛t✐♦♥ ❜② s♦❧✈✐♥❣ ✐t ❢♦r t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛❝q✉✐r✐♥❣ t❤❡ ♣♦s✐t✐♦♥ ❢r♦♠ t❤❡ ✈❡❧♦❝✐t②✿

pn+1 − pn =⇒ pn+1 = pn + vn ∆t . ∆t

vn = ■t✬s ♣✉r❡ ❛❧❣❡❜r❛✦

❚❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡ ✐s ❜② ❢❛r ♠♦r❡ ❝♦♠♣❧❡①✿

❉❡✜♥✐t✐♦♥ ✷✳✸✳✷✿ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦♥ ✐♥t❡r✈❛❧ f

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥

F

❞❡✜♥❡❞ ♦♥

I

✐s ❞❡✜♥❡❞ ♦♥ ❛♥ ✐♥t❡r✈❛❧

I✳

❚❤❡♥ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥

t❤❛t s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥✿

dF (x) = f (x) dx ❢♦r ❡✈❡r②

x✱

✐s ❝❛❧❧❡❞ ❛♥

❛♥t✐❞❡r✐✈❛t✐✈❡

♦❢

f✳

❲❡ ✉s❡ ✏❛♥✑ ❜❡❝❛✉s❡ t❤❡r❡ ♠❛② ❜❡ ♠❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡s ❢♦r ❡❛❝❤ ❢✉♥❝t✐♦♥✳ ❲❡ ❝❛♥ t❤✐♥❦ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ❛s ❛♥

●✐✈❡♥

❡q✉❛t✐♦♥✱ ❛♥ ❡q✉❛t✐♦♥ ❢♦r ❢✉♥❝t✐♦♥s✿ f,

s♦❧✈❡ ❢♦r

F

●✐✈❡♥

∆F =f ∆x

❚❤✐s ❡q✉❛t✐♦♥ ❤❛s ✐♥✜♥✐t❡❧② ♠❛♥② s♦❧✉t✐♦♥s ✇❤❡♥

❞✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡♦r❡♠✱ ✐❢ F

f,

s♦❧✈❡ ❢♦r

F

dF =f dx

f

✐s ✐♥t❡❣r❛❜❧❡✳

❋✉rt❤❡r♠♦r❡✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡

✐s ♦♥❡ ♦❢ ✐ts ❛♥t✐❞❡r✐✈❛t✐✈❡s✱ t❤❡♥ t❤❡ s❡t ♦❢ ❛❧❧ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢

{F + C : C

r❡❛❧

f

❆♥t✐✲

✐s

}.

❚❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s ❛r❡ ❧✐♥❦❡❞ t♦ ❛♥t✐❞❡r✐✈❛t✐✈❡s ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✷✳✸✳✸✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ■ ❛♥❞ ■■ ■ ❋♦r ❛♥② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ f ♦♥ [a, b]✱ t❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜② F (x) =

Z

x

f dx a

✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ f ✳ ■■ ❋♦r ❛♥② ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ f ♦♥ [a, b] ❛♥❞ ❛♥② ♦❢ ✐ts ❛♥t✐❞❡r✐✈❛t✐✈❡s F ✱ ✇❡

✷✳✸✳

■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ❤❛✈❡

✶✶✶

Z

b a

f dx = F (b) − F (a) .

❲❤② t✇♦ ♣❛rts❄ ❇❡❝❛✉s❡ ✇❡ ❞❡❛❧ ✇✐t❤ t✇♦ ♦♣❡r❛t✐♦♥s ♦❢ ❝❛❧❝❧✉❧✉s ✕ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ✐♥t❡❣r❛t✐♦♥ ✕ ❛♥❞ ✇❡ ❝❛♥ ❝♦♠♣♦s❡ t❤❡♠ ✐♥ t✇♦ ✇❛②s✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠✱ t❤❡ ♦♣❡r❛t✐♦♥s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ✐♥t❡❣r❛t✐♦♥ ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r ❛s ❢♦❧❧♦✇s✿ ❋❚❈■✿

f

→

Z

❋❚❈■■✿ F →

d  dx

→ F →

 dx d  dx

→ f

→

Z

 dx

→

f

→ F +C

❚❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s ❛r❡ ♣r♦✈❡♥ ❢r♦♠ s❝r❛t❝❤ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❛❝q✉✐r❡ t❤❡ s❛♠❡ r❡s✉❧ts ❜② ❛♣♣❧②✐♥❣ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❤❡♦r❡♠s ❛❜♦✉t t❤❡ ❞❡r✐✈❛t✐✈❡s ✭❱♦❧✉♠❡ ✷✮✳ ❲❡ st❛rt ♦✈❡r✳ ❲❡ ❝❛♥ r❡st❛t❡ t❤❡

❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡♦r❡♠ ❛s ❢♦❧❧♦✇s✿ ❈♦r♦❧❧❛r② ✷✳✸✳✹✿ ❙❡t ♦❢ ❆♥t✐❞❡r✐✈❛t✐✈❡s

❙✉♣♣♦s❡ F ✐s ❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ♦♥ ❛♥ ✐♥t❡r✈❛❧✳ ❚❤❡♥ t❤❡ s❡t ♦❢ ❛❧❧ ♦❢ ✐ts ❛♥t✐❞❡r✐✈❛t✐✈❡s ✐s {F + C : C r❡❛❧ } . ❆s ✐t ✐s ♦❢t❡♥ t❤❡ ❝❛s❡ ✇✐t❤ ❡q✉❛t✐♦♥s✱ t❤❡r❡ s❡❡♠s t♦ ❜❡ ♠❛♥② ✭✐♥✜♥✐t❡❧② ♠❛♥②✮ s♦❧✉t✐♦♥s✳ ❇✉t ❛ ✈❡r② ✐♠♣♦rt❛♥t ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t ✐t s✉✣❝❡s t♦ ✜♥❞ ❥✉st ♦♥❡ ❛♥t✐✲❞❡r✐✈❛t✐✈❡✦ ❲❛r♥✐♥❣✦

❚❤❡ ❢♦r♠✉❧❛ F + C ✇♦r❦s ♦♥❧② ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ♦♥ ❛♥ ✐♥t❡r✈❛❧✳ ❚❤✐s ✐s ✇❤❛t t❤✐s s❡t ♦❢ ❢✉♥❝t✐♦♥s ✇♦✉❧❞ ❧♦♦❦ ❧✐❦❡✿

❚❤✐s ✐s t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✐t ♠❛② ❜❡ ❝❛❧❧❡❞ ❊①❡r❝✐s❡ ✷✳✸✳✺✿

x3

❋✐♥❞ ❛❧❧ F t❤❛t s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥✿

F ′ (x) = x3 .

t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡✳

✷✳✸✳

■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠

✶✶✷

❊①❡r❝✐s❡ ✷✳✸✳✻

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥

f

✐s ❞❡✜♥❡❞ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧

✶✳ ❚❤❡ ❣r❛♣❤s ♦❢ t✇♦ ❞✐✛❡r❡♥t ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ ✷✳ ❋♦r ❡✈❡r② ♣♦✐♥t

(x, y)

✇✐t❤

x

✇✐t❤✐♥

I✱

I ✳ Pr♦✈❡ t❤❛t t❤❡ f ♥❡✈❡r ✐♥t❡rs❡❝t✳

❢♦❧❧♦✇✐♥❣✿

t❤❡r❡ ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

f

t❤❡ ❣r❛♣❤ ♦❢ ✇❤✐❝❤ ♣❛ss❡s

t❤r♦✉❣❤ ✐t✳

❚❤❡ ♣r♦❜❧❡♠ t❤❡♥ ❜❡❝♦♠❡s t❤❡ ♦♥❡ ♦❢ ✜♥❞✐♥❣ ❛ s✐♥❣❧❡ ❢✉♥❝t✐♦♥

•

❢r♦♠ ✐ts ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

•

❢r♦♠ ✐ts ❞❡r✐✈❛t✐✈❡

F✱

❡✐t❤❡r

∆F ✱ ♦r ∆x

dF ✳ dx

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ r❡❝♦♥str✉❝t t❤❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛ ✏✜❡❧❞ ♦❢ s❧♦♣❡s✑✿

❖♥❡ ❝❛♥ ✐♠❛❣✐♥❡ ❛ ✢♦✇✐♥❣ ❧✐q✉✐❞ ✇✐t❤ ✐ts ❞✐r❡❝t✐♦♥ ❦♥♦✇♥ ❛t ❡✈❡r② ❧♦❝❛t✐♦♥✳ ❍♦✇ ❞♦ ✇❡ ✜♥❞ t❤❡ ♣❛t❤ ♦❢ ❛ ♣❛rt✐❝✉❧❛r ♣❛rt✐❝❧❡❄ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦r

❚❤❡ ♣r♦❝❡ss ♦❢ r❡❝♦♥str✉❝t✐♥❣ ❛ ❢✉♥❝t✐♦♥✱

✐♥t❡❣r❛t✐♦♥✳

F✱

❢r♦♠ ✐ts ❞❡r✐✈❛t✐✈❡✱

f✱

✐s ❝❛❧❧❡❞ ❛♥t✐✲

❚❤❡ ✐♥t❡❣r❛t✐♦♥ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ s♦❧✈❡❞ ♦♥ s❡✈❡r❛❧ ♦❝❝❛s✐♦♥s ❢♦r t❤❡ ❢♦r♠❡r✱ ❞✐s❝r❡t❡ ❝❛s❡ ✕ ✈❡❧♦❝✐t② ❢r♦♠ ❛❝❝❡❧❡r❛t✐♦♥ ❛♥❞ ❧♦❝❛t✐♦♥ ❢r♦♠ ✈❡❧♦❝✐t② ✕ ✈✐❛ t❤❡s❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s✿

F (xn+1 ) = F (xn ) + f (cn )∆xn . ❋♦r ❡❛❝❤ ❧♦❝❛t✐♦♥✱ ✇❡ ❧♦♦❦ ✉♣ t❤❡ ✈❡❧♦❝✐t②✱ ✜♥❞ t❤❡ ♥❡①t ❧♦❝❛t✐♦♥✱ ❛♥❞ r❡♣❡❛t✿

■❢ t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❛r❡ ❝❧♦s❡ ❡♥♦✉❣❤ t♦ ❡❛❝❤ ♦t❤❡r✱ t❤❡s❡ ♣♦✐♥ts ❢♦r♠ ❝✉r✈❡s✿

✷✳✸✳

■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠

✶✶✸

❋♦r t❤❡ ❧❛tt❡r✱ ❝♦♥t✐♥✉♦✉s ❝❛s❡✱ t❤✐s ✐s ❛ ❝❤❛❧❧❡♥❣✐♥❣ ♣r♦❜❧❡♠✿ ❍♦✇ ❞♦❡s ♦♥❡ ♣❧♦t ❛ ❝✉r✈❡ t❤❛t ❢♦❧❧♦✇s t❤❡s❡ ✕ ✐♥✜♥✐t❡❧② ♠❛♥② ✕ t❛♥❣❡♥ts❄ ❚♦ ❜❡❣✐♥ ✇✐t❤✱ ✇❡ ❥✉st tr② t♦ r❡✈❡rs❡ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❲❡ ✇✐❧❧ tr② t♦ ❝♦♥str✉❝t ❛ t❤❡♦r② ♦❢ ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ t❤❛t ♠❛t❝❤❡s ✕ t♦ t❤❡ ❞❡❣r❡❡ ♣♦ss✐❜❧❡ ✕ t❤❛t ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❍❡r❡ ✐s ❛ s❤♦rt

❧✐st ♦❢ ❞❡r✐✈❛t✐✈❡s ♦❢ ❢✉♥❝t✐♦♥s ✭❢♦r ❛❧❧ x ❢♦r ✇❤✐❝❤ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡✮✿ ❢✉♥❝t✐♦♥ −→ ❞❡r✐✈❛t✐✈❡

xr

ex

rxr−1 1 x ex

sin x

cos x

cos x

− sin x

ln x

❛♥t✐❞❡r✐✈❛t✐✈❡ ←− ❢✉♥❝t✐♦♥ ❚♦ ✜♥❞ ❛♥t✐❞❡r✐✈❛t✐✈❡s✱ ✐✳❡✳✱ ✐♥t❡❣r❛❧s✱ r❡✈❡rs❡ t❤❡ ♦r❞❡r✿

◮ ❘❡❛❞ ❡❛❝❤ ❧✐♥❡ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✦ ❊①❛♠♣❧❡ ✷✳✸✳✼✿

sin

❛♥❞

cos

❲❤❛t ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ cos x❄ ❲❡ ♥❡❡❞ t♦ s♦❧✈❡ ❢♦r F ✿

F ′ (x) = cos x . ❏✉st ✜♥❞ cos x ✐♥ t❤❡ r✐❣❤t ❝♦❧✉♠♥ ♦❢ t❤❡ t❛❜❧❡✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❧❡❢t ✐s sin x✳ ❚❤❛t✬s t❤❡ ❛♥s✇❡r✿ F (x) = sin x✦ ❲❤❛t ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ sin x❄ ❙♦❧✈❡✿

F ′ (x) = sin x . ❏✉st ✜♥❞ sin x ✐♥ t❤❡ r✐❣❤t ❝♦❧✉♠♥✳ ■t✬s ♥♦t t❤❡r❡✳✳✳ ❜✉t − sin x ✐s✦ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❧❡❢t ✐s cos x✳ ❚❤❡♥ ✭❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✮ t❤❡ s♦❧✉t✐♦♥ ♠✉st ❜❡ − sin x✳ ■t✬s t❤❛t s✐♠♣❧❡✦ ❲❡ ♠❛② ♥❡❡❞ s♦♠❡ t✇❡❛❦✐♥❣ t♦ ♠❛❦❡ t❤❡ ❢♦r♠✉❧❛s ❛❜♦✉t t♦ ❡♠❡r❣❡ ❛s ❡❛s② t♦ ❛♣♣❧② ❛s t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡s✳ ❊①❛♠♣❧❡ ✷✳✸✳✽✿ ♣♦✇❡r ❢♦r♠✉❧❛

❋♦r ❡①❛♠♣❧❡✱ ❧❡t✬s ✜♥❞ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ xn ✳ ❯s❡ t❤❡ P♦✇❡r ❞✐✈✐❞❡ ❜② r✱ ❛♥❞ ❛♣♣❧② t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ✿ r ′

(x ) = rx

r−1

❋♦r♠✉❧❛ ❢♦r ❞✐✛❡r❡♥t✐❛t✐♦♥ ✭t❤❡ ✜rst r♦✇✮✱

1 =⇒ (xr )′ = xr−1 =⇒ r

❲❡ t❤❡♥ s✐♠♣❧✐❢② t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❜② s❡tt✐♥❣ r − 1 = s✿ ′  1 s+1 x = xs . s+1



1 r x r

′

= xr−1 .

✷✳✸✳

■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠

✶✶✹

❲❡ ♠❛❦❡ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ❛♥❞ ✇❡ ❤❛✈❡ t❤❡

P♦✇❡r ❋♦r♠✉❧❛ ❢♦r ❛♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❲❡

❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❆♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ❇✉t ✇❤❛t ✐❢

s = −1❄

xs

✐s

1 xs+1 ✱ ♣r♦✈✐❞❡❞ s+1

s 6= −1✳

❚❤❡♥ ✇❡ r❡❛❞ t❤❡ ❛♥s✇❡r ❢r♦♠ t❤❡ ♥❡①t ❧✐♥❡ ✐♥ t❤❡ t❛❜❧❡✿ ❆♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

x−1

✐s

ln |x|✳

1 ❧♦✇❡r✑ ❤❛s ❛♥ ❡①❝❡♣t✐♦♥✱ 0✲♣♦✇❡r✱ ❛♥❞ t❤❡ r✉❧❡ t❤❛t ✏t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♣♦✇❡r ❢✉♥❝t✐♦♥ ✐s ❛ ♣♦✇❡r ❢✉♥❝t✐♦♥ ♦❢ ❞❡❣r❡❡ 1 ❤✐❣❤❡r✑

❙♦✱ t❤❡ r✉❧❡ t❤❛t ✏t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♣♦✇❡r ❢✉♥❝t✐♦♥ ✐s ❛ ♣♦✇❡r ❢✉♥❝t✐♦♥ ♦❢ ❞❡❣r❡❡ t❤❡

❤❛s ❛♥ ❡①❝❡♣t✐♦♥ t♦♦✳ ❚❛❦✐♥❣ t❤❡ r❡st ♦❢ t❤❡s❡ r♦✇s✱ ✇❡ ❤❛✈❡ ❛

❧✐st ♦❢ ✐♥t❡❣r❛❧s ♦❢ ❢✉♥❝t✐♦♥s✱ ♦♥ ♦♣❡♥ ✐♥t❡r✈❛❧s✿

❢✉♥❝t✐♦♥

−→

❛♥t✐❞❡r✐✈❛t✐✈❡✴✐♥t❡❣r❛❧

xs 1 x

ln |x|

ex

ex

1 xs+1 , s+1

sin x

− cos x

cos x

sin x

❞❡r✐✈❛t✐✈❡

←−

s 6= −1

❢✉♥❝t✐♦♥

❊①❛♠♣❧❡ ✷✳✸✳✾✿ ❞♦♠❛✐♥s

❊❛❝❤ ❢♦r♠✉❧❛ ✐s ♦♥❧② ✈❛❧✐❞ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ♦♥ ✇❤✐❝❤ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ✐s ❞❡✜♥❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ✐♥t❡r♣r❡t t❤❡ s❡❝♦♥❞ r♦✇ ❛s ❢♦❧❧♦✇s✿ • ln(x) ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ x1 ♦♥ t❤❡ ✐♥t❡r✈❛❧ (0, +∞)✱ ❛♥❞ • ln(−x) ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ x1 ♦♥ t❤❡ ✐♥t❡r✈❛❧ (−∞, 0)✳

r✉❧❡s ♦❢ ✐♥t❡❣r❛t✐♦♥✳ ❙✉♠ ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s

◆❡①t✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t❤❡ ❋✐rst✱ ❝♦♥s✐❞❡r t❤❡

✭❱♦❧✉♠❡ ✷✮✿

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ s✉♠ ✐s t❤❡ s✉♠ ♦❢ t❤❡

❞❡r✐✈❛t✐✈❡s❀ ✐✳❡✳✱

(f + g)′ = f ′ + g ′ . ▲❡t✬s r❡❛❞ t❤❛t ❢♦r♠✉❧❛ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✿ ❚❤❡♦r❡♠ ✷✳✸✳✶✵✿ ❙✉♠ ❘✉❧❡ ❢♦r ■♥t❡❣r❛❧s ❆♥ ✐♥t❡❣r❛❧ ♦❢ t❤❡ s✉♠ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ✐♥t❡❣r❛❧s✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ■❢

F

✐s t❤❡ ✐♥t❡❣r❛❧ ♦❢

f

G

✐s t❤❡ ✐♥t❡❣r❛❧ ♦❢

g✱

t❤❡♥

F +G

✐s t❤❡ ✐♥t❡❣r❛❧ ♦❢

f + g✳

❛♥❞

✷✳✸✳

■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠

✶✶✺

Pr♦♦❢✳

❲❡ ❛♣♣❧② t❤❡

❙✉♠ ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s t♦ ❝♦♥✜r♠✿ (F (x) + G(x))′ = F ′ (x) + G′ (x) = f (x) + g(x) .

❊①❡r❝✐s❡ ✷✳✸✳✶✶

❲❤❛t ❛❜♦✉t t❤❡ ❝♦♥✈❡rs❡❄ ❊①❛♠♣❧❡ ✷✳✸✳✶✷✿ s✉♠s

❙♦❧✈❡ ❢♦r F ✿

F ′ (x) = x2 + sin x .

❚❤❡ ❡q✉❛t✐♦♥ ✐s s♦❧✈❡❞ ❜② s♦❧✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❡q✉❛t✐♦♥s✿ ❢♦r G : G′ (x) = x2 ❛♥❞ ❢♦r H : H ′ (x) = sin x . ❚❤❡ s♦❧✉t✐♦♥s ❛r❡ ❢♦✉♥❞ ✐♥ t❤❡ t❛❜❧❡✿ 1 G(x) = x3 ❛♥❞ H(x) = − cos x . 3

❆❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡ t❤❡ ❛♥s✇❡r ♦♥ (−∞, +∞)✿ 1 F (x) = x3 − cos x + C . 3 ❊①❡r❝✐s❡ ✷✳✸✳✶✸

❯s✐♥❣ t❤✐s r✉❧❡✱ ✜♥❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢ ln x2 ✳ ❈♦♠♣❛r❡✿ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ s✉♠ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s✳ ❚❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤❡ s✉♠ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ✐♥t❡❣r❛❧s✳ ❙✐♠✐❧❛r❧②✱ ❝♦♥s✐❞❡r t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ t❤❡ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡❀ ✐✳❡✳✱

❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s

✭❱♦❧✉♠❡ ✷✮✿ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♠✉❧t✐♣❧❡ ✐s

(cf )′ = cf ′ .

▲❡t✬s r❡❛❞ t❤❛t ❢♦r♠✉❧❛ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✿ ❚❤❡♦r❡♠ ✷✳✸✳✶✹✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ■♥t❡❣r❛❧s ❚❤❡ ✐♥t❡❣r❛❧ ♦❢ ❛ ♠✉❧t✐♣❧❡ ✐s t❤❡ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ■❢

F c

✐s t❤❡ ✐♥t❡❣r❛❧ ♦❢

f

✐s ❛ ❝♦♥st❛♥t✱

t❤❡♥

cF

✐s t❤❡ ✐♥t❡❣r❛❧ ♦❢

cf ✳

❛♥❞

✷✳✸✳ ■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠

✶✶✻

Pr♦♦❢✳

❲❡ ❛♣♣❧② t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s ✿ (cF (x))′ = cF ′ (x) = cf (x) . ❊①❡r❝✐s❡ ✷✳✸✳✶✺

❲❤❛t ❛❜♦✉t t❤❡ ❝♦♥✈❡rs❡❄ ❊①❛♠♣❧❡ ✷✳✸✳✶✻✿ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡s

❙♦❧✈❡ ❢♦r F ✿

F ′ (x) = 3 sin x .

❲❡ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ ❜② s♦❧✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿ ❢♦r G : G′ (x) = sin x . ❚❤❡ s♦❧✉t✐♦♥ ✐s ❢♦✉♥❞ ✐♥ t❤❡ t❛❜❧❡✿ G(x) = − cos x .

❆❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡ ❛♥ ❛♥s✇❡r ♦♥ (−∞, +∞)✿ F (x) = 3(− cos x) + C . ❊①❡r❝✐s❡ ✷✳✸✳✶✼

❯s✐♥❣ t❤✐s r✉❧❡✱ ✜♥❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢ ex+3 ✳ ❈♦♠♣❛r❡✿ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ✐s t❤❡ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✳ ❚❤❡ ✐♥t❡❣r❛❧ ♦❢ ❛ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ✐s t❤❡ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧✳ ❆s ✇❡ ❦♥♦✇ ❢r♦♠ t❤❡ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡♦r❡♠✱ ❡✈❡r② ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❝♦♠❡s ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ♦t❤❡rs✿ F → F + C ❢♦r ❡✈❡r② r❡❛❧ C , ♦♥ ❡✈❡r② ♦♣❡♥ ✐♥t❡r✈❛❧✳ ❚♦❣❡t❤❡r t❤❡② ❢♦r♠ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦r t❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❊①❛♠♣❧❡ ✷✳✸✳✶✽✿ ❢r❡❡ ❢❛❧❧

❲❡ ❝❛♥ ♠❛❦❡ ♦✉r ❛♥❛❧②s✐s ♦❢ ❢r❡❡ ❢❛❧❧ ♠♦r❡ s♣❡❝✐✜❝✿ ❋✉♥❝t✐♦♥s ♦❢ t✐♠❡ ❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❝♦♥st❛♥t✳ a = −g ❚❤❡ ✈❡❧♦❝✐t② ✐s ❧✐♥❡❛r✳

v = −gt + C

❚❤❡ ❧♦❝❛t✐♦♥ ✐s q✉❛❞r❛t✐❝✳ p = −gt2 /2 + Ct + K ❚❤❡ ❝♦♥st❛♥ts C ❛♥❞ K ❝♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ tr✐♣s ♦❢ t❤❡ ❜❛❧❧✳

=⇒ =⇒

✷✳✸✳

✶✶✼

■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠

❚❤✐s ✐s ❤♦✇ ✇❡ r❡✇r✐t❡ t❤❡ ❛❜♦✈❡ ❧✐st✿ Z

xs dx =

Z

Z

Z

1 s+1 x +C, s+1

❢♦r s 6= −1

1 dx = ln x + C x Z ex dx = ex + C sin x dx = − cos x + C cos x dx = sin x + C

❲❡ r❡st❛t❡ t❤❡ r✉❧❡s t♦♦✳ ❙✉♠ ❘✉❧❡✿

Z

(f + g) dx =

Z

f dx +

Z

g dx

❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✿

Z

(cf ) dx = c

Z

f dx

❊①❡r❝✐s❡ ✷✳✸✳✶✾

❙t❛t❡ t❤❡ ▲✐♥❡❛r ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ❢♦r ✐♥t❡❣r❛❧s✳ ❲✐t❤ t❤❡s❡ r✉❧❡s✱ ✇❤❡♥ ❛♣♣❧✐❝❛❜❧❡✱ ✐♥t❡❣r❛t✐♦♥ ✐s ✈❡r② s✐♠✐❧❛r t♦ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❊①❛♠♣❧❡ ✷✳✸✳✷✵✿ r✉❧❡s ♦❢ ✐♥t❡❣r❛t✐♦♥

❋✐♥❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢

3x2 + 5ex + cos x .

❖♥❡ ❝❛♥ ✐♠❛❣✐♥❡ ✇❤❛t ❤❡✬❞ ❞♦ t♦ ❞✐✛❡r❡♥t✐❛t❡ ❛♥❞ t❤❡♥ ❢♦❧❧♦✇ t❤❡ s❛♠❡ st❡♣s ❜✉t ✇✐t❤ t❤❡ ✐♥t❡❣r❛t✐♦♥ ❢♦r♠✉❧❛s ❛♥❞ r✉❧❡s ✉s❡❞ ✐♥st❡❛❞✳ ❉✐✛❡r❡♥t✐❛t✐♦♥✿

(3x2 + 5ex + cos x)′ = (3x2 )′ + 5(ex )′ + (3 sin x)′ ❙❘ = 3(x2 )′ + 5(ex )′ + 3(sin x)′ = 3 · 6x + 5ex + 3 cos x .

❈▼❘ ❚❛❜❧❡

■♥t❡❣r❛t✐♦♥✿ Z

2

x

′

Z

Z

Z

(3x ) dx + 5(e ) dx + (3 sin x)dx ❙❘ Z Z Z 2 x = 3 (x ) dx + 5 (e ) dx + 3 (sin x)dx ❈▼❘

(3x + 5e + cos x) dx =

2

x

= 3 · x3 /3 + 5ex + 3(− cos x) + C .

❚❛❜❧❡

❏✉st ❛s ✇❤❡♥ s♦❧✈✐♥❣ ❡q✉❛t✐♦♥s✱ ✇❡ ❝❛♥ ❡❛s✐❧② ❝♦♥✜r♠ t❤❛t ♦✉r ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ❝♦rr❡❝t✱ ❜② s✉❜st✐t✉✲

✷✳✸✳

■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠

t✐♦♥✳ ■♥ t❤✐s ❝❛s❡✱ ✇❡

✶✶✽

❞✐✛❡r❡♥t✐❛t❡ t❤❡ ✐♥t❡❣r❛❧ ✿ (x3 + 5ex + 3 sin x)′ = (x3 )′ + 5(ex )′ + (3 sin x)′ = 3x2 + 5ex + 3 cos x .

❚❤✐s ✐s t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥✦ ❚❤❡ ❛♥s✇❡r ❝❤❡❝❦s ♦✉t✳

❊①❡r❝✐s❡ ✷✳✸✳✷✶

❋✐♥❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢

5e3x+2 − ee ✳

❇❡❧♦✇✱ ✇❡ ❤❛✈❡ t❤❡s❡ t✇♦ ❞✐❛❣r❛♠s t♦ ✐❧❧✉str❛t❡ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♦❢ ✐♥t❡❣r❛❧s ✇✐t❤ ❛❧❣❡❜r❛✿

f, g   + y

R

←−−−− R

′

f ,g   + y

R

′

f ←−−−−   ·c y

f + g ←−−−− f ′ + g ′

cf

R

←−−−−

f′   ·c y

cf ′

❚❤❡ ❛rr♦✇s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛r❡ r❡✈❡rs❡❞✦ ❲❡ st❛rt ✇✐t❤ ❛ ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s ❛t t♦♣ r✐❣❤t✱ t❤❡♥ ✇❡ ♣r♦❝❡❡❞ ✐♥ t✇♦ ✇❛②s✿

•

▲❡❢t✿ ✐♥t❡❣r❛t❡ t❤❡♠✳ ❚❤❡♥ ❞♦✇♥✿ ❛❞❞ t❤❡ r❡s✉❧ts✳

•

❉♦✇♥✿ ❛❞❞ t❤❡♠✳ ❚❤❡♥ ❧❡❢t✿ ✐♥t❡❣r❛t❡ t❤❡ r❡s✉❧ts✳

❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡✦ ❙♦ ❢❛r✱ t❤✐s ✐s ✈❡r② s✐♠✐❧❛r t♦ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❚❤❡ str❛t❡❣② ✐s t❤❡ s❛♠❡✿ ❞✐✈✐❞❡ ❛♥❞ ❝♦♥❝✉r✳ ❙♣❧✐t ❛❞❞✐t✐♦♥ ✇✐t❤ t❤❡ ❙✉♠ ❘✉❧❡✱ t❤❡♥ ❢❛❝t♦r ♦✉t t❤❡ ❝♦❡✣❝✐❡♥ts ✇✐t❤ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡✱ t❤❡♥ ❛♣♣❧② t❤❡ t❛❜❧❡ r❡s✉❧ts t♦ t❤❡s❡ ♣✐❡❝❡s✳ ❯♥❢♦rt✉♥❛t❡❧②✱ t❤✐s ✐s ✇❤❡r❡ t❤❡ s✐♠✐❧❛r✐t✐❡s st♦♣✳ ❚❤❡r❡ ✐s ♥♦ ❛♥❛❧♦❣ ♦❢ t❤❡ Pr♦❞✉❝t ❘✉❧❡ ❢♦r t❤❡ ❉❡r✐✈❛t✐✈❡s✿ ❚❤❡

❚❤❡

❞❡r✐✈❛t✐✈❡

✐♥t❡❣r❛❧

♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❢✉♥❝t✐♦♥s

❝❛♥

❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡✐r

❞❡r✐✈❛t✐✈❡s

♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❢✉♥❝t✐♦♥s

❝❛♥♥♦t

❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡✐r

✐♥t❡❣r❛❧s

❛♥❞ t❤❡ ❢✉♥❝t✐♦♥s t❤❡♠s❡❧✈❡s✳

❛♥❞ t❤❡ ❢✉♥❝t✐♦♥s t❤❡♠s❡❧✈❡s✳

❙✐♠✐❧❛r❧② t❤❡r❡ ✐s ♥♦ ◗✉♦t✐❡♥t ❘✉❧❡✱ ♥♦r t❤❡ ❈❤❛✐♥ ❘✉❧❡✱ ❢♦r ✐♥t❡❣r❛t✐♦♥✳ ❚❤✐s ❞✐✛❡r❡♥❝❡ ❤❛s ♣r♦❢♦✉♥❞ ❝♦♥s❡q✉❡♥❝❡s✳ ❲❡ ❝❛♥ st❛rt ✇✐t❤ ❥✉st t❤❡s❡ ❢✉♥❝t✐♦♥s✿

xs , sin x, ex . ❚❤❡♥ ✕ ❜② ❛♣♣❧②✐♥❣ t❤❡ ❢♦✉r ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✱ ❝♦♠♣♦s✐t✐♦♥✱ ❛♥❞ ✐♥✈❡rt✐♥❣ ✕ ✇❡ ❝❛♥ ❝♦♥str✉❝t ❛ ❣r❡❛t ✈❛r✐❡t② ♦❢ ❢✉♥❝t✐♦♥s✳ ▲❡t✬s ❝❛❧❧ t❤❡♠ ✏❡❧❡♠❡♥t❛r② ❢✉♥❝t✐♦♥s✑✳ ❇❡❝❛✉s❡ ♦❢ t❤❡ ✇❛② t❤❡② ❛r❡ ❝♦♥str✉❝t❡❞✱

❛❧❧

♦❢

t❤❡♠ ❝❛♥ ❜❡ ❡❛s✐❧② ❞✐✛❡r❡♥t✐❛t❡❞ ✇✐t❤ t❤❡ r✉❧❡s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ t❤✉s ♣r♦❞✉❝✐♥❣ ♦t❤❡r ❡❧❡♠❡♥t❛r② ❢✉♥❝t✐♦♥s ✭❧❡❢t✮✿

✷✳✸✳

■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠

✶✶✾

❍♦✇❡✈❡r✱ ❝♦♥tr❛r② t♦ ✇❤❛t t❤❡ ❛❜♦✈❡ ❧✐st ♠✐❣❤t s✉❣❣❡st✱ ✐♥t❡❣r❛t✐♦♥ ✇✐❧❧ ♦❢t❡♥ t❛❦❡ ✉s ♦✉ts✐❞❡ ♦❢ t❤❡ r❡❛❧♠ ♦❢ ❡❧❡♠❡♥t❛r② ❢✉♥❝t✐♦♥s ✭r✐❣❤t✮✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ♥❡✇ ❢✉♥❝t✐♦♥✱ ❝❛❧❧❡❞ t❤❡ ❢♦r t❤✐s ✐♠♣♦rt❛♥t ✐♥t❡❣r❛❧✿

2 erf(x) = √ π

❚❤❡ r❡s✉❧t ✇✐❧❧ ❜❡ t❤❡ s❛♠❡ ✐❢ ✇❡

❡①❝❧✉❞❡

2

e−x dx .

❢r♦♠ t❤❡ ✏❡❧❡♠❡♥t❛r② ❢✉♥❝t✐♦♥s✑ ❡✐t❤❡r t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s

♦r t❤❡ ❡①♣♦♥❡♥t✳ ❚❤❡ r❡s✉❧t ✇✐❧❧ ❜❡ t❤❡ s❛♠❡ ✐❢ ✇❡ ❲✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡

Z

●❛✉ss ❡rr♦r ❢✉♥❝t✐♦♥✱ ♠✉st ❜❡ ❝r❡❛t❡❞

✐♥❝❧✉❞❡

❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡♦r❡♠✱

♠♦r❡ ❢✉♥❝t✐♦♥s t♦ t❤❡ ❧✐st✳

✇❡ ❝❛♥ ❝❧❛✐♠ t❤❛t ✇❡ ❤❛✈❡ ❢♦✉♥❞

♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ♦♥ ♦✉r ❧✐st✱ ♦✈❡r ♦♣❡♥ ✐♥t❡r✈❛❧s ✇✐t❤✐♥ t❤❡ ❞♦♠❛✐♥s ♦❢ ✐♥t❡❣r❛❧✱ ✐✳❡✳✱

❛❧❧ t❤❡

❛♥t✐❞❡r✐✈❛t✐✈❡s ✐♥t❡❣r❛❧ ♦❢ t❤❡

❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥✿

❉❡✜♥✐t✐♦♥ ✷✳✸✳✷✷✿ ❣❡♥❡r❛❧ ❛♥t✐❞❡r✐✈❛t✐✈❡ ❛♥❞ ✐♥❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ ❋♦r ❛ ❣✐✈❡♥ ❢✉♥❝t✐♦♥

♦✈❡r ♦♣❡♥ ✐♥t❡r✈❛❧ I

f✱

t❤❡ ❣❡♥❡r❛❧ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦r t❤❡ ✐♥❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ ♦❢ f

✐s ❞❡✜♥❡❞ ❜②✿

Z

f dx = F (x) + C ,

I ✱ ✐✳❡✳✱ F ′ = f ✱ ✉♥❞❡rst♦♦❞ ❛s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ s✉❝❤ ❢✉♥❝t✐♦♥s ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ r❡❛❧ ♥✉♠❜❡rs C ✳ ❚❤✐s ❝♦❧❧❡❝t✐♦♥ ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ✐♥❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ ♦❢ f ✳ ✇❤❡r❡

F

✐s ❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

f

♦♥

❊①❛♠♣❧❡ ✷✳✸✳✷✸✿ ❤♦✇ ♠❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡s❄ ❚❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡s✱ ❜✉t t❤❡r❡ ✐s ♠♦r❡ t♦ ✐t✳ ▲❡t✬s t❛❦❡ ❛ ♠♦r❡ ❝❛r❡❢✉❧ ❧♦♦❦ ❛t ♦♥❡ ❧✐♥❡ ♦♥ t❤❡ ❧✐st✿

Z

1 ❄❄❄ dx === ln |x| + C, x

x 6= 0 .

❚❤✐s ❢♦r♠✉❧❛ ✐s ✐♥t❡♥❞❡❞ t♦ ♠❡❛♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✶✳ ❲❡ ❤❛✈❡ ❝❛♣t✉r❡❞ ✐♥✜♥✐t❡❧② ♠❛♥② ✕ ♦♥❡ ❢♦r ❡❛❝❤ r❡❛❧ ♥✉♠❜❡r

❛❧❧ ♦❢ t❤❡♠✳ ❆♥t✐✲❞✐✛❡r❡♥t✐❛t✐♦♥ ❚❤❡♦r❡♠

C

✕ ❛♥t✐❞❡r✐✈❛t✐✈❡s✳

✷✳ ❲❡ ❤❛✈❡ ❝❛♣t✉r❡❞ ❍♦✇❡✈❡r✱ t❤❡ ❞♦♠❛✐♥ ♦❢

❛♣♣❧✐❡s ♦♥❧② t♦

1/x ❝♦♥s✐sts ♦❢ t✇♦ r❛②s (−∞, 0) ❛♥❞ (0, +∞)✳

▼❡❛♥✇❤✐❧❡✱ t❤❡

❆s ❛ r❡s✉❧t✱ ✇❡ s♦❧✈❡ t❤✐s ♣r♦❜❧❡♠

♦♥ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ✐♥t❡r✈❛❧s✳ ❚❤❡♥ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢

• ln(−x) + C ♦♥ (−∞, 0)✱ • ln(x) + C ♦♥ (0, +∞)✳

♦♥❡ ✐♥t❡r✈❛❧ ❛t ❛ t✐♠❡✳

1/x

s❡♣❛r❛t❡❧②

❛r❡✿

❛♥❞

❇✉t ✐❢ ♥♦✇ ✇❡ ✇❡r❡ t♦ ❝♦♠❜✐♥❡ ❡❛❝❤ ♦❢ t❤❡s❡ ♣❛✐rs ♦❢ ❢✉♥❝t✐♦♥s ✐♥t♦ ♦♥❡✱

F ✱ ❞❡✜♥❡❞ ♦♥ (−∞, 0)∪(0, +∞)✱

✇❡ ✇♦✉❧❞ r❡❛❧✐③❡ t❤❛t✱ ❡✈❡r② t✐♠❡✱ t❤❡ t✇♦ ❝♦♥st❛♥ts ♠✐❣❤t ❜❡ ❞✐✛❡r❡♥t✳ ❆❢t❡r ❛❧❧✱ t❤❡② ❤❛✈❡ ♥♦t❤✐♥❣ t♦ ❞♦ ✇✐t❤ ❡❛❝❤ ♦t❤❡r✦ ❲❡ ✐❧❧✉str❛t❡ t❤❡ ✇r♦♥❣ ✭✐♥❝♦♠♣❧❡t❡✮ ❛♥s✇❡r ♦♥ t❤❡ ❧❡❢t✱ ❛♥❞ t❤❡ ❝♦rr❡❝t ♦♥❡ ♦♥ t❤❡ r✐❣❤t✿

✷✳✸✳

■♥t❡❣r❛t✐♦♥ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠

✶✷✵

❚❤❡ ✐♠❛❣❡ ♦♥ t❤❡ ❧❡❢t✱ ❛s ✇❡❧❧ ❛s t❤❡ ❢♦r♠✉❧❛ ✇❡ st❛rt❡❞ ✇✐t❤✱ ♠✐❣❤t s✉❣❣❡st t❤❛t ❛❧❧ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✬s ❛♥t✐❞❡r✐✈❛t✐✈❡s ❛r❡ ❡✈❡♥ ❢✉♥❝t✐♦♥s✳ ❚❤❡ ✐♠❛❣❡ ♦♥ t❤❡ r✐❣❤t s❤♦✇s ❛ s✐♥❣❧❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ✭✐♥ r❡❞✮ ❜✉t ✐ts t✇♦ ❜r❛♥❝❤❡s ❞♦♥✬t ❤❛✈❡ t♦ ♠❛t❝❤✦ ❆❧❣❡❜r❛✐❝❛❧❧②✱ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

1 ✕ ♦♥ t❤❡ ✇❤♦❧❡ ❞♦♠❛✐♥ ✕ ✐s ❣✐✈❡♥ ❜② t❤✐s ♣✐❡❝❡✇✐s❡✲❞❡✜♥❡❞ x

❢✉♥❝t✐♦♥✿

F (x) = ■t ❤❛s

(

ln(−x) + C ln(x) + K

t✇♦ ♣❛r❛♠❡t❡rs ✐♥st❡❛❞ ♦❢ t❤❡ ✉s✉❛❧ ♦♥❡✳

❢♦r ❢♦r

x x

✐♥ ✐♥

(−∞, 0), (0, +∞).

❚❤❡ ♥✉♠❜❡r ♠❛t❝❤❡s t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ✏❝♦♠♣♦♥❡♥ts✑

♦❢ t❤❡ ❞♦♠❛✐♥✳

❊①❡r❝✐s❡ ✷✳✸✳✷✹ ❱❡r✐❢② t❤❛t t❤✐s ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

1/x✳

❊①❡r❝✐s❡ ✷✳✸✳✷✺ ■♥ ❛ s✐♠✐❧❛r ❢❛s❤✐♦♥✱ ❡①❛♠✐♥❡ t❤❡ P♦✇❡r ❋♦r♠✉❧❛ ❛❜♦✈❡ ❢♦r

s < −1✳

❊①❛♠♣❧❡ ✷✳✸✳✷✻✿ ✜♥❞ ❣r❛♣❤s ❚❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦♥ ♦✉r ❧✐st ✇❡r❡ ❞✐s❝♦✈❡r❡❞ ❜② r❡❛❞✐♥❣ t❤❡ r❡s✉❧ts ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❜❛❝❦✇❛r❞s✳ ❲❡ ❝❛♥ ❞♦ t❤❡ s❛♠❡ ❢♦r ❣r❛♣❤s✳ ❇❡❧♦✇✱ t❤❡ ❞❡r✐✈❛t✐✈❡✬s ❣r❛♣❤ ✭❣r❡❡♥✮ ✇❛s ❢♦✉♥❞ ❢r♦♠ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✭r❡❞✮ ❜② ❧♦♦❦✐♥❣ ❛t t❤❡ ′ ′ ′ ♠♦♥♦t♦♥✐❝ ❜❡❤❛✈✐♦r ♦❢ f ✭❡✐t❤❡r f > 0 ♦r f < 0✱ ❛♥❞ ❧♦❝❛❧ ❡①tr❡♠❡ ♣♦✐♥ts✿ f = 0✮✿

■♥ s✉♠♠❛r②✱ ✇❡ ❧♦♦❦ ❛t ■♥ r❡✈❡rs❡✱ ✇❡ ❧♦♦❦ ❛t

ց, ր

+, −

♦❢

♦❢

f′

f

t♦ ✜♥❞

t♦ ✜♥❞

❍❡r❡ ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢ ❤♦✇ t❤❡ ❣r❛♣❤ ♦❢

+, −

ց, ր

f

♦❢

♦❢

f ′✳

f✳

✐s ❢♦✉♥❞ ❢r♦♠ t❤❡ ❣r❛♣❤ ♦❢ ✐ts ❞❡r✐✈❛t✐✈❡✿

✷✳✹✳ ▲✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧

✶✷✶

❚♦ s❤♦✇ t❤❡ ❝♦♠♣❧❡t❡ ❛♥s✇❡r✱ ✇❡ ❤❛✈❡ t♦ s❤♦✇ ♠✉❧t✐♣❧❡ ❛♥t✐❞❡r✐✈❛t✐✈❡s✿

❲❤❡♥ t❤❡ ✐♥✐t✐❛❧ ✭♦r ✜♥❛❧✱ ♦r ♠✐❞✲✢✐❣❤t✮ st❛t❡ ✐s ❦♥♦✇♥✱ ✇❡ ♣✐❝❦ ❛ s♣❡❝✐✜❝ ❝✉r✈❡ ❢r♦♠ t❤✐s s❡t✳ ❊①❡r❝✐s❡ ✷✳✸✳✷✼

❋✐♥❞ t❤❡ ✐♥✢❡❝t✐♦♥ ♣♦✐♥ts✳

✷✳✹✳ ▲✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧

◆✉♠❡r♦✉s q✉❛♥t✐t✐❡s ❛r❡ ❞❡✜♥❡❞ ✈✐❛ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ ❛r❡❛s ❛♥❞ ❞✐s♣❧❛❝❡♠❡♥ts ❛s ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✶❀ ❛♥❞ ✇❡✐❣❤ts✱ ✈♦❧✉♠❡s✱ ❧❡♥❣t❤s✱ ✢✉①❡s✱ ✇♦r❦✱ ❛♥❞ ♠❛♥② ♠♦r❡ t♦ ❜❡ ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✸✳ ❚❤❡ ❧❛tt❡r ♣❛rt ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ❛❧❧♦✇s ✉s t♦ ❝♦♠♣✉t❡ ❛♥②t❤✐♥❣ ❞❡✜♥❡❞ t❤✐s ✇❛② ❜② ♠❡❛♥s ♦❢ ❛ s✐♠♣❧❡ s✉❜st✐t✉t✐♦♥ ✕ ❛s ❧♦♥❣ ❛s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ❝❛♥ ❜❡ ❢♦✉♥❞✦ ❲❡ ✇✐❧❧ r❡❢❡r t♦ t❤❡♠ ❛s ✐♥❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s ♦r s✐♠♣❧② ✐♥t❡❣r❛❧s✳

❍♦✇ ❞♦ ✇❡ ✜♥❞ ✐♥t❡❣r❛❧s❄ ❚❤✐s ✐s t❤❡ s✉❜❥❡❝t ♦❢ t❤❡ ♣r❡s❡♥t ❝❤❛♣t❡r✳ ■♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✇❡ s❛✇ s♦♠❡ ❢❛❝ts ❛❜♦✉t ✐♥t❡❣r❛❧s ❛s t❤❡② ❛r❡ ♠❛t❝❤❡❞ ❛❣❛✐♥st t❤♦s❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ ❢✉rt❤❡r ❡①❛♠✐♥❡ ❛♥♦t❤❡r s✉❝❤ ❢❛❝t✱ t❤❡ ▲✐♥❡❛r ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡✳ ❚❤❡ ✈❛r✐❛❜❧❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ✇❡ ❛r❡ ❝♦♥s✐❞❡r✐♥❣ ❛r❡ q✉❛♥t✐t✐❡s ✇❡ ♠❡❡t ✐♥ ❡✈❡r②❞❛② ❧✐❢❡✳ ❋r❡q✉❡♥t❧②✱ t❤❡r❡ ❛r❡ ♠✉❧t✐♣❧❡ ✇❛②s t♦ ♠❡❛s✉r❡ t❤❡s❡ q✉❛♥t✐t✐❡s✿

• ❧❡♥❣t❤ ❛♥❞ ❞✐st❛♥❝❡✿ ✐♥❝❤❡s✱ ♠✐❧❡s✱ ♠❡t❡rs✱ ❦✐❧♦♠❡t❡rs✱ ✳✳✳✱ ❧✐❣❤t ②❡❛rs

✷✳✹✳ ▲✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧

✶✷✷

• ❛r❡❛✿ sq✉❛r❡ ✐♥❝❤❡s✱ sq✉❛r❡ ♠✐❧❡s✱ ✳✳✳✱ ❛❝r❡s

• ✈♦❧✉♠❡✿ ❝✉❜✐❝ ✐♥❝❤❡s✱ ❝✉❜✐❝ ♠✐❧❡s✱ ✳✳✳✱ ❧✐t❡rs✱ ❣❛❧❧♦♥s • t✐♠❡✿ ♠✐♥✉t❡s✱ s❡❝♦♥❞s✱ ❤♦✉rs✱ ✳✳✳✱ ②❡❛rs

• ✇❡✐❣❤t✿ ♣♦✉♥❞s✱ ❣r❛♠s✱ ❦✐❧♦❣r❛♠s✱ ❦❛r❛ts

• t❡♠♣❡r❛t✉r❡✿ ❞❡❣r❡❡s ♦❢ ❈❡❧s✐✉s✱ ♦❢ ❋❛❤r❡♥❤❡✐t • ♠♦♥❡②✿ ❞♦❧❧❛rs✱ ❡✉r♦s✱ ♣♦✉♥❞s✱ ②❡♥ • ❡t❝✳

❚❤❡ ❝♦♥✈❡rs✐♦♥ ❢♦r♠✉❧❛s ❢♦r t❤❡s❡ ✉♥✐ts ❛r❡ s❡❡♥ ✐♥ ♠❛t❤❡♠❛t✐❝s ❛s ❝❤❛♥❣❡s ♦❢ ✈❛r✐❛❜❧❡s✳ ❆❧♠♦st ❛❧❧ ❝♦♥✈❡rs✐♦♥ ❢♦r♠✉❧❛s ❛r❡ ❥✉st ♠✉❧t✐♣❧✐❝❛t✐♦♥s✱ s✉❝❤ ❛s t❤✐s ♦♥❡✿ # ♦❢ ♠❡t❡rs = # ♦❢ ❦✐❧♦♠❡t❡rs · 1000 . ❲❛r♥✐♥❣✦ ❲❡ ❞♦♥✬t ❝♦♥✈❡rt ✏♣♦✉♥❞s t♦ ❦✐❧♦s✑✱ ✇❡ ❝♦♥✈❡rt t❤❡

♥✉♠❜❡r ♦❢

♣♦✉♥❞s t♦ t❤❡

♥✉♠❜❡r ♦❢

❦✐❧♦s✳

▲❡t✬s ❝♦♥s✐❞❡r ♠♦t✐♦♥ ❛s ❛♥ ❡①❛♠♣❧❡✿ • ■❢ t❤❡ ❞✐st❛♥❝❡ ✐s ♠❡❛s✉r❡❞ ✐♥

♠✐♥✉t❡✳

✐♥❝❤❡s ❛♥❞ t✐♠❡ ✐♥ ♠✐♥✉t❡s✱ t❤❡ ✈❡❧♦❝✐t② ✐s ♠❡❛s✉r❡❞ ✐♥ ✐♥❝❤❡s ♣❡r

❢❡❡t✱ t❤❡ ✈❡❧♦❝✐t② ✐s ♥♦✇ ♠❡❛s✉r❡❞ ✐♥ ❢❡❡t ♣❡r ♠✐♥✉t❡✳ • ❇✉t ✐❢ t❤❡ t✐♠❡ ✐s ♠❡❛s✉r❡❞ ✐♥ s❡❝♦♥❞s✱ t❤❡ ✈❡❧♦❝✐t② ✐s ♠❡❛s✉r❡❞ ✐♥ ✐♥❝❤❡s ♣❡r s❡❝♦♥❞✳ • ◆♦✇✱ ✐❢ t❤❡ ❞✐st❛♥❝❡ ✐s ♠❡❛s✉r❡❞ ✐♥

❲❡ ❛r❡ ❞❡❛❧✐♥❣ ✇✐t❤ t❤❡ s❛♠❡ ♠♦t✐♦♥ ❥✉st ♠❡❛s✉r❡❞ ✐♥ ❞✐✛❡r❡♥t ✉♥✐ts ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❞✐✛❡r❡♥t ❢✉♥❝t✐♦♥s✳ ❍♦✇ ❞♦ ✇❡ tr❛♥s✐t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t❤r❡❡ ❢✉♥❝t✐♦♥s❄ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ y = f (x)

❡st❛❜❧✐s❤❡s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ q✉❛♥t✐t✐❡s x ❛♥❞ y ✿ f

x −−−−→ y

◆♦✇✱ ❡✐t❤❡r ♦♥❡ ♠❛② ❜❡ r❡♣❧❛❝❡❞ ✇✐t❤ ❛ ♥❡✇ ✈❛r✐❛❜❧❡ ✭♦r ❛ ♥❡✇ ✉♥✐t✮✳ ▲❡t✬s ❝❛❧❧ t❤❡♠ t ❛♥❞ z r❡s♣❡❝t✐✈❡❧② ❛♥❞ s✉♣♣♦s❡ t❤❡s❡ r❡♣❧❛❝❡♠❡♥ts✱ ✐✳❡✳✱ s✉❜st✐t✉t✐♦♥s✱ ❛r❡ ❣✐✈❡♥ ❜② s♦♠❡ ❢✉♥❝t✐♦♥s✿ • ❈❛s❡ ✶✿ x = g(t) ❛♥❞ y = k(t) = f (g(t))

• ❈❛s❡ ✷✿ z = h(y) ❛♥❞ z = k(x) = h(f (x))

❚❤❡s❡ s✉❜st✐t✉t✐♦♥s ❝r❡❛t❡ ♥❡✇ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ ✈❛r✐❛❜❧❡s✿ g

f

❈❛s❡ ✶ : t −−−−→ x −−−−→ y f h ❈❛s❡ ✷ : x −−−−→ y −−−−→ z ❚❤❡ t✇♦ ❝❛s❡s ❛r❡ s❤♦✇♥ ✐♥ t❤❡ ♣✐❝t✉r❡ ❜❡❧♦✇✿

✷✳✹✳

▲✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧

❲❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t t❤❡s❡ ❢✉♥❝t✐♦♥s

g

✶✷✸

❛♥❞

h

❛r❡

❧✐♥❡❛r✳

❈❛s❡ ✶ ❚❤❡ ❝❤❛♥❣❡ ♦❢ ✉♥✐ts ✐s

x = g(t) = mt + b . ❘❡✲s❝❛❧✐♥❣ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s r❡q✉✐r❡s ❛♥ ❛❞❥✉st♠❡♥t ♦❢ t❤❡ ✐♥t❡❣r❛❧ ✐❢ ✐t ✐s s❡❡♥ ❛s t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤✿

❆❜♦✈❡✱ t❤❡

x✲❛①✐s

✐s s❤r✉♥❦ ❜② ❛ ❢❛❝t♦r ♦❢

❤❛✈❡ ♠✉❧t✐♣❧✐❡❞ ❜②

2

t♦ ❣❡t ♦♥❡ ❢♦r

2✱

✐✳❡✳✱

x = t/2✳

❚❤❡r❡❢♦r❡✱ t❤❡ ✐♥t❡❣r❛❧ ♦✈❡r ❛♥② ✐♥t❡r✈❛❧ ✐♥

t✳

❚❤❡♦r❡♠ ✷✳✹✳✶✿ ▲✐♥❡❛r ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ■ ❋♦r ❛♥② ♥✉♠❜❡rs

m 6= 0 Z

❛♥❞

b

❛♥❞ ❛♥② ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥

1 f (mt + b) dt = m

Z

f (x) dx

x=mt+b

f✱

✇❡ ❤❛✈❡✿

x

✇✐❧❧

✷✳✹✳ ▲✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧

✶✷✹

Pr♦♦❢✳

❲❡ t❛❦❡✿ F (x) =

Z

f (x) dx .

❲❡ ❛♣♣❧② t❤❡ ❈❤❛✐♥ ❘✉❧❡ ❛♥❞ t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❉✐✛❡r❡♥t✐❛t✐♦♥ ✿ d dt



1 F (mt + b) m



=

1 d 1 (F (mt + b)) = mF ′ (mt + b) = F ′ (mt + b) = f (mt + b) . m dt m

❊①❛♠♣❧❡ ✷✳✹✳✷✿ t✐♠❡ s❤✐❢t

❙✉♣♣♦s❡ x ✐s t❤❡ t✐♠❡✱ ❛♥❞ s✉♣♣♦s❡ ✇❡ ❝❤❛♥❣❡ t❤❡ ♠♦♠❡♥t ❢r♦♠ ✇❤✐❝❤ ✇❡ st❛rt ♠❡❛s✉r✐♥❣ t✐♠❡✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ Z Z x = t + t0 =⇒

k(x) dx =

f (t + t0 ) dt .

❊①❛♠♣❧❡ ✷✳✹✳✸✿ s❡❝♦♥❞s t♦ ♠✐♥✉t❡s

❙✉♣♣♦s❡ x ✐s t❤❡ t✐♠❡ ❛♥❞ y ✐s t❤❡ ❧♦❝❛t✐♦♥✳ ■❢ x ✐s ♠❡❛s✉r❡❞ ✐♥ s❡❝♦♥❞s✱ t❤❡♥ s✇✐t❝❤✐♥❣ t♦ t✐♠❡ t ♠❡❛s✉r❡❞ ✐♥ ♠✐♥✉t❡s ✇✐❧❧ r❡q✉✐r❡ ❛ ❢✉♥❝t✐♦♥✿ x = g(t) = 60t .

❲❡ ❦♥♦✇ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✹ t❤❛t t❤❡ ❣r❛♣❤s ♦❢ t❤❡ q✉❛♥t✐t✐❡s ❞❡s❝r✐❜✐♥❣ ♠♦t✐♦♥ ❛r❡ s✐♠♣❧② r❡✲s❝❛❧❡❞ ✈❡rs✐♦♥s ♦❢ t❤❡ ♦❧❞ ♦♥❡s✳ ▲❡t✬s r❡❝❛st t❤✐s st❛t❡♠❡♥t ✐♥ t❤❡ ✐♥t❡❣r❛❧ ❢♦r♠✳ • ❙✉♣♣♦s❡ y = q(t) ❛♥❞ y = p(x) ❛r❡ t❤❡ ❧♦❝❛t✐♦♥ ❛s ❢✉♥❝t✐♦♥s ♦❢ ♠✐♥✉t❡s ❛♥❞ s❡❝♦♥❞s r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♥ q(t) = p(60t) . ′ • ❙✉♣♣♦s❡ v(t) = q ′ (t) ❛♥❞ Z e(x) = p (x)Z❛r❡ t❤❡ ✈❡❧♦❝✐t✐❡s ❛s ❢✉♥❝t✐♦♥s ♦❢ ♠✐♥✉t❡s ❛♥❞ s❡❝♦♥❞s r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡r❡❢♦r❡✱ v dt = q ❛♥❞ e dx = p✳ ❲❡ s✉❜st✐t✉t❡ t❤❡s❡ ✐♥t♦ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥✿

Z

v dt =

❊①❡r❝✐s❡ ✷✳✹✳✹

Z

e dx

. x=60t

❊①♣r❡ss t❤❡ ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ♠✐♥✉t❡s ✐♥ t❡r♠s ♦❢ t❤❡ ✈❡❧♦❝✐t② ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ s❡❝♦♥❞s✳ ❊①❛♠♣❧❡ ✷✳✹✳✺✿ ❝♦♠♣❛r❡ t♦ ❈❤❛✐♥ ❘✉❧❡

▲❡t✬s ✜♥❞ ❜♦t❤ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥✿ k(t) = sin(3t − 1) .

❆❢t❡r ❛❧❧✱ t❤❡ ♠❛✐♥ ❝❤❛❧❧❡♥❣❡ ♠✐❣❤t ❜❡ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿ t 7→ 3t − 1 = x 7→ sin x = z

❍❡r❡✱ x ✐s t❤❡ ♥❡✇ ✈❛r✐❛❜❧❡ t❤❛t ✇❡ ❤❛✈❡ ♠❛❞❡ ✉♣✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡✿

 d d d sin(3t − 1) = 3 sin x k(t) = = 3 cos x = 3 cos(3t − 1) . dt dt dx x=3t−1 x=3t−1

❚❤❡ ✐♥t❡❣r❛❧✿ Z

1 k(t) dt = 3

Z

sin(x) dx

x=3t−1

1 1 + C = − cos(3t − 1) + C . = (− cos(x)) 3 3 x=3t−1

✷✳✹✳

▲✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧

✶✷✺

❈❛s❡ ✷ ❚❤❡ ❝❤❛♥❣❡ ♦❢ ✉♥✐ts ✐s

z = h(y) = my + b . ❘❡✲s❝❛❧✐♥❣ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s r❡q✉✐r❡s ❛♥ ❛❞❥✉st♠❡♥t ♦❢ t❤❡ ✐♥t❡❣r❛❧✿

❆❜♦✈❡✱ t❤❡ ♠✉❧t✐♣❧✐❡❞

y ✲❛①✐s ✐s s❤r✉♥❦ ❜② ❛ ❢❛❝t♦r ♦❢ 2✱ ✐✳❡✳✱ z = y/2✳ ❜② 2 t♦ ❣❡t ♦♥❡ ❢♦r z ✳

❚❤❡r❡❢♦r❡✱ t❤❡ ✐♥t❡❣r❛❧ ✇✐t❤ r❡s♣❡❝t t♦

y

✇✐❧❧ ❤❛✈❡

❚❤❡♦r❡♠ ✷✳✹✳✻✿ ▲✐♥❡❛r ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡ ■■ ❋♦r ❛♥② ♥✉♠❜❡rs

m

❛♥❞

Z

b

❛♥❞ ❛♥② ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥



mf (x) + b dx = m

Z

f✱

✇❡ ❤❛✈❡✿

f dx + bx

❚❤❡ r❡s✉❧t ✐s ❥✉st ❛♥ ✐♥st❛♥❝❡ ♦❢ t❤❡ ▲✐♥❡❛r✐t② ❘✉❧❡✳

❊①❛♠♣❧❡ ✷✳✹✳✼✿ s♣❛❝❡ s❤✐❢t ❛♥❞ ✢✐♣ ■❢

y

✐s t❤❡ ❧♦❝❛t✐♦♥ ❛♥❞ ✇❡ ❝❤❛♥❣❡ t❤❡ ♣❧❛❝❡ ❢r♦♠ ✇❤✐❝❤ ✇❡ st❛rt ♠❡❛s✉r✐♥❣✱ ✇❡ ❤❛✈❡✿

z = h(x) = y + y0 =⇒ ■❢ ✇❡ ❝❤❛♥❣❡ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡

x✲❛①✐s✱

Z

k dx =

Z

f dx + y0 x .

✇❡ ❤❛✈❡✿

z = h(x) = −y =⇒

Z

k dx = −

Z

f dx .

❊①❛♠♣❧❡ ✷✳✹✳✽✿ ♠✐❧❡s t♦ ❦✐❧♦♠❡t❡rs ❙✉♣♣♦s❡

x

✐s t❤❡ t✐♠❡ ❛♥❞

y

✐s t❤❡ ❧♦❝❛t✐♦♥✱ t❤❡♥ ❢✉♥❝t✐♦♥

h

♠❛② r❡♣r❡s❡♥t t❤❡ ❝❤❛♥❣❡ ♦❢ ✉♥✐ts ♦❢

❧❡♥❣t❤✱ s✉❝❤ ❛s ❢r♦♠ ♠✐❧❡s t♦ ❦✐❧♦♠❡t❡rs✿

z = h(y) = 1.6y . ❆s ✇❡ ❦♥♦✇✱ t❤❡ q✉❛♥t✐t✐❡s ❞❡s❝r✐❜✐♥❣ ♠♦t✐♦♥ ❛r❡ s✐♠♣❧② r❡♣❧❛❝❡❞ ✇✐t❤ t❤❡✐r

♠✉❧t✐♣❧❡s✳

❚❤❡ ♥❡✇ ❣r❛♣❤s

❛r❡ t❤❡ ✈❡rt✐❝❛❧❧② str❡t❝❤❡❞ ✈❡rs✐♦♥s ♦❢ t❤❡ ♦❧❞ ♦♥❡s✳ ▲❡t✬s r❡❝❛st t❤✐s st❛t❡♠❡♥t ✐♥ t❤❡ ✐♥t❡❣r❛❧ ❢♦r♠✿

•

■❢

a

✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ♠✐❧❡s✱ t❤❡♥ t❤❡ ✈❡❧♦❝✐t② ✇✐t❤ r❡s♣❡❝t t♦ ❦✐❧♦♠❡t❡rs ✐s

1 1.6

Z

a dx .

✷✳✹✳

▲✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧ •

■❢

v

✶✷✻

✐s t❤❡ ✈❡❧♦❝✐t② ✇✐t❤ r❡s♣❡❝t t♦ ♠✐❧❡s✱ t❤❡♥ t❤❡ ❧♦❝❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❦✐❧♦♠❡t❡rs ✐s

1 1.6

Z

v dx .

❊①❡r❝✐s❡ ✷✳✹✳✾

■❢

a

✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ♠✐❧❡s✱ t❤❡♥ ✇❤❛t ✐s t❤❡ ♣♦s✐t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❦✐❧♦♠❡t❡rs❄

❊①❡r❝✐s❡ ✷✳✹✳✶✵

Pr♦✈❡ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛s✳ ❊①❛♠♣❧❡ ✷✳✹✳✶✶✿ t✐♠❡ ❛♥❞ t❡♠♣❡r❛t✉r❡

❚❤✐s ✐s ❤♦✇

f

❜♦t❤ ❝❛s❡s ❝❛♥ ❛♣♣❡❛r✳

❘❡❝❛❧❧ t❤❡ ❡①❛♠♣❧❡ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✹ ✇❤❡♥ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥

t❤❛t r❡❝♦r❞s t❤❡ t❡♠♣❡r❛t✉r❡ ✭✐♥ ❋❛❤r❡♥❤❡✐t✮ ❛s ❛ ❢✉♥❝t✐♦♥

❛♥♦t❤❡r t❤❛t r❡❝♦r❞s t❤❡ t❡♠♣❡r❛t✉r❡ ✐♥ ❈❡❧s✐✉s ❛s ❛ ❢✉♥❝t✐♦♥

• • • •

g

f

♦❢ t✐♠❡ ✭✐♥ ♠✐♥✉t❡s✮ r❡♣❧❛❝❡❞ ✇✐t❤

♦❢ t✐♠❡ ✐♥ s❡❝♦♥❞s✿

s t✐♠❡ ✐♥ s❡❝♦♥❞s❀ m t✐♠❡ ✐♥ ♠✐♥✉t❡s❀ F t❡♠♣❡r❛t✉r❡ ✐♥ ❋❛❤r❡♥❤❡✐t❀ C t❡♠♣❡r❛t✉r❡ ✐♥ ❈❡❧s✐✉s✳

❚❤❡ ❝♦♥✈❡rs✐♦♥ ❢♦r♠✉❧❛s ❛r❡✿

m = s/60 , ❛♥❞

C = (F − 32)/1.8 . ❚❤❡s❡ ❛r❡ t❤❡ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ ❢♦✉r q✉❛♥t✐t✐❡s✿ s/60

❆♥❞ t❤✐s ✐s t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥✿

❚❤❡♥✱ ✇❡ ❤❛✈❡✿

❜② t❤❡

Z

(F −32)/1.8

f

s −−−−−−→ m −−−−→ F −−−−−−−−−−→ C

g:

 F = k(s) = f (s/60) − 32 /1.8 .  (f (s/60) − 32)/1.8 ds Z Z = f (s/60)/1.8 ds − 32/1.8 ds Z 1 f (s/60) ds − 32/1.8s = 1.8 Z 60 = − 32/1.8s , f dm 1.8

k ds =

▲✐♥❡❛r ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡✳

Z

m=s/60

❊①❡r❝✐s❡ ✷✳✹✳✶✷

Pr♦✈✐❞❡ ❛ s✐♠✐❧❛r ❛♥❛❧②s✐s ❢♦r t❤❡ s✐③❡s ♦❢ s❤♦❡s ❛♥❞ ❝❧♦t❤✐♥❣✳ ❊①❛♠♣❧❡ ✷✳✹✳✶✸✿ ❞❡❣r❡❡s t♦ r❛❞✐❛♥s

❚❤❡ ❝♦♥✈❡rs✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❞❡❣r❡❡s

y

t♦ t❤❡ ♥✉♠❜❡r ♦❢ r❛❞✐❛♥s

x=

π y. 180

x

✐s✿

✷✳✹✳

✶✷✼

▲✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧

❚❤❡♥✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ z = f (x)✱ ✇❡ ❤❛✈❡✿ Z

Z  π  180 f y dy = f dx 180 π

. π y x= 180

❇❡❝❛✉s❡ ♦❢ t❤❡ ❡①tr❛ ❝♦❡✣❝✐❡♥t✱ t❤❡ tr✐❣♦♥♦♠❡tr✐❝ ✐♥t❡❣r❛t✐♦♥ ❢♦r♠✉❧❛s✱ s✉❝❤ ❛s Z

sin x dx = − cos x + C ,

❞♦♥✬t ❤♦❧❞ ❢♦r ❞❡❣r❡❡s✳ ■♥❞❡❡❞✱ ✐❢ ✇❡ ❞❡♥♦t❡ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❢♦r ❞❡❣r❡❡s ❜② sind y ❛♥❞ cosd y r❡s♣❡❝t✐✈❡❧②✱ ✇❡ ❤❛✈❡ t✇♦ ❡♥t✐r❡❧② ♥❡✇ ❢✉♥❝t✐♦♥s✿

❚❤❡r❡❢♦r❡✱

 π   π  sind y = sin y ❛♥❞ cosd y = cos y . 180 180 Z

sin x dx π x= 180 y 180 = cos x +C π π

180 sind y dy = π

Z

x= 180 y

180 cosd y + C . = π

❚❤❡ ❢♦r♠✉❧❛ ❥✉st ❞♦❡s♥✬t ❧♦♦❦ ❛s ♥✐❝❡✦ ❊①❛♠♣❧❡ ✷✳✹✳✶✹✿ ❧♦❣❛r✐t❤♠✐❝ s❝❛❧❡

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦♥✲❧✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✉♥✐ts ✐s ❝❛❧❧❡❞ ❛ ❧♦❣❛r✐t❤♠✐❝

s❝❛❧❡ ✿

x = g(t) = 10t .

❚❤❡♥✱ ❢♦r ❛ ❢✉♥❝t✐♦♥ y = f (x)✱ s✉♣♣♦s❡ F ✐s ✐ts ❛♥t✐❞❡r✐✈❛t✐✈❡✳ ❍♦✇ ❞♦ ✇❡✱ ❛s ✇❡ ❞✐❞ ❛❜♦✈❡✱ ❡①♣r❡ss ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ y = f (10t ) ✐♥ t❡r♠s ♦❢ F ❄ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❤❛✈❡ ❛ ❢♦r♠✉❧❛✿ Z

f (10t ) dt = ...

❲❡ ♣r♦❝❡❡❞ ❛s ❜❡❢♦r❡✱ ❜② t❤❡ ❈❤❛✐♥ ❘✉❧❡✿

❚❤❡r❡❢♦r❡✱

d(F ◦ g) dF = dt dx t

x=10t

F (10 ) =

❛♥❞✱ ❢✉rt❤❡r✱

Z

Z

· 10t

′

= f (10t )10t ln 10 .

f (10t ) · 10t ln 10 dt ,

f (10t ) · 10t dt =

1 F (10t ) . ln 10

❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ ❢❛❝t♦r 10t ✐♥s✐❞❡ t❤❡ ✐♥t❡❣r❛❧ s❡❡♠s t♦ ♥♦t ❛❧❧♦✇ ✉s t♦ ✜♥✐s❤ t❤❡ ❥♦❜ ❛♥❞ ❡①♣r❡ss ❞✐r❡❝t❧② t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ y = f (10t ) ✐♥ t❡r♠s ♦❢ F ✳ ❲❡ ✇✐❧❧ ♥❡❡❞ ❢✉rt❤❡r ❛♥❛❧②s✐s✳✳✳

✷✳✺✳ ■♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥✿ ❝♦♠♣♦s✐t✐♦♥s

✶✷✽

✷✳✺✳ ■♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥✿ ❝♦♠♣♦s✐t✐♦♥s

❚❤❡ ❧✐♥❡❛r s✉❜st✐t✉t✐♦♥ ❢♦r♠✉❧❛ ❝♦♥t❛✐♥s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✜rst✱ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✿

Z

1 f (mx + b) dx = m (mx + b)′

Z

=m

f (u) du

u=mx+b

❇❡❝❛✉s❡ t❤✐s ❞❡r✐✈❛t✐✈❡ ✐s ❥✉st ❛ ♥✉♠❜❡r✱ ✐t ❝❛♥ ❜❡ ❢❛❝t♦r❡❞ ✐♥t♦ t❤❡ ✐♥t❡❣r❛❧ ❛s ✐t ✐s ♠♦✈❡❞ t♦ t❤❡ ♦t❤❡r s✐❞❡✿

(sin(3x − 1))′ = cos(3x − 1) · (3x − 1)′ = cos(3x − 1) · 3 ′  1 sin(3x − 1) = cos(3x − 1) 3 Z 1 sin(3x − 1) = cos(3x − 1) dx 3

=⇒ =⇒ ❙♦✱ ♦✉r ❢♦r♠✉❧❛ ✐s✱ ✐♥ tr✉t❤✱ t❤❡ ❢♦❧❧♦✇✐♥❣✿

Z

′

f (mx + b) · (mx + b) dx =

Z

f (u) du

u=mx+b

◆♦✇✱ ❤♦✇ ❞♦ ✇❡ ✐♥t❡❣r❛t❡ ❢✉♥❝t✐♦♥s ✇✐t❤ ❝♦♠♣♦s✐t✐♦♥s ❛♥❞ ♥♦t ❥✉st ❧✐♥❡❛r ♦♥❡s❄ ❏✉st ❛s ✇✐t❤ ♦t❤❡r ✐♥t❡❣r❛t✐♦♥ ❢♦r♠✉❧❛s ✇❡✱ ❛❣❛✐♥✱ tr② t♦ ✏r❡✈❡rs❡✑ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ▲❡t✬s t❛❦❡

sin(x2 )✳

■t ✐s ❡❛s② t♦ ❞✐✛❡r❡♥t✐❛t❡ ❜② t❤❡ ❈❤❛✐♥ ❘✉❧❡ ✿

′ sin(x2 ) = cos(x2 ) · 2x .

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❤❛✈❡ ❛ s✐♠✐❧❛r ❢♦r♠✉❧❛ ❢♦r t❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿

Z

sin(x2 ) dx =?

2 ❇✉t ✇❡ ❞♦♥✬t r❡❝♦❣♥✐③❡ sin(x ) ❛s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛♥② ❢✉♥❝t✐♦♥ ✇❡ ❦♥♦✇✳✳✳ ❈❛♥ ✇❡ s❡❡ ✇❤②❄ ■t✬s t❤❡ ❡①tr❛ ′ ❢❛❝t♦r g ✇❡ ❣❡t ❡✈❡r② t✐♠❡ ✇❡ ❛♣♣❧② t❤❡ ❈❤❛✐♥ ❘✉❧❡ t♦ ❞✐✛❡r❡♥t✐❛t❡ f ◦ g ✳ ◆♦✇✱ ✇❡ ❞♦ r❡❝♦❣♥✐③❡

cos(x2 ) · 2x

❢r♦♠ t✇♦ ❧✐♥❡s ❛❜♦✈❡✦ ❚❤❡♥✱

Z

▼♦r❡ ❡①❛♠♣❧❡s❄ ❍❡r❡ t❤❡② ❛r❡✿

Z

cos(x2 ) 2x dx = sin(x2 ) + C .

2

2

sin(x ) 2x dx = − cos(x ) + C,

Z

2

2

ex 2x dx = ex + C .

❲❤❛t ❞♦ t❤❡ t❤r❡❡ ❡①❛♠♣❧❡s ❤❛✈❡ ✐♥ ❝♦♠♠♦♥❄ ❲❡ s❡❡ ❛ ♣❛tt❡r♥✿

Z

Z

Z

Z

cos (x2 ) ·2x dx =

sin (x2 )

sin (x2 ) ·2x dx = − cos (x2 ) e

(x2 )

·2x dx =

? (x2 ) ·2x dx =

e

(x2 )

? (x2 )

✷✳✺✳

✶✷✾

■♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥✿ ❝♦♠♣♦s✐t✐♦♥s

❊✈❡r②t❤✐♥❣ ✐s t❤❡ s❛♠❡ ❡①❝❡♣t ✇❤❛t❡✈❡r ✐s ❜❡❤✐♥❞ t❤❡s❡ q✉❡st✐♦♥ ♠❛r❦s✳ ❲❡ ❦♥♦✇ ✇❤❛t ✐s ♠✐ss✐♥❣ ❛♥❞ ✇❡ r❡✇r✐t❡✿ Z

f (x2 ) · 2x dx = F (x2 ) + C ✇✐t❤ F ′ = f

■♥ ♦t❤❡r ✇♦r❞s✱ F ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ f ✳ ❙♦✱ t♦ ✐♥t❡❣r❛t❡ t❤❡s❡✱ ✇❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤✐s ♣r♦❜❧❡♠✿ ◮ ●✐✈❡♥ f ✱ ✜♥❞ F ✇✐t❤ F ′ = f ✳

❚❤✐s ✐s✱ ♦❢ ❝♦✉rs❡✱ ❛❧s♦ ❛♥ ❞❡❝♦♠♣♦s❡✿

✱ ❜✉t ♥♦t ✇✐t❤ r❡s♣❡❝t t♦ x✦ ❲❤❛t ✐s t❤✐s ✈❛r✐❛❜❧❡❄ ▲❡t✬s

✐♥t❡❣r❛t✐♦♥ ♣r♦❜❧❡♠

x 7→ x2 = u 7→ f (u) = z

❙♦✱ ❜♦t❤ f ❛♥❞ F ❛r❡ ❢✉♥❝t✐♦♥s ♦❢ s♦♠❡ u✱ ❛♥ ✐♥t❡r♠❡❞✐❛t❡ ✈❛r✐❛❜❧❡✱ t❤❛t ✇❡✬✈❡ ♠❛❞❡ ✉♣✳ ❚❤❡♥✱ t♦ ✜♥❞ F ✱ ✇❡ ✐♥t❡❣r❛t❡ f ✇✐t❤ r❡s♣❡❝t t♦ u✿ F (u) =

❚❤✐s ✐s ❛ ❝❤❛♥❣❡

♦❢ ✈❛r✐❛❜❧❡s

✦

❊①❛♠♣❧❡ ✷✳✺✳✶✿ ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤

❊✈❛❧✉❛t❡✿

Z

f (u) du

x2 Z

√ 3 x2 ·2x dx = ? |{z}

❞❡❝♦♠♣♦s❡

❚❤❡ ❦❡② st❡♣ ✐s t♦ ❜r❡❛❦ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ❛♣❛rt✱ t♦ ✜♥❞ u, f, F ✳ ❙♦✱ u = x2 , f (u) = u1/3 ✳ ❚❤❡♥✱ F (u) =

Z

√ 3

u du =

Z

1

3 4 u 3 +1 + C = u3 + C . u du == 1 4 +1 3 1 3

PF

❊✈❡♥ t❤♦✉❣❤ ✐♥t❡❣r❛t✐♦♥ ✐s ✜♥✐s❤❡❞✱ t❤✐s ✐s♥✬t t❤❡ ❛♥s✇❡r ❜❡❝❛✉s❡ ✐t ❤❛s t♦ ❜❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ x✦ ❲❡ ♥❡❡❞ t♦ s✉❜st✐t✉t❡ u = x2 ❜❛❝❦ ✐♥t♦ t❤✐s ❢✉♥❝t✐♦♥✿ F (x2 ) =

3 2  34 x +C. 4

❋♦r ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❛♥❛❧②s✐s✱ ✇❡ r❡♣❧❛❝❡ x2 ✇✐t❤ g(x)✳ ❲❡ ❛r❡ ♣r❡♣❛r❡❞ t♦ ✐♥t❡❣r❛t❡ t❤✐s ✭❛♥❞ ♥♦t❤✐♥❣ ❡❧s❡✮✿ Z

f (g(x)) · g ′ (x) dx .

❚❤❡ ❛♥s✇❡r ✐s F (g(x))✱ ✇❤❡r❡ F ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ f ✿ F′ = f . ❚❤❡♦r❡♠ ✷✳✺✳✷✿ ■♥t❡❣r❛t✐♦♥ ❜② ❙✉❜st✐t✉t✐♦♥

●✐✈❡♥ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ f ❛♥❞ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ g ✱ ✇❡ ❤❛✈❡✿ Z

f (g(x)) · g ′ (x) dx = F (g(x)) + C

✷✳✺✳ ■♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥✿ ❝♦♠♣♦s✐t✐♦♥s

✇❤❡r❡

F

✶✸✵

✐s ❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

f✿

F (u) =

Z

f (u) du

Pr♦♦❢✳

CR

(F (g(x)))′

== F ′ (g(x) · g ′ (x) = f (g(x))g ′ (x) .

❈♦♥❝❧✉s✐♦♥✿ ✇❡ ❝❛♥ ✐♥t❡❣r❛t❡ ❝♦♠♣♦s✐t✐♦♥s ✇❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s s❛t✐s✜❡❞✿

◮

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✏✐♥s✐❞❡✑ ❢✉♥❝t✐♦♥ ✐s ♣r❡s❡♥t ❛s ❛ ❢❛❝t♦r✳

■♥ ♦t❤❡r ✇♦r❞s✱ ❛s ❛ ♣r❡r❡q✉✐s✐t❡✱ ✇❡ ♥❡❡❞ t♦ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

Z

f(

g(x) |{z}

)·

✏✐♥s✐❞❡✑ ❢✉♥❝t✐♦♥

❊①❛♠♣❧❡ ✷✳✺✳✸✿ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡♦r❡♠

❊✈❛❧✉❛t❡

Z √

g ′ (x) | {z }

dx

✐ts ❞❡r✐✈❛t✐✈❡

x3 + 1 · 3x2 dx .

❖❜s❡r✈❡ ✜rst t❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐♥s✐❞❡ ✐s ♣r❡s❡♥t✿

(x3 + 1)′ = 3x2 . ❙♦✱ t❤❡ t❤❡♦r❡♠ s❤♦✉❧❞ ✇♦r❦✿ ❞❡❝♦♠♣♦s✐t✐♦♥✿

f (u) =

√

✐♥t❡❣r❛t✐♦♥✿

u

=⇒ F (u) =

u = g(x) = x3 + 1 ❜❛❝❦✲s✉❜st✐t✉t✐♦♥✿

Z

1 2 3 u 2 du = u 2 + C 3

=⇒ g ′ (x) = 3x2 3 2 F (g(x)) = (x3 + 1) 2 + C 3

◆♦t❡ ❤♦✇ ✇❡ ❝♦♥✈❡rt❡❞ t❤❡ ♦r✐❣✐♥❛❧ ✐♥t❡❣r❛❧ t♦ ❛ s✐♠♣❧❡r ♦♥❡✱ ✇✐t❤ r❡s♣❡❝t t♦

u✳

❊①❛♠♣❧❡ ✷✳✺✳✹✿ ✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥

❊✈❛❧✉❛t❡

❏✉st ♥♦t✐❝❡ t❤❛t

(x3 + 1)′ = 3x2 ✱

♥♦t

Z √ x2 ✳

x3 + 1 · x2 dx =? ❚❤❡ ❝♦♥❞✐t✐♦♥ ❞♦❡s♥✬t s❡❡♠ t♦ ❜❡ s❛t✐s✜❡❞ ❛♥②♠♦r❡✳✳✳

❍♦✇❡✈❡r✱ t❤✐s ❢✉♥❝t✐♦♥ ✐s ❥✉st ❛ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ♦♥❡ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡ ❛♥❞ s♦✱ t❤❡r❡❢♦r❡✱ ✐s t❤❡ ✐♥t❡❣r❛❧✳ ❲❡ ✇✐❧❧ ✐❣♥♦r❡ t❤✐s s❤♦rt❝✉t t❤♦✉❣❤✳ ❲❡✬❧❧ tr② t♦ ❛♣♣❧② t❤❡ ■♥t❡❣r❛t✐♦♥ ❜② ❙✉❜st✐t✉t✐♦♥ ❢♦r♠✉❧❛ ❛♥②✇❛②✿

= x3 + 1

✜rst s✉❜st✐t✉t✐♦♥✿

u

s❡❝♦♥❞ s✉❜st✐t✉t✐♦♥✿

u′ = 3x2

)

✇❡ ❝♦♥✈❡rt t❤❡ ✐♥t❡❣r❛❧ ✇✐t❤ r❡s♣❡❝t t♦ t♦ ❛ ♥❡✇ ♦♥❡ ✇✐t❤ r❡s♣❡❝t t♦

❚❤❡ ❤♦♣❡ ✐s t❤❛t t❤❡ ♥❡✇ ✐♥t❡❣r❛❧ ✇✐❧❧ ❜❡ s✐♠♣❧❡r t❤❛♥ t❤❡ ♦r✐❣✐♥❛❧✳

u

x

✷✳✺✳

✶✸✶

■♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥✿ ❝♦♠♣♦s✐t✐♦♥s

❲❡ ❛❧r❡❛❞② ❤❛✈❡ ❛❧❧ ✇❡ ♥❡❡❞ ❤❡r❡✳ ❲❡ ❜r❡❛❦ ✇❤❛t✬s ✐♥s✐❞❡ t❤❡ ✐♥t❡❣r❛❧ ❛♣❛rt ❜✉t ♥♦t ✐♥ t❤❡ ♦❜✈✐♦✉s ✇❛②✿ √

x3 + 1 =

√

1 x2 = u′ . 3

u,

◆♦✇ ✇❡ ❞❡❛❧ ✇✐t❤ t❤❡ ✐♥t❡❣r❛❧ ✐ts❡❧❢✳ Z √ x3 + 1 x2 dx | {z }

=

u ✐♥s✐❞❡

Z

1 = 3 1 PF == 3 1 = 3

❆♥s✇❡r✿

Z √

√ Z

u·

1 du 3

❆ ♥❡✇ ✐♥t❡❣r❛❧✳

1

u 2 du

2 3 ■♥t❡❣r❛t✐♦♥ ✜♥✐s❤❡❞✳ u2 + C 3 3 2 3 (x + 1) 2 + C ❇❛❝❦✲s✉❜st✐t✉t✐♦♥ u = x3 + 1 . 3

3 2 x3 + 1 x2 dx = (x3 + 1) 2 + C . 9

❊①❡r❝✐s❡ ✷✳✺✳✺

❊✈❛❧✉❛t❡

Z √

x3 + 1 x2 dx .

Z √

x4 + 1 x3 dx .

❊①❡r❝✐s❡ ✷✳✺✳✻

❊✈❛❧✉❛t❡

❲❡ ❝❛♥ r❡✲✇r✐t❡ ♦✉r t❤❡♦r❡♠ ❛s ❛ s✐♥❣❧❡ ❢♦r♠✉❧❛ ❛s ❢♦❧❧♦✇s✿ Z

f (g(x)) · g ′ (x) dx =

Z

f (u) du

u=g(x)

■♥ t❤✐s ✈❡rs✐♦♥✱ t❤❡r❡ ✐s ♥♦ ❝♦♥❢✉s✐♦♥ ❛❜♦✉t ✇❤❡t❤❡r t❤❡ ✐♥t❡❣r❛t✐♦♥ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❤❛s ❜❡❡♥ ❝❛rr✐❡❞ ♦✉t✳ ❚❤❡ s✐♠♣❧❡st ❦✐♥❞ ♦❢ ✐♥t❡❣r❛❧ ❢♦r ✇❤✐❝❤ t❤✐s ❛♣♣r♦❛❝❤ ❛❧✇❛②s ✇♦r❦s ✐♥✈♦❧✈❡s ❛ ❧✐♥❡❛r ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s✳ ■❢ g(x) = mx + b, m 6= 0 ,

♦✉r ❢♦r♠✉❧❛ ❜❡❝♦♠❡s✿

Z

f (mx + b) · m dx =

❲❡ ❝♦♥s❡q✉❡♥t❧② r❡❝♦✈❡r t❤❡ ❢❛♠✐❧✐❛r ▲✐♥❡❛r Z

Z

f (u) du

❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡

1 f (mx + b) dx = m

Z

. u=mx+b

✿

f (u) du

. u=mx+b

✷✳✺✳ ■♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥✿ ❝♦♠♣♦s✐t✐♦♥s

✶✸✷

❊①❛♠♣❧❡ ✷✳✺✳✼✿ ♦♥❡✲❧✐♥❡ ✐♥t❡❣r❛t✐♦♥ ❊✈❛❧✉❛t❡✿

Z

1 e3x dx = 3

eu du

Z

1 u = e + C 3

u=3x

❊①❛♠♣❧❡ ✷✳✺✳✽✿ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❊✈❛❧✉❛t❡✿

▲❡t✬s ❜r❡❛❦ ❞♦✇♥ t❤❡ ❝♦♠♣♦s✐t✐♦♥✱

sin(ex )✿

Z

u=3x

1 = e3x + C . 3

ex sin(ex ) dx .

u = ex ,

y = sin u .

❋✉rt❤❡r♠♦r❡✱

u′ = e x . ❯s❡ t❤❡s❡ t✇♦✿

Z

x

x

e sin(e ) dx =

Z

sin u du

= − cos u + C

❊✈❛❧✉❛t❡✳ ❙✉❜st✐t✉t❡✳

= cos ex + C . ❊①❡r❝✐s❡ ✷✳✺✳✾ ❊✈❛❧✉❛t❡✿

Z √

sin x · cos x dx .

❊①❡r❝✐s❡ ✷✳✺✳✶✵ ❊✈❛❧✉❛t❡✿

Z

ee

x +x

dx .

❊①❛♠♣❧❡ ✷✳✺✳✶✶✿ ♥♦ ❝♦♠♣♦s✐t✐♦♥

Z

tan x dx =

❲❤❛t✱ ♥♦ ❝♦♠♣♦s✐t✐♦♥❄✦

Z

sin x dx Z cos x 1 dx = sin x · cos x Z = sin x (cos x)−1 dx Z = − (cos x)′ (cos x)−1 dx Z = − (u)−1 du =

❚❤❡r❡ ✐s ❛ ❞✐✈✐s✐♦♥ t❤♦✉❣❤✳

■t✬s ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ ❢❛❝t✳

❙♦✱ t❤❡r❡ ✐s ❛ ❝♦♠♣♦s✐t✐♦♥ ❛❢t❡r ❛❧❧✦

❆♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✐♥s✐❞❡ ❢✉♥❝t✐♦♥ ✐s ♣r❡s❡♥t✳

❚❤❡ ❢♦r♠✉❧❛ ❛♣♣❧✐❡s ✇✐t❤

= − ln u + C

❲❡ ✐♥t❡❣r❛t❡✳

= − ln cos x + C .

❲❡ ❜❛❝❦✲s✉❜st✐t✉t❡✳

u = cos x.

✷✳✻✳ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧s

✶✸✸ ❲❛r♥✐♥❣✦ ❲❡ ♠❛❦❡ s✉r❡ t❤❛t t❤❡ ✐♥t❡❣r❛❧s t❤❛t ✇❡ ❢❛❝❡

❝❛♥

❜❡ ❡✈❛❧✉❛t❡❞ ✇✐t❤ t❤✐s ♠❡t❤♦❞✳

✷✳✻✳ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧s

❊①❛♠♣❧❡ ✷✳✻✳✶✿ ❢❛✐❧✉r❡ ❛❢t❡r s✉❜st✐t✉t✐♦♥

▲❡t✬s t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t t❤✐s ✐♥t❡❣r❛❧✿ Z

1 xe dx = 2 x2

Z

2

eu du = eu + C = ex + C .

■t ✇♦r❦s s♦ ✇❡❧❧✦ ❈❤❛♥❣✐♥❣ t❤❡ ♣♦✇❡r✱ x t♦ x2 ✱ r✉✐♥s t❤✐s ♥✐❝❡ ❛rr❛♥❣❡♠❡♥t✿ Z

2 x2

x e dx =

Z

ueu dx = ... ♥♦✇ ✇❤❛t❄

■♥ ❢❛❝t✱ ♥♦ ♣♦✇❡r ♦❢ x ♦t❤❡r t❤❛♥ 1 ✇✐❧❧ ❛❧❧♦✇ ✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦r♠✉❧❛✿ Z

Z

3 x2

x e dx = ?

4 x2

x e dx = ?

Z

x

Z

1/2 x2

e dx = ?

2

xπ ex dx = ?

❲❛r♥✐♥❣✦ ❉♦ ♥♦t r❡♣❧❛❝❡

x

✐♥

dx

✇✐t❤

u✱ dx 6= du✦

❲❡ st✐❧❧ ✇♦✉❧❞ ❧✐❦❡ t♦ ❜❡ ❛❜❧❡ t♦ ❝♦♥✈❡rt ❛♥ ✐♥t❡❣r❛❧ t♦ ❛ ♥❡✇ ✈❛r✐❛❜❧❡✳ ■t ✐s ❛❧✇❛②s ♣♦ss✐❜❧❡✦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ✐s ✇❤❛t t♦ ❞♦ ✇✐t❤ dx✳ ❲❡ ❤❛✈❡ t♦ ❧♦♦❦ ❛t t❤❡ ✐♥t❡❣r❛❧ ❞✐✛❡r❡♥t❧②✳ ❲❤❛t ❡①❛❝t❧② ❞♦ ✇❡ ✐♥t❡❣r❛t❡❄ ■♥ t❤❡ ✐♥t❡❣r❛❧✱ Z

k(x) dx ,

Z

✐t ❤❛s ❜❡❡♥ ❛ ❢✉♥❝t✐♦♥✱ k(x)✱ ✇❤✐❧❡ ❛♥❞ dx s❡r✈❡ ❛s ♠❡r❡ ❜r❛❝❦❡ts✳ ❚❤✐s ❞♦❡s♥✬t ✇♦r❦ ❛♥②♠♦r❡✦ ❲❡ ♥❡❡❞ t♦ ♠❛❦❡ s❡♥s❡ ♦❢ dx✳ ❙✐❣♥✐✜❝❛♥t❧②✱ ✇❡ s✇✐t❝❤ ❢r♦♠ ❛ ❢✉♥❝t✐♦♥ t♦ ❛ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠✱ k(x) · dx✳ ❆s ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✹✱ t❤❡ ❢♦r♠ ❝♦♠❡s ❢r♦♠ t❤❡ ❢♦❧❧♦✇✐♥❣✿ y = f (x) ❛t x = a =⇒

❛♥❞✱ ❢✉rt❤❡r♠♦r❡✱

dy = f ′ (a) , dx

=⇒ dy = f ′ (a) · dx .

❚❤✐s ✐s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❡①tr❛ ✈❛r✐❛❜❧❡s✱ ♥♦t❤✐♥❣ ❜✉t ♥✉♠❜❡rs✱ ♦♥❝❡ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ♦❧❞ ♦♥❡s ❤❛s ❜❡❡♥ s♣❡❝✐✜❡❞✿

✷✳✻✳

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧s

■♥ t❤❡ ❣r❛♣❤✱

y

dx

✐s t❤❡ r✉♥ ❛♥❞

dy

✶✸✹

✐s t❤❡ r✐s❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡✳ ❚❤❡② ❛r❡ ❝❛❧❧❡❞ t❤❡

❞✐✛❡r❡♥t✐❛❧s

♦❢

x

❛♥❞

r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡♠ ✈❛r✐❡s ❢r♦♠ ❧♦❝❛t✐♦♥ t♦ ❧♦❝❛t✐♦♥✳ ❚❤✐s ✐s ❢✉rt❤❡r ❞✐s❝✉ss❡❞ ✐♥

❈❤❛♣t❡r ✹✳ ❇❛❝❦ t♦ ✐♥t❡❣r❛t✐♦♥✳ ❙♦✱

dx

✐s t❤❡ ❞✐✛❡r❡♥t✐❛❧ ♦❢

x✱

❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳

✇❤✐❝❤ ✐s ❛ ✈❛r✐❛❜❧❡ s❡♣❛r❛t❡ ❢r♦♠✱ ❜✉t r❡❧❛t❡❞ t♦✱

x✳

❚❤❡♥✱

f ′ (x) · dx

✐s ❥✉st

❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♦♥ t❤❡ s❡❝♦♥❞ ✈❛r✐❛❜❧❡ ✐s ❡s♣❡❝✐❛❧❧②

s✐♠♣❧❡❀ ✐t✬s ❛ ♠✉❧t✐♣❧❡✳ ◆♦✇ ✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥✳ ❚❤❡ ❞✐✛❡r❡♥t✐❛❧ ❢r♦♠

x

t♦

u

❛♥❞

du

u ✐s ❛ ✈❛r✐❛❜❧❡ s❡♣❛r❛t❡ ❢r♦♠ dx t♦ du✳ ❍❛♥❞❧✐♥❣ t❤❡ ♦❢

❢r♦♠✱ ❜✉t r❡❧❛t❡❞ t♦✱

u✳

❙♦✱ ❝❤❛♥❣✐♥❣ ✈❛r✐❛❜❧❡s ♠❡❛♥s ❣♦✐♥❣

❞✐✛❡r❡♥t✐❛❧s ✐s ❛ s❡♣❛r❛t❡ st❡♣ ✐♥ t❤❡ ♣r♦❝❡ss ♦❢ ❝❤❛♥❣✐♥❣ t❤❡

✈❛r✐❛❜❧❡✳ ❘❡❝❛❧❧ ❤♦✇ t❤❡

❈❤❛✐♥ ❘✉❧❡✱ ✐♥ t❤❡ ▲❡✐❜♥✐③ ♥♦t❛t✐♦♥✱ ✐s ✐♥t❡r♣r❡t❡❞ ❛s✱ ❛♥❞ ✐t ✐s✱ ❛ ✏❝❛♥❝❡❧❧❛t✐♦♥✑

♦❢

du

✭✇❤❡♥

✐t✬s ♥♦t ③❡r♦✮✿

dy 6 du dy = dx 6 du dx ❆ s✐♠✐❧❛r ✐❞❡❛ ❛♣♣❧✐❡s t♦ ✐♥t❡❣r❛❧s✳ ❲❡ t❛❦❡ t❤❡ ❢♦r♠✉❧❛ ♦❢ ✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥ ❛♥❞ s✇✐t❝❤ t♦ t❤❡ ▲❡✐❜♥✐③ ♥♦t❛t✐♦♥✱ s✉❜❥❡❝t t♦ t❤❡ s✉❜st✐t✉t✐♦♥✿

Z ❛♥❞

◆♦t❡ ❤♦✇

dx

f (g(x)) · g ′ (x) dx = Z

Z

du f (u) · 6 dx = 6 dx

Z

f (u) du

✏❝❛♥❝❡❧s✑✱ t✉r♥✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ✇✐t❤ r❡s♣❡❝t t♦

u=g(x)

f (u) du . x

t♦ ♦♥❡ ✇✐t❤ r❡s♣❡❝t t♦

u✳

❲❡ t❛❦❡ t❤✐s ✐❞❡❛

♦♥❡ st❡♣ ❢✉rt❤❡r✳

❈♦r♦❧❧❛r② ✷✳✻✳✷✿ ❈❤❛♥❣❡ ♦❢ ❱❛r✐❛❜❧❡s ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❋♦r♠ ❯♥❞❡r ❛ s✉❜st✐t✉t✐♦♥

u = g(x)

✐♥ ❛♥ ✐♥t❡❣r❛❧✱ ✇❡ ❛❧s♦ s✉❜st✐t✉t❡✿

du = g ′ dx

❚❤✐s ❢♦r♠✉❧❛ ✐s ✉s❡❞ ✕ ✐♥ ❛❞❞✐t✐♦♥ t♦

u = g(x)

✕ ✐♥ ♦r❞❡r t♦ ❝♦♠♣❧❡t❡ t❤❡ s✉❜st✐t✉t✐♦♥✳

❊①❛♠♣❧❡ ✷✳✻✳✸✿ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐s t❤❡ ❣♦❛❧ ▲❡t✬s ❛♣♣❧② t❤❡ ❢♦r♠✉❧❛ t♦ t❤❡ ❡①❛♠♣❧❡ ❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥✿

Z

ex sin(ex ) dx .

✷✳✻✳

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧s

✶✸✺

❲❡ st❛rt ✇✐t❤ ❛ s✉❜st✐t✉t✐♦♥ t❤✐s t✐♠❡✿ u = ex =⇒ du = ex dx =⇒ dx =

du ex

❙✉❜st✐t✉t❡ ❜♦t❤ t❤❡ ✜rst ❛♥❞ t❤❡ ❧❛st ♦❢ t❤❡s❡ ✐♥t♦ t❤❡ ✐♥t❡❣r❛❧✿ Z

x

x

e sin(e ) dx = =

Z

Z

ex sin ex

du ❲❡ s✉❜st✐t✉t❡ t❤❡ ❧❛st ❛♥❞ ❝❛♥❝❡❧✳ ex

❲❡ s✉❜st✐t✉t❡ t❤❡ ✜rst✱ ❛♥❞ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐s ❝♦♠♣❧❡t❡✦

sin u du

❚❤❡ r❡st ✐s ❛ ❜♦♥✉s✳

= − cos u + C x

= − cos e + C .

❲✐t❤ t❤❡ ❢♦r♠✉❧❛✱ ✇❡ ❝❛♥ ❝❤❛♥❣❡ ✈❛r✐❛❜❧❡s ✐♥ ❛♥② ✐♥t❡❣r❛❧✱ ❡✈❡♥ t❤❡ ❦✐♥❞ t❤❛t✬s ♥♦t s✉❜❥❡❝t t♦ ✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥✳ ❊①❛♠♣❧❡ ✷✳✻✳✹✿ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ♦♥❧②

▲❡t✬s ❡✈❛❧✉❛t❡✿

Z

2

x2 ex dx = ?

❚❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐s t❤❡ s❛♠❡ ❛s ❜❡❢♦r❡ ✭x ≥ 0✮✿ u = x2 =⇒ du = 2x dx =⇒ dx =

du 2x

❍♦✇❡✈❡r✱ ❛♥t✐❝✐♣❛t✐♥❣ t❤❛t t❤❡ ❝❛♥❝❡❧❧❛t✐♦♥ ♠✐❣❤t ♥♦t ❜❡ ❛s ❡❛s② ❛s ❧❛st t✐♠❡✱ ✇❡ ❛❧s♦ ✜♥❞ t❤❡ ✐♥✈❡rs❡ s✉❜st✐t✉t✐♦♥ ✭❧✐t❡r❛❧❧② t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ s✉❜st✐t✉t✐♦♥ ❢✉♥❝t✐♦♥✮✿ u = x2 =⇒ x =

❲❡ ❤❛✈❡ t❤r❡❡ s✉❜st✐t✉t✐♦♥s✿ ✶✳ x2 = u du ✷✳ dx = √2x ✸✳ x = u ❙✉❜st✐t✉t❡✿

Z

2 x2

x e dx =

Z

Z

√

u.

x2 eu dx

★✶

du ★✷ 2x Z √ 2 u du = u e √ ★✸ 2 u Z 1 √ u = ue du . 2 =

x2 e u

❊✈❡♥ t❤♦✉❣❤ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❤❛s♥✬t ♠❛❞❡ t❤❡ ✐♥t❡❣r❛❧ ❡❛s✐❡r t♦ ✐♥t❡❣r❛t❡✱ t❤❡ ❝♦♥✈❡rs✐♦♥ ✐s ❝♦♠♣❧❡t❡✦ ❲❡ ❝❛♥ ❤❛✈❡

❛♥② s✉❜st✐t✉t✐♦♥ ✐♥ ❛♥② ✐♥t❡❣r❛❧✳

❊①❛♠♣❧❡ ✷✳✻✳✺✿ ❜❛❞ s✉❜st✐t✉t✐♦♥

▲❡t✬s ♣✐❝❦ ❛ ✇r♦♥❣ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ t❤❡ ❢❛♠✐❧✐❛r ✐♥t❡❣r❛❧✿ Z

2

xex dx = ?

✷✳✻✳

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧s

✶✸✻

❚❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜st✐t✉t✐♦♥ ✐s ❝❤♦s❡♥ ❡✈❡♥ t❤♦✉❣❤ ✇❡ ❞♦♥✬t ❛♥t✐❝✐♣❛t❡ t❤❛t ✐t ✇✐❧❧ s✐♠♣❧✐❢② t❤❡ ✐♥t❡❣r❛❧✿

#1. u = x3 . ❚❤❡ ❞✐✛❡r❡♥t✐❛❧ ✐s ❢♦✉♥❞✿

#2. u = x3 =⇒ du = 3x2 dx =⇒ dx =

du . 3x2

❚❤❡ ✐♥✈❡rs❡ s✉❜st✐t✉t✐♦♥ ✐s ❢♦✉♥❞✿

#3. x = u1/3 . ❙✉❜st✐t✉t❡✿

Z

x2

xe dx = =

Z

xe(u xeu

1/3 )2

2/3

Z

dx

★✸

du 3x2

★✷

du 2/3 u1/3 eu 2 3 (u1/3 ) Z 1 2/3 = u−1/3 eu du . 3 =

❖✉r ❝❤♦✐❝❡ ♦❢ ❛ ♥❡✇ ✈❛r✐❛❜❧❡ ✇❛s ✉♥✇✐s❡✳

Z

★✸

❊①❡r❝✐s❡ ✷✳✻✳✻

❈❛rr② ♦✉t t❤❡ s✉❜st✐t✉t✐♦♥

u = x4

✐♥ t❤❡ ❛❜♦✈❡ ✐♥t❡❣r❛❧✳

❊①❡r❝✐s❡ ✷✳✻✳✼

❈❛rr② ♦✉t t❤❡ s✉❜st✐t✉t✐♦♥

u=x

✐♥ t❤❡ ❛❜♦✈❡ ✐♥t❡❣r❛❧✳

❊①❡r❝✐s❡ ✷✳✻✳✽

▼❛❦❡ ✉♣ ②♦✉r ♦✇♥ ✐♥t❡❣r❛❧ ❛♥❞ ❝❛rr② ♦✉t t❤❡ s✉❜st✐t✉t✐♦♥

u = x2 ✳

❘❡♣❡❛t✳

❊①❛♠♣❧❡ ✷✳✻✳✾✿ ❝❧✉❡ ❢♦r ♥❡✇ ✈❛r✐❛❜❧❡

■❢ ✇❡ ❤♦♣❡ t♦ s✐♠♣❧✐❢② t❤❡ ✐♥t❡❣r❛❧ ❜② s✉❜st✐t✉t✐♦♥✱ t❤❡ ♥❡✇ ✈❛r✐❛❜❧❡ s❤♦✉❧❞ ❜❡ ❡q✉❛❧ t♦ t❤❡ ✏✐♥s✐❞❡✑ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ ❝♦♥s✐❞❡r✿

Z ❲❡ ❝❤♦♦s❡

u = x + 1✳ Z

√

❚❤❡♥✱

du = dx✳

√

x + 1 · x dx .

❚❤❡r❡❢♦r❡✱

Z

u1/2 (u − 1) du Z Z 1/2 = u u du + u1/2 (−1) du Z Z 3/2 = u u du − u1/2 du

x + 1 · x dx =

2 2 = u5/2 − u3/2 + C 5 3 2 2 = (x + 1)5/2 − (x + 1)3/2 + C . 5 3

■s ✐t ❛♥② ❜❡tt❡r❄

❨❡s✦

✷✳✻✳

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ✐♥t❡❣r❛❧s

✶✸✼

❊①❛♠♣❧❡ ✷✳✻✳✶✵✿ ❧♦❣❛r✐t❤♠✐❝ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s

▲❡t✬s r❡✈✐s✐t t❤❡ ✐ss✉❡ ♦❢ ❝♦♥✈❡rt✐♥❣ ✉♥✐ts t♦ ❛ ❧♦❣❛r✐t❤♠✐❝

s❝❛❧❡ ✿

x = 10t .

❚❤❡♥✱ dx = 10t ln 10 dt =⇒ dt =

❙✉❜st✐t✉t❡ ✐♥t♦ t❤❡ ✐♥t❡❣r❛❧ ❛♥❞ s✐♠♣❧✐❢②✿ Z

t

f (10 ) dt =

Z

dx . ln 10

10t

1 dx = f (x) t 10 ln 10 ln 10

Z

1 f (x) dx . x

❲❡ ❤❛✈❡ ❡①♣r❡ss❡❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢ y = f (10t ) ❛s ❛♥ ✐♥t❡❣r❛❧ ✇✐t❤ r❡s♣❡❝t t♦ x✳ ❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ ✇❤❛t ✐s ❣♦✐♥❣ ♦♥✳ ❲❡ st❛rt ✇✐t❤ ❛ ❢❛♠✐❧✐❛r ❞✐❛❣r❛♠ ❢♦r t❤❡ ❈❤❛✐♥ ❘✉❧❡ ♦❢ ❞✐✛❡r❡♥✲ t✐❛t✐♦♥✿ d 

F (g(x)) −−−dx −−−→   u=g(x) y

s✉❜st✐t✉t✐♦♥

d



−−−du −−−→

F (u)

F ′ (g(x))g ′ (x) x  ❈❘ ✇✐t❤ u=g(x)  F ′ (u)

◆♦✇ ✇❡ r❡♠❛❦❡ t❤❡ ❞✐❛❣r❛♠ ✐♥t♦ ♦♥❡ ❛❜♦✉t ✐♥t❡❣r❛t✐♦♥ ❜② r❡✈❡rs✐♥❣ t❤❡ ❤♦r✐③♦♥t❛❧ ❛rr♦✇s✿ R

 dx

R

 du

R

 dx

R

 du

F (g(x)) ←−−−−−−−   u=g(x) y

s✉❜st✐t✉t✐♦♥

F (u)

←−−−−−−−

F ′ (g(x))g ′ (x) x  ❈❘ ✇✐t❤ u=g(x)  F ′ (u)

❲❡ r❡✲♥❛♠❡ t❤❡ ❢✉♥❝t✐♦♥ ❢♦r ❝♦♥✈❡♥✐❡♥❝❡✿ F (g(x)) ←−−−−−−− x  u=g(x) 

s✉❜st✐t✉t✐♦♥

F (u)

←−−−−−−−

f (g(x))g ′ (x) x  ❈❘ ✇✐t❤ u=g(x)  f (u)

❚❤✉s t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ♠❡t❤♦❞ ♦❢ ✐♥t❡❣r❛t✐♦♥ ❣✐✈❡s ✉s ❛♥ ❛❧t❡r♥❛t✐✈❡ ✇❛② ♦❢ ❣❡tt✐♥❣ ❢r♦♠ t♦♣ r✐❣❤t t♦ t♦♣ ❧❡❢t ✭✐♥t❡❣r❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ x✮✳ ❲❡ t❛❦❡ ❛ ❞❡t♦✉r ❜② ❢♦❧❧♦✇✐♥❣ t❤❡ ❝❧♦❝❦✇✐s❡ ♣❛t❤ ❛r♦✉♥❞ t❤❡ sq✉❛r❡✿ ✶✳ t❤❡ ❈❤❛✐♥ ❘✉❧❡ ❢♦r♠✉❧❛ ✐♥ r❡✈❡rs❡ ✷✳ ✐♥t❡❣r❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ u ✸✳ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥ ❊①❛♠♣❧❡ ✷✳✻✳✶✶✿ s✉❜st✐t✉t✐♦♥ ✇✐t❤ ❞✐❛❣r❛♠

❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧ Z

cos(x2 ) 2x dx = sin(x2 ) + C

✷✳✼✳ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s

✶✸✽

✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿ R

2

 dx= ?

sin(x ) ←−−−−−−−−− x  u=x2 

s✉❜st✐t✉t✐♦♥

R

 du

←−−−−−−−

sin(u)

cos(x2 ) 2x x  ❈❘ ✇✐t❤ u=x2 

cos(u)

❊①❡r❝✐s❡ ✷✳✻✳✶✷

❊①❡❝✉t❡ t❤❡ s✉❜st✐t✉t✐♦♥ u = ex ❢♦r t❤❡ ✐♥t❡❣r❛❧ ✭❞♦♥✬t ❡✈❛❧✉❛t❡ t❤❡ r❡s✉❧t✐♥❣ ✐♥t❡❣r❛❧✮✿ Z

sin(1 + ex ) dx .

✷✳✼✳ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s ❆t t❤❡ ♥❡①t st❛❣❡✱ ✇❡ ❛rr✐✈❡ ❛t t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥✿ ◮ ❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ ✇❡ ✉s❡ s✉❜st✐t✉t✐♦♥ t♦ ❡✈❛❧✉❛t❡ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s❄

❋✐rst✱ ♥♦t❤✐♥❣ ❤❛s t♦ ❝❤❛♥❣❡✳ ❆❢t❡r ❛❧❧✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❛❧❧ ✇❡ ♥❡❡❞ ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡✳ ❙♦✱ t♦ ✜♥❞ t❤❡ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ Z

b

f (g(x))g ′ (x) dx , a

✜♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ ✜rst✱ ❛s ✇❡ ❞✐❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ H(x) =

Z

f (g(x))g ′ (x) dx ,

✐❢ ♣♦ss✐❜❧❡✳ ■❢ ✐t ✐s✱ t❤❡♥ t❤❡ ❧❛st st❡♣ ✐s ❛s s✐♠♣❧❡ ❛s ✐t ❣❡ts✿ Z

b a

f (g(x))g ′ (x) dx = H(b) − H(a) .

❊①❛♠♣❧❡ ✷✳✼✳✶✿ ✉s✐♥❣ ❋❚❈ ❢♦r ♦❧❞ ✈❛r✐❛❜❧❡

▲❡t✬s ❡✈❛❧✉❛t❡✿

Z

1

ex sin(ex ) dx . 0

❲❡ ❤❛✈❡ ❛❧r❡❛❞② ❢♦✉♥❞ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✿ Z

❚❤❡r❡❢♦r❡✱ ❉♦♥❡✦ ❆ ❜❡tt❡r q✉❡st✐♦♥ ✐s✿

Z

ex sin(ex ) dx = − cos ex + C .

1 0

ex sin(ex ) dx = − cos e1 − (− cos e0 ) = − cos e + cos 1 .

✷✳✼✳ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s

✶✸✾

◮ ❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ ✉♥❞❡r ❛ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ s✉❜st✐✲

t✉t✐♦♥s ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥❄

❚❤❡ s✉❜st✐t✉t✐♦♥ ✐s ❥✉st ❛ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ x✲❛①✐s✳ ■t✱ t❤❡r❡❢♦r❡✱ s❤r✐♥❦s✴str❡t❝❤❡s t❤❡ ❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥ ✿

❙♦✱ ✇❡ ❤❛✈❡ t♦ tr❛❝❦✱ t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ ❞♦♠❛✐♥✱ ✐✳❡✳✱ t❤❡ ❜♦✉♥❞s ♦❢ t❤❡ ✐♥t❡❣r❛❧✱ ✉♥❞❡r ♦✉r tr❛♥s❢♦r♠❛t✐♦♥✳ ❊①❛♠♣❧❡ ✷✳✼✳✷✿ ✉s✐♥❣ ❋❚❈ ❢♦r ♥❡✇ ✈❛r✐❛❜❧❡

▲❡t✬s t❛❦❡ ❛ ❝❧♦s❡r ❧♦♦❦ ❛t t❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✿ Z

Z

ex sin(ex ) dx 1 0

= − cos u + C

= − cos ex + C

ex sin(ex ) dx = − cos e1 − (− cos e0 ) = − cos e + cos 1

■t ✐s ❞♦♥❡ ✈✐❛ t❤❡ s✉❜st✐t✉t✐♦♥ u = ex ✳ ❲❡ r❡❛❧✐③❡ t❤❛t ✇❡ ❝♦✉❧❞ ❤❛✈❡ ❥✉♠♣❡❞ ❢r♦♠ − cos u + C t♦ − cos e + cos 1 ❜② ♦♠✐tt✐♥❣ t❤❡ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥ st❡♣ ✐♥ t❤❡ ✉♣♣❡r r✐❣❤t ❝♦r♥❡r✦ ■♥❞❡❡❞✿ e cos u = − cos e + cos 1 . 1

❲❡ ❥✉st ♥❡❡❞ t♦ s❡❡ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❜♦✉♥❞s ♦❢ t❤❡ t✇♦ ✐♥t❡❣r❛❧s✱ ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ u✿ x = 0 7→ u = e0 = 1 x = 1 7→ u = e1 = e

❚❤❡ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥ ❜❡❝♦♠❡s r❡❞✉♥❞❛♥t✳ ❙♦✱ ✉♥❞❡r t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡ u = g(x)✱ t❤❡ ❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥ ❝❤❛♥❣❡s ❢r♦♠ • [a, b] ❢♦r x t♦

• [g(a), g(b)] ❢♦r u✳

❊✈❡♥ t❤♦✉❣❤t t❤❡ str❡t❝❤✴s❤r✐♥❦ ♠✐❣❤t ❜❡ ♥♦♥✲✉♥✐❢♦r♠✱ ✇❡ ♦♥❧② ❝❛r❡ ❛❜♦✉t t❤❡ ❡♥❞✲♣♦✐♥ts✳ ❊①❛♠♣❧❡ ✷✳✼✳✸✿ ♥♦ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥

❋✐♥❞ t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ y = x2 cos x3 ❢r♦♠ 0 t♦ 2✳ ❲❡ ❤❛✈❡✿ ❆r❡❛ =

Z

2

x2 cos x3 dx . 0

❙✉❜st✐t✉t✐♦♥ ✜rst✿ u = x3 =⇒ du = 3x2 dx =⇒ dx =

du . 3x2

✷✳✼✳ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s ❚❤❡♥✱

Z

2

✶✹✵

3

x cos x dx =

Z

du 1 x cos u 2 = 3x 3 2

Z

cos u du .

◆♦✇✱ ✇❤❛t ✇♦✉❧❞ t❤✐s ❝♦♠♣✉t❛t✐♦♥ ❧♦♦❦ ❧✐❦❡ ❢♦r t❤❡ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧❄ ▲❡t✬s ♠❛❦❡ ✐t ❝❧❡❛r ✇❤❛t ✈❛r✐❛❜❧❡s ✇❡ ❛r❡ r❡❢❡rr✐♥❣ t♦✿ Z

x=2 2

3

x cos x dx = x=0

Z

x=2

du 1 x cos u 2 = 3x 3 2

x=0

Z

x=2

cos u du . x=0

❲❡ ❤❛✈❡ ♠✐s♠❛t❝❤❡❞ ✈❛r✐❛❜❧❡s✦ ■♥ ♦r❞❡r t♦ ✜① t❤❛t✱ ✇❡ ✜♥❞ t❤❡ ❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥ ❜② ✜♥❞✐♥❣ t❤❡ ❜♦✉♥❞s ❢♦r u ❢r♦♠ t❤❡ ❝♦rr❡✲ s♣♦♥❞✐♥❣ ❜♦✉♥❞s ❢♦r x✿ x = 0 7→ u = 03 = 0

x = 2 7→ u = 23 = 8

❙♦✱ [0, 2] ❢♦r x ❜❡❝♦♠❡s [0, 8] ❢♦r u✳ ❚❤❡♥✱ Z

x=2

1 x cos x dx = 3 2

x=0

3

Z

x=2 x=0

1 cos u du = 3

Z

u=8

cos u du . u=0

❚❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ t❤❡ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ ✐s ❝♦♠♣❧❡t❡✳ ❲❡ ❞♦♥✬t ❤❛✈❡ t♦ ❣♦ ❜❛❝❦ t♦ x ✐♥ ♦r❞❡r t♦ ✜♥✐s❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥✿ ❆r❡❛ =

1 3

Z

u=8 u=0

u=8 FTC cos u du ==== sin u = sin 8 − sin 0 = sin 8 . u=0

■♥ s✉♠♠❛r②✱ ✇❡ s❤♦✇❡❞ t❤❛t t❤❡ ❛r❡❛s ♦❢ t❤❡s❡ t✇♦ r❡❣✐♦♥s ✉♥❞❡r t❤❡s❡ ❣r❛♣❤s✱ y = x2 cos x3 ❛♥❞ y =

1 cos u , 3

❛r❡ ❡q✉❛❧ ❛♥❞ t❤❡♥ ❢♦✉♥❞ t❤❡ ❧❛tt❡r ♦♥❡✿

❚❤✐s ✐s t❤❡ s✉♠♠❛r②✿

❈♦r♦❧❧❛r② ✷✳✼✳✹✿ ❉❡✜♥✐t❡ ■♥t❡❣r❛t✐♦♥ ❜② ❙✉❜st✐t✉t✐♦♥ ❯♥❞❡r ❛ s✉❜st✐t✉t✐♦♥ u = g(x) ✐♥ ❛ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧✱ ✇❡ ❤❛✈❡✿ Z

b

g(b) = F (g(b)) − F (g(a)) , f (g(x)) · g (x) dx = F (u) ′

a

g(a)

✷✳✼✳ ❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s

✇❤❡r❡

F

✶✹✶

✐s ❛♥② ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢

f✳

Pr♦♦❢✳

❘❡❝❛❧❧ t❤❡ ❢♦r♠✉❧❛ ♦❢ ✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥ ✿ Z Z ′ f (g(x)) · g (x) dx = f (u) du

= F (g(x)) . u=g(x)

❚❤❡ ❢♦r♠✉❧❛ ✐♥ t❤❡ t❤❡♦r❡♠ ❢♦❧❧♦✇s ♥♦✇ ❢r♦♠ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✳

❊①❛♠♣❧❡ ✷✳✼✳✺✿ ♥♦ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥

▲❡t✬s ❝❛rr② ♦✉t ❛ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❛ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ ✇✐t❤♦✉t ❜❛❝❦✲s✉❜st✐t✉t✐♦♥✳ ❈♦♥s✐❞❡r Z π cos2 x sin x dx . 0

❚❤❡ ✐♥✐t✐❛❧ ♣❛rt ✕ ❝❤♦♦s✐♥❣ ❛ s✉❜st✐t✉t✐♦♥ ✕ r❡♠❛✐♥s t❤❡ s❛♠❡✳ ❲❡ ♥♦t✐❝❡ ❛ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❝❤♦♦s❡ t❤❡ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ✐♥s✐❞❡ t♦ ❜❡ t❤❡ s✉❜st✐t✉t✐♦♥ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ♥❡✇ ✈❛r✐❛❜❧❡✿

cos2 x = (cos x)2 =⇒ u = cos x . ❚❤❡ s❡❝♦♥❞ st❡♣ ✐s t♦ ✜♥❞ t❤❡ r❡st ♦❢ t❤❡ s✉❜st✐t✉t✐♦♥✿

u = cos x =⇒ du = − sin x dx =⇒ dx = ❚❤❡ ♥❡①t st❡♣ ✐s t♦ ✜♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❜♦✉♥❞s ❢♦r u✿

du − sin x

x = π =⇒ u = cos π = −1 x = 0 =⇒ u = cos 0 = 1

❲❡ ❝♦♥✈❡rt t❤❡ ✐♥t❡❣r❛❧ t♦ u ♥♦✇ ❛♥❞ t❤❡♥ ❡✈❛❧✉❛t❡ ✐t✿

Z

π 2

cos x sin x dx = 0

= =

Z

π

(cos x)2 sin x dx

Z0 −1

Z1 −1 1

Z

(u)2 sin x (u)2 −1

du −1

du − sin x

u2 du =− Z 11 u2 du = −1 u=1 1 3 = u 3 u=−1

 1 3 1 − (−1)2 3 2 = . 3 =

❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ ✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥ ✇❤❡♥ ✇❡ ♣r♦❝❡❡❞ t♦ ❞❡✜♥✐t❡ ✐♥t❡❣r❛t✐♦♥✳ ❚❤❡ ❡①tr❛ st❡♣

✷✳✽✳

❚r✐❣♦♥♦♠❡tr✐❝ s✉❜st✐t✉t✐♦♥s

✶✹✷

✐s t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✱ ❢♦r ❡✐t❤❡r ✈❛r✐❛❜❧❡✿ R

s✉❜st✐t✉t✐♦♥

❋❚❈ F (g(x)) −−−−−−−→ I   u=g(x) || y ❋❚❈ F (u) −−−−−−−→ I

dx

f (g(x))g ′ (x) −−−−−−→ x  ❈❘  R

du

−−−−−−→

f (u)

s❛♠❡✦

❚❤✉s✱ t❤❡ r❡s✉❧t ♦❢ ❞❡✜♥✐t❡ ✐♥t❡❣r❛t✐♦♥ ✕ ❛ ♥✉♠❜❡r ✕ ✐s t❤❡ s❛♠❡ ♥♦ ♠❛tt❡r ✇❤❛t ✈❛r✐❛❜❧❡ ✇❡ ❝❤♦♦s❡✳ ❍❡r❡✬s ❛ ♠♦r❡ ❡①♣❧✐❝✐t ✇❛② t♦ ✇r✐t❡ ♦✉r ❢♦r♠✉❧❛✿

Z

x=b ′

x=a

f (g(x)) · g (x) dx =

Z

u=g(b)

f (u) du u=g(a)

❚❤✐s ✐s t❤❡ ❢♦r♠✉❧❛ ♦❢ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ ✇r✐tt❡♥ ✐♥ t❤❡ ▲❡✐❜♥✐③ ♥♦t❛t✐♦♥✿

Z

b a

du f (u) · dx = dx

Z

u(b)

f (u) du u(a)

✉♥❞❡r ❛ s✉❜st✐t✉t✐♦♥ u = u(x)✳ ❊①❡r❝✐s❡ ✷✳✼✳✻

❊①❡❝✉t❡ t❤❡ s✉❜st✐t✉t✐♦♥ u = ex ❢♦r t❤❡ ✐♥t❡❣r❛❧ ✭❞♦♥✬t ❡✈❛❧✉❛t❡ t❤❡ r❡s✉❧t✐♥❣ ✐♥t❡❣r❛❧✮✿

Z

2 1

cos(1 − ex ) dx .

✷✳✽✳ ❚r✐❣♦♥♦♠❡tr✐❝ s✉❜st✐t✉t✐♦♥s

❇❛❝❦ t♦ ✐♥❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s✳✳✳ ❲❤✐❝❤ s✉❜st✐t✉t✐♦♥ t♦ ❝❤♦♦s❡ ♠✐❣❤t ♥♦t ❜❡ ❛❧✇❛②s ♦❜✈✐♦✉s✱ ❛♥❞ ✇❤❡♥ ✐t ✐s✱ ✐t ♠✐❣❤t ❧❡❛❞ t♦ ❛♥ ✐♥t❡❣r❛❧ t❤❛t ✐s♥✬t ❛♥② s✐♠♣❧❡r t❤❛t t❤❡ ♦r✐❣✐♥❛❧✳ ❚❤❡ ❧❛tt❡r ♣r♦❜❧❡♠ ✐s ❡s♣❡❝✐❛❧❧② ❝♦♠♠♦♥✳ ❈♦♥s✐❞❡r t❤❡ ❢❛♠✐❧✐❛r ✐♥t❡❣r❛❧✿ Z 1 ??? ??? eu du 2 2 տ u=ex ↑ ր u=x Z Z 1 2/3 x2 xe dx ??? ←u=sin x u−1/3 eu du u=x3 → 3

ւu=cos x

???

↓

???

u=x4

ց

1 4

Z

u−1/2 eu

1/2

du

❊✈❡♥ t❤♦✉❣❤ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❤❛s ❜❡❡♥ ❝❛rr✐❡❞ ♦✉t ❝♦rr❡❝t❧②✱ ✐t ♠✐❣❤t st✐❧❧ ❜❡ ❛ ❞❡❛❞ ❡♥❞✦ ❊①❡r❝✐s❡ ✷✳✽✳✶

❈❛rr② ♦✉t t❤❡s❡ s✉❜st✐t✉t✐♦♥s✳

✷✳✽✳

❚r✐❣♦♥♦♠❡tr✐❝ s✉❜st✐t✉t✐♦♥s

✶✹✸

❙♦♠❡t✐♠❡s ✇❡ ❤❛✈❡ t♦ ❝♦♠❡ ✉♣ ✇✐t❤ ❡♥t✐r❡❧② ♥❡✇ ✐❞❡❛s✳✳✳ ▲❡t✬s r❡✈✐s✐t t❤❡ q✉❡st✐♦♥ ❛❞❞r❡ss❡❞ ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r✱ ✇❤❛t ❆r❡❛ = 2

Z

√

R −R

✐s t❤❡ ❛r❡❛ ♦❢ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s R❄

R2 − x2 dx = πR2 .

❚♦ ♣r♦✈❡ t❤❡ ❢♦r♠✉❧❛✱ ✇❡ ♥❡❡❞ t♦ ✐♥t❡❣r❛t❡ t❤✐s✿ Z √

R2 − x2 dx .

❍♦✇❄ ❚❤❡r❡ ✐s ❛ ❝♦♠♣♦s✐t✐♦♥✳✳✳ ▲❡t✬s tr② s✉❜st✐t✉t✐♦♥✦ ❚❤❡ ♦❜✈✐♦✉s ❝❤♦✐❝❡ ✐s✿ u = R2 − x2 =⇒ du = −2x dx .

❙✉❜st✐t✉t❡✿

Z √

R2 − x2 dx =

Z

√

du u = −2x

Z

√

du 1 √ u =− 2 2 −2 R − u

Z r

R2

u du . −u

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✐s ❝♦♠♣❧❡t❡❞ ❜✉t✱ ✉♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ♥❡✇ ✐♥t❡❣r❛❧ ✐s ♥♦ s✐♠♣❧❡r t❤❛♥ t❤❡ ♦r✐❣✐♥❛❧✦ ▲❡t✬s t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t t❤❡ ✐♥t❡❣r❛♥❞✿ y=

√

R 2 − x2 .

❲❤❛t ✐s ✐ts ❣r❛♣❤❄ ■t ✐s ✉♣♣❡r ❤❛❧❢ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s R ❣✐✈❡♥ ❞✐r❡❝t❧② ✭❡①♣❧✐❝✐t❧②✮ ❛s t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✳ ❚❤❡ ❝✐r❝❧❡ ✐s ❛❧s♦ ❣✐✈❡♥ ❜② ❛ r❡❧❛t✐♦♥ ✭✐♠♣❧✐❝✐t❧②✮ ❜② x2 + y 2 = R2 . ❚❤❡r❡ ♠❛② ❜❡ ❛ t❤✐r❞ ♣♦ss✐❜✐❧✐t②✱ ✐❢ ✇❡ ❥✉st ✜♥❞ ❛ ❜❡tt❡r ✈❛r✐❛❜❧❡✳✳✳ ❖♥ t❤❡ ❧❡❢t✱ ✇❡ ✐♥t❡r♣r❡t t❤❡ ❣r❛♣❤ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ✈✐s✉❛❧✐③❡❞ ❛s ♠♦t✐♦♥ ✿ t✐♠❡ ♦♥ t❤❡ x✲❛①✐s ❛♥❞ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ y ✲❛①✐s✳ ❆s ❛ r❡s✉❧t✱ t❤❡ ❞♦ts ❛♣♣❡❛r ❛t ❡q✉❛❧ ✐♥t❡r✈❛❧s ♦❢ t✐♠❡✱ ✐✳❡✳✱ ❤♦r✐③♦♥t❛❧❧②✿

❲❤❛t ✇❡ ❝❛♥ s❡❡ ✐s ❤♦✇ ♠♦t✐♦♥ st❛rts ❢❛st✱ t❤❡♥ s❧♦✇s ❞♦✇♥ t♦ ❛❧♠♦st ③❡r♦ ✐♥ t❤❡ ♠✐❞❞❧❡✱ ❛♥❞ t❤❡♥ ❛❝❝❡❧❡r❛t❡s ❛❣❛✐♥✳ ❇✉t ✇❤❛t ✐❢ ✇❡ ❝♦♥s✐❞❡r ✐♥st❡❛❞ ❛ s✐♠♣❧❡ r♦t❛t✐♦♥ ♦♥ t❤❡ ♣❧❛♥❡❄ ❙✉❝❤ ❛ r♦t❛t✐♦♥ ✇♦✉❧❞

✷✳✽✳ ❚r✐❣♦♥♦♠❡tr✐❝ s✉❜st✐t✉t✐♦♥s

✶✹✹

♣r♦❣r❡ss t❤r♦✉❣❤ t❤❡ ❛♥❣❧❡s ❛t ❛ ❝♦♥st❛♥t r❛t❡✱ s❤♦✇♥ ♦♥ r✐❣❤t✳ ❙♦✱ ♠❛②❜❡ t❤❡ ❛♥❣❧❡✱ s❛② t✱ s❤♦✉❧❞ ❜❡ ♦✉r ♥❡✇ ✈❛r✐❛❜❧❡❄ ❚❤❡♥✱ t❤❡ ❢♦r♠✉❧❛s ❢♦r x ❛♥❞ y ❝♦♠❡ ❢r♦♠ t❤❡ ❜❛s✐❝ tr✐❣♦♥♦♠❡tr②✿ (

x = R cos t , y = R sin t .

❚❤❡ r♦t❛t✐♦♥ ✐s ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ st❛rt✐♥❣ ❢r♦♠ (R, 0) ❛♥❞ t r✉♥s ❢r♦♠ 0 t♦ π ✳ ❆ ♥❡✇ ✈❛r✐❛❜❧❡ ❤❛s ❛♣♣❡❛r❡❞ ♥❛t✉r❛❧❧②✿ x = R cos t

❚❤✐s ✐s ♦✉r s✉❜st✐t✉t✐♦♥✳ ❚❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ❢r♦♠ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡s ♦❢ ✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥ ✐s t❤❛t ✐♥st❡❛❞ ♦❢ t❤❡ ♦❧❞ ✈❛r✐❛❜❧❡ ❣✐✈❡♥ ✐♥ t❡r♠s ♦❢ t❤❡ ♥❡✇✱ ❛s ✐♥✿ u = x2 ,

❤❡r❡✱ t❤❡ ♥❡✇ ✈❛r✐❛❜❧❡ ✐s ❣✐✈❡♥ ✐♥ t❡r♠s ♦❢ t❤❡ ♦❧❞✳ ❚❤✐s ✐s ✇❤② s✉❝❤ ❛ ❢♦r♠✉❧❛ ✐s ♦❢t❡♥ ❝❛❧❧❡❞ ❛♥ ✐♥✈❡rs❡ s✉❜st✐t✉t✐♦♥✳ ◆♦ ♠❛tt❡r✦ ❲❡ ❤❛✈❡✿ t = cos−1 (x/R) ,

✇✐t❤ −π/2 ≤ t ≤ π/2 .

■♥ ❢❛❝t✱ t♦ ❝❛rr② ♦✉t t❤✐s ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s✱ ❛❧❧ ✇❡ ♥❡❡❞ ✐s t❤✐s✿ dx = −R sin t dt .

❲❡ s✉❜st✐t✉t❡ ❜✉t✱ ✐♥ ♦r❞❡r t♦ s✐♠♣❧✐❢②✱ ✇❡✬❧❧ ❛❧s♦ ♥❡❡❞ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ✭❱♦❧✉♠❡ ✶✮✿ sin2 t + cos2 t = 1 .

❚❤❡♥✱ ✇❡ ✜♥❞ ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ✇✐t❤ r❡s♣❡❝t t♦ t ❛s ❢♦❧❧♦✇s✿ Z √

R2

−

x2

dx =

Z √

Z

R2 − cos2 t · (−R sin t dt) ❯s❡ P❚✳

R sin t · (−R sin t dt) Z 2 = −R sin2 t dt Z 1 − cos 2t 2 = −R dt 2 Z   Z R2 dt − cos 2t dt =− 2   1 R2 t − sin 2t . =− 2 2 =

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✜♥✐s❤❡❞✳ ❆ tr✐❣ ❢♦r♠✉❧❛ ♥❡①t✳

✷✳✽✳

❚r✐❣♦♥♦♠❡tr✐❝ s✉❜st✐t✉t✐♦♥s

✶✹✺

■♥t❡❣r❛t✐♦♥ ✐s ✜♥✐s❤❡❞✳ ❲❡ ✇♦♥✬t ❞♦ ❜❛❝❦✲s✉❜st✐t✉t✐♦♥ ❜❡❝❛✉s❡ ♦✉r ✐♥t❡r❡st ✐s ❛ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧✳ ❲❡ ♦♥❧② ♥❡❡❞ t♦ ❦♥♦✇ t❤❡ ❜♦✉♥❞s ❢♦r t❤❡ ♥❡✇ ✈❛r✐❛❜❧❡✳

❲❡ ✜♥❞ t❤❡♠ ❢r♦♠ t❤❡ ♣✐❝t✉r❡ ❛❜♦✈❡✳ ❖r ❢r♦♠ ❛❧❣❡❜r❛✿ x = −R =⇒ t = π ❛♥❞ x = R =⇒ t = 0

❙✉❜st✐t✉t❡✿

❆r❡❛ ♦❢ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s R = 2 ·

Z

x=R

√

x=−R  2

R2 − x2 dx  t=0 1 t − sin 2t 2

R 2 t=π   1 1 2 = −R 0 − sin(2 · 0) − (π + sin 2π) 2 2

= −2 ·

= πR2 . ❊①❡r❝✐s❡ ✷✳✽✳✷

▼♦❞✐❢② t❤❡ ♣r♦♦❢ ❢♦r t❤❡ s✉❜st✐t✉t✐♦♥ x = R sin t✳ ❊①❛♠♣❧❡ ✷✳✽✳✸✿ ❞✐r❡❝t ♦r ✐♥✈❡rs❡

❘❡❝❛❧❧ ❤♦✇ ✇❡ ❡✈❛❧✉❛t❡❞ t❤❡ ✐♥t❡❣r❛❧✿

Z

2

x2 ex dx = ?

❚❤❡ s✉❜st✐t✉t✐♦♥ ✇❛s ❝❤♦s❡♥ t♦ ❜❡ u = x2 ❢r♦♠ ✇❤✐❝❤ ✇❡ ❞❡r✐✈❡❞ t❤❡ ♦t❤❡r t✇♦ ✐t❡♠s✿ 1. x2 = u du 2. dx = 2x √ 3. x = u

−→

√

u du 2. dx = √ 2 u 1. x

=

3. x2 = u

❍♦✇❡✈❡r✱ ✇❡ ❝♦✉❧❞ ❤❛✈❡ st❛rt❡❞ ✇✐t❤ ★✸ ✭t❤❡ ✐♥✈❡rs❡✮ ✇✐t❤ t❤❡ s❛♠❡ r❡s✉❧t✦ ❚❤❡ str❛t❡❣② ✐s s✉♠♠❛r✐③❡❞ ❜❡❧♦✇ ✇✐t❤ t✇♦ ❡①tr❛ ♦♣t✐♦♥s✿ ❚r✐❣♦♥♦♠❡tr✐❝ ❙✉❜st✐t✉t✐♦♥s

❙✉♣♣♦s❡ a > 0✳ ❚❤❡♥✿ √ • ❲❤❡♥ t❤❡ ✐♥t❡❣r❛♥❞ ❝♦♥t❛✐♥s a2 − x2 ✱ ♦r s♦♠❡t✐♠❡s a2 − x2 ✱ ✉s❡ t❤❡ s✉❜st✐t✉t✐♦♥✿ x = a sin t ♦r x = a cos t .

✷✳✽✳

❚r✐❣♦♥♦♠❡tr✐❝ s✉❜st✐t✉t✐♦♥s

✶✹✻

• ❲❤❡♥ t❤❡ ✐♥t❡❣r❛♥❞ ❝♦♥t❛✐♥s

√

a2 + x2 ✱ ♦r s♦♠❡t✐♠❡s a2 + x2 ✱ ✉s❡ t❤❡ s✉❜st✐t✉t✐♦♥✿ x = a tan t .

• ❲❤❡♥ t❤❡ ✐♥t❡❣r❛♥❞ ❝♦♥t❛✐♥s

√

x2 − a2 ✭♥♦t❡ t❤❡ s✐❣♥✮✱ ♦r s♦♠❡t✐♠❡s x2 − a2 ✱ ✉s❡ t❤❡ s✉❜st✐t✉t✐♦♥✿ x = a sec t .

❲❛r♥✐♥❣✦ ❚❤✐s ✐s ♥♦t ❛ t❤❡♦r❡♠ ❛♥❞ ✐t ❞♦❡s♥✬t ❣✉❛r❛♥t❡❡ s✉❝✲ ❝❡ss✳

❊①❛♠♣❧❡ ✷✳✽✳✹✿ s✐♠♣❧✐✜❝❛t✐♦♥ ▲❡t✬s s✐♠♣❧✐❢② t❤✐s✿

Z

❚❤❡ ❡①♣r❡ss✐♦♥ ♠❛t❝❤❡s ♦♣t✐♦♥ ✭✶✮✿

(4 − x2 )3/2 dx . a2

− x2

4 =⇒ a = 2

− x2

❚❤❡r❡❢♦r❡✱ ✇❡ tr②✿ x = 2 sin t .

❚❤❡♥✱ dx = 2 cos t dt .

❚❤✐s ✐s ♦✉r ❦❡② s✐♠♣❧✐✜❝❛t✐♦♥✿ 4 − x2 = 4 − (2 sin t)2 = 4 − 4 sin2 t = 4(1 − sin2 t) = 4 cos2 t .

❙✉❜st✐t✉t❡✿

Z

3 3/2

(4 − x )

dx =

Z

2

(4 cos t)

3/2

(2 cos t dt) = 16

❊①❡r❝✐s❡ ✷✳✽✳✺ ❊✈❛❧✉❛t❡ t❤✐s ✐♥t❡❣r❛❧✿

Z

1 dx . 1 + x2

❊①❡r❝✐s❡ ✷✳✽✳✻ ❊✈❛❧✉❛t❡ t❤✐s ✐♥t❡❣r❛❧✿

Z √

x2 − 1 dx .

Z

cos4 t dt .

✷✳✾✳ ■♥t❡❣r❛t✐♦♥ ❜② ♣❛rts✿ ♣r♦❞✉❝ts

✶✹✼

✷✳✾✳ ■♥t❡❣r❛t✐♦♥ ❜② ♣❛rts✿ ♣r♦❞✉❝ts

❚❤❡ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❉❡r✐✈❛t✐✈❡s ❡①♣r❡ss❡s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐♥ t❡r♠s ♦❢ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ✭❛♥❞ t❤❡ ❢✉♥❝t✐♦♥s t❤❡♠s❡❧✈❡s✮✿

(f · g)′ = f ′ · g + f · g ′ . ❚❤❡r❡ ✐s ♥♦ ✏Pr♦❞✉❝t ❘✉❧❡ ❢♦r ✐♥t❡❣r❛❧s✑ t❤❛t ✇♦✉❧❞ ❡①♣r❡ss t❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐♥ t❡r♠s ♦❢ t❤❡✐r ✐♥t❡❣r❛❧s ✭❛♥❞ t❤❡ ❢✉♥❝t✐♦♥s t❤❡♠s❡❧✈❡s✮✿

Z

(f · g) dx = ?

▲❡t✬s ♥♦♥❡t❤❡❧❡ss tr② t♦ ❣❡t ✇❤❛t❡✈❡r ✇❡ ❝❛♥ ❢r♦♠ P❘✳ ❲❡ ✐♥t❡❣r❛t❡ ✐t✿

Z

′

(f · g) dx =

f ·g

Z

Z

(f ′ · g + f · g ′ ) dx

❲❡ ✉s❡ ❋❚❈✳

(f ′ · g + f · g ′ ) dx Z Z ′ = f · g dx + f · g ′ dx . =

◆♦✇✱ t❤❡s❡ t✇♦ ✐♥t❡❣r❛❧s ❛r❡ ✈❡r② s✐♠✐❧❛r ❛♥❞ ❡✐t❤❡r ♦❢ t❤❡♠ ♠❛② ❜❡ s❡❡♥ ❛s t❤❡ ✐♥t❡❣r❛❧ ♦❢ ❛ ❝❡rt❛✐♥ ♣r♦❞✉❝t✳ ❲❡ ❞❡r✐✈❡ s♦♠❡t❤✐♥❣ ✉s❡❢✉❧ ❢r♦♠ t❤✐s✿ ❚❤❡♦r❡♠ ✷✳✾✳✶✿ ■♥t❡❣r❛t✐♦♥ ❜② P❛rts ❋♦r t✇♦ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥s

Z

f

❛♥❞

g✱

✇❡ ❤❛✈❡✿

′

f · g dx = f · g −

Z

f ′ · g dx

❲❡ ❝❛♥ ❛❧s♦ ✉s❡ t❤❡ s✉❜st✐t✉t✐♦♥ ❢♦r♠✉❧❛✱

dh = h′ (x) dx . ❚♦ ♦❜t❛✐♥ ❛ ♠♦r❡ ❝♦♠♣❛❝t ✈❡rs✐♦♥✿ ❈♦r♦❧❧❛r② ✷✳✾✳✷✿ ■♥t❡❣r❛t✐♦♥ ❜② P❛rts ❋♦r t✇♦ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥s

f Z

❛♥❞

g✱

✇❡ ❤❛✈❡

f dg = f g −

Z

g df

❚❤❡ ❢♦r♠✉❧❛ ✐s tr❛❞✐t✐♦♥❛❧❧② r❡st❛t❡❞ ✇✐t❤ t❤❡s❡✱ ❝❤❛♥❣❡❞✱ ♥❛♠❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ♦r ✈❛r✐❛❜❧❡s✿

Z

udv = uv −

Z

vdu

❲❤❡♥ ✇❡ ❛r❡ t♦ ❞❡❝✐❞❡ ✇❤✐❝❤ t❡❝❤♥✐q✉❡ ♦❢ ✐♥t❡❣r❛t✐♦♥ t♦ ✉s❡✱ ✇❡ r❡❝♦❣♥✐③❡ t❤❛t ❢♦r ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rt ✇❡ ❤❛✈❡ t♦ s❡❡ ✐♥ t❤❡ ✐♥t❡❣r❛♥❞✿

✷✳✾✳

✶✹✽

■♥t❡❣r❛t✐♦♥ ❜② ♣❛rts✿ ♣r♦❞✉❝ts

• ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❜✉t • ♥♦ ❝♦♠♣♦s✐t✐♦♥✳

❚❤❡♥✿

• ❚❤❡ ❛♣♣r♦❛❝❤ ✇♦♥✬t ✇♦r❦ ❢♦r

❜✉t ✇❡ ❤❛✈❡ s✉❜st✐t✉t✐♦♥ ❢♦r t❤❛t✳ • ◆♦r ❢♦r

❜✉t ✇❡ ❝❛♥ ❧♦♦❦ ✐t ✉♣✳ • ❚❤❡ ❛♣♣r♦❛❝❤

♠✐❣❤t

✇♦r❦ ❢♦r

Z

xex dx ,

Z

ex dx ,

Z

xex dx .

2

2

❊①❛♠♣❧❡ ✷✳✾✳✸✿ s♣❧✐t ✐♥t❡❣r❛♥❞

■❢ ✇❡ ❛r❡ t♦ ✐♥t❡❣r❛t❡ t❤✐s✱ ✇❡ ♥❡❡❞ t♦ ♠❛t❝❤ ✐t ✇✐t❤ t❤❡ ✐♥t❡❣r❛❧ ✐♥ t❤❡ ❢♦r♠✉❧❛✳ ❚❤❡s❡ t✇♦ ♠✉st ❜❡ ❡q✉❛❧✿ Z u dv

Z

x · ex dx.

❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡r❡ ❛r❡ ❛t ❧❡❛st t✇♦ ✇❛②s t♦ ♠❛t❝❤ t❤❡♠✿ • ✭❛✮ u = ex , dv = x dx✱ ❛♥❞ • ✭❜✮ u = x, dv = ex dx✳ ❲❡✬❧❧ ❤❛✈❡ t♦ ❞♦ ❜♦t❤✳ ✭❛✮ ❚♦ ✉s❡ t❤❡ ❢♦r♠✉❧❛✱ ✇❡ ♥❡❡❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ u ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢ v ✿ u = ex

=⇒ du = ex dx Z x2 dv = x dx =⇒ v = x dx = 2

■♥t❡❣r❛t✐♥❣ t♦ ✜♥❞ v ✐s t❤❡ ✜rst ♣❛rt ♦❢ ✐♥t❡❣r❛t✐♦♥✳ ❚❤❡ s❡❝♦♥❞ ♣❛rt ✐s t❤❡ ♦♥❡ ✐♥ t❤❡ ❢♦r♠✉❧❛✿ Z

udv = uv −

Z

x2 − vdu = e · 2 x

Z

x2 x · e dx . 2

❯♥❢♦rt✉♥❛t❡❧②✱ ✇❡ ❞✐s❝♦✈❡r t❤❛t t❤❡ ♥❡✇ ✐♥t❡❣r❛❧ ❧♦♦❦s ❡✈❡♥ ❝♦♠♣❧❡① t❤❛♥ t❤❡ ♦r✐❣✐♥❛❧✦ ■♥❞❡❡❞✱ t❤❡ ♣♦✇❡r ♦❢ x ✇❡♥t ✉♣✳ ❇❡❢♦r❡ ❛tt❡♠♣t✐♥❣ ♦t❤❡r t❡❝❤♥✐q✉❡s✱ ❧❡t✬s tr② t♦ r❡✈❡rs❡ t❤❡ ❝❤♦✐❝❡ ♦❢ u ❛♥❞ v ✳ ✭❜✮ ❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ♥❡❡❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ u ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢ v ✿ u=x

=⇒ du =Zdx dv = ex dx =⇒ v = ex dx = ex

❲❡ s✉❜st✐t✉t❡ t❤❡s❡ ✐♥t♦ t❤❡ ❢♦r♠✉❧❛✿ Z

udv = uv −

Z

x

vdu = x · e −

Z

ex dx .

❲❡ ♣❛✉s❡ ❤❡r❡ t♦ st♦♣ ❛♥❞ ❛♣♣r❡❝✐❛t❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ♥❡✇ ✐♥t❡❣r❛❧ ✐s s♦ ❧❡ss ❝♦♠♣❧❡① t❤❛♥ t❤❡ ♦r✐❣✐♥❛❧✦ ❚❤❛t✬s ❜❡❝❛✉s❡ t❤❡ ♣♦✇❡r ♦❢ x ✇❡♥t ❞♦✇♥✳ ❲❡ ✜♥✐s❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥✿ Z

x

x

xe dx = x · e −

Z

ex dx = xex − ex + C .

✷✳✾✳

✶✹✾

■♥t❡❣r❛t✐♦♥ ❜② ♣❛rts✿ ♣r♦❞✉❝ts

❚❤❡ ❧❡ss♦♥ s❡❡♠s t♦ ❜❡✿ • ❈❤♦♦s❡ ❢♦r u t❤❡ ♣❛rt ♦❢ t❤❡ ✐♥t❡❣r❛♥❞ t❤❛t ✇✐❧❧ ❜❡ s✐♠♣❧✐✜❡❞ ❜② ❞✐✛❡r❡♥t✐❛t✐♦♥✳

• ❈❤♦♦s❡ ❢♦r dv t❤❡ ♣❛rt ♦❢ t❤❡ ✐♥t❡❣r❛♥❞ t❤❛t ✇✐❧❧ ❜❡ s✐♠♣❧✐✜❡❞ ❜② ✐♥t❡❣r❛t✐♦♥✱ ♦r ❛t ❧❡❛st ✇✐❧❧ r❡♠❛✐♥

❛s s✐♠♣❧❡✳

❊①❛♠♣❧❡ ✷✳✾✳✹✿ s♣❧✐t ✐♥t❡❣r❛♥❞

■♥t❡❣r❛t❡✿

Z

x2 ex dx .

❖♥❝❡ ❛❣❛✐♥✱ t❤❡r❡ ❛r❡ ✭❛t ❧❡❛st✮ t✇♦ ✇❛②s t♦ ❝❤♦♦s❡ u ❛♥❞ dv ✿ • ✭❛✮ u = ex , dv = x2 dx✱ ❛♥❞ • ✭❜✮ u = x2 , dv = ex dx✳ ❲❡✬❧❧ tr② ❜♦t❤✳ ✭❛✮ ❲❡ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ u ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢ v ✿ u = ex

=⇒ du = ex dx Z x3 2 dv = x dx =⇒ v = x2 dx = 3

❊✈❡♥ t❤♦✉❣❤ du ✐s ❥✉st ❛s s✐♠♣❧❡ ❛s u✱ ✐♥t❡❣r❛t✐♦♥ ♦❢ dv ❤❛s ♠❛❞❡ t❤✐♥❣s ✇♦rs❡✳ ■♥❞❡❡❞✱ ✇❡ ❤❛✈❡✿ Z

udv = uv −

Z

x3 vdu = e · − 3 x

Z

x3 x · e dx . 3

■t✬s ♥♦t s✐♠♣❧❡r ❛s t❤❡ ♣♦✇❡r ♦❢ x ❣♦❡s ✉♣✦ ❲❡ r❡✈❡rs❡ t❤❡ ❝❤♦✐❝❡ ♦❢ u ❛♥❞ v ✳ ✭❜✮ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ u ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ♦❢ v ✿ u = x2

=⇒ du =Z2x dx dv = ex dx =⇒ v = ex dx = ex

❍❡r❡ dv ✐s s✐♠♣❧❡r t❤❛♥ u❀ t❤❛t✬s ❛ ❣♦♦❞ s✐❣♥✳ ❲❡ s✉❜st✐t✉t❡ t❤❡s❡ ✐♥t♦ t❤❡ ❢♦r♠✉❧❛✿ Z

udv = uv −

Z

2

x

vdu = x · e −

Z

ex · 2x dx .

❆❣❛✐♥✱ ✇❡ ♣❛✉s❡ t♦ ❛♣♣r❡❝✐❛t❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ✐♥t❡❣r❛t✐♦♥ t❛s❦ ❤❛s ❜❡❡♥ s✐♠♣❧✐✜❡❞✦ ❚❤❛t✬s ❜❡❝❛✉s❡ t❤❡ ♣♦✇❡r ♦❢ x ✇❡♥t ❞♦✇♥✳ ❲❡ ✜♥✐s❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ✉s✐♥❣ t❤❡ ■♥t❡❣r❛t✐♦♥ ❜② P❛rts ❢♦r♠✉❧❛ ❛♥❞ t❤❡ r❡s✉❧t ♦❢ t❤❡ ❧❛st ❡①❛♠♣❧❡✿ Z

2 x

2

x

x e dx = x · e −

Z

ex 2x dx = x2 ex − 2xex + 2ex + C .

❚❤❡ ❧❡ss♦♥ ✐s t❤❛t ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts ♠✐❣❤t ❜r✐♥❣ s✐♠♣❧✐✜❝❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ❛♥❞ ♠✐❣❤t r❡q✉✐r❡ ❛♥♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts✳ ❊①❡r❝✐s❡ ✷✳✾✳✺

❆♣♣❧② t❤❡ ❢♦r♠✉❧❛ ❢♦✉♥❞ ✐♥ t❤✐s ❡①❛♠♣❧❡ t♦ t❤❡ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡✱ ♣❛rt ✭❛✮✳ ❊①❡r❝✐s❡ ✷✳✾✳✻

■♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ tr② t❤✐s ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛♥❞✿ u = x, dv = xex ✳

✷✳✾✳ ■♥t❡❣r❛t✐♦♥ ❜② ♣❛rts✿ ♣r♦❞✉❝ts

✶✺✵

❊①❛♠♣❧❡ ✷✳✾✳✼✿ r❡❝✉rs✐♦♥ ■♥t❡❣r❛t❡✿

Z

u

❚❤❡r❡ ❛r❡ t✇♦ ✇❛②s t♦ s♣❧✐t t❤❡ ✐♥t❡❣r❛♥❞✱ ✇✐❧❧ r❡❞✉❝❡ t❤❡ ♣♦✇❡r

x✱

✐❢ ✇❡ ❝❤♦♦s❡

u=x

x3 sin x dx .

3

❛♥❞

dv ✱

❜✉t ✇❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rt

✳ ❲❡ ❛r❡ ❧❡❢t ✇✐t❤

dv = sin x✳

❚❤❡♥

u = x3

=⇒ du =Z3x2 dx dv = sin x dx =⇒ v = sin x dx = − cos x

❇② ♣❛rts✿

Z

3

x sin x dx = uv −

Z

3

vdu = −x cos x −

Z

3x2 · sin x dx .

❚❤❡ ❧❛st ✐♥t❡❣r❛❧ ✐s ❛❧♠♦st ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦r✐❣✐♥❛❧ ❜✉t t❤❡ ♣♦✇❡r ♦❢

x

✐s ❞♦✇♥ ❜②

1✳

❊①❡r❝✐s❡ ✷✳✾✳✽ ❋✐♥✐s❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ❜② ✐♥t❡❣r❛t✐♥❣ ❜② ♣❛rts t✇♦ ♠♦r❡ t✐♠❡s✳

❊①❛♠♣❧❡ ✷✳✾✳✾✿ ❝❛♥✬t s♣❧✐t ■♥t❡❣r❛t❡✿

Z

cos−1 x dx .

❚❤❡r❡ s❡❡♠s t♦ ❜❡ ♥♦t❤✐♥❣ t♦ s♣❧✐t ✐♥ t❤❡ ✐♥t❡❣r❛♥❞✦✳✳ ❚❤❡r❡ ✐s✿

1 dx u = cos−1 x =⇒ du = − √ 1 − x2 Z dv = dx =⇒ v = dx = x ❇② ♣❛rts✿

Z

cos

−1

x dx = uv −

Z

vdu Z



1 = cos x · x − x − √ 1 − x2 Z x dx = x cos−1 x + √ 1 − x2 Z 1 dz = x cos−1 x + √ z −2 1 = x cos−1 x − z −1/2 dz 2 1 z 1/2 −1 = x cos x − +C 2 1/2 √ = x cos−1 x − 1 − x2 + C. −1



dx

❊①❡r❝✐s❡ ✷✳✾✳✶✵ ❆♣♣❧② t❤❡ ■♥t❡❣r❛t✐♦♥ ❜② P❛rts ❢♦r♠✉❧❛ t♦ t❤❡ ✐♥t❡❣r❛❧✱

Z

xex dx ,

❉♦♥❡ ✇✐t❤ ♣❛rts✳

❇② s✉❜st✐t✉t✐♦♥✿

z = 1 − x2 ⇒ dz = −2xdx

❇❛❝❦✲s✉❜st✐t✉t✐♦♥✳

✷✳✶✵✳

❆♣♣r♦❛❝❤❡s t♦ ✐♥t❡❣r❛t✐♦♥

✶✺✶

✇✐t❤ t❤❡s❡ t✇♦ ❝❤♦✐❝❡s ♦❢ t❤❡ ✏♣❛rts✑✿ • ✭❛✮ x ❛♥❞ ex dx✱ • ✭❜✮ ex ❛♥❞ x dx✳

✷✳✶✵✳ ❆♣♣r♦❛❝❤❡s t♦ ✐♥t❡❣r❛t✐♦♥

▲❡t✬s s✉♠♠❛r✐③❡ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ✐♥t❡❣r❛t✐♦♥ ❛♥❞ ❝♦♠♣❛r❡ ✐t t♦ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❋✐rst✱ t❤❡ s✐♠✐❧❛r✐t✐❡s✳ ❏✉st ❛s t❤❡r❡ ✐s ❛ ❧✐st ♦❢ ❡❧❡♠❡♥t❛r② ❞❡r✐✈❛t✐✈❡s✱ ✇❡ ❤❛✈❡ ❛ ❧✐st ♦❢ ❡❧❡♠❡♥t❛r② ✐♥t❡❣r❛❧s✳ ■♥ ❢❛❝t✱ t❤❡ ❧❛tt❡r ❝♦♠❡s ❢r♦♠ t❤❡ ❢♦r♠❡r✳ ❍❡r❡ t❤❡② ❛r❡✿

s ′

(x ) = sx

Z

s−1

xs dx =

Z

1 x

1 s+1 x + C, s+1

❢♦r

s 6= −1

1 dx = ln x + C x Z x ′ x (e ) = e ex dx = ex + C Z ′ (sin x) = cos x cos x dx = sin x + C Z (cos x)′ = − sin x sin x dx = − cos x + C (ln x)′ =

■t✬s ❥✉st ❛ s❤♦rt ❧✐st ♦❢ ✐♥t❡❣r❛❧ ❢♦r♠✉❧❛s ❢♦r s♣❡❝✐✜❝ ❢✉♥❝t✐♦♥s✳ ❙♦✱ ❣✐✈❡♥ ❛ ❢✉♥❝t✐♦♥✱ ✇❡ ✜♥❞ ✐t ♦♥ t❤❡ ❧✐st ❛♥❞✱ ❛✉t♦♠❛t✐❝❛❧❧②✱ ✐ts ✐♥t❡❣r❛❧✱ ❥✉st ❧✐❦❡ ✇✐t❤ t❤❡ ❞❡r✐✈❛t✐✈❡s✳ ■♥ ❡✐t❤❡r ❝❛s❡✱ t❤❡ ❧✐st ✐s ✈❡r② s❤♦rt✦ ❚❤❡r❡ ❛r❡ ❞✐✛❡r❡♥❝❡s ❛❧r❡❛❞② ❜❡t✇❡❡♥ t❤❡ t✇♦✳ ❚❤❡ ♣r❡s❡♥❝❡ ♦❢ ✏ +C ✑ ✐♥ ❡❛❝❤ ✐♥t❡❣r❛❧ ✐♥❞✐❝❛t❡s t❤❛t t❤❡ ❛♥s✇❡r ❝♦♥t❛✐♥s ✐♥✜♥✐t❡❧② ♠❛♥② ❢✉♥❝t✐♦♥s✳ ❆❧s♦✱ t❤❡ ❢♦r♠✉❧❛s ❢♦r ✐♥t❡❣r❛❧s ♦♥❧② r❡♠❛✐♥ ✈❛❧✐❞ ✇❤❡♥ ❧✐♠✐t❡❞ t♦

✐♥t❡r✈❛❧s✳

❏✉st ❛s t❤❡r❡ ❛r❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ t❤❡r❡ ❛r❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ♦❢ ✐♥t❡❣r❛t✐♦♥✳ ❚❤❡ ❧❛tt❡r ❝♦♠❡s✱ ✐♥ ♣❛rt✱ ❢r♦♠ t❤❡ ❢♦r♠❡r✳

❙❘ ❈▼❘ ▲❈❘

′

′

(f + g) = f + g

′

Z

Z

Z

(f + g) dx = f dx + g dx Z Z ′ ′ (cf ) = cf (cf ) dx = c f dx Z Z 1 ′ ′ f (t) dt (f (mx + b)) = mf (mx + b) f (mx + b) dx = m t=mx+b

❚❤❡ ✇❛② ✇❡ ❛♣♣❧② t❤❡s❡ r✉❧❡s ✐s ✈❡r② s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ❢♦r ❞❡r✐✈❛t✐✈❡s✿

•

❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ t♦ ❜❡ ✐♥t❡❣r❛t❡❞ ✐s t❤❡ s✉♠ ♦❢ t✇♦ ✭❥✉st ❛s ✇❤❡♥ ✐t ✇❛s t♦ ❜❡ ❞✐✛❡r❡♥t✐❛t❡❞✮✱ ✇❡

•

❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ t♦ ❜❡ ✐♥t❡❣r❛t❡❞ ✐s ❛ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ♦❢ ❛♥♦t❤❡r ✭❥✉st ❛s ✇❤❡♥ ✐t ✇❛s t♦ ❜❡

s♣❧✐t t❤❡ ✐♥t❡❣r❛❧ ✐♥t♦ t✇♦ ❛♥❞ t❤❡♥ ❞❡❛❧ ✇✐t❤ ❡✐t❤❡r ✭s✐♠♣❧❡r✮ ✐♥t❡❣r❛❧ s❡♣❛r❛t❡❧②✳

❞✐✛❡r❡♥t✐❛t❡❞✮✱ ✇❡ ❢❛❝t♦r ✐t ♦✉t ❛♥❞ t❤❡♥ ❞❡❛❧ ✇✐t❤ t❤❡ r❡♠❛✐♥✐♥❣ ✭s✐♠♣❧❡r✮ ✐♥t❡❣r❛❧✳

✷✳✶✵✳

✶✺✷

❆♣♣r♦❛❝❤❡s t♦ ✐♥t❡❣r❛t✐♦♥

❚❤❡ s✐♠✐❧❛r✐t✐❡s st♦♣ ❤❡r❡✦ ❲❤❛t ❛❜♦✉t t❤❡ Pr♦❞✉❝t

❘✉❧❡

❢♦r ✐♥t❡❣r❛t✐♦♥❄ ❚❤❡r❡ ✐s ♥♦♥❡ ✐♥ t❤❡ ❛❜♦✈❡ s❡♥s❡✳

◮ ❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ t♦ ❜❡ ✐♥t❡❣r❛t❡❞ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ✭✉♥❧✐❦❡ ✇❤❡♥ ✐t ✇❛s t♦ ❜❡ ❞✐✛❡r❡♥✲

t✐❛t❡❞✮✱ ✇❡ ❝❛♥✬t s♣❧✐t t❤❡ ✐♥t❡❣r❛❧ ✐♥t♦ t✇♦ ❛♥❞ t❤❡♥ ❞❡❛❧ ✇✐t❤ ❡✐t❤❡r ✐♥t❡❣r❛❧ s❡♣❛r❛t❡❧②✳

❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❞✐✛❡r❡♥t✐❛t✐♦♥ ❝❛♥✬t ❜❡ ❡❛s✐❧② r❡✈❡rs❡❞✳✳✳ ✉♥❧❡ss ♦♥❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ✐s✱ ✐♥ ❢❛❝t✱ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ t❤❛t ✇❡ ❦♥♦✇ ♦r ❝❛♥ ✜♥❞✳ ❚❤❛t✬s t❤❡ ■♥t❡❣r❛t✐♦♥ ❜② P❛rts ❢♦r♠✉❧❛✿ Z

′

f g dx = f g −

Z

gf ′ dx

❆♥❞ ✇❤❛t ❛❜♦✉t t❤❡ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ✐♥t❡❣r❛t✐♦♥❄ ❚❤❡r❡ ✐s ♥♦♥❡✱ ✉♥❧❡ss ②♦✉ ❛r❡ ✇✐❧❧✐♥❣ t♦ ✐♥t❡r♣r❡t ❞✐✈✐s✐♦♥ ❛s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② t❤❡ r❡❝✐♣r♦❝❛❧✳ ◆♦✇✱ ✇❤❛t ❛❜♦✉t t❤❡ ❈❤❛✐♥

❘✉❧❡

❄ ❙❛♠❡ ♣r♦❜❧❡♠ ❛s ✇✐t❤ t❤❡ ♣r♦❞✉❝ts✿

◮ ❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ t♦ ❜❡ ✐♥t❡❣r❛t❡❞ ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ✭✉♥❧✐❦❡ ✇❤❡♥ ✐t ✇❛s t♦ ❜❡

❞✐✛❡r❡♥t✐❛t❡❞✮✱ ✇❡ ❝❛♥✬t s♣❧✐t t❤❡ ✐♥t❡❣r❛❧ ✐♥t♦ t✇♦ ❛♥❞ t❤❡♥ ❞❡❛❧ ✇✐t❤ ❡✐t❤❡r ✐♥t❡❣r❛❧ s❡♣❛r❛t❡❧②✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡ ❈❤❛✐♥ ❘✉❧❡ ❢♦r ❞✐✛❡r❡♥t✐❛t✐♦♥ ❝❛♥✬t ❜❡ ❡❛s✐❧② r❡✈❡rs❡❞✳✳✳ ✉♥❧❡ss t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ✐♥s✐❞❡ ✐s✱ ✐♥ ❢❛❝t✱ ♣r❡s❡♥t ❛s ❛ ❢❛❝t♦r✳ ❚❤❛t✬s t❤❡ ■♥t❡❣r❛t✐♦♥ ❜② ❙✉❜st✐t✉t✐♦♥ ❢♦r♠✉❧❛✿ Z

′

(f ◦ g) · g dx =

Z

f du

❊①❛♠♣❧❡ ✷✳✶✵✳✶✿ r♦✉t✐♥❡

❈♦♠♣✉t❡✿

Z

■❣♥♦r❡ t❤❡ ❜♦✉♥❞s ❛t ✜rst✿ Z

3

1 0



x

 x3 + 3ex − sin x dx . SR

x + 3e − sin x dx ==== CMR

====

Z

Z

3

x dx +

Z

x

Z

3e dx + sin x dx Z Z 3 x x dx + 3 e dx + sin x dx

x4 + 3 · ex − (− cos x) + C 4 1 4 simplify ==== x + 3ex + cos x + C . 4

formulas

====

❚❤❛t✬s t❤❡ ❤❛r❞ ♣❛rt✱ ✜♥❞✐♥❣ ❛♥t✐❞❡r✐✈❛t✐✈❡s✳ ◆♦✇ t❤❡ ❡❛s② ♣❛rt✿ Z

1 3

0

x



x + 3e − sin x dx

FTC

==== substitute

====

====

1 1 4 x x + 3e + cos x 4 0    1 4 1 4 1 0 1 + 3e + cos 1 + 0 + 3e + cos 0 4 4 1 + 3e + cos 1 − 0 − 3 − 1 . 4

❚❤❡ ❤❛r❞ ♣❛rt ✐s ❡❛s② ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❞✐✈✐s✐♦♥✱ ♦r ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s✳

✷✳✶✶✳

❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s

✶✺✸

❲❛r♥✐♥❣✦ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❛♣♣r♦❛❝❤ t♦ ✐♥t❡❣r❛t✐♦♥ ✐s ✉s✲ ✐♥❣ t❤❡ t❛❜❧❡ ♦❢ ✐♥t❡❣r❛❧s✦

❚❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ✐s t❤❛t

❞✐✛❡r❡♥t✐❛t✐♦♥ ♥❡✈❡r ❢❛✐❧s

❜✉t ✐♥t❡❣r❛t✐♦♥ ♠❛② ❢❛✐❧ ✐♥ t❤❡ s❡♥s❡ t❤❛t t❤❡ ✐♥t❡❣r❛❧

♠✐❣❤t t✉r♥ ♦✉t t♦ ❜❡ ❛ ❢✉♥❝t✐♦♥ ✇❡ ❤❛✈❡ ♥❡✈❡r s❡❡♥ ❜❡❢♦r❡ ♦r ❡✈❡♥ ❛ ❢✉♥❝t✐♦♥ t❤❛t ♥♦✲♦♥❡ ❤❛s s❡❡♥ ❜❡❢♦r❡✦

❊①❛♠♣❧❡ ✷✳✶✵✳✷✿ ❞❡✜♥❡ ❢✉♥❝t✐♦♥s ❛s ✐♥t❡❣r❛❧s ❲❡ ❝❛♥ ✉s❡ t❤✐s ✐❞❡❛ t♦ r❡✲❞✐s❝♦✈❡r ✏❢❛♠✐❧✐❛r✑ ❢✉♥❝t✐♦♥s ✕ st❛rt✐♥❣ ❛t t❤❡ ♦t❤❡r ❡♥❞✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♥t❡❣r❛t✐♥❣ t❤✐s r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ♣r♦❞✉❝❡s t❤❡ ❧♦❣❛r✐t❤♠✿

1 t

Z

❧❡❛❞✐♥❣ t♦

x 1

1 dt = ln x . t

❲❡ ❤❛✈❡ t❤✉s ❞❡✜♥❡❞ t❤❡ ❧♦❣❛r✐t❤♠ ✇✐t❤♦✉t r❡❢❡r❡♥❝❡ t♦ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✳ ❆❧s♦✱ ✐♥t❡❣r❛t✐♥❣ t❤✐s ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥ ♣r♦❞✉❝❡s t❤❡ ❛r❝s✐♥❡✿

1 √ 1 − t2

❧❡❛❞✐♥❣ t♦

Z

x 0

√

1 dt = sin−1 x . 2 1−t

❲❡ ❤❛✈❡ ❞❡✜♥❡❞ t❤❡ ❛r❝s✐♥❡ ✇✐t❤♦✉t r❡❢❡r❡♥❝❡ t♦ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ s✐♥❡✳

✷✳✶✶✳ ❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s ❲❡ ❝❛♥ ♦♥❧② ❝♦♠♣✉t❡ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s ♦✈❡r

❜♦✉♥❞❡❞

✐♥t❡r✈❛❧s s✉❝❤ ❛s

[a, b]✳

❊①❛♠♣❧❡ ✷✳✶✶✳✶✿ ✇❤❛t ❤❛♣♣❡♥s t♦ ❛♥t✐❞❡r✐✈❛t✐✈❡ ❈♦♥s✐❞❡r ❛❣❛✐♥ t❤✐s ❞❡❝❡♣t✐✈❡ ❢♦r♠✉❧❛✿

Z

1 ??? dx ==== ln |x| + C, x 6= 0 . x

t❤❡ t✇♦ ✐♥t❡r✈❛❧s

❚❤❡ ❢♦r♠✉❧❛ ✐s ❢❛❧s❡ ❛s st❛t❡❞✦ ■t ❤♦❧❞s ♦♥❧② ♦♥ ❡✐t❤❡r ♦❢ ✱ s❡♣❛r❛t❡❧②✱ ♦❢ t❤❡ ❞♦♠❛✐♥ 1 ♦❢ ✱ ✐✳❡✳✱ (−∞, 0) ❛♥❞ (0, ∞)✱ ❜✉t ♥♦t ♦♥ t❤❡ s❡t (−∞, 0) ∪ (0, ∞)✳ ❚❤✐s ♠❡❛♥s t❤❛t C ❝❛♥ ✈❛r② ❢r♦♠ x t❤❡ ♦♥❡ t♦ t❤❡ ♦t❤❡r✦ ■♥ ❢❛❝t✱ t❤✐s ✐s t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡✿

F (x) =

(

ln |x| + C ln |x| + K

❢♦r ❢♦r

x < 0, x > 0.

■t ✐s ❡✈❡♥ ♠♦r❡ ❞❛♥❣❡r♦✉s t♦ ✐❣♥♦r❡ t❤❡ ❣❛♣ ✐♥ t❤❡ ❞♦♠❛✐♥ ✇❤❡♥ ✇❡ ❞❡❛❧ ✇✐t❤ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s✳ ❋♦r ❡①❛♠♣❧❡✱ ♦♥❡ ♠✐❣❤t ♣r♦❞✉❝❡ t❤✐s ❢r♦♠ t❤❡ ❢♦r♠✉❧❛✿

Z

1 −1

1 1 ??? dx ==== ln |x| = ln 1 − ln | − 1| = 0 . x −1

❚❤✐s ✐s ✉♥tr✉❡ ❜❡❝❛✉s❡ t❤❡ ✐♥t❡❣r❛❧ ✐s ✐♥❞❡t❡r♠✐♥❛t❡ ❛s ❛ ❧✐♠✐t✳ ■♥❞❡❡❞✱ s✐♠♣❧② ❜❡❝❛✉s❡ ✐t✬s ✉♥❞❡✜♥❡❞ ❛t

x = 0✳

f

✐s ♥♦t ✐♥t❡❣r❛❜❧❡ ♦♥

[−1, 1]

❋✉rt❤❡r♠♦r❡✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡ ♣♦s✐t✐✈❡ ❛♥❞ t❤❡ ♥❡❣❛t✐✈❡ ❛r❡❛s

s❡❡♠ t♦ ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r✱ t❤✐s ✐s ❢❛❧s❡ ❜❡❝❛✉s❡ ❜♦t❤ ❛r❡✱ ✐♥ ❢❛❝t✱

✐♥✜♥✐t❡ ✿

✷✳✶✶✳

❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s

✶✺✹

❲❡ s❤♦✉❧❞♥✬t ❜❡❝❛s✉❛❧ ❛❜♦✉t ❞♦✐♥❣ ❛❧❣❡❜r❛ ✇✐t❤ ✐♥✜♥✐t✐❡s ✭❱♦❧✉♠❡ ✷✮✿

???

∞ − ∞ ==== 0 .

❊①❡r❝✐s❡ ✷✳✶✶✳✷ ❙❤♦✇ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ❡①❛♠♣❧❡ ✇♦♥✬t ❜❡❝♦♠❡ ✐♥t❡❣r❛❜❧❡ ✇❤❛t❡✈❡r ♥✉♠❜❡r ✇❡ ❛ss✐❣♥ t♦

❲❡ ✇✐❧❧ ♥❡①t tr② t♦ ✉♥❞❡rst❛♥❞ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❛r❡❛ ♦❢ ❛♥ ✐♥✜♥✐t❡ ♦r✱ ❜❡tt❡r✱

✉♥❜♦✉♥❞❡❞

x = 0✳

r❡❣✐♦♥✳

❊①❛♠♣❧❡ ✷✳✶✶✳✸✿ ✐♥✜♥✐t❡ ❜♦tt❧❡ ❚❤❡ ❛r❡❛ ♦❢ ❛♥ ✏✐♥✜♥✐t❡ r❡❝t❛♥❣❧❡✑✱ ❧✐❦❡ t❤❡ ♦♥❡ ❜❡❧♦✇✱ ♠✉st ❜❡ ✐♥✜♥✐t❡✿

❲❤② ♦r ✐♥ ✇❤❛t s❡♥s❡❄ ❚❤✐s r❡❣✐♦♥ ❝♦♥t❛✐♥s ❛ ❣r♦✇✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✭r❡❛❧✱ ✜♥✐t❡✮ r❡❝t❛♥❣❧❡s t❤❡ ❛r❡❛s ♦❢ ✇❤✐❝❤ ❣r♦✇ t♦ ✐♥✜♥✐t②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐t ✇♦✉❧❞ t❛❦❡ ❛♥ ✐♥✜♥✐t❡ ❛♠♦✉♥t ♦❢ ✇❛t❡r t♦ ✜❧❧ s✉❝❤ ❛ ❜♦tt❧❡✳

❲❡ ❤❛✈❡ ❝♦♠❡ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❛r❡❛s ♦❢ ✭s♦♠❡✮ ❜♦✉♥❞❡❞ r❡❣✐♦♥s✳ ❚❤✐s ✇✐❧❧ ❜❡ ♦✉r ❛♣♣r♦❛❝❤✿

•

✏❊①❤❛✉st✑ ❛♥ ✉♥❜♦✉♥❞❡❞ r❡❣✐♦♥ ✇✐t❤ ❛ s❡q✉❡♥❝❡ ♦❢ ❜♦✉♥❞❡❞ r❡❣✐♦♥s✳

•

❋✐♥❞ t❤❡✐r ❛r❡❛s✳

•

❊①❛♠✐♥❡ t❤❡

❧✐♠✐t

♦❢ t❤❡s❡ ❛r❡❛s✳

■t✬s ♥♦ ❞✐✛❡r❡♥t✱ ✐♥ ♣r✐♥❝✐♣❧❡✱ ❢r♦♠ ❡①❤❛✉st✐♥❣ ❛ ❝✐r❝❧❡ ✇✐t❤ ♣♦❧②❣♦♥s✳ ❲❡ ✇✐❧❧ r❡str✐❝t ♦✉r ❛tt❡♥t✐♦♥ t♦ r❡❣✐♦♥s✿ ✶✳ ✉♥❜♦✉♥❞❡❞ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡

x✲❛①✐s

✭✐♥✜♥✐t❡❧② ✇✐❞❡✮✱ ❛♥❞

✷✳ ✉♥❜♦✉♥❞❡❞ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡

y ✲❛①✐s

✭✐♥✜♥✐t❡❧② t❛❧❧✮✳

❆s ✇❡ ♦♥❧② ❞❡❛❧ ✇✐t❤ r❡❣✐♦♥s ❞❡t❡r♠✐♥❡❞ ❜② ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s✱ ✶✳ t❤❡ ❢♦r♠❡r ❝❛s❡ ✐s ❛❜♦✉t ❢✉♥❝t✐♦♥s ✇✐t❤

✉♥❜♦✉♥❞❡❞ ❞♦♠❛✐♥s

♦r✱ ❜❡tt❡r✱ ✉♥❜♦✉♥❞❡❞ ❞♦♠❛✐♥s ♦❢ ✐♥t❡✲

❣r❛t✐♦♥✱ ❛♥❞ ✷✳ t❤❡ ❧❛tt❡r ❛❜♦✉t ❢✉♥❝t✐♦♥s ✇✐t❤

✉♥❜♦✉♥❞❡❞ r❛♥❣❡s

✭✐✳❡✳✱ ✉♥❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s✮✳

❊✈❡♥ t❤♦✉❣❤ t❤❡s❡ t✇♦ ❝❧❛ss❡s ♦❢ ❢✉♥❝t✐♦♥s ❛r❡ ✈❡r② ❞✐✛❡r❡♥t✱ t❤❡ ✐ss✉❡ ✐s t❤❡ s❛♠❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❞❡✜♥❡s t✇♦

✐❞❡♥t✐❝❛❧

✉♥❜♦✉♥❞❡❞ r❡❣✐♦♥s✿

y = 1/x

✷✳✶✶✳ ❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s

✶✺✺

❲❡ st❛rt ✇✐t❤ t❤❡ ❢♦r♠❡r ❝❛s❡✿ ❛♥ ✉♥❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥✳ ❊①❛♠♣❧❡ ✷✳✶✶✳✹✿ ❝♦♥st❛♥t

❈♦♥s✐❞❡r ❛ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥✱

f (x) = k ♦♥ [a, ∞), k > 0 .

❚❤❡♥✱ t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❣✐♦♥ ❛❜♦✈❡ t❤❡ ✐♥t❡r✈❛❧ [a, b] ✇✐t❤ b > a ✭❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ❢r♦♠ a t♦ b✮ ✐s ❡q✉❛❧ t♦ (b − a)k ✳ ❋✉rt❤❡r♠♦r❡✱ ❆r❡❛ = (b − a)k → +∞ ❛s b → +∞ .

❚❤❡r❡❢♦r❡✱ t❤❡ ❛r❡❛ ♦❢ t❤❡ ✐♥✜♥✐t❡ str✐♣ ✐s ✐♥✜♥✐t❡✱ ❛s ❡①♣❡❝t❡❞✳ ❊①❛♠♣❧❡ ✷✳✶✶✳✺✿ r❡❝✐♣r♦❝❛❧

❈♦♥s✐❞❡r ❛❣❛✐♥

f (x) = 1/x ♦♥ [1, ∞) .

❚❤❡ ❛r❡❛ ✉♥❞❡r t❤✐s ❣r❛♣❤ ♦✈❡r t❤✐s r❛② ✐♥ t❤❡ x✲❛①✐s ✐s s❤♦✇♥ ❛❜♦✈❡✳ ❚❤✐s ✉♥❜♦✉♥❞❡❞ r❡❣✐♦♥ ✐s ❡①❤❛✉st❡❞ ❜② ❜♦✉♥❞❡❞ ♦♥❡s✳ ❍♦✇❄ ❚❤❡ ♦❜✈✐♦✉s ❛♣♣r♦❛❝❤ ✐s t♦ ✉s❡ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✳ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ✐s t❤❛t ♦♥❡ ❤❛s t♦ ❜♦t❤ ♠❛❦❡ t❤❡ r❡❝t❛♥❣❧❡s t❤✐♥♥❡r ❛♥❞ t❤✐♥♥❡r ✭❛s ❜❡❢♦r❡✮ ❛♥❞ ♠❛❦❡ t❤❡ r✐❣❤t ❡♥❞ ♦❢ t❤❡ ✐♥t❡r✈❛❧ ❡①t❡♥❞ ♠♦r❡ ❛♥❞ ♠♦r❡ t♦ t❤❡ r✐❣❤t✳

✭◆♦t❡ t❤❛t ✇❤❡♥ h = 1✱ t❤✐s s✉♠ ✐s ❝❛❧❧❡❞ ❛ s❡r✐❡s t♦ ❜❡ ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✺✳✮

✷✳✶✶✳

❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s

✶✺✻

❆♥ ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤ ✐s t♦ r❡❧② ♦♥ ✇❤❛t ✇❡ ❛❧r❡❛❞② ❦♥♦✇ ❛❜♦✉t ❛r❡❛s ✉♥❞❡r t❤❡ ❣r❛♣❤s ✇❤❡♥ t❤❡ ✐♥t❡r✈❛❧ ✐s ❜♦✉♥❞❡❞✳ ❚❤❡ ✉♥❞❡r❧②✐♥❣ r❛② ♦❢ t❤✐s r❡❣✐♦♥ ✐s ❡①❤❛✉st❡❞ ✇✐t❤ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧s✳ ❚❤❡② ❛❧❧ ❤❛✈❡ t❤❡ s❛♠❡ ❧❡❢t ❜♦✉♥❞✱

1✱

b✱

❜✉t t❤❡ r✐❣❤t ❜♦✉♥❞✱

[1, b] ❚❤❡♥✱

Z

b 1

✐s ❛♣♣r♦❛❝❤✐♥❣ ✐♥✜♥✐t②✿

[1, +∞)

❧❡❛❞✐♥❣ t♦

1 dx = ln b − ln 1 x →∞

❛s

❛s

b → +∞ .

b → +∞ .

❚❤❡ ❛r❡❛ ♦❢ t❤❡ ✉♥❜♦✉♥❞❡❞ r❡❣✐♦♥ ✐s ✐♥✜♥✐t❡✳

■♥✐t✐❛❧❧②✱ ✇❡ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥

r❛②s

❛s t❤❡ ❞♦♠❛✐♥s✿

(−∞, b] ❚❤❡ r❛②s ✇✐❧❧ ❤❛✈❡ t♦ ❜❡ ✏❡①❤❛✉st❡❞✑ ✇✐t❤

❛♥❞

[a, +∞) .

❝❧♦s❡❞ ❛♥❞ ❜♦✉♥❞❡❞

✐♥t❡r✈❛❧s✱ ♦♥ ❡❛❝❤ ♦❢ ✇❤✐❝❤ ✇❡ ❢❛❝❡ t❤❡ ✉s✉❛❧

❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s✳ ❊①❛♠♣❧❡ ✷✳✶✶✳✻✿ ♠♦r❡ r❡❝✐♣r♦❝❛❧ ♣♦✇❡rs

❲❡ ❞✐s❝♦✈❡r❡❞ t❤❛t t❤❡ ❜❛♥❞ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢

y = 1/x

✐s ♥❛rr♦✇✐♥❣ ❞♦✇♥ ❜✉t ♥♦t ❢❛st ❡♥♦✉❣❤ t♦

❛✈♦✐❞ ❣r♦✇✐♥❣ ✐ts ❛r❡❛ t♦ ✐♥✜♥✐t②✳ ❆ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛ ♥❛rr♦✇❡r str✐♣ ✇♦♥✬t ♥❡❝❡ss❛r✐❧② ❛✈♦✐❞ ❤❛✈✐♥❣ ❛♥

y = 1/(3x)✳ ■♥st❡❛❞✱ ❧❡t✬s tr②✱ ✐♥ ❝♦♥tr❛st✱ ❛ ❢✉♥❝t✐♦♥ t❤❛t ✐s ❞❡❝r❡❛s✐♥❣ ♠✉❝❤ ❢❛st❡r✳ y = 1/x2 ❄ ❲❡ ❤❛✈❡✿ Z b 1 1 dx = − +1 2 b 1 x →1 ❛s b → +∞ .

✐♥✜♥✐t❡ ❛r❡❛✿ ❛❜♦✉t

❙♦✱

1

❍♦✇

✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ ✉♥❜♦✉♥❞❡❞ r❡❣✐♦♥✳ ■t✬s ✜♥✐t❡✦

♠♦t✐♦♥

■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ❛r❡❛s✱ ✐♥t❡❣r❛❧s ❝❛♥ ❛❧s♦ ❜❡ ❡①♣❧❛✐♥❡❞ ✐♥ t❡r♠s ♦❢ ✳ ■❢ ♦✉r ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡ 2 ❜✉t ❞❡❝❧✐♥✐♥❣ ❛s 1/x ✱ ✇❤❡r❡ x ✐s t✐♠❡✱ t❤❡ ❞✐st❛♥❝❡ ✇❡ ✇✐❧❧ ❝♦✈❡r ♦✈❡r ❛♥ ✐♥✜♥✐t❡ ♣❡r✐♦❞ ♦❢ t✐♠❡ ✇✐❧❧ ♥♦t ❜❡ ✐♥✜♥✐t❡✦ ❊①❛♠♣❧❡ ✷✳✶✶✳✼✿ ❡①♣♦♥❡♥t✐❛❧

❲❤❛t ❢✉♥❝t✐♦♥ ❞❡❝r❡❛s❡s ❢❛st❡r t❤❛♥ ❛❧❧

1/xn , n = 1, 2, 3, ...❄ f (x) = e−x

♦♥

■t✬s t❤❡

❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ❢✉♥❝t✐♦♥ ✿

[1, ∞) .

❆❣❛✐♥✱ t❤❡ ✉♥❜♦✉♥❞❡❞ r❡❣✐♦♥ ✉♥❞❡r t❤✐s ❣r❛♣❤ ❛❜♦✈❡ t❤✐s r❛② ✐s ❡①❤❛✉st❡❞ ❜② ❡①❤❛✉st✐♥❣ t❤❡ ✉♥❞❡r❧②✐♥❣ r❛② ✇✐t❤ t❤❡ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧s✿

[1, b]

❧❡❛❞✐♥❣ t♦

[1, +∞)

❛s

b → +∞ .

✷✳✶✶✳

❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s

❚❤❡♥✱ t♦t❛❧ ❛r❡❛

= lim

b→+∞

✶✺✼

Z

b

e−x dx 1

= lim (−e−b − (−e−1 )) b→+∞

= ❚❤❡ ♠❛✐♥ ❞✐s❝♦✈❡r② ✐s t❤❛t

1 . e

t❤❡ ❛r❡❛ ♦❢ ❛♥ ✉♥❜♦✉♥❞❡❞ r❡❣✐♦♥ ❞♦❡s♥✬t ❤❛✈❡ t♦ ❜❡ ❜❡ ✐♥✜♥✐t❡ ✦

❊①❡r❝✐s❡ ✷✳✶✶✳✽ ❋✐♥❞ t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦✈❡r 1 ✶✳ y = 2 x 1 ✷✳ y = √ x

[1, ∞)✿

❊①❛♠♣❧❡ ✷✳✶✶✳✾✿ s✐♥✉s♦✐❞ ❲❤❛t ✐❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s♥✬t ❛❧❧ ♣♦s✐t✐✈❡❄ ❲❤❛t ✐s t❤❡ ❛r❡❛ ✕ ♦✈❡r t❤❡ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs ✕ ✉♥❞❡r ❛ s✐♥✉s♦✐❞❄

❚❤❡ ❛♥❛❧②s✐s ✐s ✐❞❡♥t✐❝❛❧✿

t♦t❛❧ ❛r❡❛

= lim

b→+∞

Z

b 0

cos x dx = lim (sin b − sin 0) b→+∞

❉◆❊

.

❚❤❡ ❧✐♠✐t ❞♦❡s♥✬t ❡①✐st ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡r❡ ✐s ♥♦ ❛r❡❛✳

❲❛r♥✐♥❣✦ ✏◆♦ ❛r❡❛✑ ✐s♥✬t t❤❡ s❛♠❡ ❛s ✏③❡r♦ ❛r❡❛✑✳

❏✉st ❛s ✇✐t❤ ❛❧❧ ❧✐♠✐ts✱ t❤❡r❡ ❛r❡ t❤r❡❡ ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s✿ ❚❤✐s ♠❛② ❜❡ ❛ ♥✉♠❜❡r✱ ♦r ✐t ♠❛② ❜❡ ✐♥✜♥✐t❡✱ ♦r ✐t ♠❛② ❜❡ ✉♥❞❡✜♥❡❞✳ ❚❤✐s ✐s t❤❡ s✉♠♠❛r②✳ ❲❡ ✏❡①❤❛✉st✑ t❤❡ ✉♥❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥✱

(−∞, b]

♦r

[a, ∞)✱

✇✐t❤ ❜♦✉♥❞❡❞ ♦♥❡s✳ ■❢ t❤❡ ❢✉♥❝t✐♦♥

✐s ✐♥t❡❣r❛❜❧❡ ♦♥ t❤❡s❡ ❝❧♦s❡❞ ❛♥❞ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥s✱ ✇❡ t❤❡♥ ✏❡①❤❛✉st✑ ❛ ♣♦ss✐❜❧② ✐♥✜♥✐t❡ ❛r❡❛ ♦❢ ♦✈❡r t❤✐s ❞♦♠❛✐♥ ✇✐t❤ ✜♥✐t❡ ♦♥❡s✿

✷✳✶✶✳ ❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s

✶✺✽

■❢ t❤❡ ❧✐♠✐t ♦❢ t❤✐s ✐♥t❡❣r❛❧ ❡①✐sts✱ ✐t ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ ■♥t❡❣r❛❧ ♦✈❡r r❛②

Z

❛♥❞

b

f (t) dt = lim

a→−∞

−∞

Z

∞

f (t) dt = lim

b→+∞

a

Z Z

b

f (t) dt a b

f (t) dt a

❚❤❡s❡ ❧✐♠✐ts✱ ♦❢ ❝♦✉rs❡✱ ❝❛♥ ❜❡ ✐♥✜♥✐t❡✳ ❲❛r♥✐♥❣✦ ❊✈❡♥ t❤♦✉❣❤ t❤❡ ♥♦t❛t✐♦♥ s✉❣❣❡sts t❤❛t t❤❡ ❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥ ✐s t❤❡ ✇❤♦❧❡ r❛②✱ t❤✐s ✐s ♥♦t❤✐♥❣ ❜✉t ❛♥ ❛❜❜r❡✈✐❛t✐♦♥ ❢♦r t❤❡ ❧✐♠✐t ♦♥ t❤❡ r✐❣❤t✳

❲❡ ❛❧s♦ ❞❡✜♥❡ t❤❡ ✐♥t❡❣r❛❧ ♦✈❡r t❤❡ ✇❤♦❧❡ r❡❛❧ ❧✐♥❡ (−∞, ∞) ✐♥ t❡r♠s ♦❢ t❤❡ ♦♥❡s ♦✈❡r r❛②s✱ ❛s t❤❡ s✉♠ ♦❢ t✇♦ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥t❡❣r❛❧s ✭❧✐♠✐ts✮ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥✿ ■♥t❡❣r❛❧ ♦✈❡r ❧✐♥❡

Z

∞

f (t) dt =

−∞

Z

0

f (t) dt + −∞

Z

∞

f (t) dt

0

■♥ t❤❡ ❝❛s❡ ♦❢ ✐♥✜♥✐t❡ ❧✐♠✐ts✱ ✇❡ ✉t✐❧✐③❡ t❤❡ ❛❧❣❡❜r❛ ♦❢ ✐♥✜♥✐t✐❡s ❛s s❤♦rt❝✉ts ✭❱♦❧✉♠❡ ✷✮✿ ( ♥✉♠❜❡r ) + (+∞) = +∞ ( ♥✉♠❜❡r ) + (−∞) = −∞ (+∞)

+ (+∞) = +∞

(−∞)

+ (−∞) = −∞

✷✳✶✶✳ ❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s

✶✺✾

❊①❛♠♣❧❡ ✷✳✶✶✳✶✵✿ ❡①♣♦♥❡♥t✐❛❧ ▲❡t✬s ❝♦♠♣✉t❡ t❤❡ ✐♥t❡❣r❛❧✱ Z

+∞

Z

Z

0

∞

ex dx e dx + 0 −∞ Z Z b 0 x ex dx e dx + lim = lim

x

e dx = −∞

x

a→−∞

b→+∞

a

0

= lim (1 − ea ) + lim (eb − 1) dx . a→−∞

b→+∞

❚❤❡ ✜rst ❧✐♠✐t ✐s 1 ❜✉t t❤❡ s❡❝♦♥❞ ❧✐♠✐t ✐s ✐♥✜♥✐t❡❀ t❤❡r❡❢♦r❡✱ ♦✉r ✐♥t❡❣r❛❧ ✐s ✐♥✜♥✐t❡✳

❊①❡r❝✐s❡ ✷✳✶✶✳✶✶ ❙❤♦✇ t❤❛t r❡♣❧❛❝✐♥❣ 0 ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ♦✈❡r (−∞, +∞) ✇✐t❤ ❛♥② r❡❛❧ c ✇✐❧❧ ♣r♦❞✉❝❡ t❤❡ s❛♠❡ r❡s✉❧t✳

❊①❡r❝✐s❡ ✷✳✶✶✳✶✷ ❙❤♦✇ t❤❛t r❡♣❧❛❝✐♥❣ t❤❡ ❧❛st ❞❡✜♥✐t✐♦♥ ✇✐t❤ Z

∞

❄

f (t) dt == lim

−∞

R→∞

Z

R

f (t) dt , −R

✇♦♥✬t ❛❧✇❛②s ♣r♦❞✉❝❡ t❤❡ s❛♠❡ r❡s✉❧t✳

❉❡✜♥✐t✐♦♥ ✷✳✶✶✳✶✸✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s ♦♥ ✐♥✜♥✐t❡ ✐♥t❡r✈❛❧s ❚❤❡ ❧✐♠✐ts ♦❢ t❤❡ ✐♥t❡❣r❛❧s ❛❜♦✈❡ ❛r❡ ❝❛❧❧❡❞ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s✳ ❲❤❡♥ s✉❝❤ ❛ ❧✐♠✐t ❡①✐sts✱ ♦r t❤❡ t✇♦ ❧✐♠✐ts ✐♥ t❤❡ ❧❛st ❝❛s❡ ❡①✐st✱ ✇❡ s❛② t❤❛t t❤❡ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧ ❝♦♥✈❡r❣❡s ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ ✐s ✐♥t❡❣r❛❜❧❡ ❀ ♦t❤❡r✇✐s❡ ✐t ❞✐✈❡r❣❡s✳ ❚❤❡ ❧❛tt❡r t❡r♠✐♥♦❧♦❣② ✐s ❜♦rr♦✇❡❞ ❢r♦♠ ❧✐♠✐ts ❛♥❞ ✐t ✇✐❧❧ ❜❡ ✉s❡❞ ❛❣❛✐♥ ✐♥ ❈❤❛♣t❡r ✺✳

❊①❛♠♣❧❡ ✷✳✶✶✳✶✹✿ ❜❡❧❧ ❝✉r✈❡ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ♦✈❡r (−∞, ∞) ❢♦❧❧♦✇s t❤❡ ❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ✐♥t❡❣r❛❧ t❤❛t ❝♦♠❡s ❢r♦♠ t❤❡ ✐❞❡❛ ♦❢ ❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ❛r❡❛s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ✇❤❛t s✉❝❤ ❛♥ ✐♥t❡❣r❛❧ ❧♦♦❦s ❧✐❦❡✿

❚❤❡ ❢✉♥❝t✐♦♥ ✐s e−x ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ✐s ❦♥♦✇♥ t♦ ❜❡ ❝♦♥✈❡r❣❡♥t✳ 2

❙♦♠❡ ✐♥t❡❣r❛❧s s❤♦✉❧❞ ❜❡ ❝♦♠♣✉t❡❞ ❛❤❡❛❞ ♦❢ t✐♠❡✳

✷✳✶✶✳ ❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s

✶✻✵

❚❤❡♦r❡♠ ✷✳✶✶✳✶✺✿ ■♠♣r♦♣❡r ■♥t❡❣r❛❧s ♦❢ ❘❡❝✐♣r♦❝❛❧s ■ ❋♦r ❛♥②

a > 0✱

✇❡ ❤❛✈❡

Z

∞ a

Pr♦♦❢✳

 1−p a 1 dx = p − 1  xp ∞

✇❤❡♥

p > 1,

✇❤❡♥

0 < p ≤ 1.

❋♦r p 6= 1✱ ✇❡ ❤❛✈❡✿ Z

∞ a

1 dx = lim b→∞ xp = lim

b→∞

Z

b

Za

1 dx xp

❆❝❝♦r❞✐♥❣ t♦ ❞❡✜♥✐t✐♦♥✳

x−p dx

❯s❡ P❋ ♥❡①t✳

b

a

b 1 x−p+1 = lim b→∞ −p + 1

a

 1 = lim b−p+1 − a−p+1 −p + 1 b→∞   1 −p+1 −p+1 lim b −a . = −p + 1 b→∞

❚❤❡ r❡♠❛✐♥✐♥❣ ❧✐♠✐t ✐s 0 ✇❤❡♥ −p + 1 < 0✱ ❛♥❞ ✐t ✐s ✐♥✜♥✐t❡ ✇❤❡♥ −p + 1 > 0✳

❊①❡r❝✐s❡ ✷✳✶✶✳✶✻ ❋✐♥✐s❤ t❤❡ ♣r♦♦❢✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✐♥t❡❣r❛❧ • ❝♦♥✈❡r❣❡s ✇❤❡♥ p > 1✱

• ❞✐✈❡r❣❡s ✇❤❡♥ 0 < p ≤ 1✳

◆♦✇✱ t❤❡ ❧❛tt❡r ❝❛s❡✿ ✉♥❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ❛♥❞ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥s ♦❢ ✐♥t❡❣r❛t✐♦♥✳

❊①❛♠♣❧❡ ✷✳✶✶✳✶✼✿ ✐♥✜♥✐t❡ ❛r❡❛ ❈♦♥s✐❞❡r

1 ♦♥ [0, 1) . 1−x ❚❤❡ ❛r❡❛ ✉♥❞❡r t❤✐s ❣r❛♣❤ ♦✈❡r t❤✐s ❤❛❧❢✲♦♣❡♥ ✐♥t❡r✈❛❧ ✐♥ t❤❡ x✲❛①✐s ✐s s❤♦✇♥ ❜❡❧♦✇✿ f (x) =

❍♦✇ ❞♦ ✇❡ ✉♥❞❡rst❛♥❞ t❤❡ ❛r❡❛ ✉♥❞❡r t❤✐s ❣r❛♣❤ ♦✈❡r t❤✐s ✐♥t❡r✈❛❧❄ ❲❡ ❝❛♥ ✉s❡ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✱ ❡①❛❝t❧② ❛s ❛❧✇❛②s✱ ❛s ❧♦♥❣ ❛s 1 ✐s ♥♦t ❛♠♦♥❣ ✐ts s❛♠♣❧❡ ♣♦✐♥ts ✿

✷✳✶✶✳

❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s

✶✻✶

❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ t❛❦❡ ❛ ❤✐♥t ❢r♦♠ t❤❡ ❛♥❛❧②s✐s ♦❢ ❝❛s❡ ✶✿ ❚❤✐s ✉♥❜♦✉♥❞❡❞ r❡❣✐♦♥ ✐s ❡①❤❛✉st❡❞ ❜② ❜♦✉♥❞❡❞ ♦♥❡s✳ ❍♦✇❄ ❚❤❡ ✉♥❞❡r❧②✐♥❣ ✐♥t❡r✈❛❧ ✐s ❡①❤❛✉st❡❞ ✇✐t❤ ❝❧♦s❡❞ ✐♥t❡r✈❛❧s✳ ❚❤❡② ❛❧❧ ❤❛✈❡ t❤❡ s❛♠❡ ❧❡❢t ❜♦✉♥❞✱

a✱

❜✉t t❤❡ r✐❣❤t ❜♦✉♥❞✱

[0, b] ❚❤❡♥✱

Z

b 0

b✱

✐s ❛♣♣r♦❛❝❤✐♥❣

❧❡❛❞✐♥❣ t♦

[0, 1)

1✿

❛s

b → 1.

1 dx = − ln(1 − b) − ln 1 1−x →∞

❛s

b → 1,

✇❤✐❝❤ ✐s t❤❡♥ t❤❡ ❛r❡❛ ♦❢ t❤❡ ✉♥❜♦✉♥❞❡❞ r❡❣✐♦♥✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ s✐♠♣❧② ❞♦❡s♥✬t ❛♣♣❧② t♦ ❛ ❢✉♥❝t✐♦♥ ✉♥❞❡✜♥❡❞ ❛t ♦♥❡ ♦❢ t❤❡ ❡♥❞s ♦❢ t❤❡ ✐♥t❡r✈❛❧✳ ■♥st❡❛❞✱ ✇❡ ❝♦♥s✐❞❡r ❛ ■♥✐t✐❛❧❧②✱ ✇❡ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥

r❡str✐❝t✐♦♥

♦❢ t❤❡ ❢✉♥❝t✐♦♥ t♦ ❛ s♠❛❧❧❡r✱ ❜✉t ❝❧♦s❡❞✱ ✐♥t❡r✈❛❧✳

❤❛❧❢✲♦♣❡♥ ✐♥t❡r✈❛❧s ✿ (c, b]

❛♥❞

[a, c) .

❆s ②♦✉ s❡❡✱ t❤❡ ❛♥❛❧②s✐s ✐s ✈❡r② s✐♠✐❧❛r t♦ t❤❡ ❢♦r♠❡r ❝❛s❡✱ ✇✐t❤ t❤✐s s✉❜st✐t✉t✐♦♥✿

[a, ∞) −→ [a, c) . ❏✉st ❛s t❤❡ ❢♦r♠❡r✱ t❤❡ ❧❛tt❡r ✇✐❧❧ ❤❛✈❡ t♦ ❜❡ ✏❡①❤❛✉st❡❞✑ ✇✐t❤

❝❧♦s❡❞ ❛♥❞ ❜♦✉♥❞❡❞

✐♥t❡r✈❛❧s✳

❊①❛♠♣❧❡ ✷✳✶✶✳✶✽✿ ✜♥✐t❡ ❛r❡❛ ▲❡t✬s ❝♦♥s✐❞❡r

f (x) = √

1 1−x

♦♥

[0, 1) .

❊✈❡♥ t❤♦✉❣❤ t❤❡✐r ❣r❛♣❤s ❧♦♦❦ ❛❧♠♦st ✐❞❡♥t✐❝❛❧✱ t❤✐s ♦♥❡ ✐♥❝r❡❛s❡s s❧♦✇❡r t❤❛♥ t❤❡ ❧❛st ♦♥❡✳ ❆❣❛✐♥✱ t❤❡ ✉♥❜♦✉♥❞❡❞ r❡❣✐♦♥ ✉♥❞❡r t❤✐s ❣r❛♣❤ ♦✈❡r t❤✐s ❤❛❧❢✲♦♣❡♥ ✐♥t❡r✈❛❧ ✐s ❡①❤❛✉st❡❞ ❜② ❡①❤❛✉st✐♥❣ t❤❡ ✉♥❞❡r❧②✐♥❣ r❛② ✇✐t❤ t❤❡ ❝❧♦s❡❞ ✐♥t❡r✈❛❧s✿

[0, b]

❛s

b → 1✳

❆r❡❛ ♦❢ ✉♥❜♦✉♥❞❡❞ r❡❣✐♦♥

❚❤❡♥ ✇❡ ❤❛✈❡✿

Z

b

1 dx b→1 0 1 − x b √ = lim −2 1 − x b→1

= lim

√

0

= 2. ❚❤❡ ❛r❡❛ ♦❢ ❛♥ ✉♥❜♦✉♥❞❡❞ r❡❣✐♦♥ ♠❛② ❜❡ ✜♥✐t❡✦

✷✳✶✶✳

❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s

✶✻✷

❊①❛♠♣❧❡ ✷✳✶✶✳✶✾✿ ❞✐✈❡r❣❡♥❝❡ ❲❤❛t ✐❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s♥✬t ❛❧❧ ♣♦s✐t✐✈❡❄ ❲❤❛t ✐s t❤❡ ❛r❡❛ ✉♥❞❡r ❛♥ ♦s❝✐❧❧❛t✐♥❣ ❣r❛♣❤✱ s✉❝❤ ❛s t❤✐s❄

y = sin

❲✐t❤ t❤❡ ❣r❛♣❤ ❧✐❦❡ t❤✐s✱ ♦♥❡ ❝❛♥

❣✉❡ss

1 x

t❤❛t t❤❡ ❧✐♠✐t ❞♦❡s♥✬t ❡①✐st ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡r❡ ✐s ♥♦ ❛r❡❛✳

❏✉st ❛s ✇✐t❤ ❛❧❧ ❧✐♠✐ts✱ t❤❡r❡ ❛r❡ t❤r❡❡ ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s ❢♦r t❤❡s❡ ❛r❡❛s✿ ❛ ♥✉♠❜❡r✱ ✐♥✜♥✐t②✱ ♦r ✉♥❞❡✜♥❡❞✳ ❚❤✐s ✐s t❤❡ s✉♠♠❛r②✳ ❲❡ ✏❡①❤❛✉st✑ t❤❡ ❤❛❧❢✲♦♣❡♥ ❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥✱

(a, b] ♦r [a, b)✱ ✇✐t❤ ❝❧♦s❡❞ ♦♥❡s✳

■❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ✐♥t❡❣r❛❜❧❡

♦♥ t❤❡s❡ ❝❧♦s❡❞ ❛♥❞ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥s✱ ✇❡ t❤❡♥ ✏❡①❤❛✉st✑ ❛ ♣♦ss✐❜❧② ✐♥✜♥✐t❡ ❛r❡❛ ♦✈❡r t❤✐s ❞♦♠❛✐♥ ✇✐t❤ ✜♥✐t❡ ♦♥❡s✳

■❢ t❤❡ ❧✐♠✐t ♦❢ t❤✐s ✐♥t❡❣r❛❧ ❡①✐sts✱ ✐t ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

■♥t❡❣r❛❧ ♦✈❡r ❤❛❧❢✲♦♣❡♥ ✐♥t❡r✈❛❧s

❛♥❞

❚❤❡s❡ ❧✐♠✐ts✱ ♦❢ ❝♦✉rs❡✱ ❝❛♥ ❜❡ ✐♥✜♥✐t❡✳

Z

b

f (t) dt = lim

a→c

c

Z

c

f (t) dt = lim a

b→c

Z Z

b

f (t) dt a b

f (t) dt a

✷✳✶✶✳ ❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s

✶✻✸

❲❛r♥✐♥❣✦ ❚❤❡ ♥♦t❛t✐♦♥ ✐s ✉♥❢♦rt✉♥❛t❡❧② ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦♥❡ ❢♦r ♣r♦♣❡r ✐♥t❡❣r❛❧s✱ ❜✉t t❤✐s ✐s ♥♦t❤✐♥❣ ❜✉t ❛♥ ❛❜✲ ❜r❡✈✐❛t✐♦♥ ❢♦r t❤❡ ❧✐♠✐t ♦♥ t❤❡ r✐❣❤t✳

❲❡ ❛❧s♦ ❞❡✜♥❡ t❤❡ ✐♥t❡❣r❛❧ ♦✈❡r ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ✐♥ t❡r♠s ♦❢ t❤❡ ♦♥❡s ♦✈❡r ❤❛❧❢✲♦♣❡♥ ✐♥t❡r✈❛❧s✱ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ❢♦r ❛♥② c ❜❡t✇❡❡♥ a ❛♥❞ b✿

■♥t❡❣r❛❧ ♦✈❡r ♦♣❡♥ ✐♥t❡r✈❛❧ Z

b

f (t) dt = a

Z

c

f (t) dt + a

Z

b

f (t) dt c

■♥ t❤❡ ❝❛s❡ ♦❢ ✐♥✜♥✐t❡ ❧✐♠✐ts✱ ✇❡ ❢♦❧❧♦✇ t❤❡ ❛❧❣❡❜r❛ ♦❢ ✐♥✜♥✐t✐❡s ❛❜♦✈❡✳ ❋♦r ❝❛s❡ ✷✱ ✇❡ r❡♣❡❛t t❤❡ ❞❡✜♥✐t✐♦♥ ❢♦r ❝❛s❡ ✶✳

❉❡✜♥✐t✐♦♥ ✷✳✶✶✳✷✵✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s ♦♥ ✜♥✐t❡ ✐♥t❡r✈❛❧s ❚❤❡ ❧✐♠✐ts ♦❢ t❤❡ ✐♥t❡❣r❛❧s ❛❜♦✈❡ ❛r❡ ✭❛❧s♦✮ ❝❛❧❧❡❞ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s✳ ❲❤❡♥ t❤❡ ❧✐♠✐t ❡①✐sts✱ ♦r t❤❡ t✇♦ ❧✐♠✐ts ✐♥ t❤❡ ❧❛st ❝❛s❡ ❡①✐st✱ ✇❡ s❛② t❤❛t t❤❡ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧ ❝♦♥✈❡r❣❡s ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ ✐s ✐♥t❡❣r❛❜❧❡ ❀ ♦t❤❡r✇✐s❡ ✐t ❞✐✈❡r❣❡s✳

❊①❡r❝✐s❡ ✷✳✶✶✳✷✶ ❙❤♦✇ t❤❛t r❡♣❧❛❝✐♥❣ t❤❡ ❧❛st ❞❡✜♥✐t✐♦♥ ✇✐t❤ Z

b a

❄

f (t) dt == lim+ ε→0

Z

b−ε

f (t) dt . a+ε

✇♦♥✬t ♣r♦❞✉❝❡ t❤❡ s❛♠❡ r❡s✉❧t✳

❊①❛♠♣❧❡ ✷✳✶✶✳✷✷✿ ❤♦❧❡ ✐♥ ❞♦♠❛✐♥ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ♦✈❡r ❛♥ ✐♥t❡r✈❛❧ ✇✐t❤ ❛ ♣♦ss✐❜❧❡ ♠✐ss✐♥❣ ♣♦✐♥t ✐♥s✐❞❡ ❢♦❧❧♦✇s ❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ✐♥t❡❣r❛❧ t❤❛t ❝♦♠❡s ❢r♦♠ t❤❡ ✐❞❡❛ ♦❢ ❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ❛r❡❛s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ✇❤❛t s✉❝❤ ❛♥ ✐♥t❡❣r❛❧ ❧♦♦❦s ❧✐❦❡✿

❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ✐s ❦♥♦✇♥ t♦ ❜❡ ❝♦♥✈❡r❣❡♥t✳

1 p , |x|

✷✳✶✶✳ ❚❤❡ ❛r❡❛s ♦❢ ✐♥✜♥✐t❡ r❡❣✐♦♥s✿ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s

✶✻✹

❚❤❡♦r❡♠ ✷✳✶✶✳✷✸✿ ■♠♣r♦♣❡r ■♥t❡❣r❛❧s ♦❢ ❘❡❝✐♣r♦❝❛❧s ■■ ❋♦r ❛♥②

a > 0✱

✇❡ ❤❛✈❡

Z Pr♦♦❢✳

b 0

 1−p b 1 dx = 1 − p  xp ∞

✇❤❡♥

0 < p < 1,

✇❤❡♥

p ≥ 1.

❋♦r p 6= 1✱ ✇❡ ❤❛✈❡✿ Z

b 0

Z b 1 1 dx = lim+ dx p p a→0 x Za x

❇② ❞❡✜♥✐t✐♦♥✳

b

= lim+ a→0

❲❡ ✉s❡ P❋ ♥❡①t✳

x−p dx

a

b 1 −p+1 x = lim+ a→0 −p + 1

a

 1 = lim+ b−p+1 − a−p+1 a→0 −p + 1   1 −p+1 −p+1 = . b − lim+ a a→0 −p + 1

❚❤❡ r❡♠❛✐♥✐♥❣ ❧✐♠✐t ✐s 0 ✇❤❡♥ −p + 1 > 0✱ ❛♥❞ ✐t ✐s ✐♥✜♥✐t❡ ✇❤❡♥ −p + 1 < 0✳ ❊①❡r❝✐s❡ ✷✳✶✶✳✷✹

❋✐♥✐s❤ t❤❡ ♣r♦♦❢✳ ❊①❡r❝✐s❡ ✷✳✶✶✳✷✺

▼❛t❝❤ t❤❡ ✐♥t❡❣r❛❧s ❛♥❞ t❤❡ ❛r❡❛s ♦❢ t❤❡ t✇♦ t❤❡♦r❡♠s ❛❜♦✉t ✐♥t❡❣r❛❧s ♦❢ t❤❡ r❡❝✐♣r♦❝❛❧s✳ ❍✐♥t✿ ■t✬s ❛❜♦✉t s②♠♠❡tr②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✐♥t❡❣r❛❧ • ❝♦♥✈❡r❣❡s ✇❤❡♥ p > 1✱

Z

b 0

• ❞✐✈❡r❣❡s ✇❤❡♥ 0 < p ≤ 1✳

1 dx✿ xp

❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ t✇♦ t②♣❡s ♦❢ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s ❢♦r t❤❡s❡ ❢✉♥❝t✐♦♥s✿

✷✳✶✷✳ Pr♦♣❡rt✐❡s ♦❢ ♣r♦♣❡r ❛♥❞ ✐♠♣r♦♣❡r ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s

✶✻✺

❊①❡r❝✐s❡ ✷✳✶✶✳✷✻

❲❤❛t ♣♦ss✐❜❧❡ ✈❛❧✉❡s ❝❛♥ t❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ❛s②♠♣t♦t❡ t❛❦❡❄ ●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ❢♦r ❡❛❝❤ ✈❛❧✉❡✳

✷✳✶✷✳ Pr♦♣❡rt✐❡s ♦❢ ♣r♦♣❡r ❛♥❞ ✐♠♣r♦♣❡r ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s ❚❤✉s✱ ✇❡ ❤❛✈❡ ❡①t❡♥❞❡❞ t❤❡ ✐❞❡❛ ♦❢ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✇✐t❤ t❤❡ ❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥✿

• ❝❧♦s❡❞ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧s✱ s✉❝❤ ❛s [a, b]✱ t♦

• ❤❛❧❢✲♦♣❡♥✱ s✉❝❤ ❛s (a, b] ❛♥❞ [a, b)✱ ❛♥❞ ❛❧s♦ ♣♦ss✐❜❧② ✐♥✜♥✐t❡✱ s✉❝❤ ❛s (−∞, b] ❛♥❞ [a, ∞)✱ ❛♥❞ ❢✉rt❤❡r t♦

• ♦♣❡♥ ✐♥t❡r✈❛❧s✱ s✉❝❤ ❛s (a, b)✱ ❛♥❞ ♣♦ss✐❜❧② ✐♥✜♥✐t❡✱ s✉❝❤ ❛s (−∞, +∞)✳

■t ✐s ♣♦ss✐❜❧❡ t♦ ❤❛✈❡ ❛ ✉♥✐❢♦r♠ tr❡❛t♠❡♥t ♦❢ t❤❡s❡ ❝❛s❡s✿

❲❡ ♦✉t❧✐♥❡ ✐t ❜❡❧♦✇✳ ■❢ ✇❡ ❞❡♥♦t❡ ❛♥ ✐♥t❡r✈❛❧ ❜② I ✱ ❛❧❧ t❤❡s❡ ✐♥t❡❣r❛❧s ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ s❛♠❡ ♥♦t❛t✐♦♥✿ ■♥t❡❣r❛❧ ♦✈❡r ✐♥t❡r✈❛❧

Z

f dx I

❚❤❡s❡ ✐♥t❡❣r❛❧s ❤❛✈❡ ✐❞❡♥t✐❝❛❧ ♣r♦♣❡rt✐❡s✳ ■♥ ❢❛❝t✱ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ♣r♦♣❡r ✐♥t❡❣r❛❧s✱ ✇❤✐❝❤ ✐♥ t✉r♥ ❝♦♠❡ ❢r♦♠ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts✳ ❆s r❡❣✐♦♥s ❛r❡ ❥♦✐♥❡❞ t♦❣❡t❤❡r ✈✐❛ ✉♥✐♦♥✱ t❤❡✐r ❛r❡❛s ❛r❡ ❛❞❞❡❞ ✕ ❡✈❡♥ t❤♦✉❣❤ t❤❡ r❡❣✐♦♥s ♠❛② ❜❡ ✉♥❜♦✉♥❞❡❞✳ ❚❤❡ ❛r❡❛ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❛❞❞✐t✐✈✐t② ✐s t❤❡ s❛♠❡ ❛s ❜❡❢♦r❡✱ ❛s ❧♦♥❣ ❛s t❤❡ ✐♥t❡❣r❛❧s ❛r❡ ❝♦♥✈❡r❣❡♥t✿

✷✳✶✷✳

Pr♦♣❡rt✐❡s ♦❢ ♣r♦♣❡r ❛♥❞ ✐♠♣r♦♣❡r ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s

✶✻✻

❚❤❡♦r❡♠ ✷✳✶✷✳✶✿ ❆❞❞✐t✐✈✐t② ♦❢ ■♥t❡❣r❛❧s ❙✉♣♣♦s❡

f

✐s ✐♥t❡❣r❛❜❧❡ ♦✈❡r ✐♥t❡r✈❛❧s

✐♥t❡❣r❛❜❧❡ ♦✈❡r

I ∪ J✱

I

❛♥❞

J

f dx =

Z

t❤❛t s❤❛r❡ ♦♥❡ ♣♦✐♥t✳ ❚❤❡♥

f

✐s

❛♥❞ ✇❡ ❤❛✈❡✿

Z

f dx + I

Z

J

f dx I∪J

❚❤❡♦r❡♠ ✷✳✶✷✳✷✿ ■♥t❡❣r❛❜✐❧✐t② ■❢

f

✐s ✐♥t❡❣r❛❜❧❡ ♦✈❡r ✐♥t❡r✈❛❧

I ✱ t❤❡♥ ✐t ✐s ❛❧s♦ ✐♥t❡❣r❛❜❧❡ ♦✈❡r ❛♥② ✐♥t❡r✈❛❧ J ⊂ I ✳

❚❤❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ❛r❡ ❛❧s♦ t❤❡ s❛♠❡✳

❚❤❡♦r❡♠ ✷✳✶✷✳✸✿ ❙✉♠ ❘✉❧❡ ❢♦r ■♥t❡❣r❛❧s ❙✉♣♣♦s❡

f

❛♥❞

g

❛r❡ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥s ♦✈❡r ✐♥t❡r✈❛❧

I✳

❚❤❡♥ s♦ ✐s

f + g✱

❛♥❞

✇❡ ❤❛✈❡✿

Z

(f + g) dx = I

Z

f dx + I

Z

g dx I

❚❤❡♦r❡♠ ✷✳✶✷✳✹✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ■♥t❡❣r❛❧s ❙✉♣♣♦s❡

f

✐s ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ ♦✈❡r ✐♥t❡r✈❛❧

I✳

❚❤❡♥ s♦ ✐s

c·f

❢♦r ❛♥② r❡❛❧

✷✳✶✷✳

Pr♦♣❡rt✐❡s ♦❢ ♣r♦♣❡r ❛♥❞ ✐♠♣r♦♣❡r ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s

✶✻✼

c✱ ❛♥❞ ✇❡ ❤❛✈❡✿ Z

I

(c · f ) dx = c ·

Z

f dx I

❊①❡r❝✐s❡ ✷✳✶✷✳✺ Pr♦✈❡ t❤❡ t✇♦ t❤❡♦r❡♠s✳ ❍✐♥t✿ ❯s❡ t❤❡ r✉❧❡s ♦❢ ❧✐♠✐ts✳

❊①❡r❝✐s❡ ✷✳✶✷✳✻ ❲❤❛t ✐s t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❢♦r ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s❄

❚❤❡ ♠❛✐♥ ♣❛rt ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧ ✐s ♦♥❡ ♦❢ t❤❡s❡ ❧✐♠✐ts✿

lim

x→b

Z

x

dx

❛♥❞

a

lim

x→∞

Z

x

dx . a

❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ✐s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❧✐♠✐t✳

❚❤❡r❡ ✐s ❛ ✇❛② t♦ ♣r❡❞✐❝t ✇❤❛t ❤❛♣♣❡♥s

✇✐t❤♦✉t ❡✈❛❧✉❛t✐♥❣ t❤❡ ❧✐♠✐t✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ♠❛ ❥♦r t❤❡♦r❡♠ ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✶✮✿

❚❤❡♦r❡♠ ✷✳✶✷✳✼✿ ▼♦♥♦t♦♥❡ ❈♦♥✈❡r❣❡♥❝❡

❊✈❡r② ♠♦♥♦t♦♥❡ ❛♥❞ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡s✳ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥ ❝♦♥✈❡r❣❡s ❛t ✐♥✜♥✐t②✳

❊✈❡r② ♠♦♥♦t♦♥❡ ❛♥❞

❲❡ ❛❧s♦ ❦♥♦✇ ❢r♦♠ ❱♦❧✉♠❡ ✷✿

❚❤❡♦r❡♠ ✷✳✶✷✳✽✿ ▼♦♥♦t♦♥✐❝✐t② ❚❤❡♦r❡♠

❆♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s ♠♦♥♦t♦♥❡ ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❡✐t❤❡r ❛❧❧ ♣♦s✐✲ t✐✈❡ ♦r ❛❧❧ ♥❡❣❛t✐✈❡✳ ▲❡t✬s ❡①❝❧✉❞❡ t❤❡ r❡st ♦❢ t❤❡ ❢✉♥❝t✐♦♥s✱ s✉❝❤ ❛s

sin

❛♥❞

cos✿

❚❤❡♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✷✳✶✷✳✾✿ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ◆♦♥✲♥❡❣❛t✐✈❡ ■♥t❡❣r❛❧

■❢ ❛ ❢✉♥❝t✐♦♥ ✐s ♥♦♥✲♥❡❣❛t✐✈❡✱ ✐ts ✐♥t❡❣r❛❧s ❛r❡ ❡✐t❤❡r ❝♦♥✈❡r❣❡♥t ♦r ✐♥✜♥✐t❡✳ ❊①❡r❝✐s❡ ✷✳✶✷✳✶✵ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❚❤❡♥✱ t♦ ❡st❛❜❧✐s❤ ❝♦♥✈❡r❣❡♥❝❡✱ ✇❡ ❝❛♥ ✉s❡ ❛ ❞✐r❡❝t ❝♦♠♣❛r✐s♦♥ ✇✐t❤ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥✱ ❛ ❢✉♥❝t✐♦♥ t❤❛t ❤❛s ❛ ❝♦♥✈❡r❣❡♥t ✐♥t❡❣r❛❧✳ ❚❤❡ ♦t❤❡r ❢✉♥❝t✐♦♥ s❤♦✉❧❞ ❜❡ ❧❛r❣❡r✿

✷✳✶✷✳

Pr♦♣❡rt✐❡s ♦❢ ♣r♦♣❡r ❛♥❞ ✐♠♣r♦♣❡r ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s

✶✻✽

❙✐♠✐❧❛r❧②✱ t♦ ❡st❛❜❧✐s❤ ❞✐✈❡r❣❡♥❝❡✱ ✇❡ ❝❛♥ ✉s❡ ❛ ❞✐r❡❝t ❝♦♠♣❛r✐s♦♥ ✇✐t❤ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥✱ ❛ ❢✉♥❝t✐♦♥ t❤❛t ❤❛s ❛ ❞✐✈❡r❣❡♥t ✐♥t❡❣r❛❧✳ ❚❤❡ ♦t❤❡r ❢✉♥❝t✐♦♥ s❤♦✉❧❞ ❜❡ s♠❛❧❧❡r✳

❊①❛♠♣❧❡ ✷✳✶✷✳✶✶✿ ❝♦♠♣❛r✐s♦♥ ❈♦♥s✐❞❡r t❤❡s❡ t✇♦ ❢❛❝ts✿ ❚❤❡ ✐♥t❡❣r❛❧

❚❤❡ ✐♥t❡❣r❛❧

Z

Z

∞

1 x1/2

1 ∞ 1

dx

1 dx x3

❞✐✈❡r❣❡s

❝♦♥✈❡r❣❡s

❜❡❝❛✉s❡

❜❡❝❛✉s❡

Z

Z

∞ 1 ∞ 1

1 x1/3

dx

1 dx x2

❞✐✈❡r❣❡s✳

❝♦♥✈❡r❣❡s✳

❚❤❡s❡ ❝♦♥❝❧✉s✐♦♥s ❝♦♠❡ ❢r♦♠ t❤❡ ✐♥❡q✉❛❧✐t✐❡s ❜❡❧♦✇✿

1/3

≤ 1/2

≤

2

≤3

=⇒

x1/3

≤ x1/2

≤

x2

≤ x3

=⇒

1

Z

1 1 1 ≥ 1/2 ≥ ≥ 3 =⇒ 2 Z ∞x Z ∞x Z ∞x 1 1 1 1 dx ≥ dx = ∞ > dx ≥ dx 1/3 1/2 2 x x x x3 1 1 1

x1/3 ∞ 1

❊①❡r❝✐s❡ ✷✳✶✷✳✶✷ ❲❤❛t ❞♦❡s t❤❡ ♠✐❞❞❧❡ ✐♥❡q✉❛❧✐t② ❣✐✈❡ ✉s❄

❚❤❡ ✐❞❡❛ ❛♣♣❧✐❡s t♦ ❛❧❧ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s✿

❚❤❡♦r❡♠ ✷✳✶✷✳✶✸✿ ❈♦♠♣❛r✐s♦♥ ❢♦r ■♠♣r♦♣❡r ■♥t❡❣r❛❧s ❙✉♣♣♦s❡

I

✐s ❛♥ ✐♥t❡r✈❛❧✱ ❛♥❞

0 ≤ f (x) ≤ g(x) ❢♦r ❛❧❧

• •

x

✐♥

I✳

❚❤❡♥✱ ❢♦r ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s ♦✈❡r

■❢ t❤❡ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧ ♦❢ ♦❢

f

I✱

✇❡ ❤❛✈❡✿

❞✐✈❡r❣❡s✱ t❤❡♥ s♦ ❞♦❡s t❤❡ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧

g✳

■❢ t❤❡ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧ ♦❢

g

❝♦♥✈❡r❣❡s✱ t❤❡♥ s♦ ❞♦❡s t❤❡ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧

✷✳✶✷✳

Pr♦♣❡rt✐❡s ♦❢ ♣r♦♣❡r ❛♥❞ ✐♠♣r♦♣❡r ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s ♦❢

f

❛♥❞ ✇❡ ❤❛✈❡✿

0≤

✶✻✾

Z

I

f dx ≤

Z

g dx . I

❊①❡r❝✐s❡ ✷✳✶✷✳✶✹ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❙✉♣♣♦s❡ t❤❡ ❢✉♥❝t✐♦♥s ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡✳

❆❝❝♦r❞✐♥❣ t♦ t❤❡

❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠✱

t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥

♠❛❦❡s s❡♥s❡✿

❈♦♥✈❡r❣❡♥❝❡ ♦❢ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧

Z

I

f dx < ∞

❉✐✈❡r❣❡♥❝❡ ♦❢ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧

Z ❚❤❡♥ t❤❡

❈♦♠♣❛r✐s♦♥ ❚❤❡♦r❡♠

I

f dx = ∞

❛❜♦✈❡ ❝❛♥ ❜❡ r❡❛❞ ❢r♦♠ t❤❡s❡ s✐♠♣❧❡ ✐♥❡q✉❛❧✐t✐❡s✿

Z ❛♥❞

Z

I

I

f dx ≥

Z

f dx ≤

Z

I

I

g dx = ∞ g dx < ∞ .

❊①❡r❝✐s❡ ✷✳✶✷✳✶✺ ❲❤❛t ✐❢ ✇❡ ✉s❡

str✐❝t

✐♥❡q✉❛❧✐t✐❡s ✐♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡❄

❊①❡r❝✐s❡ ✷✳✶✷✳✶✻ ❲❤❛t ❝❛♥ ✇❡ ❞❡r✐✈❡ ❛❜♦✉t t❤❡ ❝♦♥✈❡r❣❡♥❝❡✴❞✐✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s ♦❢ t❤❡ r❡❝✐♣r♦❝❛❧ ♣♦✇❡rs ❜❛s❡❞ ❡♥t✐r❡❧② ♦♥ t❤❛t ♦❢

1/x❄

❍✐♥t✿

❊①❡r❝✐s❡ ✷✳✶✷✳✶✼ ❙t❛t❡ ❛♥❞ ♣r♦✈❡ t❤❡ ❙q✉❡❡③❡ ❚❤❡♦r❡♠ ❢♦r ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s✳

✷✳✶✸✳ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

✶✼✵

✷✳✶✸✳ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

▲❡t✬s r❡✈✐❡✇ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ❢r❡❡ ✢✐❣❤t ❢r♦♠ ❱♦❧✉♠❡ ✷✳ ❆ s♦❝❝❡r ❜❛❧❧ r♦❧❧✐♥❣ ♦♥ ❛ ❤♦r✐③♦♥t❛❧ ♣❧❛♥❡ ✇✐❧❧ ❤❛✈❡ ❛ ❝♦♥st❛♥t ✈❡❧♦❝✐t②✿

❆ ♣✐♥❣✲♣♦♥❣ ❜❛❧❧ t❤r♦✇♥ ✉♣ ✐♥ t❤❡ ❛✐r ❣♦❡s ✉♣✱ s❧♦✇s ❞♦✇♥ ✉♥t✐❧ ✐t st♦♣s ❢♦r ❛♥ ✐♥st❛♥t✱ ❛♥❞ t❤❡♥ ❛❝❝❡❧❡r❛t❡s t♦✇❛r❞ t❤❡ s✉r❢❛❝❡✿

❲❤❛t ✐❢ ✇❡ ❞♦ ❜♦t❤✿ ❲❡ r♦❧❧ ❛ s♦❝❝❡r ❜❛❧❧ ❤♦r✐③♦♥t❛❧❧② ❛♥❞ t❤r♦✇ ❛ ♣✐♥❣✲♣♦♥❣ ❜❛❧❧ ✈❡rt✐❝❛❧❧②❄ ▲❡t✬s tr② t♦ ❢♦❧❧♦✇ ❜♦t❤ ❜❛❧❧s ❛t t❤❡ s❛♠❡ t✐♠❡✿

❲❡✬❞ ❤❛✈❡ t♦ ✢② t❤r♦✉❣❤ t❤❡ ❛✐r ❛s ✐❢ t❤r♦✇♥ ❛t ❛♥ ❛♥❣❧❡✦ ❖✉r ✉♥❞❡rst❛♥❞✐♥❣ ✐s t❤❛t ❛ t❤r♦✇♥ ❜❛❧❧ ♠♦✈❡s ✐♥ ❜♦t❤ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ ❞✐r❡❝t✐♦♥s✱ s✐♠✉❧t❛♥❡♦✉s❧② ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t❧②✿

✷✳✶✸✳ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

✶✼✶

❚❤❡ ❞②♥❛♠✐❝s ✐s ✈❡r② ❞✐✛❡r❡♥t✿ ✶✳ ■♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❞✐r❡❝t✐♦♥✱ ❛s t❤❡r❡ ✐s ♥♦ ❢♦r❝❡ ❝❤❛♥❣✐♥❣ t❤❡ ✈❡❧♦❝✐t②✱ t❤❡ ❧❛tt❡r r❡♠❛✐♥s ❝♦♥st❛♥t✳ ✷✳ ▼❡❛♥✇❤✐❧❡✱ t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t❧② ❝❤❛♥❣❡❞ ❜② t❤❡ ❢♦r❝❡ ♦❢ ❣r❛✈✐t②✳ ▲❡t✬s ♥♦✇ ✉s❡ t❤❡s❡ ❞❡s❝r✐♣t✐♦♥s t♦ r❡♣r❡s❡♥t t❤❡ ♠♦t✐♦♥ ♠❛t❤❡♠❛t✐❝❛❧❧②✳ ❘❡❝❛❧❧ ❤♦✇ ✇❡ ✉s❡❞ t❤❡s❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts t♦ ✜♥❞ ✈❡❧♦❝✐t② ❛♥❞ t❤❡♥ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥✿ pn

7→

vn =

pn+1 − pn ∆p = ∆t h

7→

an =

∆v vn+1 − vn = ∆t h

❍❡r❡ h = ∆t ✐s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡✳ ❚❤❡s❡ ❢♦r♠✉❧❛s ❤❛✈❡ ❛❧s♦ ❜❡❡♥ s♦❧✈❡❞ ❢♦r pn+1 ❛♥❞ vn+1 r❡s♣❡❝t✐✈❡❧② ✐♥ ♦r❞❡r t♦ ♠♦❞❡❧ ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✿ an

7→

vn+1 = vn + han

7→

pn+1 = pn + hvn

❚❤❡s❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❛r❡ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ✿ ❚❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ❛r❡ ❜❡✐♥❣ ❛❞❞❡❞ ♦♥❡ ❛t ❛ t✐♠❡✳ ❚❤✐s ✐s ✇❤❛t t❤❡ r❡s✉❧ts ♠✐❣❤t ❧♦♦❦ ❧✐❦❡✿

❚❤✐s t✐♠❡✱ ✇❡ ❤❛✈❡ t✇♦ s✉❝❤ s❡q✉❡♥❝❡s✿ ♦♥❡ ❢♦r ❤♦r✐③♦♥t❛❧ ❛♥❞ ♦♥❡ ❢♦r ✈❡rt✐❝❛❧✳ ❲❡ ❝♦♥str✉❝t t❤❡ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐♥ t❤❡ ♠♦st ❝♦♥✈❡♥✐❡♥t ✇❛②✿ • ❚❤❡ x✲❛①✐s ✐s ❤♦r✐③♦♥t❛❧✳

• ❚❤❡ y ✲❛①✐s ✐s ✈❡rt✐❝❛❧✳

✷✳✶✸✳

✶✼✷

❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

❍♦✇❡✈❡r✱ ✇❡ ❛❜❛♥❞♦♥ t❤❡ ❢❛♠✐❧✐❛r y = f (x) s❡t✉♣✦ ❲❡ ❤❛✈❡ t❤r❡❡

✈❛r✐❛❜❧❡s

♥♦✇✿

✶✳ t ✐s t✐♠❡✳ ✷✳ x ✐s t❤❡ ❤♦r✐③♦♥t❛❧ ❞✐♠❡♥s✐♦♥✱ t❤❡ ❞❡♣t❤✳ ✸✳ y ✐s t❤❡ ✈❡rt✐❝❛❧ ❞✐♠❡♥s✐♦♥✱ t❤❡ ❤❡✐❣❤t✳ ❊✐t❤❡r ♦❢ t❤❡ t✇♦ s♣❛t✐❛❧ ✈❛r✐❛❜❧❡s ❞❡♣❡♥❞s ♦♥ t❤❡ t❡♠♣♦r❛❧ ✈❛r✐❛❜❧❡✳ ❚❤❡✐r ❣r❛♣❤s ❛r❡ ♣❧♦tt❡❞ ❜❡❧♦✇ ✭❧❡❢t✮✿

▼❡❛♥✇❤✐❧❡✱ t❤❡ ❛❝t✉❛❧ ♣❛t❤ ♦❢ t❤❡ ❜❛❧❧ t❤r♦✉❣❤ s♣❛❝❡ ✇✐❧❧ ❛♣♣❡❛r t♦ ❛♥ ♦❜s❡r✈❡r ❛s ❛ ❝✉r✈❡ ✐♥ t❤❡✱ ✈❡rt✐❝❛❧❧② ❛❧✐❣♥❡❞✱ xy ✲♣❧❛♥❡ ✭r✐❣❤t✮✳ ❆s t❤❡r❡ ✐s ♥♦ t✲❛①✐s✱ ✇❡ ♣r♦✈✐❞❡ t❤❡ t✐♠❡s ❜② ❧❛❜❡❧✐♥❣ ❛ ❢❡✇ ♣♦✐♥ts ♦❢ t❤❡ tr❛❥❡❝t♦r②✳ ❍✐st♦r✐❝❛❧❧②✱ ♦♥❡ ♦❢ t❤❡ ✈❡r② ✜rst ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❝❛❧❝✉❧✉s ✇❛s ✐♥ ❜❛❧❧✐st✐❝s✳ ❇❡❢♦r❡ ❝❛❧❝✉❧✉s✱ ♦♥❡ ❤❛❞ t♦ r❡s♦rt t♦ tr✐❛❧ ❛♥❞ ❡rr♦r ❛♥❞ ✇❛t❝❤✐♥❣ ✇❤❡r❡ t❤❡ ❝❛♥♥♦♥❜❛❧❧s ✇❡r❡ ❧❛♥❞✐♥❣✳ ❆ ✇❡❧❧✲❞❡s✐❣♥❡❞ t❡st ♠❛② ♣r♦✈✐❞❡ ♦♥❡ ✇✐t❤ ❛ t❛❜❧❡ ✭✐✳❡✳✱ ❛ ❢✉♥❝t✐♦♥✮ t❤❛t ❣✐✈❡s t❤❡ s❤♦t ❧❡♥❣t❤ ❢♦r ❡❛❝❤ ❛♥❣❧❡ ♦❢ t❤❡ ❜❛rr❡❧✳ ❍♦✇❡✈❡r✱ s✉❝❤ ❛ r❡❢❡r❡♥❝❡ t❛❜❧❡ ♠❛② ♣r♦✈❡ ✉s❡❧❡ss ✇❤❡♥ ♦♥❡ ✐s t♦ s❤♦♦t ❢r♦♠ ❛♥ ❡❧❡✈❛t❡❞ ♣♦s✐t✐♦♥✱ ♦r ❛t ❛♥ ❡❧❡✈❛t❡❞ t❛r❣❡t✱ ♦r ♦✈❡r ❛♥ ♦❜st❛❝❧❡✳ ❚❤❡r❡ ❛r❡ ❥✉st t♦♦ ♠❛♥② ♣❛r❛♠❡t❡rs✦ ▲❡t✬s ❝♦♥s✐❞❡r ♦♥❡ ♦❢ s✉❝❤ ♣r♦❜❧❡♠s✳ P❘❖❇▲❊▼✿ ❋r♦♠ ❛ 200✲❢♦♦t ❡❧❡✈❛t✐♦♥✱ ❛ ❝❛♥♥♦♥ ✐s ✜r❡❞ ❤♦r✐③♦♥t❛❧❧② ❛t 200 ❢❡❡t ♣❡r s❡❝♦♥❞✳ ❍♦✇ ❢❛r ✇✐❧❧ t❤❡ ❝❛♥♥♦♥❜❛❧❧ ❣♦❄

◮

❲❡ ✇✐❧❧ ✜♥❞ t❤❡ ✇❤♦❧❡ ♣❛t❤✦ ▲❡t h = ∆t ❜❡ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡✳ ❲❡ ❤❛✈❡ t❤❡s❡ s✐① s❡q✉❡♥❝❡s ✇✐t❤ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❝♦♠♣✉t❡❞ ❢♦✉r t✐♠❡s✿ ❤♦r✐③♦♥t❛❧ ✈❡rt✐❝❛❧ ♣♦s✐t✐♦♥

xn

✈❡❧♦❝✐t②

vn =

❛❝❝❡❧❡r❛t✐♦♥

xn+1 − xn h vn+1 − vn an = h

yn yn+1 − yn h un+1 − un bn = h un =

❉◗ ❉◗

✷✳✶✸✳ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

✶✼✸

◆♦✇✱ ❢r♦♠ t❤❡ ♣✉r♣♦s❡ ♦❢ ♠♦❞❡❧✐♥❣ ❛♥❞ s✐♠✉❧❛t✐♦♥✱ t❤❡ ❞❡r✐✈❛t✐♦♥ s❤♦✉❧❞ ❣♦ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✳ ❲❡ ❣♦ ✐♥ r❡✈❡rs❡✿ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡♥ t❤❡ ❧♦❝❛t✐♦♥ ❢r♦♠ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳ ❲❤❡♥ ✇❡ s♦❧✈❡ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥s✱ ✇❡ ❡♥❞ ✉♣ ✇✐t❤ t❤❡s❡ ❢♦✉r r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ✭t❤❡ ❘✐❡♠❛♥♥ s✉♠s✮ ❢♦r ♦✉r s✐① s❡q✉❡♥❝❡s✿ ❤♦r✐③♦♥t❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ✈❡❧♦❝✐t② ♣♦s✐t✐♦♥

✈❡rt✐❝❛❧

an

bn

vn+1 = vn + han

un+1 = un + hbn ❘❙

xn+1 = xn + hvn yn+1 = yn + hun ❘❙

❊①❛♠♣❧❡ ✷✳✶✸✳✶✿ ❤♦✇ ❢❛r

◆♦✇ ✐♥ t❤❡ s♣❡❝✐✜❝ ❝❛s❡ ♦❢ ❢r❡❡ ❢❛❧❧✱ t❤❡r❡ ✐s ❥✉st ♦♥❡ ❢♦r❝❡✱ t❤❡ ❣r❛✈✐t②✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❤♦r✐③♦♥t❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ③❡r♦ ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❝♦♥st❛♥t ✭❢❡❡t ♣❡r s❡❝♦♥❞ sq✉❛r❡❞✮✿ a = 0, b = −32 .

❲❡ ❝❤♦♦s❡ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡✿ h = 0.1 .

◆❡①t✱ ✇❡ ❛❝q✉✐r❡ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ x

y

✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✿ x0 = 0 y0 = 200 ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✿ v0 = 200 u0 = 0 ❚❤❡s❡ ❢♦✉r ♥✉♠❜❡rs s❡r✈❡ ❛s t❤❡ ✐♥✐t✐❛❧ t❡r♠s ♦❢ ♦✉r ❢♦✉r s❡q✉❡♥❝❡s✿ ❤♦r✐③♦♥t❛❧

t✐♠❡ t0

t1 = t0 + h

❛❝❝❡❧❡r❛t✐♦♥ ✈❡❧♦❝✐t② ♣♦s✐t✐♦♥ ❛❝❝❡❧❡r❛t✐♦♥ ✈❡❧♦❝✐t② ♣♦s✐t✐♦♥

t2 = t1 + h ...

a0 = 0

✈❡rt✐❝❛❧ b0 = −32

v1 = 200 + 0.1 · 0 u1 = 0 + 0.1 · (−32) x1 = 0 + 0.1 · 200 y1 = 200 + 0.1 · 0 a1 = 0

v2 = v1 + ha1

b1 = −32

u2 = u1 + hb1

x2 = x1 + hv1

y3 = y1 n + hu1

...

...

❚❤❡ ❢♦✉r ❢♦r♠✉❧❛s ❛r❡ ✐❞❡♥t✐❝❛❧ ❥✉st ❛s ❜❡❢♦r❡✿ ❂❘❬✲✶❪❈✰❘❈❬✲✶❪✯❘✶❈✶

❲❡ ✉s❡ t❤❡♠ t♦ ❡✈❛❧✉❛t❡ t❤❡ ❧♦❝❛t✐♦♥ ❡✈❡r② h = 0.1 s❡❝♦♥❞✳ ❲❡ t❛❦❡ t❤❡ s♣r❡❛❞s❤❡❡t ♣r❡s❡♥t❡❞ ❛❜♦✈❡✱ ❝♦♣② ❛♥❞ ♣❛st❡ t❤❡ ❝♦❧✉♠♥s ❢♦r ❛❝❝❡❧❡r❛t✐♦♥✱ ✈❡❧♦❝✐t②✱ ❛♥❞ ♣♦s✐t✐♦♥✿

✷✳✶✸✳

❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

✶✼✹

❖❢ ❝♦✉rs❡✱ ❢♦r t❤❡ ❤♦r✐③♦♥t❛❧ ✈❛❧✉❡s✱ ✇❡ r❡♣❧❛❝❡ ❛❝❝❡❧❡r❛t✐♦♥ ✇✐t❤

a = 0✳

❚♦ ✜♥❞ ✇❤❡♥ ❛♥❞ ✇❤❡r❡ t❤❡ ❜❛❧❧ ❤✐ts t❤❡ ❣r♦✉♥❞✱ ✇❡ s❝r♦❧❧ ❞♦✇♥ t♦ ✜♥❞ t❤❡ r♦✇ ✇✐t❤

■t ❤❛♣♣❡♥s ❛❜♦✉t

t = 3.5✳

❲❡ ❝❛♥ ❛❧s♦ ❝♦♠❜✐♥❡ t❤❡

❚❤❡♥✱ t❤❡ ✈❛❧✉❡ ♦❢

x✲❝♦❧✉♠♥

❛♥❞ t❤❡

x

❛t t❤❡ t✐♠❡ ✐s ❛❜♦✉t

y ✲❝♦❧✉♠♥

y

❛♥❣❧❡

❚❤❡ ✈❡❧♦❝✐t② ♦❢

0✳

x = 700✳

t♦ ♣❧♦t t❤❡ ♣❛t❤ ♦❢ t❤❡ ❝❛♥♥♦♥❜❛❧❧✿

❲✐t❤ t❤❡ s♣r❡❛❞s❤❡❡t✱ ✇❡ ❝❛♥ ❛s❦ ❛♥❞ ❛♥s✇❡r ❛ ✈❛r✐❡t② ♦❢ q✉❡st✐♦♥s ❛❜♦✉t s✉❝❤ ♠♦t✐♦♥✳ ✐♥tr♦❞✉❝❡ t❤❡

❝❧♦s❡ t♦

❇✉t ✜rst✱ ❧❡t✬s

♦❢ t❤❡ ❜❛rr❡❧ ♦❢ t❤❡ ❝❛♥♥♦♥ ✐♥t♦ t❤❡ ♠♦❞❡❧✳

200 ❢❡❡t ♣❡r s❡❝♦♥❞ ✇❡ ❤❛✈❡ ❜❡❡♥ ✉s✐♥❣ ✐s t❤❡ ✏♠✉③③❧❡ ✈❡❧♦❝✐t②✑✱ ✐✳❡✳✱ t❤❡ s♣❡❡❞✱ s✱ ✇✐t❤ ✇❤✐❝❤ α✱ ✐s✳ ❚❤❛t✬s ✇❤❡r❡ t❤❡ ✐♥✐t✐❛❧ ❤♦r✐③♦♥t❛❧ ❛♥❞

t❤❡ ❝❛♥♥♦♥❜❛❧❧ ❧❡❛✈❡s t❤❡ ♠✉③③❧❡ ✕ ♥♦ ♠❛tt❡r ✇❤❛t t❤❡ ❛♥❣❧❡✱ t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t✐❡s ❝♦♠❡ ❢r♦♠✿

✷✳✶✸✳

❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

✶✼✺

❚❤❡s❡ ❢♦r♠✉❧❛s ❝♦♠❡ ❢r♦♠ tr✐❣♦♥♦♠❡tr② ✭❱♦❧✉♠❡ ✶✮✿

v0 = s cos α

❛♥❞

u0 = s sin α .

❲❡ ✉s❡ t❤❡♠ ❜❡❧♦✇ t♦ ♣r♦✈✐❞❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡s ♦❢ t❤❡ t✇♦ ✈❡❧♦❝✐t✐❡s✿

❲❡ ❝❛♥ ❢r❡❡❧② ❡♥t❡r t❤❡ ❞❛t❛ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ✭❤✐❣❤❧✐❣❤t❡❞ ✐♥ ❣r❡❡♥✮✿

•

t❤❡ ✐♥✐t✐❛❧ s♣❡❡❞

•

t❤❡ ✐♥✐t✐❛❧ ❛♥❣❧❡

•

t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥

•

❛❧❧ ❛❝❝❡❧❡r❛t✐♦♥s

❚❤❡ r❡st ✐s ❝♦♠♣✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s❛♠❡ ❢♦r♠✉❧❛s ❛s ❜❡❢♦r❡✳

❊①❛♠♣❧❡ ✷✳✶✸✳✷✿ ❧♦♥❣❡st s❤♦t ■t ✐s r❡❛❧❧② tr✉❡ t❤❛t

45

❞❡❣r❡❡s ✐s t❤❡ ❜❡st ❛♥❣❧❡ t♦ s❤♦♦t ❢♦r ❛ ❧♦♥❣❡r ❞✐st❛♥❝❡✿

■t ❛♣♣❡❛rs t❤❛t t❤❡ ♦♥❡ ✐♥ t❤❡ ♠✐❞❞❧❡ ✐s t❤❡ ❜❡st✱ ❜✉t ✇❡ ❝❛♥✬t ♣r♦✈❡ t❤✐s ✇✐t❤ ❥✉st t❤❡ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s✳ ◆♦✇✱ ✇❤❛t ✐❢ ✇❡ tr② t♦ s❤♦♦t ❢r♦♠ ❛ ❤✐❧❧ ❛❣❛✐♥✱ s❛②✱

500

❢❡❡t ❤✐❣❤❄

✷✳✶✸✳ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

✶✼✻

■t✬s ♥♦t t❤❡ ❜❡st ❛♥②♠♦r❡✦ ❊①❡r❝✐s❡ ✷✳✶✸✳✸

❙❤♦✇ t❤❛t t❤❡ ❜❡st s❤♦t ✇✐❧❧ ❜❡❝♦♠❡ ♠♦r❡ ❛♥❞ ♠♦r❡ ✢❛t ❛s t❤❡ ❡❧❡✈❛t✐♦♥ ❣r♦✇s✳ ❊①❛♠♣❧❡ ✷✳✶✸✳✹✿ ✈❛r✐❛❜❧❡ ❣r❛✈✐t②

❲❤❛t ❤❛♣♣❡♥s ✐❢ t❤❡ ❣r❛✈✐t② s✉❞❞❡♥❧② ❞✐s❛♣♣❡❛rs❄ ■♥ t❤❡ ❝♦❧✉♠♥ ❢♦r t❤❡ ✈❡rt✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥✱ ✇❡ ❥✉st r❡♣❧❛❝❡ −32 ✇✐t❤ 0 ❛❢t❡r ❛ ❢❡✇ r♦✇s✿

❚❤❡ ❝❛♥♥♦♥❜❛❧❧ ✢✐❡s ♦✛ ♦♥ ❛ t❛♥❣❡♥t✳ ❊①❛♠♣❧❡ ✷✳✶✸✳✺✿ ✈❛r✐❛❜❧❡ ❣r❛✈✐t②

❲❤❛t ❤❛♣♣❡♥s ✐❢ t❤❡ ❣r❛✈✐t② st❛rts t♦ ✐♥❝r❡❛s❡❄ ▲❡t✬s ✐♥❝r❡❛s❡ t❤❡ ❞♦✇♥✇❛r❞ ❛❝❝❡❧❡r❛t✐♦♥ 1 ❢♦♦t ♣❡r s❡❝♦♥❞ sq✉❛r❡❞ ♣❡r s❡❝♦♥❞✿

❚❤❡ tr❛❥❡❝t♦r② ❧♦♦❦s st❡❡♣❡r ❛♥❞ st❡❡♣❡r✱ ❜✉t ✐s t❤❡r❡ ❛ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡ ❄ ❲❡ ❝❛♥✬t ❛♥s✇❡r ✇✐t❤ ❥✉st t❤❡ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s✳

✷✳✶✸✳ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

✶✼✼

❊①❛♠♣❧❡ ✷✳✶✸✳✻✿ ❤♦r✐③♦♥t❛❧ ❣r❛✈✐t②

❲❤❛t ❤❛♣♣❡♥s ✐❢ t❤❡ ❣r❛✈✐t② ✐s ❤♦r✐③♦♥t❛❧ ✐♥st❡❛❞❄ ❚❤❡ ♠♦t✐♦♥ ✇✐❧❧ ❜❡ ❛❧♦♥❣ ❛ ♣❛r❛❜♦❧❛ t❤❛t ❧✐❡s ♦♥ ✐ts s✐❞❡✱ ♦❢ ❝♦✉rs❡✳ ❇✉t ✇❤❛t ✐❢ t❤❡r❡ ❛r❡ ❜♦t❤ ✈❡rt✐❝❛❧ ✭❞♦✇♥✮ ❛♥❞ ❤♦r✐③♦♥t❛❧ ✭❧❡❢t✮ ❢♦r❝❡s ♦❢ ❣r❛✈✐t②❄ ▲❡t✬s ♠♦❞✐❢② t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❝♦❧✉♠♥s ❛❝❝♦r❞✐♥❣❧② ❜② r❡♣❧❛❝✐♥❣ 0✬s ✇✐t❤ −32 ✐♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ❝♦❧✉♠♥✿

■s t❤✐s ❛ ♣❛r❛❜♦❧❛❄ ❊①❡r❝✐s❡ ✷✳✶✸✳✼

❊①♣❧❛✐♥ t❤❡ r❡s✉❧ts ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✳ ■♥ s♣✐t❡ ♦❢ t❤❡s❡ ♥✉♠❡r♦✉s ❡①❛♠♣❧❡s✱ ✇❡ ❝❛♥ ♦♥❧② ❞♦ ♦♥❡ ❛t ❛ t✐♠❡✦ ❚❤❡ ❝♦♥❝❧✉s✐♦♥s ✇❡ ❞r❛✇ ❛r❡ ❛❧s♦ s♣❡❝✐✜❝ t♦ t❤❡s❡ s✐t✉❛t✐♦♥s ✭❛❝❝❡❧❡r❛t✐♦♥s✱ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✱ ❡t❝✳✮✳ ❊①❛♠♣❧❡ ✷✳✶✸✳✽✿ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s

❇❡❝❛✉s❡ ❡✈❡r②t❤✐♥❣ ✐s r❡❝✉rs✐✈❡✱ ✇❡ ❤❛✈❡ t♦ r✉♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ❞✐r❡❝t❧② t♦ s❡❡ ✇❤❛t ❤❛♣♣❡♥s✱ ❛s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡✿ ❛❝❝❡❧❡r❛t✐♦♥ an ✈❡❧♦❝✐t② vn+1 = vn + han ♣♦s✐t✐♦♥ xn+1 = xn + hvn ■s t❤❡r❡ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r t❤❡ ♣♦s✐t✐♦♥❄ ■♥ ♦t❤❡r ✇♦r❞s✱ ❝❛♥ ✇❡ ❡①♣r❡ss xn ✐♥ t❡r♠s ♦❢ n❄ ■❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ③❡r♦✱ ✐t✬s ❡❛s②✿ an = 0 =⇒ vn = v0 = v =⇒ xn+1 = xn + hv .

❆❞❞✐♥❣ t❤❡ s❛♠❡ ♥✉♠❜❡r ✐s ❥✉st ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ xn = x0 + hvn .

■❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ♥♦♥✲③❡r♦ ❜✉t ❝♦♥st❛♥t✱ ✐t✬s ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞✿ an = a =⇒ vn+1 = vn + ha .

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ vn = v0 + han .

✷✳✶✸✳ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥ ◆❡①t✿

✶✼✽

xn+1 = xn + hvn = xn + h(v0 + han) = xn + hv0 + h2 an .

❲❡ ❛r❡ ❛❞❞✐♥❣ ❝♦♥s❡❝✉t✐✈❡ ✐♥t❡❣❡rs✦ ❊①❡r❝✐s❡ ✷✳✶✸✳✾

❋✐♥✐s❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥✳ ❍✐♥t✿ ❯s❡ ❛ ❢♦r♠✉❧❛ ❢r♦♠ ❈❤❛♣t❡r ✶✳ ❊①❡r❝✐s❡ ✷✳✶✸✳✶✵

❲❤❛t ✐❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ✐♥❝r❡❛s✐♥❣ ❧✐♥❡❛r❧②❄ ❍✐♥t✿ ❯s❡ ❛ ❢♦r♠✉❧❛ ❢r♦♠ ❈❤❛♣t❡r ✶✳ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ✜♥❞✐♥❣ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s ❢♦r s✉♠s ♦❢ s❡q✉❡♥❝❡s ✐s ❛❞❞r❡ss❡❞ ✐♥ ❈❤❛♣t❡r ✺✳ ❚❤✐s ✐s ✇❤② ✇❡ ♥♦✇ t✉r♥ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✱ ✐✳❡✳✱ ✇❡ t❛❦❡ t❤❡ ❧✐♠✐t ♦❢ ❡✈❡r②t❤✐♥❣ ❛❜♦✈❡✿ h = ∆t → 0

❚❤❡ ❞✐s❛♣♣❡❛r❛♥❝❡ ♦❢ h ♠❛❦❡s ❛❧❣❡❜r❛ s✐♠♣❧❡r✦ ❚❤✐s t✐♠❡✱ ✐♥st❡❛❞ ♦❢ s✐① s❡q✉❡♥❝❡s✱ ✇❡ ❤❛✈❡ t❤❡s❡ s✐① ❢✉♥❝t✐♦♥s ♦❢ t✐♠❡✿ x✱ t❤❡ ❞❡♣t❤✱ t❤❡ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥

v = x′ ✱ t❤❡ ❤♦r✐③♦♥t❛❧ ✈❡❧♦❝✐t②

a = v ′ ✱ t❤❡ ❤♦r✐③♦♥t❛❧ ❛❝❝❡❧❡r❛t✐♦♥

y ✱ t❤❡ ❤❡✐❣❤t✱ t❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥

u = y ′ ✱ t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t②

b = u′ ✱ t❤❡ ✈❡rt✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥

❚❤❡r❡ ✐s ♥♦ t✐♠❡ ✐♥❝r❡♠❡♥t ❛s ❛ ♣❛r❛♠❡t❡r ❛♥②♠♦r❡✦ ◆♦✇ t❤❡ s♣❡❝✐✜❝ ❝❛s❡ ♦❢ ❢r❡❡ ❢❛❧❧✿ a = 0, b = −g .

❋r♦♠ ❱♦❧✉♠❡ ✷✱ ✇❡ ❦♥♦✇✿ ✶✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ❧✐♥❡❛r✳ ❆♥❞ t❤❡ ♦♥❧② ❢✉♥❝t✐♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤✐❝❤ ✐s ❧✐♥❡❛r ✐s ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✳ ✷✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ✐s ❝♦♥st❛♥t✳ ❆♥❞ t❤❡ ♦♥❧② ❢✉♥❝t✐♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤✐❝❤ ✐s ❝♦♥st❛♥t ✐s ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧✳ ❲❡ ❝♦♥❝❧✉❞❡ ❛❜♦✉t ❢r❡❡ ❢❛❧❧ ✿ ✶✳ ❚❤❡ ❤♦r✐③♦♥t❛❧ ♣♦s✐t✐♦♥ x = x(t) ✐s ❧✐♥❡❛r✳ ✷✳ ❚❤❡ ✈❡rt✐❝❛❧ ♣♦s✐t✐♦♥ y = y(t) ✐s q✉❛❞r❛t✐❝✳ ❲❤❛t ♠❛❦❡s t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s s♣❡❝✐✜❝ ❛r❡ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✿ x0 ✱ t❤❡ ✐♥✐t✐❛❧ ❞❡♣t❤✱ x0 = x(0) y0 ✱ t❤❡ ✐♥✐t✐❛❧ ❤❡✐❣❤t✱ y0 = y(0)

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

dx dt t=0 dy u0 ✱ t❤❡ ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ✈❡❧♦❝✐t②✱ u(0) = dt t=0 v0 ✱ t❤❡ ✐♥✐t✐❛❧ ❤♦r✐③♦♥t❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ✈❡❧♦❝✐t②✱ v(0) =

x = x0 +v0 t 1 y = y0 +u0 t − gt2 2

❚❤❡s❡ t✇♦ ❡q✉❛t✐♦♥s ❛❧❧♦✇ ✉s t♦ s♦❧✈❡ ❛ ✈❛r✐❡t② ♦❢ ♣r♦❜❧❡♠s ❛❜♦✉t ♠♦t✐♦♥✳ ❲❡ ❝❛rr② t❤✐s ♦✉t ❢♦r x ❛♥❞ y s❡♣❛r❛t❡❧② ❛♥❞ t❤❡ r❡s✉❧ts ❛r❡ s❤♦✇♥ ✐♥ t❤❡ s♣r❡❛❞s❤❡❡t✿

✷✳✶✸✳ ❙❤♦♦t✐♥❣ ❛ ❝❛♥♥♦♥

✶✼✾

❊①❛♠♣❧❡ ✷✳✶✸✳✶✶✿ ❤♦✇ ❢❛r

▲❡t✬s r❡✈✐s✐t t❤❡ ♣r♦❜❧❡♠ ❛❜♦✉t ❛ s♣❡❝✐✜❝ s❤♦t ✇❡ s♦❧✈❡❞ ♥✉♠❡r✐❝❛❧❧②✳ ❖✉r ❡q✉❛t✐♦♥s ❜❡❝♦♠❡✿ x =

200t

y = 200

−16t2

◆♦✇✱ ❛♥❛❧②t✐❝❛❧❧②✱ t❤❡ ❤❡✐❣❤t ❛t t❤❡ ❡♥❞ ✐s 0✱ s♦ t♦ ✜♥❞ ✇❤❡♥ ✐t ❤❛♣♣❡♥❡❞✱ ✇❡ s❡t y = 0✱ ♦r 200 − 16t2 = 0 ,

❛♥❞ s♦❧✈❡ ❢♦r t✳ ❚❤❡♥✱ t❤❡ t✐♠❡ ♦❢ ❧❛♥❞✐♥❣ ✐s✿ t1 =

r

√ 5 2 200 = . 16 2

❚♦ ✜♥❞ ✇❤❡r❡ ✐t ❤❛♣♣❡♥❡❞✱ ✇❡ s✉❜st✐t✉t❡ t❤✐s ✈❛❧✉❡ ♦❢ t ✐♥t♦ x❀ t❤❡ ❧♦❝❛t✐♦♥ ✐s✿ √ 5 2 ≈ 707 . x1 = 200t1 = 200 2

❚❤❡ r❡s✉❧t ♠❛t❝❤❡s ♦✉r ❡st✐♠❛t❡✦ ❊①❛♠♣❧❡ ✷✳✶✸✳✶✷✿ ✈❛r✐❛❜❧❡ ❣r❛✈✐t②

❲❤❛t ❤❛♣♣❡♥s ✐❢ t❤❡ ❣r❛✈✐t② ✐s ❞❡❝r❡❛s✐♥❣❄ ❙✉♣♣♦s❡ ✐t ✐s ❞❡❝r❡❛s✐♥❣ 1 ❢♦♦t ♣❡r s❡❝♦♥❞ sq✉❛r❡❞ ♣❡r s❡❝♦♥❞✳ ❲✐t❤ t❤❡ t♦♦❧s ❞❡✈❡❧♦♣❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✱ ✐t✬s ❡❛s②✿ b = −g − t =⇒ u = u0 − gt −

t2 gt2 t3 =⇒ y = y0 + u0 t − − . 2 2 6

❲❡ ❝♦♥✜r♠ t❤❛t t❤❡ tr❛❥❡❝t♦r② ✇✐❧❧ ❜❡❝♦♠❡ st❡❡♣❡r ❛♥❞ st❡❡♣❡r✳ ❲❡ ❛❧s♦ ❞✐s❝♦✈❡r t❤❛t t❤❡r❡ ✐s ♥♦ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡✦

❈❤❛♣t❡r ✸✿ ❲❤❛t ✇❡ ❝❛♥ ❞♦ ✇✐t❤ ✐♥t❡❣r❛❧ ❝❛❧❝✉❧✉s

❈♦♥t❡♥ts

✸✳✶ ❚❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t✇♦ ❣r❛♣❤s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ❚❤❡ ❧✐♥❡❛r ❞❡♥s✐t② ❛♥❞ t❤❡ ♠❛ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹ ❚❤❡ r❛❞✐❛❧ ❞❡♥s✐t② ❛♥❞ t❤❡ ♠❛ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✺ ❚❤❡ ✢♦✇ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ✢✉① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✻ ❚❤❡ ❢♦r❝❡ ❛♥❞ t❤❡ ✇♦r❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✼ ❚❤❡ t♦t❛❧ ❛♥❞ t❤❡ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✸✳✽ ◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✾ ▲❡♥❣t❤s ♦❢ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✵ ❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ t❤r❡❡ ✳ ✸✳✶✶ ❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ✈✐❛ t❤❡✐r ❝r♦ss✲s❡❝t✐♦♥s ✳ ✳ ✸✳✶✷ ❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ♦❢ r❡✈♦❧✉t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✶✽✵ ✶✽✽ ✶✾✷ ✷✵✶ ✷✵✹ ✷✵✽ ✷✶✺ ✷✶✾ ✷✸✵ ✷✸✻ ✷✹✹ ✷✺✶

✸✳✶✳ ❚❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t✇♦ ❣r❛♣❤s

❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ✇❛②s t♦ ❡♥t❡r ❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s✿ t❤r♦✉❣❤ t❤❡ st✉❞② ♦❢ ❣❡♦♠❡tr② ✭✜♥❞✐♥❣ s❡❝❛♥t ❛♥❞ t❛♥❣❡♥t ❧✐♥❡s ♦❢ ❝✉r✈❡s✮ ❛♥❞ t❤r♦✉❣❤ t❤❡ st✉❞② ♦❢ ♠♦t✐♦♥ ✭✜♥❞✐♥❣ ✈❡❧♦❝✐t② ❛♥❞ ❛❝❝❡❧❡r❛t✐♦♥ ❢r♦♠ ❧♦❝❛t✐♦♥✮✳ ❙✐♠✐❧❛r❧②✱ t❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ✇❛②s t♦ ❡♥t❡r ✐♥t❡❣r❛❧ ❝❛❧❝✉❧✉s✿ t❤r♦✉❣❤ t❤❡ st✉❞② ♦❢ ❣❡♦♠❡tr② ✕ ✜♥❞✐♥❣ ❛r❡❛s ✉♥❞❡r ❝✉r✈❡s ✕ ❛♥❞ t❤r♦✉❣❤ t❤❡ st✉❞② ♦❢ ♠♦t✐♦♥ ✕ ✜♥❞✐♥❣ ❧♦❝❛t✐♦♥ ❢r♦♠ ✈❡❧♦❝✐t② ❛♥❞ ❛❝❝❡❧❡r❛t✐♦♥✳ ❚❤❡s❡ ❛r❡ t✇♦ ✈❡r② ❞✐st✐♥❝t ❡①❛♠♣❧❡s ♦❢ r❡❝♦❣♥✐③✐♥❣ ❘✐❡♠❛♥♥ s✉♠s✳ ❚❤r♦✉❣❤♦✉t t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐❧❧ ❝❛rr② ♦✉t t❤✐s ❦❡② st❡♣ ✐♥ ❛ ✈❛r✐❡t② ♦❢ ❡♥t✐r❡❧② ♥❡✇ s✐t✉❛t✐♦♥s✳ ❇✉t ✇❡ ✇✐❧❧ st❛rt ✇✐t❤ s♦♠❡t❤✐♥❣ ❢❛♠✐❧✐❛r✳ ❊①❛♠♣❧❡ ✸✳✶✳✶✿ ❛r❡❛ ♦❢ ❝✐r❝❧❡

■♥ ❈❤❛♣t❡r ✶✱ ✇❡ ❝♦♥✜r♠❡❞ t❤❛t t❤❡ ❛r❡❛ ♦❢ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s r ✐s A = πr2 ✉s✐♥❣ ♥♦t❤✐♥❣ ❜✉t ❛ s♣r❡❛❞✲ s❤❡❡t✳ ❆♥❞ ❧❛t❡r ✐♥ t❤❡ ❝❤❛♣t❡r✱ ✇❡ ✉s❡❞ ✐♥t❡❣r❛t✐♦♥ t♦ ♣r♦✈✐❞❡ ❛ ♣r❡❝✐s❡ ❛♥s✇❡r✳ ❚❤❡ s♦❧✉t✐♦♥✱ ❤♦✇❡✈❡r✱ ✇❛s♥✬t ❢✉❧❧② s❛t✐s❢❛❝t♦r② ❜❡❝❛✉s❡ ✇❡ r❡❧✐❡❞ ♦♥ t❤❡ s②♠♠❡tr② ♦❢ t❤❡ ❝✐r❝❧❡ t♦ ❝♦♠♣✉t❡ t❤❡ ❛r❡❛ ♦❢ ✐ts ❤❛❧❢ s♦ t❤❛t t❤❡ ❛r❡❛ ♦❢ t❤❡ ✇❤♦❧❡ ❝✐r❝❧❡ ✐s t❤❡♥ t✇✐❝❡ t❤✐s ♥✉♠❜❡r✳ ❚❤✐s ✐s t♦♦ ❧✐♠✐t✐♥❣✳ ▲❡t✬s st❛rt ♦✈❡r✳ ❚❤❡r❡ ❛r❡

t✇♦ ❢✉♥❝t✐♦♥s t❤✐s t✐♠❡✱ ❢♦r t❤❡ t♦♣ ❛♥❞ t❤❡ ❜♦tt♦♠ ♦❢ t❤❡ ❝✐r❝❧❡✿ f (x) =

❲✐t❤ t❤❡ ❢♦r♠✉❧❛s✿

√

√ 1 − x2 ❛♥❞ g(x) = − 1 − x2 ,

−1 ≤ x ≤ 1 .

❂❙◗❘❚✭✶✲❘❈❬✲✷❪✂ ✷✮ ❛♥❞ ❂✲❙◗❘❚✭✶✲❘❈❬✲✷❪✂ ✷✮

✸✳✶✳

❚❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t✇♦ ❣r❛♣❤s

✶✽✶

✇❡ ♣❧♦t ❜♦t❤✿

❲❡ ❧❡t t❤❡ ✈❛❧✉❡s ♦❢ x r✉♥ ❢r♦♠ −1 t♦ 1 ❡✈❡r② 0.1 ❛♥❞ ❝♦✈❡r❡❞✱ ❜❡st ✇❡ ❝❛♥✱ t❤✐s ❝✐r❝❧❡ ✇✐t❤ ✈❡rt✐❝❛❧ ❜❛rs ❜❛s❡❞ ♦♥ t❤❡s❡ s❡❣♠❡♥ts✳ ❚❤❡♥ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ s✉♠ ♦❢ t❤❡ ❛r❡❛s ♦❢ t❤❡ ❜❛rs✿ ❲❡ ❛❞❞ ❛ ❝♦❧✉♠♥ ♦❢ t❤❡ ✇✐❞t❤s ♦❢ t❤❡ ❜❛rs✱ ♠✉❧t✐♣❧② t❤❡♠ ❜② t❤❡ ❤❡✐❣❤ts✱ ♣❧❛❝❡ t❤❡ r❡s✉❧t ✐♥ t❤❡ ❧❛st ❝♦❧✉♠♥✱ ❛♥❞ ✜♥❛❧❧② ❛❞❞ ❛❧❧ ❡♥tr✐❡s ✐♥ t❤✐s ❝♦❧✉♠♥✿

❚❤❡ ❤❡✐❣❤t ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ❧♦❝❛t❡❞ ❛t x ✐s f (x) − g(x)✱ ❛♥❞ ✐ts ❛r❡❛ ✐s (f (x) − g(x)) · 0.1✳ ❲❡ ❝♦♠♣✉t❡ t❤❡s❡ ✐♥ t❤❡ ♥❡①t ❝♦❧✉♠♥ ❛♥❞ t❤❡♥ ❛❞❞ t❤❡♠✿ ❛♣♣r♦①✐♠❛t❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ = 3.1 . ■t ✐s ❝❧♦s❡ t♦ t❤❡ t❤❡♦r❡t✐❝❛❧ r❡s✉❧t ✇❡ ❡st❛❜❧✐s❤❡❞ ✐♥ ❈❤❛♣t❡r ✶✿ ❡①❛❝t ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ = π = 3.14159... ❖❢ ❝♦✉rs❡✱ ✇❡ r❡❛❧✐③❡ t❤❛t ✇❡ ❝♦✉❧❞ ♣r♦❞✉❝❡ t❤❡ s❛♠❡ r❡s✉❧t ✐❢ ✇❡ t❛❦❡ t❤❡ ❞❛t❛ ❢r♦♠ t❤❡ ✜rst s♣r❡❛❞✲ s❤❡❡t✱ X i

f (ci ) · 0.1 ,

❛♥❞ t❤❡♥ s✉❜tr❛❝t t❤❡ ❞❛t❛ ❢♦r t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥✱ X i

❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡ X i

f (ci )0.1 −

X i

g(ci ) · 0.1 .

g(ci ) · 0.1 =

X i

 f (ci ) − g(ci ) · 0.1 .

❚❤❡ ❝♦♠♠♦♥ s❡♥s❡ ❛❜♦✉t ❤♦✇ t❤❡ ✭✉♥s✐❣♥❡❞✮ ❧❡♥❣t❤s ♦❢ ✐♥t❡r✈❛❧s ❜❡❤❛✈❡ ✐s t❤❛t t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✉♥✐♦♥ ♦❢ t✇♦ ✐♥t❡r✈❛❧s ✐s t❤❡ s✉♠ ♦❢ t❤❡ t✇♦ ❧❡♥❣t❤s ♠✐♥✉s t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥✿ ❧❡♥❣t❤ ♦❢ P ∪ Q = ❧❡♥❣t❤ ♦❢ P + ❧❡♥❣t❤ ♦❢ Q − ❧❡♥❣t❤ ♦❢ P ∩ Q .

✸✳✶✳

❚❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t✇♦ ❣r❛♣❤s

■t ✐s ❝❛❧❧❡❞ t❤❡ ♣♦✐♥t✳

✶✽✷

❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ❧❡♥❣t❤✳

❚❤❡ ❧❛st t❡r♠ ❞✐s❛♣♣❡❛rs ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♦✈❡r❧❛♣ ♦r ✐t ✐s ❥✉st ❛

■❢ ✇❡ ❜✉✐❧❞ r❡❝t❛♥❣❧❡s ♦♥ t♦♣ ♦❢ t❤❡s❡ ✐♥t❡r✈❛❧s✱ ✇❡ ❛r❡ ✐♥ ❛ s✐♠✐❧❛r s✐t✉❛t✐♦♥ ✕ ❢♦r t❤❡ ✭✉♥s✐❣♥❡❞✮ ❛r❡❛s✿

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❛r❡❛ ♦❢ t❤❡ ✉♥✐♦♥ ♦❢ t✇♦ r❡❣✐♦♥s ✐s t❤❡ s✉♠ ♦❢ t❤❡ t✇♦ ❛r❡❛s ♠✐♥✉s t❤❡ ❛r❡❛ ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥✿ ❛r❡❛ ♦❢ P ∪ Q = ❛r❡❛ ♦❢ P + ❛r❡❛ ♦❢ Q − ❛r❡❛ ♦❢ P ∩ Q .

■t ✐s ❝❛❧❧❡❞ t❤❡

❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ❛r❡❛✳

❚❤❡ ❧❛st t❡r♠ ❞✐s❛♣♣❡❛rs ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♦✈❡r❧❛♣ ♦r ✐t ✐s ❥✉st ❛ ❝✉r✈❡✳

❚❤❡ ✐❞❡❛ ✐s t❤❡♥ t♦ ❜❡ ❛♣♣❧✐❡❞ t♦ ❝✉r✈❡❞ r❡❣✐♦♥s✿

❍♦✇❡✈❡r✱ ♦✉r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❛r❡❛s ✐s ❧✐♠✐t❡❞ t♦ t❤♦s❡ ♦❢ r❡❣✐♦♥s ✉♥❞❡r ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s✳ ❊✈❡♥ t❤❡♥✱ t❤❡ ❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ❛r❡❛s ♦❢ t❤♦s❡ r❡❣✐♦♥s ❤❛s ❜❡❡♥ ♦♥❧② ❞❡♠♦♥str❛t❡❞ ❢♦r t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ✇❤❡♥ s✉❝❤ ❛ r❡❣✐♦♥ ✐s ❝✉t ❜② ❛ ✈❡rt✐❝❛❧ ❧✐♥❡✿ Z Z Z c

c

b

a

f dx .

f dx =

f dx +

b

❚❤✐s ❝❛s❡ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇ ✭t❤❡ ❧✐♥❡ ✐s x = b✮✿

❲❤❛t ✐❢ t❤❡ r❡❣✐♦♥ ✉♥❞❡r t❤❡ ❣r❛♣❤ ✐s ❝✉t ❜② ❛♥♦t❤❡r ❣r❛♣❤❄

a

✸✳✶✳

❚❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t✇♦ ❣r❛♣❤s

✶✽✸

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✜♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❣✐♦♥ ❜❡t✇❡❡♥

t❤❡ ❣r❛♣❤s ✿

❚❤❡ ✐♥t❡❣r❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ✐s ❡❛s② t♦ s❡❡✿ ■❢ f (x) ≥ g(x) ❢♦r ❛❧❧ x ✐♥ [a, b]✱ t❤❡♥✿ P =R−Q=

Z

b a

f dx −

Z

b

g dx = a

❲❡ ❤❛✈❡ ❛ss✉♠❡❞ t❤❡ ❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ❛r❡❛s ❛♥❞ ✉s❡❞ t❤❡ ❙✉♠

Z

b a

(f − g) dx .

❘✉❧❡ ❢♦r ✐♥t❡❣r❛❧s✳ ❍♦✇❡✈❡r✱ ❡✈❡r② t❡r♠ ✐♥ t❤❡ ❢♦r♠✉❧❛ ✐s t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤✳ ■♥ ♦r❞❡r t♦ ❥✉st✐❢② t❤❡ ❛❞❞✐t✐✈✐t② ❢♦r t❤❡ ❛r❡❛s ❜❡t✇❡❡♥ t❤❡ ❣r❛♣❤s✱ ✇❡ ♥❡❡❞ t♦ st❛rt ❢r♦♠ s❝r❛t❝❤✳ ❇❛❝❦ t♦ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❲❡ st❛rt✱ ❛s ❜❡❢♦r❡✱ ✇✐t❤ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡r✈❛❧ [a, b] ✐♥t♦ n ✐♥t❡r✈❛❧s ♦❢ ♣♦ss✐❜❧② ❞✐✛❡r❡♥t ❧❡♥❣t❤s✿ [x0 , x1 ], [x1 , x2 ], ..., [xn−1 , xn ] ,

✇✐t❤ x0 = a, xn = b✳

❚❤❡ ♣r✐♠❛r② ♥♦❞❡s ♦❢ P ❛r❡✿ x0 < x1 < x2 < ... < xn−1 < xn

❚❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ✐♥t❡r✈❛❧s ❛r❡✿ ❚❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ P ❛r❡✿

∆xi = xi − xi−1 , i = 1, 2, ..., n .

c1 ✐♥ [x0 , x1 ], c2 ✐♥ [x1 , x2 ], ..., cn ✐♥ [xn−1 , xn ] .

✸✳✶✳ ❚❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t✇♦ ❣r❛♣❤s

✶✽✹

❚❤✐s t✐♠❡✱ ✇❡ ❢❛❝❡ t✇♦ ❢✉♥❝t✐♦♥s✳ ❲❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t❤❡ ❣r❛♣❤s ✇✐t❤ r❡❝t❛♥❣❧❡s ✇✐t❤ t❤❡s❡ ✇✐❞t❤s✿

▲❡t✬s t❛❦❡ ❛ ❧♦♦❦ ❛t t❤❡ it❤ r❡❝t❛♥❣❧❡✳ ■ts ✇✐❞t❤ ✐s✱ ❛s ❜❡❢♦r❡✱ ∆xi ✳ ◆♦✇✱ ✐ts t♦♣ ✐s f (ci ) ❛♥❞ t❤❡ ❜♦tt♦♠ ✐s g(ci ) ✭✐♥st❡❛❞ ♦❢ t❤❡ x✲❛①✐s✮✳ ❚❤❡r❡❢♦r❡✱ ✐ts ❤❡✐❣❤t ✐s f (ci ) − g(ci )✳ ❚❤❡♥✱ ✐ts ❛r❡❛ ✐s (f (ci ) − g(ci ))∆x2 ✳ ❍❡♥❝❡✱ t❤❡ t♦t❛❧ ❛r❡❛ ♦❢ t❤❡ r❡❝t❛♥❣❧❡s ✐s✿ (f (c1 ) − g(c1 ))∆x1 + (f (c2 ) − g(c2 ))∆x2 + ... + (f (cn ) − g(cn ))∆xn .

❚❤❡ ❦❡② st❡♣ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❲❡ r❡❝♦❣♥✐③❡ t❤✐s ❡①♣r❡ss✐♦♥ ❛s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤✐s ♥❡✇ ❢✉♥❝t✐♦♥✱ f − g ✱ t❤❡ ❞✐✛❡r❡♥❝❡ ✿ Σ (f − g) · ∆x =

n X i=1

|

(f − g)(ci )∆xi . {z

}

❛r❡❛s ♦❢ t❤❡ r❡❝t❛♥❣❧❡s

■♥❞❡❡❞✱ ✐t ✐s ❝♦♥✈❡♥✐❡♥t t♦ t❤✐♥❦ ♦❢ ❡❛❝❤ t❡r♠ ❛s ✐❢ ✐t r❡❢❡rs t♦ ❛ s✐♥❣❧❡ ❢✉♥❝t✐♦♥✱ f − g ✿ f (ck ) − g(ck ) = (f − g)(ck ) .

❚❤❡ r❡❝t❛♥❣❧❡s ✇❡ st❛rt❡❞ ✇✐t❤ ❛r❡ s❤♦✇♥ ♦♥ t❤❡ ❧❡❢t ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ♦♥ t❤❡ r✐❣❤t✿

❲❡ ❝❛♥ st✐❧❧ ❣♦ ❜❛❝❦ ❛♥❞ ❡①♣❧❛✐♥ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ ♥❡✇ ❢✉♥❝t✐♦♥ ✐♥ t❡r♠s ♦❢ ❛r❡❛s✳ ■t✬s ❛s ✐❢ t❤❡ r❡❝t❛♥❣❧❡s ❛r❡ ✜rst ❛❧✐❣♥❡❞ ✇✐t❤ y = f (x)✱ t❤❡♥ ❝✉t ❢r♦♠ ❜❡❧♦✇ ✇✐t❤ y = g(x)✱ s✉s♣❡♥❞❡❞ ✐♥ t❤❡ ❛✐r✱ ❛♥❞ t❤❡♥ ❞r♦♣♣❡❞ ♦♥ t❤❡ x✲❛①✐s✱ ❧✐❦❡ t❤✐s✿

✸✳✶✳

❚❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t✇♦ ❣r❛♣❤s

✶✽✺

❲❤❛t ✇❡ s❡❡ ✐s t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ f − g ✳ ❲❡ ❞❡✜♥❡ t❤❡ ❛r❡❛ ❛❝❝♦r❞✐♥❣❧②✿

❉❡✜♥✐t✐♦♥ ✸✳✶✳✷✿ ❛r❡❛ ❜❡t✇❡❡♥ ❣r❛♣❤s ❙✉♣♣♦s❡ f ❛♥❞ g ❛r❡ t✇♦ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ ✐♥t❡r✈❛❧ [a, b] ✇✐t❤ f (x) ≥ g(x) ❢♦r ❛❧❧ x ✐♥ [a, b]✳ ❚❤❡♥ t❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t❤❡ ❣r❛♣❤s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ♦✈❡r ✐♥t❡r✈❛❧ [a, b] ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❧✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ♦❢ t❤❡✐r ❞✐✛❡r❡♥❝❡ ✇✐t❤ t❤❡ ♠❡s❤ ♦❢ t❤❡✐r ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥s Pk ❛♣♣r♦❛❝❤✐♥❣ 0 ❛s k → ∞✱ ✇❤❡♥ ❛❧❧ t❤❡s❡ ❧✐♠✐ts ❡①✐st ❛♥❞ ❛r❡ ❛❧❧ ❡q✉❛❧ t♦ ❡❛❝❤ ♦t❤❡r✿ ❆r❡❛ ❜❡t✇❡❡♥ t❤❡ ❣r❛♣❤s ♦❢ f, g = lim Σ (f − g) · ∆x . k→∞

❲❛r♥✐♥❣✦ ❯♥❧✐❦❡ t❤❡ ❛r❡❛ ✏✉♥❞❡r✑ t❤❡ ❣r❛♣❤✱ t❤✐s ♥✉♠❜❡r ❝❛♥✲ ♥♦t ❜❡ ♥❡❣❛t✐✈❡ ❛s ❞❡✜♥❡❞✳

❊①❡r❝✐s❡ ✸✳✶✳✸ ■❢ f ❛♥❞ g r❡♣r❡s❡♥t t❤❡ ✈❡❧♦❝✐t✐❡s ♦❢ t✇♦ ♦❜❥❡❝ts✱ ✇❤❛t ❞♦❡s t❤❡ ❛r❡❛ r❡♣r❡s❡♥t❄ ❚❤❡ ❞❡✜♥✐t✐♦♥ r❡♣❡❛ts t❤❛t ♦❢ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✿

❚❤❡♦r❡♠ ✸✳✶✳✹✿ ❆r❡❛ ❇❡t✇❡❡♥ ●r❛♣❤s ❛s ■♥t❡❣r❛❧ ❙✉♣♣♦s❡ f ❛♥❞ g ❛r❡ t✇♦ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ ✐♥t❡r✈❛❧ [a, b] ✇✐t❤ f (x) ≥ g(x) ❢♦r ❛❧❧ x ✐♥ [a, b]✳ ■❢ f − g ✐s ✐♥t❡❣r❛❜❧❡✱ t❤❡♥ t❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t❤❡ ❣r❛♣❤s ♦❢ f ❛♥❞ g ✐s ❡q✉❛❧ t♦ ❆r❡❛ ❜❡t✇❡❡♥ t❤❡ ❣r❛♣❤s ♦❢ f, g =

Z

b a

(f − g) dx

❲❡ ❤❛✈❡ ❛ ✈❛r✐❡t② ♦❢ r❡❣✐♦♥s ✇❡ ✉s❡❞ t♦ ❜❡ ✉♥❛❜❧❡ t♦ ❝♦♠♣✉t❡✳

❊①❛♠♣❧❡ ✸✳✶✳✺✿ ❛r❡❛ ❜❡t✇❡❡♥ ♣❛r❛❜♦❧❛s ❊✈❛❧✉❛t❡ t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ ♣❛r❛❜♦❧❛s y = x2 ❛♥❞ y = 2x2 + 1 ❜❡t✇❡❡♥ x = 0 ❛♥❞ x = 1✳ ■t ✐s ❝❧❡❛r t❤❛t g(x) = x2 ❛♥❞ f (x) = 2x2 + 1✱ ❛s ✇❡❧❧ ❛s a = 0 ❛♥❞ b = 1✳ ❚❤❡ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥t✐♥✉♦✉s ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐♥t❡❣r❛❜❧❡✳ ❇❡❢♦r❡ ✇❡ ❛♣♣❧② t❤❡ ❢♦r♠✉❧❛✱ ✇❡ ❥✉st ♥❡❡❞ t♦ ❝♦♥✜r♠ t❤❛t t❤❡ ❣r❛♣❤ ♦❢ f ✐s ❛❜♦✈❡ t❤❡ ❣r❛♣❤ ♦❢ g ✿

✸✳✶✳

❚❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t✇♦ ❣r❛♣❤s

✶✽✻

❋♦r ❡✈❡r② x ❜❡t✇❡❡♥ 0 ❛♥❞ 1✱ ✇❡ ❤❛✈❡ x2 < 2x2 + 1 ❜❡❝❛✉s❡ 0 < x2 + 1✳ ❚❤✉s✱ ❆r❡❛ =

Z

b a

(f − g) dx =

Z

1

(2x2 + 1) − x

0

 2

❙♦♠❡t✐♠❡s t❤❡ ✐♥t❡r✈❛❧ ✐s ♥♦t ♣r♦✈✐❞❡❞✳

dx =

Z

1 0

1 1 1 4 2 3 (x + 1) dx = x + x = + 1 = . 3 3 3 0

❊①❛♠♣❧❡ ✸✳✶✳✻✿ ❛r❡❛ ♣❛r❛❜♦❧❛ ❛♥❞ ❛ ❧✐♥❡

❊✈❛❧✉❛t❡ t❤✐s ❛r❡❛ ♦❢ t❤❡ r❡❣✐♦♥ ✐s ❜♦✉♥❞❡❞ ❜② t❤❡ ♣❛r❛❜♦❧❛ y = x2 ❛♥❞ t❤❡ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ y = 3✳ ❲❡ ✇✐❧❧ ♥❡❡❞ s♦♠❡ ❛❧❣❡❜r❛ t❤✐s t✐♠❡✱ t♦ ✜♥❞ a, b✿

√

❚❤❡ ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥ts (x, y) s❛t✐s❢②✿ y = 3 = x2 ✳ ❚❤❡♥ a = − 3, b = s❦❡t❝❤ t❤❛t f (x) = 3 ❛♥❞ g(x) = x2 ✳ ❚❤❡♥✱ ❆r❡❛

=

Z

Z

√

3✳ ❲❡ ❛❧s♦ r❡❛❧✐③❡ ❢r♦♠ t❤❡

b

(f − g) dx

a√

3

 3 − x2 dx − 3 √3 1 = 3x − x3 3 √ − 3     √ √ 1√ 3 1 √ 3 = 3 3− 3 − −3 3 − (− 3) 3 3   √ √ 1 3 =2 3 3− 3 3  √ √  =2 3 3− 3 =

√

√ = 4 3.

❊①❛♠♣❧❡ ✸✳✶✳✼✿ ❛r❡❛ ❜❡t✇❡❡♥

x2

❛♥❞

x3

❊✈❛❧✉❛t❡ t❤❡ ❛r❡❛ ❜❡t✇❡❡♥ y = x2 ❛♥❞ y = x3 ✳ ❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ✜♥❞ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥ts ❜② s♦❧✈✐♥❣ x2 = x3 ✳ ❲❡ ❤❛✈❡ a = 0 ❛♥❞ b = 1✱ ✇❤✐❝❤ ❝♦♥✜r♠s t❤❡ s❦❡t❝❤ ❛♥❞ t❤❡ ❢❛❝t t❤❛t x3 < x2 ✿

✸✳✶✳

❚❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t✇♦ ❣r❛♣❤s

✶✽✼

❚❤❡♥✱ ✇❡ ❤❛✈❡✿ ❆r❡❛

=

Z

Z

b a

(f − g) dx 1

 x2 − x3 dx 0 1 1 3 1 4 = x − x 3 4

=

0

1 1 − 3 4 1 = . 12 =

❊①❛♠♣❧❡ ✸✳✶✳✽✿ ❝✐r❝❧❡ ▲❡t✬s r❡✈✐s✐t t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡✿

❚❤✐s t✐♠❡ ✇❡ ❞♦♥✬t ❤❛✈❡ t♦ s♣❧✐t ✐t ✐♥ ❤❛❧❢ ❛♥❞ r❡❧② ♦♥ ✐ts s②♠♠❡tr②❀ t❤❡ ❝✐r❝❧❡ ✐s t❤❡ r❡❣✐♦♥ ❜❡t✇❡❡♥ t✇♦ ❣r❛♣❤s✿

y= ❚❤❡ ❢♦r♠❡r ✐s

f

❛♥❞ t❤❡ ❧❛tt❡r ✐s

❆r❡❛

=

Z

R −R

√

g✳

R2

√

R 2 − x2

❆❧s♦✱

−

x2

+

❛♥❞

√ y = − R 2 − x2 .

a = −R, b = R✳ √

R2

−

x2



❚❤❡♥✱

dx = 2

Z

R −R

√

R2 − x2 dx = πR2 .

❚❤❡ ✐♥t❡❣r❛❧ ✐s ❡✈❛❧✉❛t❡❞ ✈✐❛ ❛ tr✐❣ s✉❜st✐t✉t✐♦♥✱ ❥✉st ❛s ❜❡❢♦r❡✳

❊①❡r❝✐s❡ ✸✳✶✳✾ ❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ t✇♦ r❡❣✐♦♥s ❜♦✉♥❞❡❞ ❜② t❤❡ ❝✐r❝❧❡s

1✳

x2 +y 2 = 1 ❛♥❞ (x−1)2 +y 2 =

✸✳✷✳

❚❤❡ ❧✐♥❡❛r ❞❡♥s✐t② ❛♥❞ t❤❡ ♠❛ss

✶✽✽

❊①❡r❝✐s❡ ✸✳✶✳✶✵

❋✐♥❞ t❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t❤❡ ❝✉r✈❡s

x = y2

❛♥❞

x = y4✳

✸✳✷✳ ❚❤❡ ❧✐♥❡❛r ❞❡♥s✐t② ❛♥❞ t❤❡ ♠❛ss

❚❤❡ ♠❡t❤♦❞ t❤❛t st❛rts t♦ s❤❛♣❡ ✉♣ ✐s ❛s ❢♦❧❧♦✇s✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ q✉❛♥t✐t②

Q

✏❝♦♥t❛✐♥❡❞✑ ✐♥ ❛ s♣❛❝❡ r❡❣✐♦♥

R✿

❛r❡❛✱ ✈♦❧✉♠❡✱ ♠❛ss✱ ♣❛rt✐❝✉❧❛r ♠❛t❡r✐❛❧✱

❝❤❛r❣❡✱ ❡t❝✳ ❚❤❡♥✿ ✶✳ ❲❡ r❡♣r❡s❡♥t t❤❡ t♦t❛❧ q✉❛♥t✐t②

Q

❛s t❤❡ s✉♠ ♦❢ ✐ts ✈❛❧✉❡s

Qi

♦✈❡r s✐♠♣❧❡r✱ ❛♥❞ s♠❛❧❧❡r✱ ♣❛rts ♦❢

R✳

✷✳ ❲❡ r❡♣r❡s❡♥t✱ ♦r ❛♣♣r♦①✐♠❛t❡✱ ❡❛❝❤ ♦❢ t❤❡s❡ ✈❛❧✉❡s ✈✐❛ ❛ ❢❛♠✐❧✐❛r q✉❛♥t✐t②✱ ❡✳❣✳✱ ❛r❡❛ ✈✐❛ ❧❡♥❣t❤✱ ✈♦❧✉♠❡ ✈✐❛ ❛r❡❛✱ ❡t❝✳ ✸✳ ❲❡ r❡❝♦❣♥✐③❡ t❤❡ s✉♠ ❛s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥ t❤❛t r❡♣r❡s❡♥ts s♦♠❡ ♦t❤❡r q✉❛♥t✐t②

q

s♣r❡❛❞

♦✈❡r t❤❡ r❡❣✐♦♥✳ ✹✳ ❚❤❡ q✉❛♥t✐t②

Q

✐s ❡q✉❛❧ t♦ t❤❡ ✐♥t❡❣r❛❧ ♦❢

q✳

❚❤❡ ❧❛st st❡♣ ✐s ♥❡❝❡ss❛r② ♦♥❧② ✇❤❡♥ ✇❡ ❛♣♣r♦①✐♠❛t❡ ❛♥ ✐❞❡❛❧✐③❡❞ s✐t✉❛t✐♦♥✳ ❲❡ ✇✐❧❧ ✐❧❧✉str❛t❡ t❤❡ ♠❡t❤♦❞ ✇✐t❤ ♦♥❡ ♠♦r❡ ❡①❛♠♣❧❡✳ ▲❡t✬s r❡❝❛❧❧ ❤♦✇ t❤❡

❧✐♥❡❛r ❞❡♥s✐t② ✇❛s ❞❡✜♥❡❞ ✐♥ ❱♦❧✉♠❡ ✷✳

❲❡ ❛r❡ ❣✐✈❡♥ ❛ ♠❡t❛❧ r♦❞✿

❚❤❡ r♦❞ ♠✐❣❤t ❜❡ ♥♦♥✲✉♥✐❢♦r♠✱ ✐✳❡✳✱ t❤❡ ❞❡♥s✐t② ✈❛r✐❡s ❜✉t ♦♥❧② ✐♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❞✐r❡❝t✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ✇❤❛t ♠✐❣❤t ❤❛♣♣❡♥ ✇❤❡♥ t✇♦ ♠❡t❛❧s ❛r❡ ✭✐♠♣❡r❢❡❝t❧②✮ ♠❡❧t❡❞ ✐♥t♦ ❛ ♣✐❡❝❡ ♦❢ ❛❧❧♦②✿

❆♥♦t❤❡r ❡①❛♠♣❧❡ ✐s ♣❛rt✐❝❧❡s s✉s♣❡♥❞❡❞ ✐♥ ❛ ❧✐q✉✐❞ t❤❛t s❡tt❧❡s ✕ ❜❡❝❛✉s❡ ♦❢ ❣r❛✈✐t② ✕ ✐♥ ❛ ♣❛tt❡r♥ t❤❛t ✐s ❞❡♥s❡r ❛t t❤❡ ❜♦tt♦♠✿

✸✳✷✳

❚❤❡ ❧✐♥❡❛r ❞❡♥s✐t② ❛♥❞ t❤❡ ♠❛ss

✶✽✾

x✲❛①✐s✮ ✇✐t❤ ♥♦ ❝❤❛♥❣❡ ✐♥ ❞❡♥s✐t② ✐♥ t❤❡ ❞✐r❡❝t✐♦♥s ♣❡r♣❡♥❞✐❝✉❧❛r x ❞❡s✐❣♥❛t✐♥❣ ❧✐♥❡❛r ❞❡♥s✐t② y = l(x)✳

■♥ ❡✐t❤❡r ❝❛s❡✱ t❤❡r❡ ✐s ❛ ❧✐♥❡ ✭✇❡ ❝❛❧❧ ✐t t❤❡

t♦ ✐t✳ ❲❡ t❤❡♥ ✐❣♥♦r❡ t❤♦s❡ ❞✐r❡❝t✐♦♥s ❛♥❞ t❤❡ ❞❡♥s✐t② ❜❡❝♦♠❡s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❛ s✐♥❣❧❡ ♥✉♠❜❡r t❤❡ ❧♦❝❛t✐♦♥ ❛❧♦♥❣ t❤✐s ❧✐♥❡❀ ❤❡♥❝❡ t❤❡

❚❛❦❡ ❛ s♠❛❧❧ ♣✐❡❝❡ ♦❢ t❤❡ r♦❞ ❛t ❧♦❝❛t✐♦♥

x✱ ∆x

❧♦♥❣✱ ❛♥❞ ❧❡t✬s ❝❛❧❧ ✐ts ♠❛ss

❤❛✈❡✿ ▲✐♥❡❛r ❞❡♥s✐t②

=

♠❛ss ❧❡♥❣t❤

=

∆m✳

❚❤❡♥✱ ❢♦r t❤✐s ♣✐❡❝❡✱ ✇❡

∆m m(x + ∆x) − m(x) = . ∆x ∆x

▲❡t✬s r❡✈❡rs❡ t❤✐s ❛♥❛❧②s✐s✳ ❙✉♣♣♦s❡ t❤✐s t✐♠❡ t❤❡ ❧✐♥❡❛r ❞❡♥s✐t②

l

✐s ❣✐✈❡♥✱ ✇❤❛t ✐s t❤❡ ♠❛ss ♦❢ t❤❡ r♦❞❄

❊①❛♠♣❧❡ ✸✳✷✳✶✿ t✇♦ ♣✐❡❝❡s ❙✉♣♣♦s❡ t❤❡ t✇♦ ♠❡t❛❧s ❤❛✈❡♥✬t ♠❡r❣❡❞ ❛t ❛❧❧✿

❚❤❡r❡❢♦r❡✱ t❤❡ ♠❛ss ✐s s✐♠♣❧② t❤❡ s✉♠ ♦❢ t❤❡ t✇♦✿ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ♦✈❡r

l✱

1·1+2·1 = 3✳

■t ✐s ❛❧s♦ t❤❡ ❛r❡❛ ♦❢ t❤❡ t✇♦ r❡❝t❛♥❣❧❡s

✇❤✐❝❤ ✐s ❛ st❡♣✲❢✉♥❝t✐♦♥✱ ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ✐♥t❡❣r❛❧ ♦❢

[0, 2]✳

❊①❡r❝✐s❡ ✸✳✷✳✷ ❲❤❛t ✐❢ t❤❡ t✇♦ r♦❞s ❤❛✈❡ ❧❡♥❣t❤s

■♥st❡❛❞ ♦❢ ❥✉st ♣♦✐♥t✐♥❣ ♦✉t ✇❤❛t

m

❲❡ ❤❛✈❡ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥

P✿

0.5

❛♥❞

1.5❄

✐s✱ ❧❡t✬s st❛rt ❢r♦♠ s❝r❛t❝❤✳

a = x0 ≤ c1 ≤ x1 ≤ ... ≤ cn ≤ xn = b ,

l

✸✳✷✳

❚❤❡ ❧✐♥❡❛r ❞❡♥s✐t② ❛♥❞ t❤❡ ♠❛ss

✶✾✵

✇✐t❤ t❤❡s❡ ❧❡♥❣t❤s ♦❢ s❡❣♠❡♥ts✿

∆xi = xi − xi−1 . ❲❡ ✜rst ✐♠❛❣✐♥❡ t❤❛t t❤❡ r♦❞ ✐s ❞✐✈✐❞❡❞ ✐♥t♦ s♠❛❧❧❡r ♣✐❡❝❡s s♦ t❤❛t t❤❡ ❞❡♥s✐t② ♦❢ ❡❛❝❤ ✐s ❢♦✉♥❞ s❡♣❛r❛t❡❧②✿ F1 , F2 , ..., Fn ✿

❚❤❡♥ t❤❡ t♦t❛❧ ✇❡✐❣❤t ✐s s✐♠♣❧② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚♦t❛❧ ✇❡✐❣❤t = F1 ∆x1 + F2 ∆x2 + ... + Fn ∆xn . ❚❤❡ ❢♦r♠✉❧❛ ✐s s✉✣❝✐❡♥t ❢♦r ❛♣♣❧✐❝❛t✐♦♥s ✇❤❡♥ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s s✉✣❝✐❡♥t✳ ❍❡r❡✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ❞❡♥s✐t② ✐s ❝❤❛♥❣✐♥❣ ❝♦♥t✐♥✉♦✉s❧② ❛♥❞ ✇❡ ❝✉t t❤❡ r♦❞ ✐♥t♦ t❤❡s❡ s♠❛❧❧ s❡❣♠❡♥ts ❜② t❤❡ ♣❧❛♥❡s st❛rt✐♥❣ ❛t x = xi ❛♥❞ t❤❡♥ s❛♠♣❧❡ ✐ts ❞❡♥s✐t② ❛t t❤❡ ♣♦✐♥ts ci ✿

❚❤❡♥ t❤❡ ❞❡♥s✐t② ♦❢ ❡❛❝❤ s❡❣♠❡♥t ✕ ✐❢ ✉♥✐❢♦r♠ ✕ ✐s ❣✐✈❡♥ ❜② l(ci ) ❛♥❞ ✇❡ ❤❛✈❡✿ ▼❛ss ♦❢ it❤ s❡❣♠❡♥t = ❞❡♥s✐t② · ❧❡♥❣t❤ = l(ci ) · ∆xi . ❚❤❡♥✱ ❚♦t❛❧ ♠❛ss =

n X i=1

❲❡

l(ci ) · ∆xi .

r❡❝♦❣♥✐③❡ t❤✐s ❡①♣r❡ss✐♦♥ ❛s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ ❧✐♥❡❛r ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ♦✈❡r t❤✐s ♣❛rt✐t✐♦♥✿ ❉❡✜♥✐t✐♦♥ ✸✳✷✳✸✿ ♠❛ss ■❢ ❛ ❢✉♥❝t✐♦♥ l ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ s❡❣♠❡♥t [a, b] ✐s ❝❛❧❧❡❞ ❧✐♥❡❛r ❞❡♥s✐t②✱ t❤❡♥ ✐ts ❘✐❡♠❛♥♥ s✉♠

✐s ❝❛❧❧❡❞ t❤❡

♠❛ss ♦❢ t❤❡ s❡❣♠❡♥t✳

Σ l · ∆x

◆♦✇✱ ✇❤❛t ✐❢ t❤❡ ❞❡♥s✐t② ✈❛r✐❡s ❝♦♥t✐♥✉♦✉s❧②❄

❊①❛♠♣❧❡ ✸✳✷✳✹✿ ❧✐♥❡❛r ❞❡♣❡♥❞❡♥❝❡ ❙✉♣♣♦s❡ t❤❡ ❞❡♥s✐t② ♦❢ ❛ r♦❞ ♦❢ ❧❡♥❣t❤ 2 ✐s ❝❤❛♥❣✐♥❣ ❧✐♥❡❛r❧②✿ ❢r♦♠ 1 t♦ 2✳ ❚❤❡♥ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❛✈❡r❛❣❡ ❞❡♥s✐t② ✐s ❝❧❡❛r❀ ✐t ✐s 1.5✳ ❲❡ ❥✉st ❛✈❡r❛❣❡✳ ▲❡t✬s ♠❛❦❡ s✉r❡ t❤❛t t❤❡ ❞❡✜♥✐t✐♦♥ ♠❛❦❡s s❡♥s❡✿

✸✳✷✳

❚❤❡ ❧✐♥❡❛r ❞❡♥s✐t② ❛♥❞ t❤❡ ♠❛ss

✶✾✶

■t ❢♦❧❧♦✇s t❤❛t t❤❡ ♠❛ss ✐s 1.5 · 2 = 3✳ ■t ✐s ❛❧s♦ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr❛♣❡③♦✐❞ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ l(x) = 1 + x/2 ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ♦✈❡r [0, 2]✳ ■❢ t❤❡ ❞❡♥s✐t② ✐s ✈❛r✐❛❜❧❡✱ t❤❡♥ t❤❡ ♠❛ss ♦❢ ❡❛❝❤ s❡❣♠❡♥t ✕ ✇❤❡♥ s❤♦rt ❡♥♦✉❣❤ ✕ ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ♠❛ss ♦❢ s✉❝❤ ❛ s❡❣♠❡♥t ♠❛❞❡ ❡♥t✐r❡❧② ♦❢ ♠❛t❡r✐❛❧ ♦❢ ❞❡♥s✐t② l(ci )✿ ▼❛ss ♦❢ it❤ s❡❣♠❡♥t ≈ ❞❡♥s✐t② · ❧❡♥❣t❤ = l(ci ) · ∆xi , ❛♥❞ ❚♦t❛❧ ♠❛ss ≈ ❚❤❡♥✱ ✇❡ ❞❡✜♥❡ t❤❡ ✐♥t❡❣r❛❧ ♦❢ l✿

♠❛ss

n X i=1

l(ci ) · ∆xi = Σ l · ∆x .

♦❢ t❤❡ r♦❞ ❛s t❤❡ ❧✐♠✐t✱ ✐❢ ✐t ❡①✐sts✱ ♦❢ t❤❡s❡ ❘✐❡♠❛♥♥ s✉♠s✱ ✐✳❡✳✱ t❤❡ ❘✐❡♠❛♥♥

❉❡✜♥✐t✐♦♥ ✸✳✷✳✺✿ ♠❛ss ■❢ ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ l ♦♥ s❡❣♠❡♥t [a, b] ✐s ❝❛❧❧❡❞ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ Z b l dx ✐s ❝❛❧❧❡❞ t❤❡

♠❛ss ♦❢ t❤❡ s❡❣♠❡♥t✳

❧✐♥❡❛r ❞❡♥s✐t②✱

t❤❡♥ ✐ts

a

❊①❛♠♣❧❡ ✸✳✷✳✻✿ q✉❛❞r❛t✐❝ ❞❡♣❡♥❞❡♥❝❡ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛❣❛✐♥ ❛ r♦❞ ♦❢ ❧❡♥❣t❤ 2 ✇✐t❤ t❤❡ ❞❡♥s✐t② ❝❤❛♥❣✐♥❣ ❢r♦♠ 1 t♦ 2✱ ❜✉t q✉❛❞r❛t✐❝❛❧❧②✱ l(x) = x2 + 1✳ ❚❤❡ ♠❛ss ✐s ✐♠♣♦ss✐❜❧❡ t♦ ❣✉❡ss✿

❲❡ ❝♦♠♣✉t❡ t❤❡ ✐♥t❡❣r❛❧✿ ▼❛ss =

Z

b

l dx = a

Z

1

1 x3 4 (x + 1) dx = + x = . 3 3 0 2

0

✸✳✸✳

❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss

✶✾✷

❍❡r❡ ✐s ❛♥♦t❤❡r ✇❛② t♦ ❡①♣❧❛✐♥ ♦✉r ❞❡✜♥✐t✐♦♥✳ ❲❡ r❡❛❧✐③❡ t❤❛t ❡✈❡r② ❧♦❝❛t✐♦♥ ✇✐t❤ ❤✐❣❤❡r ❞❡♥s✐t② s✐♠♣❧② ❝♦♥t❛✐♥s ♠♦r❡ ♠❛t❡r✐❛❧ ❛♥❞ ✇❡ ❝❛♥ ❥✉st s♣r❡❛❞ ✐t ♦✉t ✕ ✈❡rt✐❝❛❧❧② ✕ ♠❛❦✐♥❣ ❛ ♣❧❛t❡ t❤❛t ✐s ✇✐❞❡r ❛t t❤✐s s♣♦t ❛♥❞ t❤✐♥♥❡r ❛t t❤❡ ❧♦❝❛t✐♦♥ ✇✐t❤ ❛ ❧♦✇❡r ❞❡♥s✐t②✿

■♥ r❡✈❡rs❡✱ ✐♠❛❣✐♥❡ t❤❛t t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤ ✐s ♠❛❞❡ ♦❢ ❛ s❤❡❡t ♦❢ ♠❡t❛❧✱ ✇❤✐❝❤ ✐s t❤❡♥ ♥♦♥✲✉♥✐❢♦r♠ r♦❞✳

r♦❧❧❡❞

✐♥t♦ ❛

❊①❡r❝✐s❡ ✸✳✷✳✼

❋✐♥❞ ❤♦✇ t❤❡ ♠❛ss ♦❢ ❛ r♦❞ ✇✐t❤ ❛♥ ❡①♣♦♥❡♥t✐❛❧❧② ❣r♦✇✐♥❣ ❞❡♥s✐t② ❣r♦✇s✳ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ♣r♦✈✐❞❡s ❢✉rt❤❡r ✐♥s✐❣❤t✳ ❙✉♣♣♦s❡ m(x) ✐s t❤❡ ✇❡✐❣❤t ♦❢ t❤❡ r♦❞ ❢r♦♠ a t♦ x✳ ❚❤❡♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s t❤❡ ❞❡♥s✐t②✿

m′ (x) = l(x) . ❊①❡r❝✐s❡ ✸✳✷✳✽

■s ✐t ♠❡❛♥✐♥❣❢✉❧ t♦ s♣❡❛❦ ♦❢ t❤❡ ♠❛ss ♦❢ ❛♥ ✐♥✜♥✐t❡❧② ❧♦♥❣ r♦❞❄

✸✳✸✳ ❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss

❈❛♥ ✇❡ ♥♦✇ ❜❛❧❛♥❝❡ t❤✐s ♥♦♥✲✉♥✐❢♦r♠ r♦❞ ♦♥ ❛ s✐♥❣❧❡ ♣♦✐♥t ♦❢ s✉♣♣♦rt❄ ❚r✐❛❧ ❛♥❞ ❡rr♦r s✉❣❣❡st t❤✐s✿

❚❤❡ q✉❡st✐♦♥ ✐s ✐♠♣♦rt❛♥t ❜❡❝❛✉s❡ t❤✐s ♣♦✐♥t✱ ❝❛❧❧❡❞ t❤❡ ♦❜❥❡❝t✳

❝❡♥t❡r ♦❢ ♠❛ss✱

✐s t❤❡ ❝❡♥t❡r ♦❢ r♦t❛t✐♦♥ ♦❢ t❤❡

❚❤❡ ❛♥❛❧②s✐s st❛rts ✇✐t❤ ❛ s✐♠♣❧❡st ❝❛s❡✱ s❡❡s❛✇✳ ❚✇♦ ♣❡rs♦♥s ♦❢ ❡q✉❛❧ ✇❡✐❣❤t ✇✐❧❧ ❜❡ ✐♥ ❛ st❛t❡ ♦❢ ❜❛❧❛♥❝❡ ✇❤❡♥ ❧♦❝❛t❡❞ ❛t ❡q✉❛❧ ❞✐st❛♥❝❡s ❢r♦♠ t❤❡ ♣♦✐♥t ♦❢ s✉♣♣♦rt✿

✸✳✸✳

❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss

✶✾✸

◆♦✇✱ ✇❤❛t ❝❛♥ ❜❡ ❝❤❛♥❣❡❞❄ ❲❤❛t ✐❢ ♦♥❡ ♣❡rs♦♥ ✐s ❤❡❛✈✐❡r t❤❛♥ t❤❡ ♦t❤❡r❄ ❋r♦♠ ❡①♣❡r✐❡♥❝❡✱ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ❢♦r♠❡r ♣❡rs♦♥ s❤♦✉❧❞ s✐t ❢❛rt❤❡r ❢r♦♠ t❤❡ ❝❡♥t❡r ✐♥ ♦r❞❡r t♦ ❜❛❧❛♥❝❡ t❤❡ ❜❡❛♠✿

■♥ ❢❛❝t✱ ✐❢ t❤❡ ♣❡rs♦♥ ✐s t✇✐❝❡ ❛s ❤❡❛✈② ❛s t❤❡ ♦t❤❡r✱ t❤❡ ❞✐st❛♥❝❡ ❢♦r t❤❡ ♦t❤❡r s❤♦✉❧❞ ❜❡ t✇✐❝❡ ❛s ❧♦♥❣✦ ❈♦♥✈❡rs❡❧②✱ ✐❢ ♦♥❡ ♣❡rs♦♥ s✐ts ❢❛rt❤❡r ❢r♦♠ t❤❡ ❝❡♥t❡r t❤❛♥ t❤❡ ♦t❤❡r ♦❢ t❤❡ s❛♠❡ ✇❡✐❣❤t✱ t❤❡ ❢♦r♠❡r ♣❡rs♦♥ s❤♦✉❧❞ ❜❡ ❥♦✐♥❡❞ ❜② ❛♥♦t❤❡r ✐♥ ♦r❞❡r t♦ ❜❛❧❛♥❝❡ t❤❡ ❜❡❛♠✳ ❙✉♣♣♦s❡ t❤❡ s❤♦rt❡r ❞✐st❛♥❝❡ ✐s ❛♥❞ ✇❡✐❣❤ts ❛r❡

2m

❛♥❞

m✳

a

❛♥❞ t❤❡ s♠❛❧❧❡r ✇❡✐❣❤t ✐s

❲❡ ❡①♣r❡ss t❤✐s ❞❛t❛ ✈✐❛ t❤❡

m✳

❚❤❡♥✱ ❝♦♠❜✐♥❡❞✱ t❤❡ ❞✐st❛♥❝❡s ❛r❡

❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥

a

❛♥❞

2a

✿

(a)(2m) = (2a)(m) .

■♥ ♦t❤❡r ✇♦r❞s✱ t❤✐s ❡①♣r❡ss✐♦♥✿ ❞✐st❛♥❝❡ ❝❛❧❧❡❞

❧❡✈❡r

t❤❡ ♠♦♠❡♥t

·

✇❡✐❣❤t

✱ ✐s t❤❡ s❛♠❡ t♦ t❤❡ ❧❡❢t ❛♥❞ t♦ t❤❡ r✐❣❤t ♦❢ t❤❡ s✉♣♣♦rt✳ ❚❤✐s ❞✐st❛♥❝❡ ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡

✳

▲❡t✬s ❛❞❞ t❤❡

x✲❛①✐s✳

❲❡ t❤❡♥ r❡❛❧✐③❡ t❤❛t ✐t ✐s t❤❡

s✐❣♥❡❞ ❞✐st❛♥❝❡

✱ ✐✳❡✳✱ t❤❡

x✲❝♦♦r❞✐♥❛t❡✱

r❡✲✇r✐t❡ t❤❡ ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥✿

(−a)(2m) + (2a)(m) = 0 .

♦❢ t❤❡ ♦❜❥❡❝t t❤❛t ♠❛tt❡rs✳ ❲❡ s✐♠♣❧②

✸✳✸✳

❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss

✶✾✹

❚❤❡♥✱

♠♦♠❡♥t = ❝♦♦r❞✐♥❛t❡ · ✇❡✐❣❤t

❋✉rt❤❡r♠♦r❡✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❡r❡ ✐s ❛♥ ♦❜❥❡❝t ❛t ❡✈❡r② ❧♦❝❛t✐♦♥ ❜✉t t❤❡ r❡st ♦❢ t❤❡♠ ❤❛✈❡ 0 ♠❛ss✳ ❚❤❡ ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s✿ ... + (−2a)(0) + (−a)(2m) + (0)(0) + (a)(0) + (2a)(m) + ... = 0 .

❚❤✐s ❛♥❛❧②s✐s ❜r✐♥❣s ✉s t♦ t❤❡ ✐❞❡❛ ♦❢ ❝♦♠❜✐♥✐♥❣ t❤❡ ✇❡✐❣❤ts ❛♥❞ t❤❡ ❞✐st❛♥❝❡s ✐♥ ❛ ♣r♦♣♦rt✐♦♥❛❧ ♠❛♥♥❡r ✐♥ ♦r❞❡r t♦ ❡✈❛❧✉❛t❡ t❤❡ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ ❛ ♣❛rt✐❝✉❧❛r ✇❡✐❣❤t t♦ t❤❡ ♦✈❡r❛❧❧ ❜❛❧❛♥❝❡✳ ❚❤❡ ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥ s✐♠♣❧② s❛②s t❤❛t t❤❡ s✉♠ ♦❢ ❛❧❧ ♠♦♠❡♥ts ✐s 0✳

❉❡✜♥✐t✐♦♥ ✸✳✸✳✶✿ ✇❡✐❣❤ts ❲❡ ❝❛❧❧ ❛ s②st❡♠ ♦❢ ✇❡✐❣❤ts ❛♥② ❝♦❧❧❡❝t✐♦♥ ♦❢ ♥♦♥✲♥❡❣❛t✐✈❡ ♥✉♠❜❡rs m1 , ..., mn ❝❛❧❧❡❞ ✇❡✐❣❤ts ❛ss✐❣♥❡❞ t♦ n ❧♦❝❛t✐♦♥s ✇✐t❤ ❝♦♦r❞✐♥❛t❡s a1 , ..., an ♦♥ t❤❡ x✲❛①✐s✳

❉❡✜♥✐t✐♦♥ ✸✳✸✳✷✿ t♦t❛❧ ♠♦♠❡♥t

❚❤❡ t♦t❛❧ ♠♦♠❡♥t ♦❢ t❤❡ s②st❡♠ ♦❢ ✇❡✐❣❤ts ✇✐t❤ r❡s♣❡❝t t♦ ❜❡ t❤❡ s✉♠ ♦❢ t❤❡ ♠♦♠❡♥ts ♦❢ t❤❡ ✇❡✐❣❤ts✱ ✐✳❡✳✱ X

t♦ t❤❡ ♦r✐❣✐♥ ✐s ❞❡✜♥❡❞

mi ai .

i

❚❤❡ ❜❛❧❛♥❝❡

❡q✉❛t✐♦♥ ♦❢ t❤❡ s②st❡♠ st❛t❡s t❤❛t ✐ts t♦t❛❧ ♠♦♠❡♥t ✐s ③❡r♦✳

❲❡ ♥♦✇ ❣♦ ❜❛❝❦ t♦ t❤❡ ♦r✐❣✐♥❛❧ ♣r♦❜❧❡♠✿ ◮ ❙✉♣♣♦s❡ ❞✐✛❡r❡♥t ✇❡✐❣❤ts ❛r❡ ❧♦❝❛t❡❞ ♦♥ ❛ ❜❡❛♠✱ ✇❤❡r❡ ❞♦ ✇❡ ♣✉t t❤❡ s✉♣♣♦rt ✐♥ ♦r❞❡r t♦

❜❛❧❛♥❝❡ ✐t❄

■t ✇❛s ❡♥t✐r❡❧② ♦✉r ❞❡❝✐s✐♦♥ t♦ ♣❧❛❝❡ t❤❡ ♦r✐❣✐♥ ♦❢ ♦✉r x✲❛①✐s ❛t t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss✳ ❚❤❡ r❡s✉❧t ✇❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ s❤♦✉❧❞ ❜❡ ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ t❤❛t ❝❤♦✐❝❡ ❛♥❞ ✇❡ ❝❛♥ ♠♦✈❡ t❤❡ ♦r✐❣✐♥ ❛♥②✇❤❡r❡✳

❲❡ ❥✉st ♥❡❡❞ t♦ ❡①❡❝✉t❡ ❛ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s✳ ❙✉♣♣♦s❡ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ✭❛♥❞ t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ ♦❧❞ ❝♦♦r❞✐♥❛t❡ s②st❡♠✮ ✐s ❧♦❝❛t❡❞ ❛t t❤❡ ♣♦✐♥t ✇✐t❤ ❝♦♦r❞✐♥❛t❡ c ♦❢ t❤❡ ♥❡✇ ❝♦♦r❞✐♥❛t❡ s②st❡♠✳ ❚❤❡♥✱ t❤❡ ♥❡✇ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ it❤ ♦❜❥❡❝t ✐s ci = ai + c .

❚❤❡r❡❢♦r❡✱ t❤❡ ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥ ❤❛s t❤✐s ❢♦r♠✿ X i

mi (ci − c) = 0 .

✸✳✸✳

❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss

❲❡ r❡✇r✐t❡✿

✶✾✺

X

mi ci = c

i

❚❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♠❛② ❜❡ s❡❡♥ ❛s ❢♦❧❧♦✇s✿

X

mi .

i

◮ ❚❤❡ ✇❤♦❧❡ ✇❡✐❣❤t ✐s ❝♦♥❝❡♥tr❛t❡❞ ❛t c✳

❍❡♥❝❡ t❤❡ ♥❛♠❡✳

❉❡✜♥✐t✐♦♥ ✸✳✸✳✸✿ ♠♦♠❡♥t ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s②st❡♠ ♦❢ ✇❡✐❣❤ts m1 , ..., mn ❧♦❝❛t❡❞ ❛t c1 , ..., cn ♦♥ t❤❡ x✲❛①✐s✳ ❋♦r ❛ ❣✐✈❡♥ ♣♦✐♥t c ❛♥❞ ❢♦r ❡❛❝❤ i✱ t❤❡ ♣r♦❞✉❝t mi (ci − c)

✐s ❝❛❧❧❡❞ t❤❡ it❤ ✇❡✐❣❤t✬s ♠♦♠❡♥t

✇✐t❤ r❡s♣❡❝t t♦ c✳

X i

❚❤❡ s✉♠ ♦❢ t❤❡ ♠♦♠❡♥ts✱

mi (ci − c) ,

✐s ❝❛❧❧❡❞ t❤❡ t♦t❛❧ ♠♦♠❡♥t ✇✐t❤ r❡s♣❡❝t t♦ c✳ ❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ t❤✐s s②st❡♠ ♦❢ ✇❡✐❣❤ts ✐s s✉❝❤ ❛ ♣♦✐♥t c t❤❛t t❤❡ t♦t❛❧ ♠♦♠❡♥t ✇✐t❤ r❡s♣❡❝t t♦ c ✐s ③❡r♦✳ ❖❢ ❝♦✉rs❡✱ ✐❢ c = 0✱ ✇❡ ❤❛✈❡ t❤❡ ♦❧❞ ❞❡✜♥✐t✐♦♥✳

❊①❛♠♣❧❡ ✸✳✸✳✹✿ ❝❡♥t❡r ♦❢ ♠❛ss ❜② tr✐❛❧ ❛♥❞ ❡rr♦r ❋♦❧❧♦✇✐♥❣ t❤✐s ✐♥s✐❣❤t✱ ❧❡t✬s ✜♥❞ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ ❛♥ ♦❜❥❡❝t✳ ❚❤❡ ♠❡t❤♦❞ ❛♠♦✉♥ts t♦ tr✐❛❧ ❛♥❞ ❡rr♦r✳ ❲❡ ❥✉st ♠♦✈❡ c ✇❤✐❧❡ ✇❛t❝❤✐♥❣ t❤❡ t♦t❛❧ ♠♦♠❡♥t✿

✸✳✸✳

❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss

✶✾✻

❊①❡r❝✐s❡ ✸✳✸✳✺

❲❤❛t ✐❢ ✇❡ ❛❧❧♦✇ t❤❡ ✈❛❧✉❡s ♦❢ mi t♦ ❜❡ ♥❡❣❛t✐✈❡❄ ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ s②st❡♠ ❛♥❞ ♦❢ c❄ ❚♦ ♠❛❦❡ ♦✉r t❛s❦ ❡❛s✐❡r✱ ✇❡ s♦❧✈❡ t❤❡ ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥ ❢♦r c✳ ❚❤❡♦r❡♠ ✸✳✸✳✻✿ ❈❡♥t❡r ♦❢ ▼❛ss ■❢

c

✐s t❤❡ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ t❤❡ s②st❡♠ ♦❢ ✇❡✐❣❤ts✱ t❤❡♥ ✇❡

❤❛✈❡✿

P mi ci c = Pi i mi ❊①❡r❝✐s❡ ✸✳✸✳✼

Pr♦✈❡ t❤❡ ❢♦r♠✉❧❛✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ❈❡♥t❡r ♦❢ ♠❛ss =

t♦t❛❧ ♠♦♠❡♥t t♦t❛❧ ♠❛ss

❊①❛♠♣❧❡ ✸✳✸✳✽✿ ❝❡♥t❡r ♦❢ ♠❛ss ❢r♦♠ t❤❡ ❢♦r♠✉❧❛

❆r♠❡❞ ✇✐t❤ t❤✐s ❢♦r♠✉❧❛✱ ✇❡ ❝❛♥ q✉✐❝❦❧② ✜♥❞ t❤❡ ❝❡♥t❡rs ♦❢ ♠❛ss ♦❢ ♦❜❥❡❝ts✳ ❇❡❧♦✇ ✐s t❤❡ s❤❛♣❡ ❢r♦♠ t❤❡ ❧❛st ❡①❛♠♣❧❡✿

❊①❛♠♣❧❡ ✸✳✸✳✾✿ t✇♦ ♦❜ ❥❡❝ts

▲❡t✬s t❡st t❤❡ ❢♦r♠✉❧❛ ❢♦r ❛ s②st❡♠ ♦❢ ❥✉st t✇♦ ♦❜❥❡❝ts✳ ❋✐rst✱ s✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ ✐❞❡♥t✐❝❛❧ ✇❡✐❣❤ts ❧♦❝❛t❡❞ ❛t a ❛♥❞ b✳ ❚❤❡♥ c=

ma + mb a+b = . m+m 2

❙♦✱ ♥♦ ♠❛tt❡r ✇❤❛t t❤❡ ✇❡✐❣❤t ✐s✱ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ❧✐❡s ❤❛❧❢✇❛② ❜❡t✇❡❡♥ t❤❡ t✇♦ ♦❜❥❡❝ts✱ ❛s ❡①♣❡❝t❡❞✳ ❲❤❛t ✐❢ t❤❡ ✇❡✐❣❤ts ❛r❡ ❞✐✛❡r❡♥t❄ ❲❡ ❝❛♥ ❣✉❡ss t❤❛t t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ✇✐❧❧ ❜❡ ❝❧♦s❡r t♦ t❤❡ ❤❡❛✈✐❡r ♦❜❥❡❝t✳ ❇✉t ❜② ❤♦✇ ♠✉❝❤❄ ❙✉♣♣♦s❡ t❤❡s❡ ❛r❡ m ❛♥❞ 2m✳ ❲❡ ❝♦♠♣✉t❡✿ c=

ma + 2mb a + 2b 1 2 = = a + b. m + 2m 3 3 3

■t✬s t✇✐❝❡ ❛s ❝❧♦s❡ t♦ t❤❡ ❤❡❛✈✐❡r ♦❜❥❡❝t ✭❜♦tt♦♠ ❧❡❢t✮✿

✸✳✸✳

❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss

■♥ ❣❡♥❡r❛❧✱

✶✾✼

t❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ t❤❡ ❞✐st❛♥❝❡ ✐s t❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ t❤❡ ✇❡✐❣❤t✳

❊①❡r❝✐s❡ ✸✳✸✳✶✵

■❢ α ❛♥❞ β ❛r❡ t❤❡ s❤❛r❡s ♦❢ t❤❡ t♦t❛❧ ✇❡✐❣❤t✱ ✇❤❡r❡ ✐s t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ t❤✐s t✇♦✲♦❜❥❡❝t s②st❡♠❄ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ t❤❡ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ t❤❡ s②st❡♠ ❝❛♥ ❜❡ r❡✲✇r✐tt❡♥✿ P mi ci X mi P = ci . c = Pi i mi j mj i ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳

❈♦r♦❧❧❛r② ✸✳✸✳✶✶✿ ❲❡✐❣❤t❡❞ ❆✈❡r❛❣❡ ■❢ ❛t

c ✐s t❤❡ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ ❛ s②st❡♠ ci ✱ ✇❡✐❣❤ts mi ✱ ❛♥❞ t❤❡ t♦t❛❧ ✇❡✐❣❤t M ✱ t❤❡♥ X c= µi c i ,

♦❢ ✇❡✐❣❤ts ✇✐t❤ ❧♦❝❛t✐♦♥s

i

✇❤❡r❡

µi

❛r❡ t❤❡

r❡❧❛t✐✈❡ ✇❡✐❣❤ts

✿

µi =

mi . M

❲❡ st❛rt t♦ ♥♦t✐❝❡ t❤❛t t❤❡ ♥✉♠❡r♦✉s ❜❧♦❝❦s ♣❧❛❝❡❞ ♦♥ t❤❡ ❜❛r st❛rt t♦ ❧♦♦❦ ❧✐❦❡ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥✦ ❚❤❡ ✈❛❧✉❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ ❜❧♦❝❦s ♣❧❛❝❡❞ ❛t t❤❛t ❧♦❝❛t✐♦♥✳ ❲❡ ❦♥♦✇ t❤❛t t❤✐s ❢✉♥❝t✐♦♥ ❝❛♥ ❛❧s♦ ❜❡ s❡❡♥ ❛s t❤❡ ❧✐♥❡❛r ❞❡♥s✐t② ♦❢ ❛ r♦❞✳ ◆❡①t✱ ❧❡t✬s ✐♠❛❣✐♥❡ t❤❛t t❤❡ ❞❡♥s✐t② ✈❛r✐❡s ✐♥ ❛ ♠♦r❡ ✉♥♣r❡❞✐❝t❛❜❧❡ ✇❛②✳ ❲❡ ❝♦♥t✐♥✉❡ ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ✕ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ P ♦❢ ✐♥t❡r✈❛❧ [a, b] ✐s ❣✐✈❡♥✿

a = x0 ≤ c1 ≤ x1 ≤ ... ≤ xn−1 ≤ cn ≤ xn = b ❚❤❡♥ t❤❡ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ l ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✳ ❚❤❡♥ t❤❡ t❡r♠s l(ci )∆xi r❡♣r❡s❡♥t✐♥❣ t❤❡ ✇❡✐❣❤t ♦❢ ❡❛❝❤ ✐♥t❡r✈❛❧ ❛r❡ ❢♦r♠❡❞✳✳✳ ❜✉t ♥♦t s✐♠♣❧② ❛❞❞❡❞ t❤✐s t✐♠❡✿

✸✳✸✳

❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss

✶✾✽

❊❛❝❤ ♦❢ t❤❡s❡ t❡r♠s ✐s ❛ ✇❡✐❣❤t ♣❧❛❝❡❞ ♦♥ t♦♣ ♦❢ t❤❡ ✐♥t❡r✈❛❧ ✈✐s✉❛❧✐③❡❞ ❛s ❛ r❡❝t❛♥❣❧❡✳ ❍♦✇❡✈❡r✱ ✐t ✐s ❛ss✉♠❡❞ t❤❛t t❤❡ ✇❡✐❣❤t ♦❢ t❤❡

it❤

r❡❝t❛♥❣❧❡ ✐s ❝♦♥❝❡♥tr❛t❡❞ ❛t

ci ✳

❚❤❡ ❧❡✈❡r ♦❢ ❡❛❝❤ ✇❡✐❣❤t ✐s ❛❧s♦ s❤♦✇♥✳ ❚❤❡♥ t❤❡

t♦t❛❧ ♠♦♠❡♥t ♦❢ t❤✐s s②st❡♠ ♦❢ ✇❡✐❣❤ts ✇✐t❤ r❡s♣❡❝t t♦ s♦♠❡

X i

mi (ci − c) =

X i

c

l(ci ) · ∆xi (ci − c) =

✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

X i

l(ci )(ci − c) · ∆xi .

❍❛✈❡ ✇❡ ♣r♦❞✉❝❡❞ ❛ ❘✐❡♠❛♥♥ s✉♠ ❛s ❜❡❢♦r❡❄ ❲❡❧❧✱ t❤✐s ✐s♥✬t t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ✭❞❡♣❡♥❞❡♥t ♦♥ ♦✉r ❝❤♦✐❝❡ ♦❢

l✦

▲❡t✬s tr② t❤✐s ❢✉♥❝t✐♦♥

c✮✿ f (x) = l(x)(x − c) .

❚❤❡♥✱ ✐♥❞❡❡❞✱ ✇❡ ❢❛❝❡ ✐ts ❘✐❡♠❛♥♥ s✉♠✿

X i

mi (ci − c) = Σ f · ∆x .

❏✉st ❛s ❛❜♦✈❡✱ t❤❡ s②st❡♠ ♦❢ ✇❡✐❣❤ts t❤❛t ♠❛❦❡s ✉♣ t❤❡ r♦❞ ✐s ❜❛❧❛♥❝❡❞ ✇❤❡♥ t❤❡ t♦t❛❧ ♠♦♠❡♥t ✐s ③❡r♦✿

Σ l(x)(x − c) · ∆x = 0 . ❲❡ ❛rr✐✈❡ t♦ ❛ s✐♠✐❧❛r ❝♦♥❝❧✉s✐♦♥ ❜❡❧♦✇✳

❚❤❡♦r❡♠ ✸✳✸✳✶✷✿ ❈❡♥t❡r ♦❢ ▼❛ss ✕ ❉✐s❝r❡t❡ ❈❛s❡ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ y = l(x) ✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ci , i = 1, 2, ..., n✱ ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✐♥t❡r✈❛❧ [a, b]✳ ❚❤❡♥ t❤❡ s②st❡♠ ♦❢ ✇❡✐❣❤ts l(ci )∆xi , i = 1, 2, ..., n✱ ❤❛s ✐ts ❝❡♥t❡r ♦❢ ♠❛ss ❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦✐♥t✿ c=

Σ l(x)x · ∆x Σ l(x) · ∆x

❲❤❛t ✇❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞ ✐s t❤❛t t❤❡ ♣r♦❜❧❡♠ ♦❢ ❜❛❧❛♥❝✐♥❣ ❛ r♦❞ ✇✐t❤ ❛ ✈❛r✐❛❜❧❡ ❞❡♥s✐t② ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❜❛❧❛♥❝✐♥❣ t❤❡ r❡❣✐♦♥ ❜❡❧♦✇ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✿

✸✳✸✳

❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss

✶✾✾

❊①❛♠♣❧❡ ✸✳✸✳✶✸✿ ♣✐❡❝❡ ♦❢ ❝✐r❝❧❡ ▲❡t✬s t❡st t❤✐s ❢♦r♠✉❧❛ ♦♥ s♦♠❡ r❡❣✐♦♥s ❝✉t ❢r♦♠ t❤❡ ✉♥✐t ❝✐r❝❧❡✿

■♥ ❈❤❛♣t❡r ✹✱ ✇❡ ✇✐❧❧ ♦✛❡r ♠♦r❡ ❝♦♠♣❧❡① ❡①❛♠♣❧❡s✳

❊①❡r❝✐s❡ ✸✳✸✳✶✹ Pr♦✈❡ t❤❛t ✐❢ t❤❡ ❞❡♥s✐t② ♦❢ t❤❡ r♦❞ ✐s str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ✭♦r ❞❡❝r❡❛s✐♥❣✮✱ ✐ts ❝❡♥t❡r ♦❢ ♠❛ss ❝❛♥♥♦t ❜❡ ✐♥ t❤❡ ❝❡♥t❡r✳ ❚❤❡ ♥❡①t st❡♣ ✐s t♦ t❤✐♥❦ ♦❢ t❤❡ ✇❡✐❣❤ts ❛ss✐❣♥❡❞ t♦ ❞✐str✐❜✉t✐♦♥ ♦❢ ✇❡✐❣❤t ✐s ♥♦ ❧♦♥❣❡r ✐♥❝r❡♠❡♥t❛❧✳

❡✈❡r② ❧♦❝❛t✐♦♥ ♦♥ t❤❡ x✲❛①✐s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡

❲❤❛t ✇❡ ❤❛✈❡ ❧❡❛r♥❡❞ ✐s t❤❛t t❤❡ t♦t❛❧ ♠♦♠❡♥t ♦❢ t❤❡ r❡❣✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ s♦♠❡ c ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❛t ♦❢ t❤✐s s②st❡♠ ♦❢ ✇❡✐❣❤ts✱ ✇❤✐❝❤ ✐s t❤❡ ❘✐❡♠❛♥♥ s✉♠✱ X i

♦❢ t❤❡ ❢✉♥❝t✐♦♥

mi (ci − c) =

X i

l(ci ) · ∆xi (ci − c) = Σ f · ∆x ,

f (x) = l(x)(x − c) .

❚❤❡ ❜❡❛♠ ❞♦❡s♥✬t ❤❛✈❡ t♦ ❜❡ ❜❛❧❛♥❝❡❞ ❛♥❞ t❤❡ t♦t❛❧ ♠♦♠❡♥t ❞♦❡s♥✬t ❤❛✈❡ t♦ ❜❡ ③❡r♦ ❢♦r ❡❛❝❤ ♣❛rt✐t✐♦♥✱ ❜✉t ✐t ❞♦❡s ❤❛✈❡ t♦ ❞✐♠✐♥✐s❤ t♦ ③❡r♦ ❛s ✇❡ r❡✜♥❡ t❤❡ ♣❛rt✐t✐♦♥s✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s ③❡r♦✳

❉❡✜♥✐t✐♦♥ ✸✳✸✳✶✺✿ ❝❡♥t❡r ♦❢ ♠❛ss ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ❢✉♥❝t✐♦♥ y = l(x) ✐♥t❡❣r❛❜❧❡ ♦♥ s❡❣♠❡♥t [a, b] ❝❛❧❧❡❞ t❤❡ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳ ❚❤❡♥✱ ❢♦r ❛ ❣✐✈❡♥ ♣♦✐♥t c✱ t❤❡ ✐♥t❡❣r❛❧ Z

b a

l(x)(x − c) dx

✐s ❝❛❧❧❡❞ t❤❡ t♦t❛❧ ♠♦♠❡♥t ♦❢ t❤❡ s❡❣♠❡♥t ✇✐t❤ r❡s♣❡❝t t♦ c✳ ❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ t❤❡ s❡❣♠❡♥t ✇✐t❤ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ l ✐s s✉❝❤ ❛ ♣♦✐♥t c t❤❛t t❤❡ t♦t❛❧ ♠♦♠❡♥t

✇✐t❤ r❡s♣❡❝t t♦ c ✐s ③❡r♦✳

❏✉st ❛s ✐♥ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✱ t❤❡ ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥ ❝❛♥ ❜❡ s♦❧✈❡❞ ❢♦r c✿

✸✳✸✳

❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss

✷✵✵

❚❤❡♦r❡♠ ✸✳✸✳✶✻✿ ❈❡♥t❡r ♦❢ ▼❛ss ✕ ❈♦♥t✐♥✉♦✉s ❈❛s❡ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ❢✉♥❝t✐♦♥

y = l(x)

✐♥t❡❣r❛❜❧❡ ♦♥ ✐♥t❡r✈❛❧

[a, b]✳

■❢ t❤❡ ♠❛ss ♦❢ t❤❡ s❡❣♠❡♥t ✐s ♥♦t ③❡r♦✱ t❤❡♥ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ✐s✿

Rb

l(x)x dx c = Ra b l(x) dx a

Pr♦♦❢✳ ❋✐rst✱ ✇❡ ♥♦t❡ t❤❛t ❢♦❧❧♦✇✐♥❣✿

y = l(x)(x − c)

0= ◆♦✇ s♦❧✈❡ ❢♦r

t♦t❛❧ ♠♦♠❡♥t

✐s ✐♥t❡❣r❛❜❧❡ ❜② P❘✳ ❚❤❡♥ ✇❡ ✉s❡

=

c✳

Z

b a

l(x)(x − c) dx =

Z

❙❘

b

l(x)x dx + c a

❛♥❞

Z

❈▼❘

t♦ ❝♦♠♣✉t❡ t❤❡

b

l(x) dx . a

❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ❤❛✈❡✿

❈❡♥t❡r ♦❢ ♠❛ss

=

t♦t❛❧ ♠♦♠❡♥t t♦t❛❧ ♠❛ss

❊①❡r❝✐s❡ ✸✳✸✳✶✼ ❙❤♦✇ t❤✐s t❤❡♦r❡♠ ✐♠♣❧✐❡s t❤❡ ♣r❡✈✐♦✉s ♦♥❡✳

❊①❛♠♣❧❡ ✸✳✸✳✶✽✿ ❧✐♥❡❛r ❞❡♣❡♥❞❡♥❝❡ ❙✉♣♣♦s❡ t❤❡ ❞❡♥s✐t② ♦❢ ❛ r♦❞ ♦❢ ❧❡♥❣t❤

❚❤❡♥✱ t❤❡ ♠❛ss ✐s

3✳

2

✐s ❝❤❛♥❣✐♥❣ ❧✐♥❡❛r❧②✿ ❢r♦♠

1

t♦

2✱

✐✳❡✳✱

l(x) = x/2 + 1✳

■t ✇❛s ❢♦✉♥❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ❜❛s❡❞ ♦♥ ❛ ❝♦♠♠♦♥ s❡♥s❡ ❛♥❛❧②s✐s✳ ❚❤❛t✬s t❤❡

❞❡♥♦♠✐♥❛t♦r ♦❢ t❤❡ ❢r❛❝t✐♦♥✳ ◆♦✇✱ t❤❡ ♥✉♠❡r❛t♦r✳ ▼❡r❡ ❝♦♠♠♦♥ s❡♥s❡ ✇♦♥✬t ❤❡❧♣ t❤✐s t✐♠❡❀ ✇❡ ♥❡❡❞ t♦ ✐♥t❡❣r❛t❡✿

Z

2

l(x)x dx = 0

Z

2

(x/2 + 1)x dx

Z0 2

(x2 /2 + x) dx 0 2 = x3 /6 + x2 /2

=

0

= 8/6 + 4/2

= 10/3 .

✸✳✹✳

❚❤❡ r❛❞✐❛❧ ❞❡♥s✐t② ❛♥❞ t❤❡ ♠❛ss

✷✵✶

❚❤❡r❡❢♦r❡✱ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ✐s c=

❙❧✐❣❤t❧② t♦ t❤❡ r✐❣❤t ♦❢ t❤❡ ❝❡♥t❡r✳✳✳

10 10 ÷3= . 3 9

❊①❡r❝✐s❡ ✸✳✸✳✶✾

❋✐♥❞ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ ❛ r♦❞ ✇✐t❤ ❛ ❧✐♥❡❛r❧② ✐♥❝r❡❛s✐♥❣ ❞❡♥s✐t②✳ ❊①❡r❝✐s❡ ✸✳✸✳✷✵

❋✐♥❞ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ ❛ ♣❧❛t❡ ❝✉t ❢r♦♠ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s 1 ❝❡♥t❡r❡❞ ❛t t❤❡ ♦r✐❣✐♥ ❜② t❤❡ ❧✐♥❡s x = a ❛♥❞ x = b✳ ❊①❡r❝✐s❡ ✸✳✸✳✷✶

❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ ❛♥ ✐♥✜♥✐t❡ ♦❜❥❡❝t❄

✸✳✹✳ ❚❤❡ r❛❞✐❛❧ ❞❡♥s✐t② ❛♥❞ t❤❡ ♠❛ss

❙✉♣♣♦s❡ ♥❡①t ✇❡ ❤❛✈❡ ❛♥ ❛❧❧♦② t❤❛t ✐s r♦t❛t❡❞ ❛s ✐t ❤❛r❞❡♥s✳ ❚❤❡♥ ✐ts ❞❡♥s✐t② ❞❡♣❡♥❞s ✭♦♥❧②✮ ♦♥ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❝❡♥t❡r✳

❚❤❡ s❛♠❡ ❡✛❡❝t ✐s ♣r♦❞✉❝❡❞ ❜② st✐rr✐♥❣ ❛ ❧✐q✉✐❞✳ ■♥ ❡✐t❤❡r ❝❛s❡✱ ✇❡ ✐❣♥♦r❡ t❤❡ ❞❡♣t❤ ❛♥❞ ❛❧❧ ✇❡ s❡❡ ✐s ❛ ❞✐s❦✳ ❚❤❡♥✱ ❢♦r ❛♥② r❛❞✐❛❧ ❧✐♥❡ ✭✇❡ ♣✐❝❦ ♦♥❡ ❛♥❞ ❝❛❧❧ ✐t t❤❡ x✲❛①✐s✮ t❤❡r❡ ✐s ♥♦ ❝❤❛♥❣❡ ✐♥ ❞❡♥s✐t② ✐♥ t❤❡ ❞✐r❡❝t✐♦♥s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ✐t✳ ❲❡ t❤❡♥ ✐❣♥♦r❡ t❤♦s❡ ❞✐r❡❝t✐♦♥s ❛♥❞ t❤❡ ❞❡♥s✐t② ❜❡❝♦♠❡s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❛ s✐♥❣❧❡ ♥✉♠❜❡r x ❞❡s✐❣♥❛t✐♥❣ t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ❝❡♥t❡r ❛❧♦♥❣ t❤✐s ❧✐♥❡❀ ❤❡♥❝❡ t❤❡ r❛❞✐❛❧ ❞❡♥s✐t② y = r(x)✳ ❍❡r❡ ❛r❡ ❛ ❢❡✇ ❡①❛♠♣❧❡s ♦❢ t❤✐s ❢✉♥❝t✐♦♥✿

✸✳✹✳

❚❤❡ r❛❞✐❛❧ ❞❡♥s✐t② ❛♥❞ t❤❡ ♠❛ss

✷✵✷

❲❡ ✇✐❧❧ ♣r♦✈✐❞❡ ❛♥❛❧②s✐s s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ t♦ ❞❡✜♥❡ t❤❡ ♠❛ss ♦❢ s✉❝❤ ❛♥ ♦❜❥❡❝t✳ ❙✉♣♣♦s❡ t❤❡ r❛❞✐❛❧ ❞❡♥s✐t② r ✐s ❣✐✈❡♥✱ ✇❤❛t ✐s t❤❡ ♠❛ss ♦❢ t❤❡ ❞✐s❦❄ ❊①❛♠♣❧❡ ✸✳✹✳✶✿ t✇♦ ♣✐❡❝❡s

❙✉♣♣♦s❡ t❤❡ t✇♦ ♠❡t❛❧s ✇✐t❤ ❞❡♥s✐t✐❡s 2 ♦♥ t❤❡ ✐♥s✐❞❡ ❛♥❞ 1 ♦♥ t❤❡ ♦✉ts✐❞❡ ❤❛✈❡♥✬t ♠❡r❣❡❞ ❛t ❛❧❧✳ ❚❤❡ ♦❜❥❡❝t ✐s s✐♠♣❧② ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❛ ❞✐s❦ ♦❢ r❛❞✐✉s 1 ❛♥❞ ❛ ✇❛s❤❡r ❛r♦✉♥❞ ✐t ♦❢ t❤✐❝❦♥❡ss 1✿

❚❤❡♥✱ t❤❡ ♠❛ss ✐s s✐♠♣❧② t❤❡ s✉♠ ♦❢ t❤❡ ♠❛ss ♦❢ t❤❡ ❞✐s❦ ❛♥❞ t❤❡ ♠❛ss ♦❢ t❤❡ ✇❛s❤❡r✿ ▼❛ss

= 2 · ❛r❡❛ ♦❢ t❤❡ ❞✐s❦ =2·π·1

2

+1 · ❛r❡❛ ♦❢ t❤❡ ✇❛s❤❡r +1 · (π · 22 − π · 12 ) .

■t✬s 5π ✳ ❲❡ ❝❛♥ ❥✉st r❡♣❧❛❝❡ t❤❡ ❞✐s❦ t❤❛t ❤❛s ❛ ❝♦♥st❛♥t t❤✐❝❦♥❡ss ❛♥❞ ❛ ✈❛r✐❛❜❧❡ ❞❡♥s✐t② ✇✐t❤ ♦♥❡ t❤❛t ❤❛s ❛ ✈❛r✐❛❜❧❡ t❤✐❝❦♥❡ss ❛♥❞ ❛ ❝♦♥st❛♥t ❞❡♥s✐t②✳ ❚❤❡♥ ✇❡ ❝❛♥ ✉s❡ t❤❡ r❡s✉❧ts ♦❢ t❤❡ ❧❛st s❡❝t✐♦♥✳ ■♥st❡❛❞ ✇❡ st❛rt ❢r♦♠ s❝r❛t❝❤✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ P ♦❢ t❤❡ r❛❞✐✉s✿

a = x0 ≤ c1 ≤ x1 ≤ ... ≤ cn ≤ xn = b ❍❡r❡✱ ✇❡ ❝✉t t❤❡ ❞✐s❦ ✐♥t♦ s♠❛❧❧ ✇❛s❤❡rs ❜② t❤❡ ❝②❧✐♥❞❡rs st❛rt✐♥❣ ❛t x = xi ❛♥❞ t❤❡♥ s❛♠♣❧❡ ✐ts ❞❡♥s✐t② ❛t t❤❡ ♣♦✐♥ts ci ✿

❚❤❡♥ t❤❡ ❞❡♥s✐t② ♦❢ ❡❛❝❤ ✇❛s❤❡r ✕ ✇❤❡♥ ✉♥✐❢♦r♠ ✕ ✐s r(ci ) ❛♥❞ ✇❡ ❤❛✈❡✿

 ▼❛ss ♦❢ it❤ ✇❛s❤❡r = ❞❡♥s✐t② · ❛r❡❛ = r(ci ) · πx2i − πx2i−1 ,

s✐♥❝❡ t❤❡ ✐♥s✐❞❡ r❛❞✐✉s ♦❢ t❤❡ ✇❛s❤❡r ✐s xi−1 ❛♥❞ t❤❡ ♦✉ts✐❞❡ ✐s xi ✳

✸✳✹✳

❚❤❡ r❛❞✐❛❧ ❞❡♥s✐t② ❛♥❞ t❤❡ ♠❛ss

✷✵✸

❚❤❡♥ ✇❡ ❤❛✈❡✿ ▼❛ss ♦❢ t❤❡ ❞✐s❦ =

n X i=1

❚❤✐s ❢♦r♠✉❧❛ ✐s ✜♥❡ ❢♦r ❝♦♠♣✉t❛t✐♦♥s ❜✉t ✐t ✐s

♥♦t

r(ci ) · π x2i − x2i−1



t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛♥② ❢✉♥❝t✐♦♥✦

❆ ❝❧❡✈❡r tr✐❝❦ ✐s t♦ ❝❤♦♦s❡ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s t♦ ❜❡ t❤❡ ♠✐❞✲♣♦✐♥ts✿

1 ci = (xi + xi−1 ) . 2 ❚❤❡♥✱ ✇❡ ❝❛♥ ❢❛❝t♦r t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t✇♦ sq✉❛r❡s ❛♥❞ s✐♠♣❧✐❢②✿ n X

▼❛ss ♦❢ t❤❡ ❞✐s❦ =

i=1

❚❤✐s

r(ci ) · π(xi + xi−1 )(xi − xi−1 ) = 2π

✐s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ s✐♠♣❧❡ ❢✉♥❝t✐♦♥✿

n X i=1

r(ci )ci · ∆xi .

▼❛ss ♦❢ t❤❡ ❞✐s❦ = 2πΣ xr(x) · ∆x .

❞❡♥s✐t② ✈❛r✐❡s ❝♦♥t✐♥✉♦✉s❧② ❄ ❚❤❡♥ t❤❡ ♠❛ss ♦❢ ❡❛❝❤ ✇❛s❤❡r ✕ ✇❤❡♥ t❤✐♥ ❡♥♦✉❣❤ ✕ ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ♠❛ss ♦❢ s✉❝❤ ❛ ✇❛s❤❡r ♠❛❞❡ ❲✐t❤ t❤✐s ❞✐s❝♦✈❡r②✱ ✇❡ ❝❛♥ ❛❞❞r❡ss t❤❡ q✉❡st✐♦♥✿ ❲❤❛t ✐❢ t❤❡ ❡♥t✐r❡❧② ♦❢ ♠❛t❡r✐❛❧ ♦❢ ❞❡♥s✐t② r(ci )✿

 ▼❛ss ♦❢ it❤ ✇❛s❤❡r ≈ ❞❡♥s✐t② · ❛r❡❛ = r(ci ) · πx2i − πx2i−1 .

❚❤❡♥ ✇❡ ❣♦ t❤r♦✉❣❤ t❤❡ s❛♠❡ ❛❧❣❡❜r❛✿ ❚♦t❛❧ ♠❛ss ≈ ❚❤❡♥✱ ✇❡ ❞❡✜♥❡ t❤❡

n X i=1

r(ci ) · π

x2i

−

x2i−1



= 2π

n X i=1

r(ci )ci · ∆xi .

♠❛ss ♦❢ t❤❡ ❞✐s❦ ❛s t❤❡ ❧✐♠✐t ♦❢ t❤❡s❡ ❘✐❡♠❛♥♥ s✉♠s❀ ✐✳❡✳✱ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✿ ❉❡✜♥✐t✐♦♥ ✸✳✹✳✷✿ ♠❛ss ♦❢ ❞✐s❦ ■❢ ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ r ♦♥ s❡❣♠❡♥t [0, b] ✐s ❝❛❧❧❡❞ ❛ r❛❞✐❛❧ ❢♦❧❧♦✇✐♥❣ ✐♥t❡❣r❛❧ ✐s ❝❛❧❧❡❞ t❤❡ ♠❛ss ♦❢ t❤❡ ❞✐s❦ ♦❢ r❛❞✐✉s b✿ ▼❛ss = 2π

Z

❞❡♥s✐t②✱ t❤❡♥ t❤❡

b

xr(x) dx . 0

❖♥❝❡ ❛❣❛✐♥✱ ✇❡ r❡❛❧✐③❡ t❤❛t ❡❛❝❤ ❧♦❝❛t✐♦♥ ✇✐t❤ ❤✐❣❤❡r ❞❡♥s✐t② s✐♠♣❧② ❝♦♥t❛✐♥s ♠♦r❡ ♠❛t❡r✐❛❧ ❛♥❞ ✇❡ ❝❛♥ ❥✉st s♣r❡❛❞ ✐t ♦✉t ✕ ✈❡rt✐❝❛❧❧② ✕ ♠❛❦✐♥❣ t❤❡ ❞✐s❦ t❤✐❝❦❡r ❛t t❤✐s s♣♦t ❛♥❞ t❤✐♥♥❡r ❛t t❤❡ ❧♦❝❛t✐♦♥ ♦❢ ❧♦✇❡r ❞❡♥s✐t②✳

✸✳✺✳ ❚❤❡ ✢♦✇ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ✢✉①

✷✵✹

❊①❛♠♣❧❡ ✸✳✹✳✸✿ ❧✐♥❡❛r ❞❡♣❡♥❞❡♥❝❡

❙✉♣♣♦s❡ t❤❡ ❞❡♥s✐t② ♦❢ ❛ ❞✐s❦ ♦❢ r❛❞✐✉s 2 ✐s ❝❤❛♥❣✐♥❣ ❧✐♥❡❛r❧②✿ ❢r♦♠ 1 t♦ 2✳ ❚❤❡♥ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❛✈❡r❛❣❡ ❞❡♥s✐t② ❞❡♣❡♥❞s ♦♥ t❤❡ r❡s♣❡❝t✐✈❡ ❛r❡❛s✱ ❛s s❤♦✇♥ ❛❜♦✈❡✳

❚❤❡ ♠❛ss ♠✉st ❤❛✈❡ s♦♠❡t❤✐♥❣ t♦ ❞♦ ✇✐t❤ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤✐s s✉r❢❛❝❡ ♦❢ r❡✈♦❧✉t✐♦♥✳✳✳ ▲❡t✬s ✐♥t❡❣r❛t❡✿ ▼❛ss

Z

b

xr(x) dx = 2π Z 2a x(2 − x/2) dx =π 0 Z 2 (2x − x2 /2) dx =π 0 2 2 3 = π(x − x /6) 0

2

3

= π(2 − 2 /6) =

8π . 3

✸✳✺✳ ❚❤❡ ✢♦✇ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ✢✉① ❙✉♣♣♦s❡ ✇❛t❡r ✢♦✇s ✐♥ ❛ ❝❛♥❛❧✿

❍♦✇ ♠✉❝❤ ✇❛t❡r ✐s ❝r♦ss✐♥❣ t❤❡ ❣✐✈❡♥ ❧✐♥❡ ♣❡r ✉♥✐t ♦❢ t✐♠❡❄ ❲❡ ✇✐❧❧ ✐❣♥♦r❡ t❤❡ ❞❡♣t❤ ❛♥❞ ❝♦♥s✐❞❡r t❤✐s ✈✐❡✇ ❢r♦♠ ❛❜♦✈❡✿

❲❤❡♥ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ✇❛t❡r ✐s t❤❡ s❛♠❡ ❛t ❛❧❧ ❧♦❝❛t✐♦♥s✱ t❤❡ t♦t❛❧ ❛♠♦✉♥t ♦❢ t❤❡ ✇❛t❡r t❤❛t ❤❛s ❝r♦ss❡❞

✸✳✺✳ ❚❤❡ ✢♦✇ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ✢✉①

✷✵✺

t❤❡ ❧✐♥❡✱ ❝❛❧❧❡❞ t❤❡ ✢✉① F ✱ ✐s t❤❡ ✈❡❧♦❝✐t② v t✐♠❡s t❤❡ ✇✐❞t❤ W ♦❢ t❤❡ ❝r♦ss✲s❡❝t✐♦♥✿

F =v·W . ❚❤❡ ✢♦✇ ✈❡❧♦❝✐t② ♠❛② ✈❛r② ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❧♦❝❛t✐♦♥ ✭♥♦t t✐♠❡✦✮✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ✈❡❧♦❝✐t② ✐s t❤❡ s❛♠❡ ❛❧♦♥❣ t❤❡ ❧✐♥❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ ✇❛❧❧s ♦❢ t❤❡ ❝❛♥❛❧✳ ❲❡ ✈✐s✉❛❧✐③❡ t❤❡ ♣r♦❝❡ss ❜② ✐♠❛❣✐♥✐♥❣ t❤❛t ❛ ♥❛rr♦✇ str✐♣ ♦❢ r❡❞ ❞②❡ ✐s ❛♣♣❧✐❡❞ ❛❝r♦ss t❤❡ ❝❛♥❛❧ ❛♥❞ t❤❡♥ ❛❢t❡r✱ s❛②✱ ♦♥❡ ♠✐♥✉t❡ ✇❡ s❡❡ ❤♦✇ t❤❡ ❞✐❡ ❤❛s ♣r♦❣r❡ss❡❞✿

❲❤❛t ✐s t❤❡ ✢✉① t❤❡♥❄ ❚♦ ❜❡❣✐♥ ✇✐t❤✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ✢♦✇ ✈❡❧♦❝✐t② ❞❡♣❡♥❞s ♦♥ ❛ s✐♥❣❧❡ ✈❛r✐❛❜❧❡ ✭♦♥❡ ❞❡❣r❡❡ ♦❢ ❢r❡❡❞♦♠ ❛❣❛✐♥✮✱ t❤❡ ❧♦❝❛t✐♦♥ ❞✐st❛♥❝❡ ❛❝r♦ss t❤❡ ❝❛♥❛❧✳ ❚❤❡♥✱ t❤❡r❡ ✐s ❛ ❧✐♥❡ ✕ ✇❡ ❝❤♦♦s❡ ✐t t♦ ❜❡ ✐♥t❡r✈❛❧ [a, b] ♦♥ t❤❡ x✲❛①✐s ✕ ✇✐t❤ ♥♦ ❝❤❛♥❣❡ ✐♥ ✈❡❧♦❝✐t② ✐♥ t❤❡ ❞✐r❡❝t✐♦♥s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ✐t✳ ❚❤❡♥ t❤❡ ✈❡❧♦❝✐t② ✐s ❛ ❢✉♥❝t✐♦♥ y = v(x) ♦❢ ❛ s✐♥❣❧❡ ♥✉♠❜❡r x ✐♥ [a, b]✳ ❙✉♣♣♦s❡ t❤✐s t✐♠❡ t❤❡ ✈❡❧♦❝✐t② v ✐s ❣✐✈❡♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❧♦❝❛t✐♦♥✱ ✇❤❛t ✐s t❤❡ ✢✉①❄ ❊①❛♠♣❧❡ ✸✳✺✳✶✿ t✇♦ ❣❛t❡s

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ s❡♣❛r❛t❡ ❝❛♥❛❧s s✐❞❡ ❜② s✐❞❡✱ ✇✐t❤ t❤❡ ✈❡❧♦❝✐t✐❡s 1 ❛♥❞ 2 ❛♥❞ t❤❡ s❛♠❡ ✇✐❞t❤ 1✿

❚❤❡r❡❢♦r❡✱ t❤❡ ✈♦❧✉♠❡ ✐s s✐♠♣❧② t❤❡ s✉♠ ♦❢ t❤❡ t✇♦✿ 1 · 1 + 2 · 1 = 3✳ ■t ✐s ❛❧s♦ t❤❡ ❛r❡❛ ♦❢ t❤❡ t✇♦ r❡❝t❛♥❣❧❡s ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✈❡❧♦❝✐t② ❢✉♥❝t✐♦♥ v ✱ ✇❤✐❝❤ ✐s ❛ st❡♣✲❢✉♥❝t✐♦♥✱ ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ✐♥t❡❣r❛❧ ♦❢ l ♦✈❡r [0, 2]✳ ■♥st❡❛❞ ♦❢ ❥✉st ♣♦✐♥t✐♥❣ ♦✉t t❤❛t t❤❡ ✢✉① ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ s♣❡❡❞ ✭✇✐t❤ r❡s♣❡❝t t♦ ❧♦❝❛t✐♦♥ ♥♦t t✐♠❡✦✮✱ ❧❡t✬s st❛rt ❢r♦♠ s❝r❛t❝❤✳ ❲❡ ❤❛✈❡ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ P ✿

a = x0 ≤ c1 ≤ x1 ≤ ... ≤ cn ≤ xn = b ❲❡ ✜rst ✐♠❛❣✐♥❡ t❤❛t t❤❡ ❝❛♥❛❧ ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ❝❤❛♥♥❡❧s ♦r ❧♦❝❦s s♦ t❤❛t t❤❡ ✢♦✇ ✈❡❧♦❝✐t② t❤r♦✉❣❤ ❡❛❝❤ ✐s ❢♦✉♥❞ s❡♣❛r❛t❡❧②✿ F1 , F2 , ..., Fn ✿

✸✳✺✳ ❚❤❡ ✢♦✇ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ✢✉①

✷✵✻

❚❤❡♥ t❤❡ t♦t❛❧ ✢♦✇ ✐s s✐♠♣❧② t♦t❛❧ ✈♦❧✉♠❡ = F1 ∆x1 + F2 ∆x2 + ... + Fn ∆xn . ❚❤❡ ❢♦r♠✉❧❛ ✐s s✉✣❝✐❡♥t ❢♦r ♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s✳ ❚♦ ✜♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r ❛♥ ✐❞❡❛❧✐③❡❞ s✐t✉❛t✐♦♥✱ ✇❡ ❝♦♥t✐♥✉❡ ♦♥✳ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✜♥❞ ❛ ❘✐❡♠❛♥♥ s✉♠ ❤❡r❡✳ ❲❡ ✐♠❛❣✐♥❡ t❤❛t t❤❡ ✢♦✇ ✈❡❧♦❝✐t② ✈❛r✐❡s ✐♥❝r❡♠❡♥t❛❧❧② ♦✈❡r t❤❡ ❣❛t❡s t❤❛t ❞✐✈✐❞❡ t❤❡ ❝❛♥❛❧✬s ❝r♦ss✲s❡❝t✐♦♥✳ ❚❤❡ ❝❛♥❛❧ ✐s ❝✉t ✐♥t♦ s❡❣♠❡♥ts ❜② t❤❡ ❧✐♥❡ st❛rt✐♥❣ ❛t x = xi ❛♥❞ s❛♠♣❧❡❞ ✈❡❧♦❝✐t② ❛t t❤❡ ♣♦✐♥ts ci ✐s v(ci )✿

❚❤❡♥ ✇❡ ❤❛✈❡✿ ❚❤❡♥✱

❋❧✉① t❤♦✉❣❤ it❤ s❡❣♠❡♥t = ✈❡❧♦❝✐t② · ✇✐❞t❤ = v(ci ) · ∆xi . ❚♦t❛❧ ✢✉① =

n X i=1

v(ci ) · ∆xi .

❲❡ r❡❝♦❣♥✐③❡ t❤✐s ❡①♣r❡ss✐♦♥ ❛s t❤❡ ❘✐❡♠❛♥♥ s✉♠✱ Σ v · ∆x✱ ♦❢ t❤❡ ✈❡❧♦❝✐t② ❢✉♥❝t✐♦♥ ♦✈❡r t❤✐s ♣❛rt✐t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✸✳✺✳✷✿ ✢♦✇ ✈❡❧♦❝✐t②

■❢ ❛ ❢✉♥❝t✐♦♥ v ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ s❡❣♠❡♥t [a, b] ✐s ❝❛❧❧❡❞ ❛ ✢♦✇ ✈❡❧♦❝✐t②✱ t❤❡♥ ✐ts ❘✐❡♠❛♥♥ s✉♠ ✐s ❝❛❧❧❡❞ t❤❡ ✢✉① ✿ ❋❧✉① = Σ v · ∆x . ❲❤❛t ✐❢ t❤❡ ✢♦✇ ✈❡❧♦❝✐t② ✈❛r✐❡s ❝♦♥t✐♥✉♦✉s❧②❄

❊①❛♠♣❧❡ ✸✳✺✳✸✿ ❧✐♥❡❛r ❞❡♣❡♥❞❡♥❝❡ ❙✉♣♣♦s❡ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❝❛♥❛❧ ♦❢ ✇✐❞t❤ 2 ✐s ❝❤❛♥❣✐♥❣ ❧✐♥❡❛r❧②✿ ❢r♦♠ 1 t♦ 2✳ ❚❤❡♥ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t② ✐s ❝❧❡❛r❀ ✐t ✐s 1.5✳

❚❤❡r❡❢♦r❡✱ t❤❡ ✈♦❧✉♠❡ ✐s 1.5 · 1 = 1.5✳ ■t ✐s ❛❧s♦ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✈❡❧♦❝✐t② ❢✉♥❝t✐♦♥ v(x) = 1 + x ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ✐♥t❡❣r❛❧ ♦❢ v ♦✈❡r [0, 2]✳ ❚❤❡♥ t❤❡ ✢✉① t❤r♦✉❣❤ ❡❛❝❤ s❡❣♠❡♥t ✕ ✇❤❡♥ s❤♦rt ❡♥♦✉❣❤ ✕ ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ✈♦❧✉♠❡ ✇✐t❤ t❤❡ ✇❛t❡r ♠♦✈✐♥❣ ❡♥t✐r❡❧② ❛t t❤❡ ✈❡❧♦❝✐t② v(ci )✿ ❱♦❧✉♠❡ ♦❢ it❤ s❡❣♠❡♥t ≈ ✈❡❧♦❝✐t② · ✇✐❞t❤ = v(ci ) · ∆xi .

✸✳✺✳

❚❤❡ ✢♦✇ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ✢✉①

✷✵✼

❚❤❡♥✱ ❋❧✉①

❲❡ ❞❡✜♥❡ t❤❡

✢✉①

= ❚♦t❛❧

✈♦❧✉♠❡

≈

n X i=1

v(ci ) · ∆xi .

♦❢ t❤❡ r♦❞ ❛s t❤❡ ❧✐♠✐t✱ ✐❢ ✐t ❡①✐sts✱ ♦❢ t❤❡s❡ ❘✐❡♠❛♥♥ s✉♠s✱ ✐✳❡✳✱ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♦❢

v✳

❉❡✜♥✐t✐♦♥ ✸✳✺✳✹✿ ✢✉① ■❢ ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥

v

♦♥ s❡❣♠❡♥t

t❤❡♥ ✐ts ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✐s ❝❛❧❧❡❞ t❤❡

❋❧✉①

=

[a, b]

✢✉① Z

✐s ❝❛❧❧❡❞ ❛

✢♦✇ ✈❡❧♦❝✐t②

♦❢ t❤❡ ✢♦✇✿

b

v dx . a

❍❡r❡ ✐s ❛♥♦t❤❡r ✇❛② t♦ ❡①♣❧❛✐♥ t❤✐s r❡s✉❧t✳ ❲❡ ❝❛♥ t❛❦❡ ♦✉r ❝❛♥❛❧✱ ✇✐t❤ ❛ ✈❛r✐❛❜❧❡ ✇❛t❡r

❝♦♥st❛♥t ✈❡❧♦❝✐t② ❛s ♦♥❡ t❤❛t ❤❛s ♠♦r❡ ✇❛t❡r✳ ❛ ❝❛♥❛❧ ✇✐t❤ t❤❡ s❛♠❡ ✢✉① ❜✉t ❛

✐♥ ❛ ❝❛♥❛❧✱

✈❡❧♦❝✐t②✱ ❛♥❞ ✐♠❛❣✐♥❡

✈❡❧♦❝✐t②✳ ❍♦✇ ✐s ✐t ♣♦ss✐❜❧❡❄ ❲❡ t❤✐♥❦ ♦❢ ❡❛❝❤ ❧♦❝❛t✐♦♥ ✇✐t❤ ❤✐❣❤❡r

❚❤❡ ✜rst ❛♣♣r♦❛❝❤ ✐s t♦ s♣r❡❛❞ t❤❡ ✇❛t❡r ♦✉t ✕ ✈❡rt✐❝❛❧❧② ✕ ♠❛❦✐♥❣ t❤❡ ❝❛♥❛❧

❞❡❡♣❡r

❛t t❤✐s s♣♦t ❛♥❞ s❤❛❧❧♦✇❡r

❛t t❤❡ ❧♦❝❛t✐♦♥ ✇✐t❤ ❛ ❧♦✇❡r ✈❡❧♦❝✐t②✿

❚❤❡ s❡❝♦♥❞ ❛♣♣r♦❛❝❤ ✐s t♦ t❤✐♥❦ ♦❢ ❡❛❝❤ ❧♦❝❛t✐♦♥ ✇✐t❤ ❤✐❣❤❡r ✈❡❧♦❝✐t② ❛s s✐♠♣❧② ♦♥❡ ✇✐t❤

❞❡♥s❡r

❧✐q✉✐❞✳

❊①❡r❝✐s❡ ✸✳✺✳✺ ❲❤❛t ✐❢ t❤✐s ✐s ❛♥ ♦❝❡❛♥✱ ✐✳❡✳✱ t❤❡ ❝r♦ss✲s❡❝t✐♦♥ ♦❢ ♦✉r ✏❝❛♥❛❧✑ ✐s ✐♥✜♥✐t❡❧② ✇✐❞❡❄

❆ ✈❛r✐❛t✐♦♥ ♦❢ t❤✐s ❛♥❛❧②s✐s ✐s ❛s ❢♦❧❧♦✇s✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t t❤❡ ✇❛t❡r ✢♦✇s t❤r♦✉❣❤ ❛

❙✉♣♣♦s❡ t❤❡ ✢♦✇ ✈❡❧♦❝✐t② ✈❛r✐❡s ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡

♣✐♣❡ ✿

❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥ t♦ t❤❡ ♣✐♣❡✬s ✇❛❧❧✳

❋♦r ❡①❛♠♣❧❡✱

t❤❡ ✇❛t❡r ♠❛② ❣♦ s❧♦✇❡r ♥❡①t t♦ t❤❡ ✇❛❧❧ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❢r✐❝t✐♦♥✳ ❲❡ ❤❛✈❡ ❛ ❝✐r❝✉❧❛r ♣❛tt❡r♥ ❛❣❛✐♥✳✳✳

❉❡✜♥✐t✐♦♥ ✸✳✺✳✻✿ ✢✉① ■❢ ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ ♣✐♣❡ ♦❢ r❛❞✐✉s

R✱

v

♦♥ s❡❣♠❡♥t

[0, R]

t❤❡♥ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

✐s ❝❛❧❧❡❞ ❛ Z R

✢♦✇ ✈❡❧♦❝✐t②

xv(x) dx

2π

0

t❤r♦✉❣❤ ❛

✐s ❝❛❧❧❡❞ t❤❡

✢✉①✳

✸✳✻✳

❚❤❡ ❢♦r❝❡ ❛♥❞ t❤❡ ✇♦r❦

✷✵✽

❊①❡r❝✐s❡ ✸✳✺✳✼

❋♦❧❧♦✇✐♥❣ t❤❡ ✐❞❡❛s ❞❡✈❡❧♦♣❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✱ ❥✉st✐❢② t❤❡ ❛❜♦✈❡ ❞❡✜♥✐t✐♦♥✳

✸✳✻✳ ❚❤❡ ❢♦r❝❡ ❛♥❞ t❤❡ ✇♦r❦

❙✉♣♣♦s❡ ❛ ❜❛❧❧ ✐s ❞r♦♣♣❡❞ ♦♥ t❤❡ ❣r♦✉♥❞ ❢r♦♠ ❛ ❝❡rt❛✐♥ ❤❡✐❣❤t✿

❚❤✐s ♣❤❡♥♦♠❡♥♦♥ ✐s t❤❡ r❡s✉❧t ♦❢ t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❢♦r❝❡✳ ❚❤✐s ❢♦r❝❡ ✐s ❞✐r❡❝t❡❞ ❞♦✇♥✱ ❥✉st ❛s t❤❡ ♠♦✈❡♠❡♥t ♦❢ t❤❡ ❜❛❧❧✳ ❚❤❡ ✇♦r❦ ❞♦♥❡ ♦♥ t❤❡ ❜❛❧❧ ❜② t❤✐s ❢♦r❝❡ ❛s ✐t ❢❛❧❧s ✐s ❡q✉❛❧ t♦ t❤❡ ✭s✐❣♥❡❞✮ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❢♦r❝❡✱ ✐✳❡✳✱ t❤❡ ✇❡✐❣❤t ♦❢ t❤❡ ❜❛❧❧✱ ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ ✭s✐❣♥❡❞✮ ❞✐st❛♥❝❡ t♦ t❤❡ ❣r♦✉♥❞✱ ✐✳❡✳✱ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✳ ❆❧❧ ❤♦r✐③♦♥t❛❧ ♠♦t✐♦♥ ✐s ✐❣♥♦r❡❞ ❛s ✉♥r❡❧❛t❡❞ t♦ t❤❡ ❣r❛✈✐t②✳ ❚❤❡ ♥❡❡❞ ❢♦r ✉s✐♥❣ t❤❡ s✐❣♥❡❞ ❞✐st❛♥❝❡ D ❛♥❞ ❢♦r❝❡ F ✐s r❡✈❡❛❧❡❞ ❜② t❤❡ ❡①❛♠♣❧❡ ♦❢ ♠♦✈✐♥❣ ❛♥ ♦❜❥❡❝t ✉♣ ❢r♦♠ t❤❡ ❣r♦✉♥❞✳ ❚❤❡♥ t❤❡ ✇♦r❦ W ♣❡r❢♦r♠❡❞ ❜② t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❢♦r❝❡ ✐s ♥❡❣❛t✐✈❡✦

❖❢ ❝♦✉rs❡✱ t❤❡ s✐❣♥ ✐♥ ❡✐t❤❡r ❝❛s❡ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❛①✐s ✇❡ ❛ss✐❣♥ t♦ t❤❡ ❧✐♥❡ ♦❢ ♠♦t✐♦♥✳ ❙✉♣♣♦s❡ ✇❡ ❛r❡ t♦ ♠♦✈❡ ❢r♦♠ ♣♦✐♥t a ♦♥ t❤❡ x✲❛①✐s t♦ ♣♦✐♥t b > a✳ ❲❤❡♥ t❤❡ ❢♦r❝❡ F ✐s ❝♦♥st❛♥t✱ t❤❡ ✇♦r❦ W ✐s ❡q✉❛❧ t♦ t❤❡ ❢♦r❝❡ F t✐♠❡s t❤❡ ❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❜❡t✇❡❡♥ a ❛♥❞ b✿ W = F · (b − a) .

❚❤❡ ❢♦r❝❡ ♠❛② ✈❛r② ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❧♦❝❛t✐♦♥ ❜❡t✇❡❡♥ a ❛♥❞ b✳ ❊①❛♠♣❧❡ ✸✳✻✳✶✿ ♣❤②s✐❝s

❚❤❡ ❡①❛♠♣❧❡s ♦❢ ✈❛r✐❛❜❧❡ ❢♦r❝❡s ♠❛② ❜❡ t❤❡s❡✿ s♣r✐♥❣✱ ❣r❛✈✐t❛t✐♦♥✱ ❛✐r ♣r❡ss✉r❡✳

✸✳✻✳

✷✵✾

❚❤❡ ❢♦r❝❡ ❛♥❞ t❤❡ ✇♦r❦

■♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ♦❜❥❡❝t ❛tt❛❝❤❡❞ t♦ ❛ s♣r✐♥❣✱ t❤❡ ❢♦r❝❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✭s✐❣♥❡❞✮ ❞✐st❛♥❝❡ ♦❢ t❤❡ ♦❜❥❡❝t t♦ ✐ts ❡q✉✐❧✐❜r✐✉♠ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❍♦♦❦❡✬s ▲❛✇ ✿

F (x) = −kx .

❆✇❛② ❢r♦♠ t❤❡ ❣r♦✉♥❞✱ t❤❡ ❣r❛✈✐t② ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ r❡❝✐♣r♦❝❛❧ ♦❢ t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ ♦❜❥❡❝t t♦ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ♣❧❛♥❡t ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ◆❡✇t♦♥✬s ▲❛✇ ♦❢ ●r❛✈✐t❛t✐♦♥ ✿

F (x) = − ❚❤❡

♣r❡ss✉r❡

k . x2

❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ♠❡❞✐✉♠✬s r❡s✐st❛♥❝❡ t♦ ♠♦t✐♦♥ ♠❛② ❝❤❛♥❣❡ ❛r❜✐tr❛r✐❧②✳

❊①❛♠♣❧❡ ✸✳✻✳✷✿ tr❛❝t✐♦♥

❙✉♣♣♦s❡ t❤❡ ❢♦r❝❡ ✐s ♦t❤❡r r♦✉❣❤❡r✳

✳ ❙✉♣♣♦s❡ t❤❛t t❤❡r❡ ❛r❡ t✇♦ ❞✐st✐♥❝t str✐♣s✿ ♦♥❡ ✐s s♠♦♦t❤❡r ❛♥❞ t❤❡

tr❛❝t✐♦♥

❚❤❡ ❢♦r❝❡ t❛❦❡s ✕ ❜❡t✇❡❡♥ a = 0 ❛♥❞ b = 2 ✕ ♦♥❧② t✇♦ ❞✐✛❡r❡♥t ✈❛❧✉❡s 1 ❛♥❞ 2 s✇✐t❝❤✐♥❣ ❛t c = 1✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✇♦r❦ ✐s s✐♠♣❧② t❤❡ s✉♠ ♦❢ t❤❡ t✇♦ ♦✈❡r ❡✐t❤❡r ♦❢ t❤❡ s❡❣♠❡♥ts✿ 1 · 1 + 2 · 1 = 3✳ ■t ✐s ❛❧s♦ t❤❡ ❛r❡❛ ♦❢ t❤❡ t✇♦ r❡❝t❛♥❣❧❡s ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢♦r❝❡ ❢✉♥❝t✐♦♥ F ✱ ✇❤✐❝❤ ✐s ❛ st❡♣✲❢✉♥❝t✐♦♥✱ ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ✐♥t❡❣r❛❧ ♦❢ l ♦✈❡r [0, 2]✳ ❋♦r t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✱ ✇❡ ❤❛✈❡ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ P ✿

a = x0 ≤ c1 ≤ x1 ≤ ... ≤ cn ≤ xn = b

✸✳✻✳

❚❤❡ ❢♦r❝❡ ❛♥❞ t❤❡ ✇♦r❦

✷✶✵

❚❤❡ ♣❛t❤ ✐s ❞✐✈✐❞❡❞ ✐♥t♦ s♠❛❧❧ s❡❣♠❡♥ts ❜②

x = xi ❛♥❞ t❤❡♥ t❤❡ ❢♦r❝❡ F (ci ) ❛♥❞ ✇❡ ❤❛✈❡✿

✐s s❛♠♣❧❡❞ ❛t t❤❡ ♣♦✐♥ts

ci ✳

❚❤❡♥ t❤❡

❢♦r❝❡ ♦♥ ❡❛❝❤ s❡❣♠❡♥t ✕ ✐❢ ❝♦♥st❛♥t ✕ ✐s ❡q✉❛❧ t♦ ❲♦r❦ ♦♥

it❤

s❡❣♠❡♥t

=

❢♦r❝❡

❚❤❡♥✱

=

❚♦t❛❧ ✇♦r❦

❖♥❝❡ ❛❣❛✐♥✱ ✇❡

r❡❝♦❣♥✐③❡

n X i=1

·

❧❡♥❣t❤

= F (ci ) · ∆xi .

F (ci ) · ∆xi .

t❤✐s ❡①♣r❡ss✐♦♥ ❛s t❤❡ ❘✐❡♠❛♥♥ s✉♠✱

♣❛rt✐t✐♦♥✳

Σ F · ∆x✱

♦❢ t❤❡ ❢♦r❝❡ ❢✉♥❝t✐♦♥ ♦✈❡r t❤✐s

❉❡✜♥✐t✐♦♥ ✸✳✻✳✸✿ ✇♦r❦ ■❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞ ❛

F

✐s ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛ s❡❣♠❡♥t

❢♦r❝❡ ❢✉♥❝t✐♦♥✱ t❤❡♥ ✐ts ❘✐❡♠❛♥♥ s✉♠ ✐s ❝❛❧❧❡❞ t❤❡ ✇♦r❦

♦✈❡r ✐♥t❡r✈❛❧

[a, b]

♦❢ t❤❡ ❢♦r❝❡

[a, b]✿

❲♦r❦

= Σ F · ∆x .

❲❤❛t ✐❢ t❤❡ ❢♦r❝❡ ✈❛r✐❡s ✏❝♦♥t✐♥✉♦✉s❧②✑❄

❊①❛♠♣❧❡ ✸✳✻✳✹✿ ❧✐♥❡❛r ❞❡♣❡♥❞❡♥❝❡ ❙✉♣♣♦s❡ t❤❡ ❢♦r❝❡ ✐s ❝❤❛♥❣✐♥❣ ❧✐♥❡❛r❧② ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ ❛✈❡r❛❣❡ ❢♦r❝❡ ✐s ❝❧❡❛r❀ ✐t ✐s

[0, 2]✿

❢r♦♠

1

t♦

2✳

❚❤❡♥ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡

1.5✳

1.5 · 2 = 3✳ ■t ✐s ❛❧s♦ t❤❡ ❛r❡❛ F (x) = 1 + x/2 ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ✐♥t❡❣r❛❧ ♦❢

❚❤❡r❡❢♦r❡✱ t❤❡ ✇♦r❦ ✐s

♦❢ t❤❡ tr✐❛♥❣❧❡ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢♦r❝❡

❢✉♥❝t✐♦♥

t❤✐s ❢✉♥❝t✐♦♥ ♦✈❡r

[0, 2]✳

❲❤❡♥ t❤❡ ❝❤❛♥❣❡ ♦❢

t❤❡ ❢♦r❝❡ ✐s ♥♦♥✲❧✐♥❡❛r✱ t❤❡ ❛r❣✉♠❡♥t ❢❛✐❧s✳

❚❤❡ ✇♦r❦ ♦♥ ❡❛❝❤ s❡❣♠❡♥t ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ✇♦r❦ ✇✐t❤ t❤❡ ❢♦r❝❡ ❜❡✐♥❣ ❝♦♥st❛♥t❧② ❡q✉❛❧ t♦ ❲♦r❦ ♦♥

it❤

s❡❣♠❡♥t

≈

❢♦r❝❡

❚❤❡♥✱ ❚♦t❛❧ ✇♦r❦

≈

n X i=1

·

❧❡♥❣t❤

F (ci )✿

= F (ci ) · ∆xi .

F (ci ) · ∆xi .

❲❡ ❞❡✜♥❡ t❤❡ ✇♦r❦ ♦❢ t❤❡ ❢♦r❝❡ ❛s t❤❡ ❧✐♠✐t✱ ✐❢ ✐t ❡①✐sts✱ ♦❢ t❤❡s❡ ❘✐❡♠❛♥♥ s✉♠s✱ ✐✳❡✳✱ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♦❢

F✳

❉❡✜♥✐t✐♦♥ ✸✳✻✳✺✿ ✇♦r❦ ■❢ ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥

F

♦♥ s❡❣♠❡♥t

❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✐s ❝❛❧❧❡❞ t❤❡

✇♦r❦

❲♦r❦

[a, b]

✐s ❝❛❧❧❡❞ ❛

❢♦r❝❡ ❢✉♥❝t✐♦♥✱ t❤❡♥ ✐ts

♦❢ t❤❡ ❢♦r❝❡ ♦✈❡r ✐♥t❡r✈❛❧

=

Z

b

F dx . a

[a, b]✿

✸✳✻✳

❚❤❡ ❢♦r❝❡ ❛♥❞ t❤❡ ✇♦r❦

✷✶✶

❊①❡r❝✐s❡ ✸✳✻✳✻

❍♦✇ ♠✉❝❤ ✇♦r❦ ❞♦❡s ✐t t❛❦❡ t♦ ♠♦✈❡ ❛♥ ♦❜❥❡❝t ❛tt❛❝❤❡❞ t♦ ❛ s♣r✐♥❣ s ✉♥✐ts ❢r♦♠ t❤❡ ❡q✉✐❧✐❜r✐✉♠❄ ❊①❡r❝✐s❡ ✸✳✻✳✼

❍♦✇ ♠✉❝❤ ✇♦r❦ ❞♦❡s ✐t t❛❦❡ t♦ ♠♦✈❡ ❛♥ ♦❜❥❡❝t s ✉♥✐ts ❢r♦♠ t❤❡ ❝❡♥t❡r ♦❢ ❛ ♣❧❛♥❡t❄ ❆s ❛ s✉♠♠❛r②✱ ✇❡ ❤❛✈❡ s♦❧✈❡❞ t❤❡ ♣r♦❜❧❡♠ ♦❢ ✜♥❞✐♥❣ ❛ ❝❡rt❛✐♥ q✉❛♥t✐t② W ✕ ✇♦r❦✱ ✢♦✇✱ ❛♥❞ ♠❛ss ✕ ✐♥ ❛♥ ✐❞❡♥t✐❝❛❧ ♠❛♥♥❡r✳ ❲❡ ❤❛✈❡ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ P ✿

a = x0 ≤ c1 ≤ x1 ≤ ... ≤ cn ≤ xn = b ❲❡ ❞✐✈✐❞❡ t❤❡ ♣❛t❤ ✐♥t♦ s♠❛❧❧ s❡❣♠❡♥ts ❜② x = xi ❛♥❞ t❤❡♥ s❛♠♣❧❡ q✉❛♥t✐t② F ✕ t❤❡ ❢♦r❝❡✱ ♦r t❤❡ ✢♦✇ s♣❡❡❞✱ ♦r t❤❡ ❧✐♥❡❛r ❞❡♥s✐t② ✕ ❛t t❤❡ ♣♦✐♥ts ci ✿

❚❤❡♥ t❤✐s q✉❛♥t✐t②✱ F (ci )✱ ♦♥ ❡❛❝❤ s❡❣♠❡♥t ✐s ✉s❡❞ t♦ ✜♥❞ t❤❡ ✈❛❧✉❡ ♦❢ W ✿

Wi = F (ci ) · ∆xi .

❚❤❡♥✱ t❤❡ t♦t❛❧ ❛♣♣r♦①✐♠❛t❡❞ ✈❛❧✉❡ ♦❢ W ♦✈❡r t❤❡ ✇❤♦❧❡ s❡❣♠❡♥t ✐s

X i

Wi =

n X i=1

F (ci ) · ∆xi ,

✇❤✐❝❤ ✐s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ F ♦✈❡r t❤✐s ♣❛rt✐t✐♦♥✳ ❚❤❡ ❡①❛❝t ✈❛❧✉❡ ♦❢ t❤❡ t♦t❛❧ ♦❢ W ✐s t❤❡ ❧✐♠✐t ♦❢ t❤❡s❡ ❘✐❡♠❛♥♥ s✉♠s✱ ✐✳❡✳✱ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡r✈❛❧ ♦❢ F ✿ Z b F dx . W = a

❲❛r♥✐♥❣✦

❈♦♥tr❛r② t♦ t❤❡ ✉♥✐✜❡❞ ❛♣♣r♦❛❝❤ ♣r❡s❡♥t❡❞ ❤❡r❡✱ t❤❡ tr❡❛t♠❡♥ts ♦❢ t❤❡ t❤r❡❡ ✐♥t❡❣r❛❧s ❛r❡ s✉❜st❛♥✲ t✐❛❧❧② ❞✐✛❡r❡♥t ✐♥ ❞✐♠❡♥s✐♦♥ 3 ✭❈❤❛♣t❡r ✹❍❉✲✺ ❛♥❞ ✹❍❉✲✻✮✿ • ❚❤❡ ✇♦r❦ ✐s ❛♥ ✐♥t❡❣r❛❧ ♦✈❡r ❛ ❝✉r✈❡✳ • ❚❤❡ ✢♦✇ ✐s ❛♥ ✐♥t❡❣r❛❧ ♦✈❡r ❛ s✉r❢❛❝❡✳ • ❚❤❡ ♠❛ss ✐s ❛♥ ✐♥t❡❣r❛❧ ♦✈❡r ❛ s♦❧✐❞✳

✸✳✻✳

❚❤❡ ❢♦r❝❡ ❛♥❞ t❤❡ ✇♦r❦

✷✶✷

❲❡ ♥♦✇ ❝♦♥s✐❞❡r ❛ ❞✐✛❡r❡♥t s❡t✉♣✳✳✳ ❲❡ ❛rr✐✈❡❞ ❛t t❤❡ ✐♥t❡❣r❛❧ ❢♦r♠✉❧❛ ❛❜♦✈❡ ❜❡❝❛✉s❡ ♦❢ ❛ s✐♠♣❧❡ ✭✏❛❞❞✐t✐✈❡✑✮ ♣r♦♣❡rt② ♦❢ ✇♦r❦✿

◮

❲❤❡♥ t❤❡r❡ ❛r❡ t✇♦ s❡❣♠❡♥ts ♦❢ t❤❡ tr✐♣✱ t❤❡ ✇♦r❦ t♦ ♠♦✈❡ t❤r♦✉❣❤ t❤❡ t✇♦ ✐s ❡q✉❛❧ t♦ t❤❡

✇♦r❦ r❡q✉✐r❡❞ t♦ ♠♦✈❡ t❤r♦✉❣❤ t❤❡ ✜rst ♣❧✉s t❤❡ ✇♦r❦ r❡q✉✐r❡❞ t♦ ♠♦✈❡ t❤r♦✉❣❤ t❤❡ s❡❝♦♥❞✳ ❲❡ ❛r❡ t♦ ❝♦♥s✐❞❡r ❛ s✐t✉❛t✐♦♥ ✇❤❡♥

◮

❚✇♦ ♦❜❥❡❝ts✱ ♣♦ss✐❜❧② ✐❞❡♥t✐❝❛❧✱ ✉♥❞❡r ❛ ❢♦r❝❡✱ ♣♦ss✐❜❧② ❝♦♥st❛♥t✱ ❤❛✈❡ t♦ ❜❡ ♠♦✈❡❞

❞✐st❛♥❝❡s✳

❞✐✛❡r❡♥t

❚❤❡♥ t❤❡r❡ ✐s ♥♦ s✉❝❤ ❛ s❤♦rt❝✉t ❢♦r♠✉❧❛✳ ❊①❛♠♣❧❡ ✸✳✻✳✽✿ ❜r✐❝❦s

❆♥ ❡①❛♠♣❧❡ ♦❢ s✉❝❤ ❛ t❛s❦ ✐s st❛❝❦✐♥❣ ❜r✐❝❦s✿

❚❤❡♥ t❤❡ ✇♦r❦ ✕ ♦❢ t❤❡ ♣❡rs♦♥ ❛❝t✐♥❣

❛❣❛✐♥st

t❤❡ ❣r❛✈✐t② ✕ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

W = M · 0 · h + M · 1 · h + M · 2 · h + M · 3 · h, ✇❤❡r❡

M

✐s t❤❡ ✇❡✐❣❤t ♦❢ t❤❡ ❜r✐❝❦ ❛♥❞

h

✐s ✐ts ❤❡✐❣❤t✳

❋✉rt❤❡r♠♦r❡✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ♣♦ss✐❜✐❧✐t② t❤❛t t❤❡ ❢♦r❝❡ ✐s ❝♦♥st❛♥t ❜✉t t❤❡ ♦❜❥❡❝t ❝❛♥✬t ❜❡ t❤♦✉❣❤t ♦❢ ❛s ❛ ♣♦✐♥t ❛♥②♠♦r❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❞✐✛❡r❡♥t ♣❛rts ♦❢ t❤❡ ♦❜❥❡❝t ✇✐❧❧ tr❛✈❡❧ ❞✐✛❡r❡♥t ❞✐st❛♥❝❡s✳ ❚❤✐s s✐t✉❛t✐♦♥ ✐s♥✬t ❝♦✈❡r❡❞ ❜② t❤❡ ❛❜♦✈❡ ❞❡✜♥✐t✐♦♥ ♦❢ ✇♦r❦✳ ❊①❛♠♣❧❡ ✸✳✻✳✾✿ ❝✉❜✐❝❛❧ t❛♥❦

❙✉♣♣♦s❡ ✇❡ ❛r❡ t♦ ✜❧❧ ❛ t❛♥❦ ✇✐t❤

w×w

❜❛s❡ ❛♥❞ ❤❡✐❣❤t

❲❤❛t ✐s t❤❡ ✇♦r❦ r❡q✉✐r❡❞ ❛ss✉♠✐♥❣ t❤❛t t❤❡ ❞❡♥s✐t② ✐s

h

✇✐t❤ ✇❛t❡r ✕ ❢r♦♠ t❤❡ ❜♦tt♦♠✿

1❄

❲❡ ✐♠❛❣✐♥❡ t❤❛t ✇❛t❡r ❛♣♣❡❛rs ❛t t❤❡ ❜♦tt♦♠ ✐♥ t❤✐♥ s❧✐❝❡s ❛♥❞ t❤❡♥ ❡❛❝❤ ✐s ❞❡❧✐✈❡r❡❞ t♦ t❤❡ ❛♣♣r♦♣r✐❛t❡

[0, h]✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ x✲❛①✐s ✐s ✈❡rt✐❝❛❧✳ ❚❤❡ it❤ s❧✐❝❡ ✐s ❛ sq✉❛r❡ ❜❡t✇❡❡♥ t❤❡ ♣❧❛♥❡s x = xi−1 ❛♥❞ x = xi ✳ ■ts t❤✐❝❦♥❡ss ✐s ∆xi = xi − xi−1 ❛♥❞ 2 ✐ts ✇❡✐❣❤t ✐s w · ∆xi ✳ ◆♦✇✱ t❤❡ it❤ s❧✐❝❡ ✐s ❞❡❧✐✈❡r❡❞ t♦ ❤❡✐❣❤t ci ✳ ❚❤❡ ✇♦r❦ t♦ ❞♦ s♦ ✐s ❤❡✐❣❤t✳ ❚❤❡② ❝♦♠❡ ❢r♦♠ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥

P

♦❢

w2 ∆xi · ci .

✸✳✻✳

❚❤❡ ❢♦r❝❡ ❛♥❞ t❤❡ ✇♦r❦

✷✶✸

❚❤❡♥ t❤❡ t♦t❛❧ ✇♦r❦ ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜②✿ ❲♦r❦ ≈ ❚❤✐s ✐s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ ✐♥t❡❣r❛❧✿ ❲♦r❦ = w

2

n X

w2 ci · ∆xi .

Z

x dx = w2

i=1

h 0

h2 . 2

❚❤❡ r❡s✉❧t ♠❛t❝❤❡s t❤❡ ✐❞❡❛ t❤❛t t❤❡ ✇♦r❦ r❡q✉✐r❡❞ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ✇♦r❦ t♦ ♠♦✈❡ t❤❡ ✇❤♦❧❡ ❛♠♦✉♥t ♦❢ ✇❛t❡r✱ ✈♦❧✉♠❡ w2 h✱ ❢r♦♠ t❤❡ ❜♦tt♦♠ t♦ t❤❡ ❛✈❡r❛❣❡ ❤❡✐❣❤t ✇✐t❤✐♥ t❤❡ t❛♥❦✱ h/2✳

❊①❡r❝✐s❡ ✸✳✻✳✶✵

❙✉♣♣♦s❡ ✇❡ ❛r❡ t♦ ✜❧❧ ❛ ❝②❧✐♥❞r✐❝❛❧ t❛♥❦ ✇✐t❤ ❜❛s❡ ♦❢ r❛❞✐✉s R ❛♥❞ ❤❡✐❣❤t h ✇✐t❤ ✇❛t❡r ❢r♦♠ t❤❡ ❜♦tt♦♠✳ ❲❤❛t ✐s t❤❡ ✇♦r❦ r❡q✉✐r❡❞❄ ❊①❡r❝✐s❡ ✸✳✻✳✶✶

❲❤❛t ✐❢ t❤❡ ❤♦r✐③♦♥t❛❧ ❝r♦ss✲s❡❝t✐♦♥s ♦❢ t❤❡ t❛♥❦ ❤❛✈❡ ❛r❜✐tr❛r② ✭❜✉t ✐❞❡♥t✐❝❛❧✮ s❤❛♣❡❄ ❊①❡r❝✐s❡ ✸✳✻✳✶✷

❙✉♣♣♦s❡ ❛ ❝❤❛✐♥ ♦❢ ✇❡✐❣❤t M ❛♥❞ ❧❡♥❣t❤ h ✐s t♦ ❜❡ ♣✉❧❧❡❞ ❛❧❧ t❤❡ ✇❛② ✉♣ ❢r♦♠ t❤❡ ❣r♦✉♥❞❄ ❲❤❛t ✐s t❤❡ ✇♦r❦ r❡q✉✐r❡❞❄ ■♥ t❤❡ ❡①❛♠♣❧❡s ❛❜♦✈❡✱ t❤❡ ✇♦r❦ ✐s r❡♣❡t✐t✐✈❡✳ ❲❤❛t ✐❢ t❤❡ ❝r♦ss✲s❡❝t✐♦♥ ✈❛r✐❡s ✐♥ s❤❛♣❡ ❛♥❞ s✐③❡❄ ❊①❛♠♣❧❡ ✸✳✻✳✶✸✿ s♣❤❡r✐❝❛❧ t❛♥❦

❙✉♣♣♦s❡ ✇❡ ❛r❡ t♦ ✜❧❧ ❛ s♣❤❡r✐❝❛❧ t❛♥❦ ♦❢ r❛❞✐✉s R ✇✐t❤ ✇❛t❡r ❢r♦♠ t❤❡ ❜♦tt♦♠✿

❲❤❛t ✐s t❤❡ ✇♦r❦ r❡q✉✐r❡❞❄ ❲❡ ✐♠❛❣✐♥❡ t❤❛t ✇❛t❡r ❛♣♣❡❛rs ❛t t❤❡ ❜♦tt♦♠ ✐♥ t❤✐♥ s❧✐❝❡s ❛♥❞ t❤❡♥ ❡❛❝❤ ✐s ❞❡❧✐✈❡r❡❞ t♦ t❤❡

✸✳✻✳

❚❤❡ ❢♦r❝❡ ❛♥❞ t❤❡ ✇♦r❦

❛♣♣r♦♣r✐❛t❡ ❤❡✐❣❤t✳

it❤

P ♦❢ [−R, R]✳ ❚❤❡ it❤ s❧✐❝❡ ✐s ❛ ❞✐s❦ ∆xi = xi − xi−1 ✱ r❛❞✐✉s ri ✭t♦ ❜❡ ❢♦✉♥❞✮✱ ❛♥❞

❚❤❡② ❝♦♠❡ ❢r♦♠ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥

❜❡t✇❡❡♥ t❤❡ ♣❧❛♥❡s x = 2 ✐ts ✇❡✐❣❤t ✐s πri · ∆xi ✳ ◆♦✇✱ t❤❡

✷✶✹

xi−1

❛♥❞

x = xi ✳

■ts t❤✐❝❦♥❡ss ✐s

s❧✐❝❡ ✐s ❞❡❧✐✈❡r❡❞ t♦ ❧♦❝❛t✐♦♥

❞✐s♣❧❛❝❡♠❡♥t ✐s✱ t❤❡r❡❢♦r❡✱

R + ci

ci

✭❞❡♣✐❝t❡❞ ♥❡❣❛t✐✈❡✮✱ ❝♦✈❡r✐♥❣ t❤❡ ✐♥t❡r✈❛❧

[−R, ci ]✳

❚❤❡

❛♥❞ t❤❡ ✇♦r❦ t♦ ❞♦ s♦ ✐s

πri2 ∆xi · (R + ci ) . ❚❤❡♥ t❤❡ t♦t❛❧ ✇♦r❦ ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿

❲♦r❦

≈

n X i=1

πri2 (R + ci ) · ∆xi .

▲❡t✬s ✜♥❞ t❤❡ r❛❞✐✉s ♦❢ t❤❡ s❧✐❝❡✳ ❋r♦♠ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✱ ✇❡ ❤❛✈❡✿

ri2 = R2 − c2i . ❚❤❡♥ t❤❡ ❛❜♦✈❡ ❡①♣r❡ss✐♦♥ ✐s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡❣r❛❧✿

✇♦r❦

=π =π

Z

R

Z−R R

−R

(R2 − x2 )(R + x) dx  R3 − x2 R + R2 x − x3 dx

 R 1 1 1 = π R 3 x − x3 R + R 2 x2 − x4 3 2 4   −R  1 4 1 4 1 4 1 4 4 4 41 41 = π R − R + R − R − π −R + R + R − R 3 2 4 3 2 4 4 = πR4 . 3 

❚❤❡ r❡s✉❧t ♠❛t❝❤❡s t❤❡ ✐❞❡❛ t❤❛t t❤❡ ✇♦r❦ r❡q✉✐r❡❞ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ✇♦r❦ t♦ ♠♦✈❡ t❤❡ ✇❤♦❧❡ ❜❛❧❧ ♦❢ 3 4 ✇❛t❡r✱ ✈♦❧✉♠❡ πR ✱ s♦ t❤❛t ✐ts ❝❡♥t❡r ♦❢ ♠❛ss ♠♦✈❡s ❢r♦♠ −R t♦ 0✳ 3 ❊①❡r❝✐s❡ ✸✳✻✳✶✹

❙✉♣♣♦s❡ ✇❡ ❛r❡ t♦ ✜❧❧ ❛ ✏♣❛r❛❜♦❧♦✐❞✑ t❛♥❦ ❛❝q✉✐r❡❞ ❜② r♦t❛t✐♥❣ t❤❡ ❣r❛♣❤ ♦❢

y = x2

❛r♦✉♥❞ t❤❡

x✲❛①✐s✱

✇❤✐❝❤ ✐s ✈❡rt✐❝❛❧✱ ❢r♦♠ t❤❡ ❜♦tt♦♠✳ ❲❤❛t ✐s t❤❡ ✇♦r❦ r❡q✉✐r❡❞❄

❊①❡r❝✐s❡ ✸✳✻✳✶✺

❲❤❛t ✇♦r❦ ✐s ♥❡❡❞❡❞ t♦ ♣✉❧❧ ❛❧❧ t❤❡ ✇❛② ✉♣ ❛ ❝❤❛✐♥ ❤❛♥❣✐♥❣ ❞♦✇♥ ✐❢ ✐t ✐s

10

❢❡❡t ❧♦♥❣ ❛♥❞

20

♣♦✉♥❞s

❤❡❛✈②❄

❊①❡r❝✐s❡ ✸✳✻✳✶✻

❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ✇♦r❦ r❡q✉✐r❡❞ ♦✈❡r ❛♥ ✐♥✜♥✐t❡❧② ❧♦♥❣ tr✐♣❄

❊①❡r❝✐s❡ ✸✳✻✳✶✼

❙❤♦✇ t❤❛t t❤❡ ✇♦r❦ r❡q✉✐r❡❞ t♦ ♠♦✈❡ ❛♥ ♦❜❥❡❝t ♣✐❡❝❡ ❜② ♣✐❡❝❡✱ ❛s ❛❜♦✈❡✱ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ✇♦r❦ r❡q✉✐r❡❞ t♦ ♠♦✈❡ t❤❡ ✇❤♦❧❡ ♦❜❥❡❝t ❛s ✐❢ ✐ts t♦t❛❧ ♠❛ss ✐s ❝♦♥❝❡♥tr❛t❡❞ ❛t ❛ s✐♥❣❧❡ ♣♦✐♥t✱ ✐ts ❝❡♥t❡r ♦❢ ♠❛ss✳

✸✳✼✳

❚❤❡ t♦t❛❧ ❛♥❞ t❤❡ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✶✺

✸✳✼✳ ❚❤❡ t♦t❛❧ ❛♥❞ t❤❡ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

❲❤❛t ❞♦ t❤❡s❡ ❡①❛♠♣❧❡s ❤❛✈❡ ✐♥ ❝♦♠♠♦♥❄ ❆ ❝❡rt❛✐♥ q✉❛♥t✐t②✱

f ✱ ✐s ✏s♣r❡❛❞✑

❛r♦✉♥❞ ❧♦❝❛t✐♦♥s ✐♥ s♣❛❝❡❀ ❢♦r ♥♦✇✱ ✐t ✐s ❛♥ ✐♥t❡r✈❛❧ ✇✐t❤✐♥ t❤❡

x✲❛①✐s✳

❚❤✐s

q✉❛♥t✐t② ♠❛② ❜❡✿ ❧❡♥❣t❤✱ ❛r❡❛✱ ❞❡♥s✐t②✱ ✈❡❧♦❝✐t②✱ ❢♦r❝❡✳ ❲❤❡♥ t❤❡ q✉❛♥t✐t② ✐s ❝♦♥st❛♥t ✇✐t❤✐♥ ❛ s❡❣♠❡♥t ♦❢ t❤❡ ✐♥t❡r✈❛❧✱ ♠✉❧t✐♣❧②✐♥❣ t❤✐s ✈❛❧✉❡ ❜② t❤❡ ❧❡♥❣t❤ ♦❢ t❤✐s ♣✐❡❝❡✱

❲❤❡♥ t❤❡ q✉❛♥t✐t②

f

f✳

❛r❡❛

t♦t❛❧ ❛r❡❛

❧✐♥❡❛r ❞❡♥s✐t②

♠❛ss

t♦t❛❧ ♠❛ss

✢♦✇ r❛t❡

✢✉①

t♦t❛❧ ✢✉①

❢♦r❝❡

✇♦r❦

t♦t❛❧ ✇♦r❦

t♦t❛❧ ✈❛❧✉❡

♦❢

f

✐s t❤❡ s✉♠ ♦❢ t❤❡ t❡r♠s

❲❤❡♥ t❤✐s ❝❤❛♥❣❡ ✐s ❝♦♥t✐♥✉♦✉s✱ t❤❡

❛t t❤❡ ❧✐♠✐t✱ ✐t ✐s t❤❡ ✐♥t❡❣r❛❧ ♦❢ ◆♦✇✱ t❤❡

❧❡♥❣t❤

✈❛r✐❡s ❢r♦♠ s❡❣♠❡♥t t♦ s❡❣♠❡♥t ♦✈❡r t❤❡ ✐♥t❡r✈❛❧✱ ✐t ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ ❢✉♥❝t✐♦♥✳ ❲❤❡♥

t❤✐s ❝❤❛♥❣❡ ✐s ✐♥❝r❡♠❡♥t❛❧✱ t❤❡ ❢✉♥❝t✐♦♥

f · ∆x Σf · ∆x

f

q✉❛♥t✐t②

❛✈❡r❛❣❡

❘❡❝❛❧❧ t❤❛t t❤❡

f

♦✈❡r

t♦t❛❧ ✈❛❧✉❡

♦❢

f

f · ∆x✱

✐✳❡✳✱ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡

✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤✐s ❘✐❡♠❛♥♥ s✉♠ ❛♥❞✱

[a, b]✳

✈❛❧✉❡✳

♠❡❛♥

✭♦r t❤❡ ❛✈❡r❛❣❡✮ ♦❢ ❛ q✉❛♥t✐t② ❣✐✈❡♥ ❜②

❢♦❧❧♦✇✐♥❣✿ ▼❡❛♥

=

✐♥✜♥✐t❡❧② ♠❛♥②

▲❡t✬s st❛rt ✇✐t❤ t❤❡ ✐❞❡❛ ♦❢ ❛

n

♥✉♠❜❡rs

y1 , ..., yn

✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡

y1 + y2 + ... + yn . n

❍♦✇ s❤♦✉❧❞ ✇❡ ✉♥❞❡rst❛♥❞ t❤❡ ♠❡❛♥ ♦❢ ❛ q✉❛♥t✐t② t❤❛t ✐s ❚❤❡ ♥✉♠❡r❛t♦r ✇♦✉❧❞ ❤❛✈❡

m1 , ..., mn

∆x✱ ❣✐✈❡s ✉s ❛ ♥❡✇ ❜✉t st✐❧❧ ❢❛♠✐❧✐❛r q✉❛♥t✐t②✿

❝♦♥t✐♥✉♦✉s❧②

s♣r❡❛❞ ♦✈❡r ❛ ❧✐♥❡ s❡❣♠❡♥t✱ s❛②

[a, b]❄

t❡r♠s✦

✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡✳

❲❡ ❛ss✉♠❡ t❤❛t ✇❡ ❤❛✈❡

n ✇❡✐❣❤ts✱

✐✳❡✳✱

n

♣♦s✐t✐✈❡ ♥✉♠❜❡rs

✇✐t❤

m1 + ... + mn = 1 . ❚❤❡♥ ❢♦r ❛♥② ❣✐✈❡♥

n

♥✉♠❜❡rs

y1 , ..., yn ✱

✇❡ ❞❡✜♥❡ t❤❡✐r ✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ❛s ❢♦❧❧♦✇s✿

❲❡✐❣❤t❡❞ ❛✈❡r❛❣❡

= m1 y1 + m2 y2 + ... + mn yn =

n X

mi yi .

i=1

❊①❡r❝✐s❡ ✸✳✼✳✶

❙❤♦✇ t❤❛t t❤❡ ♠❡❛♥ ✐s t❤❡ ✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ✇✐t❤

mi = 1/n

❢♦r ❛❧❧

i✳

❊①❛♠♣❧❡ ✸✳✼✳✷✿ s❝♦r❡s

❚❤❡ ✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ♠❛② ❛♣♣❡❛r ✇❤❡♥ ♦♥❡ ❝♦♠♣✉t❡s t❤❡ t♦t❛❧ s❝♦r❡ ✐♥ ❛ ❝❧❛ss ❛❢t❡r s❡✈❡r❛❧ ❛ss✐❣♥♠❡♥ts ♦❢ ❞✐✛❡r❡♥t

• • • •

✇❡✐❣❤ts✳

❋♦r ❡①❛♠♣❧❡✱ t❤✐s ♠❛② ❜❡ t❤❡ ❣r❛❞❡ ❜r❡❛❦❞♦✇♥✿

20% 30% ♠✐❞t❡r♠✿ 20% ✜♥❛❧ ❡①❛♠✿ 30%

♣❛rt✐❝✐♣❛t✐♦♥✿ q✉✐③③❡s✿

❚❤❡♥ t❤❡ t♦t❛❧ s❝♦r❡ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ♦❢ t❤❡ ✜✈❡ s❝♦r❡s✿ ❚❖❚❆▲

= .20 × P + .30 × Q + .20 × M + .30 × F .

✸✳✼✳

❚❤❡ t♦t❛❧ ❛♥❞ t❤❡ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✶✻

❊①❛♠♣❧❡ ✸✳✼✳✸✿ ❝❡♥t❡r ♦❢ ♠❛ss❄ ❘❡❝❛❧❧ t❤❛t ✐❢ c ✐s t❤❡ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ ❛ s②st❡♠ ♦❢ ✇❡✐❣❤ts ✇✐t❤ ❧♦❝❛t✐♦♥s ❛t yi ❛♥❞ r❡❧❛t✐✈❡ ✇❡✐❣❤ts mi ✱ t❤❡♥ X c=

mi yi .

i

❚❤❡r❡❢♦r❡✱ t❤❡ ✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ✐s t❤❡ s❛♠❡✱ ✐♥ t❤✐s ❝❛s❡✱ ❛s t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ t❤❡ s②st❡♠✳

❚❤❡ ♥❡✇ s❡t✉♣ ✐s ❛s ❢♦❧❧♦✇s✿ ◮ ❆ s✉❜st❛♥❝❡ ✐s ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ♦✈❡r ❛ s❡❣♠❡♥t ♦❢ t❤❡ ❧✐♥❡✳

❚❤❡♥✱ t❤❡ ✇❡✐❣❤t ♦❢ t❤❡ s❡❣♠❡♥t ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ ✐ts ✇✐❞t❤✳ ❲❡ ❛r❡✱ t❤❡♥✱ ❥✉st✐✜❡❞ t♦ ✉s❡ t❤❡s❡ ✇✐❞t❤s ❛s s✉❜st✐t✉t❡s ❢♦r t❤❡ t❤❡ ♠❛✐♥ ✐❞❡❛✿ ◮ ❊❛❝❤ ✇❡✐❣❤t mi ✐s t❤❡

r❡❧❛t✐✈❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✐♥t❡r✈❛❧

✇❡✐❣❤ts

✐♥ t❤❡ ✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡✳ ❚❤✐s ✐s

✇❤❡r❡ yi ♦❢ t❤❡ q✉❛♥t✐t② ✐s ❧♦❝❛t❡❞✳

❲❡ st❛rt ✇✐t❤ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ P ♦❢ t❤❡ ✐♥t❡r✈❛❧✿ a = x0 ≤ c1 ≤ x1 ≤ ... ≤ cn ≤ xn = b

❚❤❡♥ ✇❡ ✇r✐t❡ t❤❡ r❡❧❛t✐✈❡ ❧❡♥❣t❤s✿ mi =

▲❡t✬s s✉❜st✐t✉t❡✿

∆xi . b−a

n n X 1 X ∆xi yi = yi · ∆xi . ❲❡✐❣❤t❡❞ ❛✈❡r❛❣❡ = b−a b − a i=1 i=1

❋✉rt❤❡r♠♦r❡✱ ✐❢ t❤❡s❡ ♥✉♠❜❡rs ❛r❡ ❣✐✈❡♥ ❜② ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ f (ci ) = yi ,

t❤❡♥ ✇❡ ❤❛✈❡✿

n

1 X ❲❡✐❣❤t❡❞ ❛✈❡r❛❣❡ = f (ci ) · ∆xi . b − a i=1

❚❤✐s s✉♠ ✐s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✸✳✼✳✹✿ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❢✉♥❝t✐♦♥ ❚❤❡ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r✈❛❧ [a, b] ✐s ❞❡♥♦t❡❞ ❛♥❞ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿ f¯ =

1 Σ f · ∆x . b−a

■t ✐s✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ t♦t❛❧ ✈❛❧✉❡ ♦❢ f ♣❡r ✉♥✐t ♦❢ ❧❡♥❣t❤✳

✸✳✼✳

❚❤❡ t♦t❛❧ ❛♥❞ t❤❡ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✶✼

❲❛r♥✐♥❣✦ ❚❤❡ ❛✈❡r❛❣❡ ♠❛ss ♦❢ ❛ s②st❡♠ ♦❢ ✇❡✐❣❤ts ✐s ♥♦t t❤❡ s❛♠❡ ❛s t❤❡ ❛✈❡r❛❣❡

❧♦❝❛t✐♦♥

♦❢ t❤❡ ✇❡✐❣❤ts✳

❲❤❛t ✐❢ t❤❡ ❢✉♥❝t✐♦♥ ❞♦❡s♥✬t ❝❤❛♥❣❡ ✐♥❝r❡♠❡♥t❛❧❧② ❜✉t ✏❝♦♥t✐♥✉♦✉s❧②✑❄

❊①❛♠♣❧❡ ✸✳✼✳✺✿ ❧✐♥❡❛r ❞❡♣❡♥❞❡♥❝❡ f 1.5✳

❙✉♣♣♦s❡ t❤❡ ❢✉♥❝t✐♦♥ ✈❛❧✉❡ ✐s ❝❧❡❛r❀ ✐t ✐s

✐s ❧✐♥❡❛r✱ ❢r♦♠

1

t♦

2

♦✈❡r t❤❡ ✐♥t❡r✈❛❧

[0, 2]✳

❚❤❡♥ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❛✈❡r❛❣❡

❲❤❡r❡ ❞♦❡s ✐t ❝♦♠❡s ❢r♦♠❄ ❚❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ ♦❢

f

♦✈❡r

[0, 2]✱

❞✐✈✐❞❡❞ ❜② ✐ts ❧❡♥❣t❤✱

f✱

✐✳❡✳✱

3✱

✇❤✐❝❤ ✐s t❤❡ ✐♥t❡❣r❛❧

2✳

❚❤❡♥ ✇❡ t❤✐♥❦ ♦❢ t❤❡ ❢r❛❝t✐♦♥ ❛❜♦✈❡ ❛s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❛✈❡r❛❣❡✳ ❚❤✐s ❛♥❛❧②s✐s ❥✉st✐✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥✿

❉❡✜♥✐t✐♦♥ ✸✳✼✳✻✿ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❢✉♥❝t✐♦♥ ❚❤❡

❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ f

❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿

f¯ =

1 b−a

❚♦ ✐❧❧✉str❛t❡✱ ❝♦♥s✐❞❡r ❤♦✇ ♦♥❡ ❧❡✈❡❧s ❛♥ ✉♥❡✈❡♥ s✉r❢❛❝❡ ♦❢ s❛♥❞✿

❚❤❡ ❛♠♦✉♥t ♦❢ s❛♥❞ ✐s t❤❡ s❛♠❡✳ ❚❤❡ ❛✈❡r❛❣❡ ❞❡♣t❤ ♦❢ ❛ ❝❛♥❛❧ ✐s ❛♥♦t❤❡r ✐♥t❡r♣r❡t❛t✐♦♥✿

Z

♦✈❡r ✐♥t❡r✈❛❧

b

f dx . a

[a, b]

✐s ❞❡♥♦t❡❞ ❛♥❞

✸✳✼✳

❚❤❡ t♦t❛❧ ❛♥❞ t❤❡ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥

✷✶✽

❇♦t❤ ❝❛♥❛❧s ❤❛✈❡ t❤❡ s❛♠❡ ❛♠♦✉♥t ♦❢ ✇❛t❡r✳ ❚❤✉s✱ ❜② ✏❛✈❡r❛❣✐♥❣✑ ✇❡ ♠❡❛♥ r❡♣❧❛❝✐♥❣ ❛♥② ❢✉♥❝t✐♦♥✱

y = f (x)✱

t❤❛t t❤❡ t✇♦ ❤❛✈❡ t❤❡ s❛♠❡ ✐♥t❡❣r❛❧✿

f¯ · (b − a) =

Z

✇✐t❤ ❛

❝♦♥st❛♥t ❢✉♥❝t✐♦♥✱ y = f¯✱ ❝❤♦s❡♥ s♦

b

f dx . a

❊①❡r❝✐s❡ ✸✳✼✳✼

Pr♦✈❡ t❤❡ ❛❜♦✈❡ st❛t❡♠❡♥t✳

❚❤❡♦r❡♠ ✸✳✼✳✽✿ Pr♦♣❡rt✐❡s ♦❢ ❆✈❡r❛❣❡ ❖✈❡r ❛ ❣✐✈❡♥ ✐♥t❡r✈❛❧✱ ✇❡ ❤❛✈❡✿

•

❚❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ s✉♠ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❛✈❡r❛❣❡s✿

f + g = f¯ + g¯ . •

❚❤❡ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❛✈❡r❛❣❡ ✐s t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡✿

cf = cf¯ . ❊①❡r❝✐s❡ ✸✳✼✳✾

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❊①❡r❝✐s❡ ✸✳✼✳✶✵

❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t t❤❡ ❛✈❡r❛❣❡ ♦❢ ✭❛✮ ❛♥ ♦❞❞ ❢✉♥❝t✐♦♥ ♦✈❡r

[−r, r]✱

[−r, r]✱

✭❜✮ ❛♥ ❡✈❡♥ ❢✉♥❝t✐♦♥ ♦✈❡r

✭❝✮ ❛ ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥❄

❲❡ ❝❛♥ r❡✇r✐t❡ t❤❡ t❛❜❧❡ ✇✐t❤ ✇❤✐❝❤ ✇❡ st❛rt❡❞ t❤❡ s❡❝t✐♦♥✿

f

Z

b

f dx a

1 b−a

Z

b

f dx a

❧❡♥❣t❤

t♦t❛❧ ❛r❡❛

❛✈❡r❛❣❡ ❧❡♥❣t❤

❧✐♥❡❛r ❞❡♥s✐t②

t♦t❛❧ ♠❛ss

❛✈❡r❛❣❡ ❧✐♥❡❛r ❞❡♥s✐t②

✢✉①

t♦t❛❧ ✢✉①

❛✈❡r❛❣❡ ✢✉①

❢♦r❝❡

t♦t❛❧ ✇♦r❦

❛✈❡r❛❣❡ ❢♦r❝❡

❆❧❧ ❤❛✈❡ ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ❝❤❛♣t❡r ❡①❝❡♣t ❢♦r t❤❡ ✜rst ✐t❡♠ t❤❛t ❝♦♠❡s ❢r♦♠ ❱♦❧✉♠❡ ✷✳

✸✳✽✳

◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥

✷✶✾

❊①❡r❝✐s❡ ✸✳✼✳✶✶

❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❛✈❡r❛❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ ❛♥ ✐♥✜♥✐t❡ ✐♥t❡r✈❛❧❄

✸✳✽✳ ◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥

❚♦ ❛♣♣❧② t❤❡ ✐♥t❡❣r❛❧ ❢♦r♠✉❧❛s ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ♥❡❡❞ t♦ ❡✈❛❧✉❛t❡ t❤♦s❡ ✐♥t❡❣r❛❧s✳ ❚❤✐s ✐s ❛♥ ✐❞❡❛❧ ♦✉t❝♦♠❡✿ ❆r❡❛ = ❲❡ ❤❛✈❡ ❛♥ ❡①❛❝t

♥✉♠❜❡r✳

Z

2

1 1 x3 13 03 − = . x dx = = 3 0 3 3 3 2

0

❍♦✇❡✈❡r✱ s✉❝❤ ❛♥ ♦✉t❝♦♠❡ ✐s ❛♥ ❡①❝❡♣t✐♦♥✱ ♥♦t ❛ r✉❧❡✦ ❙♦♠❡ ✐♥t❡❣r❛❧s ❞♦ ♥♦t ♣r♦❞✉❝❡ ❢❛♠✐❧✐❛r ❢✉♥❝t✐♦♥s s♦ t❤❛t ✇❡ ❝❛♥ ❥✉st ♣❧✉❣ ✐♥ t❤❡ t✇♦ ✈❛❧✉❡s✳ ❈♦♥✈❡rs❡❧②✱ s♦♠❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ❤❛✈❡ ❜❡❡♥ ❞❡✜♥❡❞ ❛s ✐♥t❡❣r❛❧s ♦♥❧②✱ s✉❝❤ ❛s✿ Z 2 erf(x) = √ π

2

e−x dx .

❚❤❡r❡ ✐s ♥♦ ♦t❤❡r ❢♦r♠✉❧❛✦

❲❤❛t ❞♦ ✇❡ ❞♦❄ ❚❤❡ ❛♥s✇❡r ✐s ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✳ ■t ✐s ❞❡✜♥❡❞ ✈✐❛ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡s❡ s✉♠s s❡r✈❡ ❛s ✐ts ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❲❡ ✇✐❧❧ ❛ss✉♠❡ ❜❡❧♦✇ t❤❛t ❛❧❧ ❢✉♥❝t✐♦♥s ❛r❡ ✐♥t❡❣r❛❜❧❡✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t ❛♥② ❝❤♦✐❝❡ ♦❢ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ❢♦r t❤❡ ❘✐❡♠❛♥♥ s✉♠s ✐s ❡q✉❛❧❧② ✈❛❧✐❞✳ ❊①❛♠♣❧❡ ✸✳✽✳✶✿ ♣❛rt✐t✐♦♥ s❝❤❡♠❡s

▲❡t✬s r❡✈✐❡✇ t❤❡ ✇❛②s ✇❡ ❡st✐♠❛t❡ t❤✐s ✐♥t❡❣r❛❧ ♦❢ f (x) = x2 ♦✈❡r [0, 1]✿ Z

1

f dx . 0

❲❡ ❝❤♦♦s❡ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥t❡r✈❛❧s t♦ ❜❡ n = 4 ✇✐t❤ ❡q✉❛❧ ✐♥t❡r✈❛❧s ♦❢ ❧❡♥❣t❤ h = 1/4✳ ❚❤❡♥ ✇❡ ❝❤♦♦s❡✱ ❛s t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✱ t❤❡ ❧❡❢t✲❡♥❞ ♦r t❤❡ r✐❣❤t✲❡♥❞ ♦❢ ❡❛❝❤ ✐♥t❡r✈❛❧✿

✸✳✽✳

◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥

✷✷✵

❆t t❤♦s❡ ♣♦✐♥ts✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❡✈❛❧✉❛t❡❞✳ ❚❤✐s ✐s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❧❡❢t✲❡♥❞ ❘✐❡♠❛♥♥ s✉♠

•

x x

−−

0

2

0

|

1/4

−−

1/16

L4 0 · 1/4

+

1/16 · 1/4

|

−− |

1/2

+

−− •

3/4

1/4

1

9/16

1/4 · 1/4 +

1

9/16 · 1/4

❲❡✱ ❢✉rt❤❡r♠♦r❡✱ r❡❛❧✐③❡ t❤❛t ✇❡ ❛r❡ ❝♦♠♣✉t✐♥❣ t❤❡ ❘✐❡♠❛♥♥ s✉♠

L4 ✿

❢✉♥❝t✐♦♥

≈ 0.22 ❢♦r t❤✐s ❛✉❣♠❡♥t❡❞ ♣❛rt✐✲

t✐♦♥✳ ■ts ❢♦✉r ✈❛❧✉❡s ❛r❡ s❤♦✇♥ ❛t ❜♦tt♦♠ ♦❢ t❤❡ t❛❜❧❡✳

❲❛r♥✐♥❣✦ ■t ✐s ❜❡tt❡r t♦ ❛✈♦✐❞ ✏❛♣♣r♦①✐♠❛t✐♥❣ ✇✐t❤ ❛♣♣r♦①✐✲ ♠❛t✐♦♥s✑ ❛♥❞ r❡♣❧❛❝❡ t❤❡ ❧❛st ♥✉♠❜❡r ✇✐t❤ ✐ts ❡①❛❝t ✈❛❧✉❡✿

L4 =

7 = .21875 . 32

❊①❡r❝✐s❡ ✸✳✽✳✷ ❈r❡❛t❡ ❛ t❛❜❧❡ ♦❢ ✈❛❧✉❡s ❢♦r t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ r✐❣❤t ❡♥❞s✳

❊①❛♠♣❧❡ ✸✳✽✳✸✿ ♠✐❞✲♣♦✐♥t ❘✐❡♠❛♥♥ s✉♠s ❲❡ ❝❛♥ ❛❧s♦ ❝❤♦♦s❡ t❤❡ ♠✐❞✲♣♦✐♥ts ❛s t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s✿

❚❤✐s ✐s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ♠✐❞✲♣♦✐♥t ❘✐❡♠❛♥♥ s✉♠

x

•

−− 1/8

(1/8)2

f (x) = x2 M4

|

−− 3/8

(3/8)2

M4

|

❢♦r t❤❡ s❛♠❡ ✐♥t❡❣r❛❧✿

−− 5/8

(5/8)2

|

−−

7/8

•

(7/8)2

(1/8)2 · 1/4 + (3/8)2 · 1/4 + (5/8)2 · 1/4 + (7/8)2 · 1/4 = 0.328125

■t ✐s ♠✉❝❤ ❝❧♦s❡r t❤❛♥

L4

t♦ t❤❡ tr✉❡ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧✱

1/3✳

✸✳✽✳

◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥

✷✷✶

❊①❡r❝✐s❡ ✸✳✽✳✹

❲❡ ❤❛✈❡ ♣r❡✈✐♦✉s❧② ✉s❡❞ ❛ s♣r❡❛❞s❤❡❡t t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❢✉♥❝t✐♦♥ ❢♦r Ln ✳ ❈r❡❛t❡ ❛ s♣r❡❛❞s❤❡❡t t♦ ❛✉t♦♠❛t❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❢✉♥❝t✐♦♥ ❢♦r Rn ❛♥❞ Mn ✳ ❚❤✐s ✐s ✇❤❛t ❛❧❧ ❘✐❡♠❛♥♥ s✉♠s ❤❛✈❡ ✐♥ ❝♦♠♠♦♥✿ ❲❡ ❝❤♦♦s❡ ❛ ♣♦✐♥t ♦♥ t❤❡ ❣r❛♣❤ ❛♥❞ t❤❡♥ ❛♣♣r♦①✐♠❛t❡s ✐ts ♣✐❡❝❡ ✇✐t❤ ❛ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥t✳ ❚❤❡ t❤r❡❡ ♠❛✐♥ ❝❤♦✐❝❡s ❛r❡ s❤♦✇♥ ❜❡❧♦✇✿

❲❤❛t ✐❢ ✇❡ ❝❤♦♦s❡ t✇♦ ♣♦✐♥ts ✕ ❛t t❤❡ ❡♥❞ ❛♥❞ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ✐♥t❡r✈❛❧ ✕ ❛♥❞ ❛♣♣r♦①✐♠❛t❡ t❤✐s ♣✐❡❝❡ ♦❢ t❤❡ ❣r❛♣❤ ✇✐t❤ ❛ s❧♦♣❡❞ ❧✐♥❡❄ ■t ✐s✱ ✐♥ ❢❛❝t✱ t❤❡ ❢❛♠✐❧✐❛r s❡❝❛♥t ❧✐♥❡✦ ❚❤✐s t❤✐r❞ ✇❛② t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❛r❡❛ ✐s s❤♦✇♥ ♦♥ t❤❡ ❢❛r r✐❣❤t✳ ■♥st❡❛❞ ♦❢ ❛ r❡❝t❛♥❣❧❡✱ ✇❡ ❤❛✈❡ ❛ tr❛♣❡③♦✐❞✳ ■ts ❛r❡❛ ✐s t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ t✇♦ ❜❛s❡s ✭✈❡rt✐❝❛❧✮ ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ ❤❡✐❣❤t ✭❤♦r✐③♦♥t❛❧✮✿

❚❤❡♥ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr❛♣❡③♦✐❞ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ [xk−1 , xk ] ✐s ❡q✉❛❧ t♦

f (xk−1 ) + f (xk ) h. 2 ❚❤❡ s✉♠ ♦❢ ❛❧❧ n ♦❢ t❤❡s❡ ✐s ❝❛❧❧❡❞ t❤❡

tr❛♣❡③♦✐❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ❛♥❞ ❞❡♥♦t❡❞ ❜② Tn✳

❊①❡r❝✐s❡ ✸✳✽✳✺

❙❤♦✇ t❤❛t Tn ✐s t❤❡ ❛✈❡r❛❣❡ ♦❢ Ln ❛♥❞ Rn ✳

✸✳✽✳

◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥

❊①❛♠♣❧❡ ✸✳✽✳✻✿

✷✷✷

Tn

▲❡t✬s ❝♦♠♣✉t❡ s✉♠

T4

❢♦r t❤❡ s❛♠❡ ✐♥t❡❣r❛❧✳ ❲❡ ✉s❡ t❤❡ s❛♠❡ ❞❛t❛ ❛♥❞ t❤❡♥ ❛❞❞ t❤❡ ❢♦❧❧♦✇✐♥❣ t❡r♠s✿

f (xk−1 )h + f (xk )h ❢♦r ❡❛❝❤ ✐♥t❡r✈❛❧✿

x x2

• 0

−−

|

1/4

0 0 · 1/4

−−

1/16

−− |

1/2

1/16 · 1/4

+

−− •

3/4

1/4

1/16 · 1/4

+

|

1

9/16

1 ≈ 0.016

1/4 · 1/4

1/4 · 1/4 +

≈ 0.079

9/16 · 1/4

≈ 0.203

9/16 · 1/4 +

1 · 1/4 ≈ 0.391 s✉♠

T4

❤❛❧❢

≈ 0.689 ≈ 0.345

❲❛r♥✐♥❣✦ ❚❤❡ r❡s✉❧t ✐s

♥♦t

❛ ❘✐❡♠❛♥♥ s✉♠✳

❊①❡r❝✐s❡ ✸✳✽✳✼ ❈r❡❛t❡ ❛ s♣r❡❛❞s❤❡❡t t♦ ❛✉t♦♠❛t❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢

Tn ✳

❚❤❡s❡ ❛r❡ t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❢♦✉r ❛♣♣r♦①✐♠❛t✐♦♥s✿

Ln =

n X i=1

Rn =

n X

 f a + (i − 1)h h  f a + ih h

i=1 n X

 f a + (i − 1)h + h/2 h i=1   n X   1 f a + (i − 1)h + f a + ih h Tn = 2 i=1 Mn =

❚❤❡ ❡①♣r❡ss✐♦♥s ❛♣♣r♦①✐♠❛t❡ t❤❡ ✐♥t❡❣r❛❧ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✳

❚❤❡♦r❡♠ ✸✳✽✳✽✿ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❘✐❡♠❛♥♥ ❙✉♠s ■❢

f

✐s ✐♥t❡❣r❛❜❧❡ ♦♥

[a, b]✱

t❤❡♥ t❤❡ s❡q✉❡♥❝❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡

tr❛♣❡③♦✐❞ s✉♠ ❝♦♥✈❡r❣❡ t♦ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♦❢

Ln , R n , M n , T n → ❖♥❧② t❤❡ ❧❛st ♣❛rt ♥❡❡❞s ♣r♦♦❢✳

❊①❡r❝✐s❡ ✸✳✽✳✾ Pr♦✈❡ t❤❡ ♠✐ss✐♥❣ ♣❛rt✳ ❍✐♥t✿ ❚❤❡ ❙q✉❡❡③❡ ❚❤❡♦r❡♠✳

Z

f✿

b

f dx a

❛s

n → ∞.

✸✳✽✳

◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥

✷✷✸

❍♦✇ ✇❡❧❧ ❞♦ t❤❡s❡ ❢♦✉r ♣❡r❢♦r♠❄ • ◗✉❡st✐♦♥✿ ❋♦r ❛ ❣✐✈❡♥ n✱ ❤♦✇ ❝❧♦s❡ ❛r❡ ✇❡ t♦ t❤❡ tr✉❡ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧❄ • ❆♥s✇❡r✿ ❲❡ ❞♦♥✬t ❦♥♦✇✱ ❛♥❞ ✇❡ ❝❛♥✬t ❦♥♦✇✱ ✇✐t❤♦✉t s♦♠❡

❛ ♣r✐♦r✐ ❦♥♦✇❧❡❞❣❡✳

■❢ ✇❡ ❦♥❡✇ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧✱ ✇❡ ✇♦✉❧❞♥✬t ♥❡❡❞ t♦ ❛♣♣r♦①✐♠❛t❡ ✐t✦ ❱❛r✐♦✉s ❜❡❤❛✈✐♦rs ♦❢ f ❛r❡ s❤♦✇♥ ❛❧♦♥❣ ✇✐t❤ t❤❡ ❡rr♦r ✭②❡❧❧♦✇✮ ♦❢ Ln ✿

❙✐♥❝❡ ❛❧❧ ❢♦✉r ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ s♣❡❝✐✜❝ ❛r❡❛s✱ t❤❡ ❡rr♦rs ❛r❡ ❛❧s♦ s❡❡♥ ❛s ❝❡rt❛✐♥ ❛r❡❛s✿

❚❤❡ s✐♠♣❧❡st ❛

♣r✐♦r✐ ❦♥♦✇❧❡❞❣❡ ❛❜♦✉t ❛ ❢✉♥❝t✐♦♥ ✐s ✐ts ♠♦♥♦t♦♥✐❝✐t②✳ ❚❤❡♦r❡♠ ✸✳✽✳✶✵✿ ❘✐❡♠❛♥♥ ❙✉♠s ♦❢ ▼♦♥♦t♦♥❡ ❋✉♥❝t✐♦♥s ✶✳ ■❢

f

✐s ✐♥❝r❡❛s✐♥❣ ♦♥

[a, b]✱

t❤❡ ❧❡❢t✲❡♥❞ ❘✐❡♠❛♥♥ s✉♠ ✉♥❞❡r❡st✐♠❛t❡s t❤❡

✐♥t❡❣r❛❧✱ ✇❤✐❧❡ t❤❡ r✐❣❤t✲❡♥❞ s✉♠ ♦✈❡r❡st✐♠❛t❡s ✐t✿

f ր =⇒ Ln ≤

Z

b

f dx ≤ Rn .

a

✷✳ ❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞❡❝r❡❛s✐♥❣✱ t❤❡ ✐♥❡q✉❛❧✐t✐❡s ❛r❡ r❡✈❡rs❡❞✿

f ց =⇒ Ln ≥

Z

b a

f dx ≥ Rn .

■♥ ❡✐t❤❡r ❝❛s❡✱ t❤❡ tr✉❡ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧ I ❧✐❡s ❜❡t✇❡❡♥ Rn ❛♥❞ Ln ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ ❈♦r♦❧❧❛r② ✸✳✽✳✶✶✿ ■♥t❡r✈❛❧ ❢♦r ■♥t❡❣r❛❧ ♦❢ ▼♦♥♦t♦♥❡ ❋✉♥❝t✐♦♥ ■❢ ❛ ❢✉♥❝t✐♦♥

f

✐s ♠♦♥♦t♦♥❡✱ t❤❡♥

Z

b

f dx a

❧✐❡s ✇✐t❤✐♥ ❡✐t❤❡r

[Ln , Rn ]

♦r

[Rn , Ln ] .

✸✳✽✳

◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥

✷✷✹

❚❤❡ st❛t❡♠❡♥t ✐s t❤❡ ❛❜s♦❧✉t❡✱ ♥♦t ❛♣♣r♦①✐♠❛t❡✱ tr✉t❤✦ ❲❤✐❧❡ t❤❡ ❧❛st r❡s✉❧t r❡❧✐❡s ♦♥ ♠♦♥♦t♦♥✐❝✐t② ✭❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f ✮✱ ❢♦r t❤❡ ♦t❤❡r t✇♦ ❛♣♣r♦①✐♠❛t✐♦♥s✱ ❛ s✐♠✐❧❛r r❡s✉❧t r❡❧✐❡s ♦♥ ❝♦♥❝❛✈✐t② ✭❛♥❞ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ f ✮✳ ❚❤❡♦r❡♠ ✸✳✽✳✶✷✿ ❘✐❡♠❛♥♥ ❙✉♠s ♦❢ ❈♦♥✈❡① ❋✉♥❝t✐♦♥s ✶✳ ■❢

f

✐s ❝♦♥❝❛✈❡ ❞♦✇♥ ♦♥

[a, b]✱

t❤❡ tr❛♣❡③♦✐❞ s✉♠ ✉♥❞❡r❡st✐♠❛t❡s t❤❡ ✐♥t❡✲

❣r❛❧✿

f ⌢ =⇒ Tn ≤

Z

b

f dx . a

✷✳ ▼❡❛♥✇❤✐❧❡✱ ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥❝❛✈❡ ✉♣✱ t❤❡ ✐♥❡q✉❛❧✐t② ✐s r❡✈❡rs❡❞✿

f ⌣ =⇒ Tn ≥

Z

b

f dx . a

❲❡ t❤✉s ❤❛✈❡ ❞✐s❝♦✈❡r❡❞ ❤♦✇ t❤❡s❡ ❛♣♣r♦①✐♠❛t✐♦♥s ❡rr ✐♥ ❞✐✛❡r❡♥t ❞✐r❡❝t✐♦♥s ✉♥❞❡r ❞✐✛❡r❡♥t ❝✐r❝✉♠st❛♥❝❡s✳ ❍♦✇❡✈❡r✱ t❤❡ tr✉❡ ♠❡❛s✉r❡ ♦❢ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s t❤❡ ❛❝t✉❛❧ ❞✐✛❡r❡♥❝❡✱ ✐✳❡✳✱ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ✐♥t❡❣r❛❧✿

❊rr♦r = ■♥t❡❣r❛❧ − ❆♣♣r♦①✐♠❛t✐♦♥

❙✐♥❝❡ ✇❡ ❞♦♥✬t ❦♥♦✇ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧✱ ✇❡ ❞♦♥✬t ❦♥♦✇ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❡rr♦r❀ ✇❡ ❝❛♥ ♦♥❧② ❡st✐♠❛t❡ ✐t✳ ❲✐t❤ t❤✐s ❡st✐♠❛t❡✱ ✇❡ ❝❛♥ ❡st❛❜❧✐s❤ t❤❛t ✇❡ ❤❛✈❡♥✬t ❞❡✈✐❛t❡❞ ❢r♦♠ t❤❡ tr✉t❤ t♦♦ ❢❛r✳ ❊①❡r❝✐s❡ ✸✳✽✳✶✸

❊st✐♠❛t❡ t❤❡ ❡rr♦r ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♠♦♥♦t♦♥❡✳ ▲❡t✬s t❛❦❡ ❛ ❧♦♦❦ ❛t t❤❡ ❧❡❢t✲❡♥❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦♥ ❛ s✐♥❣❧❡ ✐♥t❡r✈❛❧✳ ❙✉♣♣♦s❡ t❤❡ ♦♥❧② ✈❛❧✉❡ t❤❛t ♠❛tt❡rs✱ f (xk )✱ ✐s ❦♥♦✇♥✳ ❇❡②♦♥❞ t❤❛t✱ t❤❡ ❢✉♥❝t✐♦♥ ❝❛♥ ❡①❤✐❜✐t ❛ ✈❛r✐❡t② ♦❢ ❜❡❤❛✈✐♦rs✱ ✐♥❝❧✉❞✐♥❣ ❢❛st ❣r♦✇t❤✳ ❚❤❡ ❢❛st❡r f ❣r♦✇s ♣❛st xk ✱ t❤❡ ❧❛r❣❡r ✐s t❤❡ ❡rr♦r ♦❢ Ln ✳ ❆s t❤❡ r❛t❡ ♦❢ t❤✐s ❣r♦✇t❤ ✐s ❧✐♠✐t❧❡ss✱ s♦ ✐s ♦✉r ❡rr♦r ✭❧❡❢t✮✿

❈❛♥ ✇❡ ❝♦♥tr♦❧ t❤❡ s✐③❡ ♦❢ t❤❡ ❡rr♦r❄ ❨❡s✱ ✐❢ ✇❡ ❛r❡ ❛✇❛r❡ ♦❢ ✕ ❛ ♣r✐♦r✐ ✕ t❤❡ ❧✐♠✐t ♦♥ t❤❡ r❛t❡ ♦❢ ❣r♦✇t❤ ♦❢ f ✱ ✐✳❡✳✱ ✐ts ❞❡r✐✈❛t✐✈❡✳ ■❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ❧❡ss t❤❛♥ s♦♠❡ ♥✉♠❜❡r K ✱ t❤❡♥ t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f ✐s ❧❡ss t❤❛♥ K ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❣r❛♣❤ ✇✐❧❧ ❤❛✈❡ t♦ st❛② ✉♥❞❡r t❤❡ ❧✐♥❡ ✇✐t❤ s❧♦♣❡ K ✭r✐❣❤t✮✳ ❚❤✐s ❧✐♥❡ ✐s t❤❡ ✇♦rst✲❝❛s❡ s❝❡♥❛r✐♦✳ ◆♦t❡ t❤❛t s✉❝❤ ❛ r❡str✐❝t✐♦♥ ✐s ❡①♣❡❝t❡❞ t♦ ❜❡ ♣♦ss✐❜❧❡ ✇❤❡♥ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ ✐s ❝♦♥t✐♥✉♦✉s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❊①tr❡♠❡ ❱❛❧✉❡ ❚❤❡♦r❡♠ ✭❱♦❧✉♠❡ ✷✮✳ ❇❡❧♦✇✱ ✇❡ ♣r♦✈✐❞❡ ❛ ❜♦✉♥❞ ❢♦r t❤❡ ❡rr♦r✿

✸✳✽✳

◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥

✷✷✺

❚❤❡♦r❡♠ ✸✳✽✳✶✹✿ ❊rr♦r ❇♦✉♥❞ ♦❢ ❘✐❡♠❛♥♥ ❙✉♠s ■

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ ❢♦r ❛❧❧ x ✐♥ [a, b] ✇❡ ❤❛✈❡✿ |f ′ (x)| ≤ K1 ,

❢♦r s♦♠❡ ♥✉♠❜❡r K1 ✳ ❚❤❡♥ Z b K1 (b − a)2 ≤ Sn − , f dx 2n a

✇❤❡t❤❡r Sn ✐s t❤❡ ❧❡❢t Ln (f ) ♦r t❤❡ r✐❣❤t Rn (f ) ❘✐❡♠❛♥♥ s✉♠ ♦❢ f ✳ Pr♦♦❢✳

■❢ ✇❡ ❤❛✈❡ ❛♥ ❡st✐♠❛t❡ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡✱ f ′ (x) ≤ K ,

♦♥ t❤❡ ✇❤♦❧❡ ❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥ [a, b]✱ ✇❡ ❤❛✈❡ ❛♥ ❡st✐♠❛t❡ ❢♦r t❤❡ ❡rr♦r ♦♥ ❡❛❝❤ ✐♥t❡r✈❛❧ ✜rst✿ Z xk 1 1 f dx ≤ (K∆x) · ∆x = K∆x2 , f (xk )∆x − 2 2 xk−1

❛s t❤❡ ❛r❡❛ ♦❢ t❤✐s tr✐❛♥❣❧❡✳ ❚❤❡♥ ✇❡ ❝♦♠♣✉t❡ ❛♥ ❡st✐♠❛t❡ ❢♦r t❤❡ ❡rr♦r ♦❢ t❤❡ ❧❡❢t✲❡♥❞ ❘✐❡♠❛♥♥ s✉♠ ♦✈❡r t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✿ En

Z b f dx = Ln − a n−1 n−1 Z xk X X = f (xk )∆x − f dx k=0 xk−1 k=0 ! Z n−1 X xk = f (xk )∆x − f dx xk−1 k=0 Z xk n−1 X ≤ f dx f (xk )∆x − xk−1

❇② t❤❡ ❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t② ✐♥ ❈❤❛♣t❡r ✶P❈✲✷.

k=0

≤ =

n−1 X 1

k=0 n−1 X k=0

=

2

1 K 2

n−1 X 1 k=0

K∆x2

2

K



b−a n

2

(b − a)2 n2

1 (b − a)2 . = K 2 n

❙♦✱ ❛♥ ❛

♣r✐♦r✐ ❜♦✉♥❞ ♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ❣✐✈❡s ❛♥ ❛ ♣r✐♦r✐ ❜♦✉♥❞ ♦♥ t❤❡ ❡rr♦r✳

■❢ ♥♦✇ ✇❡ ♥❡❡❞ t♦ s❛② s♦♠❡t❤✐♥❣ s♣❡❝✐✜❝ ❛❜♦✉t t❤❡ ✉♥❦♥♦✇♥ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧✱ ✇❡ ❝❛♥✿ ❈♦r♦❧❧❛r② ✸✳✽✳✶✺✿ ■♥t❡r✈❛❧ ❢♦r ❘✐❡♠❛♥♥ ■♥t❡❣r❛❧

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ ❢♦r ❛❧❧ x ✐♥ [a, b] ✇❡ ❤❛✈❡✿ |f ′ (x)| ≤ K1 ,

✸✳✽✳

◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥

✷✷✻

❢♦r s♦♠❡ ♥✉♠❜❡r

K1 ✳

▲❡t

En = ❚❤❡♥✱ t❤❡ ✐♥t❡❣r❛❧

K1 (b − a)2 . 2n

Z

b

f dx a

❧✐❡s ✇✐t❤✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡r✈❛❧✿

✇❤❡t❤❡r

Sn

✐s t❤❡ ❧❡❢t

Ln (f )



 Sn − En , Sn + En ,

♦r t❤❡ r✐❣❤t

Rn (f )

❘✐❡♠❛♥♥ s✉♠ ♦❢

f✳

❊①❛♠♣❧❡ ✸✳✽✳✶✻✿ ❡rr♦r ❜♦✉♥❞ ♦❢ ❘✐❡♠❛♥♥ s✉♠s

▲❡t✬s t❡st t❤✐s t❤❡♦r❡♠ ♦♥ t❤❡ ✐♥t❡❣r❛❧ Z

1

x2 dx = 1/3 , 0

✇✐t❤ L4 = 0.22 ❝♦♠♣✉t❡❞ ♣r❡✈✐♦✉s❧②✳ ❋✐rst✱ ✇❡ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡✿ f (x) = x2 =⇒ f ′ (x) = 2x .

❚❤❡♥✱ ✇❡ ❝❤♦♦s❡✱ ♦❢ ❝♦✉rs❡✱ K1 = 2 .

◆❡①t✱ E4 =

2 2(1 − 0)2 = = 0.25 . 2·4 8

❚❤❡♥ t❤❡ ✐♥t❡❣r❛❧✬s ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ✇✐t❤✐♥ t❤❡ ✐♥t❡r✈❛❧✿

[L4 − E4 , L4 + E4 ] = [0.22 − 0.25, 0.22 + 0.25] = [−0.03, 0.45] .

❆ ✈❡r② ❝r✉❞❡ ❜✉t ❝♦rr❡❝t ❡st✐♠❛t❡✦ ❲❡ ❝❛♥ ❛♣♣❧② t❤❡ t❤❡♦r❡♠ ❛❣❛✐♥✱ t♦ t❤❡ r✐❣❤t✲❡♥❞ ❛♣♣r♦①✐♠❛t✐♦♥✱ r❡s✉❧t✐♥❣ ✐♥ ❛♥ ✐♥t❡r✈❛❧ ♦❢ t❤❡ s❛♠❡ s✐③❡ ❜✉t ❝❡♥t❡r❡❞ ❛r♦✉♥❞ R4 ✿ [R4 − E4 , R4 + E4 ] = [0.47 − 0.25, 0.47 + 0.25] = [0.22, 0.72] .

❋✉rt❤❡r♠♦r❡✱ s✐♥❝❡ t❤❡ ✈❛❧✉❡ I ♦❢ t❤❡ ✐♥t❡❣r❛❧ ❜❡❧♦♥❣s t♦ ❜♦t❤ ✐♥t❡r✈❛❧s✱ ✐t ❜❡❧♦♥❣s t♦ t❤❡✐r ✐♥t❡rs❡❝t✐♦♥✿ [−0.03, 0.45] ∩ [0.22, 0.72] = [0.22, 0.45] .

❙✐♠✐❧❛r❧②✱ ✇❡ ❤❛✈❡ I ✇✐t❤✐♥ [L10 − E10 , L10 + E10 ] = [0.29 − 0.1, 0.29 + 0.1] = [0.19, 0.39] . ❊①❡r❝✐s❡ ✸✳✽✳✶✼

❋✐♥❞ t❤❡ ✐♥t❡r✈❛❧ ❢♦r t❤❡ ❛❜♦✈❡ ✐♥t❡❣r❛❧ ✉s✐♥❣ R10 ✳ ❋♦r ❝♦♥tr❛st✱ ❧❡t✬s t❛❦❡ ❛ ❧♦♦❦ ❛t t❤❡ ♠✐❞✲♣♦✐♥t ❛♣♣r♦①✐♠❛t✐♦♥✳ ❋✐rst s✉♣♣♦s❡ t❤❛t f ✐s ❧✐♥❡❛r✿

✸✳✽✳

◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥

✷✷✼

❊✈❡♥ t❤♦✉❣❤ t❤❡ s❧♦♣❡s ❛r❡ ❞✐✛❡r❡♥t✱ t❤❡ ❡rr♦r ✐s t❤❡ s❛♠❡✱ ③❡r♦✳ ■t ❛♣♣❡❛rs t❤❡♥ t❤❛t t❤❡

❞❡r✐✈❛t✐✈❡ ❞♦❡s♥✬t

♠❛tt❡r✦ ▲❡t✬s ♥♦✇ ❛❞❞ ❝♦♥❝❛✈✐t②✿

❚❤❡ ❡rr♦r ✐s♥✬t ③❡r♦ ❛s ✐♥ t❤❡ ❢♦r♠❡r ❝❛s❡✳ ❚❤✐s ♦❜s❡r✈❛t✐♦♥ s✉❣❣❡sts t❤❛t t❤❡ ❡rr♦r ✐s ✏❝r❡❛t❡❞✑ ❜② t❤❡

❞❡r✐✈❛t✐✈❡ ♦❢ f ✳

s❡❝♦♥❞

❊①❡r❝✐s❡ ✸✳✽✳✶✽

❲❤❛t ❞✐✛❡r❡♥❝❡ ❞♦❡s ✐t ♠❛❦❡ ✐❢

f

✐s ❝♦♥❝❛✈❡ ❞♦✇♥ ✐♥st❡❛❞ ♦❢ ✉♣❄

❚❤❡ ✐❞❡❛ ♦❢ t❤❡ ❧❛st t❤❡♦r❡♠ ✇❛s t♦ ✉s❡ ❛ ❜♦✉♥❞ ❢♦r t❤❡ ❞❡✈✐❛t❡ t♦♦ ❢❛r ❢r♦♠ ✐ts

❞❡r✐✈❛t✐✈❡ t♦ ♠❛❦❡ s✉r❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ❞♦❡s♥✬t

❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ✭♦♥ ❡❛❝❤ ✐♥t❡r✈❛❧✮✳ ❙✐♠✐❧❛r❧②✱ t❤❡ ✐❞❡❛ ♦❢ t❤❡ ♥❡①t t❤❡♦r❡♠ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ t♦ ♠❛❦❡ s✉r❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ❞♦❡s♥✬t ❞❡✈✐❛t❡ t♦♦ ❢❛r ❢r♦♠ ✐ts

✐s t♦ ✉s❡ ❛ ❜♦✉♥❞ ❢♦r t❤❡

❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✳ ❲❡ ❛❝❝❡♣t t❤❡ r❡s✉❧t ✇✐t❤♦✉t ♣r♦♦❢✳

❚❤❡♦r❡♠ ✸✳✽✳✶✾✿ ❊rr♦r ❇♦✉♥❞ ♦❢ ❘✐❡♠❛♥♥ ❙✉♠s ■■ ❙✉♣♣♦s❡ ❢♦r ❛❧❧

x

✐♥

[a, b]✱

✇❡ ❤❛✈❡

|f ′′ (x)| ≤ K2 ,

✸✳✽✳

◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥

✷✷✽

❢♦r s♦♠❡ r❡❛❧

K2 ✳

❚❤❡♥

Z b K2 (b − a)3 ≤ Mn (f ) − f dx , 24n2 a

❛♥❞

❙♦✱ ❛♥ ❛

Z b K2 (b − a)3 ≤ Tn (f ) − f dx . 12n2 a

♣r✐♦r✐ ❜♦✉♥❞ ♦♥ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ❣✐✈❡s ❛♥ ❛ ♣r✐♦r✐ ❜♦✉♥❞ ♦♥ t❤❡ ❡rr♦r✳

❊①❡r❝✐s❡ ✸✳✽✳✷✵

❙✉❣❣❡st ❛ s✐♠✐❧❛r t❤❡♦r❡♠ ❢♦r Ln ❛♥❞ Rn ✳ ❍✐♥t✿ ❲❤❛t ✐s t❤❡ ✇♦rst✲❝❛s❡ s❝❡♥❛r✐♦❄ ❚❤✉s✱ t❤❡ tr✉❡ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ❧✐❡s ✇✐t❤✐♥ t❤✐s ✐♥t❡r✈❛❧✿ [Mn − En , Mn + En ] ,

✇❤❡r❡

En =

K2 (b − a)3 . 24n2

❊①❛♠♣❧❡ ✸✳✽✳✷✶✿ ❡rr♦r ❜♦✉♥❞ ♦❢ ❘✐❡♠❛♥♥ s✉♠s✱ ❝♦♥t✐♥✉❡❞

▲❡t✬s ❝♦♥✜r♠ t❤✐s r❡s✉❧t ❢♦r

Z

1

x2 dx = 1/3 0

❛♥❞ M4 = 0.328125 ❝♦♠♣✉t❡❞ ♣r❡✈✐♦✉s❧②✳ ❋✐rst✱ ✇❡ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡s✿ f (x) = x2 =⇒ f ′ (x) = 2x =⇒ f ′′ (x) = 2 .

❚❤❡♥✱ ✇❡ ❝❤♦♦s❡✱ ♦❢ ❝♦✉rs❡✱ K2 = 2 .

◆❡①t✱ E4 =

2(1 − 0)3 2 = = 0.0052083333... 2 24 · 4 24 · 16

❚❤❡♥ t❤❡ ✐♥t❡❣r❛❧✬s ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ✇✐t❤✐♥ t❤❡ ✐♥t❡r✈❛❧✿

[M4 − E4 , M4 + E4 ] = [0.328125 − 0.0052083..., 0.328125 + 0.0052083...] = [0.329166..., 0.333333...] .

■t ❤❛♣♣❡♥s t♦ ❜❡ ❡①❛❝t❧② t❤❡ r✐❣❤t ❡♥❞ ♦❢ t❤❡ ✐♥t❡r✈❛❧✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t K2 ✐s♥✬t ❛♥ ❡st✐♠❛t❡ ❜✉t t❤❡ ❡①❛❝t ✈❛❧✉❡ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✳ ❊①❛♠♣❧❡ ✸✳✽✳✷✷✿ ♠♦r❡ ❝♦♠♣❧❡① ❡rr♦r ❜♦✉♥❞

❆ ♠♦r❡ ❝♦♠♣❧❡① ❡①❛♠♣❧❡ ✐s✿

Z

1

x3 dx = 1/4 . 0

❋✐rst✱ t❤❡ ❡st✐♠❛t❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ✇✐t❤ n = 4✿ M4 = (1/8)3 · 1/4 + (3/8)3 · 1/4 + (5/8)3 · 1/4 + (7/8)3 · 1/4 = 0.2421875 .

❚❤❡♥✱ ✇❡ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡s✿

f (x) = x3 =⇒ f ′ (x) = 3x2 =⇒ f ′′ (x) = 6x .

❲❡ ♥❡❡❞ K2 t♦ s❛t✐s❢②✿

K2 ≥ |f ′′ (x)| = 6x, ❢♦r ❛❧❧ 0 ≤ x ≤ 1 .

✸✳✽✳

◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥

✷✷✾

❚❤❡ ❝❤♦✐❝❡ ✐s t❤❡♥ ♦❜✈✐♦✉s✿ K2 = 6 .

◆❡①t✱ t❤❡ ❡rr♦r ❜♦✉♥❞✿ E4 =

K2 (b − a)3 6(1 − 0)3 6 = = = 0.015625 . 2 2 24n 24 · 4 24 · 16

❚❤❡♥ t❤❡ ✐♥t❡❣r❛❧✬s ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ✇✐t❤✐♥ t❤❡ ✐♥t❡r✈❛❧✿

[M4 − E4 , M4 + E4 ] = [0.242 − .016, 0.242 + 0.016] = [0.226, 0.258] .

■t ✐s✳ ◆♦t❡ t❤❛t t❤❡ ❡①✐st❡♥❝❡ ♦❢ K2 ✐s ❣✉❛r❛♥t❡❡❞ ❜② t❤❡ ❊①tr❡♠❡

❱❛❧✉❡ ❚❤❡♦r❡♠ ♣r♦✈✐❞❡❞ f ′′ ✐s ❝♦♥t✐♥✉♦✉s✳

❊①❛♠♣❧❡ ✸✳✽✳✷✸✿ ❤♦✇ t♦ ❣✉❛r❛♥t❡❡ ❛❝❝✉r❛❝②

❆t t❤❡ ♥❡①t✱ ♠♦r❡ ♣r❛❝t✐❝❛❧✱ ❧❡✈❡❧✱ ✇❡ ❛r❡ ❛s❦❡❞ t♦ ❡st✐♠❛t❡ ❛♥ ✐♥t❡❣r❛❧ ✇✐t❤ ❛ ❡①❛♠♣❧❡✱ s✉♣♣♦s❡ ✇❡ ♥❡❡❞ t♦ ❦♥♦✇ ✇✐t❤✐♥ 0.1 t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧✿ Z

❣✐✈❡♥ ❛❝❝✉r❛❝②✳ ❋♦r

1

x3 dx . 0

❚❤❡♥ t❤❡ ❛♥s✇❡r ❛❜♦✈❡ ❛♣♣❧✐❡s ❛s E = 0.015625 < 0.1✳ ❲❤❛t ✐❢ t❤❡ ❛❝❝✉r❛❝② ♥❡❡❞s t♦ ❜❡ 0.01❄ ❚❤❡♥ n = 4 ✐s t♦♦ s♠❛❧❧✦ ▲❡t✬s tr② n = 5✳ ❲❡ ❤❛✈❡✿ E5 =

K2 (b − a)3 6(1 − 0)3 6 = = = 0.01 . 24 · 52 24 · 52 24 · 25

❋✉rt❤❡r♠♦r❡✱ ✇❡ ♦❜s❡r✈❡ t❤❛t ✐♥ ♦r❞❡r t♦ ❡♥s✉r❡ t❤❛t t❤❡ ❡rr♦r ✐s ❧❡ss t❤❛♥ s♦♠❡ ε > 0✱ ✇❡ s✐♠♣❧② ♥❡❡❞ t♦ ✜♥❞ n t❤❛t s❛t✐s✜❡s✿ 3 6(1 − 0) ≤ ε. 24 · n2

■♥ ❣❡♥❡r❛❧✱ ✇❡ ❛r❡ s♦❧✈✐♥❣ t❤❡ ✐♥❡q✉❛❧✐t②✿ En =

K2 (b − a)3 ≤ ε. 24n2

❈♦r♦❧❧❛r② ✸✳✽✳✷✹✿ ❊st✐♠❛t✐♦♥ ♦❢ ❊rr♦r ♦❢ ◆✉♠❡r✐❝❛❧ ■♥t❡❣r❛t✐♦♥ ❙✉♣♣♦s❡ ❢♦r ❛❧❧

x

✐♥

[a, b]

✇❡ ❤❛✈❡

|f ′′ (x)| ≤ K2 , ❢♦r s♦♠❡ r❡❛❧

K2 ✳

❚❤❡♥✱ ❢♦r ❛♥② ❣✐✈❡♥

Z ❧✐❡s ✇✐t❤✐♥

ε

❢r♦♠

Mn

ε > 0✱

t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧

b

f dx a

♣r♦✈✐❞❡❞

n≥

r

K2 (b − a)3 . 24ε

✸✳✾✳

▲❡♥❣t❤s ♦❢ ❝✉r✈❡s

✷✸✵

❊①❡r❝✐s❡ ✸✳✽✳✷✺ ❈r❡❛t❡ ❛ s♣r❡❛❞s❤❡❡t t♦ ❛✉t♦♠❛t❡ t❤❡s❡ ❝♦♠♣✉t❛t✐♦♥s✳

✸✳✾✳ ▲❡♥❣t❤s ♦❢ ❝✉r✈❡s

❲❡ ❤❛✈❡ s✉❝❝❡ss❢✉❧❧② ✉s❡❞ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❝♦♥str✉❝t✐♦♥ t♦ ❛♣♣r♦①✐♠❛t❡ ❛♥❞✱ ❛t t❤❡ ❧✐♠✐t✱ ❝♦♠♣✉t❡ t❤❡

❛r❡❛s

✉♥❞❡r t❤❡ ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s✳ ■t ✇♦✉❧❞ ❜❡✱ ❤♦✇❡✈❡r✱ ❛ ❣r❛✈❡ ♠✐st❛❦❡ t♦ t❤✐♥❦ t❤❛t t❤❡ st❡♣ ❢✉♥❝t✐♦♥

♣r♦❞✉❝❡❞ ❜② t❤✐s ❝♦♥str✉❝t✐♦♥ ❝❛♥ s❡r✈❡ ❛s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐ts❡❧❢✿

❚❤❡ r❡❛s♦♥ ✐s r❡✈❡❛❧❡❞ ✇❤❡♥ ✇❡ ✇❛t❝❤ ❤♦✇ s♣❡❝t❛❝✉❧❛r❧② t❤✐s ✐❞❡❛ ❢❛✐❧s ✇❤❡♥ ❛♣♣❧✐❡❞ t♦ ❝♦♠♣✉t✐♥❣ t❤❡

❧❡♥❣t❤s

♦❢ ❝✉r✈❡s✳

❊①❛♠♣❧❡ ✸✳✾✳✶✿ str❛✐❣❤t ❧✐♥❡ ▲❡t✬s ❝♦♥s✐❞❡r ❛ ✈❡r② s✐♠♣❧❡ ❝❛s❡ ♦❢

y = f (x) = x

♦✈❡r

[0, 1]✳

❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❝✉r✈❡ ✇✐t❤ t❤❡

❤♦r✐③♦♥t❛❧ s❡❣♠❡♥ts ❧♦♦❦s ❥✉st ❛s ❣♦♦❞ ❛s t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤✿

❚❤❡ r❡s✉❧t ✐s ✐❧❧✉str❛t❡❞ ❢♦r ❛ ♣❛rt✐t✐♦♥ ✇✐t❤

n = 10

✐♥t❡r✈❛❧s ♦❢ ❡q✉❛❧ ❧❡♥❣t❤ ❛♥❞ t❤❡ ❧❡❢t ❡♥❞s ❛s

s❡❝♦♥❞❛r② ♥♦❞❡s✳

❆ ♣r♦❜❧❡♠ ❛♣♣❡❛rs ✇❤❡♥ ✇❡ ❧♦♦❦ ❛t t❤❡ ❛❝t✉❛❧ ♥✉♠❜❡rs✳ t❤❡

P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✳

❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❣r❛♣❤ ✐s

√

2

❜②

▼❡❛♥✇❤✐❧❡✱ t❤❡ t♦t❛❧ ❧❡♥❣t❤ ♦❢ t❤❡ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥ts t❤❛t ♠❛❦❡ ✉♣ t❤❡

❣r❛♣❤ ♦❢ t❤❡ r❡s✉❧t✐♥❣ st❡♣ ❢✉♥❝t✐♦♥ ✐s

1❀

✐t✬s s✐♠♣❧② t❤❡ ❜♦tt♦♠ ♦❢ t❤❡ ❜✐❣ tr✐❛♥❣❧❡✳ ❚♦♦ ❧♦✇✦

✸✳✾✳

▲❡♥❣t❤s ♦❢ ❝✉r✈❡s

✷✸✶

❖♥❡ ♠❛② tr② t♦ ✜① t❤❡ ♣r♦❜❧❡♠ ❜② ❛❞❞✐♥❣ t❤❡ ✈❡rt✐❝❛❧ s❡❣♠❡♥ts t♦ ♦✉r ❡st✐♠❛t❡ ♦❢ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧✳ ❚❤❡♥✱ t❤❡ ❡st✐♠❛t❡ ❜❡❝♦♠❡s 2❀ ✐t✬s s✐♠♣❧② t❤❡ s✉♠ ♦❢ t❤❡ ♦t❤❡r t✇♦ s✐❞❡s ♦❢ t❤❡ ❜✐❣ tr✐❛♥❣❧❡✳ ❚♦♦ ❤✐❣❤✦ ■t ✐s ✐♠♣♦rt❛♥t t❤❛t t❤❡ ♥✉♠❜❡rs ✇♦♥✬t ❝❤❛♥❣❡ ❡✈❡♥ ✐❢ ✇❡ st❛rt t♦ r❡✜♥❡ t❤❡ ♣❛r✲ t✐t✐♦♥✳ ■♥ ❝♦♥tr❛st✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❛r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡✱ Ln ✱ ✐s ❣❡tt✐♥❣ ❜❡tt❡r ❛s ✇❡ ✐♥❝r❡❛s❡ n✳ ❚♦p ✉♥❞❡rst❛♥❞ t❤❡ r❡❛s♦♥ ❢♦r t❤✐s ❞✐s❝r❡♣❛♥❝②✱ ❧❡t✬s ❝♦♥s✐❞❡r t❤❡ ❧✐♥❡ y = g(x) = x/2✳ ■ts ❛❝t✉❛❧ ❧❡♥❣t❤ ✐s 12 + (1/2)2 ≈ 1.19 ❜② t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✇✐t❤ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥ts ✐s st✐❧❧ ❡q✉❛❧ t♦ 1 ❛♥❞ t❤❡ ♦♥❡ ✇✐t❤ ❜♦t❤ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥ts ✐s 1.5✳ ❚❤❡ ❡st✐♠❛t❡s ❛r❡ st✐❧❧ ♦✛ ❜✉t t❤❡② ❛r❡ ❝❧♦s❡r t♦ t❤❡ tr✉t❤✦

❲❤❛t ❡①♣❧❛✐♥s t❤❡ ❞✐✛❡r❡♥❝❡❄ ❚❤❡ s❧♦♣❡✳ ❚♦ ❝♦♥✜r♠ t❤✐s ✐❞❡❛✱ ❥✉st t❛❦❡ t❤❡ ❧✐♥❡ ✇✐t❤ ❚❤❡♥ t❤❡ ❡st✐♠❛t❡ ✐s ❡q✉❛❧ t♦ ✐ts ❛❝t✉❛❧ ❧❡♥❣t❤✦ ■♥ ❢❛❝t✱ t❤❡ ❝❛s❡ ♦❢ ❛ ❧✐♥❡❛r f ✐s ✈❡r② s✐♠♣❧❡✿ q ❜❛s❡ 2 + ❤❡✐❣❤t ▲❡♥❣t❤ =

2

=

q

③❡r♦

s❧♦♣❡✳

❜❛s❡ 2 + ( ❜❛s❡ · s❧♦♣❡ )2 .

❊①❡r❝✐s❡ ✸✳✾✳✷

❙❤♦✇ t❤❛t t❤❡ ❝♦♥❝❧✉s✐♦♥s r❡♠❛✐♥ ✈❛❧✐❞ ♥♦ ♠❛tt❡r ✇❤❛t ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ ♦❢ [0, 1] ✇❡ ❝❤♦♦s❡✳ ❊①❡r❝✐s❡ ✸✳✾✳✸

❙❤♦✇ t❤❛t t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛ st❡♣ ❢✉♥❝t✐♦♥ ♦✈❡r ❛♥② ♣❛rt✐t✐♦♥ ♦❢ [a, b] ✐s b − a✳ ❚❤❡ ❧❡ss♦♥ ✐s t❤❛t t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❣r❛♣❤ ✕ ✉♥❧✐❦❡ t❤❡ ♦♥❡ ❢♦r t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ❣r❛♣❤ ✕ s❤♦✉❧❞ ❞❡♣❡♥❞ ♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❊①❛♠♣❧❡ ✸✳✾✳✹✿ ❧❡♥❣t❤ ♦❢ ❝✐r❝❧❡

❇✉t ✜rst✱ ❧❡t✬s ❝♦♠♣✉t❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝✐r❝❧❡ ❛s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥✳ ❲❡ ❝♦♠♣✉t❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✉♣♣❡r ❤❛❧❢ ♦❢ t❤❡ ✉♥✐t ❝✐r❝❧❡ ❜② ✜rst r❡♣r❡s❡♥t✐♥❣ ✐t ❛s t❤❡ ❣r❛♣❤ ♦❢ ❛ s✐♠♣❧❡ ❢✉♥❝t✐♦♥✿ √ f (x) = 1 − x2 .

✸✳✾✳

▲❡♥❣t❤s ♦❢ ❝✉r✈❡s

✷✸✷

❚❤❡ ✐❞❡❛ ✐s s✐♠♣❧❡✿

P❧❛❝❡ ♣♦✐♥ts ♦♥ t❤❡ ❝✉r✈❡✱ ❝♦♥♥❡❝t t❤❡♠ ❝♦♥s❡❝✉t✐✈❡❧② ❜② ❡❞❣❡s✱ ❛♥❞ t❤❡♥

❛♣♣r♦①✐♠❛t❡ t❤❡ ❝✉r✈❡ ✇✐t❤ ❛ ❝♦♥t✐♥✉♦✉s ❝✉r✈❡ ♠❛❞❡ ♦❢ t❤❡s❡ ❡❞❣❡s✳ ❲❡ ❤❛✈❡ ❛ ❧✐st ♦❢ t❤❡ ✈❛❧✉❡s ♦❢

x

✐♥ t❤❡ ✜rst ❝♦❧✉♠♥✿

x0 , x1 , ..., xn , ❛♥❞ t❤❡ ❧✐st ♦❢ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡s ♦❢

y

✐♥ t❤❡ ♥❡①t ❝♦❧✉♠♥✿

y0 = f (x0 ), y1 = f (x1 ), ..., yn = f (xn ) . ■♥ t❤❡ t❤✐r❞ ❝♦❧✉♠♥✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ s❡❣♠❡♥ts ✈✐❛ t❤❡

lk = ❲❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛✿

p

❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ✿

(xk+1 − xk )2 + (yk+1 − yk )2 .

❂❙◗❘❚✭✭❘❈❬✲✷❪✲❘❬✲✶❪❈❬✲✷❪✮✂ ✷✰✭❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪✮✂ ✷✮

❆s ✇❡ ✐♥❝r❡❛s❡ t❤❡ ♥✉♠❜❡r ♦❢ s❡❣♠❡♥ts✱

n✱ t❤❡ r❡s✉❧t t❤❛t ✇❡ ❦♥♦✇ t♦ ❜❡ ❝♦rr❡❝t✱ π ✱ ✐s ❜❡✐♥❣ ❛♣♣r♦❛❝❤❡❞✳

❲❡ ✇✐❧❧ s❡❡ ✐♥ ❈❤❛♣t❡r ✹ ❛ ❜❡tt❡r ✇❛② t♦ r❡♣r❡s❡♥t ❝✉r✈❡s ❛♥❞ ❡s♣❡❝✐❛❧❧② t❤❡ ❝✐r❝❧❡✳

❲❡✱ ❥✉st ❛s ❛❧✇❛②s✱ st❛rt ✇✐t❤ ❛ ❞✐s❝r❡t❡ s✐t✉❛t✐♦♥✳ ❲❡ s✐♠♣❧② ❤❛✈❡ ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡✳ ❙✉❝❤ ❛ s❡q✉❡♥❝❡ ✐s s❡❡♥ ❛s ❛ ✏❝✉r✈❡✑ ✐❢ ✇❡ ♣r♦❝❡❡❞ ❢r♦♠ ♣♦✐♥t t♦ ♣♦✐♥t ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡✳ ❚❤❡ ❧❡♥❣t❤s ♦❢ t❤❡s❡ s❡❣♠❡♥ts ❛r❡ ❢♦✉♥❞ ❜② t❤❡

❉✐st❛♥❝❡ ❋♦r♠✉❧❛✱ ❥✉st

❛s ✐♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✳ ■t ✐s t❤✐s s✐♠♣❧❡✦ ◆♦✇✱ s♦♠❡t❤✐♥❣ ♠♦r❡ s♣❡❝✐✜❝✳ ❲❤❛t ✐❢ t❤❡s❡ ♣♦✐♥ts ❢♦r♠ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♥♦❞❡s ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢

(xk−1 , f (xk−1 ))

■t ✐s t❤❡

t♦

❞❡✜♥❡❞ ❛t t❤❡

[a, b]❄

❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ ❡❛❝❤ ✐♥t❡r✈❛❧ ❢r♦♠

y = f (x)

(xk , f (xk ))✳

[xk−1 , xk ], k = 1, 2, ..., n ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳

❚❤❡ ❣r❛♣❤ ♦❢

f

❣♦❡s ✭❥✉♠♣s✮

❲❡ t❤❡♥ ❝♦♥str✉❝t ❛ s❧♦♣❡❞ s❡❣♠❡♥t ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ♣♦✐♥ts✿

s❡❝❛♥t ❧✐♥❡ ✦ ❆ r✐❣❤t tr✐❛♥❣❧❡ ✐s ❢♦r♠❡❞ ❜② t❤❡s❡ t✇♦ s❡❣♠❡♥ts✿

✸✳✾✳ ▲❡♥❣t❤s ♦❢ ❝✉r✈❡s

✷✸✸

• ❤♦r✐③♦♥t❛❧ [xk−1 , xk ]✱ ❛♥❞

• ✈❡rt✐❝❛❧ ❢r♦♠ f (xk−1 ) t♦ f (xk )✱ ♦r ✈✐❝❡ ✈❡rs❛✳

❚❤❡ ❧❡♥❣t❤s ♦❢ t❤❡s❡ s✐❞❡s ❛r❡✿

• ❤♦r✐③♦♥t❛❧ ✭❜❛s❡✱ t❤❡ r✉♥✮✿ h = xk − xk−1 = ∆xk ✱ ❛♥❞

• ✈❡rt✐❝❛❧ ✭❤❡✐❣❤t✱ t❤❡ r✐s❡✮✿ |f (xk ) − f (xk−1 )| = ∆yk ✳

❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❡❞❣❡ ✭t❤❡ ❤②♣♦t❡♥✉s❡ ♦❢ t❤❡ tr✐❛♥❣❧❡✮ ✐s t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ q q ∆x2k + ∆yk2 = ∆x2k + (f (xk ) − f (xk−1 ))2 .

❚❤✉s✱ t❤❡ ❢✉❧❧ ❧❡♥❣t❤ ♦❢ t❤❡ tr✐♣ ❛❧♦♥❣ t❤❡s❡ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤ ♦❢ f ✐s ❡q✉❛❧ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿ n q n q X X 2 2 ❚♦t❛❧ ❧❡♥❣t❤ = ∆xk + ∆yk = ∆x2k + (f (xk ) − f (xk−1 ))2 k=1

k=1

❊①❛♠♣❧❡ ✸✳✾✳✺✿ ❧❡♥❣t❤ ♦❢ ♣❛r❛❜♦❧❛

❲❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❧❡♥❣t❤ ♦❢ ❛ ♣❛r❛❜♦❧❛ y = x2 , 0 ≤ x ≤ 1✱ ❜❡❧♦✇✿

❲❤❛t ✐❢ ♥♦✇ ✇❡ ❤❛✈❡ ❛ ❝♦♥t✐♥✉♦✉s ❝✉r✈❡✱ t❤❡ ❣r❛♣❤ ♦❢ y = f (x) ❞❡✜♥❡❞ ♦♥ t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧ [a, b]❄ ❚❤❡s❡ ❡st✐♠❛t❡s ❛r❡ ❡①❛❝t ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛ ❧✐♥❡❛r f ✳ ❲❡ ✇✐❧❧✱ ❥✉st ❛s ❜❡❢♦r❡✱ ✉s❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡r✈❛❧ t♦ s♣❧✐t t❤❡ ❝✉r✈❡ ✐♥t♦ s♠❛❧❧❡r ♣✐❡❝❡s ❜✉t t❤❡♥ ✇❡ ✇✐❧❧ ❛♣♣r♦①✐♠❛t❡ t❤❡s❡ ♣✐❡❝❡s ♥♦t ✇✐t❤ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥ts ❜✉t ✇✐t❤ s❡❝❛♥t ❧✐♥❡s✳

✸✳✾✳ ▲❡♥❣t❤s ♦❢ ❝✉r✈❡s

✷✸✹

▲❡t✬s ❞❡✜♥❡ ❛♥❞ t❤❡♥ ❝♦♠♣✉t❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ y = f (x) ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ [a, b]✳ ❚❤❡♥✱ t❤❡ ❢✉❧❧ ❧❡♥❣t❤ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ s✉♠ ♦❢ ❛❧❧ n ♦❢ t❤♦s❡✱ ❛s ❢♦❧❧♦✇s✿ ❧❡♥❣t❤ ≈ Ln =

n q X k=1

∆x2k + (f (xk ) − f (xk−1 ))2 .

❚❤❡ ❧✐♠✐t ♦❢ t❤✐s s❡q✉❡♥❝❡ ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡✦ ❲❡ ✇♦✉❧❞ ♣r❡❢❡r✱ ❤♦✇❡✈❡r✱ t♦ ❝♦♥♥❡❝t t❤✐s ✐❞❡❛ ❜❛❝❦ t♦ t❤❡ ✐❞❡❛ ♦❢ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ❛♥❞ t♦ t❤❡ ♠❛❝❤✐♥❡r② t❤❛t ✇❡ ❤❛✈❡ ❞❡✈❡❧♦♣❡❞✳ ❯♥❢♦rt✉♥❛t❡❧②✱ t❤✐s ❡①♣r❡ss✐♦♥ ❞♦❡s♥✬t ❧♦♦❦ ❧✐❦❡ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥✦ ❲❤❛t ✐s ♠✐ss✐♥❣ ✐s ∆xk ❛s ❛ ♠✉❧t✐♣❧❡ ✐♥ ❡❛❝❤ ♦❢ t❤❡ t❡r♠s✳ ❲❡ ✇✐❧❧ ❤❛✈❡ t♦ ❝r❡❛t❡ ✐t ❜② ♠❛♥✐♣✉❧❛t✐♥❣ t❤❡ ❢♦r♠✉❧❛✳ ❲❡ ❝❛♥ ✉s❡ t❤❡ ✐♥s✐❣❤t ❢r♦♠ t❤❡ ❡❛r❧✐❡r ❞✐s❝✉ss✐♦♥✿ ❚❤❡r❡ ♠✉st ❜❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f ♣r❡s❡♥t✳ ❚❤✐s ♠❡❛♥s t❤❛t ✇❡ ♠✉st s❡❡ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ✐♥ t❤❡ ❢♦r♠✉❧❛✦ ❲❤❡r❡ ✐s ✐t❄ ❲❡ s❡❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜✉t ♥♦t t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳ ❲❡ ✇✐❧❧ ♥❡❡❞ t♦ ❝r❡❛t❡ ✐t ❜② ♠❛♥✐♣✉❧❛t✐♥❣ t❤❡ ❢♦r♠✉❧❛✳ ❚❤❡ t✇♦ ❣♦❛❧s ♠❛t❝❤ ✉♣✿ ❲❡ ❞✐✈✐❞❡ ❛♥❞ ♠✉❧t✐♣❧② ❡❛❝❤ t❡r♠ ❜② ∆xk ✳ ❚✇♦ ❜✐r❞s ✇✐t❤ ♦♥❡ st♦♥❡✿ ❙✉♠ ♦❢ ❧❡♥❣t❤s

n q X ∆x2k + (f (xk ) − f (xk−1 ))2 = k=1

n q X ∆xk = ∆x2k + (f (xk ) − f (xk−1 ))2 · ∆xk k=1 s n X 1 (∆x2k + (f (xk ) − f (xk−1 ))2 ) · ∆xk = 2 ∆x k k=1 s 2  n X f (xk ) − f (xk−1 ) · ∆xk . = 1+ ∆x k k=1

❍❡r❡ ✐s ∆xk . ❍❡r❡ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✳

❇✉t t❤✐s ✐s st✐❧❧ ♥♦t t❤❡ ❘✐❡♠❛♥♥ s✉♠✳ ❚❤❡ ❡①♣r❡ss✐♦♥ t❤❛t ♣r❡❝❡❞❡s ∆xk ✇♦✉❧❞ ❤❛✈❡ t♦ ❜❡ t❤❡ ✈❛❧✉❡ ♦❢ s♦♠❡ ❢✉♥❝t✐♦♥ ❡✈❛❧✉❛t❡❞ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❲❡ ❤❛✈❡♥✬t s♣❡❝✐✜❡❞ t❤♦s❡ ②❡t ❛♥❞ t❤❛t✬s ❛ ❣♦♦❞ ♥❡✇s ❜❡❝❛✉s❡ ♥♦✇ ✐t ✐s ♦✉r ❝❤♦✐❝❡✦ ❲❡ ❛♣♣❧②✱ ❛s ✇❡✬✈❡ ❞♦♥❡ ♠❛♥② t✐♠❡s ❜❡❢♦r❡✱ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠✳ ❚❤❡r❡ ✐s s♦♠❡ ck ✐♥ t❤❡ ✐♥t❡r✈❛❧ [xk−1 , xk ] s✉❝❤ t❤❛t t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛t t❤❛t ❧♦❝❛t✐♦♥ ✐s ❡q✉❛❧ t♦ t❤❡ s❧♦♣❡ ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧✿ ❚❤❡r❡❢♦r❡✱

f (xk ) − f (xk−1 ) = f ′ (ck ) . ∆xk

n q X ❙✉♠ ♦❢ ❧❡♥❣t❤s = 1 + (f ′ (ck ))2 · ∆xk . k=1

❋✐♥❛❧❧②✱ t❤✐s ✐s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦✈❡r t❤❡ ♣❛rt✐t✐♦♥ ♦❢ [a, b] ✇✐t❤ t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s c1 , ..., cn ❀ t❤❡ ❢✉♥❝t✐♦♥ ✐s✿ q g(x) =

1 + (f ′ (x))2 .

❏✉st ❛s ❢♦r t❤❡ ❛r❡❛ ✭♠❛ss✱ ✇♦r❦✱ ❡t❝✳✮✱ t❤❡ ❛♥❛❧②s✐s ❛❜♦✈❡ r❡✈❡❛❧s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ♥❡✇ ❝♦♥❝❡♣t✳

❉❡✜♥✐t✐♦♥ ✸✳✾✳✻✿ ❧❡♥❣t❤ ♦❢ ❝✉r✈❡ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡ ❣✐✈❡♥ ❜② t❤❡ ❣r❛♣❤ y = f (x) ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ♦✈❡r ✐♥t❡r✈❛❧ [a, b] ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡❣r❛❧✿ Z bq ▲❡♥❣t❤ = 1 + (f ′ )2 dx a

✸✳✾✳

▲❡♥❣t❤s ♦❢ ❝✉r✈❡s

✷✸✺

✐❢ ✐t ❡①✐sts✳

◆♦t❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥

f

✐ts❡❧❢ ✐s

❛❜s❡♥t

❢r♦♠ t❤❡ ❢♦r♠✉❧❛✦ ❚❤❛t✬s ✉♥❞❡rst❛♥❞❛❜❧❡ ❜❡❝❛✉s❡ ♦♥❧② t❤❡ s❤❛♣❡

✭❣✐✈❡♥ ❜② t❤❡ ❞❡r✐✈❛t✐✈❡✮ ❛♥❞ ♥♦t t❤❡ ❧♦❝❛t✐♦♥ ♠❛tt❡rs ❢♦r t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡✳ ❈❤❛♣t❡r ✷❉❈✲✺ t❤❛t ✐❢

f ′ = g′✱

t❤❡♥

f =g+C

■♥ ❢❛❝t✱ ✇❡ ❦♥♦✇ ❢r♦♠

❛♥❞ ✐ts ❣r❛♣❤ ❤❛s t❤❡ s❛♠❡ ❧❡♥❣t❤✳

❚❤❡♦r❡♠ ✸✳✾✳✼✿ ▲❡♥❣t❤ ♦❢ ❈✉r✈❡

■❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ✐s ❝♦♥t✐♥✉♦✉s✱ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡ ❣✐✈❡♥ ❜② t❤❡ ❣r❛♣❤ y = f (x) ♦✈❡r [a, b] ✐s ❞❡✜♥❡❞✳ Pr♦♦❢✳ ❲❡ ♥❡❡❞ t❤❡ ❡①tr❛ ❝♦♥❞✐t✐♦♥ t♦ ❡♥s✉r❡ t❤❛t t❤❡

▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❛♣♣❧✐❡s ❛♥❞ t❤❡ r❡s✉❧t✐♥❣ ❢✉♥❝t✐♦♥

✐s ✐♥t❡❣r❛❜❧❡✳

❊①❛♠♣❧❡ ✸✳✾✳✽✿ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ ❝✐r❝❧❡ ■t ✐s t✐♠❡ t♦ ♣r♦✈❡ t❤❛t t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

R

✐s

2πR✳

❲❡ r❡♣r❡s❡♥t✱ ❛❣❛✐♥✱ t❤❡ ✉♣♣❡r ❤❛❧❢ ♦❢ t❤❡ ❝✐r❝❧❡ ❜② t❤❡ ❣r❛♣❤✿

y = f (x) =

√

❚❤❡♥✱

f ′ (x) = − √ ❲❡ ❛♣♣❧② t❤❡ ❢♦r♠✉❧❛✿

❍❛❧❢ ♦❢ t❤❡ ❧❡♥❣t❤

R 2 − x2 . x . − x2

R2

Z bq = 1 + (f ′ (x))2 dx a s 2  Z R x dx 1 + −√ = 2 − x2 R −R Z Rr x2 1+ 2 = dx R − x2 −R r Z R R2 = dx R 2 − x2 −R Z R 1 √ dx =R· R 2 − x2 −R = ... = R·π.

❱✐❛ tr✐❣ s✉❜st✐t✉t✐♦♥✳

✸✳✶✵✳

❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ t❤r❡❡

✷✸✻

❊①❡r❝✐s❡ ✸✳✾✳✾

❋✐♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❡❣♠❡♥t ♦❢ t❤❡ ♣❛r❛❜♦❧❛ y = x2 ❢r♦♠ (0, 0) t♦ (1, 1)✳ ❊①❡r❝✐s❡ ✸✳✾✳✶✵

❋✐♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❡❣♠❡♥t ♦❢ t❤❡ ❝✉r✈❡ y = x3 ❢r♦♠ (0, 0) t♦ (1, 1)✳ ❊①❡r❝✐s❡ ✸✳✾✳✶✶

❋✐♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❡❣♠❡♥t ♦❢ t❤❡ ❝✉r✈❡ y = sin x ❛❜♦✈❡ t❤❡ ✐♥t❡r✈❛❧ [0, π]✳

✸✳✶✵✳ ❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ t❤r❡❡

❲❡ ♣✉rs✉❡❞ t❤❡ ✐❞❡❛ ♦❢ ❛ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐♥ ♦r❞❡r t♦ tr❛♥s✐t✐♦♥ ❢r♦♠

• •

❣❡♦♠❡tr② ✿ ♣♦✐♥ts✱ ❧✐♥❡s✱ tr✐❛♥❣❧❡s✱ ❝✐r❝❧❡s✱ ♣❧❛♥❡s✱ ❝✉❜❡s✱ s♣❤❡r❡s✱ ❡t❝✳✱ t♦ ❛❧❣❡❜r❛ ✿ ♥✉♠❜❡rs✱ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ♥✉♠❜❡rs✱ ❢✉♥❝t✐♦♥s✱ ❡t❝✳

❚❤✐s ❛♣♣r♦❛❝❤ ❛❧❧♦✇s ✉s t♦ s♦❧✈❡ ❣❡♦♠❡tr✐❝ ♣r♦❜❧❡♠s ✕ s✉❝❤ ❛s ✜♥❞✐♥❣ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ✕ ✇✐t❤♦✉t ♠❡❛s✉r✐♥❣✳ ❘❡❝❛❧❧ ❤♦✇✱ ❢♦r ❞✐♠❡♥s✐♦♥ 2✱ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s ❜✉✐❧t✿

❲❡ ❤❛✈❡ ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡✿ ❧♦❝❛t✐♦♥ P ←→ ♣❛✐r (x, y). ❚❤✐s ✐s ❤♦✇ ✐t ✇♦r❦s✿

✸✳✶✵✳

❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ t❤r❡❡

❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤

❞✐♠❡♥s✐♦♥ 3✳

✷✸✼

❚❤❡r❡ ✐s ♠✉❝❤ ♠♦r❡ ❣♦✐♥❣ ♦♥✿

■t ✐s ❜✉✐❧t ✐♥ s❡✈❡r❛❧ st❛❣❡s✿ ✶✳ ❚❤r❡❡

❝♦♦r❞✐♥❛t❡ ❛①❡s ❛r❡ ❝❤♦s❡♥✿

t❤❡ x✲❛①✐s✱ t❤❡ y ✲❛①✐s✱ ❛♥❞ t❤❡ z ✲❛①✐s✳

✷✳ ❚❤❡ t✇♦ ❛①❡s ❛r❡ ♣✉t t♦❣❡t❤❡r ❛t t❤❡✐r ♦r✐❣✐♥s s♦ t❤❛t ✐t ✐s ❛ 90✲❞❡❣r❡❡ t✉r♥ ❢r♦♠ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ♦❢ ♦♥❡ ❛①✐s t♦ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ♥❡①t ✕ ❢r♦♠ x t♦ y t♦ z t♦ x✳ ✸✳ ❯s❡ t❤❡ ♠❛r❦s ♦♥ t❤❡ ❛①✐s t♦ ❞r❛✇ ❛ ❣r✐❞✳

❆❧t❡r♥❛t✐✈❡❧②✱ t❤❡ s②st❡♠ ✐s ❜✉✐❧t ❢r♦♠ t❤r❡❡ ❝♦♣✐❡s ♦❢ t❤❡ ❈❛rt❡s✐❛♥ ♣❧❛♥❡✿ t❤❡ xy ✲♣❧❛♥❡✱ t❤❡ yz ✲♣❧❛♥❡✱ ❛♥❞ t❤❡ zx✲♣❧❛♥❡✳ ❚❤❡② ❛r❡ ❝❛❧❧❡❞ t❤❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s✳

✸✳✶✵✳

❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ t❤r❡❡

✷✸✽

❲❡ ❤❛✈❡✱ ❛s ❜❡❢♦r❡✱ ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ❧♦❝❛t✐♦♥s ❛♥❞ t❤❡✐r ❛❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥s✿ ❧♦❝❛t✐♦♥ P ←→ tr✐♣❧❡ (x, y, z) ■t ✇♦r❦s ✐♥

❜♦t❤ ❞✐r❡❝t✐♦♥s✳

❋♦r ❡①❛♠♣❧❡✱ s✉♣♣♦s❡ P ✐s ❛ ❧♦❝❛t✐♦♥ ✐♥ t❤✐s s♣❛❝❡✳ ❲❡ t❤❡♥ ✜♥❞ t❤❡ ❞✐st❛♥❝❡s ❢r♦♠ t❤❡ t❤r❡❡ ♣❧❛♥❡s t♦ t❤❛t ❧♦❝❛t✐♦♥ ✕ ♣♦s✐t✐✈❡ ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥ ✕ ❛♥❞ t❤❡ r❡s✉❧t ✐s t❤❡ t❤r❡❡ ❝♦♦r❞✐♥❛t❡s ♦❢ P ✱ s♦♠❡ ♥✉♠❜❡rs x✱ y ✱ ❛♥❞ z ✿

❚❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ yz ✲♣❧❛♥❡ ✐s ♠❡❛s✉r❡❞ ❛❧♦♥❣ t❤❡ x✲❛①✐s✱ ❡t❝✳ ❲❡ ✉s❡ t❤❡ ♥❡❛r❡st ♠❛r❦ t♦ s✐♠♣❧✐❢② t❤❡ t❛s❦✳ ❈♦♥✈❡rs❡❧②✱ s✉♣♣♦s❡ x, y, z ❛r❡

♥✉♠❜❡rs✳

• ❋✐rst✱ ✇❡ ♠❡❛s✉r❡ x ❛s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ yz ✲♣❧❛♥❡ ✕ ♣♦s✐t✐✈❡ ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥ t❤❡ ♥❡❣❛t✐✈❡ ❞✐r❡❝t✐♦♥ ✕ ❛❧♦♥❣ t❤❡ x✲❛①✐s ❛♥❞ ❝r❡❛t❡ ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ yz ✲♣❧❛♥❡✳ • ❙❡❝♦♥❞✱ ✇❡ ♠❡❛s✉r❡ y ❛s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ xz ✲♣❧❛♥❡ ❛❧♦♥❣ t❤❡ y ✲❛①✐s ❛♥❞ ❝r❡❛t❡ ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ xz ✲♣❧❛♥❡✳ • ❚❤✐r❞✱ ✇❡ ♠❡❛s✉r❡ z ❛s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ xy ✲♣❧❛♥❡ ❛❧♦♥❣ t❤❡ z ✲❛①✐s ❛♥❞ ❝r❡❛t❡ ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ xy ✲♣❧❛♥❡✳ ❚❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡s❡ t❤r❡❡ ♣❧❛♥❡s ✕ ❛s ✐❢ t❤❡s❡ ✇❡r❡ t❤❡ t✇♦ ✇❛❧❧s ❛♥❞ t❤❡ ✢♦♦r ✐♥ ❛ r♦♦♠ ✕ ✐s ❛ P = (x, y, z) ✐♥ t❤❡ s♣❛❝❡✿

❧♦❝❛t✐♦♥

✸✳✶✵✳

❚❤✐s

❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ t❤r❡❡

3✲❞✐♠❡♥s✐♦♥❛❧

❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s ❝❛❧❧❡❞

✷✸✾

t❤❡ ❈❛rt❡s✐❛♥ s♣❛❝❡

✱ ♦r t❤❡

3✲s♣❛❝❡✳

❖♥❝❡ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s ✐♥ ♣❧❛❝❡✱ ✐t ✐s ❛❝❝❡♣t❛❜❧❡ t♦ t❤✐♥❦ ♦❢ ❧♦❝❛t✐♦♥ ❛s tr✐♣❧❡s ♦❢ ♥✉♠❜❡rs ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ■♥ ❢❛❝t✱ ✇❡ ❝❛♥ ✇r✐t❡✿

P = (x, y, z) . ❖♥❡ ❝❛♥ t❤✐♥❦ ♦❢ t❤❡

3✲s♣❛❝❡

❛s ❛

st❛❝❦ ♦❢ ♣❧❛♥❡s

✱ ❡❛❝❤ ♦❢ ✇❤✐❝❤ ✐s ❥✉st ❛ ❝♦♣② ♦❢ ♦♥❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡

♣❧❛♥❡s✿

❲❡ ❝❛♥ ✉s❡ t❤✐s ✐❞❡❛ t♦ r❡✈❡❛❧ t❤❡ ✐♥t❡r♥❛❧ str✉❝t✉r❡ ♦❢ t❤❡ s♣❛❝❡✳

❚❤❡♦r❡♠ ✸✳✶✵✳✶✿ P❧❛♥❡s P❛r❛❧❧❡❧ t♦ ❈♦♦r❞✐♥❛t❡ P❧❛♥❡s

L ✐s ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ xy ✲♣❧❛♥❡✱ t❤❡♥ ❛❧❧ ♣♦✐♥ts ♦♥ L ❤❛✈❡ t❤❡ s❛♠❡ z ✲❝♦♦r❞✐♥❛t❡✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ ❛ ❝♦❧❧❡❝t✐♦♥ L ♦❢ ♣♦✐♥ts ❝♦♥s✐sts ♦❢ ❛❧❧ ♣♦✐♥ts ✇✐t❤ t❤❡ s❛♠❡ z ✲❝♦♦r❞✐♥❛t❡✱ L ✐s ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ xy ✲♣❧❛♥❡✳ ■❢ L ✐s ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ yz ✲♣❧❛♥❡✱ t❤❡♥ ❛❧❧ ♣♦✐♥ts ♦♥ L ❤❛✈❡ t❤❡ s❛♠❡ x✲❝♦♦r❞✐♥❛t❡✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ ❛ ❝♦❧❧❡❝t✐♦♥ L ♦❢ ♣♦✐♥ts ❝♦♥s✐sts ♦❢ ❛❧❧ ♣♦✐♥ts ✇✐t❤ t❤❡ s❛♠❡ x✲❝♦♦r❞✐♥❛t❡✱ L ✐s ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ yz ✲♣❧❛♥❡✳ ■❢ L ✐s ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ zx✲♣❧❛♥❡✱ t❤❡♥ ❛❧❧ ♣♦✐♥ts ♦♥ L ❤❛✈❡ t❤❡ s❛♠❡ y ✲❝♦♦r❞✐♥❛t❡✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ ❛ ❝♦❧❧❡❝t✐♦♥ L ♦❢ ♣♦✐♥ts ❝♦♥s✐sts ♦❢ ❛❧❧ ♣♦✐♥ts ✇✐t❤ t❤❡ s❛♠❡ y ✲❝♦♦r❞✐♥❛t❡✱ L ✐s ❛ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ zx✲♣❧❛♥❡✳

✶✳ ■❢

✷✳

✸✳

✸✳✶✵✳

❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ t❤r❡❡

✷✹✵

❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛ ❝♦♠♣❛❝t ✇❛② t♦ r❡♣r❡s❡♥t t❤❡s❡ ♣❧❛♥❡s✿

x = k, y = k, ❢♦r s♦♠❡ r❡❛❧

❘❡❧❛t✐♦♥s ♥✉♠❜❡rs

♦r

z = k,

k✳

❛r❡ ✉s❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s ❜❡❢♦r❡ ❜✉t ✇✐t❤ ♠♦r❡ ✈❛r✐❛❜❧❡s✳

(x, y, z)

❆ r❡❧❛t✐♦♥ ♣r♦❝❡ss❡s ❛ tr✐♣❧❡ ♦❢

❛s t❤❡ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡s ❛♥ ♦✉t♣✉t✱ ✇❤✐❝❤ ✐s✿ ❨❡s ♦r ◆♦✳ ■❢ ✇❡ ❛r❡ t♦ ♣❧♦t t❤❡

❛ r❡❧❛t✐♦♥✱ t❤✐s ♦✉t♣✉t ❜❡❝♦♠❡s✿ ❛ ♣♦✐♥t ♦r ♥♦ ♣♦✐♥t✳ ❋♦r ❡①❛♠♣❧❡✱ ❧❡t✬s tr② ♦♥❡ ♦❢ t❤❡ r❡❧❛t✐♦♥s ❛❜♦✈❡✿

✐♥♣✉t

r❡❧❛t✐♦♥

(x, y, z) 7→

x = 2?   ❋❆▲❙❊ y

♦✉t♣✉t

❚❘❯❊

−−−−−−→

P❧♦t

(x, y, z) .

❉♦♥✬t ♣❧♦t✳

❖♥❧② t❤❡ ♣♦✐♥ts ✇✐t❤ t❤❡

x✲❝♦♦r❞✐♥❛t❡

❡q✉❛❧ t♦

2

✇✐❧❧ ❜❡ ♣❧♦tt❡❞✳

❊①❛♠♣❧❡ ✸✳✶✵✳✷✿ ♣❧❛♥❡ ❈♦♥s✐❞❡r t❤✐s r❡❧❛t✐♦♥✿

✐♥♣✉t

r❡❧❛t✐♦♥

(x, y, z) 7→

x + y + z = 2?   ❋❆▲❙❊ y

♦✉t♣✉t

❚❘❯❊

−−−−−−→

P❧♦t

❉♦♥✬t ♣❧♦t✳

❲❡ ❝❛♥ ❞♦ ✐t ❜② ❤❛♥❞✿

❲❡ ❝❛♥ ✉s❡✱ ❛s ❜❡❢♦r❡✱ t❤❡

s❡t✲❜✉✐❧❞✐♥❣ ♥♦t❛t✐♦♥ ✿

{(x, y, z) :

❛ ❝♦♥❞✐t✐♦♥ ♦♥

❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❛❜♦✈❡ r❡❧❛t✐♦♥ ✐s ❛ s✉❜s❡t ♦❢

x, y, z} .

R2

❣✐✈❡♥ ❜②✿

{(x, y, z) : x + y + z = 2} . ❲❡ ❛❝❝❡♣t t❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐t❤♦✉t ♣r♦♦❢✿

(x, y, z) .

❣r❛♣❤

♦❢

✸✳✶✵✳

✷✹✶

❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ t❤r❡❡

❚❤❡♦r❡♠ ✸✳✶✵✳✸✿ P❧❛♥❡ ❊✈❡r② ♣❧❛♥❡ t❤r♦✉❣❤ ❛ ♣♦✐♥t

(h, k, l)

✐s ❣✐✈❡♥ ❜② t❤❡ r❡❧❛t✐♦♥✿

A(x − h) + B(y − k) + C(z − l) = 0 . ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❤❛✈❡✿ ✶✳ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ yz ✲♣❧❛♥❡✿ B = C = 0 ✷✳ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ zx✲♣❧❛♥❡✿ A = C = 0 ✸✳ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ xy ✲♣❧❛♥❡✿ A = B = 0 ❊①❛♠♣❧❡ ✸✳✶✵✳✹✿ ❡q✉❛❧ ❞✐st❛♥❝❡

❍♦✇ ❞♦ ✇❡ ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢ ❛ ♠♦r❡ ❝♦♠♣❧❡① r❡❧❛t✐♦♥❄ ▲❡t✬s ❝♦♥s✐❞❡r t❤✐s✿ ✐♥♣✉t

(x, y, z) 7→

r❡❧❛t✐♦♥

x2 + y 2 + z 2 = 1?   ❋❆▲❙❊ y

♦✉t♣✉t ❚❘❯❊

−−−−−−→

P❧♦t (x, y, z) .

❉♦♥✬t ♣❧♦t✳

❲❡ t❡st ❡❛❝❤ ♦❢ t❤❡s❡ tr✐♣❧❡s (x, y, z) ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ ❛ s♣r❡❛❞s❤❡❡t✳ ❏✉st ❛s ❜❡❢♦r❡✱ ✐♥st❡❛❞ ♦❢ t❡st✐♥❣ ✇❤❡t❤❡r x2 + y 2 + z 2 ✐s ❡q✉❛❧ t♦ 1✱ ✇❡ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ✐s ✇✐t❤✐♥ ❛ s♠❛❧❧ ✜①❡❞ ♥✉♠❜❡r✱ s✉❝❤ ❛s 0.001✱ ❢r♦♠ 1 ❜❡❢♦r❡ ✇❡ ♣❧♦t ✐t✳ ❚❤❡ s♣r❡❛❞s❤❡❡t ✐s ❡✈❛❧✉❛t❡❞ s❡♣❛r❛t❡❧② ❢♦r s❡✈❡r❛❧ ❞✐st✐♥❝t ✈❛❧✉❡s ♦❢ z ✿

❚❤❡♥ ✇❡ ♣✉t t❤❡s❡ t♦❣❡t❤❡r ❛s s❤❡❡ts ♦❢ ♣❛♣❡r ✭❢❛r r✐❣❤t✮✳ ❚❤❡ r❡s✉❧t ❧♦♦❦s ❧✐❦❡ ❛ s✉r❢❛❝❡❀ ✇❡ ✇✐❧❧ ❞❡♠♦♥str❛t❡ t❤❛t ✇❡ ❤❛✈❡ ❛ s♣❤❡r❡✳ ◆♦✇ t❤❛t ❡✈❡r②t❤✐♥❣ ✐s ❝♦♦r❞✐♥❛t❡s✳

♣r❡✲♠❡❛s✉r❡❞✱

✇❡ ❝❛♥ s♦❧✈❡ t❤❡ ❣❡♦♠❡tr✐❝ ♣r♦❜❧❡♠s ❜② ❛❧❣❡❜r❛✐❝❛❧❧② ♠❛♥✐♣✉❧❛t✐♥❣

❚❤❡ ✜rst ❣❡♦♠❡tr✐❝ t❛s❦ ✐s ✜♥❞✐♥❣ t❤❡ ❞✐st❛♥❝❡✳ ❲❤❛t ✐s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❧♦❝❛t✐♦♥s P ❛♥❞ Q ✐♥ t❡r♠s ♦❢ t❤❡✐r ❝♦♦r❞✐♥❛t❡s (x, y, z) ❛♥❞ (x′ , y ′ , z ′ )❄ ❋♦r ❞✐♠❡♥s✐♦♥ 2✱ ✇❡ ✉s❡❞ t❤❡ ❞✐st❛♥❝❡ ❢♦r♠✉❧❛ ❢r♦♠ t❤❡ 1✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✳ ❲❡ ❢♦✉♥❞ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡ ❛s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ✕ ✇✐t❤ ✐ts s✐❞❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ✕ t❤❛t ❤❛s t❤❡s❡ ♣♦✐♥ts ❛t t❤❡ ♦♣♣♦s✐t❡ ❝♦r♥❡rs✿

✸✳✶✵✳

❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ t❤r❡❡

✷✹✷

❙✐♠✐❧❛r❧②✱ ✇❡ ✜♥❞ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ✐♥ s♣❛❝❡ ❛s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ t❤❡

❜♦①

✕ ✇✐t❤

✐ts ❡❞❣❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s ❛♥❞ s✐❞❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s ✕ t❤❛t ❤❛s t❤❡s❡ ♣♦✐♥ts ❛t t❤❡ ♦♣♣♦s✐t❡ ❝♦r♥❡rs✿

❚❤❡♦r❡♠ ✸✳✶✵✳✺✿ ❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r ❉✐♠❡♥s✐♦♥ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♣♦✐♥ts ❣✐✈❡♥ ❜② t❤❡✐r ❝♦♦r❞✐♥❛t❡s

(x′ , y ′ , z ′ )

3 P = (x, y, z)

❛♥❞

Q=

✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

d(P, Q) =

p (x − x′ )2 + (y − y ′ )2 + (z − z ′ )2

Pr♦♦❢✳

❲❡ ✉s❡ t❤❡ ❞✐st❛♥❝❡ ❢♦r♠✉❧❛ ❢r♦♠ t❤❡

1✲❞✐♠❡♥s✐♦♥❛❧

❝❛s❡ s❡♣❛r❛t❡❧② ❢♦r ❡❛❝❤ ♦❢ t❤❡ t❤r❡❡ ❛①❡s✱ ❛s

❢♦❧❧♦✇s✳ ❚❤❡ ❞✐st❛♥❝❡

x ❛♥❞ x′ ♦♥ t❤❡ x✲❛①✐s ✐s |x − x′ |✱ ′ ′ ❜❡t✇❡❡♥ y ❛♥❞ y ♦♥ t❤❡ y ✲❛①✐s ✐s |y − y |✱ ❛♥❞ ′ ′ ❜❡t✇❡❡♥ z ❛♥❞ z ♦♥ t❤❡ z ✲❛①✐s ✐s |z − z |✳ ′ ′ ′ ❚❤❡♥✱ t❤❡ s❡❣♠❡♥t ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts P = (x, y, z) ❛♥❞ Q = (x , y , z ) ✐s t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ t❤✐s ✏❜♦①✑✳ ′ ′ ′ ■ts s✐❞❡s ❛r❡✿ |x−x |✱ |y −y |✱ ❛♥❞ |z −z |✳ ❖✉r ❝♦♥❝❧✉s✐♦♥ ❜❡❧♦✇ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ • • •

❜❡t✇❡❡♥

❛♣♣❧✐❡❞ t✇✐❝❡✿ ❲❡ ✜rst ✜♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ t❤❡ ♦♣♣♦s✐t❡ ❢❛❝❡ ♦❢ t❤❡ ❜♦① ❛♥❞ t❤❡♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♠❛✐♥ ❞✐❛❣♦♥❛❧✱ ❛s ❢♦❧❧♦✇s✿

d(P, A) = |x − x′ |

d(A, B) = |y − y ′ | =⇒ d(P, B)2 = (x − x′ )2 + (y − y ′ )2

d(P, B)2 = (x − x′ )2 + (y − y ′ )2 d(B, Q) = |z − z ′ | =⇒ d(P, Q)2 = d(P, B)2 + d(B, Q)2

= (x − x′ )2 + (y − y ′ )2 + (z − z ′ )2

❊①❡r❝✐s❡ ✸✳✶✵✳✻

Pr♦✈❡ t❤❛t ✐♥ t❤❡ ❧❛tt❡r ❝❛s❡ t❤❡ tr✐❛♥❣❧❡ ✐s ✐♥❞❡❡❞ ❛ r✐❣❤t tr✐❛♥❣❧❡✳

❆ tr❡❛t♠❡♥t ♦❢ t❤❡ s❡❝♦♥❞ ❣❡♦♠❡tr✐❝ t❛s❦✱

❞✐r❡❝t✐♦♥s✱ ✐s ♣♦st♣♦♥❡❞ ✉♥t✐❧ ❈❤❛♣t❡r ✹❍❉✲✶✳

❚❤✐s ✐s ♦✉r ❝♦♥❝❧✉s✐♦♥ ❛❜♦✉t t❤❡ r❡❧❛t✐♦♥ ❝♦♥s✐❞❡r❡❞ ❛❜♦✈❡✿

✸✳✶✵✳

❚❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❢♦r ❞✐♠❡♥s✐♦♥ t❤r❡❡

✷✹✸

❚❤❡♦r❡♠ ✸✳✶✵✳✼✿ ❙♣❤❡r❡ ❚❤❡ s♣❤❡r❡ ♦❢ r❛❞✐✉s

R

♦❢ ❛❧❧ ♣♦✐♥ts

R>0

❝❡♥t❡r❡❞ ❛t ❛ ♣♦✐♥t

✉♥✐ts ❛✇❛② ❢r♦♠

(h, k, l)✱

(h, k, l)✱

✇❤✐❝❤ ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥

✐s ❣✐✈❡♥ ❜② t❤❡ r❡❧❛t✐♦♥✿

(x − h)2 + (y − k)2 + (z − l)2 = R2 . Pr♦♦❢✳

■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡

❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r ❞✐♠❡♥s✐♦♥ 3✳

❚❤❡♦r❡♠ ✸✳✶✵✳✽✿ ❈②❧✐♥❞❡r ❚❤❡ ❝②❧✐♥❞❡r ♦❢ r❛❞✐✉s ♦❢ ❛❧❧ ♣♦✐♥ts

R

R>0

❝❡♥t❡r❡❞ ❛r♦✉♥❞ t❤❡

z ✲❛①✐s✱

✇❤✐❝❤ ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥

✉♥✐ts ❛✇❛② ❢r♦♠ t❤❡ ❛①✐s ♠❡❛s✉r❡❞ ❤♦r✐③♦♥t❛❧❧②✱ ✐s ❣✐✈❡♥ ❜② t❤❡

r❡❧❛t✐♦♥✿

x2 + y 2 = R 2 . Pr♦♦❢✳

■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡

❉✐st❛♥❝❡ ❋♦r♠✉❧❛ ❢♦r ❉✐♠❡♥s✐♦♥ 2✳

❘❡♣r❡s❡♥t❛t✐♦♥s ♦❢ s❡ts ♦❢ ❛❧❧ ♣♦✐♥ts

t❤❡ s♣❛❝❡ n = 1, 2, 3✿

1

❞✐♠❡♥s✐♦♥✿ ❞✐st❛♥❝❡

1

2

x2 + y 2 = 1 x2 + y 2 + z 2 = 1

|x| = 1

s❡t✿

t✇♦ ♣♦✐♥ts

❝✐r❝❧❡

s♣❤❡r❡

◆✴❆

❧❡♥❣t❤

❛r❡❛

x2 + y 2 ≤ 1 x2 + y 2 + z 2 ≤ 1

≤1

|x| ≤ 1

s❡t✿

✐♥t❡r✈❛❧

❞✐s❦

❜❛❧❧

❧❡♥❣t❤

❛r❡❛

✈♦❧✉♠❡

✐ts ✏s✐③❡✑✿

❚❤❡ ❧❛tt❡r ❧✐st ✐s ❛ ❧✐st ♦❢ t❤❡ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ❝❛❧❧❡❞ ✏❝❡❧❧s✑✳ ❚❤❡② ❛r❡ ♣r❡s❡♥t❡❞ ❜❡❧♦✇✱ ❢♦r t❤❡

t❤❡ ❝❡❧❧s m = 0, 1, 2, 3✿

❞✐♠❡♥s✐♦♥s ♦❢

3

=1

✐ts ✏s✐③❡✑✿ ❞✐st❛♥❝❡

✉♥✐t ❛✇❛② ❢r♦♠ t❤❡ ♦r✐❣✐♥ ❛r❡ ♣r❡s❡♥t❡❞ ❜❡❧♦✇✱ ❢♦r t❤❡

❞✐♠❡♥s✐♦♥s ♦❢

✸✳✶✶✳

✷✹✹

❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ✈✐❛ t❤❡✐r ❝r♦ss✲s❡❝t✐♦♥s

❚❤❡ tr❛♥s✐t✐♦♥ ❢r♦♠ r❡❧❛t✐♦♥s ♦❢ t❤r❡❡ ✈❛r✐❛❜❧❡s t♦

❢✉♥❝t✐♦♥s

♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✇✐❧❧ ❜❡ ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✹✳

❲❡ s❡❡ ❤♦✇ ♠✉❝❤ ❤❛r❞❡r ✐t ✐s t♦ ✈✐s✉❛❧✐③❡ t❤✐♥❣s ✐♥ t❤❡ 3✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳ ❚❤❛t✬s ✇❤② ✇❡ ✇✐❧❧ ♥❡❡❞ ❛ ❢✉rt❤❡r ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝ tr❡❛t♠❡♥t ♦❢ t❤❡s❡ ❣❡♦♠❡tr✐❝ ✐❞❡❛s ✐♥ ❱♦❧✉♠❡ ✹ ✭❈❤❛♣t❡r ✹P❈✲✷✮✳

✸✳✶✶✳ ❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ✈✐❛ t❤❡✐r ❝r♦ss✲s❡❝t✐♦♥s

❲❡ ❤❛✈❡ ❝♦♠❡ t♦ ✉♥❞❡rst❛♥❞

❛r❡❛s ✐♥ t❡r♠s ♦❢ ❧❡♥❣t❤s✳

■♥❞❡❡❞✱ ✐❢ ✇❡ r❡❛rr❛♥❣❡ t❤❡s❡ ♣❡♥❝✐❧s ❜② ♠♦✈✐♥❣ ❡❛❝❤ ✉♣ ♦r ❞♦✇♥✱ t❤❡② ✇✐❧❧ st✐❧❧ ❝♦✈❡r t❤❡ s❛♠❡ ❛r❡❛✿

❚❤✐s ❢❛❝t ✐s ♠❡❛♥t t♦ ✐❧❧✉str❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❢♦✉r ❢✉♥❝t✐♦♥s f, g, F, G t❤❛t ❤❛✈❡ ♥♦t❤✐♥❣ t♦ ❞♦ ✇✐t❤ ❡❛❝❤ ♦t❤❡r ❡①❝❡♣t t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❣r❛♣❤s ✕ ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✕ ✐s t❤❡ s❛♠❡✿

f (x) − g(x) = F (x) − G(x) , ❢♦r ❛❧❧ x ✐♥ [a, b]✳ ▲❡t✬s ❝♦♠♣❛r❡✿ t❤❡ ❛r❡❛ ❜❡t✇❡❡♥ f ❛♥❞ g ✈s✳ t❤❡ ❛r❡❛ ❜❡t✇❡❡♥ F ❛♥❞ G✳ ❊❛❝❤ ♣❛✐r ♦❢ ❝♦rr❡s♣♦♥❞✐♥❣ r❡❝t❛♥❣❧❡s ✐♥ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ t❤❡ t✇♦ ❛r❡❛s ♦✈❡r s♦♠❡ ♣❛rt✐t✐♦♥ ❤❛✈❡ t❤❡ s❛♠❡ ❤❡✐❣❤t ✭s❛♠❡ ♣❡♥❝✐❧✮✳

❚❤❛t ✐s ✇❤② t❤❡ ❘✐❡♠❛♥♥ s✉♠s ♦✈❡r ❛ ♣❛rt✐t✐♦♥ ♦❢ [a, b] t❤❛t ❛♣♣r♦①✐♠❛t❡ t❤❡ ❛r❡❛s ❜❡t✇❡❡♥ ❡✐t❤❡r ♣❛✐r ♦❢ ❣r❛♣❤s ❛r❡ t❤❡ s❛♠❡✿ Σ (f − g) · ∆x = Σ (F − G) · ∆x ,

✸✳✶✶✳ ❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ✈✐❛ t❤❡✐r ❝r♦ss✲s❡❝t✐♦♥s

♦✈❡r ❛♥② ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥

P✳

✷✹✺

❚❤❡r❡❢♦r❡✱ t❤❡ ✐♥t❡❣r❛❧s ✕ t❤❡ ❛r❡❛s ❜❡t✇❡❡♥ t❤❡ ❣r❛♣❤s ✕ ❛r❡ ❡q✉❛❧ t♦♦✿

Z

b a

(f − g) dx =

Z

b a

(F − G) dx .

❊①❛♠♣❧❡ ✸✳✶✶✳✶✿ ❡q✉❛❧ ❛r❡❛s

❚❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t❤❡ ❣r❛♣❤s ♦❢

y = x2 + 1

❛♥❞

y = x2 + 2

✐s t❤❡ s❛♠❡ ❛s t❤❛t ♦❢ t❤❡ sq✉❛r❡ ❜❡❧♦✇✿

❈♦♥❝❧✉s✐♦♥✿

◮ ❚❤❡ ✈❡rt✐❝❛❧ ❝r♦ss✲s❡❝t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② y = f (x) ❛♥❞ y = g(x) ✐s t❤❡ ✈❡rt✐❝❛❧ s❡❣♠❡♥t [g(x), f (x)] ❢♦r ❡❛❝❤ x✱ ❛♥❞ ♦♥❧② t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡s❡ s❡❣♠❡♥ts✱ f (x) − g(x)✱ ❛✛❡❝t t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❣✐♦♥✳

▲❡t✬s ♥♦✇ ❣♦ ✉♣ ✐♥ ❞✐♠❡♥s✐♦♥ ❛♥❞ ❡①❛♠✐♥❡ t❤❡ ❝r♦ss✲s❡❝t✐♦♥s ♦❢ ❛ s♦❧✐❞ ❛♥❞ ✜♥❞ ♦✉t ✇❤❛t t❤❡② t❡❧❧ ✉s ❛❜♦✉t t❤❡ ✈♦❧✉♠❡ ♦❢ t❤✐s s♦❧✐❞✳ ❇✉t ✜rst✱ ✇❤❛t ✐s ✈♦❧✉♠❡ ❄ ❚❤❡ q✉❡st✐♦♥ ✇✐❧❧ ❜❡ ❛❞❞r❡ss❡❞ ✐♥ ❢✉❧❧ ♦♥❧② ✐♥ ❱♦❧✉♠❡ ✹ ✭❈❤❛♣t❡r ✹❍❉✲✺✮✳ ❋♦r ♥♦✇✱ ✇❡ ✇✐❧❧ r❡❧② ♦♥ ❛ s✐♠♣❧✐❢②✐♥❣ ❛ss✉♠♣t✐♦♥✳

V = w · d · h✱ ✇❤❡r❡ w 2 ❝②❧✐♥❞❡r✱ V = πR h✱ ✇❤❡r❡ R

❲❡ ❞♦ ✉♥❞❡rst❛♥❞ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ✈♦❧✉♠❡ ♦❢ s✉❝❤ ❛ s✐♠♣❧❡ s♦❧✐❞ ❛s ❛ ❜♦①✳ ■t ✐s t❤❡ ✇✐❞t❤✱

d

t❤❡ ❞❡♣t❤✱ ❛♥❞

✐ts r❛❞✐✉s ❛♥❞

h

h

t❤❡ ❤❡✐❣❤t✳ ❲❡ ❛❧s♦ ✏❦♥♦✇✑ t❤❡ ✈♦❧✉♠❡ ♦❢ ❛

✐s ✐s

✐s t❤❡ ❤❡✐❣❤t✳

❲❡ ❝❛♥ ❣❛✐♥ ✐♥s✐❣❤t ❢r♦♠ t❤✐s✿

❱♦❧✉♠❡

=

■♥❞❡❡❞✱ t❤❡ ❛r❡❛ ♦❢ t❤❡ ❜❛s❡ ✐s✱ r❡s♣❡❝t✐✈❡❧②✱ ❲❤❛t ❞♦ t❤❡② ❛❧❧ ❤❛✈❡ ✐♥ ❝♦♠♠♦♥❄ ✭❛♥ ✐♥t❡❣r❛❧✮✳

❛r❡❛ ♦❢ t❤❡ ❜❛s❡

A = wd

❛♥❞

❤❡✐❣❤t

A = πR2 ✳

❚❤❡ s❛♠❡ ✐s tr✉❡ ❢♦r t❤❡ ♣r✐s♠✳

❚❤❡ ❜❛s❡ ✐s ❛ r❡❣✐♦♥ ♦♥ t❤❡ ♣❧❛♥❡ ❛♥❞ ✇❡ ♠✐❣❤t ❦♥♦✇ ✐ts ❛r❡❛

❚❤✐s r❡❣✐♦♥ ✐s ❧✐❢t❡❞ ♦✛ t❤❡ ♣❧❛♥❡ t♦ t❤❡ ❤❡✐❣❤t

❝②❧✐♥❞❡r✲❧✐❦❡ s♦❧✐❞✿

·

h✳

A

❇❡t✇❡❡♥ t❤❡s❡ t✇♦ ♣❧❛♥❡ r❡❣✐♦♥s ❧✐❡s ❛

✸✳✶✶✳ ❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ✈✐❛ t❤❡✐r ❝r♦ss✲s❡❝t✐♦♥s

✷✹✻

❲❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t ✐ts ✈♦❧✉♠❡ ✐s✿

V = A · h. ❏✉st ❛s ✇❡ ❤❛✈❡ ❜❡❡♥ ✉s✐♥❣ r❡❝t❛♥❣❧❡s t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ s❧✐❝❡s ♦❢ ♣❧❛♥❡ r❡❣✐♦♥s✱ t❤❡s❡ ✏s❤❡❧❧s✑ ✇✐❧❧ ❛♣♣r♦①✲ ✐♠❛t❡ s❧✐❝❡s ♦❢ s♦❧✐❞s✳ ❙✉♣♣♦s❡ t❤❛t✱ ✐♥st❡❛❞ ♦❢ ❛ st❛❝❦ ♦❢ ♣❡♥❝✐❧s✱ ✇❡ ❤❛✈❡ ❛ st❛❝❦ ♦❢ ❝♦✐♥s✳ ■❢ ✇❡ r❡❛rr❛♥❣❡ t❤❡s❡ ❝♦✐♥s ❜② ♠♦✈✐♥❣ t❤❡♠ s✐❞❡ t♦ s✐❞❡✱ t❤❡ t♦t❛❧ ✈♦❧✉♠❡ ✇✐❧❧ r❡♠❛✐♥ t❤❡ s❛♠❡✿

❲❡ r❡❛❧✐③❡ t❤❛t ✇❡ s❤♦✉❧❞ tr② t♦ ✉♥❞❡rst❛♥❞ ✈♦❧✉♠❡s ✐♥ t❡r♠s ♦❢ ❛r❡❛s✳ ■t ✐s ❝❛❧❧❡❞ t❤❡ ❈❛✈❛❧✐❡r✐ ♣r✐♥❝✐♣❧❡ ✿

■❢ t❤❡ ✈❡rt✐❝❛❧ ❝r♦ss✲s❡❝t✐♦♥s ♦❢ t✇♦ s♦❧✐❞s ✐♥ t❤❡ s♣❛❝❡ ❤❛✈❡ ❡q✉❛❧ ❛r❡❛s✱ t❤❡♥ t❤❡✐r ✈♦❧✉♠❡s ❛r❡ ❛❧s♦ ❡q✉❛❧✳

❚♦ ❝♦♥✜r♠ t❤❛t t❤✐s ♣r✐♥❝✐♣❧❡ ♠❛❦❡s s❡♥s❡✱ ✇❡ ❝❛♥ ♠❛t❝❤ ✐t ✇✐t❤ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t t❤❡ ✐❞❡❛ ♦❢ ❛r❡❛ ✐♥ t❡r♠s ♦❢ ❧❡♥❣t❤s✿

■❢ t❤❡ ✈❡rt✐❝❛❧ ❝r♦ss✲s❡❝t✐♦♥s ♦❢ t✇♦

❤❛✈❡ ❡q✉❛❧

❙✉♣♣♦s❡ ♦✉r s♦❧✐❞

S

−−



❧❡♥❣t❤s ❛r❡❛s



− −✱

−−



r❡❣✐♦♥s ✐♥ t❤❡ ♣❧❛♥❡ s♦❧✐❞s ✐♥ t❤❡ s♣❛❝❡

t❤❡♥ t❤❡✐r

✐s ❧♦❝❛t❡❞ ✐♥ t❤❡ ❈❛rt❡s✐❛♥

3✲s♣❛❝❡✳

−−



❛r❡❛s ✈♦❧✉♠❡s





−−

−−

❛r❡ ❛❧s♦ ❡q✉❛❧✳

■ts ❝r♦ss✲s❡❝t✐♦♥s ❛r❡ t❤❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢

S

✇✐t❤

t❤❡ ✈❛r✐♦✉s ♣❧❛♥❡s✱ ❡s♣❡❝✐❛❧❧② t❤❡ ♦♥❡s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ♣❧❛♥❡s✳ ❲❡ ❝❤♦♦s❡ t❤♦s❡ ♣❛r❛❧❧❡❧ t♦ t❤❡

yz ✲♣❧❛♥❡

❛♥❞✱ t❤❡r❡❢♦r❡✱ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡

s♦❧✐❞ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛❧❧ ✈❛❧✉❡s t❤❡

x✲❛①✐s✳

x

x✲❛①✐s✳

❚❤✉s✱ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r ❛❧❧ ✈❡rt✐❝❛❧ ❝r♦ss✲s❡❝t✐♦♥ ♦❢ t❤✐s

❛s t❤❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢

S

✇✐t❤ t❤❡ ✈❡rt✐❝❛❧ ♣❧❛♥❡s t❤r♦✉❣❤ t❤❡ ♣♦✐♥t

x

♦♥

❊❛❝❤ ♦❢ t❤❡♠ ✐s ❛ ♣❧❛♥❡ r❡❣✐♦♥ ❛♥❞✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❈❛✈❛❧✐❡r✐ ♣r✐♥❝✐♣❧❡✱ ♦♥❧② ✐ts ❛r❡❛ ❛✛❡❝ts t❤❡

✈♦❧✉♠❡ ♦❢ t❤❡ r❡❣✐♦♥✿

✸✳✶✶✳

✷✹✼

❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ✈✐❛ t❤❡✐r ❝r♦ss✲s❡❝t✐♦♥s

❲❡ ❞❡♥♦t❡ t❤✐s ❛r❡❛ ♦❢ t❤❡ ❝r♦ss✲s❡❝t✐♦♥ ❛t x ❜② A(x)✳ ■t ✐s s✐♠♣❧② ❛ ❢✉♥❝t✐♦♥ ♦❢ x✳ ❊①❛♠♣❧❡ ✸✳✶✶✳✷✿ ❝②❧✐♥❞❡r

❲❤❛t ✐s t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ ❝②❧✐♥❞❡r ♦❢ r❛❞✐✉s R ❛♥❞ ❤❡✐❣❤t h❄

■t ✐s ❧♦❝❛t❡❞ ✐♥ ♦✉r 3✲s♣❛❝❡✱ ❜✉t ❛❧❧ ✇❡ ♥❡❡❞ t♦ ❦♥♦✇ ✐s ✐ts ❞✐♠❡♥s✐♦♥s✳ ❲❡ ❤❛✈❡ A(x) = πR2 .

❇② t❤❡ ❈❛✈❛❧✐❡r✐ ♣r✐♥❝✐♣❧❡✱ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤✐s ❝②❧✐♥❞❡r ✐s t❤❡ s❛♠❡ ❛s t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ ❜♦① s❡❝t✐♦♥ ♦❢ ✇❤✐❝❤ ✐s ❛ sq✉❛r❡ ✇✐t❤ ❛r❡❛ πR2 ❛♥❞ t❤❡ s❛♠❡ ❤❡✐❣❤t✿

t❤❡

❝r♦ss✲

❱♦❧✉♠❡ = πR2 · h . ▲❡t✬s ❝♦♥✜r♠ t❤❡ ✐❞❡❛ ♦❢ t❤❡ ❈❛✈❛❧✐❡r✐ ♣r✐♥❝✐♣❧❡ ✈✐❛ ❘✐❡♠❛♥♥ s✉♠s✳ ❲❡ ♣❧❛❝❡ t❤❡ x✲❛①✐s s♦♠❡❤♦✇ ❛❧♦♥❣ t❤❡ s♦❧✐❞✳ ❙✉♣♣♦s❡ t❤❡ s♦❧✐❞ S ❧✐❡s ❡♥t✐r❡❧② ❜❡t✇❡❡♥ s♦♠❡ ✈❡rt✐❝❛❧ ♣❧❛♥❡s x = a ❛♥❞ x = b✳ ❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥ P ♦❢ t❤❡ ✐♥t❡r✈❛❧ [a, b]✿ a = x0 ≤ c1 ≤ x1 ≤ ... ≤ xn = b

❚❤❡ ✈❡rt✐❝❛❧ ♣❧❛♥❡s x = xi ❝✉t t❤❡ s♦❧✐❞ ✐♥t♦ n s❧✐❝❡s✳ ❚❤❡ it❤ s❧✐❝❡ ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡ ❝r♦ss✲s❡❝t✐♦♥ ♦❢ S ❝r❡❛t❡❞ ❜② t❤❡ ✈❡rt✐❝❛❧ ♣❧❛♥❡ x = ci ✐s ❛ ♣❧❛♥❡ r❡❣✐♦♥❀ ✐ts ❛r❡❛ ✐s A(ci )✿

✸✳✶✶✳

✷✹✽

❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ✈✐❛ t❤❡✐r ❝r♦ss✲s❡❝t✐♦♥s

❲❡ ❝♦♥str✉❝t ❛ ♥❡✇ s♦❧✐❞ ❢r♦♠ t❤✐s ♣❧❛♥❡ r❡❣✐♦♥ ❜② ❣✐✈✐♥❣ ✐t ❛ t❤✐❝❦♥❡ss ❡q✉❛❧ t♦ ∆xi = xi − xi−1 ✳ ❚❤❡♥ ✐ts ✈♦❧✉♠❡ ✐s A(ci ) · ∆xi ✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ ❚♦t❛❧ ✈♦❧✉♠❡ =

n X i=1

❚❤✐s ✈❛❧✉❡s ✐s t❤❡♥ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✳

r❡❝♦❣♥✐③❡❞

A(ci ) · ∆xi .

❛s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ y = A(x) ♦✈❡r [a, b]✿ Σ A · ∆x✳ ❚❤❡✐r ❧✐♠✐t ✐s t❤❡

❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✸✿ ✈♦❧✉♠❡ ♦❢ s♦❧✐❞ ❚❤❡

✈♦❧✉♠❡ ♦❢ ❛ s♦❧✐❞

✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ✐♥t❡❣r❛❧ Z

b

A(x) dx , a

✐❢ ✐t ❡①✐sts✱ ✇❤❡r❡ A(c) ✐s t❤❡ ❛r❡❛ ✭✐❢ ✐t ❡①✐sts✮ ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ s♦❧✐❞ ❛♥❞ t❤❡ ♣❧❛♥❡ x = c✳

❲❛r♥✐♥❣✦ ❚❤❡ ❛r❡❛

A(x) ✐ts❡❧❢✱ ❢♦r ❡❛❝❤ x✱ ✐s ✉♥❞❡rst♦♦❞✱ ❛♥❞

♠❛② ❤❛✈❡ t♦ ❜❡ ❝♦♠♣✉t❡❞✱ ❛s ❛ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✳

❚❤✉s✱ t❤❡ ✈♦❧✉♠❡ ✐s t❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤❡ ❛r❡❛✿

✸✳✶✶✳

✷✹✾

❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ✈✐❛ t❤❡✐r ❝r♦ss✲s❡❝t✐♦♥s

❊①❛♠♣❧❡ ✸✳✶✶✳✹✿ s♣❤❡r❡

❚❤❡ ❝r♦ss✲s❡❝t✐♦♥s ♦❢ t❤❡ s♣❤❡r❡ ❛r❡ ❝✐r❝❧❡s✿

▼♦r❡ ♣r❡❝✐s❡❧②✱ t❤❡ ❝r♦ss✲s❡❝t✐♦♥s ♦❢ t❤❡ ❜❛❧❧ ❛r❡ ❞✐s❦s ❛♥❞ ✐t ✐s t❤❡ ❛r❡❛s ♦❢ t❤❡s❡ ❞✐s❦s t❤❛t ✇❡ ♥❡❡❞ t♦ ✜♥❞✳ ❙✉♣♣♦s❡ t❤❡ r❛❞✐✉s ♦❢ t❤✐s ❝✐r❝❧❡ ❛t x ✐s r✳ ❲❤❛t ✐s ✐t❄ ▲❡t✬s t❛❦❡ ❛ s✐❞❡ ✈✐❡✇✿

❚❤❡♥

x2 + r 2 = R 2 .

❚❤❡♥ t❤❡ ❛r❡❛ ♦❢ t❤✐s ❝✐r❝❧❡ ✐s✿ A(x) = π

❚❤❡r❡❢♦r❡✱ ❱♦❧✉♠❡ =

Z

R

A(x) dx = π −R

Z

√

R 2 − x2

R −R

R2 − x

2

 2

= π(R2 − x2 ) .



1 dx = π R2 x − x3 3

 R 4 = πR3 . 3 −R

❲❡ ❤❛✈❡ ❞♦♥❡ ❛❧❧ ♣r❡❧✐♠✐♥❛r② ✇♦r❦ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✿ ❲❤❛t ✐s t❤❡ ✈♦❧✉♠❡ ✇❤❡♥ t❤❡ ❝r♦ss✲s❡❝t✐♦♥s ❛r❡ ❝✐r❝❧❡s t❤❛t ❝❤❛♥❣❡ ❢r♦♠ s❧✐❝❡ t♦ s❧✐❝❡❄

✸✳✶✶✳

✷✺✵

❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ✈✐❛ t❤❡✐r ❝r♦ss✲s❡❝t✐♦♥s

❉❡✜♥✐t✐♦♥ ✸✳✶✶✳✺✿ s♦❧✐❞ ♦❢ r❡✈♦❧✉t✐♦♥ ❙✉♣♣♦s❡ y = f (x) s❛t✐s✜❡s f (x) ≥ 0 ❢♦r ❛❧❧ x ✐♥ [a, b]✳ ❚❤❡♥✱ t❤❡ s♦❧✐❞ ♦❢ r❡✈♦❧✉t✐♦♥ ♦❢ f ❛❜♦✉t t❤❡ x✲❛①✐s ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ✐♥ t❤❡ xyz ✲s♣❛❝❡✿ {(x, y, z) :

p

y 2 + z 2 ≤ f (x) } .

❚❤❡♦r❡♠ ✸✳✶✶✳✻✿ ❱♦❧✉♠❡ ♦❢ ❙♦❧✐❞ ♦❢ ❘❡✈♦❧✉t✐♦♥ ❙✉♣♣♦s❡ y = f (x) s❛t✐s✜❡s f (x) ≥ 0 ❢♦r ❛❧❧ x ✐♥ [a, b]✳ ❚❤❡♥✱ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ s♦❧✐❞ ♦❢ r❡✈♦❧✉t✐♦♥ ♦❢ f ❛❜♦✉t t❤❡ x✲❛①✐s ✐s✿ V =

Z

b

πf (x)2 dx . a

❊①❡r❝✐s❡ ✸✳✶✶✳✼ Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳

❲❛r♥✐♥❣✦ ❊✈❡♥ ✇❤❡♥ t❤❡ ❝r♦ss✲s❡❝t✐♦♥s ❛r❡ ❝✐r❝❧❡s✱ t❤❡② ♠❛② ❝❤❛♥❣❡ ❢r♦♠ s❧✐❝❡ t♦ s❧✐❝❡ ✐♥ ✇❛②s t❤❛t ❛r❡ s♦ ❝♦♠✲ ♣❧❡① t❤❛t ✇❡ ♠❛② ❤❛✈❡ t♦ t✉r♥ t♦ ♥✉♠❡r✐❝❛❧ ✐♥t❡✲ ❣r❛t✐♦♥✳

■♥ ❣❡♥❡r❛❧✱ ❝r♦ss✲s❡❝t✐♦♥s ❝❛♥ ❤❛✈❡ ❛♥② ❣❡♦♠❡tr②

♦r t♦♣♦❧♦❣② ✿

❊①❡r❝✐s❡ ✸✳✶✶✳✽ ❉❡s❝r✐❜❡ t❤❡ ❝r♦ss✲s❡❝t✐♦♥s ♦❢ t❤❡s❡ s✉r❢❛❝❡s✳

❊①❛♠♣❧❡ ✸✳✶✶✳✾✿ ♣②r❛♠✐❞ ▲❡t✬s ✜♥❞ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ r✐❣❤t ♣②r❛♠✐❞ ✭✐✳❡✳✱ ♦♥❡ ✇✐t❤ ✐ts ❤❡✐❣❤t ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ✐ts ❜❛s❡✮ t❤❛t ❤❛s ❛ sq✉❛r❡ ❜❛s❡ ✇✐t❤ s✐❞❡ 2h ❛♥❞ ❤❡✐❣❤t h✳ ■ts ❝r♦ss✲s❡❝t✐♦♥s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❜❛s❡ ❛r❡ sq✉❛r❡s ✿

✸✳✶✷✳

❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ♦❢ r❡✈♦❧✉t✐♦♥

✷✺✶

❚❤❡ s✐❞❡ ♦❢ t❤❡ sq✉❛r❡ ❧♦❝❛t❡❞ x ✉♥✐ts ❢r♦♠ t❤❡ ❜❛s❡ ✐s 2(h − x)❀ t❤❡r❡❢♦r❡✱ ❱♦❧✉♠❡ =

Z

h

A(x) dx = 0

Z

h 0

❊①❡r❝✐s❡ ✸✳✶✶✳✶✵

h 2 2 2 3 2(h − x) dx = − (h − x) = h3 . 3 3 0

▼♦❞✐❢② t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡ t♦ ✜♥❞ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ r✐❣❤t ♣②r❛♠✐❞ ✇✐t❤ sq✉❛r❡ ❜❛s❡ ✇✐t❤ s✐❞❡ Q ❛♥❞ ❤❡✐❣❤t h✳ ❊①❡r❝✐s❡ ✸✳✶✶✳✶✶

❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ r✐❣❤t ❝♦♥❡ ✇✐t❤ ❛ ❝✐r❝✉❧❛r ❜❛s❡ ♦❢ r❛❞✐✉s R ❛♥❞ ❤❡✐❣❤t h✳ ❲❡ ❞❡✜♥❡❞ ✐♥ t❤✐s s❡❝t✐♦♥ t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ s♦❧✐❞ ✈✐❛ ✐ts ❝r♦ss✲s❡❝t✐♦♥s✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ r❡❧✐❡s ♦♥ t❤❡ ❝②❧✐♥❞r✐❝❛❧ s❧✐❝❡s t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ s♦❧✐❞✳ ❚❤❡s❡ ❝♦♠♣❧❡① ♦❜❥❡❝ts ❛r❡ t♦ ❜❡ r❡♣❧❛❝❡❞ ✇✐t❤ tr✉❡ ❡❧❡♠❡♥t❛r② ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ♦❢ s♦❧✐❞s ✕ ❜r✐❝❦s ❛♥❞ ❜♦①❡s ✕ t♦ ❢♦❧❧♦✇ t❤❡ ✐❞❡❛ ♦❢ t❤❡ ❘✐❡♠❛♥♥ s✉♠✳ ❚❤❡ ❣❡♥❡r❛❧ ❞❡✜♥✐t✐♦♥ ✐s ♣r❡s❡♥t❡❞ ✐♥ ❱♦❧✉♠❡ ✹ ✭❈❤❛♣t❡r ✹❍❉✲✺✮✳ ❊①❡r❝✐s❡ ✸✳✶✶✳✶✷

✭❛✮ Pr♦✈❡ t❤❛t t❤❡ ✇♦r❦ ♥❡❡❞❡❞ t♦ ✜❧❧ ✕ ❢r♦♠ t❤❡ ❜♦tt♦♠ ✕ ❛ t❛♥❦ ❧♦❝❛t❡❞ ❜❡t✇❡❡♥ t❤❡ ♣❧❛♥❡s x = 0 ❛♥❞ x = h ✭t❤❡ x✲❛①✐s ✐s ✈❡rt✐❝❛❧✮ ❛♥❞ ✇✐t❤ t❤❡ ❛r❡❛ ♦❢ ✐ts ❤♦r✐③♦♥t❛❧ ❝r♦ss✲s❡❝t✐♦♥ ❛t ❤❡✐❣❤t x ❡q✉❛❧ Z h A(x)x dx✳ ✭❜✮ ❙❤♦✇ t❤❛t t❤✐s ✇♦r❦ ✐s ❡q✉❛❧ t♦ t❤❡ ✇♦r❦ ♥❡❡❞❡❞ t♦ ♠♦✈❡ t❤✐s ♠❛ss ❢r♦♠ t♦ A(x) ✐s 0

❤❡✐❣❤t 0 t♦ t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ t❤❡ t❛♥❦✳ ❊①❡r❝✐s❡ ✸✳✶✶✳✶✸

❙❡t ✉♣ ❜✉t ❞♦ ♥♦t ❡✈❛❧✉❛t❡ t❤❡ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ❢♦r t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ ❜♦① W × D × H ✐♥ t❡r♠s ♦❢ ✐ts ❝r♦ss✲s❡❝t✐♦♥s✳

✸✳✶✷✳ ❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ♦❢ r❡✈♦❧✉t✐♦♥

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r ♠♦r❡ ❝♦♠♣❧❡① s✉r❢❛❝❡s ♦❢ r❡✈♦❧✉t✐♦♥✳ ❈♦♥s✐❞❡r ❛♥ ♦❜❥❡❝t t❤❛t ✐s r♦t❛t❡❞ ❛s ✐t ❤❛r❞❡♥s✿

✸✳✶✷✳

❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ♦❢ r❡✈♦❧✉t✐♦♥

✷✺✷

❚❤❡ s❛♠❡ ❡✛❡❝t ✐s ♣r♦❞✉❝❡❞ ❜② ✉s✐♥❣ ❛ ❝✉tt✐♥❣ t♦♦❧ ♦♥ ❛ ❤❛r❞ ♦❜❥❡❝t ❛s ✐t ✐s ❜❡✐♥❣ r♦t❛t❡❞✳ ▲❡t✬s r♦t❛t❡ ❛ ❝✉r✈❡✳ ■❢ t❤✐s ❝✉r✈❡ ✐s ❛ ❝✐r❝❧❡✱ t❤❡ r❡s✉❧t ♦❢ t❤❡ r♦t❛t✐♦♥ ✐s s✐♠✐❧❛r t♦ ❛ s❧✐♥❦②✿

▼❛t❤❡♠❛t✐❝❛❧❧②✱ ✇❡ ❤❛✈❡ ❛ ❝✉r✈❡ ❛♥❞ ❛ ❧✐♥❡ ♦♥ t❤❡ xy ✲♣❧❛♥❡✱ ✇❡ ❛❞❞ t❤❡ z ✲❛①✐s✱ t❤❡♥ ✇❡ r♦t❛t❡ t❤❡ ❝✉r✈❡ ❛r♦✉♥❞ t❤❡ ❧✐♥❡ ✐♥ t❤❡ r❡s✉❧t✐♥❣ 3✲s♣❛❝❡✱ ♦♥❡ ♣♦✐♥t ❛t ❛ t✐♠❡✳

❊❛❝❤ ♦❢ t❤❡ ♣♦✐♥ts ♦♥ t❤❡ ❝✉r✈❡ ♣r♦❞✉❝❡s ❛ ❝✐r❝❧❡✳ ❚♦❣❡t❤❡r t❤❡s❡ ❝✐r❝❧❡s ❢♦r♠ ❛ s✉r❢❛❝❡✳ ❚❤✐s s✉r❢❛❝❡ ❜♦✉♥❞s ❛ s♦❧✐❞✳ ❲❤❛t ✐s t❤❡ ✈♦❧✉♠❡ ♦❢ t❤✐s s♦❧✐❞❄ ❙✉♣♣♦s❡ t❤✐s ❝✉r✈❡ ✐s s✐♠♣❧② t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ y = f (x) ≥ 0, a ≤ x ≤ b ,

❛♥❞ s✉♣♣♦s❡ t❤❡ ❧✐♥❡ ✐s t❤❡ x✲❛①✐s ♦r t❤❡ y ✲❛①✐s✿

✸✳✶✷✳

❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ♦❢ r❡✈♦❧✉t✐♦♥

✷✺✸

❆s t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ x✲❛①✐s ✐s ❡❛s✐❧② ❛❞❞r❡ss❡❞ ❜② t❤❡ ❈❛✈❛❧✐❡r✐ ♣r✐♥❝✐♣❧❡✱ ✇❡ ❝❤♦♦s❡ t❤❡ y ✲❛①✐s✳ ▲❡t✬s ❜❡ ❝❧❡❛r ✇❤❛t ✇❡ ❛r❡ t❛❧❦✐♥❣ ❛❜♦✉t✳ ❚❤❡ s✉r❢❛❝❡ ❝r❡❛t❡❞ ❜② ❛ r♦t❛t❡❞ ❝✉r✈❡ ❤❛s ♥♦ ✈♦❧✉♠❡❀ t❤❡ s♦❧✐❞ ✐t ✕ ♣❛rt✐❛❧❧② ✕ ❜♦✉♥❞s ❞♦❡s✳ ❋♦r t❤❡ ❝❛s❡ ♦❢ ❛ ❞❡❝r❡❛s✐♥❣ f ✱ t❤✐s s♦❧✐❞ ❝♦♥t❛✐♥s ❡✈❡r② ♣♦✐♥t (x, y, z) t❤❛t s❛t✐s✜❡s✿

• ■ts ❞✐st❛♥❝❡ ✭♠❡❛s✉r❡❞ ❤♦r✐③♦♥t❛❧❧②✮ ❢r♦♠ t❤❡ y ✲❛①✐s ✐s ❜❡t✇❡❡♥ a ❛♥❞ x ✉♥✐ts✳

• ■ts ❞✐st❛♥❝❡ ✭♠❡❛s✉r❡❞ ✈❡rt✐❝❛❧❧②✮ ❢r♦♠ t❤❡ xz ✲♣❧❛♥❡ ✐s ❜❡t✇❡❡♥ f (b) ❛♥❞ f (x) ✉♥✐ts✳

❊①❡r❝✐s❡ ✸✳✶✷✳✶

❉❡s❝r✐❜❡ t❤❡ s♦❧✐❞ ❢♦r t❤❡ ❝❛s❡ ♦❢ ❛♥ ✐♥❝r❡❛s✐♥❣ f ✳ ❚❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ✐❞❡❛ ♦❢ ✈♦❧✉♠❡ ❢♦❧❧♦✇✐♥❣ t❤❡ ❈❛✈❛❧✐❡r✐ ♣r✐♥❝✐♣❧❡ ✐s ❜❛s❡❞ ♦♥ ❝✉tt✐♥❣ t❤❡ s♦❧✐❞ ✐♥t♦ ❞✐s❦s✳ ❖❢ ❝♦✉rs❡✱ ✐t ❝❛♥ ❜❡ ✉s❡❞ ❢♦r ❡✐t❤❡r ❝❛s❡✳ ■♥st❡❛❞✱ ✇❡ st❛rt ❢r♦♠ s❝r❛t❝❤ ❛♥❞ ♣✉rs✉❡ t❤❡ ✐❞❡❛ ♦❢ ❝✉tt✐♥❣ t❤❡ s♦❧✐❞ ✐♥t♦ ✇❛s❤❡rs ✭r✐♥❣s✮✳ ❲❡ ✇✐❧❧ ✉s❡✱ ❤♦✇❡✈❡r✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝t ♣r❡✈✐♦✉s❧② ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❈❛✈❛❧✐❡r✐ ♣r✐♥❝✐♣❧❡✿ ❚❤❡♦r❡♠ ✸✳✶✷✳✷✿ ❱♦❧✉♠❡ ♦❢ ❲❛s❤❡r

❚❤❡ ✈♦❧✉♠❡ ♦❢ ❛ ✇❛s❤❡r ✇✐t❤ t❤❡ ✐♥♥❡r r❛❞✐✉s r✱ t❤❡ ♦✉t❡r r❛❞✐✉s R✱ ❛♥❞ t❤✐❝❦♥❡ss h ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ✈♦❧✉♠❡s ♦❢ t❤❡ t✇♦ ❝②❧✐♥❞❡rs✿ ❱♦❧✉♠❡ = πR2 h − πr2 h = πh(R2 − r2 ) . ❲❡ ❥✉st s✉❜tr❛❝t t❤❡ ✈♦❧✉♠❡s ♦❢ t❤❡s❡ t✇♦ ❝②❧✐♥❞❡rs✿

❊①❛♠♣❧❡ ✸✳✶✷✳✸✿ ♣❡❞❡st❛❧

❙✉♣♣♦s❡ t❤❡ ♦❜❥❡❝t ✐s s✐♠♣❧② t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❛ ❞✐s❦ ♦❢ r❛❞✐✉s 1 ❛♥❞ t❤❡ ✇❛s❤❡r ❛r♦✉♥❞ ✐t ♦❢ t❤✐❝❦♥❡ss 1✿

✸✳✶✷✳

❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ♦❢ r❡✈♦❧✉t✐♦♥

✷✺✹

❚❤❡♥✱ t❤❡ ✈♦❧✉♠❡ ✐s s✐♠♣❧② t❤❡ s✉♠ ♦❢ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ❞✐s❦ ❛♥❞ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ✇❛s❤❡r✿

= 2 · ❛r❡❛

❱♦❧✉♠❡

+1 · ❛r❡❛

♦❢ t❤❡ ❞✐s❦

= 2 · π · 12

♦❢ t❤❡ ✇❛s❤❡r

+1 · (π · 22 − π · 12 ) .

❊①❛♠♣❧❡ ✸✳✶✷✳✹✿ ②✉rt ❙✉♣♣♦s❡ t❤❡ t❤✐❝❦♥❡ss ✐s ❝❤❛♥❣✐♥❣ ❧✐♥❡❛r❧② ❢r♦♠

1

t♦

2✿

❲❤❛t ✐s t❤❡ ✈♦❧✉♠❡ ♦❢ t❤✐s ♦❜❥❡❝t❄ ❊✈❡♥ t❤♦✉❣❤ ✇❡ ❦♥♦✇ t❤❡ ❛♥s✇❡r ❢r♦♠ t❤❡ ❈❛✈❛❧✐❡r✐ ♣r✐♥❝✐♣❧❡✱ ✇❡ ✇✐❧❧ ❤❛✈❡ t♦ st❛rt ✇✐t❤ ❛♣♣r♦①✐♠❛t✐♦♥s ❛❣❛✐♥✳✳✳

❲❡ ❤❛✈❡ ❛♥ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥

P

♦❢ t❤❡ r❛❞✐✉s✿

a = x0 ≤ c1 ≤ x1 ≤ ... ≤ cn ≤ xn = b ❚❤❡s❡ ❛r❡ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ s❡❣♠❡♥ts✿

∆xi = xi − xi−1 . ❍❡r❡✱ ✇❡ ❝✉t t❤❡ s♦❧✐❞ ✐♥t♦ t❤✐♥ ✇❛s❤❡rs ❜② t❤❡ ❝②❧✐♥❞❡rs st❛rt✐♥❣ ❛t ♣♦✐♥ts s❛♠♣❧❡ ✐ts ❤❡✐❣❤t ❛t t❤❡ ♣♦✐♥ts

ci ✿

x = xi

♦♥ t❤❡

x✲❛①✐s

❛♥❞ t❤❡♥

✸✳✶✷✳

❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ♦❢ r❡✈♦❧✉t✐♦♥

✷✺✺

❚❤❡♥ t❤❡ ❤❡✐❣❤t ♦❢ ❡❛❝❤ ✇❛s❤❡r ✐s f (ci )✱ ❛♥❞ ✇❡ ❤❛✈❡✿

 ▼❛ss ♦❢ it❤ ✇❛s❤❡r = r❛❞✐✉s · ❛r❡❛ = f (ci ) · πx2i − πx2i−1 ,

s✐♥❝❡ t❤❡ ✐♥s✐❞❡ r❛❞✐✉s ♦❢ t❤❡ ✇❛s❤❡r ✐s xi−1 ❛♥❞ t❤❡ ♦✉ts✐❞❡ ✐s xi ✳

❚❤❡♥✱ ✇❡ ❤❛✈❡✿

❚♦t❛❧ ✈♦❧✉♠❡ =

n X i=1

f (ci ) · π x2i − x2i−1



❲❡ ❝❛♥ ✉s❡ t❤✐s ❢♦r♠✉❧❛ ❢♦r ❝♦♠♣✉t❛t✐♦♥s✳ ❲❤❛t ✐❢ t❤❡ s♦❧✐❞ ✐s♥✬t ❛❝t✉❛❧❧② ♠❛❞❡ ♦❢ ✇❛s❤❡rs ❛♥❞ ✐ts t❤✐❝❦♥❡ss ✈❛r✐❡s ❝♦♥t✐♥✉♦✉s❧②❄ ❚❤❡♥ t❤❡ ✈♦❧✉♠❡ ♦❢ ❡❛❝❤ ✇❛s❤❡r ✕ ✇❤❡♥ t❤✐♥ ❡♥♦✉❣❤ ✕ ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ✈♦❧✉♠❡ ♦❢ s✉❝❤ ❛ ✇❛s❤❡r ✇✐t❤ t❤❡ ❝♦♥st❛♥t ❤❡✐❣❤t f (ci )✿  ♠❛ss ♦❢ it❤ ✇❛s❤❡r ≈ r❛❞✐✉s · ❛r❡❛ = f (ci ) · πx2i − πx2i−1 . ❚❤❡♥✱ ✇❡ ❤❛✈❡✿

❚♦t❛❧ ✈♦❧✉♠❡ ≈

n X i=1

 f (ci ) · π x2i − x2i−1 .

❚❤✐s ✐s t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ✇❛s❤❡rs ❜✉✐❧t ♦♥ t♦♣ ♦❢ t❤❡ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥✳ ❚❤✐s t✐♠❡✱ ❥✉st ❛s ♦♥ s❡✈❡r❛❧ ♦❝❝❛s✐♦♥s ❜❡❢♦r❡✱ ✇❡ ❞♦ ♥♦t r❡❝♦❣♥✐③❡ t❤✐s ❡①♣r❡ss✐♦♥ ❛s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛♥② ❢✉♥❝t✐♦♥ ♦✈❡r t❤✐s ♣❛rt✐t✐♦♥✱ ✇❤✐❝❤ ✐s s✉♣♣♦s❡❞ t♦ ❜❡✿ n X g(ci ) · ∆xi i=1

❢♦r s♦♠❡ ❢✉♥❝t✐♦♥ g ✳

❋❛❝t♦r✐♥❣ t❛❦❡s ✉s ♦♥❡ st❡♣ ❝❧♦s❡r t♦ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥✿ t♦t❛❧ ✈♦❧✉♠❡ ≈

n X i=1

πf (ci )(xi + xi−1 ) · ∆xi .

❲❡ ❥✉st ♥❡❡❞ t♦ ❞♦ s♦♠❡t❤✐♥❣ ❛❜♦✉t t❤❡ t❡r♠ (xi + xi−1 )✳✳✳ ❲❡ ❜❛❝❦ ✉♣ ❛ ❜✐t❀ ✇❡ ❤❛✈❡♥✬t ❝❤♦s❡♥ s❡❝♦♥❞❛r② ♥♦❞❡s✦ ▲❡t✬s ❛ss✉♠❡ t❤❛t ❢✉♥❝t✐♦♥ f ✐s ✐♥t❡❣r❛❜❧❡✳ ❚❤❡♥ t❤❡ ❝❤♦✐❝❡ ♦❢ s❡❝♦♥❞❛r② ♥♦❞❡s ✐s ♦✉rs✳ ▲❡t✬s ❝❤♦♦s❡ t❤❡ ♠✐❞✲♣♦✐♥ts✿

1 ci = (xi + xi−1 ) . 2 ❚❤❡♥✱ ❚♦t❛❧ ✈♦❧✉♠❡ ≈ 2π

n X i=1

f (ci )ci · ∆xi .

✸✳✶✷✳

✷✺✻

❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ♦❢ r❡✈♦❧✉t✐♦♥

❚❤✐s t✐♠❡✱ ✇❡ ❞♦ r❡❝♦❣♥✐③❡ t❤✐s ❡①♣r❡ss✐♦♥ ❛s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ s✐♠♣❧❡ ❢✉♥❝t✐♦♥✳ ❚❤❡♥✱ ✇❡ ❞❡✜♥❡ t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ s♦❧✐❞ ❛s t❤❡ ❧✐♠✐t ♦❢ t❤❡s❡ ❘✐❡♠❛♥♥ s✉♠s✿ 2πΣxf (x) ∆x .

■t ✐s ✐♠♣♦rt❛♥t t♦ ❝♦♥✜r♠ t❤❛t t❤✐s ♥❡✇ ❞❡✜♥✐t✐♦♥ ♦❢ ✈♦❧✉♠❡ ♠❛t❝❤❡s t❤❡ ♦❧❞ ♦♥❡✳ ❚❤❡♦r❡♠ ✸✳✶✷✳✺✿ ❱♦❧✉♠❡ ♦❢ ❙♦❧✐❞ ♦❢ ❘❡✈♦❧✉t✐♦♥ ●✐✈❡♥ ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥

f

♦♥ s❡❣♠❡♥t

[a, b]✱

t❤❡ ❛❜♦✈❡ ✐♥t❡❣r❛❧ ✐s ❡q✉❛❧ t♦

t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ s♦❧✐❞ ♦❢ r❡✈♦❧✉t✐♦♥ ♦❜t❛✐♥❡❞ ❜② r♦t❛t✐♥❣ ♦❢ t❤❡ ❣r❛♣❤ ♦❢

❱♦❧✉♠❡

= 2π

Z

f✿

b

xf (x) dx . a

Pr♦♦❢✳

❋♦r s✐♠♣❧✐❝✐t②✱ ✇❡ ❛ss✉♠❡ t❤❛t f ✐s ❞❡❝r❡❛s✐♥❣✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ✈♦❧✉♠❡ ♦❢ t❤❡ ❈❛✈❛❧✐❡r✐ ♣r✐♥❝✐♣❧❡✳ ❚❤❡ ❝r♦ss✲s❡❝t✐♦♥s ♦❢ t❤❡ s♦❧✐❞ ❛❧♦♥❣ t❤❡ y ✲❛①✐s ❛r❡ ❝✐r❝❧❡s❀ s♣❡❝✐✜❝❛❧❧②✱ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ s✉r❢❛❝❡ ✇✐t❤ t❤❡ ♣❧❛♥❡ y = q ✐s ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s f −1 (q)✳

▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ✇❤♦❧❡ s♦❧✐❞ s✇❡♣t ❜② t❤✐s ❝✉r✈❡✳ ❲❡ ❦♥♦✇ t❤✐s✿ ❱♦❧✉♠❡ = π ❲❡ ❛♣♣❧② ■♥t❡❣r❛t✐♦♥

❜② ❙✉❜st✐t✉t✐♦♥

Z

f −1 (a)

f −1 (y) f −1 (b)

✇✐t❤ x = f −1 (y)✳ ❚❤❡♥✿

❱♦❧✉♠❡ = π ❲❡ ❛♣♣❧② ■♥t❡❣r❛t✐♦♥

❜② P❛rts

Z

2

dy .

a

x2 f ′ (x) dx . b

✇✐t❤ u = x2 , dv = f ′ dx✳ ❚❤❡♥✿

❱♦❧✉♠❡

a Z ! a 2xf (x) dx = π x2 f (x) − b b Z b 2 2 xf (x) dx . = πa f (a) − πb f (b) + 2π a

❚❤❡ ❡①tr❛ t❡r♠s ❝♦♠❡ ❢r♦♠ t❤❡ ❞✐s❦ ❛t t❤❡ ❜♦tt♦♠ ❛♥❞ t❤❡ ❝②❧✐♥❞❡r ✐♥ t❤❡ ♠✐❞❞❧❡ t♦ ❜❡ r❡♠♦✈❡❞✿

✸✳✶✷✳

❱♦❧✉♠❡s ♦❢ s♦❧✐❞s ♦❢ r❡✈♦❧✉t✐♦♥

✷✺✼

❊①❡r❝✐s❡ ✸✳✶✷✳✻ ▼♦❞✐❢② t❤❡ ♣r♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠ ❢♦r t❤❡ ❝❛s❡ ♦❢ ❛♥ ✐♥❝r❡❛s✐♥❣

f✳

❈❤❛♣t❡r ✹✿ ❙❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

❈♦♥t❡♥ts

✹✳✶ ❆ ❜❛❧❧ ✐s t❤r♦✇♥✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✷ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥ t❤❡ ♣❧❛♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✸ ❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✹ ■♥tr♦❞✉❝t✐♦♥ t♦ ❝❛❧❝✉❧✉s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✹✳✺ ❚❤❡ ❝❡♥tr♦✐❞ ♦❢ ❛ ✢❛t ♦❜❥❡❝t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✻ ❈♦♦r❞✐♥❛t❡ s②st❡♠s❀ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✼ ❱❡❝t♦rs ♦♥ t❤❡ ♣❧❛♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✽ ❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✾ ❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2 ✳ ✳ ✹✳✶✵ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿ C ✐s♥✬t ❥✉st R2 ✹✳✶✶ ❉✐s❝r❡t❡ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✶✷ ❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✷✺✽ ✷✻✹ ✷✼✹ ✷✽✺ ✷✾✸ ✷✾✼ ✸✵✼ ✸✶✷ ✸✶✾ ✸✷✸ ✸✷✽ ✸✸✼

✹✳✶✳ ❆ ❜❛❧❧ ✐s t❤r♦✇♥✳✳✳

▲❡t✬s r❡✈✐❡✇ ✇❤❛t ✇❡ ❧❡❛r♥❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✸ ❛❜♦✉t ♠♦t✐♦♥ ♦❢ ❛ ❜❛❧❧ ✭♦r ❛ ❝❛♥♥♦♥❜❛❧❧✮✳ ❲❤❡♥ ❛ ❜❛❧❧ ✐s t❤r♦✇♥ ✐♥ t❤❡ ❛✐r ✉♥❞❡r ❛♥ ❛♥❣❧❡✱ ✐t ♠♦✈❡s ✐♥ ❜♦t❤ ✈❡rt✐❝❛❧ ❛♥❞ ❤♦r✐③♦♥t❛❧ ❞✐r❡❝t✐♦♥s✱ s✐♠✉❧t❛♥❡♦✉s❧② ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t❧②✳ ❚❤❡ ❞②♥❛♠✐❝s ✐s ✈❡r② ❞✐✛❡r❡♥t✳ ■♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❞✐r❡❝t✐♦♥✱ ❛s t❤❡r❡ ✐s ♥♦ ❢♦r❝❡ ❝❤❛♥❣✐♥❣ t❤❡ ✈❡❧♦❝✐t②✱ t❤❡ ❧❛tt❡r r❡♠❛✐♥s ❝♦♥st❛♥t✿

▼❡❛♥✇❤✐❧❡✱ t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t② ✐s ❝♦♥st❛♥t❧② ❝❤❛♥❣❡❞ ❜② t❤❡ ❣r❛✈✐t②✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❤❡✐❣❤t ♦♥ t❤❡ t✐♠❡ ✐s q✉❛❞r❛t✐❝✿

✹✳✶✳

❆ ❜❛❧❧ ✐s t❤r♦✇♥✳✳✳

❲❡ ❤❛✈❡

✷✺✾

t❤r❡❡ ✈❛r✐❛❜❧❡s✿

• t ✲ t✐♠❡✳

• x ✲ t❤❡ ❤♦r✐③♦♥t❛❧ ❞✐♠❡♥s✐♦♥✱ t❤❡ ❞❡♣t❤✱ t❤❛t ❞❡♣❡♥❞s ♦♥ t✐♠❡✳

• y ✲ t❤❡ ✈❡rt✐❝❛❧ ❞✐♠❡♥s✐♦♥✱ t❤❡ ❤❡✐❣❤t✱ t❤❛t ❛❧s♦ ❞❡♣❡♥❞s ♦♥ t✐♠❡✳

❚❤❡ ♣❛t❤ ♦❢ t❤❡ ❜❛❧❧ ✇✐❧❧ ❛♣♣❡❛r t♦ ❛♥ ♦❜s❡r✈❡r ✕ ❢r♦♠ t❤❡ r✐❣❤t ❛♥❣❧❡ ✕ ❛s ❛ ❝✉r✈❡✳ ■t ✐s ♣❧❛❝❡❞ ✐♥ t❤❡ xy ✲♣❧❛♥❡ ♣♦s✐t✐♦♥❡❞ ✈❡rt✐❝❛❧❧②✿

❋✐rst✱ t❤❡

s❡q✉❡♥❝❡s✳

❲❡ ✉s❡❞ t❤❡s❡

❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts t♦ ✜♥❞ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡♥ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥✿ ❤♦r✐③♦♥t❛❧ ♣♦s✐t✐♦♥ ✈❡❧♦❝✐t② ❛❝❝❡❧❡r❛t✐♦♥

xn

xn+1 − xn vn = h vn+1 − vn an = h

✈❡rt✐❝❛❧

yn

yn+1 − yn h un+1 − un bn = h un =

✇❤❡r❡ h ✐s t❤❡ ✐♥❝r❡♠❡♥t ♦❢ t✐♠❡✳ ❚❤❡s❡ ❢♦r♠✉❧❛s ❝❛♥ ♥♦✇ ❜❡ s♦❧✈❡❞ ✐♥ ♦r❞❡r t♦ ❜❡ ❛❜❧❡ t♦ ♠♦❞❡❧ t❤❡ ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳ ❚❤❡ r❡s✉❧t ✐s t❤❡s❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❘✐❡♠❛♥♥ s✉♠s ✿ ❤♦r✐③♦♥t❛❧

✈❡rt✐❝❛❧

❛❝❝❡❧❡r❛t✐♦♥

an

bn

✈❡❧♦❝✐t②

vn+1 = vn + han

un+1 = un + hbn

♣♦s✐t✐♦♥

xn+1 = xn + hvn yn+1 = yn + hun

Pr♦❜❧❡♠✿ ❋r♦♠ ❛ 200✲❢❡❡t ❡❧❡✈❛t✐♦♥✱ ❛ ❝❛♥♥♦♥ ✐s ✜r❡❞ ❤♦r✐③♦♥t❛❧❧② ❛t 200 ❢❡❡t ♣❡r s❡❝♦♥❞✳ ❍♦✇ ❢❛r ✇✐❧❧ t❤❡ ❝❛♥♥♦♥❜❛❧❧ ❣♦❄

✹✳✶✳

✷✻✵

❆ ❜❛❧❧ ✐s t❤r♦✇♥✳✳✳

❚❤❡ ♣❤②s✐❝s ✐s ❛s ❢♦❧❧♦✇s✿ • ❍♦r✐③♦♥t❛❧✿ ❚❤❡r❡ ✐s ♥♦ ❢♦r❝❡✱ ❤❡♥❝❡ an = 0 ❢♦r ❛❧❧ n✳

• ❱❡rt✐❝❛❧✿ ❚❤❡ ❢♦r❝❡ ✐s ❝♦♥st❛♥t ❛♥❞ bn = −g ❢♦r ❛❧❧ n✳ ❍❡r❡ g ✐s t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❝♦♥st❛♥t✿ g = 32 ❢t✴s❡❝2 .

◆❡①t✱ ✇❡ ❛❝q✉✐r❡ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ • ❚❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✿ x0 = 0 ❛♥❞ y0 = 200✳

• ❚❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✿ v0 = 200 ❛♥❞ u0 = 0✳ ❊①❛♠♣❧❡ ✹✳✶✳✶✿ ❤♦✇ ❢❛r

❚♦ ✜♥❞ ✇❤❡♥ ❛♥❞ ✇❤❡r❡ t❤❡ ❜❛❧❧ ❤✐ts t❤❡ ❣r♦✉♥❞✱ ✇❡ s❝r♦❧❧ ❞♦✇♥ t♦ ✜♥❞ t❤❡ r♦✇ ✇✐t❤ y ❝❧♦s❡st t♦ 0✿

■t ❤❛♣♣❡♥s s♦♠❡t✐♠❡ ❜❡t✇❡❡♥ t = 3.5 ❛♥❞ t = 3.6 s❡❝♦♥❞s✱ s❛② t1 = 3.55 s❡❝♦♥❞s✳ ❙❡❝♦♥❞✱ t❤❡ ✈❛❧✉❡s ♦❢ x ❞✉r✐♥❣ t❤✐s t✐♠❡ ♣❡r✐♦❞ ❛r❡ ❜❡t✇❡❡♥ x = 700 ❛♥❞ x = 720 ❢❡❡t✱ s❛②✱ x1 = 710 ❢❡❡t✳ ❲❡ ❛❧s♦ ♣❧♦t t❤❡ ❣r❛♣❤s ♦❢ x ❛♥❞ y ❛s ❢✉♥❝t✐♦♥s ♦❢ t ♦♥ t❤❡ r✐❣❤t✳ ❚❤❡ s♣r❡❛❞s❤❡❡t ✐s ❝♦♥str✉❝t❡❞ ❢♦r x ❛♥❞ y s❡♣❛r❛t❡❧②✱ ❛s ❢♦❧❧♦✇s✳ ❚❤❡ t✐♠❡ ✐s ✐♥ t❤❡ ✜rst ❝♦❧✉♠♥ ♣r♦❣r❡ss✐♥❣ ❢r♦♠ 0 ❡✈❡r② 0.05✳ ❚❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ✐s ✐♥ t❤❡ ♥❡①t✱ 0 ❛♥❞ −32✱ r❡s♣❡❝t✐✈❡❧②✳ ■♥ t❤❡ ♥❡①t ❝♦❧✉♠♥✱ t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② ✐s ❡♥t❡r❡❞ ✐♥ t❤❡ t♦♣ ❝❡❧❧✱ 200 ❛♥❞ 0 r❡s♣❡❝t✐✈❡❧②✳ ❇❡❧♦✇✱ t❤❡ ✈❡❧♦❝✐t② ✐s ❝♦♠♣✉t❡❞ ❛s ❛ ❘✐❡♠❛♥♥ s✉♠ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♣r❡✈✐♦✉s ❝♦❧✉♠♥✱ ✇✐t❤ t❤❡ s❛♠❡ ❢♦r♠✉❧❛✿ ❂❘❬✲✶❪❈✰✭❘❈❬✲✷❪✲❘❬✲✶❪❈❬✲✷❪✮✯❘❬✲✶❪❈❬✲✶❪

■♥ t❤❡ ♥❡①t ❝♦❧✉♠♥✱ t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥ ✐s ❡♥t❡r❡❞ ✐♥ t❤❡ t♦♣ ❝❡❧❧✱ 0 ❛♥❞ 200 r❡s♣❡❝t✐✈❡❧②✳ ❇❡❧♦✇✱ t❤❡ ❧♦❝❛t✐♦♥ ✐s ❝♦♠♣✉t❡❞ ❛s ❛ ❘✐❡♠❛♥♥ s✉♠ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♣r❡✈✐♦✉s ❝♦❧✉♠♥✱ ✇✐t❤ t❤❡ s❛♠❡ ❢♦r♠✉❧❛✿ ❂❘❬✲✶❪❈✰✭❘❈❬✲✸❪✲❘❬✲✶❪❈❬✲✸❪✮✯❘❈❬✲✶❪

❚❤❡ r❡s✉❧ts ❛r❡ s❤♦✇♥ ❜❡❧♦✇✿

✹✳✶✳

✷✻✶

❆ ❜❛❧❧ ✐s t❤r♦✇♥✳✳✳

❚♦ ✜♥❞ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ♣r♦❜❧❡♠ ❢r♦♠ t❤✐s ❞❛t❛✱ ✇❡ ✜♥❞ t❤❡ ✐♥t❡r✈❛❧ ❞✉r✐♥❣ ✇❤✐❝❤ t❤❡ ❝❛♥♥♦♥❜❛❧❧ ❤✐t t❤❡ ❣r♦✉♥❞✱ ✐✳❡✳✱ y = 0✳ ❲❡ ❣♦ ❞♦✇♥ t❤❡ y ❝♦❧✉♠♥ ✉♥t✐❧ ✇❡ ✜♥❞ t❤❡ ✈❛❧✉❡ ❝❧♦s❡st t♦ 0❀ ✐t ✐s y = 1.2✳ ❲❡ t❤❡♥ ✜♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡ ♦❢ x❀ ✐t ✐s x = 700✳ P❧♦tt✐♥❣ x ❛❣❛✐♥st y ♣r♦❞✉❝❡s t❤❡ ♣❛t❤ ♦❢ t❤❡ ❝❛♥♥♦♥❜❛❧❧✿

❊①❡r❝✐s❡ ✹✳✶✳✷

❯♥❞❡r t❤❡ s❛♠❡ ❝♦♥❞✐t✐♦♥s✱ s♦❧✈❡ ♥✉♠❡r✐❝❛❧❧② t❤❡ ♣r♦❜❧❡♠ ♦❢ ❤✐tt✐♥❣ ❛ t❛r❣❡t 500 ❢❡❡t ❛✇❛②✳ ❲❡ st❛rt ✇✐t❤ t❤❡

❝♦♥t✐♥✉♦✉s ❝❛s❡

♥♦✇✿

• ❤♦r✐③♦♥t❛❧✿ x′′ = 0 • ✈❡rt✐❝❛❧✿ y ′′ = −g

❲❡ st❛rt ❛t t❤❡ s❛♠❡ ♣❧❛❝❡ ❛s ❛❜♦✈❡✿ x′′ = 0,

x′ (0) = 200, x(0) = 0

y ′′ = −g, y ′ (0) = 0,

y(0) = 200

❙✐♥❝❡ t❤❡ ✈❡❧♦❝✐t② ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✱ ✇❡ ✐♥t❡❣r❛t❡ t❤❡s❡✳ ❚❤❡♥ ❢♦r ❤♦r✐③♦♥t❛❧✱ ✇❡ ❤❛✈❡✿ ′

x =

✇❤❡r❡ Cx ✐s ❛♥② ❝♦♥st❛♥t✳ ◆❡①t✱ ❢♦r t❤❡ ✈❡rt✐❝❛❧✱ ′

y =

✇❤❡r❡ Cy ✐s ❛♥② ❝♦♥st❛♥t✳

Z

Z

0 dt = Cx ,

−g dt = −gt + Cy ,

❙✐♥❝❡ t❤❡ ❧♦❝❛t✐♦♥ ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✈❡❧♦❝✐t②✱ ✇❡ ✐♥t❡❣r❛t❡ t❤❡s❡✳ ❚❤❡♥ ❢♦r ❤♦r✐③♦♥t❛❧✱ ✇❡ ❤❛✈❡✿ x=

Z

′

x dt =

Z

Cx dt = Cx t + Kx ,

✹✳✶✳

✷✻✷

❆ ❜❛❧❧ ✐s t❤r♦✇♥✳✳✳

✇❤❡r❡ Kx ✐s ❛♥② ❝♦♥st❛♥t✳ ◆❡①t✱ ❢♦r t❤❡ ✈❡rt✐❝❛❧✱ y=

✇❤❡r❡ Ky ✐s ❛♥② ❝♦♥st❛♥t✳ ❚❤✉s✱ t❤❡ ❣❡♥❡r❛❧

s♦❧✉t✐♦♥

Z

′

y dx =

Z

(−gt + Cy ) dt = − 12 gt2 + Cy t + Ky ,

♦❢ t❤✐s s②st❡♠ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✐s✿ x = y =

C x t + Kx , − 12 gt2

+ C y t + Ky .

❆♥② ♣♦ss✐❜❧❡ ❞②♥❛♠✐❝s ✐s ❢♦✉♥❞ ❜② s♣❡❝✐❢②✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢♦✉r ❝♦♥st❛♥ts✿ Cx , Cy , , Kx , Ky .

❚❤❡ ♣❤②s✐❝s ♦❢ t❤❡ s✐t✉❛t✐♦♥ ❛❧❧♦✇s ✉s t♦ ❛ss✐❣♥ ♠❡❛♥✐♥❣s t♦ t❤❡s❡ ❢♦✉r ❝♦♥st❛♥ts✳ ❋✐rst✱ x′ =

Cx =⇒ x′ (0) = Cx ,

y ′ = −gt + Cy =⇒ y ′ (0) = Cy .

❚❤❡r❡❢♦r❡✱ • Cx ✐s t❤❡ ✭❝♦♥st❛♥t✮ ❤♦r✐③♦♥t❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ✈❡❧♦❝✐t②❀ • Cy ✐s t❤❡ ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ✈❡❧♦❝✐t②✳

◆❡①t✱

x =

Cx t + Kx =⇒ x(0) = Kx ,

y = − 21 gt2 + Cy t + Ky =⇒ y(0) = Ky .

❚❤❡r❡❢♦r❡✱

• Kx ✐s t❤❡ ✐♥✐t✐❛❧ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥ ✭❞❡♣t❤✮❀ • Ky ✐s t❤❡ ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥ ✭❤❡✐❣❤t✮✳

❚❤✉s✱ ✇❡ ❤❛✈❡✿

❉❡♣t❤ = ❍❡✐❣❤t =

✐♥✐t✐❛❧ ❞❡♣t❤ ✐♥✐t✐❛❧ ❤❡✐❣❤t

+ +

✐♥✐t✐❛❧ ❤♦r✐③♦♥t❛❧ ✈❡❧♦❝✐t② ✐♥✐t✐❛❧ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t②

· t✐♠❡

· t✐♠❡

− 21 g · t✐♠❡

2

❲❡ ✉s❡❞ t❤❡s❡ t✇♦ ❡q✉❛t✐♦♥s t♦ s♦❧✈❡ ❛ ✈❛r✐❡t② ♦❢ ♣r♦❜❧❡♠s ❛❜♦✉t ♠♦t✐♦♥✳ ❊①❛♠♣❧❡ ✹✳✶✳✸✿ ❤♦✇ ❢❛r

❋r♦♠ ❛ 200✲❢❡❡t ❡❧❡✈❛t✐♦♥✱ ❛ ❝❛♥♥♦♥ ✐s ✜r❡❞ ❤♦r✐③♦♥t❛❧❧② ❛t 200 ❢❡❡t ♣❡r s❡❝♦♥❞✳ ❍♦✇ ❢❛r ✇✐❧❧ t❤❡ ❝❛♥♥♦♥❜❛❧❧ ❣♦❄

❚❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ • t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✿ 0 ❛♥❞ 200✳ • t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t②✿ 200 ❛♥❞ 0✳

✹✳✶✳

✷✻✸

❆ ❜❛❧❧ ✐s t❤r♦✇♥✳✳✳

❚❤❡♥ ♦✉r ❡q✉❛t✐♦♥s ❜❡❝♦♠❡✿ x =

200t , −16t2 .

y = 200

Pr❡✈✐♦✉s❧② ✇❡ s♦❧✈❡❞ t❤❡ ♣r♦❜❧❡♠ ❛❧❣❡❜r❛✐❝❛❧❧② ❛s ❢♦❧❧♦✇s✳ ❚❤❡ ❤❡✐❣❤t ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ✢✐❣❤t ✐s y1 = 0✱ s♦ t♦ ✜♥❞ t❤❡ t✐♠❡✱ ✇❡ s❡t y = 200 − 16t2 = 0 ❛♥❞ s♦❧✈❡ ❢♦r t✿ t1 =

r

200 ≈ 3.54 . 16

❲❡ s✉❜st✐t✉t❡ t❤✐s ✈❛❧✉❡ ♦❢ t ✐♥t♦ x t♦ ✜♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡♣t❤✿ √ 5 2 ≈ 707 . x1 = 200t1 = 200 2

❲❤❛t ❛❜♦✉t t❤❡ ✈❡❧♦❝✐t② ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡❄ ❲❡ ❤❛✈❡✿ dx = vx , dt dy = vy −gt . dt

❆❞❞✐♥❣ t❤❡s❡ t✇♦ ❡q✉❛t✐♦♥s t♦ t❤❡ ❢♦r♠❡r t✇♦ ❛❧❧♦✇s ✉s t♦ s♦❧✈❡ ♠♦r❡ ♣r♦❢♦✉♥❞ ♣r♦❜❧❡♠s✳ ❊①❛♠♣❧❡ ✹✳✶✳✹✿ ✐♠♣❛❝t

■♥ t❤❡ s❡tt✐♥❣ ♦❢ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ❤♦✇ ❤❛r❞ ❞♦❡s t❤❡ ❜❛❧❧ ❤✐t t❤❡ ❣r♦✉♥❞❄ ❋✐rst✱ ✇❡ ❡①❛♠✐♥❡ t❤❡ s♣r❡❛❞s❤❡❡t✳ ■♥st❡❛❞ ♦❢ t❤❡ ❢♦r♠✉❧❛s✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t✐❡s ✭✐✳❡✳✱ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✮ t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ✈❡❧♦❝✐t✐❡s✳ ❚❤❡ ❢♦r♠✉❧❛ ❢♦r x′ ✐s✿ ❂✭❘❈❬✲✷❪✲❘❬✲✶❪❈❬✲✷❪✮✴✭❘❈❬✲✸❪✲❘❬✲✶❪❈❬✲✸❪✮

❛♥❞ t❤❡ ❢♦r♠✉❧❛ ❢♦r y ′ ✐s✿ ❂✭❘❈❬✲✷❪✲❘❬✲✶❪❈❬✲✷❪✮✴✭❘❈❬✲✹❪✲❘❬✲✶❪❈❬✲✹❪✮

❚❤❡ ❞❡♥♦♠✐♥❛t♦rs r❡❢❡r t♦ t❤❡ ❝♦❧✉♠♥ t❤❛t ❝♦♥t❛✐♥s t❤❡ t✐♠❡✱ ❛♥❞ t❤❡ ♥✉♠❡r❛t♦r r❡❢❡rs t♦ t❤❡ ❝♦❧✉♠♥s t❤❛t ❝♦♥t❛✐♥ x ❛♥❞ y r❡s♣❡❝t✐✈❡❧②✳

▲♦♦❦✐♥❣ ❛t t❤❡ s❛♠❡ r♦✇ ❛s ❜❡❢♦r❡✱ ✇❡ s❡❡ t❤❛t t❤❡ ✈❡rt✐❝❛❧ ✈❡❧♦❝✐t② ❛t t❤❡ ♠♦♠❡♥t ♦❢ ✐♠♣❛❝t ✐s ❜❡t✇❡❡♥ −110.4 ❛♥❞ −113.6 ❢❡❡t ♣❡r s❡❝♦♥❞✳ ◆♦✇✱ t❤❡ ❛❧❣❡❜r❛✳ ❚❤❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ✈❡❧♦❝✐t✐❡s t❛❦❡ t❤✐s ❢♦r♠✿ dx = 200 , dt dy = −32t . dt

▲❡t✬s ✜♥❞ t❤❡ ✈❡❧♦❝✐t② ❛t t❤❡ t✐♠❡ ♦❢ ❝♦♥t❛❝t✳ ❲❡ s✉❜st✐t✉t❡ t❤❡ t✐♠❡ ✇❡✬✈❡ ❢♦✉♥❞✱ √ 5 2 t1 = , 2

✹✳✷✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥ t❤❡ ♣❧❛♥❡

✷✻✹

✐♥t♦ t❤❡ ❢♦r♠✉❧❛s ❢♦r ✈❡❧♦❝✐t②✿ dx = 200, dt t=t1 √ 5 2 dy ≈ −112 . = −32t1 = −32 dt t=t1 2

❚❤❡ ❛♥s✇❡r ♠❛t❝❤❡s ♦✉r ❡st✐♠❛t❡✳

❇✉t ✇❤✐❝❤ ♦♥❡ ♦❢ t❤❡ t✇♦ ♥✉♠❜❡rs r❡♣r❡s❡♥t ❤♦✇ ❢❛st t❤❡ ❜❛❧❧ ❤✐ts t❤❡ ❣r♦✉♥❞❄ ■t ✐s t❤❡ ❧❛tt❡r ✐❢ t❤❡ ❜❛❧❧ ❤✐ts t❤❡ ✭❤♦r✐③♦♥t❛❧✮ s✉r❢❛❝❡✱ ❛♥❞ ✐t ✐s t❤❡ ❢♦r♠❡r ✐❢ t❤✐s ✐s ❛ ✇❛❧❧✳ ❚❤❡♥✱ t❤❡ ❣❡♥❡r❛❧ ❛♥s✇❡r s❤♦✉❧❞ ❜❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦✳ ❚❤✐s ✐s ❤♦✇ t❤❡② s❤♦✉❧❞ ❜❡ ❝♦♠❜✐♥❡❞ ✈✐❛ t❤❡ P②t❤❛❣♦r❡❛♥ ❚❤❡♦r❡♠ ✿

❚❤❡♥✱ t❤❡ ✐♠♣❛❝t ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤✐s ♥✉♠❜❡r✿ p 2002 + (−112)2 ≈ 229 .

✹✳✷✳ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥ t❤❡ ♣❧❛♥❡

❊①❛♠♣❧❡ ✹✳✷✳✶✿ ❜❛❧❧

■♠❛❣✐♥❡ ❛ ♣❡rs♦♥ ♦❜s❡r✈✐♥❣ t❤❡ ✢✐❣❤t ♦❢ ❛ t❤r♦✇♥ ❜❛❧❧ ❛s ✐t ♣❛ss❡s ❜②✿

■s t❤❡r❡ ❛♥♦t❤❡r ✇❛② t♦ ❝❛♣t✉r❡ t❤✐s ✢✐❣❤t❄ ■♠❛❣✐♥❡ t❤❡r❡ ❛r❡ t✇♦ ♠♦r❡ ♦❜s❡r✈❡rs✿ • ❚❤❡ ✜rst ♦♥❡ ✭r❡❞✮ ✐s ♦♥ t❤❡ ❣r♦✉♥❞ ✉♥❞❡r t❤❡ ♣❛t❤ ♦❢ t❤❡ ❜❛❧❧ ❛♥❞ ❝❛♥ ♦♥❧② s❡❡ t❤❡ ❢♦r✇❛r❞ ♣r♦❣r❡ss ♦❢ t❤❡ ❜❛❧❧✳ • ❚❤❡ s❡❝♦♥❞ ♦♥❡ ✭❣r❡❡♥✮ ✐s ❜❡❤✐♥❞ t❤❡ t❤r♦✇ ❛♥❞ ❝❛♥ s❡❡ ♦♥❧② t❤❡ r✐s❡ ❛♥❞ ❢❛❧❧ ♦❢ t❤❡ ❜❛❧❧✳

✹✳✷✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥ t❤❡ ♣❧❛♥❡

✷✻✺

■❢ t❤❡ t✇♦ ♠❛❦❡ r❡❝♦r❞s ♦❢ ✇❤❡r❡ t❤❡ ❜❛❧❧ ✇❛s ❛t ✇❤❛t t✐♠❡✱ t❤❡② ❝❛♥ ✉s❡ t❤❡ t✐♠❡ st❛♠♣s t♦ ♠❛t❝❤ t❤❡ t✇♦ ❝♦♦r❞✐♥❛t❡s ❛♥❞ t❤❡♥ ♣❧♦t t❤✐s ♣♦✐♥t ♦♥ t❤❡ xy ✲♣❧❛♥❡✳ ❚❤❡s❡ ♣♦✐♥ts ✇✐❧❧ ❢♦r♠ t❤❡ ❜❛❧❧✬s tr❛❥❡❝t♦r②✱ ✇❤❛t t❤❡ ✜rst ♦❜s❡r✈❡r s❛✇✳ ■t ✐s ❝❛❧❧❡❞ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ❈✉r✈❡s ❛r❡♥✬t r❡♣r❡s❡♥t❡❞ ❛s ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s✳ ■♥ ❢❛❝t✱ y ❞♦❡s♥✬t ❞❡♣❡♥❞ ♦♥ x ❛♥②♠♦r❡✱ ❜✉t t❤❡② ❛r❡

r❡❧❛t❡❞ t♦ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡ ❧✐♥❦ ✐s ❡st❛❜❧✐s❤❡❞ ❜② ♠❡❛♥s ♦❢ ❛♥♦t❤❡r ✈❛r✐❛❜❧❡✱ t✳ ❙♦✱ ✇❡ ❤❛✈❡ t✇♦ ❢✉♥❝t✐♦♥s

t❤❛t ❤❛✈❡ ♥♦t❤✐♥❣ t♦ ❞♦ ✇✐t❤ ❡❛❝❤ ♦t❤❡r ❡①❝❡♣t t❤❡ ✐♥♣✉ts ❝❛♥ ❜❡ ♠❛t❝❤❡❞✳

❉❡✜♥✐t✐♦♥ ✹✳✷✳✷✿ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❆ ♣❛r❛♠❡tr✐❝ ✈❛r✐❛❜❧❡✿

❝✉r✈❡ ♦♥ t❤❡ ♣❧❛♥❡ ✐s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ s❛♠❡ (

x = f (t) y = g(t)

❲❡ ❝❛♥ ❛❧s♦ ✉s❡ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ♦❢ ♣♦✐♥ts t♦ r❡♣r❡s❡♥t t❤✐s ❝✉r✈❡✿ (x, y) = f (t), g(t)



❊①❡r❝✐s❡ ✹✳✷✳✸ ❊①♣❧❛✐♥ ❤♦✇ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❛ r❡❧❛t✐♦♥✳

❊①❛♠♣❧❡ ✹✳✷✳✹✿ ♣❧♦tt❡r ❆ ❝✉r✈❡ ♠❛② ❜❡ ♣❧♦tt❡❞ ♦♥ ❛ ♣✐❡❝❡ ♦❢ ♣❛♣❡r ❜② ❤❛♥❞ ♦r ❜② ❛ ❝♦♠♣✉t❡r ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❡t❤♦❞✳ ❆ ♣❡♥ ✐s ❛tt❛❝❤❡❞ t♦ ❛ r✉♥♥❡r ♦♥ ❛ ✈❡rt✐❝❛❧ ❜❛r✱ ✇❤✐❧❡ t❤❛t ❜❛r s❧✐❞❡s ❛❧♦♥❣ ❛ ❤♦r✐③♦♥t❛❧ r❛✐❧ ❛t t❤❡ ❜♦tt♦♠ ❡❞❣❡ ♦❢ t❤❡ ♣❛♣❡r✿

✹✳✷✳ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥ t❤❡ ♣❧❛♥❡

✷✻✻

❚❤❡ ❝♦♠♣✉t❡r ❝♦♠♠❛♥❞s t❤❡ ♥❡①t ❧♦❝❛t✐♦♥ ♦❢ ❜♦t❤ ❛s ❢♦❧❧♦✇s✳ ❆t ❡❛❝❤ ♠♦♠❡♥t ♦❢ t✐♠❡ t✱ ✇❡ ❤❛✈❡✿ ✶✳ ❚❤❡ ❤♦r✐③♦♥t❛❧ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ✈❡rt✐❝❛❧ ❜❛r ✭❛♥❞ t❤❡ ♣❡♥✮ ✐s ❣✐✈❡♥ ❜② x = f (t)✳ ✷✳ ❚❤❡ ✈❡rt✐❝❛❧ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ♣❡♥ ✐s ❣✐✈❡♥ ❜② y = g(t)✳ ❲❛r♥✐♥❣✦ ❚❤✐s ✈✐❡✇ ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐s ♠♦st ✉s❡❢✉❧ ✇✐t❤✐♥ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s ❛♥❞ ✈❡❝✲ t♦rs✳

❊①❛♠♣❧❡ ✹✳✷✳✺✿ str❛✐❣❤t ❧✐♥❡s

▲❡t✬s ❡①❛♠✐♥❡ ♠♦t✐♦♥ ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡ ♦♥ t❤❡ xy ✲♣❧❛♥❡✳ ❋✐rst ✇❡ ❣♦ ❛❧♦♥❣ t❤❡ x✲❛①✐s✳ ❚❤❡ ♠♦t✐♦♥ ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ ❢❛♠✐❧✐❛r ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✿ x = 2t + 1 .

❲❡ ❛r❡ ♠♦✈✐♥❣ 2 ❢❡❡t ♣❡r s❡❝♦♥❞ t♦ t❤❡ r✐❣❤t st❛rt✐♥❣ ❛t x = 1✳ ❚❤❡s❡ ❛r❡ ❛ ❢❡✇ ♦❢ t❤❡ ❧♦❝❛t✐♦♥s✿

❙❡❝♦♥❞ ✇❡ ❣♦ ❛❧♦♥❣ t❤❡ y ✲❛①✐s✳ ❚❤❡ ♠♦t✐♦♥ ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛♥♦t❤❡r ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✿ y = 3t + 2 .

❲❡ ❛r❡ ♠♦✈✐♥❣ 3 ❢❡❡t ♣❡r s❡❝♦♥❞ ✉♣ st❛rt✐♥❣ ❛t y = 2✳ ❚❤❡s❡ ❛r❡ ❛ ❢❡✇ ♦❢ t❤❡ ❧♦❝❛t✐♦♥s✿

◆♦✇✱ ✇❤❛t ✐❢ t❤❡s❡ t✇♦ ❛r❡ ❥✉st t✇♦ ❞✐✛❡r❡♥t ✈✐❡✇s ♦❢ t❤❡ s❛♠❡ ♠♦t✐♦♥ ❢r♦♠ t✇♦ ❞✐✛❡r❡♥t ♦❜s❡r✈❡rs❄ ❚❤❡♥ ✇❡ ❤❛✈❡✿ ( x = 2t + 1 , y = 3t + 2 .

❚❤❡s❡ ❛r❡ ❛ ❢❡✇ ♦❢ t❤❡ ❧♦❝❛t✐♦♥s✿

✹✳✷✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥ t❤❡ ♣❧❛♥❡

✷✻✼

❊①❡r❝✐s❡ ✹✳✷✳✻

❊①♣❧❛✐♥ ✇❤② t❤❡s❡ ♣♦✐♥ts ❧✐❡ ♦♥ ❛ str❛✐❣❤t ❧✐♥❡✳ ❍✐♥t✿ tr✐❛♥❣❧❡s✳ ❖❢ ❝♦✉rs❡✱ t❤❡ ♠♦t✐♦♥ ♠❡t❛♣❤♦r ✕ x ❛♥❞ y ❛r❡ ❝♦♦r❞✐♥❛t❡s ✐♥ t❤❡ s♣❛❝❡ ✕ ✇✐❧❧ ❜❡ s✉♣❡rs❡❞❡❞✳ ■♥ ❝♦♥tr❛st t♦ t❤✐s ❛♣♣r♦❛❝❤✱ ✇❡ ❧♦♦❦ ❛t t❤❡ t✇♦ q✉❛♥t✐t✐❡s ❛♥❞ t✇♦ ❢✉♥❝t✐♦♥s t❤❛t ♠✐❣❤t ❤❛✈❡ ♥♦t❤✐♥❣ t♦ ❞♦ ✇✐t❤ ❡❛❝❤ ♦t❤❡r ✭❡①❝❡♣t ❢♦r t✱ ♦❢ ❝♦✉rs❡✮✳ ❊①❛♠♣❧❡ ✹✳✷✳✼✿ ❝♦♠♠♦❞✐t✐❡s tr❛❞❡r

❙✉♣♣♦s❡ ❛ ❝♦♠♠♦❞✐t✐❡s tr❛❞❡r ❢♦❧❧♦✇s t❤❡ ♠❛r❦❡t✳ ❲❤❛t ❤❡ s❡❡s ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ • t ✲ t✐♠❡ • x ✲ t❤❡ ♣r✐❝❡ ♦❢ ✇❤❡❛t ✭s❛②✱ ✐♥ ❞♦❧❧❛rs ♣❡r ❜✉s❤❡❧✮ • y ✲ t❤❡ ♣r✐❝❡ ♦❢ s✉❣❛r ✭s❛②✱ ✐♥ ❞♦❧❧❛rs ♣❡r t♦♥✮ ❲❡ s✐♠♣❧② ❤❛✈❡ t✇♦ ❢✉♥❝t✐♦♥s ❛♥❞ ✇❡ ✕ ✐♥✐t✐❛❧❧② ✕ ❧♦♦❦ ❛t t❤❡♠ s❡♣❛r❛t❡❧②✳ ❋✐rst✱ ❧❡t✬s ✐♠❛❣✐♥❡ t❤❛t t❤❡ ♣r✐❝❡ ♦❢

✇❤❡❛t

✐s ❞❡❝r❡❛s✐♥❣✿

x ց ❚❤❡ ❞❛t❛ ❝♦♠❡s t♦ t❤❡ ♦❜s❡r✈❡r ✐♥ ❛ ♣✉r❡✱ ♥✉♠❡r✐❝❛❧ ❢♦r♠✳ ❚♦ s✐♠✉❧❛t❡ t❤✐s s✐t✉❛t✐♦♥ ❛♥❞ t♦ ♠❛❦❡ t❤✐s s♣❡❝✐✜❝✱ ♦♥❡ ❝❛♥ ❝❤♦♦s❡ ❛ ❢♦r♠✉❧❛✱ ❢♦r ❡①❛♠♣❧❡✿

x = f (t) =

1 . t+1

❚♦ s❤♦✇ s♦♠❡ ❛❝t✉❛❧ ❞❛t❛✱ ✇❡ ❡✈❛❧✉❛t❡ x ❢♦r s❡✈❡r❛❧ ✈❛❧✉❡s ♦❢ t✿

t

x

0 1.00 1 0.50 2 0.33 ❲✐t❤ ♠♦r❡ ♣♦✐♥ts ❛❝q✉✐r❡❞ ✐♥ ❛ s♣r❡❛❞s❤❡❡t✱ ✇❡ ❝❛♥ ♣❧♦t t❤❡ ❣r❛♣❤ ♦♥ t❤❡ tx✲♣❧❛♥❡✿

✹✳✷✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥ t❤❡ ♣❧❛♥❡

✷✻✽

❆t t❤✐s ♣♦✐♥t✱ ✇❡ ❝♦✉❧❞✱ ✐❢ ♥❡❡❞❡❞✱ ❛♣♣❧② t❤❡ ❛✈❛✐❧❛❜❧❡ ❛♣♣❛r❛t✉s t♦ st✉❞② t❤❡ s②♠♠❡tr✐❡s✱ t❤❡ ♠♦♥♦t♦♥✐❝✐t②✱ t❤❡ ❡①tr❡♠❡ ♣♦✐♥ts✱ ❡t❝✳ ♦❢ t❤✐s ❢✉♥❝t✐♦♥✳ ❙❡❝♦♥❞✱ s✉♣♣♦s❡ t❤❛t t❤❡ ♣r✐❝❡ ♦❢

s✉❣❛r

✐s ✐♥❝r❡❛s✐♥❣ ❛♥❞ t❤❡♥ ❞❡❝r❡❛s✐♥❣✿

y րց ❚♦ ♠❛❦❡ t❤✐s s♣❡❝✐✜❝✱ ✇❡ ❝❛♥ ❝❤♦♦s❡ ❛♥ ✉♣s✐❞❡✲❞♦✇♥ ♣❛r❛❜♦❧❛✿

y = g(t) = −(t − 1)2 + 2 . ❲❡ t❤❡♥ ❛❣❛✐♥ ❡✈❛❧✉❛t❡ y ❢♦r s❡✈❡r❛❧ ✈❛❧✉❡s ♦❢ t✿

t

y

0 1.00 1 2.00 2 1.00 ❲✐t❤ ♠♦r❡ ♣♦✐♥ts ❛❝q✉✐r❡❞ ✐♥ ❛ s♣r❡❛❞s❤❡❡t✱ ✇❡ ♣❧♦t t❤❡ ❣r❛♣❤ ♦♥ t❤❡ ty ✲♣❧❛♥❡✿

❲❤❛t ✐❢ t❤❡ tr❛❞❡r ✐s ✐♥t❡r❡st❡❞ ✐♥ ✜♥❞✐♥❣ ❤✐❞❞❡♥ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡s❡ ❝♦♠❜✐♥❡ t❤❡ ❞❛t❛ ✜rst✿

t

x

0 1.00 1 0.50 2 0.33

t ❛♥❞

y

0 1.00 1 2.00 2 1.00

t −→

x

t✇♦ ❝♦♠♠♦❞✐t✐❡s✳

▲❡t✬s

y

0 1.00 1.00 1 0.50 2.00 2 0.33 1.00

❙✐♥❝❡ t❤❡ ✐♥♣✉t t ✐s t❤❡ s❛♠❡✱ ✇❡ ❣✐✈❡ ✐t ❛ s✐♥❣❧❡ ❝♦❧✉♠♥✳ ❚❤❡r❡ s❡❡♠s t♦ ❜❡ t✇♦ ♦✉t♣✉ts✳ ❆ ❜❡tt❡r ✐❞❡❛ ✐s t♦ s❡❡ ♣❛✐rs (x, y)✿ , y ) t ( x

0 ( 1.00 , 1.00 ) 1 ( 0.50 , 2.00 ) 2 ( 0.33 , 1.00 ) ❆ ✈❛❧✉❡ ♦❢ x ✐s ♣❛✐r❡❞ ✉♣ ✇✐t❤ ❛ ✈❛❧✉❡ ♦❢ y ✇❤❡♥ t❤❡② ❛♣♣❡❛r ❛❧♦♥❣ t❤❡ s❛♠❡ t ✐♥ ❜♦t❤ ♣❧♦ts✳ ❍♦✇ ❞♦ ✇❡ ❝♦♠❜✐♥❡ t❤❡ t✇♦ ♣❧♦ts ❄ ❆s t❤❡ t✇♦ ♣❧♦ts ❛r❡ ♠❛❞❡ ♦❢ ✭✐♥✐t✐❛❧❧②✮ ❞✐s❝♦♥♥❡❝t❡❞ ♣♦✐♥ts ✕ (t, x) ❛♥❞ (t, y) ✕ s♦ ✐s t❤❡ ♥❡✇ ♣❧♦t✳ ❚❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ ❡❛❝❤ ♣❛✐r✿

✹✳✷✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥ t❤❡ ♣❧❛♥❡

✷✻✾

❚❤❡r❡ ✐s ♥♦ t✦ ❆s t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ✐s t❤❡ s❛♠❡ ❢♦r ❜♦t❤ ❢✉♥❝t✐♦♥s✱ ♦♥❧② t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s ❛♣♣❡❛r✳ ■♥st❡❛❞ ♦❢ ♣❧♦tt✐♥❣ ❛❧❧ ♣♦✐♥ts

(x, y)

♦♥ t❤❡

xy ✲♣❧❛♥❡

✕ ❢♦r ❡❛❝❤

❚❤❡ ❞✐r❡❝t✐♦♥ ♠❛tt❡rs✦ ❙✐♥❝❡

t

t✳

(t, x, y)✱

✇❤✐❝❤ ❜❡❧♦♥❣ t♦ t❤❡

3✲❞✐♠❡♥s✐♦♥❛❧

s♣❛❝❡✱ ✇❡ ❥✉st ♣❧♦t

■t✬s ❛ ✏s❝❛tt❡r ♣❧♦t✑ ❝♦♥♥❡❝t❡❞ t♦ ♠❛❦❡ ❛ ❝✉r✈❡✿

✐s ♠✐ss✐♥❣✱ ✇❡ ❤❛✈❡ t♦ ♠❛❦❡ s✉r❡ ✇❡ ❦♥♦✇ ✐♥ ✇❤✐❝❤ ❞✐r❡❝t✐♦♥ ✇❡ ❛r❡

♠♦✈✐♥❣ ❛♥❞ ✐♥❞✐❝❛t❡ t❤❛t ✇✐t❤ ❛♥ ❛rr♦✇✳ ■❞❡❛❧❧②✱ ✇❡ ❛❧s♦ ❧❛❜❡❧ t❤❡ ♣♦✐♥ts ✐♥ ♦r❞❡r t♦ ✐♥❞✐❝❛t❡ ♥♦t ♦♥❧② ✏✇❤❡r❡✑ ❜✉t ❛❧s♦ ✏✇❤❡♥✑✿

❚❤✉s✱ t❤✐s ✐s ♠♦t✐♦♥✱ ❥✉st ❛s ❜❡❢♦r❡✱ ❜✉t t❤r♦✉❣❤ ✇❤❛t s♣❛❝❡❄ ❆♥ ❛❜str❛❝t

s♣❛❝❡ ♦❢ ♣r✐❝❡s

♠❛❞❡ ✉♣✳ ❚❤❡ s♣❛❝❡ ✐s ❝♦♠♣r✐s❡❞ ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ♣r✐❝❡s✱ ✐✳❡✳✱ ❛ ♣♦✐♥t ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ ♣r✐❝❡s✿

x

❢♦r ✇❤❡❛t ❛♥❞

y

(x, y)

t❤❛t ✇❡✬✈❡ st❛♥❞s ❢♦r

❢♦r s✉❣❛r✳

❍♦✇ ♠✉❝❤ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ t✇♦ ♣r✐❝❡s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥s ❝❛♥ ✇❡ r❡❝♦✈❡r ❢r♦♠ t❤❡ ♥❡✇ ❣r❛♣❤❄ ❆ ❧♦t✳ ❲❡ ❝❛♥ s❤r✐♥❦ t❤❡ ❣r❛♣❤ ✈❡rt✐❝❛❧❧② t♦ ❞❡✲❡♠♣❤❛s✐③❡ t❤❡ ❝❤❛♥❣❡ ♦❢

y

❛♥❞ t♦ r❡✈❡❛❧ t❤❡

q✉❛❧✐t❛t✐✈❡

❜❡❤❛✈✐♦r ♦❢

x✱

❛♥❞ ✈✐❝❡ ✈❡rs❛✿

✹✳✷✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥ t❤❡ ♣❧❛♥❡

✷✼✵

❲❡ s❡❡ t❤❡ ❞❡❝r❡❛s❡ ♦❢ x ❛♥❞ t❤❡♥ t❤❡ ✐♥❝r❡❛s❡ ❢♦❧❧♦✇❡❞ ❜② t❤❡ ❞❡❝r❡❛s❡ ♦❢ y ✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ❞❡♥s✐t② ♦❢ t❤❡ ♣♦✐♥ts ✐♥❞✐❝❛t❡s t❤❡ s♣❡❡❞ ♦❢ t❤❡ ♠♦t✐♦♥✳ ❊①❛♠♣❧❡ ✹✳✷✳✽✿ ❛❜str❛❝t

❲❡ ❝❛♥ ❞♦ t❤✐s ✐♥ ❛ ❢✉❧❧② ❛❜str❛❝t s❡tt✐♥❣✳ ❲❤❡♥ t✇♦ ❢✉♥❝t✐♦♥s✱ f, g ✱ ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡✐r r❡s♣❡❝t✐✈❡ ❧✐sts ♦❢ ✈❛❧✉❡s ✭✐♥st❡❛❞ ♦❢ ❢♦r♠✉❧❛s✮✱ t❤❡② ❛r❡ ❡❛s✐❧② ❝♦♠❜✐♥❡❞ ✐♥t♦ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱ F ✳ ❲❡ ❥✉st ♥❡❡❞ t♦ ❡❧✐♠✐♥❛t❡ t❤❡ r❡♣❡❛t❡❞ ❝♦❧✉♠♥ ♦❢ ✐♥♣✉ts✳ ❙✉♣♣♦s❡ ✇❡ ♥❡❡❞ t♦ ❝♦♠❜✐♥❡ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✿

t x = f (t)

t y = g(t)

0

1

0

5

1

2

1

2

3

−1

3

0

3

3

4

1

4

0

&

2

= ?

2

❲❡ r❡♣❡❛t t❤❡ ✐♥♣✉ts ❝♦❧✉♠♥ ✕ ♦♥❧② ♦♥❝❡ ✕ ❛♥❞ t❤❡♥ r❡♣❡❛t t❤❡ ♦✉t♣✉ts ♦❢ ❡✐t❤❡r ❢✉♥❝t✐♦♥✳ ❋✐rst r♦✇✿

f : 0 7→ 1 & g : 0 7→ 5

=⇒ F : 0 7→ (0, 5)

❙❡❝♦♥❞ r♦✇✿

f : 1 7→ 2 & g : 1 7→ −1

❆♥❞ s♦ ♦♥✳ ❚❤✐s ✐s t❤❡ ✇❤♦❧❡ s♦❧✉t✐♦♥✿

=⇒ F : 1 7→ (2, −1)

t x = f (t)

t y = g(t)

t P = (f (t) , g(t))

0

1

0

5

0

(1

,

5)

1

2

1

1

(2

,

2

3

−1

2

(3

,

−1)

3

0

3

3

3

(0

,

3)

4

1

4

0

4

(1

,

0)

❛♥❞

2

2

−→

2)

❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡r❡ ❛r❡ ♥♦ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ❝❛rr✐❡❞ ♦✉t ❛♥❞ t❤❡r❡ ✐s ♥♦ ♥❡✇ ❞❛t❛✱ ❥✉st t❤❡ ♦❧❞ ❞❛t❛ ❛rr❛♥❣❡❞ ✐♥ ❛ ♥❡✇ ✇❛②✳ ❍♦✇❡✈❡r✱ ✐t ✐s ❜❡❝♦♠✐♥❣ ❝❧❡❛r t❤❛t t❤❡ ❧✐st ✐s ❛❧s♦ ❛ ❢✉♥❝t✐♦♥ ♦❢ s♦♠❡ ❦✐♥❞✳ ❲❛r♥✐♥❣✦ ❚❤❡ ❡♥❞ r❡s✉❧t ✐s♥✬t t❤❡ ❣r❛♣❤ ♦❢ ❛♥② ❢✉♥❝t✐♦♥✳

✹✳✷✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥ t❤❡ ♣❧❛♥❡

✷✼✶

❊①❛♠♣❧❡ ✹✳✷✳✾✿ s♣r❡❛❞s❤❡❡t

❚❤✐s ✐s ❛ s✉♠♠❛r② ♦❢ ❤♦✇ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❢♦r♠❡❞ ❢r♦♠ t✇♦ ❢✉♥❝t✐♦♥s ♣r♦✈✐❞❡❞ ✐♥ ❛ s♣r❡❛❞s❤❡❡t✳ ❚❤❡ t❤r❡❡ ❝♦❧✉♠♥s ✕

❚❤✐s ❝❤❛rt ✐s t❤❡

t ✱ x✱

♣❛t❤

❛♥❞

y

✕ ❛r❡ ❝♦♣✐❡❞ ❛♥❞ t❤❡♥ t❤❡ ❧❛st t✇♦ ❛r❡ ✉s❡❞ t♦ ❝r❡❛t❡ ❛ ❝❤❛rt✿

✕ ♥♦t t❤❡ ❣r❛♣❤ ✕ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ◆♦t❡ ❛❧s♦ t❤❛t t❤❡ ❝✉r✈❡ ✐s♥✬t t❤❡

❣r❛♣❤ ♦❢ ❛♥② ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ ❛s t❤❡

❱❡rt✐❝❛❧ ▲✐♥❡ ❚❡st

✐s ✈✐♦❧❛t❡❞✳

❊①❛♠♣❧❡ ✹✳✷✳✶✵✿ ♣❛tt❡r♥

P❧♦tt✐♥❣ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♠❛② r❡✈❡❛❧ ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ q✉❛♥t✐t✐❡s✿

P❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛r❡ ❢✉♥❝t✐♦♥s ✦ ❚❤✐s ✐❞❡❛ ❝♦♠❡s ✇✐t❤ ❝❡rt❛✐♥ ♦❜❧✐❣❛t✐♦♥s ✭❱♦❧✉♠❡ ✶✮✳

❋✐rst✱ ✇❡ ❤❛✈❡ t♦

♥❛♠❡

✐t✱ s❛②

F✳

❙❡❝♦♥❞✱ ❛s ✇❡

❝♦♠❜✐♥❡ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✱ ✇❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ❢♦r t❤✐s ♦♣❡r❛t✐♦♥✿ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

F = (f, g) :

(

x = f (t) y = g(t)

✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❄ ■t ✐s t✳ ❆❢t❡r ❛❧❧✱ t❤✐s ✐s t❤❡ ✐♥♣✉t ♦❢ ❜♦t❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ✐♥✈♦❧✈❡❞✳ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❄ ■t ✐s t❤❡ ✏❝♦♠❜✐♥❛t✐♦♥✑ ♦❢ t❤❡ ♦✉t♣✉ts ♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✱ ✐✳❡✳✱ x ❛♥❞ y✳

◆❡①t✱ ✇❤❛t ✐s t❤❡ ❲❤❛t ✐s t❤❡

❲❡ ❦♥♦✇ ❤♦✇ t♦ ❝♦♠❜✐♥❡ t❤❡s❡❀ ✇❡ ❢♦r♠ ❛ ♣❛✐r✱

P = (x, y)✳

❚❤✐s

P

✐s ❛ ♣♦✐♥t ♦♥ t❤❡

xy ✲♣❧❛♥❡✦

❚♦ s✉♠♠❛r✐③❡✱ ✇❡ ❞♦ ✇❤❛t ✇❡ ❤❛✈❡ ❞♦♥❡ ♠❛♥② t✐♠❡s ❜❡❢♦r❡ ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❡t❝✳✮ ❛ ♥❡✇ ❢✉♥❝t✐♦♥ ❢r♦♠ t✇♦ ♦❧❞ ❢✉♥❝t✐♦♥s✳

❲❡ r❡♣r❡s❡♥t ❛ ❢✉♥❝t✐♦♥

f

❞✐❛❣r❛♠♠❛t✐❝❛❧❧② ❛s ❛

✕ ✇❡ ❝r❡❛t❡

❜❧❛❝❦ ❜♦①

t❤❛t

✹✳✷✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥ t❤❡ ♣❧❛♥❡

✷✼✷

♣r♦❝❡ss❡s t❤❡ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡s t❤❡ ♦✉t♣✉t✿ ✐♥♣✉t

→

t ◆♦✇✱ ✇❤❛t ✐❢ ✇❡ ❤❛✈❡ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥

❢✉♥❝t✐♦♥

x

g✿ ❢✉♥❝t✐♦♥

→

t F = (f, g)❄

→

f

✐♥♣✉t

❍♦✇ ❞♦ ✇❡ r❡♣r❡s❡♥t

♦✉t♣✉t

♦✉t♣✉t

→

g

y

❚♦ r❡♣r❡s❡♥t ✐t ❛s ❛ s✐♥❣❧❡ ❢✉♥❝t✐♦♥✱ ✇❡ ♥❡❡❞ t♦ ✏✇✐r❡✑ t❤❡✐r ❞✐❛❣r❛♠s

t♦❣❡t❤❡r s✐❞❡ ❜② s✐❞❡✿

t →

f

t →

g

|| ■t ✐s ♣♦ss✐❜❧❡ ❜❡❝❛✉s❡ t❤❡ ✐♥♣✉t ♦❢

f

→ x l

→ y

✐s t❤❡ s❛♠❡ ❛s t❤❡ ✐♥♣✉t ♦❢

g✳

❋♦r t❤❡ ♦✉t♣✉ts✱ ✇❡ ❝❛♥ ❝♦♠❜✐♥❡ t❤❡♠

❡✈❡♥ ✇❤❡♥ t❤❡② ❛r❡ ♦❢ ❞✐✛❡r❡♥t ♥❛t✉r❡✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ❛ ❞✐❛❣r❛♠ ♦❢ ❛ ♥❡✇ ❢✉♥❝t✐♦♥✿

(f, g) : t →

❲❡ s❡❡ ❤♦✇ t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡

t

t

ր t →

f

ց t →

g

→ x ց

(x, y)

→ y ր

→ P

✐s ❝♦♣✐❡❞ ✐♥t♦ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✱ ♣r♦❝❡ss❡❞ ❜② t❤❡♠

✐♥ ♣❛r❛❧❧❡❧✱ ❛♥❞ ✜♥❛❧❧②

t❤❡ t✇♦ ♦✉t♣✉ts ❛r❡ ❝♦♠❜✐♥❡❞ t♦❣❡t❤❡r t♦ ♣r♦❞✉❝❡ ❛ s✐♥❣❧❡ ♦✉t♣✉t✳ ❚❤❡ r❡s✉❧t ❝❛♥ ❜❡ s❡❡♥ ❛❣❛✐♥ ❛s ❛ ❜❧❛❝❦ ❜♦①✿

t →

→ P

F

❚❤❡ ❞✐✛❡r❡♥❝❡ ❢r♦♠ ❛❧❧ t❤❡ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡ s❡❡♥ s♦ ❢❛r ✐s t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ♦✉t♣✉t✳ ❲❤❛t ❛❜♦✉t t❤❡ ♦✉t♣✉ts ♦❢

F✳

✐♠❛❣❡

✭t❤❡ r❛♥❣❡ ♦❢ ✈❛❧✉❡s✮ ♦❢

F = (f, g)❄

■t ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ❛ r❡❝♦r❞✐♥❣ ♦❢ ❛❧❧ ♣♦ss✐❜❧❡

❚❤❡ t❡r♠✐♥♦❧♦❣② ✉s❡❞ ✐s ♦❢t❡♥ ❞✐✛❡r❡♥t t❤♦✉❣❤✳

❉❡✜♥✐t✐♦♥ ✹✳✷✳✶✶✿ ♣❛t❤ ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❚❤❡

♣❛t❤

♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

P = (f (t), g(t))

♦♥ t❤❡

x = f (t), y = g(t)

✐s t❤❡ s❡t ♦❢ ❛❧❧ s✉❝❤ ♣♦✐♥ts

xy ✲♣❧❛♥❡✳

❚❤❡ ♣❛t❤ ✐s t②♣✐❝❛❧❧② ❛ ❝✉r✈❡✳ ❲❡ ♣❧♦t s❡✈❡r❛❧ ♦❢ t❤❡♠ ❜❡❧♦✇✳

❊①❛♠♣❧❡ ✹✳✷✳✶✷✿ ♣❛t❤ ■♥ ❣❡♥❡r❛❧✱ t❤❡ t✇♦ ♣r♦❝❡ss❡s✱

x = x(t) ❛♥❞ y = y(t)✱ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳ ❲❤❡♥ ✇❡ ❝♦♠❜✐♥❡ (x, y) ❢♦r ❡❛❝❤ t✱ t❤❡ r❡s✉❧t ♠❛② ❜❡ ✉♥❡①♣❡❝t❡❞✿

t❤❡ ♣❛t❤ ♦❢ t❤❡ ♦❜❥❡❝t ❜② ♣❧♦tt✐♥❣

t❤❡♠ t♦ s❡❡

✹✳✷✳

P❛r❛♠❡tr✐❝ ❝✉r✈❡s ♦♥ t❤❡ ♣❧❛♥❡

✷✼✸

❲❤❛t ❛❜♦✉t t❤❡ ❣r❛♣❤ ♦❢ F = (f, g)❄ ❆s ✇❡ ❦♥♦✇ ❢r♦♠ ❈❤❛♣t❡r ✶✱ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ❛ r❡❝♦r❞✐♥❣ ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ♦❢ F ✳ ❲❤❛t ✐❢ t❤❡ ♦✉t♣✉ts ❛r❡ 2✲❞✐♠❡♥s✐♦♥❛❧❄

❉❡✜♥✐t✐♦♥ ✹✳✷✳✶✸✿ ❣r❛♣❤ ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ x = f (t), y = g(t) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts ♦❢ t❤❡ ❢♦r♠✿ (t, x, y) = (t, f (t), g(t))

✐♥ t❤❡ txy ✲s♣❛❝❡✳ ❚❤❡ ❣r❛♣❤ ✐s ❜✉✐❧t ❢r♦♠ t❤❡s❡ t✇♦✿ • t❤❡ ❣r❛♣❤ ♦❢ x = f (t) ♦♥ t❤❡ tx✲♣❧❛♥❡ ✭t❤❡ ✢♦♦r✮✱ ❛♥❞

• t❤❡ ❣r❛♣❤ ♦❢ y = g(t) ♦♥ t❤❡ ty ✲♣❧❛♥❡ ✭t❤❡ ✇❛❧❧ ❢❛❝✐♥❣ ✉s✮✳

■t ✐s ❛ ❝✉r✈❡ ✐♥ s♣❛❝❡✱ ❛❦✐♥ t♦ ❛ ♣✐❡❝❡ ♦❢ ✇✐r❡✿

❚❤❡♥ t❤❡ s❤❛❞♦✇ ♦❢ t❤✐s ✇✐r❡ ♦♥ t❤❡ ✢♦♦r ✐s t❤❡ ❣r❛♣❤ x = f (t) ✭❧✐❣❤t ❢r♦♠ ❛❜♦✈❡✮✳ ■❢ t❤❡ ❧✐❣❤t ✐s ❜❡❤✐♥❞ ✉s✱ t❤❡ s❤❛❞♦✇ ♦♥ t❤❡ ✇❛❧❧ ✐♥ ❢r♦♥t ✐s t❤❡ ❣r❛♣❤ y = g(t)✳ ■♥ ❛❞❞✐t✐♦♥✱ ♣♦✐♥t✐♥❣ ❛ ✢❛s❤❧✐❣❤t ❢r♦♠ r✐❣❤t t♦ ❧❡❢t

✹✳✸✳

❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

✷✼✹

✇✐❧❧ ♣r♦❞✉❝❡ t❤❡ ♣❛t❤ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ♦♥ t❤❡ xy ✲♣❧❛♥❡✳ ❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ t❡r♠✐♥♦❧♦❣②✿ t②♣❡s ♦❢ ❢✉♥❝t✐♦♥s✿

❣❡♥❡r❛❧ ❢✉♥❝t✐♦♥s ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ♠♦t✐♦♥

t❤❡ s❡t ♦❢ ❛❧❧ ♦✉t♣✉ts✿ ✐♠❛❣❡

r❛♥❣❡

♣❛t❤

tr❛❥❡❝t♦r②

✹✳✸✳ ❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

❆♥② ❢♦r♠✉❧❛ ✇✐t❤ t✇♦ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s ❛♥❞ ♦♥❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❝❛♥ ❜❡ st✉❞✐❡❞ ✐♥ t❤✐s ♠❛♥♥❡r✿

a = wd ♦r z = x + y . ❙✉❝❤ ❛♥ ❡①♣r❡ss✐♦♥ ✐s ❝❛❧❧❡❞ ❛

❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳

❚❤❡ ♥♦t❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿

g(w, d) = wd ♦r f (x, y) = x + y . ❊①❛♠♣❧❡ ✹✳✸✳✶✿ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

▲❡t

f (x, y) = x + y . ❲❡ ✐❧❧✉str❛t❡ t❤✐s ♥❡✇ ❢✉♥❝t✐♦♥ ❜❡❧♦✇✳ ❋✐rst✱ ❜② ❝❤❛♥❣✐♥❣ ✕ ✐♥❞❡♣❡♥❞❡♥t❧② ✖ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s✱ ✇❡ ❝r❡❛t❡ ❛ t❛❜❧❡ ♦❢ ♥✉♠❜❡rs ✭❧❡❢t✮✳ ❲❡ ❝❛♥✱ ❢✉rt❤❡r♠♦r❡✱ ❝♦❧♦r t❤✐s ❛rr❛② ♦❢ ❝❡❧❧s ✭♠✐❞❞❧❡✮ s♦ t❤❛t t❤❡ ❝♦❧♦r ♦❢ t❤❡ (x, y)✲❝❡❧❧ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ✈❛❧✉❡ ♦❢ z ✿

❚❤❡ ✈❛❧✉❡ ♦❢ z ❝❛♥ ❛❧s♦ ❜❡ ✈✐s✉❛❧✐③❡❞ ❛s t❤❡ ❡❧❡✈❛t✐♦♥ ♦❢ ❛ ♣♦✐♥t ❛t t❤❛t ❧♦❝❛t✐♦♥ ✭r✐❣❤t✮✳ ❙♦✱ t❤❡ ♠❛✐♥ ♠❡t❛♣❤♦r ❢♦r ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✇✐❧❧ ❜❡

t❡rr❛✐♥ ✿

❊❛❝❤ ❧✐♥❡ ✐♥❞✐❝❛t❡s ❛ ❝♦♥st❛♥t ❡❧❡✈❛t✐♦♥✳ ❊①❛♠♣❧❡ ✹✳✸✳✷✿ ❞✐st❛♥❝❡

❚❤❡ ❞✐st❛♥❝❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❈❛rt❡s✐❛♥ ♣❧❛♥❡ ❝r❡❛t❡s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❚❤✐s ✐s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ❛ ♣♦✐♥t (x, y) t♦ t❤❡ ♦r✐❣✐♥✿ p z = x2 + y 2 .

✹✳✸✳ ❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

✷✼✺

❙❧✐❣❤t❧② s✐♠♣❧❡r ✐s t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ❛ ♣♦✐♥t (x, y) t♦ t❤❡ ♦r✐❣✐♥✿ z = x2 + y 2 .

❲❡ ❝r❡❛t❡ ❛ t❛❜❧❡ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❡①♣r❡ss✐♦♥ ♦♥ t❤❡ ❧❡❢t ✐♥ ❛ s♣r❡❛❞s❤❡❡t ✇✐t❤ t❤❡ ❢♦r♠✉❧❛✿ ❂❘❈✶✂ ✷✰❘✶❈✂ ✷

❲❡ t❤❡♥ ❝♦❧♦r t❤❡ ❝❡❧❧s✿

❚❤❡ ♥❡❣❛t✐✈❡ ✈❛❧✉❡s ♦❢ z ❛r❡ ✐♥ ❜❧✉❡ ❛♥❞ t❤❡ ♣♦s✐t✐✈❡ ❛r❡ ✐♥ r❡❞✳ ❚❤❡ ❝✐r❝✉❧❛r ♣❛tt❡r♥ ✐s ❝❧❡❛r✳ ❚❤❡ ♣❛tt❡r♥ s❡❡♠s t♦ ❜❡ ♠❛❞❡ ❢r♦♠ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s ✇✐t❤ t❤❡ r❛❞✐✐ t❤❛t ✈❛r② ✇✐t❤ z ✿

❋♦r ❡❛❝❤ z ✱ ✇❡ ❤❛✈❡ ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ x ❛♥❞ y ✳ ❲❡ ❛❧s♦ r❡♣r❡s❡♥t ❛ ❢✉♥❝t✐♦♥ p ❞✐❛❣r❛♠♠❛t✐❝❛❧❧② ❛s ❛ ❜❧❛❝❦ ❜♦① t❤❛t ♣r♦❝❡ss❡s t❤❡ ✐♥♣✉ts ❛♥❞ ♣r♦❞✉❝❡s t❤❡ ♦✉t♣✉t✿ ✐♥♣✉ts ❢✉♥❝t✐♦♥ ♦✉t♣✉t x ց

→

p

ր

y

z

■♥st❡❛❞✱ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ s❡❡ ❛ s✐♥❣❧❡ ✐♥♣✉t ✈❛r✐❛❜❧❡✱ (x, y)✱ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t✇♦✱ x ❛♥❞ y ✱ t♦ ❜❡ ♣r♦❝❡ss❡❞ ❜② t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❡ s❛♠❡ t✐♠❡ ✿ (x, y) →

p

→ z

❚❤❡ ❞✐✛❡r❡♥❝❡ ❢r♦♠ ❛❧❧ t❤❡ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡ s❡❡♥ s♦ ❢❛r ✐s t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ✐♥♣✉t✳ ❙♦✱ ❡✈❡♥ t❤♦✉❣❤ ✇❡ s♣❡❛❦ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✱ t❤❡ ✐❞❡❛ ♦❢ ❢✉♥❝t✐♦♥ r❡♠❛✐♥s t❤❡ s❛♠❡✿ ◮ ❚❤❡r❡ ✐s ❛ s❡t ✭❞♦♠❛✐♥✮ ❛♥❞ ❛♥♦t❤❡r ✭❝♦❞♦♠❛✐♥✮ ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ ❛ss✐❣♥s t♦ ❡❛❝❤ ❡❧❡♠❡♥t ♦❢

t❤❡ ❢♦r♠❡r ❛♥ ❡❧❡♠❡♥t ♦❢ t❤❡ ❧❛tt❡r✳

❚❤❡ ✐❞❡❛ ✐s r❡✢❡❝t❡❞ ✐♥ t❤❡ ♥♦t❛t✐♦♥ ✇❡ ✉s❡✿ ♦r

F :X→Z F

X −−−−→ Z

✹✳✸✳ ❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

✷✼✻

❆ ❝♦♠♠♦♥ ✇❛② t♦ ✈✐s✉❛❧✐③❡ t❤❡ ❝♦♥❝❡♣t ♦❢ ❢✉♥❝t✐♦♥ ✕ ❡s♣❡❝✐❛❧❧② ✇❤❡♥ t❤❡ s❡ts ❝❛♥♥♦t ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ♠❡r❡ ❧✐sts ✕ ✐s t♦ ❞r❛✇ s❤❛♣❡❧❡ss ❜❧♦❜s ❝♦♥♥❡❝t❡❞ ❜② ❛rr♦✇s✿

■♥ ❝♦♥tr❛st t♦ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥✱ ❤♦✇❡✈❡r✱ t❤❡ ❞♦♠❛✐♥ ✐s ❛ s✉❜s❡t ♦❢ t❤❡ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡ ❢♦r

(x, y)✲♣❧❛♥❡✳

f (x, y) = x + y ✿

(0, 0) → 0, (0, 1) → 1, (1, 0) → 1, (1, 1) → 2, (1, 2) → 3, (2, 1) → 3, ... ❊❛❝❤ ❛rr♦✇ ❝❧❡❛r❧② ✐❞❡♥t✐✜❡s t❤❡ ✐♥♣✉t ✭❛♥ ❡❧❡♠❡♥t ♦❢ ✭❛♥ ❡❧❡♠❡♥t ♦❢

Z✮

X✮

♦❢ t❤✐s ♣r♦❝❡❞✉r❡ ❜② ✐ts ❜❡❣✐♥♥✐♥❣ ❛♥❞ t❤❡ ♦✉t♣✉t

❜② ✐ts ❡♥❞✳

❚❤✐s ✐s t❤❡ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ♦✉t♣✉t ♦❢ ❛ ❢✉♥❝t✐♦♥

F

✇❤❡♥ t❤❡ ✐♥♣✉t ✐s

x✿

■♥♣✉t ❛♥❞ ♦✉t♣✉t ♦❢ ❢✉♥❝t✐♦♥ F (x, y) = z ♦r

F : (x, y) → z ■t r❡❛❞s✿ ✏ F ♦❢

(x, y)

✐s

z ✑✳

❲❡ st✐❧❧ ❤❛✈❡✿

F(

✐♥♣✉t

)=

✐♥♣✉t

→

♦✉t♣✉t

❛♥❞

F :

♦✉t♣✉t

.

❋✉♥❝t✐♦♥s ❛r❡ ❡①♣❧✐❝✐t r❡❧❛t✐♦♥s✳ ❚❤❡r❡ ❛r❡ t❤r❡❡ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ t♦ ❡❛❝❤ ♦t❤❡r✱ ❜✉t t❤✐s r❡❧❛t✐♦♥ ✐s ✉♥❡q✉❛❧✿ ❚❤❡ t✇♦ ✐♥♣✉t ✈❛r✐❛❜❧❡s ❝♦♠❡ ✜rst ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ♦✉t♣✉t ✐s ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ✐♥♣✉t✳ ❚❤❛t ✐s ✇❤② ✇❡ s❛② t❤❛t t❤❡ ✐♥♣✉ts ❛r❡ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s ✇❤✐❧❡ t❤❡ ♦✉t♣✉t ✐s t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳

❊①❛♠♣❧❡ ✹✳✸✳✸✿ ✢♦✇❝❤❛rts r❡♣r❡s❡♥t ❢✉♥❝t✐♦♥s ❋♦r ❡①❛♠♣❧❡✱ ❢♦r ❛ ❣✐✈❡♥ ✐♥♣✉t

(x, y)✱

✇❡ ✜rst s♣❧✐t ✐t✿

x

❛♥❞

y

❛r❡ t❤❡ t✇♦ ♥✉♠❡r✐❝❛❧ ✐♥♣✉ts✳ ❚❤❡♥ ✇❡

❞♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥s❡❝✉t✐✈❡❧②✿

• • •

❛❞❞

x

❛♥❞

y✱ 2✱

♠✉❧t✐♣❧② ❜②

❛♥❞ t❤❡♥

sq✉❛r❡✳

❙✉❝❤ ❛ ♣r♦❝❡❞✉r❡ ❝❛♥ ❜❡ ❝♦♥✈❡♥✐❡♥t❧② ✈✐s✉❛❧✐③❡❞ ✇✐t❤ ❛ ✏✢♦✇❝❤❛rt✑✿

(x, y) →

x+y

→ u →

u·2

→ z →

z2

→ v

❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❝♦♠❡ ❢r♦♠ ♠❛♥② s♦✉r❝❡s ❛♥❞ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ ❞✐✛❡r❡♥t ❢♦r♠s✿

•

❛ ❧✐st ♦❢ ✐♥str✉❝t✐♦♥s ✭❛♥ ❛❧❣♦r✐t❤♠✮

•

❛♥ ❛❧❣❡❜r❛✐❝ ❢♦r♠✉❧❛

•

❛ ❧✐st ♦❢ ♣❛✐rs ♦❢ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts

•

❛ ❣r❛♣❤

✹✳✸✳

❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

• ❆♥

✷✼✼

❛ tr❛♥s❢♦r♠❛t✐♦♥

❛❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥

❣✐✈❡ ✐t ❛ ♥❛♠❡✱ s❛②

f✱

✐s ❡①❡♠♣❧✐✜❡❞ ❜②

z = x2 y ✳

■♥ ♦r❞❡r t♦ ♣r♦♣❡r❧② ✐♥tr♦❞✉❝❡ t❤✐s ❛s ❛ ❢✉♥❝t✐♦♥✱ ✇❡

❛♥❞ ✇r✐t❡✿

f (x, y) = x2 y . ▲❡t✬s ❡①❛♠✐♥❡ t❤✐s ♥♦t❛t✐♦♥✿ ❋✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

z ♥❛♠❡✿

=

f

↑

(

x, y

↑

❞❡♣❡♥❞❡♥t

↑ ↑

❢✉♥❝t✐♦♥

✈❛r✐❛❜❧❡

x2 y

) =

↑↑

✐♥❞❡♣❡♥❞❡♥t

✐♥❞❡♣❡♥❞❡♥t

✈❛r✐❛❜❧❡s

✈❛r✐❛❜❧❡s

❊①❛♠♣❧❡ ✹✳✸✳✹✿ ♣❧✉❣ ✐♥ ✈❛❧✉❡s

■♥s❡rt ♦♥❡ ✐♥♣✉t ✈❛❧✉❡ ✐♥ ❛❧❧ ♦❢ t❤❡s❡ ❜♦①❡s ❛♥❞ t❤❡ ♦t❤❡r ✐♥ t❤♦s❡ ❝✐r❝❧❡s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ❢✉♥❝t✐♦♥✿

f (x) =

2x2 y − 3y + 7 , y 3 + 2x + 1

❝❛♥ ❜❡ ✉♥❞❡rst♦♦❞ ❛♥❞ ❡✈❛❧✉❛t❡❞ ✈✐❛ t❤✐s ❞✐❛❣r❛♠✿

f () = ❚❤✐s ✐s ❤♦✇

f (3, 0)

22 −3 +7 .

3 + 2 + 1

✐s ❡✈❛❧✉❛t❡❞✿

f



3

✵ ,



2

✵ − 3 ✵ +7 23 = . 3 ✵ +2 3 +1

■♥ s✉♠♠❛r②✱

◮ ✏ x✑

❛♥❞ ✏ y ✑ ✐♥ ❛ ❢♦r♠✉❧❛ s❡r✈❡ ❛s

♣❧❛❝❡❤♦❧❞❡rs

❢♦r✿ ♥✉♠❜❡rs✱ ✈❛r✐❛❜❧❡s✱ ❛♥❞ ✇❤♦❧❡ ❢✉♥❝t✐♦♥s✳

❍♦✇ ❞♦ ✇❡ st✉❞② ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s❄ ❲❡ ✉s❡ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ❢✉♥❝t✐♦♥s ♦❢ s✐♥❣❧❡ ✈❛r✐❛❜❧❡✳ ❆❜♦✈❡ ✇❡ ❧♦♦❦❡❞ ❛t t❤❡ ❝✉r✈❡s ♦❢ ❝♦♥st❛♥t ❡❧❡✈❛t✐♦♥ ♦❢ t❤❡ s✉r❢❛❝❡s✳

❆♥ ❛❧t❡r♥❛t✐✈❡ ✐❞❡❛ ✐s ❛ s✉r✈❡②✐♥❣

♠❡t❤♦❞✿

◮

■♥ ♦r❞❡r t♦ st✉❞② ❛ t❡rr❛✐♥✱ ✇❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ t✇♦ ♠❛✐♥ ❞✐r❡❝t✐♦♥s✳

■♠❛❣✐♥❡ t❤❛t ✇❡ ❞r✐✈❡ s♦✉t❤✲♥♦rt❤ ❛♥❞ ❡❛st✲✇❡st s❡♣❛r❛t❡❧②✱ ✇❛t❝❤✐♥❣ ❤♦✇ t❤❡ ❡❧❡✈❛t✐♦♥ ❝❤❛♥❣❡s✿

❲❡ ❝❛♥ ❡✈❡♥ ✐♠❛❣✐♥❡ t❤❛t ✇❡ ❞r✐✈❡ ❛r♦✉♥❞ ❛ ❝✐t② ♦♥ ❛ ❤✐❧❧ ❛♥❞ t❤❡s❡ tr✐♣s ❢♦❧❧♦✇ t❤❡ str❡❡t ❣r✐❞✿

✹✳✸✳

❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

✷✼✽

❊❛❝❤ ♦❢ t❤❡s❡ tr✐♣s ❝r❡❛t❡s ❛ ❢✉♥❝t✐♦♥ ♦❢ s✐♥❣❧❡ ✈❛r✐❛❜❧❡✱ ❚♦ ✈✐s✉❛❧✐③❡✱ ❝♦♥s✐❞❡r t❤❡ ♣❧♦t ♦❢

F (x, y) = sin(xy)

x

♦r

y✳

♦♥ t❤❡ ❧❡❢t✿

❲❡ ♣❧♦t t❤❡ s✉r❢❛❝❡ ❛s ❛ ✏✇✐r❡✲❢r❛♠❡✑ ♦♥ t❤❡ r✐❣❤t✳ ❊❛❝❤ ✇✐r❡ ✐s ❛ s❡♣❛r❛t❡ tr✐♣✳ ❚❤❡ ❣r❛♣❤s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛r❡ t❤❡ s❧✐❝❡s ❝✉t ❜② t❤❡ ✈❡rt✐❝❛❧ ♣❧❛♥❡s ❛❧✐❣♥❡❞ ✇✐t❤ t❤❡ ❛①❡s ❢r♦♠ t❤❡ s✉r❢❛❝❡ t❤❛t ✐s t❤❡ ❣r❛♣❤ ♦❢

F✿

❊①❛♠♣❧❡ ✹✳✸✳✺✿ s❛❞❞❧❡

▲❡t✬s ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

z = xy .

✹✳✸✳ ❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

✷✼✾

❚❤✐s ✐s ✇❤❛t t❤❡ ❣r❛♣❤s ♦❢ t❤❡s❡ r❡❧❛t✐♦♥s ❧♦♦❦ ❧✐❦❡ ✇❤❡♥ ♣❧♦tt❡❞ ❢♦r ✈❛r✐♦✉s z ✬s❀ t❤❡② ❛r❡ ❝✉r✈❡s ❝❛❧❧❡❞ ❤②♣❡r❜♦❧❛s ✿

■♥st❡❛❞✱ ✇❡ ✜① ♦♥❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❛t ❛ t✐♠❡✳ ❲❡ ✜① y ✜rst✿

♣❧❛♥❡

❡q✉❛t✐♦♥ ❝✉r✈❡ y=2 z = x · 2 ❧✐♥❡ ✇✐t❤ y=1 z = x · 1 ❧✐♥❡ ✇✐t❤ y=0 z = x · 0 = 0 ❧✐♥❡ ✇✐t❤ y = −1 z = x · (−1) ❧✐♥❡ ✇✐t❤ y = −2 z = x · (−2) ❧✐♥❡ ✇✐t❤

❚❤❡ ✈✐❡✇ s❤♦✇♥ ❜❡❧♦✇ ✐s ❢r♦♠ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ y ✲❛①✐s✿

s❧♦♣❡ 2 s❧♦♣❡ 1 s❧♦♣❡ 0 s❧♦♣❡ 1 s❧♦♣❡ − 2

❚❤❡ ❞❛t❛ ❢♦r ❡❛❝❤ ❧✐♥❡ ❝♦♠❡s ❢r♦♠ t❤❡ x✲❝♦❧✉♠♥ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ❛♥❞ ♦♥❡ ♦❢ t❤❡ z ✲❝♦❧✉♠♥s✳ ❚❤❡s❡ ❧✐♥❡s ❣✐✈❡ t❤❡ ❧✐♥❡s ♦❢ ❡❧❡✈❛t✐♦♥ ♦❢ t❤✐s t❡rr❛✐♥ ✐♥ ❛ ♣❛rt✐❝✉❧❛r✱ s❛②✱ ❡❛st✲✇❡st ❞✐r❡❝t✐♦♥✳ ❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ ❝✉tt✐♥❣ t❤❡ ❣r❛♣❤ ❜② ❛ ✈❡rt✐❝❛❧ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ xz ✲♣❧❛♥❡✳ ❲❡ ✜① x s❡❝♦♥❞✿

♣❧❛♥❡

❡q✉❛t✐♦♥ ❝✉r✈❡ x=2 z = 2 · y ❧✐♥❡ ✇✐t❤ x=1 z = 1 · y ❧✐♥❡ ✇✐t❤ x=0 z = 0 · y = 0 ❧✐♥❡ ✇✐t❤ x = −1 z = (−1) · y ❧✐♥❡ ✇✐t❤ x = −2 z = (−2) · y ❧✐♥❡ ✇✐t❤

s❧♦♣❡ 2 s❧♦♣❡ 1 s❧♦♣❡ 0 s❧♦♣❡ 1 s❧♦♣❡ − 2

❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ ❝✉tt✐♥❣ t❤❡ ❣r❛♣❤ ❜② ❛ ✈❡rt✐❝❛❧ ♣❧❛♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ yz ✲♣❧❛♥❡✳ ❚❤❡ ✈✐❡✇ s❤♦✇♥ ❜❡❧♦✇ ✐s ❢r♦♠ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ x✲❛①✐s✿

❚❤❡ ❞❛t❛ ❢♦r ❡❛❝❤ ❧✐♥❡ ❝♦♠❡s ❢r♦♠ t❤❡ y ✲r♦✇ ♦❢ t❤❡ s♣r❡❛❞s❤❡❡t ❛♥❞ ♦♥❡ ♦❢ t❤❡ z ✲r♦✇s✳ ❚❤❡s❡ ❧✐♥❡s ❣✐✈❡ t❤❡ ❧✐♥❡s ♦❢ ❡❧❡✈❛t✐♦♥ ♦❢ t❤✐s t❡rr❛✐♥ ✐♥ ❛ ♣❛rt✐❝✉❧❛r✱ s❛②✱ ♥♦rt❤✲s♦✉t❤ ❞✐r❡❝t✐♦♥✳

✹✳✸✳

❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

✷✽✵

❊①❡r❝✐s❡ ✹✳✸✳✻

Pr♦✈✐❞❡ ❛ s✐♠✐❧❛r ❛♥❛❧②s✐s ❢♦r f (x, y) = 3x + 2y ✳ ❊①❛♠♣❧❡ ✹✳✸✳✼✿ ❜❛❦❡r

❲❡ ✇✐❧❧ t❛❦❡ ❛ ❧♦♦❦ ❛t t❤❡ ❡①❛♠♣❧❡ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ❢r♦♠ ❛ ❞✐✛❡r❡♥t ❛♥❣❧❡✳ ❚❤❡ t✐♠❡ t ✐s ♥♦t ❛ ♣❛rt ♦❢ ♦✉r ❝♦♥s✐❞❡r❛t✐♦♥ ❛♥②♠♦r❡ ❜✉t ✇❡ r❡t❛✐♥ t❤❡ t✇♦ ✈❛r✐❛❜❧❡s r❡♣r❡s❡♥t✐♥❣ t❤❡ t✇♦ ❝♦♠♠♦❞✐t✐❡s ✿ • x ✐s t❤❡ ♣r✐❝❡ ♦❢ ✇❤❡❛t✳ • y ✐s t❤❡ ♣r✐❝❡ ♦❢ s✉❣❛r✳ ❲❡ ❛❧s♦ ❛❞❞ ❛ ♣r♦❞✉❝t t♦ t❤❡ s❡t✉♣✿ • z ✐s t❤❡ ♣r✐❝❡ ♦❢ ❛ ❧♦❛❢ ♦❢ ❜r❡❛❞✳ ❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡s❡ t❤r❡❡❄ ❆s t❤♦s❡ t✇♦ ❛r❡ t❤❡ t✇♦ ♠❛❥♦r ✐♥❣r❡❞✐❡♥ts ✐♥ ❜r❡❛❞✱ ✇❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t ◮ z ❞❡♣❡♥❞s ♦♥ x ❛♥❞ y ✳ ❖♥❡ ❝❛♥ ✐♠❛❣✐♥❡ ❛ ❜❛❦❡r ✇❤♦ ❡✈❡r② ♠♦r♥✐♥❣✱ ✉♣♦♥ r❡❝❡✐✈✐♥❣ t❤❡ ✉♣❞❛t❡❞ ♣r✐❝❡s ♦❢ ✇❤❡❛t ❛♥❞ s✉❣❛r✱ ✉s❡s ❛ t❛❜❧❡ t❤❛t ❤❡ ♠❛❞❡ ✉♣ ✐♥ ❛❞✈❛♥❝❡ t♦ ❞❡❝✐❞❡ ♦♥ t❤❡ ♣r✐❝❡ ♦❢ ❤✐s ❜r❡❛❞ ❢♦r t❤❡ r❡st ♦❢ t❤❡ ❞❛②✳ ▲❡t✬s s❡❡ ❤♦✇ s✉❝❤ ❛ t❛❜❧❡ ♠✐❣❤t ❝♦♠❡ ❛❜♦✉t✳ ❲❤❛t ❦✐♥❞ ♦❢ ❞❡♣❡♥❞❡♥❝✐❡s ❛r❡ t❤❡s❡❄ ■♥❝r❡❛s✐♥❣ ♣r✐❝❡s ♦❢ t❤❡ ✐♥❣r❡❞✐❡♥ts ✐♥❝r❡❛s❡s t❤❡ ❝♦st ❛♥❞ ✉❧t✐♠❛t❡❧② t❤❡ ♣r✐❝❡ ♦❢ t❤❡ ♣r♦❞✉❝t✿ x ր =⇒ z ր

y ր =⇒ z ր

❆t ✐ts s✐♠♣❧❡st✱ s✉❝❤ ❛♥ ✐♥❝r❡❛s❡ ✐s ❧✐♥❡❛r✳ ■♥ ❛❞❞✐t✐♦♥ t♦ s♦♠❡ ✜①❡❞ ❝♦sts✿ • ❊❛❝❤ ✐♥❝r❡❛s❡ ♦❢ x ❧❡❛❞s t♦ ❛ ♣r♦♣♦rt✐♦♥❛❧ ✐♥❝r❡❛s❡ ♦❢ z ✳ • ❊❛❝❤ ✐♥❝r❡❛s❡ ♦❢ y ❧❡❛❞s t♦ ❛ ♣r♦♣♦rt✐♦♥❛❧ ✐♥❝r❡❛s❡ ♦❢ z ✳ ■♥❞❡♣❡♥❞❡♥t❧②✦ ❆ s✐♠♣❧❡ ❢♦r♠✉❧❛ t❤❛t ❝❛♣t✉r❡s t❤✐s ❞❡♣❡♥❞❡♥❝❡ ♠❛② ❜❡ t❤✐s✿

z = p(x, y) = 2x + y + 1 . ■♥ ♦r❞❡r t♦ ✈✐s✉❛❧✐③❡ t❤✐s ❢✉♥❝t✐♦♥✱ ✇❡ ❝♦♠♣✉t❡ ❛ ❢❡✇ ♦❢ ✐ts ✈❛❧✉❡s✿ • p(0, 0) = 1 • p(0, 1) = 2 • p(0, 2) = 3 • p(1, 0) = 3 • p(1, 1) = 4 • ❡t❝✳ ❊✈❡♥ t❤♦✉❣❤ t❤✐s ✐s ❛ ❧✐st✱ ✇❡ r❡❛❧✐③❡ t❤❛t t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡s ❞♦♥✬t ✜t ✐♥t♦ ❛ ❧✐st ❝♦♠❢♦rt❛❜❧② ✕ t❤❡② ❢♦r♠ ❛ t❛❜❧❡ ✦ (0, 0) (1, 0) (2, 0) ...

(0, 1) (1, 1) (2, 1) ... (0, 2) (1, 2) (2, 2) ... ...

...

...

...

■♥ ❢❛❝t✱ ✇❡ ❝❛♥ ❛❧✐❣♥ t❤❡s❡ ♣❛✐rs ✇✐t❤ x ✐♥ ❡❛❝❤ ❝♦❧✉♠♥ ❛♥❞ y ✐♥ ❡❛❝❤ r♦✇✿

y\x

0

1

2

...

0

(0, 0) (1, 0) (2, 0) ...

1

(0, 1) (1, 1) (2, 1) ...

2

(0, 2) (1, 2) (2, 2) ...

...

...

...

...

...

✹✳✸✳

❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

✷✽✶

◆♦✇✱ t❤❡ ✈❛❧✉❡s✱ z = p(x, y)✿

y\x

0

1

2 ...

0

1

3

5 ...

1

2

4

6 ...

2

3

5

7 ...

...

... ... ... ...

❚❤❛t✬s ✇❤❛t t❤❡ ❜❛❦❡r✬s t❛❜❧❡ ♠✐❣❤t ❧♦♦❦ ❧✐❦❡✳ ▲❡t✬s ❜r✐♥❣ t❤❡s❡ t✇♦ t♦❣❡t❤❡r✿

y\x 0

0

1

2

...

(0, 0)

(1, 0)

(2, 0)

...

ց 1

1

(0, 1)

(0, 2)

...

...

... 6 ...

(2, 2) ց

3

5 ... ...

ց

4

(1, 2) ց

...

(2, 1) ց

2

ց

3

(1, 1) ց

2

ց

ց

5

...

... ... 7 ...

...

...

■♥ t❤❡ ♣❛st✱ ✇❡ ❤❛✈❡ ✈✐s✉❛❧✐③❡❞ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s ❜② ♣✉tt✐♥❣ ❜❛rs ♦♥ t♦♣ ♦❢ t❤❡ x✲❛①✐s✳ ◆♦✇✱ ✇❡ ✈✐s✉❛❧✐③❡ t❤❡ ✈❛❧✉❡s ❜② ❜✉✐❧❞✐♥❣ ❝♦❧✉♠♥s ✇✐t❤ ❛♣♣r♦♣r✐❛t❡ ❤❡✐❣❤ts ♦♥ t♦♣ ♦❢ t❤❡ xy ✲♣❧❛♥❡✿

◆♦t✐❝❡ t❤❛t ❜② ✜①✐♥❣ ♦♥❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ✕ x = 0, 1, 2 ♦r y = 0, 1, 2 ✕ ✇❡ ❝r❡❛t❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦t❤❡r ✈❛r✐❛❜❧❡✳ ❲❡ ✜① x ❜❡❧♦✇ ❛♥❞ ❡①tr❛❝t t❤❡ ❝♦❧✉♠♥s ❢r♦♠ t❤❡ t❛❜❧❡✿

y z x=0:

0 1 1 2

y z x=1:

2 3 ❆

♣❛tt❡r♥

0 3 1 4

x 0 1 2 z 1 3 5

x=2:

2 5

✐s ❝❧❡❛r✿ ❣r♦✇t❤ ❜② 1✳ ❲❡ ♥❡①t ✜① y ❛♥❞ ❡①tr❛❝t t❤❡

y=0:

y z

y=1:

x 0 1 2 z 2 4 6

0 5 1 6 2 7

r♦✇s

❢r♦♠ t❤❡ t❛❜❧❡✿

y=2:

x 0 1 2 z 3 5 7

✹✳✸✳

❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

❆

♣❛tt❡r♥

✐s ❝❧❡❛r✿ ❣r♦✇t❤ ❜②

✷✽✷

2✳

❲❡ ❤❛✈❡ t❤❡ t♦t❛❧ ♦❢ s✐① ✭❧✐♥❡❛r✮ ❢✉♥❝t✐♦♥s✦

▲❡t✬s ❞♦ t❤❡ s❛♠❡ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✳ ❚❤✐s ✐s t❤❡ ❞❛t❛✿

❚❤❡ ✈❛❧✉❡ ✐♥ ❡❛❝❤ ❝❡❧❧ ✐s ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡ ♦❢ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❛❧✉❡ ♦❢

y

x

✭❛❧❧ t❤❡ ✇❛② ✉♣✮ ❛♥❞ ❢r♦♠ t❤❡

✭❛❧❧ t❤❡ ✇❛② ❧❡❢t✮✳ ❚❤✐s ✐s t❤❡ ❢♦r♠✉❧❛✿

❂✷✯❘✸❈✰❘❈✷✰✶ ❚❤❡ s✐♠♣❧❡st ✇❛② t♦ ✈✐s✉❛❧✐③❡ ✐s ❜② ❝♦❧♦r✐♥❣ t❤❡ ❝❡❧❧ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✈❛❧✉❡s ✭❝♦♠♠♦♥ ✐♥ ❝❛rt♦❣r❛♣❤②✿ ❡❧❡✈❛t✐♦♥✱ t❡♠♣❡r❛t✉r❡✱ ❤✉♠✐❞✐t②✱ ♣r❡❝✐♣✐t❛t✐♦♥✱ ♣♦♣✉❧❛t✐♦♥ ❞❡♥s✐t②✱ ❡t❝✳✿

❚❤❡ ❣r♦✇t❤ ✐s ✈✐s✐❜❧❡✿ ■t ❣r♦✇s t❤❡ ♠♦st ✐♥ s♦♠❡ ❞✐❛❣♦♥❛❧ ❞✐r❡❝t✐♦♥ ❜✉t ✐t✬s

♥♦t

45

❞❡❣r❡❡s✳

❲❡ ❝❛♥ ❛❧s♦ ✈✐s✉❛❧✐③❡ ✇✐t❤ ❛ ❜❛r ❝❤❛rt✱ ❥✉st ❛s ❜❡❢♦r❡✿

■❢ ✇❡ ✉s❡❞ ❜❛rs t♦ r❡♣r❡s❡♥t t❤❡ ❘✐❡♠❛♥♥ s✉♠s t♦ ❝♦♠♣✉t❡ t❤❡

❛r❡❛✱ ❤❡r❡ ✇❡ ❛r❡ ❛❢t❡r t❤❡ ✈♦❧✉♠❡✳✳✳

❚❤❡ ♠♦st ❝♦♠♠♦♥ ✇❛②✱ ❤♦✇❡✈❡r✱ t♦ ✈✐s✉❛❧✐③❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✐♥ ♠❛t❤❡♠❛t✐❝s ✐s ✇✐t❤ ✐ts

❣r❛♣❤✱ ✇❤✐❝❤✱ ✐♥ t❤✐s ❝❛s❡✱ ✐s ❛ s✉r❢❛❝❡✿

✹✳✸✳ ❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

✷✽✸

■♥ t❤✐s ♣❛rt✐❝✉❧❛r ❝❛s❡✱ t❤✐s ✐s ❛ ♣❧❛♥❡✳ ❚❤❡ s❡❝♦♥❞ ❣r❛♣❤ ✐s t❤❡ s❛♠❡ s✉r❢❛❝❡ ❜✉t ❞✐s♣❧❛②❡❞ ❛s ❛ ✇✐r❡✲ ❢r❛♠❡ ✭♦r ❡✈❡♥ ❛ ✇✐r❡✲❢❡♥❝❡✮✳ ❚❤❡ ✇✐r❡s ❛r❡ t❤❡ ❣r❛♣❤s ♦❢ t❤♦s❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ ❝r❡❛t❡❞ ❢r♦♠ ♦✉r ❢✉♥❝t✐♦♥ ✇❤❡♥ ✇❡ ✜① ♦♥❡ ✈❛r✐❛❜❧❡ ❛t ❛ t✐♠❡✳ ❊❛❝❤ ♦❢ t❤❡s❡ ✇✐r❡s ❝♦♠❡s ❢r♦♠ ❝❤♦♦s✐♥❣ ❡✐t❤❡r✿

• •

t❤❡ r♦✇ ♦❢

x✬s

t❤❡ ❝♦❧✉♠♥ ♦❢

✭t♦♣✮ ❛♥❞ ♦♥❡ ♦t❤❡r r♦✇ ✐♥ t❤❡ t❛❜❧❡✱ ♦r

y ✬s

✭❧❡❢t♠♦st✮ ❛♥❞ ♦♥❡ ♦t❤❡r ❝♦❧✉♠♥ ✐♥ t❤❡ t❛❜❧❡✳

❊①❡r❝✐s❡ ✹✳✸✳✽

Pr♦✈✐❞❡ ❛ s✐♠✐❧❛r ❛♥❛❧②s✐s ❢♦r ✭❛✮ t❤❡ ✇✐♥❞✲❝❤✐❧❧ ❛♥❞ ✭❜✮ t❤❡ ❤❡❛t ✐♥❞❡①✳

❚❤❡ ❢✉♥❝t✐♦♥s ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ ❝r❡❛t❡❞ ❢r♦♠ ♦✉r ❢✉♥❝t✐♦♥

z = p(x, y)

✇❤❡♥ ✇❡ ✜① ♦♥❡ ✈❛r✐❛❜❧❡ ❛t ❛ t✐♠❡ ❛r❡✿

y = b −→ fb (x) = p(x, b)

x = a −→ ga (y) = p(a, y)

❚❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♦❢ t❤❡♠✳ ❚❤❡✐r ❣r❛♣❤s ❛r❡ t❤❡ s❧✐❝❡s ✕ ❛❧♦♥❣ t❤❡ ❛①❡s ✕ ♦❢ t❤❡ s✉r❢❛❝❡ t❤❛t ✐s t❤❡ ❣r❛♣❤ ♦❢

F✳

❚❤❡r❡❢♦r❡✱ t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s t❡❧❧s ✉s ❛❜♦✉t t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢

p ✕ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡

❛①❡s✦ ❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡ ❢✉♥❝t✐♦♥s✳

❚❤✐s ✐❞❡❛ ❝♦♠❡s ✇✐t❤ ❝❡rt❛✐♥ q✉❡st✐♦♥s t♦ ❜❡ ❛♥s✇❡r❡❞✳ ❲❤❛t ✐s t❤❡ ✐♥♣✉t✱ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❄ ❚❛❦✐♥❣ ❛ ❝❧✉❡ ❢r♦♠ ♦✉r ❛♥❛❧②s✐s ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✱ ✇❡ ❛♥s✇❡r✿ ■t ✐s t❤❡ ✏❝♦♠❜✐♥❛t✐♦♥✑ ♦❢ t❤❡ t✇♦ ✐♥♣✉ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱ ✐✳❡✳✱

x

❛♥❞

y

t❤❛t ❢♦r♠ ❛ ♣❛✐r✱

t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❄ ■t ✐s

z✳

X = (x, y)✱

✇❤✐❝❤ ✐s ❛ ♣♦✐♥t ♦♥ t❤❡

xy ✲♣❧❛♥❡✳

❲❤❛t ✐s t❤❡ ♦✉t♣✉t✱

✹✳✸✳ ❋✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

✷✽✹

❲❡ r❡♣r❡s❡♥t ❛ ❢✉♥❝t✐♦♥ p ❞✐❛❣r❛♠♠❛t✐❝❛❧❧② ❛s ❛ ❜❧❛❝❦ ❜♦① t❤❛t ♣r♦❝❡ss❡s t❤❡ ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡s t❤❡ ♦✉t♣✉t✿ ✐♥♣✉ts

❢✉♥❝t✐♦♥

♦✉t♣✉t

x ց

y

7→

p

ր

z

■♥st❡❛❞✱ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ s❡❡ ❛ s✐♥❣❧❡ ✐♥♣✉t ✈❛r✐❛❜❧❡✱ (x, y)✱ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t✇♦✱ x ❛♥❞ y ✱ t♦ ❜❡ ♣r♦❝❡ss❡❞ ❜② t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❡ s❛♠❡ t✐♠❡ ✿ (x, y) →

p

→ z

❚❤❡ ❞✐✛❡r❡♥❝❡ ❢r♦♠ ❛❧❧ t❤❡ ❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡ s❡❡♥ s♦ ❢❛r ✐s t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ✐♥♣✉t✳ ◆❡①t✱ ✇❤❛t ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ p❄ ■t ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ❛ r❡❝♦r❞✐♥❣ ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ✐♥♣✉ts✱ ✐✳❡✳✱ ❛❧❧ ♣❛✐rs (x, y) ❢♦r ✇❤✐❝❤ t❤❡ ♦✉t♣✉t z = p(x, y) ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♠❛❦❡s s❡♥s❡✳ ❚❤✐s r❡q✉✐r❡♠❡♥t ❝r❡❛t❡s ❛ s✉❜s❡t ♦❢ t❤❡ xy ✲♣❧❛♥❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ x ❛♥❞ y ✳ ❲❤❛t ❛❜♦✉t t❤❡ ✐♠❛❣❡✱ ✐✳❡✳✱ t❤❡ r❛♥❣❡ ♦❢ ✈❛❧✉❡s ♦❢ p❄ ■t ✐s ❛ r❡❝♦r❞✐♥❣ ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ♦✉t♣✉ts ♦❢ p✳

❉❡✜♥✐t✐♦♥ ✹✳✸✳✾✿ ✐♠❛❣❡ ♦❢ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ❚❤❡ ✐♠❛❣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s z = p(x, y) ✐s t❤❡ s❡t ♦❢ ❛❧❧ s✉❝❤ ✈❛❧✉❡s z ♦♥ t❤❡ z ✲❛①✐s✳ ❲❤❛t ❛❜♦✉t t❤❡ ❣r❛♣❤ ♦❢ p = (f, g)❄ ■t ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ❛ r❡❝♦r❞✐♥❣ ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ♦❢ F ✳

❉❡✜♥✐t✐♦♥ ✹✳✸✳✶✵✿ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

❊①❡r❝✐s❡ ✹✳✸✳✶✶

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s z = p(x, y) ✐s t❤❡ s❡t ♦❢ ❛❧❧ s✉❝❤ ♣♦✐♥ts x, y, p(x, y) ✐♥ t❤❡ xyz ✲s♣❛❝❡✳

❙❦❡t❝❤ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿

x = t2 − 1, y = 2t2 + 3 .

❊①❡r❝✐s❡ ✹✳✸✳✶✷ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ r❡♣r❡s❡♥ts t❤❡ ♠♦t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ♦♥ t❤❡ ♣❧❛♥❡✿ x = 3t − 1, y = t2 − 1 .

✭❛✮ ❲❤❡♥ ❞♦❡s t❤❡ ♦❜❥❡❝t ❝r♦ss t❤❡ x✲❛①✐s❄ ✭❜✮ ❲❤❡♥ ❞♦❡s t❤❡ ♦❜❥❡❝t ❝r♦ss t❤❡ y ✲❛①✐s❄

❊①❡r❝✐s❡ ✹✳✸✳✶✸ ❘❡♣r❡s❡♥t ❛s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ t❤❡ r♦t❛t✐♦♥ ♦❢ ❛ r♦❞ ♦❢ ❧❡♥❣t❤ 2 t❤❛t ♠❛❦❡s ♦♥❡ ❢✉❧❧ t✉r♥ ❡✈❡r② 3 s❡❝♦♥❞s✳

✹✳✹✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ❝❛❧❝✉❧✉s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✷✽✺

✹✳✹✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ❝❛❧❝✉❧✉s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

❲❤❡♥ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❢♦r♠❡❞ ❢r♦♠ t✇♦ ❢✉♥❝t✐♦♥s ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡✿ x = f (t), y = g(t) ,

❛r❡ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ f ❛♥❞ g ✈✐s✐❜❧❡ ✐♥ t❤❡ s❤❛♣❡ ✭❛♥❞ t❤❡ s❧♦♣❡✮ ♦❢ t❤❡ ♣❛t❤❄ ❈♦♥✈❡rs❡❧②✱ ❝❛♥ t❤❡ ❞❡r✐✈❛t✐✈❡s ❜❡ ❞❡❞✉❝❡❞ ❢r♦♠ t❤❡ s❧♦♣❡s ♦r ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ♣❛t❤❄ ❚❤❡ s❧♦♣❡s ♦❢ t❤❡ ❣r❛♣❤s ♦❢ f ❛♥❞ g ♣r♦❞✉❝❡ t❤❡ s❧♦♣❡ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛❝❝♦r❞✐♥❣ t♦ ❛ s✐♠♣❧❡ r✉❧❡ ✇❤✐❝❤ ✐s ❡❛s② t♦ ❞✐s❝♦✈❡r ❢r♦♠ t❤❡ ❝❛s❡ ✇❤❡♥ ❜♦t❤ ❢✉♥❝t✐♦♥s ❛r❡ ❧✐♥❡❛r✿

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ m ❛♥❞ n ❛r❡ t❤❡ s❧♦♣❡s ♦❢ f ❛♥❞ g r❡s♣❡❝t✐✈❡❧②✱ t❤❡♥ t❤❡ s❧♦♣❡ ♦❢ (f, g) ✐s mn ✳ ■♥❞❡❡❞✱ ❝❤❛♥❣❡ ♦❢ y = ❝❤❛♥❣❡ ♦❢ y/❝❤❛♥❣❡ ♦❢ t . ❝❤❛♥❣❡ ♦❢ x ❝❤❛♥❣❡ ♦❢ x/❝❤❛♥❣❡ ♦❢ t ❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥s ❛r❡ ♥♦♥✲❧✐♥❡❛r✱ t❤❡ r✉❧❡ ✐s t❤❡ s❛♠❡ ❜✉t ✐t ✐s ❛♣♣❧✐❡❞ ♦♥❡ ♣♦✐♥t ❛t ❛ t✐♠❡✿ s❧♦♣❡ ❛t (a, b)

=

g ′ (b) . f ′ (a)

❚♦ s❡❡ ✇❤②✱ ✐t s✉✣❝❡s t♦ ③♦♦♠ ✐♥ ♦♥❡ ❛ ♣♦✐♥t ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛s ✇❡❧❧ ❛s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥ts ♦❢ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✿

✹✳✹✳

■♥tr♦❞✉❝t✐♦♥ t♦ ❝❛❧❝✉❧✉s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✷✽✻

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ t❤✐s✿

dy dy = dx dt ❚❤❡ ❢♦r♠✉❧❛ r❡s❡♠❜❧❡s t❤❡



dx dt

❈❤❛✐♥ ❘✉❧❡✱ ♥♦t ❜② ❝♦✐♥❝✐❞❡♥❝❡✳

❊①❡r❝✐s❡ ✹✳✹✳✶

Pr♦✈❡ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝❛s❡ ✇❤❡♥

f

✐s ♦♥❡✲t♦✲♦♥❡✳

❚❤❡ t✇♦ s♣❡❝✐❛❧ ❝❛s❡s ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

• f ′ (a) = 0 =⇒

• g ′ (b) = 0 =⇒

t❤❡ s❧♦♣❡ ✐s ✈❡rt✐❝❛❧✱ ❛♥❞ t❤❡ s❧♦♣❡ ✐s ❤♦r✐③♦♥t❛❧✳

❚❤❡ ❢♦r♠❡r ❝❛s❡ ✇❛s s❡❡♥ ❛s ✏❡①tr❡♠❡✑ ✐♥ ❝❛❧❝✉❧✉s ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡✳ ■t✬s ♥♦t ❡①tr❡♠❡ ✐♥ ❝❛❧❝✉❧✉s ♦❢ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✦ ❊①❛♠♣❧❡ ✹✳✹✳✷✿ ❝♦♠♠♦♦❞✐t✐❡s tr❛❞❡r

❘❡❝❛❧❧ t❤❛t t❤❡ ♣r✐❝❡ ♦❢ ✇❤❡❛t ❛♥❞ t❤❡ ♣r✐❝❡ ♦❢ s✉❣❛r ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② t✇♦ ❢✉♥❝t✐♦♥s ❝♦♠❜✐♥❡❞ ✐♥t♦ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ❲✐t❤ t❤❡s❡ ❢✉♥❝t✐♦♥s s❛♠♣❧❡❞✱ ✇❡ ❝♦♠♣✉t❡ t❤❡✐r r❛t❡s ♦❢ ❝❤❛♥❣❡✿

t

x

x′

0 1.00 ↓

↓

1 0.50 ↓

↓

t

y

y′

0 1.00 0.5 − 1.0 = −.5 1−0

0.33 − 0.5 2 0.33 = −.17 2−1

↓

↓

1 2.00 ↓

↓

2 1.00

2.00 − 1.00 =1 1−0 1.00 − 2.00 = −1 2−1

◆♦t❡ t❤❛t t❤❡r❡ ❛r❡ ❢❡✇❡r ♥✉♠❜❡rs t❤❛♥ ✐♥ t❤❡ ♦r✐❣✐♥❛❧ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ❢❡✇❡r s❡❣♠❡♥ts t❤❛♥ ♣♦✐♥ts✳

✹✳✹✳

✷✽✼

■♥tr♦❞✉❝t✐♦♥ t♦ ❝❛❧❝✉❧✉s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

▲❡t✬s ♥♦✇ ❝♦♥✜r♠ t❤✐s r❡s✉❧t ✈✐❛ ❛❝t✉❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ♦✉r ❢✉♥❝t✐♦♥s✿ x = f (t) =

1 t+1

=⇒ 2

y = g(t) = −(x − 1) + 2

x′ = f ′ (t) = −

1 (t + 1)2

y ′ = g ′ (t) = −2(x − 1)

❲❡ ❤❛✈❡ ❝♦♠♣✉t❡❞ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ t✇♦ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s✳ ❈♦♠❜✐♥❡❞✱ t❤❡② ❛❧s♦ ❢♦r♠ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✦ ❚❤❡ s✐❣♥s ♦❢ t❤❡ t✇♦ ♥❡✇ ❢✉♥❝t✐♦♥s t❡❧❧ ✉s t❤❡ ✐♥❝r❡❛s✐♥❣✴❞❡❝r❡❛s✐♥❣ ❜❡❤❛✈✐♦r ♦❢ t❤❡ t✇♦ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥s ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ❋♦r ❡①❛♠♣❧❡✱ x′ < 0 s❤♦✇s t❤❛t t❤❡ ❝✉r✈❡ ♠♦✈❡s t♦ t❤❡ ❧❡❢t ❛♥❞✳✳✳ ♠♦✈❡s ❞♦✇♥ ✐♥✐t✐❛❧❧② ❜❡❝❛✉s❡ y ′ < 0✳ ❚❤❡ ❝✉r✈❡ ❛❧s♦ ♠♦✈❡s ✉♣ ❜❡❝❛✉s❡ y ′ > 0✳ ▲❡t✬s ✈✐s✉❛❧✐③❡ ❛♥❞ ❝♦♥✜r♠ t❤❡s❡ r❡s✉❧ts ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t ✭✇✐t❤♦✉t ✉s✐♥❣ t❤❡ ❝♦♠♣✉t❡❞ ❞❡r✐✈❛t✐✈❡s ❛❜♦✈❡✮✿

■♥ ♦r❞❡r t♦ ❛♣♣r♦①✐♠❛t❡ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❞❡r✐✈❛t✐✈❡s✱ ✇❡ ✉s❡ t❤❡ s❧♦♣❡ ♦❢ t❤❡ s❡❣♠❡♥t ❜❡t✇❡❡♥ t✇♦ ❛❞❥❛❝❡♥t ♣♦✐♥ts✳ ■t ✐s t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✭❛❧s♦ ❦♥♦✇♥ ❛s t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t✱ t❤❡ s❧♦♣❡ ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡✱ ❡t❝✳✮✿ ❝❤❛♥❣❡ ♦❢ x ❝❤❛♥❣❡ ♦❢ y ❛♥❞ . ❝❤❛♥❣❡ ♦❢ t ❝❤❛♥❣❡ ♦❢ t ❚❤❡ ❝❤❛♥❣❡ ♦❢ t ✐s ✜①❡❞ ❛s h = ∆t✳ ❚❤✐s ✈❛❧✉❡ ❢♦r ❡✐t❤❡r x ❛♥❞ y ✐s ❝♦♠♣✉t❡❞ ✈✐❛ t❤❡ s❛♠❡ s♣r❡❛❞s❤❡❡t ❢♦r♠✉❧❛ ❛s ❜❡❢♦r❡✿ ❂✭❘❈❬✲✶❪✲❘❬✲✶❪❈❬✲✶❪✮✴❘✷❈✶

◆♦t❡ t❤❛t t❤❡r❡ ❛r❡ ♦♥❡ ❢❡✇❡r ❝❡❧❧s ✐♥ t❤✐s ❝♦❧✉♠♥ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ♦♥❡ ❢❡✇❡r s❡❣♠❡♥ts t❤❛♥ ♣♦✐♥ts✳

■♥ ❛❞❞✐t✐♦♥ t♦ x′ < 0 =⇒ x ց✱ ✇❡ ❛❧s♦ ❣❡t x′ ր =⇒ x ⌣ ✭❝♦♥❝❛✈❡ ✉♣✮✳ ❙✐♠✐❧❛r❧②✱ y ′ > 0 =⇒ x ր ❛♥❞ y ′ ց =⇒ y ⌢ ✭❝♦♥❝❛✈❡ ❞♦✇♥✮ ✐♥✐t✐❛❧❧② ❛♥❞ t❤❡♥ t❤❡ ♦♣♣♦s✐t❡✳ ❆❧s♦✱ t❤❡ ❛♣♣❛r❡♥t ❧✐♥❡❛r✐t② ♦❢ y ′

✹✳✹✳

✷✽✽

■♥tr♦❞✉❝t✐♦♥ t♦ ❝❛❧❝✉❧✉s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✐♥❞✐❝❛t❡ t❤❛t y ♠✐❣❤t ❜❡ q✉❛❞r❛t✐❝✳✳✳ ❋r♦♠ t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ✐♥✐t✐❛❧❧② t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❣♦❡s տ ❛♥❞ t❤❡♥ ւ t♦ ❝♦♥✜r♠ t❤❡ ♣✐❝t✉r❡✳ ❊①❡r❝✐s❡ ✹✳✹✳✸

❲❤❛t ❝♦♥❝❧✉s✐♦♥s ❛❜♦✉t t❤❡ s❤❛♣❡ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❝❛♥ ②♦✉ ❞r❛✇ ❢r♦♠ t❤❡ ❝♦♥❝❛✈✐t② ♦❢ ✐ts ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s❄ ❚❤✉s✱ ✇❡ ❝❛♥ s❛② ❛❜♦✉t ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ t❤❛t ✐ts ❞❡r✐✈❛t✐✈❡ ✐s ♠❛❞❡ ✉♣ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ✐♥✈♦❧✈❡❞✳ ❚❤❡r❡ ❛r❡ ♦♥❧② t✇♦ ❢♦r ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✕ ❛♥❞ ✐♥✜♥✐t❡❧② ♠❛♥② ❢♦r ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✦ ❊①❛♠♣❧❡ ✹✳✹✳✹✿ ❜❛❦❡r

❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ♣r✐❝❡ ♦❢ ❜r❡❛❞ ♦♥ t❤❡ ♣r✐❝❡s ♦❢ ✇❤❡❛t ❛♥❞ s✉❣❛r ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❢✉♥❝t✐♦♥ z = p(x, y) ❜❡❧♦✇✳ ❆s ✐t ✐s s❛♠♣❧❡❞✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ r❛t❡s ♦❢ ❝❤❛♥❣❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ ❞✐r❡❝t✐♦♥s✿

y\x 0 → 1 → 2 1 → 3 → 5

0 ♦✈❡r x :

♦✈❡r y :

2 → 4 → 6

1

3 → 5 → 7

2

y\x 0

1

2

0

1

3

5

↓

↓

↓

↓

↓

↓

↓

↓

1 2

2

4

3

5

6 7

❘❡❝❛❧❧ t❤❛t ❜② ✜①✐♥❣ ♦♥❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s✱ ✇❡ ❝r❡❛t❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦t❤❡r ✈❛r✐❛❜❧❡✳ ◆♦✇ ✇❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❥✉st ❛s ❜❡❢♦r❡✱ ✈✐❛ t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✿ ❝❤❛♥❣❡ ♦❢ z ❝❤❛♥❣❡ ♦❢ z ❛♥❞ . ❝❤❛♥❣❡ ♦❢ x ❝❤❛♥❣❡ ♦❢ y ❋✐rst✱ ✇❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ y ✿

y (x = 0)

z′

z

y (x = 1)

z′

z

y (x = 2)

z′

z

0 ↓

1 ↓

0 ↓

3 ↓

0 ↓

5 ↓

1 ↓

2 ↓

1 ↓

4 ↓

1 ↓

6 ↓

2 ↓

2−1 =1 1−0 3−2 3 ↓ =1 2−1

4−3 =1 1−0 5−4 5 ↓ =1 2−1

2 ↓

6−5 =1 2−1 7−6 7 ↓ =1 2−1

2 ↓

❆❧❧ 1s✳ ◆♦t❡ t❤❛t t❤❡r❡ ❛r❡ ❢❡✇❡r ♥✉♠❜❡rs t❤❛♥ ✐♥ t❤❡ ♦r✐❣✐♥❛❧ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ❢❡✇❡r s❡❣♠❡♥ts t❤❛♥ ♣♦✐♥ts✳ ❙✐♠✐❧❛r❧②✱ ✇❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ x✿

(y = 0) x 0 → 1

→ 2

(y = 1) x 0 1 2

(y = 0) x 0 1 2

z 2 4 6

z 3 5 7

z′

z′

z 1 → 3 → 5 3 − 1 5−3 z′ =2 =2 1−0 2−1

2 2

2 2

❆❧❧ 2s✳ ❲❡ ♣✉t t❤❡s❡ ♦♥❡✲✈❛r✐❛❜❧❡ ❢✉♥❝t✐♦♥s t♦❣❡t❤❡r❀ t❤❡♥ t❤❡ r❛t❡s ♦❢ ❝❤❛♥❣❡ ♦❢ F ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ y ❛r❡ t❤❡s❡ ♥❡✇ ❢✉♥❝t✐♦♥s ♦❢ t✇♦ ✈❛r✐❛❜❧❡s r❡s♣❡❝t✐✈❡❧②✿

y\x 0 1 2 ❧❡❛❞✐♥❣ t♦

0

2 2

1

2 2

2

2 2

y\x 0 1 2 &

0 1

1 1 1

2

1 1 1

y\x 0 ❧❡❛❞✐♥❣ t♦

1

2

0 1

(2, 1) (2, 1)

2

(2, 1) (2, 1)

✹✳✹✳

✷✽✾

■♥tr♦❞✉❝t✐♦♥ t♦ ❝❛❧❝✉❧✉s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

❚❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❢✉rt❤❡r ❝♦♠❜✐♥❡❞ ♦♥ t❤❡ r✐❣❤t✳ ❆s ✇❡ s❤❛❧❧ s❡❡ ❧❛t❡r✱ ❣♦✐♥❣ 2 ❤♦r✐③♦♥t❛❧❧② ❛♥❞ 1 ✈❡rt✐❝❛❧❧② ✐s t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❢❛st❡st ❣r♦✇t❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ p✳ ▲❡t✬s ♥♦✇ ❝♦♥✜r♠ t❤✐s r❡s✉❧t ✈✐❛ ❛❝t✉❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ♦✉r ❢✉♥❝t✐♦♥✿ p(x, y) = 2x + y + 1 .

❏✉st ❛s ❜❡❢♦r❡✱ ✇❡ ✜① ♦♥❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ❛♥❞ ❞✐✛❡r❡♥t✐❛t❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦t❤❡r✳ ❲❡ ❝❛❧❧ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ p ✇✐t❤ r❡s♣❡❝t t♦ x ❛♥❞ y r❡s♣❡❝t✐✈❡❧②✳ ❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ t②♣❡s ♦❢ ♥♦t❛t✐♦♥ ✭❢♦❧❧♦✇✐♥❣ ▲❡✐❜♥✐③ ❛♥❞ ▲❛❣r❛♥❣❡ ❛s ❜❡❢♦r❡✮✳ ❋♦r x✱ ✇❡ ❞❡❝❧❛r❡ y ✜①❡❞ ❛♥❞ ❞✐✛❡r❡♥t✐❛t❡ ♦✈❡r x ∂ ∂ ∂p = p′x = (2x + y + 1) = (2x) + 0 + 0 = 2 . ∂x ∂x ∂x

❋♦r y ✱ ✇❡ ❞❡❝❧❛r❡ x ✜①❡❞ ❛♥❞ ❞✐✛❡r❡♥t✐❛t❡ ♦✈❡r y ✿ ∂p ∂ ∂ = p′y = (2x + y + 1) = 0 + (y) + 0 = 1 . ∂y ∂y ∂y

❚❤❡ ❝♦♥❝❧✉s✐♦♥ ♠✐❣❤t s♦✉♥❞ ❢❛♠✐❧✐❛r✿ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✐s ❝♦♥st❛♥t✦ ❚❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ✇✐❧❧ ❜❡ s❡❡♥ ❛s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ p ❝❛❧❧❡❞ t❤❡ ❣r❛❞✐❡♥t ♦❢ p✳ ❚❤✐s ✐s ❛ ♥❡✇ ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥ t♦ ❜❡ ❞✐s❝✉ss❡❞ ✐♥ ❱♦❧✉♠❡ ✹ ✭❈❤❛♣t❡rs ✹❍❉✲✹ ❛♥❞ ✹❍❉✲✺✮✳ ❲❡ ❝❛♥ ❝♦♥✜r♠ t❤❡s❡ r❡s✉❧ts ❜② ❡①❛♠✐♥✐♥❣ t❤❡ s♣r❡❛❞s❤❡❡t✳ ❊❛❝❤ ❧✐♥❡ ✭✇✐r❡✮ ❜❡❧♦✇ ♦♥ t❤❡ r✐❣❤t ✐s t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡✿

❆♥❞ ❡❛❝❤ ❤❛s ✐ts ♦✇♥ ❞❡r✐✈❛t✐✈❡✦ ❆s ✇❡ ♠♦✈❡ ❤♦r✐③♦♥t❛❧❧②✱ t❤❡ ✈❛❧✉❡s ♦❢ x ❣r♦✇ ❜② 0.1 ✇❤✐❧❡ t❤❡ ✈❛❧✉❡s ♦❢ z ❣r♦✇ ❜② 0.2✳ ❚❤❡r❡❢♦r❡✱ p′x = 2✳ ❙✐♠✐❧❛r❧②✱ ❛s ✇❡ ♠♦✈❡ ✈❡rt✐❝❛❧❧②✱ t❤❡ ✈❛❧✉❡s ♦❢ y ❣r♦✇ ❜② 0.1 ❛♥❞ s♦ ❞♦ t❤❡ ✈❛❧✉❡s ♦❢ z ✳ ❚❤❡r❡❢♦r❡✱ p′y = 1✳ ❲❛r♥✐♥❣✦

❉♦ ♥♦t ❝♦♥❢✉s❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✇✐t❤ ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥t✐❛t✐♦♥ t❤❛t ❝♦♠❡s ✉♥❞❡r r❡❧❛t❡❞ r❛t❡s ✿ ∂ (xy) = y ∂x

✈s✳

d dy (xy) = y + . dx dx

❚❤❡ ❡①tr❛ t❡r♠ ♦♥ t❤❡ r✐❣❤t ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t t❤❡ t✇♦ ✈❛r✐❛❜❧❡s ❛r❡ r❡❧❛t❡❞✳

✹✳✹✳

■♥tr♦❞✉❝t✐♦♥ t♦ ❝❛❧❝✉❧✉s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✷✾✵

❊①❛♠♣❧❡ ✹✳✹✳✺✿ ♥♦♥✲❧✐♥❡❛r

▲❡t✬s ❝♦♥s✐❞❡r t❤✐s ❢✉♥❝t✐♦♥ ❛❣❛✐♥✿ q(x, y) = sin(xy) .

❈♦♠♣✉t❡ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❜② t❤❡ ❈❤❛✐♥ ❘✉❧❡✳ ❋✐rst ✇❡ ❞❡❝❧❛r❡ y ❛♥ ✉♥❦♥♦✇♥ ❛♥❞ ✉♥s♣❡❝✐✜❡❞ ❜✉t ✜①❡❞ ♣❛r❛♠❡t❡r ❛♥❞ ❝❛rr② ♦✉t ❞✐✛❡r❡♥t✐❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ x✿  ∂ ∂ ∂q sin(xy) = cos(xy) · = (xy) = cos(xy)y . ∂x ∂x ∂x

❚❤✐s t✐♠❡✱ x ✐s t❤❡ ♣❛r❛♠❡t❡r✿

 ∂q ∂ ∂ sin(xy) = cos(xy) · = (xy) = cos(xy)x . ∂y ∂y ∂y

▲❡t✬s ❝♦♥✜r♠ t❤❡s❡ r❡s✉❧ts ❜② ❡①❛♠✐♥✐♥❣ t❤❡ ❣r❛♣❤ ♦❢ q ♣❧♦tt❡❞ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✿

◆♦t❡ t❤❛t t❤❡ ❡❞❣❡ ♦❢ t❤❡ s✉r❢❛❝❡ ✐s ❛ ❝✉r✈❡ ❛♥❞ ✐t ✐s t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ✐♥ t❤❡ ✈❡r② ❧❛st r♦✇ ♦❢ t❤❡ t❛❜❧❡✳ ❲❡ ❛❧s♦ ♥♦t✐❝❡ t❤❛t✿ ∂q = 0 ❢♦r y = 0✳ • ❚❤❡ s✉r❢❛❝❡ ✐s ✢❛t ❛❧♦♥❣ t❤❡ x✲❛①✐s✱ ❜❡❝❛✉s❡ ∂x ∂q • ❚❤❡ s✉r❢❛❝❡ ✐s ✢❛t ❛❧♦♥❣ t❤❡ y ✲❛①✐s✱ ❜❡❝❛✉s❡ ∂x = 0 ❢♦r x = 0✳ ❲❡ ❛♣♣r♦①✐♠❛t❡ t❤❡s❡ ❞❡r✐✈❛t✐✈❡s ❥✉st ❛s ❜❡❢♦r❡✿❚❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✇✐t❤ t❤❡ ❝❤❛♥❣❡ ♦❢ x ❛♥❞ y ✐s ✜①❡❞ ❛s h = ∆x = ∆y ✳ ■♥ ❛ s♣r❡❛❞s❤❡❡t✱ t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ z ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ x ❛♥❞ y ✐s ❝♦♠♣✉t❡❞ ✈✐❛ t❤❡ s❛♠❡ ❢♦r♠✉❧❛ ❜✉t ❛♣♣❧✐❡❞ ❤♦r✐③♦♥t❛❧❧② ❛♥❞ ✈❡rt✐❝❛❧❧② r❡s♣❡❝t✐✈❡❧②✿ ❂✭❘❬✲✷✸❪❈✲❘❬✲✷✸❪❈❬✲✶❪✮✴❘✶❈✶ ❛♥❞ ❂✭❘❈❬✲✷✸❪✲❘❬✲✶❪❈❬✲✷✸❪✮✴❘✶❈✶

❚❤❡ ❢♦r♠✉❧❛s ♣r♦❞✉❝❡ t❤❡ t✇♦ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s✿

✹✳✹✳

✷✾✶

■♥tr♦❞✉❝t✐♦♥ t♦ ❝❛❧❝✉❧✉s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

❆t t❤❡ ♣♦✐♥t (0.1, 0.1)✱ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ t✇♦ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❡q✉❛❧✱ ✇❤✐❝❤ ✐s ✇❤② t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❢❛st❡st ❣r♦✇t❤ ♦❢ q ✐s ❛t 45 ❞❡❣r❡❡s✳ ◆♦t❡ ❛❧s♦ t❤❛t t❤❡ ❤✐❣❤❡st ❧♦❝❛t✐♦♥s ❢♦r♠ ❛ r✐❞❣❡❀ ✐t ✐s ✇❤❡r❡ ❜♦t❤ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❡q✉❛❧ t♦ 0✳ ❚♦ ✜♥❞ t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ r✐❞❣❡✱ ✇❡ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥

cos(xy) = 0 =⇒ xy = π/2 . ■t✬s ❛ ❤②♣❡r❜♦❧❛✳ ❚❤❡ ✐❞❡❛ t❤❛t t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ✐♥❞✐❝❛t❡ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❢❛st❡st ❣r♦✇t❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ✐❧❧✉str❛t❡❞ ❛♣♣r♦①✐♠❛t❡❧② ❛s ❢♦❧❧♦✇s✿

\x qx′ > 0 qx′ < 0

y

qy′ > 0

ր

qy′ < 0

ց

տ ւ

❊①❛♠♣❧❡ ✹✳✹✳✻✿ ❜r❡❛❞ ❜✉②❡rs

❲❡ ✇✐❧❧ t❛❦❡ t❤❡ t✇♦ ❡①❛♠♣❧❡s ✕ t❤❡ ❝♦♠♠♦❞✐t② tr❛❞❡r ❛♥❞ t❤❡ ❜❛❦❡r ✕ ❢r♦♠ t❤❡ ❧❛st t✇♦ s❡❝t✐♦♥s ❛♥❞ ❛s❦✱ ✇❤❛t ♣r✐❝❡ ♦❢ ❜r❡❛❞ ❤❛✈❡ ❞❛✐❧② ✈✐s✐t♦rs t♦ t❤❡ ❜❛❦❡r② s❤♦♣ s❡❡♥ ♦✈❡r t✐♠❡❄ ❚❤❡s❡ ❛r❡ t❤❡ ✈❛r✐❛❜❧❡s✿ • t ✐s t✐♠❡✳ ❚✇♦ ✈❛r✐❛❜❧❡s r❡♣r❡s❡♥t✐♥❣ t❤❡s❡ t✇♦ ❝♦♠♠♦❞✐t✐❡s ✿ • x ✐s t❤❡ ♣r✐❝❡ ♦❢ ✇❤❡❛t✳ • y ✐s t❤❡ ♣r✐❝❡ ♦❢ s✉❣❛r✳ ❆♥❞ ❛ ♣r♦❞✉❝t ✿ • z ✐s t❤❡ ♣r✐❝❡ ♦❢ ❛ ❧♦❛❢ ♦❢ ❜r❡❛❞✳ ❚❤❡ ✈✐s✐t♦rs s❡❡ ❤♦✇ z ❞❡♣❡♥❞s ♦♥ t✱ ✈✐❛ s♦♠❡ ❢✉♥❝t✐♦♥ ♦❢ s✐♥❣❧❡ ✈❛r✐❛❜❧❡ ✿

z = h(t) . ❲❤❛t ✐s ✐t❄ ❚❤✐s ✐s t❤❡ s✉♠♠❛r② ♦❢ t❤❡ s❡t✉♣✿ ❊①❛♠♣❧❡ ✶ ✭tr❛❞❡r✮

t

−→

❲❡ r❡❛❧✐③❡ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s ❛❜♦✉t

(x, y)



−→

z

❊①❛♠♣❧❡ ✷ ✭❜❛❦❡r✮

❝♦♠♣♦s✐t✐♦♥s ✦

❘❡❝❛❧❧ t❤❛t t❤❡ ♣r✐❝❡ ♦❢ ✇❤❡❛t ❛♥❞ t❤❡ ♣r✐❝❡ ♦❢ s✉❣❛r ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿

x = f (t) =

1 , y = g(t) = −(x − 1)2 + 2 . t+1

❋✉rt❤❡r♠♦r❡✱ t❤❡ ♣r✐❝❡ ♦❢ ❜r❡❛❞ ✐s ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ♦t❤❡r t✇♦ ♣r✐❝❡s ❜② t❤❡ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✿

z = p(x, y) = 2x + y + 1 . ❚❤❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ✈✐s✉❛❧✐③❡❞ ❛s ❢♦❧❧♦✇s✿

✹✳✹✳

■♥tr♦❞✉❝t✐♦♥ t♦ ❝❛❧❝✉❧✉s ♦❢ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

❚❤❡♥✱ ♦❢ ❝♦✉rs❡✱

h

✷✾✷

✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡s❡ t✇♦✿

t 7→ (x, y) 7→ z , ❝♦♠♣✉t❡❞ ✈✐❛ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜st✐t✉t✐♦♥✿

 h(t) = p f (t), g(t) .

❚♦ ✈✐s✉❛❧✐③❡ ✇❤❛t ❤❛♣♣❡♥s✱ ✐♠❛❣✐♥❡ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✕ ♦♥ t❤❡ ❣r❛♣❤ ♦❢

xy ✲♣❧❛♥❡

✕ ❜❡✐♥❣ ✏❧✐❢t❡❞✑ t♦ t❤❡

p✿

❚❤❡ ❡❧❡✈❛t✐♦♥ ✐s t❤❡♥ t❤❡ ✈❛❧✉❡ ♦❢

h✳

❚❤❡ ❡♥❞ r❡s✉❧t ✐s ❜❡❧♦✇✿

■♥ t❤❡ ♣❛st✱ ✇❡ ❤❛✈❡ ❢♦✉♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ❜② t❤❡

❈❤❛✐♥ ❘✉❧❡ ✿

❲❡

❡①♣r❡ss❡❞ ✐t ✐♥ t❡r♠s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s ✐♥✈♦❧✈❡❞ ✭t❤❡✐r ♣r♦❞✉❝t✮✳ ❲❡ t❤❡♥ ❝♦♥❥❡❝t✉r❡ t❤❛t ✐♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ❛❜♦✈❡✱ ✇❡ ♥❡❡❞ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣✿

•

t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✱ ❛♥❞

✹✳✺✳

❚❤❡ ❝❡♥tr♦✐❞ ♦❢ ❛ ✢❛t ♦❜❥❡❝t

•

✷✾✸

t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳

✹✳✺✳ ❚❤❡ ❝❡♥tr♦✐❞ ♦❢ ❛ ✢❛t ♦❜❥❡❝t ❲❡ ❝♦♥t✐♥✉❡ ✇✐t❤ ♦✉r ❛♥❛❧②s✐s ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss st❛rt❡❞ ✐♥ ❈❤❛♣t❡r ✸ ❜✉t ✐♥ t❤❡

2✲❞✐♠❡♥s✐♦♥❛❧

s❡tt✐♥❣✳

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣❧❛t❡ ✇✐t❤ ✉♥✐❢♦r♠ ❛ ❞❡♥s✐t② ❛♥❞ ✐❞❡♥t✐❝❛❧ t❤✐❝❦♥❡ss ✭✐t ✐s ❦♥♦✇♥ ❛s ❛ ✏❧❛♠✐♥❛✑✮✳ ❍♦✇ ❝❛♥ ✇❡ ❜❛❧❛♥❝❡ ✐t ♦♥ ❛ s✐♥❣❧❡ s✉♣♣♦rt ❝❛❧❧❡❞ t❤❡

❝❡♥tr♦✐❞ ❄

❚❤❡r❡ ❛r❡ ❛ ❢❡✇ ❤❡✉r✐st✐❝s t❤❛t ❤❡❧♣✳ ■❢ t❤❡ ♦❜❥❡❝t ❤❛s ❛ ✏❝❡♥t❡r✑✱ s✉❝❤ ❛s ❛ ❝✐r❝❧❡ ♦r ❛ sq✉❛r❡✱ t❤✐s ✐s ✐t✳

❆❧s♦✱ ❛♥② ❛①✐s ♦❢ s②♠♠❡tr② ✇✐❧❧ ❤❛✈❡ t♦ ❝♦♥t❛✐♥ t❤❡ ❝❡♥tr♦✐❞✳ ❚❤❡ ✐❞❡❛ ♦❢ ❝❡♥tr♦✐❞ ✐s r❡❧❛t❡❞ t♦ t❤❡ ❝♦♥❝❡♣t ♦❢ t❤❡

❝❡♥t❡r ♦❢ ♠❛ss

✇❤✐❝❤ ✐s t❤❡ ❝❡♥t❡r ♦❢ r♦t❛t✐♦♥ ♦❢ t❤❡

♦❜❥❡❝t ✇❤❡♥ s✉❜❥❡❝t❡❞ t♦ ❛ ❢♦r❝❡✳ ❲❡ st✉❞✐❡❞ t❤✐s ❝♦♥❝❡♣t ♣r❡✈✐♦✉s❧② ❜✉t ✇✐t❤ t❤❡ ✇❡✐❣❤t ❞✐str✐❜✉t❡❞ ✇✐t❤✐♥ ❛ str❛✐❣❤t s❡❣♠❡♥t✱ s✉❝❤ ❛s ❛ s❡❡s❛✇✿

❲❡ ❢♦✉♥❞ t❤❛t ✐❢ ♦♥❡ ♣❡rs♦♥ ✐s ❤❡❛✈✐❡r t❤❛♥ t❤❡ ♦t❤❡r✱ t❤❡ ❧❛tt❡r ♣❡rs♦♥ s❤♦✉❧❞ s✐t ❢❛rt❤❡r ❢r♦♠ t❤❡ ❝❡♥t❡r ✐♥ ♦r❞❡r t♦ ❜❛❧❛♥❝❡ t❤❡ ❜❡❛♠✳ ■♥ ❢❛❝t✱ t❤❡ ❞✐st❛♥❝❡ s❤♦✉❧❞ ❜❡ t✇✐❝❡ ❛s ❧♦♥❣✦ ❈♦♥s✐❞❡r ❛ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡

s❡❡s❛✇✳

♣♦✐♥t ♦❢ s✉♣♣♦rt ✐♥ t❤❡ ♠✐❞❞❧❡✿

■t ✐s ♠❛❞❡ ♦❢ t✇♦ ❜❡❛♠s ♥❛✐❧❡❞ t♦❣❡t❤❡r t♦ ❢♦r♠ ❛ ❝r♦ss ✇✐t❤ ❛ s✐♥❣❧❡

✹✳✺✳

❚❤❡ ❝❡♥tr♦✐❞ ♦❢ ❛ ✢❛t ♦❜❥❡❝t

✷✾✹

■t ❛♣♣❡❛rs t❤❛t ❢♦✉r ♣❡rs♦♥s ♦❢ ❡q✉❛❧ ✇❡✐❣❤t ✇✐❧❧ ❜❡ ✐♥ ❜❛❧❛♥❝❡ ✇❤❡♥ ❧♦❝❛t❡❞ ❛t ❡q✉❛❧ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ♣♦✐♥t ♦❢ s✉♣♣♦rt✳ ❇✉t t❤❡r❡ ✐s ♠♦r❡✿ ❚❤❡② ✇✐❧❧ ❜❡ ❜❛❧❛♥❝❡❞ ❛s ❧♦♥❣ ❛s ❡✐t❤❡r ✐♥ ❜❛❧❛♥❝❡✦ ❲❡ ❝❛♥ t❤❡♥ ✉s❡ ✇❤❛t ✇❡ ❤❛✈❡ ❧❡❛r♥❡❞ ❢r♦♠ t❤❡

1✲❞✐♠❡♥s✐♦♥❛❧

❲❡ ❡①♣❧♦r❡ t❤✐s ✐❞❡❛ ❜② r❡♣❧❛❝✐♥❣ t❤✐s ❝♦♥str✉❝t✐♦♥ ✇✐t❤ ❛ sq✉❛r❡✳ ♣r❡✈✐♦✉s❧② ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s t❤✐s sq✉❛r❡

♣❛✐r

❜❛❧❛♥❝❡❞ ♦♥ ❛ ❜❛r

♦❢ ♣❡rs♦♥s ❢❛❝✐♥❣ ❡❛❝❤ ♦t❤❡r ❛r❡ ❝❛s❡✳

❚❤❡♥ t❤❡ s❡❡s❛✇s t❤❛t ✇❡ ❝♦♥s✐❞❡r❡❞

t❤❛t ❣♦❡s ❛❧❧ t❤❡ ✇❛② ❛❝r♦ss✿

❲❡ ❝❛♥ s♣r❡❛❞ t❤❡ ✇❡✐❣❤t ❛❧♦♥❣ t❤❡ ❧✐♥❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ ❜❛r ❜❡❝❛✉s❡ ♦♥❧② t❤❡ ❞✐st❛♥❝❡ t♦ t❤✐s ❜❛r ♠❛tt❡rs ❢♦r t❤❡ ❧❡✈❡r❛❣❡ ♦❢ ❡❛❝❤ ✇❡✐❣❤t✳

❖♥❝❡ ✇❡ ❛❞❞ t❤❡

x✲

❛♥❞

y ✲❛①✐s

t♦ t❤❡ ♣✐❝t✉r❡✱ t❤✐s ❞✐st❛♥❝❡ ✐s s✐♠♣❧② t❤❡

x✲❝♦♦r❞✐♥❛t❡✿

◆♦✇✱ ♦✉r ♣r♦❜❧❡♠ ✐s t❤❛t ♦❢ ❜❛❧❛♥❝✐♥❣ t❤❡ r❡❣✐♦♥ ❜❡❧♦✇ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥✳

✹✳✺✳ ❚❤❡ ❝❡♥tr♦✐❞ ♦❢ ❛ ✢❛t ♦❜❥❡❝t

✷✾✺

▲❡t✬s r❡✈✐❡✇ ❤♦✇ ✇❡ ❞♦ t❤✐s✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ❢✉♥❝t✐♦♥ y = f (x) ✐♥t❡❣r❛❜❧❡ ♦♥ s❡❣♠❡♥t [a, b]✳ ❋♦r ❛ ❣✐✈❡♥ ♣♦✐♥t c✱ t❤❡ ✐♥t❡❣r❛❧ Z b

a

f (x)(x − c) dx

✐s ❝❛❧❧❡❞ t❤❡ t♦t❛❧ ♠♦♠❡♥t ♦❢ t❤❡ r❡❣✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ c✳ ❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ t❤❡ r❡❣✐♦♥ ✐s s✉❝❤ ❛ ♣♦✐♥t c t❤❛t t❤❡ t♦t❛❧ ♠♦♠❡♥t ✇✐t❤ r❡s♣❡❝t t♦ c ✐s ③❡r♦✿ Rb f (x)x dx c = Ra b . f (x) dx a ❊①❛♠♣❧❡ ✹✳✺✳✶✿ ❜❛❧❛♥❝❡ ❛ tr✐❛♥❣❧❡

▲❡t✬s r❡✈✐❡✇ ❤♦✇ ✇❡ ❝❛♥ ❜❛❧❛♥❝❡ ❛ tr✐❛♥❣❧❡ ♦♥ ✐ts ❤♦r✐③♦♥t❛❧ ❡❞❣❡✳ ❙✉♣♣♦s❡ ✐t ✐s t❤❡ r❡❣✐♦♥ ✉♥❞❡r t❤❡ ❣r❛♣❤ y = f (x) = x ❢r♦♠ 0 t♦ 1✿

❲❡ ❝♦♠♣✉t❡ t❤❡ t♦t❛❧ ♠♦♠❡♥t ♦❢ t❤❡ ♦❜❥❡❝t✿

Z

1

f (x)x dx = 0

Z

1

Z0

x · x dx 1

x2 dx 0 1 3 = x /3

=

0

= 1/3 .

▼❡❛♥✇❤✐❧❡✱ t❤❡ ♠❛ss ✐s s✐♠♣❧② 1/2✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ✕ ♦♥ t❤❡ x✲❛①✐s ✕ ✐s

c1 =

1 1 2 ÷ = . 3 2 3

❲❤❛t ✐❢ ✇❡ ✇❛♥t t♦ ❜❛❧❛♥❝❡ t❤❡ tr✐❛♥❣❧❡ ♦♥ ✐ts ♦t❤❡r ❡❞❣❡❄ ❲❡ ♣❧❛❝❡ t❤❡ x✲❛①✐s ❛❧♦♥❣ t❤❛t ❡❞❣❡✱ t❤❡♥ t❤❡ s❧❛♥t❡❞ ❡❞❣❡s ✐s ❣✐✈❡♥ ❜② y = g(x) = 1 − x✳ ❲❡ ❝♦♠♣✉t❡ t❤❡ t♦t❛❧ ♠♦♠❡♥t ♦❢ t❤❡ ♦❜❥❡❝t✿

Z

1

g(x)x dx = 0

Z

Z0

1

(1 − x)x dx 1

(x − x2 ) dx 0 1 2 3 = y /2 − y /3

=

0

= 1/2 − 1/3

= 1/6 .

✹✳✺✳ ❚❤❡ ❝❡♥tr♦✐❞ ♦❢ ❛ ✢❛t ♦❜❥❡❝t

✷✾✻

▼❡❛♥✇❤✐❧❡✱ t❤❡ ♠❛ss ✐s st✐❧❧ 1/2✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ✕ ❛❧♦♥❣ t❤✐s ❡❞❣❡ ✕ ✐s

c2 =

1 1 1 ÷ = . 6 2 3

❲❡ ❝❛♥ ❜❛❧❛♥❝❡ t❤❡ tr✐❛♥❣❧❡ ♦♥ ❡✐t❤❡r ♦❢ t✇♦ ❜❛rs✳ ◆♦✇ ✇❡ r❡♠♦✈❡ t❤❡ ❜❛rs ❛♥❞ r❡♣❧❛❝❡ t❤❡♠ ✇✐t❤ ❛ s✐♥❣❧❡ s✉♣♣♦rt ♣❧❛❝❡❞ ❛t t❤❡✐r ✐♥t❡rs❡❝t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✹✳✺✳✷✿ t♦t❛❧ ♠♦♠❡♥t ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ y = f (x) ✐s ✐♥t❡❣r❛❜❧❡ ♦♥ [a, b]✳ ❚❤❡♥ t❤❡ t♦t❛❧ ♠♦♠❡♥t ♦❢ t❤❡ r❡❣✐♦♥ ✉♥❞❡r t❤❡ ❣r❛♣❤ ♦❢ f ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❧✐♥❡ x = c ✐s ❞❡✜♥❡❞ t♦ ❜❡✿

Z

b a

(x − c)f (x) dx

❙✉❝❤ ❛ ❧✐♥❡ ✐s ❛♥ ❛①✐s ♦❢ t❤❡ r❡❣✐♦♥ ✐❢ t❤❡ t♦t❛❧ ♠♦♠❡♥t ✐s ③❡r♦✳

❊①❛♠♣❧❡ ✹✳✺✳✸✿ ❤❛❧❢✲❝✐r❝❧❡ ▲❡t✬s tr② t❤❡ ❤❛❧❢✲❝✐r❝❧❡✳ ❖♥❡ ♦❢ t❤❡ ❛①❡s ✇✐❧❧ ❣♦ t❤r♦✉❣❤ t❤❡ ❝❡♥t❡r ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❞✐❛♠❡t❡r✳ ❲❡ ❥✉st ♥❡❡❞ t♦ ✜♥❞ t❤❡ ♦t❤❡r✳ ❚❤❛t✬s ✇❤② ✇❡ ♣❧❛❝❡ t❤❡ q✉❛rt❡r ♦❢ t❤❡ ❞✐s❦ ❛❞❥❛❝❡♥t t♦ t❤❡ ♦r✐❣✐♥✿

❚❤❡♥ t❤❡ t♦t❛❧ ♠♦♠❡♥t ✐s✿

❚❤❡r❡❢♦r❡✱

Z Z

1 0

1 0

√

√ (x − c) 1 − x2 dx = 0 .

x 1−

x2

dx = c

Z

1

√

0

1 − x2 dx .

❚❤❡ ✐♥t❡❣r❛❧ ♦♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s s✐♠♣❧② t❤❡ ❛r❡❛ ♦❢ t❤❡ q✉❛rt❡r ❝✐r❝❧❡✱ ❛♥❞ t❤❡ ♦♥❡ ♦♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ✐s ❡❛s✐❧② ❡✈❛❧✉❛t❡❞ ❜② s✉❜st✐t✉t✐♦♥ ✭u = 1 − x2 ✮✿ Z 0 Z 1 √ 1√ − u du x 1 − x2 dx = 2 1 0 0 1 2 3/2 =− u 23 1

1 = . 3

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

1 4 = cπ/4 =⇒ c = ≈ .42 . 3 3π

✹✳✻✳

✷✾✼

❈♦♦r❞✐♥❛t❡ s②st❡♠s❀ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s

❊①❡r❝✐s❡ ✹✳✺✳✹ ❋✐♥❞ t❤❡ ❛①❡s ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ ❣r❛♣❤ y =

1 − 0.5 ❛♥❞ t❤❡ ❛①❡s✳ x + 0.5

❲❡ ❤❛✈❡ ❜❡❡♥ ❛❜❧❡ t♦ ✉s❡ ♦♥❧② t❤✐s ❞❡✜♥✐t✐♦♥ t♦ ✜♥❞ t❤❡ ❛①❡s ♦❢ r❡❣✐♦♥s ✇✐t❤ s②♠♠❡tr✐❡s✳ ❋♦r t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✱ ✇❡ ✇✐❧❧ ❤❛✈❡ t♦ ✉s❡ t❤❡ x✲ ❛♥❞ y ✲❛①❡s ❛✈❛✐❧❛❜❧❡ t♦ ✉s✿

❉❡✜♥✐t✐♦♥ ✹✳✺✳✺✿ ❝❡♥tr♦✐❞ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❞❡❝r❡❛s✐♥❣ ♦♥ [0, A] ❛♥❞ f (0) = B > 0✳ ❚❤❡♥ t❤❡ ❝❡♥tr♦✐❞ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ ❣r❛♣❤ ♦❢ f ✱ t❤❡ x✲❛①✐s✱ ❛♥❞ t❤❡ y ✲❛①✐s ✐s ❛ ♣♦✐♥t (cx , cy ) s✉❝❤ t❤❛t t❤❡ t♦t❛❧ ♠♦♠❡♥ts ♦❢ t❤❡ r❡❣✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❧✐♥❡s x = cx ❛♥❞ y = cy ❛r❡ ③❡r♦❀ ✐✳❡✳✱ Z

❛♥❞

Z

A 0

(x − cx )f (x) dx = 0 ,

B 0

(y − cy )f −1 (y) dy = 0 .

❚❤❡♥✱ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❝❡♥tr♦✐❞ ❛r❡✿ 1 cx = A

Z

A 0

1 xf (x) dx ❛♥❞ cy = A

Z

B

yf −1 (y) 0

✇❤❡r❡ A ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❣✐♦♥✳

❊①❡r❝✐s❡ ✹✳✺✳✻ ❋✐♥❞ t❤❡ ❝❡♥tr♦✐❞ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ ❣r❛♣❤ ♦❢ y = 1 − x2 ✳ ❚❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ♦❢ ❛ ♣❧❛t❡ ♦❢ ❛♥ ❛r❜✐tr❛r② s❤❛♣❡ ✇✐❧❧ ❜❡ ❛❞❞r❡ss❡❞ ✐♥ ❱♦❧✉♠❡ ✹✳

✹✳✻✳ ❈♦♦r❞✐♥❛t❡ s②st❡♠s❀ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s

■❢ ✇❡ ✇❛♥t t♦ st✉❞② t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡✱ ❛♥❞ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr②✱ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✇❡ s✉♣❡r✐♠♣♦s❡ t❤❡ ❈❛rt❡s✐❛♥ ❣r✐❞ ♦✈❡r t❤✐s ♣❧❛♥❡✿

✹✳✻✳ ❈♦♦r❞✐♥❛t❡ s②st❡♠s❀ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s

✷✾✽

❲❡ ❝❛♥ ♣❧❛❝❡ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ♦♥ t♦♣ ♦❢ ♦✉r ♣❤②s✐❝❛❧ s♣❛❝❡ ✐♥ ❛ ♥✉♠❜❡r ♦❢ ✇❛②s✳

▼❡❛♥✇❤✐❧❡ t❤❡

❣❡♦♠❡tr② ♦♥ t❤❡ ♣✐❡❝❡ ♦❢ ♣❛♣❡r ❞❡t❡r♠✐♥❡s ✇❤❛t ✐s ❣♦✐♥❣ ♦♥✱ ♥♦t ❛ ♣❛rt✐❝✉❧❛r ❝❤♦✐❝❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠✳ ❲❡ st❛rt ✇✐t❤ ❞✐♠❡♥s✐♦♥

1✳

❚❤❡ ❧✐♥❡ ❝❛♥ ❤❛✈❡ ❞✐✛❡r❡♥t ❝♦♦r❞✐♥❛t❡ s②st❡♠s ❛ss✐❣♥❡❞ t♦ ✐t✳ ❲❡ ❝❛♥ ✐♠❛❣✐♥❡ t❤❛t ✇❡ ❤❛✈❡ t❤r❡❡ ♣❡♥❝✐❧s✿ ♦♥❡ ✉♥♠❛r❦❡❞ ❛♥❞ t✇♦ ✇✐t❤ t❤❡ ✇❤♦❧❡

x✲❛①✐s

✐s ❞r❛✇♥ ♦♥ t❤❡♠✳ ❚❤❡ ✜rst r❡♣r❡s❡♥ts t❤❡ ✏r❡❛❧✐t②✑✱❛♥❞ t❤❡

♦t❤❡r t✇♦ r❡♣r❡s❡♥t t✇♦ ❞✐✛❡r❡♥t ❝♦♦r❞✐♥❛t❡ s②st❡♠s t♦ ❜❡ ✉s❡❞ t♦ r❡❝♦r❞ t❤❡ ❧♦❝❛t✐♦♥s ♦♥ t❤❡ ✜rst✿

❚❤❡♥ ✇❡ ❤❛✈❡✿ ✶✳ P♦✐♥t

A ❤❛s ❝♦♦r❞✐♥❛t❡ 1 ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✜rst ❈❛rt❡s✐❛♥ s②st❡♠ ❜✉t −2 ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s❡❝♦♥❞✳

✷✳ P♦✐♥t

B

❤❛s ❝♦♦r❞✐♥❛t❡

2 ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✜rst ❈❛rt❡s✐❛♥ s②st❡♠ ❜✉t −1 ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s❡❝♦♥❞✳

✸✳ ❆♥❞ s♦ ♦♥✳ ❚❤❡s❡ ❈❛rt❡s✐❛♥ s②st❡♠s ❛r❡ r❡❧❛t❡❞ t♦ ❡❛❝❤ ♦t❤❡r ✈✐❛ s♦♠❡ ❜❛s✐❝ tr❛♥s❢♦r♠❛t✐♦♥s✱ s✉❝❤ ❛s ❛ s❤✐❢t✿

❆❜♦✈❡ ②♦✉ s❡❡ t✇♦ ✇❛②s t♦ ✐♥t❡r♣r❡t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥✿ ✶✳ ❚❤❡ ❛rr♦✇s ❛r❡ ❜❡t✇❡❡♥ t❤❡ ✷✳ ❲❡ ♠♦✈❡ t❤❡

y ✲❛①✐s

s♦ t❤❛t

x✲❛①✐s

❛♥❞ t❤❡ ✐♥t❛❝t

y = f (x)

y ✲❛①✐s✳

✐s ❛❧✐❣♥❡❞ ✇✐t❤

x✳

❲❡ ❢♦❧❧♦✇❡❞ t❤❡ ❢♦r♠❡r ✐♥ ❈❤❛♣t❡r ✶P❈✲✷ ❛♥❞ ✇❡ ✇✐❧❧ ❢♦❧❧♦✇ t❤❡ ❧❛tt❡r ✐♥ t❤✐s s❡❝t✐♦♥✳ ❚❤❡s❡ ❛r❡ t❤❡ t❤r❡❡ ♠❛✐♥ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ ❛♥ ❛①✐s✿ s❤✐❢t✱ ✢✐♣✱ ❛♥❞ str❡t❝❤ ✭❧❡❢t✮✳ ❆♥❞ t❤✐s ✐s ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ♣♦✐♥ts ✭r✐❣❤t✮✿

✹✳✻✳

✷✾✾

❈♦♦r❞✐♥❛t❡ s②st❡♠s❀ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s

❲❛r♥✐♥❣✦ ❚❤❡ ♣♦✐♥ts ❞♦♥✬t ♠♦✈❡✳

❚❤✐s ✐s t❤❡ ❛❧❣❡❜r❛ ❢♦r t❤❡ ❜❛s✐❝ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❛①✐s✱ t❤❡ ♦❧❞ ❛♥❞ t❤❡ ♥❡✇ ❝♦♦r❞✐♥❛t❡s✿ s❤✐❢t ❜② k

1. t −−−−−−−−−→ ✢✐♣

2. t −−−−−−→

str❡t❝❤ ❜② k

x=t−k x = −t

3. t −−−−−−−−−−→ x = t/k

❚❤❡ t❤r❡❡ ♠❛② ❝♦♠❡ r❡s♣❡❝t✐✈❡❧② ❢r♦♠✿ ✶✳ ❝❤❛♥❣✐♥❣ t❤❡ st❛rt✐♥❣ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ♠✐❧❡st♦♥❡s ✷✳ r❡♣❧❛❝✐♥❣ ❡❛st ✇✐t❤ ✇❡st ❛s t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥ ✸✳ s✇✐t❝❤✐♥❣ ❢r♦♠ ♠✐❧❡s t♦ ❦✐❧♦♠❡t❡rs ◆♦✇ ❞✐♠❡♥s✐♦♥ 2✱ t❤❡

✳

♣❧❛♥❡

❇♦t❤ x✲ ❛♥❞ y ✲❛①✐s ❝❛♥ ❜❡ s✉❜❥❡❝t❡❞ t♦ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ❛❜♦✈❡✳ ❚❤❡ ❝❤❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s ✉♥❞❡r t❤❡ r❡s✉❧t✐♥❣ s✐① ❜❛s✐❝ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ xy ✲♣❧❛♥❡ ✐s s❤♦✇♥ ❜❡❧♦✇✿ ✈❡rt✐❝❛❧ s❤✐❢t✿

( x , y

❤♦r✐③♦♥t❛❧ s❤✐❢t✿

( x

)

( x , y−k ) , y )

( x−k , y )

✢✐♣✿

( x , y

✢✐♣✿

( x

)

( x , y · (−1) )

, y )

( x · (−1) , y )

str❡t❝❤✿ str❡t❝❤✿

( x , y

)

( x , y/k ) ( x

, y )

( x/k , y )

.

✹✳✻✳ ❈♦♦r❞✐♥❛t❡ s②st❡♠s❀ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s

✸✵✵

❇✉t s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s ❝❛♥♥♦t ❜❡ r❡❞✉❝❡❞ t♦ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡s❡ s✐①✱ s✉❝❤ ❛s t❤❡ r♦t❛t✐♦♥✳ ❊①❛♠♣❧❡ ✹✳✻✳✶✿ ♦t❤❡r tr❛♥s❢♦r♠❛t✐♦♥s

❘❡❝❛❧❧ ❢r♦♠ ❈❤❛♣t❡r ✶P❈✲✸✱ t❤❛t ✐♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥✱ ✇❡ ❡①❡❝✉t❡ ❛ ✢✐♣ ❛❜♦✉t t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ t❤❡ ♣❧❛♥❡✳ ❲❡ ❣r❛❜ t❤❡ ❡♥❞ ♦❢ t❤❡ x✲❛①✐s ✇✐t❤ t❤❡ r✐❣❤t ❤❛♥❞ ❛♥❞ ❣r❛❜ t❤❡ ❡♥❞ ♦❢ t❤❡ y ✲❛①✐s ✇✐t❤ t❤❡ ❧❡❢t ❤❛♥❞ t❤❡♥ ✐♥t❡r❝❤❛♥❣❡ t❤❡♠✿

❲❡ ❢❛❝❡ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡ ♦❢ t❤❡ ♣❛♣❡r t❤❡♥✱ ❜✉t t❤❡ ❣r❛♣❤ ✐s st✐❧❧ ✈✐s✐❜❧❡✿ ❚❤❡ x✲❛①✐s ✐s ♥♦✇ ♣♦✐♥t✐♥❣ ✉♣ ❛♥❞ t❤❡ y ✲❛①✐s r✐❣❤t✳ ❚❤❡ ❛①❡s ❝❛♥ ❛❧s♦ ❜❡ s❦❡✇❡❞ ✿

❊✈❡♥ t❤❡♥✱ t❤❡ t✇♦ ♥✉♠❜❡rs ✐♥❞✐❝❛t✐♥❣ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t✇♦ ❧✐♥❡s ✇✐❧❧ ✉♥❛♠❜✐❣✉♦✉s❧② ❞❡t❡r♠✐♥❡ ❛ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ ♣❧❛♥❡✳ ❆♥❞ s♦ ♦♥✳✳✳ ❋✉rt❤❡r ❛♥❛❧②s✐s ✐s ♣r❡s❡♥t❡❞ ✐♥ ❱♦❧✉♠❡ ✹ ✭❈❤❛♣t❡r ✹❉❊✲✷✮✳ ❚❤❡r❡ ❛r❡ ❛❧s♦ ❛❧t❡r♥❛t✐✈❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠s✳ ◆♦t ♦♥❧② t❤❡ ❛①❡s ❛r❡ ♥♦t r❡❝t❛♥❣✉❧❛r❀ t❤❡② ❛r❡ ❛❧s♦ ❝✉r✈❡❞ ✦ ❚❤❡ ❝✐r❝❧❡ ✐s ❛ ✈❡r② s♣❡❝✐❛❧ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✳ ❚❤✐s ❝✉r✈❡ ✇✐❧❧ ❛❧s♦ s✉♣♣❧② ✉s ✇✐t❤ ❛ ♥❡✇ ✇❛② t♦ r❡❝♦r❞ ❧♦❝❛t✐♦♥s ♦♥ t❤❡ ♣❧❛♥❡✳ ■t ✐s ❝❛❧❧❡❞ t❤❡ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡ s②st❡♠✳ ❲❤❛t ♠❛❦❡s ❝♦♦r❞✐♥❛t❡ s②st❡♠s ♣♦ss✐❜❧❡ ✐s ♦✉r ❛❜✐❧✐t② t♦ ✉♥❛♠❜✐❣✉♦✉s❧② ❛ss✐❣♥ ❡❛❝❤ ♣♦✐♥t t♦ ❝❡rt❛✐♥ ♣r❡✲ ❞❡t❡r♠✐♥❡❞ s❡ts✳ P❛r❛❧❧❡❧ ❧✐♥❡s ❞♦♥✬t ✐♥t❡rs❡❝t ❛♥❞ t❤❡② ❛❧s♦ ❝♦✈❡r t❤❡ ✇❤♦❧❡ ♣❧❛♥❡✳ ❚❤❛t✬s ✇❤② ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✶✳ ❊✈❡r② ♣♦✐♥t ❜❡❧♦♥❣s t♦ ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ❤♦r✐③♦♥t❛❧ ❧✐♥❡✳ ✷✳ ❊✈❡r② ♣♦✐♥t ❜❡❧♦♥❣s t♦ ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ✈❡rt✐❝❛❧ ❧✐♥❡✳

✹✳✻✳ ❈♦♦r❞✐♥❛t❡ s②st❡♠s❀ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s

✸✵✶

Pr♦✈✐❞✐♥❣ ♦♥❡ ❢r♦♠ ❡✐t❤❡r ❝❧❛ss ❢♦r ❡❛❝❤ ♣♦✐♥t ✐s ❤♦✇ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ✇♦r❦s✿

◆♦✇✱ ✇❡ ❛❧s♦ ❤❛✈❡✿ ✶✳ ❈♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s ❞♦♥✬t ✐♥t❡rs❡❝t ❛♥❞ t❤❡②✱ ♣❧✉s t❤❡ ❝❡♥t❡r✱ ❛❧s♦ ❝♦✈❡r t❤❡ ✇❤♦❧❡ ♣❧❛♥❡✳

❚❤❛t✬s ✇❤②

❡✈❡r② ♣♦✐♥t ❜❡❧♦♥❣s t♦ ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ♦❢ t❤❡s❡ ❝✐r❝❧❡s✱ ♦r t❤❡ ❝❡♥t❡r✳ ✷✳ ▲✐♥❡s t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥ ❤❛✈❡ ♦♥❧② ♦♥❡ ♣♦✐♥t ✐♥ ❝♦♠♠♦♥ ❛♥❞ t❤❡② ❛❧s♦ ❝♦✈❡r t❤❡ ✇❤♦❧❡ ♣❧❛♥❡✳ ❚❤❛t✬s ✇❤② ❡✈❡r② ♣♦✐♥t✱ ♦t❤❡r t❤❛♥ t❤❡ ♦r✐❣✐♥✱ ❜❡❧♦♥❣s t♦ ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ♦❢ t❤❡s❡ ❧✐♥❡s✳ Pr♦✈✐❞✐♥❣ ♦♥❡ ❢r♦♠ ❡✐t❤❡r ❝❧❛ss ❢♦r ❡❛❝❤ ♣♦✐♥t ✐s ❤♦✇ t❤❡ ♣♦❧❛r s②st❡♠ ✇♦r❦s✳ ◆♦✇✱ t❤❡ ♥✉♠❡r✐❝❛❧ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❧♦❝❛t✐♦♥s✱ ✐✳❡✳✱ t❤❡✐r ❝♦♦r❞✐♥❛t❡s✳ ❚❤❡ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s ❜❛s❡❞ ♦♥ t❤❡ ✐❞❡❛ ♦❢ ♠❡❛s✉r✐♥❣ t❤❡ ❞✐st❛♥❝❡s ❢r♦♠ t❤❡ ♣♦✐♥t t♦ t❤❡ ❛①❡s✳

❚❤❡ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s ❜❛s❡❞ ♦♥ t❤❡s❡ t✇♦ ✐❞❡❛s✿

•

♠❡❛s✉r✐♥❣ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ♣♦✐♥t t♦ t❤❡ ♦r✐❣✐♥ ❛♥❞

•

♠❡❛s✉r✐♥❣ t❤❡ ❛♥❣❧❡ ✇✐t❤ t❤❡

❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡ t❤❡s❡✿

x✲❛①✐s✳

✹✳✻✳

✸✵✷

❈♦♦r❞✐♥❛t❡ s②st❡♠s❀ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s

❚❤❡ ❞❡✜♥✐t✐♦♥ ✐s ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ❈❛rt❡s✐❛♥ s②st❡♠✿

❉❡✜♥✐t✐♦♥ ✹✳✻✳✷✿ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s ❙✉♣♣♦s❡ ❛ ♣♦✐♥t O ❝❛❧❧❡❞ t❤❡ ♣♦❧❡ ❛♥❞ ❛ r❛② L ❝❛❧❧❡❞ t❤❡ ♣♦❧❛r ❛①✐s st❛rt✐♥❣ ❛t O ❛r❡ ❣✐✈❡♥ ♦♥ t❤❡ ♣❧❛♥❡✳ ❚❤❡♥ ❢♦r ❛♥② ♣♦✐♥t P ✱ ✐ts ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s ❛r❡ t❤❡ t✇♦ ♥✉♠❜❡rs✱ θ ❛♥❞ r✱ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿ • θ ✐s t❤❡ ❛♥❣❧❡ ❢r♦♠ L t♦ t❤❡ ❧✐♥❡ OP ✐♥ t❤❡ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ❞✐r❡❝t✐♦♥✳ • r ✐s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ P t♦ O✳ ❚❤❡② ❝♦♠♠♦♥❧② ❝♦✲❡①✐st t❤♦✉❣❤✳

❉❡✜♥✐t✐♦♥ ✹✳✻✳✸✿ ❛ss♦❝✐❛t❡❞ ♣♦❧❛r ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s ❆ ♣♦❧❛r ❛♥❞ ❛ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡ s②st❡♠s ❛r❡ ❝❛❧❧❡❞ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❡❛❝❤ ✇❤❡♥ t❤❡ ♣♦❧❡ ♦❢ t❤❡ ❢♦r♠❡r ✐s t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ ❧❛tt❡r ❛♥❞ t❤❡ ♣♦❧❛r ❛①✐s ♦❢ t❤❡ ❢♦r♠❡r ✐s t❤❡ x✲❛①✐s ♦❢ t❤❡ ❧❛tt❡r✳

❲❛r♥✐♥❣✦ ◆♦ ♠❛tt❡r ✇❤❛t

θ

✐s✱ ✐t✬s

O

✐❢

r = 0✳

❊①❛♠♣❧❡ ✹✳✻✳✹✿ ♣♦❧❛r ♣♦✐♥ts ▲❡t✬s ❝♦♥s✐❞❡r t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❛ss♦❝✐❛t❡❞ ♣♦❧❛r ❛♥❞ ❈❛rt❡s✐❛♥ s②st❡♠s✳ ❋♦r ❡❛❝❤ ♦❢ t❤❡s❡ ♣❛✐rs (θ, r)✱ ✇❡ ❝♦♠♣✉t❡ ✐ts ❝♦✉♥t❡r♣❛rt ♦♥ t❤❡ xy ✲♣❧❛♥❡✿ θ 0.00 1.00 1.00 3.14

r →

x

y

1.00

0.00

0.00

0.00

0.54

0.84

1.00 → −1.00

0.00

1.00 →

0.00 →

1.00 →

1.57 −1.50 →

❲❡ ♣❧♦t t❤❡♠ ❤❡r❡✿

0.00 −1.50

✹✳✻✳

✸✵✸

❈♦♦r❞✐♥❛t❡ s②st❡♠s❀ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s

❲❡ ✉s❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❧❛st t✇♦ ❝♦❧✉♠♥s✿ ❂❘❈❬✲✾❪✯❈❖❙✭❘❈❬✲✶✵❪✮ ❛♥❞ ❂❘❈❬✲✶✵❪✯❙■◆✭❘❈❬✲✶✶❪✮ ❚❤❡♦r❡♠ ✹✳✻✳✺✿ ❈♦♥✈❡rs✐♦♥ ❇❡t✇❡❡♥ P♦❧❛r ❆♥❞ ❈❛rt❡s✐❛♥ ✶✳ ❆ ♣♦✐♥t

P

✇✐t❤ t❤❡ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s

r ❛♥❞ θ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♦r❞✐♥❛t❡s

✐♥ t❤❡ ❛ss♦❝✐❛t❡❞ ❈❛rt❡s✐❛♥ s②st❡♠✿

x = r cos θ, y = r sin θ ✷✳ ❆ ♣♦✐♥t

P

✇✐t❤ t❤❡ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s

❝♦♦r❞✐♥❛t❡s ✐♥ t❤❡ ❛ss♦❝✐❛t❡❞ ♣♦❧❛r s②st❡♠✿

θ = arctan

y x

, r=

x 6= 0

❛♥❞

y

❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣

p x2 + y 2

❊①❡r❝✐s❡ ✹✳✻✳✻

Pr♦✈❡ t❤❡ t❤❡♦r❡♠✳ ❚❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ❢r♦♠ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ✐s ❛s ❢♦❧❧♦✇s✳ ❚❤❡ ✐❞❡❛ ♦❢ ❛ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s t♦ ❛ss♦❝✐❛t❡ ❛ ♣❛✐r ✭♦r tr✐♣❧❡✱ ❡t❝✳✮ ♦❢ ♥✉♠❜❡rs t♦ ❡✈❡r② ❧♦❝❛t✐♦♥ ✐♥ ❛♥ ✉♥❛♠❜✐❣✉♦✉s ✇❛②✳ ■t✬s ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡✿ ❧♦❝❛t✐♦♥ ←→ ❛ ♣❛✐r ♦❢ ♥✉♠❜❡rs ■♥ t❤❡ ❢♦r✇❛r❞ ❞✐r❡❝t✐♦♥✱ → ✳ ❈❛rt❡s✐❛♥✿

❙✉♣♣♦s❡ P ✐s ❛ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ ♣❧❛♥❡✳ ❲❡ t❤❡♥ ❞r❛✇ ❛ ✈❡rt✐❝❛❧ ❧✐♥❡ t❤r♦✉❣❤ P ✉♥t✐❧ ✐t ✐♥t❡rs❡❝ts t❤❡ x✲❛①✐s✳ ❚❤❡ ♠❛r❦✱ x✱ ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ✇❤❡r❡ t❤❡② ❝r♦ss ✐s t❤❡ x✲❝♦♦r❞✐♥❛t❡ ♦❢ P ✳ ❲❡ ♥❡①t ❞r❛✇ ❛ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ t❤r♦✉❣❤ P ✉♥t✐❧ ✐t ✐♥t❡rs❡❝ts t❤❡ y ✲❛①✐s✳ ❚❤❡ ♠❛r❦✱ y ✱ ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ✇❤❡r❡ t❤❡② ❝r♦ss ✐s t❤❡ y ✲❝♦♦r❞✐♥❛t❡ ♦❢ P ✳ ❲❡ ❡♥❞ ✉♣ ✇✐t❤ r ≥ 0 ❛♥❞ 0 ≤ θ < 2π ✳

✹✳✻✳

✸✵✹

❈♦♦r❞✐♥❛t❡ s②st❡♠s❀ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s

P♦❧❛r✿

❙✉♣♣♦s❡ P ✐s ❛ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ ♣❧❛♥❡✳ ❲❡ t❤❡♥ ❞r❛✇ ❛ ❧✐♥❡ ❢r♦♠ O t❤r♦✉❣❤ P ✳ ❲❡ ♠❡❛s✉r❡ t❤❡ ❛♥❣❧❡ ♦❢ OP ✇✐t❤ t❤❡ ♣♦❧❛r ❛①✐s ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✳ ❚❤❛t✬s θ✳ ❲❡ ♠❡❛s✉r❡ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ O t♦ P ✳ ❚❤❛t✬s r✳ ■♥ t❤❡ ❜❛❝❦✇❛r❞ ❞✐r❡❝t✐♦♥✱ ← ✳ ❈❛rt❡s✐❛♥✿

❙✉♣♣♦s❡ x ❛♥❞ y ❛r❡ ♥✉♠❜❡rs✳ ❋✐rst✱ ✇❡ ✜♥❞ t❤❡ ♠❛r❦ x ♦♥ t❤❡ x✲❛①✐s ❛♥❞ ❞r❛✇ ❛ ✈❡rt✐❝❛❧ ❧✐♥❡ t❤r♦✉❣❤ t❤✐s ♣♦✐♥t✳ ❙❡❝♦♥❞✱ ✇❡ ✜♥❞ t❤❡ ♠❛r❦ y ♦♥ t❤❡ y ✲❛①✐s ❛♥❞ ❞r❛✇ ❛ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ t❤r♦✉❣❤ t❤✐s ♣♦✐♥t✳ ❚❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡s❡ t✇♦ ❧✐♥❡s ✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❧♦❝❛t✐♦♥ P ♦♥ t❤❡ ♣❧❛♥❡✳ P♦❧❛r✿

❙✉♣♣♦s❡ θ ❛♥❞ r ❛r❡ ♥✉♠❜❡rs✳ ❋✐rst✱ ✇❡ r♦t❛t❡ t❤❡ ♣♦❧❛r ❛①✐s θ r❛❞✐❛♥s ✐♥ t❤❡ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡ ❞✐r❡❝t✐♦♥✳ ❙❡❝♦♥❞✱ ✇❡ ✜♥❞ t❤❡ ♠❛r❦ r ♦♥ t❤❡ ♣♦❧❛r ❛①✐s ❛♥❞ ❞r❛✇ ❛ ❝✐r❝❧❡ ❝❡♥t❡r❡❞ ❛t O t❤r♦✉❣❤ t❤✐s ♣♦✐♥t✳ ❚❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡s❡ t✇♦ ✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❧♦❝❛t✐♦♥ P ♦♥ t❤❡ ♣❧❛♥❡✳ ❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡r❡ ❛r❡ ♠♦r❡ ❞✐✛❡r❡♥t ♣❛✐rs (θ, r) t❤❛t ♣r♦❞✉❝❡ ✭✐♥ ❛❞❞✐t✐♦♥ t♦ (θ, 0)✮ ✐❞❡♥t✐❝❛❧ ❧♦❝❛t✐♦♥s✳ ❯♥❧✐❦❡ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠✱ t❤❡ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❞♦❡s ♥♦t ❣✐✈❡ ❛♥ ✉♥❛♠❜✐❣✉♦✉s r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❡✈❡r② ❧♦❝❛t✐♦♥ ♦♥ t❤❡ ♣❧❛♥❡✿ (θ + 2π, r) = (θ, r),

(θ, −r) = (θ + π, r) .

✹✳✻✳

✸✵✺

❈♦♦r❞✐♥❛t❡ s②st❡♠s❀ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s

❲❡ ❛❞❞r❡ss t❤✐s ♣r♦❜❧❡♠ ❜② ✉s✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✳ ❘❡❝❛❧❧ t❤❛t t❤❡ x✲❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ♣♦✐♥t ♦♥ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s r ❛t ❛♥❣❧❡ θ ✇✐t❤ t❤❡ x✲❛①✐s ✐s r cos θ ❛♥❞ ✐ts y ✲❝♦♦r❞✐♥❛t❡ ✐s r sin θ✿

❊①❡r❝✐s❡ ✹✳✻✳✼

❙✉❣❣❡st ♦t❤❡r ❡①❛♠♣❧❡s ♦❢ ❤♦✇ t✇♦ ❞✐✛❡r❡♥t ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s ♣r♦❞✉❝❡ t❤❡ s❛♠❡ ♣♦✐♥t✳ ❊①❡r❝✐s❡ ✹✳✻✳✽

❘❡♣r❡s❡♥t ✐♥ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s t❤❡s❡ ♣♦✐♥ts ❣✐✈❡♥ ❜② t❤❡✐r ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s✿ ✭❛✮ (1, 2)❀ ✭❜✮ (−1, −1)❀ ✭❝✮ (0, 0)✳ ❊①❡r❝✐s❡ ✹✳✻✳✾

❘❡♣r❡s❡♥t ✐♥ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s t❤❡s❡ ♣♦✐♥ts ❣✐✈❡♥ ❜② t❤❡✐r ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s✿ ✭❛✮ θ = 0, r = −1❀ ✭❜✮ θ = π/4, r = 2❀ ✭❝✮ θ = 1, r = 0✳ ❲❡ ❤❛✈❡ ❛ ❢❛♠✐❧② ♦❢ ❝♦♥❝❡♥tr✐❝ ❝✐r❝❧❡s ♣❛r❛♠❡tr✐③❡❞ ✐♥ ❛ ✉♥✐❢♦r♠ ✇❛②✿

❚❤❡r❡❢♦r❡✱ t❤❡ ♣❛✐r ♦❢ ♥✉♠❜❡rs✱ θ −∞ < θ < +∞

❛♥❞

r −∞ < r < +∞,

✹✳✻✳

✸✵✻

❈♦♦r❞✐♥❛t❡ s②st❡♠s❀ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s

❞❡t❡r♠✐♥❡s ✭❜✉t ♥♦t ✉♥❛♠❜✐❣✉♦✉s❧②✮ ❛ ❧♦❝❛t✐♦♥ ♦♥ t❤❡ ♣❧❛♥❡✿ • ❚❤❡ ❧❛tt❡r ♥✉♠❜❡r✱ r✱ ❞❡t❡r♠✐♥❡s

• ❚❤❡ ❢♦r♠❡r ♥✉♠❜❡r✱ θ✱ t❡❧❧s

✇❤✐❝❤

❤♦✇ ❢❛r

❝✐r❝❧❡ ✇❡ ♣✐❝❦✳

✇❡ ❣♦ ❛❧♦♥❣ t❤✐s ❝✐r❝❧❡✳

❚❤✐s ❛♠❜✐❣✉✐t② ♦❢ t❤❡ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❛❧❧♦✇s ✉s t♦ ❡✛❡❝t✐✈❡❧② r❡♣r❡s❡♥t s♦♠❡ ❝♦♠♣❧❡① ❝✉r✈❡s✳ ❊①❛♠♣❧❡ ✹✳✻✳✶✵✿ ♣♦❧❛r ❝✉r✈❡s

❙✐♠♣❧❡ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ r ❛♥❞ θ ♣r♦❞✉❝❡ ❝✉r✈❡s✿ ♦♥ t❤❡ θr✲♣❧❛♥❡ ❛♥❞ ♦♥ t❤❡ xy ✲♣❧❛♥❡✳ ❚❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s R ✐s ✈❡r② s✐♠♣❧❡✱ ❜② ❞❡s✐❣♥✿ r = R.

❙♦✱ r ✐s ✜①❡❞ ✇❤✐❧❡ θ ✈❛r✐❡s✿

◆❡①t✱ θ ✐s ✜①❡❞ ✇❤✐❧❡ r ✈❛r✐❡s❀ ✐t✬s ❛ r❛②✿

■❢ ❜♦t❤ ✈❛r②✱ ✐❞❡♥t✐❝❛❧❧②✱ ✇❡ ❤❛✈❡ t❤✐s s♣✐r❛❧✿

❊①❛♠♣❧❡ ✹✳✻✳✶✶✿ s♣✐r❛❧

❙✉♣♣♦s❡ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛ ♣♦❧❛r r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤✐s s♣✐r❛❧✱ ✇✐♥❞✐♥❣ ♦♥t♦ t❤❡ ♦r✐❣✐♥✿

✹✳✼✳

✸✵✼

❱❡❝t♦rs ♦♥ t❤❡ ♣❧❛♥❡

❲❡ r❡❛❧✐③❡ t❤❛t ✇❡ ♥❡❡❞ r t♦ ❛♣♣r♦❛❝❤ 0 ❛s θ ❣♦❡s t♦ ✐♥✜♥✐t②✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✇✐❧❧ ❞♦✿ r = 1/θ, θ > 0 . ❊①❡r❝✐s❡ ✹✳✻✳✶✷

❋✐♥❞ ❛ ♣♦❧❛r r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤✐s s♣✐r❛❧✿

❚❤❡ r❡❝t❛♥❣✉❧❛r ❣r✐❞ ♦❢ t❤❡ ❈❛rt❡s✐❛♥ s②st❡♠ ✐s r❡♣❧❛❝❡❞ ✇✐t❤ t❤✐s✿

❊①❡r❝✐s❡ ✹✳✻✳✶✸

❘❡♣r❡s❡♥t ♦♥❡ ♦❢ t❤❡ ❝✉r✈❡❞ r❡❝t❛♥❣❧❡s ❛s ❛ s❡t ✉s✐♥❣ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s✳

✹✳✼✳ ❱❡❝t♦rs ♦♥ t❤❡ ♣❧❛♥❡

▲❡t✬s t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t t❤❡ ♣r♦❜❧❡♠ ❢r♦♠ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❝❤❛♣t❡r✳ Pr♦❜❧❡♠✿

❋r♦♠ ❛ 200✲❢❡❡t ❡❧❡✈❛t✐♦♥✱ ❛ ❝❛♥♥♦♥ ✐s ✜r❡❞ ❤♦r✐③♦♥t❛❧❧② ❛t 200 ❢❡❡t ♣❡r s❡❝♦♥❞✳ ❍♦✇ ❢❛r ✇✐❧❧ t❤❡

✹✳✼✳

❱❡❝t♦rs ♦♥ t❤❡ ♣❧❛♥❡

✸✵✽

❝❛♥♥♦♥❜❛❧❧ ❣♦❄

❚❤❡ r❡s✉❧t ✇❛s t❤❡s❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❝♦♠♣✉t❡❞ ❛s t❤❡ ❘✐❡♠❛♥♥ s✉♠s✿ ❤♦r✐③♦♥t❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ✈❡❧♦❝✐t② ♣♦s✐t✐♦♥

✈❡rt✐❝❛❧

an

bn

vn+1 = vn + han

un+1 = un + hbn

xn+1 = xn + hvn yn+1 = yn + hun

❚❤❡ ♥❡①t st❡♣ ✐s ♦❜✈✐♦✉s✿ ◮ ❲❡ ✐♥t❡r♣r❡t t❤❡ ❧♦❝❛t✐♦♥s ❛s

♣♦✐♥ts ♦♥ t❤❡ ♣❧❛♥❡✳

❚❤❡② ❛r❡ ❝♦♠♣✉t❡❞ r❡❝✉rs✐✈❡❧②✱ ❥✉st ❛s t❤❡✐r ❝♦♦r❞✐♥❛t❡s ❛r❡✿ ❧♦❝❛t✐♦♥ ❤♦r✐③♦♥t❛❧ ✈❡rt✐❝❛❧ (xn , yn )

✈❡❧♦❝✐t② ❞✐s♣❧❛❝❡♠❡♥t ❤♦r✐③♦♥t❛❧ ✈❡rt✐❝❛❧ ❤♦r✐③♦♥t❛❧ ✈❡rt✐❝❛❧ < vn , u n >

< hvn , hun >

❚❤❡ ♥❡①t q✉❡st✐♦♥ ✐s t❤❡♥✿ ◮ ❍♦✇ ❞♦ ✇❡ ✐♥t❡r♣r❡t t❤❡ ✈❡❧♦❝✐t✐❡s ❛♥❞ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts❄

❚❤❡② ❛r❡ ✈❡❝t♦rs✳ ❏✉st ❛s ✇❡ ❝♦♠❜✐♥❡ ♣❛✐rs ♦❢ ♥✉♠❜❡rs ✐♥t♦ ♣♦✐♥ts ❢♦r ❧♦❝❛t✐♦♥s✱ Pn = (xn , yn ) ,

✇❡ ♥♦✇ ❝♦♠❜✐♥❡ t❤❡ ♣❛✐rs ♦❢ ♥✉♠❜❡rs ✐♥t♦ ✈❡❝t♦rs ❢♦r ✈❡❧♦❝✐t✐❡s ❛♥❞ ❞✐s♣❧❛❝❡♠❡♥ts✿ Vn =< vn , un >, Dn =< hvn , hun > .

❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s ✐♥ ❛❧❣❡❜r❛✳✳✳ ❇✉t ✜rst t❤❡ ❞❡✜♥✐t✐♦♥s✳

❉❡✜♥✐t✐♦♥ ✹✳✼✳✶✿ ✈❡❝t♦r ✐♥ ❞✐♠❡♥s✐♦♥ 2 ■❢ ❛ s❡❣♠❡♥t✬s st❛rt✐♥❣ ♣♦✐♥t ✐s t❤❡ ♦r✐❣✐♥✱ ✐✳❡✳✱ ✐t✬s OP ❢♦r s♦♠❡ P ✱ ✐t ✐s ❝❛❧❧❡❞ ❛ ✭2✲❞✐♠❡♥s✐♦♥❛❧✮ ✈❡❝t♦r ✐♥ R2 ✳

✹✳✼✳

❱❡❝t♦rs ♦♥ t❤❡ ♣❧❛♥❡

✸✵✾

❉❡✜♥✐t✐♦♥ ✹✳✼✳✷✿ ✈❡❝t♦r ✐♥ xy✲♣❧❛♥❡ ❚❤❡

❝♦♠♣♦♥❡♥ts

♦❢ ✈❡❝t♦r

OP

❛r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ✐ts t❡r♠✐♥❛❧ ♣♦✐♥t

P✱

❛❝✲

❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥✿

P = (a, b) ⇐⇒ OP =< a, b >

❲❛r♥✐♥❣✦ ■t ✐s ❛❧s♦ ❝♦♠♠♦♥ t♦ ✉s❡

(a, b) t♦ ❞❡♥♦t❡ t❤❡ ✈❡❝t♦r✳

▲❡t✬s ❝♦♥s✐❞❡r t❤❡ ❛❧❣❡❜r❛✳ ❚❤✐s ✐s ✇❤❛t ✇❡ ❤❛✈❡ ✇❤❡♥ ✇❡ tr❡❛t t❤❡ t✇♦ ❞✐r❡❝t✐♦♥s s❡♣❛r❛t❡❧②✿

xn+1 = xn + hvn , yn+1 = yn + hun . ❆❢t❡r ❝♦♠❜✐♥✐♥❣ t❤❡s❡ ♣❛✐rs ♦❢ ♥✉♠❜❡rs ✐♥t♦ ♣♦✐♥ts ❛♥❞ ✐♥t♦ ✈❡❝t♦rs✱ ✇❡ ❤❛✈❡✿

(xn+1 , yn+1 ) = (xn , yn )+ < hvn , hun > . ❚❤❡ ❛❞❞✐t✐♦♥ ♦❢ ♣♦✐♥ts ❛♥❞ ✈❡❝t♦rs ✐s ❞✐s❝✉ss❡❞ ✐♥ ❱♦❧✉♠❡ ✹✳ ❲❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ♦♣❡r❛t✐♦♥s✳ ❋✐rst✱ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ❛r❡ t❤❡ ✈❡❧♦❝✐t✐❡s ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥t

h✿

Dn = hVn . ❚❤✐s ✐s

s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ✈❡❝t♦rs

✳

❙❡❝♦♥❞✱ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ❛r❡ ❛❞❞❡❞ ❝♦♥s❡❝✉t✐✈❡❧②✿

Pn+1 = Pn + Dn = (Pn−1 + Dn−1 ) + Dn = Pn−1 + (Dn−1 + Dn ) . ❚❤✐s ✐s

✈❡❝t♦r ❛❞❞✐t✐♦♥

✳

❊①❛♠♣❧❡ ✹✳✼✳✸✿ ✈❡❧♦❝✐t② ♦❢ str❡❛♠ ■❢ ✇❡ ❧♦♦❦ ❛t t❤❡ ✈❡❧♦❝✐t✐❡s ♦❢ ♣❛rt✐❝❧❡s ✐♥ ❛ str❡❛♠✱ t❤❡② ♠❛② ❛❧s♦ ❜❡ ❝♦♠❜✐♥❡❞ ✇✐t❤ t❤❡ s♣❡❡❞ ♦❢ r♦✇✐♥❣ ♦❢ t❤❡ ❜♦❛t✿

❊①❡r❝✐s❡ ✹✳✼✳✹ ❲✐t❤ t❤❡ ✈❡❧♦❝✐t✐❡s ❛s s❤♦✇♥✱ ✇❤❛t ✐s t❤❡ ❜❡st str❛t❡❣② t♦ ❝r♦ss t❤❡ ❝❛♥❛❧❄

✹✳✼✳

❱❡❝t♦rs ♦♥ t❤❡ ♣❧❛♥❡

✸✶✵

❊①❛♠♣❧❡ ✹✳✼✳✺✿ ❢♦r❝❡s

▲❡t✬s ❛❧s♦ ❧♦♦❦ ❛t ❢♦r❝❡s r❡s♣❡❝t✐✈❡ ❞✐r❡❝t✐♦♥s✿

❛s ✈❡❝t♦rs✳

❋♦r ❡①❛♠♣❧❡✱ s♣r✐♥❣s ❛tt❛❝❤❡❞ t♦ ❛♥ ♦❜❥❡❝t ✇✐❧❧ ♣✉❧❧ ✐t ✐♥ t❤❡✐r

❲❡ ❛❞❞ t❤❡s❡ ✈❡❝t♦rs t♦ ✜♥❞ t❤❡ ❝♦♠❜✐♥❡❞ ❢♦r❝❡ ❛s ✐❢ ♣r♦❞✉❝❡❞ ❜② ❛ s✐♥❣❧❡ s♣r✐♥❣✳ ❚❤❡ ❢♦r❝❡s ❛r❡ ✈❡❝t♦rs t❤❛t st❛rt ❛t t❤❡ s❛♠❡ ❧♦❝❛t✐♦♥✳ ❊①❛♠♣❧❡ ✹✳✼✳✻✿ ❞✐s♣❧❛❝❡♠❡♥ts

❲❡ ❝❛♥ ✐♥t❡r♣r❡t t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts✱ t♦♦✱ ❛s ✈❡❝t♦rs ❛❧✐❣♥❡❞ t♦ t❤❡✐r st❛rt✐♥❣ ♣♦✐♥ts✳ ■♠❛❣✐♥❡ ✇❡ ❛r❡ ❝r♦ss✐♥❣ ❛ r✐✈❡r 3 ♠✐❧❡s ✇✐❞❡ ❛♥❞ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ❝✉rr❡♥t t❛❦❡s ✉s 2 ♠✐❧❡s ❞♦✇♥str❡❛♠✳ ❚❤❡r❡ ❛r❡ t❤r❡❡ ❞✐✛❡r❡♥t ✇❛②s t❤✐s ❝❛♥ ❤❛♣♣❡♥✿ ✶✳ ❛ tr✐♣ 3 ♠✐❧❡s ♥♦rt❤ ❢♦❧❧♦✇❡❞ ❜② ❛ tr✐♣ 2 ♠✐❧❡s ❡❛st❀ ♦r ✷✳ ❛ tr✐♣ 2 ♠✐❧❡s ❡❛st ❢♦❧❧♦✇❡❞ ❜② ❛ tr✐♣ 3 ♠✐❧❡s ♥♦rt❤❀ ❜✉t ❛❧s♦ ✸✳ ❛ tr✐♣ ❛❧♦♥❣ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ ❛ r❡❝t❛♥❣❧❡ ✇✐t❤ ♦♥❡ s✐❞❡ ❣♦✐♥❣ 3 ♠✐❧❡s ♥♦rt❤ ❛♥❞ ❛♥♦t❤❡r 2 ♠✐❧❡s ❡❛st✳ ❚❤❡ t❤r❡❡ ♦✉t❝♦♠❡s ❛r❡ t❤❡ s❛♠❡✿

❚❤❡② ❛r❡ t❤❡ s❛♠❡✳ ❚❤✐s ✐s ✈❡❝t♦r

❛❧❣❡❜r❛ ✇❡ ♥❡❡❞ t♦ ❧❡❛r♥✳

❙♦✱ ✇❡ ❤❛✈❡ ❛ ❝♦♦r❞✐♥❛t❡✇✐s❡ ❛❞❞✐t✐♦♥✿

✹✳✼✳

❱❡❝t♦rs ♦♥ t❤❡ ♣❧❛♥❡

✸✶✶

❍❡r❡✿ < 1, 2 > + < 2, −1 >=< 1 + 2, 2 + (−1) >=< 3, 1 > .

●❡♦♠❡tr✐❝❛❧❧②✱ t♦ ❛❞❞ t✇♦ ✈❡❝t♦rs✱ ✇❡ ❢♦❧❧♦✇ ❡✐t❤❡r✿ ✶✳ ❚❤❡ ❤❡❛❞✲t♦✲t❛✐❧✿ t❤❡ tr✐❛♥❣❧❡ ❝♦♥str✉❝t✐♦♥✳ ✷✳ ❚❤❡ t❛✐❧✲✇✐t❤✲t❛✐❧✿ t❤❡ ♣❛r❛❧❧❡❧♦❣r❛♠ ❝♦♥str✉❝t✐♦♥✳ ❚❤❡② ❤❛✈❡ t♦ ♣r♦❞✉❝❡ t❤❡ s❛♠❡ r❡s✉❧t✦ ❚❤❡② ❞♦✱ ❛s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❲❡ ❛❧s♦ ❤❛✈❡ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ❜② t❤❡ s❛♠❡ ♥✉♠❜❡r✿

❍❡r❡✿ 2· < 2, 1 >=< 2 · 2, 2 · 1 >=< 4, 2 > .

■♥ s✉♠♠❛r②✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❉❡✜♥✐t✐♦♥ ✹✳✼✳✼✿ ✈❡❝t♦r ♦♣❡r❛t✐♦♥s ✶✳ ❆♥② t✇♦ ✈❡❝t♦rs < a, b > ❛♥❞ < u, v > ❝❛♥ ❜❡ ❛❞❞❡❞✱ ♣r♦❞✉❝✐♥❣ ❛ ♥❡✇ ✈❡❝t♦r ❝❛❧❧❡❞ t❤❡✐r s✉♠ ✿ < a, b > + < u, v >=< a + u, b + v >

✷✳ ❆♥② ✈❡❝t♦r < a, b > ❝❛♥ ❜❡ ♠✉❧t✐♣❧✐❡❞ ❜② ❛ ♥✉♠❜❡r k ✱ ♣r♦❞✉❝✐♥❣ ❛ ♥❡✇ ✈❡❝t♦r ❝❛❧❧❡❞ t❤❡ s❝❛❧❛r ♣r♦❞✉❝t ✿ k < a, b >=< ka, kb >

❆❧♦♥❣ t❤✐s ❛❧❣❡❜r❛✱ t❤❡r❡ ✐s s✐❧❧ ❣❡♦♠❡tr② t♦♦✳ ❆ ✈❡❝t♦r ❤❛s ❛ ❞✐r❡❝t✐♦♥✱ ✇❤✐❝❤ ✐s ♦♥❡ ♦❢ t❤❡ t✇♦ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❧✐♥❡ ✐t ❞❡t❡r♠✐♥❡s✱ ❛♥❞ ❛ ♠❛❣♥✐t✉❞❡✱ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✳

✹✳✽✳

❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

✸✶✷

❉❡✜♥✐t✐♦♥ ✹✳✼✳✽✿ ♠❛❣♥✐t✉❞❡ ♦❢ ✈❡❝t♦r ❚❤❡

♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✈❡❝t♦r OP

✐ts t✐♣

P✱

=< a, b >

✐s ❞❡✜♥❡❞ ❛s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠

O

t♦

❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

|| < a, b > || = ❆❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr② ♦❢ ✈❡❝t♦rs ❝❛♥ ✐♥t❡r❛❝t✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡

√

a2 + b2

❚r✐❛♥❣❧❡ ■♥❡q✉❛❧✐t② t❛❦❡ t❤✐s ❢♦r♠✿

||A + B|| ≤ ||A|| + ||B||

❊①❡r❝✐s❡ ✹✳✼✳✾ ❲❤② ♥♦t ✏ 0 =⇒ x2 + 1 6= 0 . ■❢ ✇❡ tr② t♦ s♦❧✈❡ ✐t t❤❡ ✉s✉❛❧ ✇❛②✱ ✇❡ ❣❡t t❤❡s❡✿

x= ❚❤❡r❡ ❛r❡ ♥♦ s✉❝❤

r❡❛❧ ♥✉♠❜❡rs✳

√

−1

❛♥❞

√ x = − −1 .

❍♦✇❡✈❡r✱ ❧❡t✬s ✐❣♥♦r❡ t❤✐s ❢❛❝t ❢♦r ❛ ♠♦♠❡♥t✳ ▲❡t✬s s✉❜st✐t✉t❡ ✇❤❛t ✇❡ ❤❛✈❡ ❜❛❝❦ ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✕ ❜❧✐♥❞❧② ✕ ❢♦❧❧♦✇ t❤❡ r✉❧❡s ♦❢ ❛❧❣❡❜r❛✳ ❲❡ ✏❝♦♥✜r♠✑ t❤❛t t❤✐s ✏♥✉♠❜❡r✑ ✐s ❛ ✏s♦❧✉t✐♦♥✑✿

√ x2 + 1 = ( −1)2 + 1 = (−1) + 1 = 0 . ❲❡ ❝❛❧❧ t❤✐s ❡♥t✐t② t❤❡

✐♠❛❣✐♥❛r② ✉♥✐t✱ ❞❡♥♦t❡❞ ❜② i✳

❲❡ ❥✉st ❛❞❞ t❤✐s ✏♥✉♠❜❡r✑ t♦ t❤❡ s❡t ♦❢ ♥✉♠❜❡rs ✇❡ ❞♦ ❛❧❣❡❜r❛ ✇✐t❤✿

✹✳✽✳

❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

✸✶✸

❆♥❞ s❡❡ ✇❤❛t ❤❛♣♣❡♥s✳✳✳ ▼❛❦✐♥❣

i

❛ ♣❛rt ♦❢ ❛❧❣❡❜r❛ ✇✐❧❧ ♦♥❧② r❡q✉✐r❡ t❤✐s t❤r❡❡✲♣❛rt ❝♦♥✈❡♥t✐♦♥✿

i 6= 0✮✱

✶✳

i

✷✳

i ❝❛♥ ♣❛rt✐❝✐♣❛t❡ ✐♥ t❤❡ ✭❢♦✉r✮ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✇✐t❤ r❡❛❧ ♥✉♠❜❡rs ❜② ❢♦❧❧♦✇✐♥❣ t❤❡ s❛♠❡ r✉❧❡s❀

✸✳

i2 = −1✳

✐s ♥♦t ❛ r❡❛❧ ♥✉♠❜❡r ✭❛♥❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱

❜✉t ❛❧s♦

❲❤❛t ❛❧❣❡❜r❛✐❝ r✉❧❡s ❛r❡ t❤♦s❡❄ ❆ ❢❡✇ ✈❡r② ❜❛s✐❝ ♦♥❡s✿

x + y = y + x, x · y = y · x, x(y + z) = xy + xz, ❲❡ ❛❧❧♦✇ ♦♥❡ ♦r s❡✈❡r❛❧ ♦❢ t❤❡s❡ ♣❛r❛♠❡t❡rs t♦ ❜❡

i✳

❡t❝✳

❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡✿

i + y = y + i, i · y = y · i, i(y + z) = iy + iz,

❡t❝✳

❲❤❛t ♠❛❦❡s t❤✐s ❡①tr❛ ❡✛♦rt ✇♦rt❤✇❤✐❧❡ ✐s ❛ ♥❡✇ ❧♦♦❦ ❛t q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s ❤♦✇ ✇❡ ♠❛② ❢❛❝t♦r ♦♥❡✿

x2 − 1 = (x − 1)(x + 1) . ❚❤❡♥

x=1

❛♥❞

x = −1

❛r❡ t❤❡

❇✉t s♦♠❡ ♣♦❧②♥♦♠✐❛❧s✱ ❝❛❧❧❡❞

x✲✐♥t❡r❝❡♣ts

♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧✿

✐rr❡❞✉❝✐❜❧❡✱ ❝❛♥♥♦t ❜❡ ❢❛❝t♦r❡❞❀ t❤❡r❡ ❛r❡ ♥♦ a, b s✉❝❤ t❤❛t✿ x2 + 1 = (x − a)(x − b) .

❚❤❡r❡ ❛r❡ ♥♦

r❡❛❧ a, b✱ t❤❛t ✐s✦ ❯s✐♥❣ ♦✉r r✉❧❡s✱ ✇❡ ❞✐s❝♦✈❡r✿

(x − i)(x + i) = x2 − ix + ix − i2 = x2 + 1 . ❖❢ ❝♦✉rs❡✱ t❤❡ ♥✉♠❜❡r

i

✐s

♥♦t ❛♥ x✲✐♥t❡r❝❡♣t ♦❢ f (x) = x2 + 1 ❛s t❤❡ x✲❛①✐s ✭✏t❤❡ r❡❛❧ ❧✐♥❡✑✮ ❝♦♥s✐sts ♦❢ ♦♥❧②

✭❛♥❞ ❛❧❧✮ r❡❛❧ ♥✉♠❜❡rs✳ ❙♦✱ ♠✉❧t✐♣❧❡s ♦❢

i

❛♣♣❡❛r ✐♠♠❡❞✐❛t❡❧② ❛s ✇❡ st❛rt ❞♦✐♥❣ ❛❧❣❡❜r❛ ✇✐t❤ ✐t✳

❉❡✜♥✐t✐♦♥ ✹✳✽✳✶✿ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs ❚❤❡ r❡❛❧ ♠✉❧t✐♣❧❡s ♦❢ t❤❡ ✐♠❛❣✐♥❛r② ✉♥✐t✱ ✐✳❡✳✱

z = ri, r ❛r❡ ❝❛❧❧❡❞

✐♠❛❣✐♥❛r② ♥✉♠❜❡rs✳

r❡❛❧,

✹✳✽✳ ❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

✸✶✹

❲❡ ❤❛✈❡ ❝r❡❛t❡❞ ❛ ✇❤♦❧❡ ❝❧❛ss ♦❢ ♥♦♥✲r❡❛❧ ♥✉♠❜❡rs✦ ❖❢ ❝♦✉rs❡✱

ri✱

✇❤❡r❡

r

✐s r❡❛❧✱ ❝❛♥✬t ❜❡ r❡❛❧✿

(ri)2 = r2 i2 = −r2 < 0 . ❚❤❡ ♦♥❧② ❡①❝❡♣t✐♦♥ ✐s

0i = 0❀

✐t✬s r❡❛❧✦

❚❤❡r❡ ❛r❡ ❛s ♠❛♥② ♦❢ t❤❡♠ ❛s t❤❡ r❡❛❧ ♥✉♠❜❡rs✿

❊①❛♠♣❧❡ ✹✳✽✳✷✿ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s

❚❤❡ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs ♠❛② ❛❧s♦ ❝♦♠❡ ❢r♦♠ s♦❧✈✐♥❣ t❤❡ s✐♠♣❧❡st q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❡q✉❛t✐♦♥

x2 + 4 = 0 ❣✐✈❡s ✉s ✈✐❛ ♦✉r s✉❜st✐t✉t✐♦♥✿

■♥❞❡❡❞✱ ✐❢ ✇❡ s✉❜st✐t✉t❡

p √ √ √ x = ± −4 = ± 4(−1) = ± 4 −1 = ±2i .

x = 2i

✐♥t♦ t❤❡ ❡q✉❛t✐♦♥✱ ✇❡ ❤❛✈❡✿

(2i)2 + 4 = (2)2 (i)2 + 4 = 4(−1) + 4 = 0 . ▼♦r❡ ❣❡♥❡r❛❧ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳

■♠❛❣✐♥❛r② ♥✉♠❜❡rs ♦❜❡② t❤❡ ❧❛✇s ♦❢ ❛❧❣❡❜r❛ ❛s ✇❡ ❦♥♦✇ t❤❡♠✦ ■❢ ✇❡ ♥❡❡❞ t♦ s✐♠♣❧✐❢② t❤❡ ❡①♣r❡ss✐♦♥✱ ✇❡ tr② t♦ ♠❛♥✐♣✉❧❛t❡ ✐t ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t r❡❛❧ ♥✉♠❜❡rs ❛r❡ ❝♦♠❜✐♥❡❞ ✇✐t❤ r❡❛❧ ✇❤✐❧❡ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ❥✉st ❢❛❝t♦r

i

i

✐s ♣✉s❤❡❞ ❛s✐❞❡✳

♦✉t ♦❢ ❛❧❧ ❛❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥✿

5i + 3i = (5 + 3)i = 8i . ■t ❧♦♦❦s ❡①❛❝t❧② ❧✐❦❡ ♠✐❞❞❧❡ s❝❤♦♦❧ ❛❧❣❡❜r❛✿

5x + 3x = (5 + 3)x = 8x . ❆❢t❡r ❛❧❧✱

x

❝♦✉❧❞ ❜❡ i✳ ❆♥♦t❤❡r s✐♠✐❧❛r✐t② ✐s ✇✐t❤ t❤❡ ❛❧❣❡❜r❛ ♦❢ q✉❛♥t✐t✐❡s t❤❛t ❤❛✈❡ ✉♥✐ts✿

5

✐♥✳

+3

✐♥✳

= (5 + 3)

✐♥✳

=8

✐♥✳ .

❙♦✱ t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ✉♥✐t ❞♦❡s♥✬t ♠❛tt❡r ✭✐❢ ✇❡ ❝❛♥ ♣✉s❤ ✐t ❛s✐❞❡✮✳ ❊✈❡♥ s✐♠♣❧❡r✿

5

❛♣♣❧❡s

+3

❛♣♣❧❡s

= (5 + 3)

❛♣♣❧❡s

=8

❛♣♣❧❡s

.

■t✬s ✏ 8 ❛♣♣❧❡s✑ ♥♦t ✏ 8✑✦ ❆♥❞ s♦ ♦♥✳ ❚❤✐s ✐s ❤♦✇ ✇❡ ♠✉❧t✐♣❧② ❛♥ ✐♠❛❣✐♥❛r② ♥✉♠❜❡r ❜② ❛ r❡❛❧ ♥✉♠❜❡r✿

2 · (3i) = (2 · 3)i = 6i . ❲❡ ❤❛✈❡ ❛ ♥❡✇ ✐♠❛❣✐♥❛r② ♥✉♠❜❡r✳ ❍♦✇ ❞♦ ✇❡ ♠✉❧t✐♣❧② t✇♦ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs❄ ■t✬s ❞✐✛❡r❡♥t❀ ❛❢t❡r ❛❧❧✱ ✇❡ ❞♦♥✬t ✉s✉❛❧❧② ♠✉❧t✐♣❧② ❛♣♣❧❡s ❜② ❛♣♣❧❡s✦ ■♥ ❝♦♥tr❛st t♦ t❤❡ ❛❜♦✈❡✱ ❡✈❡♥ t❤♦✉❣❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ❢♦❧❧♦✇ t❤❡ s❛♠❡ r✉❧❡ ❛s ❛❧✇❛②s✱ ✇❡ ❝❛♥✱ ✇❤❡♥ ♥❡❝❡ss❛r②✱ ❛♥❞ ♦❢t❡♥ ❤❛✈❡ t♦✱ s✐♠♣❧✐❢② t❤❡ ♦✉t❝♦♠❡ ♦❢ ♦✉r ❛❧❣❡❜r❛ ✉s✐♥❣ ♦✉r ❢✉♥❞❛♠❡♥t❛❧ ✐❞❡♥t✐t② ✿

i2 = −1 .

✹✳✽✳

❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

✸✶✺

❋♦r ❡①❛♠♣❧❡✿

(5i) · (3i) = (5 · 3)(i · i) = 15i2 = 15(−1) = −15 . ■t✬s r❡❛❧✦ ❲❡ ❛❧s♦ s✐♠♣❧✐❢② t❤❡ ♦✉t❝♦♠❡ ❜② ✉s✐♥❣ t❤❡ ♦t❤❡r

❢✉♥❞❛♠❡♥t❛❧ ❢❛❝t

❛❜♦✉t t❤❡ ✐♠❛❣✐♥❛r② ✉♥✐t✿

i 6= 0 . ❲❡ ❝❛♥ ❞✐✈✐❞❡ ❜②

i✦

❋♦r ❡①❛♠♣❧❡✱

5i 5 5 5i = = ·1= . 3i 3i 3 3

❆s ②♦✉ ❝❛♥ s❡❡✱ ❞♦✐♥❣ ❛❧❣❡❜r❛ ✇✐t❤ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs ✇✐❧❧ ♦❢t❡♥ ❜r✐♥❣ ✉s ❜❛❝❦ t♦ r❡❛❧ ♥✉♠❜❡rs✳ ❚❤❡s❡ t✇♦ ❝❧❛ss❡s ♦❢ ♥✉♠❜❡rs ❝❛♥♥♦t ❜❡ s❡♣❛r❛t❡❞ ❢r♦♠ ❡❛❝❤ ♦t❤❡r✦ ❚❤❡② ❛r❡♥✬t✳ ▲❡t✬s t❛❦❡ ❛♥♦t❤❡r ❧♦♦❦ ❛t q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s✳ ❚❤❡ ❡q✉❛t✐♦♥

ax2 + bx + c = 0, a 6= 0 ,

✐s s♦❧✈❡❞ ✇✐t❤ t❤❡ ❢❛♠✐❧✐❛r

◗✉❛❞r❛t✐❝ ❋♦r♠✉❧❛ ✿ x=

−b ±

√

b2 − 4ac . 2a

▲❡t✬s ❝♦♥s✐❞❡r

x2 + 2x + 10 = 0 . ❚❤❡♥ t❤❡ r♦♦ts ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡✿

√

22 − 4 · 10 √2 −2 ± −36 = 2 √ = −1 ± −9 √ √ = −1 ± 9 −1

x =

−2 ±

❚❤❡r❡ ✐s ♥♦ r❡❛❧ s♦❧✉t✐♦♥✦ ❇✉t ✇❡ ❣♦ ♦♥✳

= −1 ± 3i . ❲❡ ❡♥❞ ✉♣ ❛❞❞✐♥❣ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs✦ ❆s t❤❡r❡ ✐s ♥♦ ✇❛② t♦ s✐♠♣❧✐❢② t❤✐s✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

◮

❆ ♥✉♠❜❡r

a + bi✱

✇❤❡r❡

a, b 6= 0

❛r❡ r❡❛❧✱ ✐s ♥❡✐t❤❡r r❡❛❧ ♥♦r ✐♠❛❣✐♥❛r②✳

❊①❡r❝✐s❡ ✹✳✽✳✸

❊①♣❧❛✐♥ ✇❤②✳

❚❤✐s ❛❞❞✐t✐♦♥ ✐s ♥♦t ❧✐t❡r❛❧✳ ■t✬s ❧✐❦❡ ✏❛❞❞✐♥❣✑ ❛♣♣❧❡s t♦ ♦r❛♥❣❡s✿

5 ■t✬s ♥♦t

❛♣♣❧❡s

+3

♦r❛♥❣❡s

= ...

8 ❛♥❞ ✐t✬s ♥♦t 8 ❢r✉✐t ❜❡❝❛✉s❡ ✇❡ ✇♦✉❧❞♥✬t ❜❡ ❛❜❧❡ t♦ r❡❛❞ t❤✐s ❡q✉❛❧✐t② ❜❛❝❦✇❛r❞s✳

❤♦✇❡✈❡r✱ ❜❡ ♠❡❛♥✐♥❣❢✉❧✿

(5a + 3o) + (2a + 4o) = (5 + 3)a + (3 + 4)o = 8a + 7o . ■t ✐s ❛s ✐❢ ✇❡ ❝♦❧❧❡❝t

s✐♠✐❧❛r t❡r♠s✱ ❧✐❦❡ t❤✐s✿ (5 + 3x) + (2 + 4x) = (5 + 2) + (3 + 4)x = 8 + 7x .

❚❤❡ ❛❧❣❡❜r❛ ✇✐❧❧✱

✹✳✽✳

❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

✸✶✻

❚❤✐s ✐❞❡❛ ❡♥❛❜❧❡s ✉s t♦ ❞♦ t❤✐s✿

(5 + 3i) + (2 + 4i) = (5 + 3) + (3 + 4)i = 8 + 7i . ❊❛❝❤ ♦❢ t❤❡ ♥✉♠❜❡rs ✇❡ ❛r❡ ❢❛❝✐♥❣ ❝♦♥t❛✐♥ ❜♦t❤ r❡❛❧ ♥✉♠❜❡rs ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts✳ ❚❤✐s ❢❛❝t ♠❛❦❡s t❤❡♠ ✏❝♦♠♣❧❡①✑✳✳✳

❉❡✜♥✐t✐♦♥ ✹✳✽✳✹✿ ❝♦♠♣❧❡① ♥✉♠❜❡r ❆♥② s✉♠ ♦❢ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs ✐s ❝❛❧❧❡❞ ❛

❝♦♠♣❧❡① ♥✉♠❜❡r✳ ❚❤❡ s❡t ♦❢

❛❧❧ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✐s ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

C = {z = a + bi : a, b

r❡❛❧}

❲❛r♥✐♥❣✦ ❆❧❧ r❡❛❧ ♥✉♠❜❡rs ❛r❡ ❝♦♠♣❧❡① ✭b

= 0✮✳

❆❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥ ❛r❡ ❡❛s②❀ ✇❡ ❥✉st ❝♦♠❜✐♥❡ s✐♠✐❧❛r t❡r♠s ❥✉st ❧✐❦❡ ✐♥ ♠✐❞❞❧❡ s❝❤♦♦❧✳ ❋♦r ❡①❛♠♣❧❡✱ (1 + 5i) + (3 − i) = 1 + 5i + 3 − i = (1 + 3) + (5i − i) = 4 + 4i . ❚♦ s✐♠♣❧✐❢②

♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ✇❡ ❡①♣❛♥❞ ❛♥❞ t❤❡♥ ✉s❡ i2 = −1✱ ❛s ❢♦❧❧♦✇s✿ (1 + 5i) · (3 − i) = 1 · 3 + 5i · 3 + 1 · (−i) + 5i · (−i) = 3 + 15i − i − 5i2

= (3 + 5) + (15i − i)

= 8 + 14i . ■t✬s ❛ ❜✐t tr✐❝❦✐❡r ✇✐t❤

❞✐✈✐s✐♦♥ ✿

1 + 5i 3 + i 1 + 5i = 3−i 3−i 3+i (1 + 5i)(3 + i) = (3 − i)(3 + i) −2 + 8i = 2 3 − i2 −2 + 8i = 2 3 +1 1 = (−2 + 8i) 10 = −0.2 + 0.8i .

❚❤❡ s✐♠♣❧✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r ✐s ♠❛❞❡ ♣♦ss✐❜❧❡ ❜② t❤❡ tr✐❝❦ ♦❢ ♠✉❧t✐♣❧②✐♥❣ ❜②

3 + i✳

■t ✐s t❤❡ s❛♠❡

tr✐❝❦ ✇❡ ✉s❡❞ ✐♥ ❱♦❧✉♠❡ ✶ t♦ s✐♠♣❧✐❢② ❢r❛❝t✐♦♥s ✇✐t❤ r♦♦ts t♦ ❝♦♠♣✉t❡ t❤❡✐r ❧✐♠✐ts✿

√ √ 1+ x 1 1 1+ x √ = √ √ = . 1−x 1− x 1− x1+ x

❉❡✜♥✐t✐♦♥ ✹✳✽✳✺✿ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ❚❤❡

❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ♦❢ z = a + bi ✐s ❞❡✜♥❡❞ ❛♥❞ ❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿ z¯ = a + bi = a − bi .

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❝r✉❝✐❛❧✳

✹✳✽✳

❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

✸✶✼

❚❤❡♦r❡♠ ✹✳✽✳✻✿ ❆❧❣❡❜r❛ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡rs ❚❤❡ r✉❧❡s ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ✐❞❡♥t✐❝❛❧ t♦ t❤♦s❡ ♦❢ r❡❛❧ ♥✉♠✲ ❜❡rs✿

• • • • •

z+u=u+z (z + u) + v = z + (u + v) ❈♦♠♠✉t❛t✐✈✐t② ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ z · u = u · z ❆ss♦❝✐❛t✐✈✐t② ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ (z · u) · v = z · (u · v) ❉✐str✐❜✉t✐✈✐t②✿ z · (u + v) = z · u + z · v ❈♦♠♠✉t❛t✐✈✐t② ♦❢ ❛❞❞✐t✐♦♥✿ ❆ss♦❝✐❛t✐✈✐t② ♦❢ ❛❞❞✐t✐♦♥✿

❚❤✐s ✐s t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠ ❀ ✐t ❢♦❧❧♦✇s t❤❡ r✉❧❡s ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡r s②st❡♠ ❜✉t ❛❧s♦ ❝♦♥t❛✐♥s ✐t✳ ❚❤✐s t❤❡♦r❡♠ ✇✐❧❧ ❛❧❧♦✇ ✉s t♦ ❜✉✐❧❞ ❝❛❧❝✉❧✉s ❢♦r ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s t❤❛t ✐s ❛❧♠♦st ✐❞❡♥t✐❝❛❧ t♦ ❝❛❧❝✉❧✉s ❢♦r r❡❛❧ ❢✉♥❝t✐♦♥s ❜✉t ❛❧s♦ ❝♦♥t❛✐♥s ✐t✳

❉❡✜♥✐t✐♦♥ ✹✳✽✳✼✿ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡r ❊✈❡r② ❝♦♠♣❧❡① ♥✉♠❜❡r x ❤❛s t❤❡ st❛♥❞❛r❞

r❡♣r❡s❡♥t❛t✐♦♥ ✿

z = a + bi ,

✇❤❡r❡ a ❛♥❞ b ❛r❡ t✇♦ r❡❛❧ ♥✉♠❜❡rs✳ ❚❤❡ t✇♦ ❝♦♠♣♦♥❡♥ts ❛r❡ ♥❛♠❡❞ ❛s ❢♦❧❧♦✇s✿ • a ✐s t❤❡ r❡❛❧ ♣❛rt ♦❢ z ✱ ✇✐t❤ ♥♦t❛t✐♦♥✿ a = Re(z) ; • bi ✐s t❤❡

✐♠❛❣✐♥❛r② ♣❛rt ♦❢ z ✱ ✇✐t❤ ♥♦t❛t✐♦♥✿ b = Im(z) .

❚❤❡♥✱ t❤❡ ♣✉r♣♦s❡ ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ❛❜♦✈❡ ✇❛s t♦ ✜♥❞ t❤❡ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r t❤❛t ❝♦♠❡s ❢r♦♠ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✇✐t❤ ♦t❤❡r ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❚❤❡② ✇❡r❡ ❧✐t❡r❛❧❧② s✐♠♣❧✐✜❝❛t✐♦♥s✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♠❛❦❡s s❡♥s❡ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✿

❚❤❡♦r❡♠ ✹✳✽✳✽✿ ❙t❛♥❞❛r❞ ❋♦r♠ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡r ❚✇♦ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ❡q✉❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❜♦t❤ t❤❡✐r r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts ❛r❡ ❡q✉❛❧✳

❙♦✱ ✇❡ ❤❛✈❡✿ z = Re(z) + Im(z)i .

■♥ ♦r❞❡r t♦ s❡❡ t❤❡ ❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ✇❡ ♥❡❡❞ t♦ ❝♦♠❜✐♥❡ t❤❡ r❡❛❧ ♥✉♠❜❡r ❧✐♥❡ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♥✉♠❜❡r ❧✐♥❡✳ ❍♦✇❄ ❲❡ r❡❛❧✐③❡ t❤❛t t❤❡② ❤❛✈❡ ♥♦t❤✐♥❣ ✐♥ ❝♦♠♠♦♥✳✳✳ ❡①❝❡♣t 0 = 0i ❜❡❧♦♥❣s t♦ ❜♦t❤✿

❲❡ ❝❛♥ tr② t♦ ❝♦♠❜✐♥❡ t❤❡♠ ❧✐❦❡ t❤❛t✱ ♦r ❧✐❦❡ t❤✐s✿

✹✳✽✳

❍♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❡♠❡r❣❡

✸✶✽

❖r ✇❡ ❝❛♥ tr② t♦ ❝♦♠❜✐♥❡ t❤❡♠ ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r ✇❡ ❜✉✐❧t t❤❡

xy ✲♣❧❛♥❡✿

❚❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ❤❡❧♣s ✉s ✉♥❞❡rst❛♥❞ t❤❡ ♠❛✐♥ ✐❞❡❛✿

◮ ■❢

❈♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤❡ r❡❛❧ ✉♥✐t✱

z = a + bi✱

t❤❡♥

a

❛♥❞

b

1✱

❛r❡ t❤♦✉❣❤t ♦❢ ❛s t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ✈❡❝t♦r

❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ✉♥✐t✱

z

i✳

✐♥ t❤❡ ♣❧❛♥❡✳ ❲❡ ❤❛✈❡ ❛ ♦♥❡✲t♦✲♦♥❡

❝♦rr❡s♣♦♥❞❡♥❝❡✿

C ←→ R2 , ❣✐✈❡♥ ❜②

a + bi ←→ < a, b > . ❚❤❡♥ t❤❡

x✲❛①✐s

■t ✐s ❝❛❧❧❡❞ t❤❡

♦❢ t❤✐s ♣❧❛♥❡ ❝♦♥s✐sts ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡rs ❛♥❞ t❤❡

y ✲❛①✐s

♦❢ t❤❡ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs✳

❝♦♠♣❧❡① ♣❧❛♥❡

✳

❲❛r♥✐♥❣✦ ❚❤✐s ✐s ❥✉st ❛ ✈✐s✉❛❧✐③❛t✐♦♥✳

❚❤❡♥ t❤❡ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ♦❢

z

✐s t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r ✇✐t❤ t❤❡ s❛♠❡ r❡❛❧ ♣❛rt ❛s

♣❛rt ✇✐t❤ t❤❡ ♦♣♣♦s✐t❡ s✐❣♥✿

Re(¯ z ) = Re(z)

❛♥❞

Im(¯ z ) = − Im(z) .

z

❛♥❞ t❤❡ ✐♠❛❣✐♥❛r②

✹✳✾✳

❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2

✸✶✾

❲❛r♥✐♥❣✦ ❆❧❧ ♥✉♠❜❡rs ✇❡ ❤❛✈❡ ❡♥❝♦✉♥t❡r❡❞ s♦ ❢❛r ❛r❡ r❡❛❧ ♥♦♥✲❝♦♠♣❧❡①✱ ❛♥❞ s♦ ❛r❡ ❛❧❧ q✉❛♥t✐t✐❡s ♦♥❡ ❝❛♥ ❡♥✲ ❝♦✉♥t❡r ✐♥ ❞❛②✲t♦✲❞❛② ❧✐❢❡ ♦r s❝✐❡♥❝❡✿ t✐♠❡✱ ❧♦❝❛t✐♦♥✱ ❧❡♥❣t❤✱ ❛r❡❛✱ ✈♦❧✉♠❡✱ ♠❛ss✱ t❡♠♣❡r❛t✉r❡✱ ♠♦♥❡②✱ ❡t❝✳

✹✳✾✳ ❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡

C

✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡

R2

♥✉♠❜❡rs✱ t❤❡② ♠✉st ❜❡ s✉❜ ❥❡❝t t♦ s♦♠❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✳ 2 ❲❡ ✇✐❧❧ ✐♥✐t✐❛❧❧② ❧♦♦❦ ❛t t❤❡♠ t❤r♦✉❣❤ t❤❡ ❧❡♥s ♦❢ ✈❡❝t♦r ❛❧❣❡❜r❛ ♦❢ t❤❡ ♣❧❛♥❡ R ✳ ❆ ❝♦♠♣❧❡① ♥✉♠❜❡r z ❤❛s t❤❡ st❛♥❞❛r❞ r❡♣r❡s❡♥t❛t✐♦♥ ✿ ■❢ ✇❡ ❝❛❧❧ ❝♦♠♣❧❡① ♥✉♠❜❡r

z = a + bi , ✇❤❡r❡

a

❛♥❞

b

❛r❡ t✇♦ r❡❛❧ ♥✉♠❜❡rs✳

❚❤❡s❡ t✇♦ ❝❛♥ ❜❡ s❡❡♥ ✐♥ t❤❡

❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥

♦❢ ❝♦♠♣❧❡①

♥✉♠❜❡rs✿

❚❤❡r❡❢♦r❡✱

a

❛♥❞

b

❛r❡ t❤♦✉❣❤t ♦❢ ❛s t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢

✐s ♥♦t ♦♥❧② ❛ ♣♦✐♥t ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ ❜✉t ❛❧s♦ ❛

z

❛s ❛

✈❡❝t♦r✳

♣♦✐♥t

♦♥ t❤❡ ♣❧❛♥❡✳ ❇✉t ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r

❲❡ ❤❛✈❡ ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡✿

C ←→ R2 , ❣✐✈❡♥ ❜②

a + bi ←→ < a, b > ❚❤❡r❡ ✐s ♠♦r❡ t♦ t❤✐s t❤❛♥ ❥✉st ❛ ♠❛t❝❤❀ t❤❡ ❛❧❣❡❜r❛ ♦❢ ✈❡❝t♦rs ✐♥

R2

❛♣♣❧✐❡s✦

✹✳✾✳

❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2

✸✷✵

❲❛r♥✐♥❣✦

■♥ s♣✐t❡ ♦❢ t❤✐s ❢✉♥❞❛♠❡♥t❛❧ ❝♦rr❡s♣♦♥❞❡♥❝❡✱ ✇❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ t❤✐♥❦ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛s ♥✉♠❜❡rs ✭❛♥❞ ✉s❡ t❤❡ ❧♦✇❡r ❝❛s❡ ❧❡tt❡rs✮✳ ▲❡t✬s s❡❡ ❤♦✇ t❤✐s ❛❧❣❡❜r❛ ♦❢ ♥✉♠❜❡rs ✇♦r❦s ✐♥ ♣❛r❛❧❧❡❧ ✇✐t❤ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❋✐rst✱ t❤❡ ❛❞❞✐t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✐s ❞♦♥❡

(a + bi)

2✲✈❡❝t♦rs✳

❝♦♠♣♦♥❡♥t✇✐s❡ ✿

+ (c + di)

= (a + c)

+ (b + d)i

< a, b > + < c, d > = < a + c ,

b+d>

■t ❝♦rr❡s♣♦♥❞s t♦ ❛❞❞✐t✐♦♥ ♦❢ ✈❡❝t♦rs✿

❙❡❝♦♥❞✱ ✇❡ ❝❛♥ ❡❛s✐❧② ♠✉❧t✐♣❧② ❝♦♠♣❧❡① ♥✉♠❜❡rs ❜② r❡❛❧ ♦♥❡s✿

(a + bi)

c = (ac)

+ (bc)i

< a, b > c = < ac ,

bc >

■t ❝♦rr❡s♣♦♥❞s t♦ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ✈❡❝t♦rs✳

❲❛r♥✐♥❣✦

❱❡❝t♦r ❛❧❣❡❜r❛ ♦❢ R2 ✐s ❝♦♠♣❧❡① ❛❧❣❡❜r❛✱ ❜✉t ♥♦t ✈✐❝❡ ✈❡rs❛✳ ❈♦♠♣❧❡① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ✇❤❛t ♠❛❦❡s ✐t ❞✐✛❡r❡♥t✳ ❊①❛♠♣❧❡ ✹✳✾✳✶✿ ❝✐r❝❧❡ ❲❡ ❝❛♥ ❡❛s✐❧② r❡♣r❡s❡♥t ❝✐r❝❧❡s ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✿

z = r cos θ + r sin θ · i .

❖✉r st✉❞② ♦❢ ❝❛❧❝✉❧✉s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs st❛rts ✇✐t❤ t❤❡ st✉❞② ♦❢ t❤❡ t♦♣♦❧♦❣② ✐s t❤❡ s❛♠❡ ❛s t❤❛t ♦❢

t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡

R

2

✦

t♦♣♦❧♦❣②

♦❢ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✳ ❚❤✐s

✹✳✾✳

❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2

❏✉st ❛s ❜❡❢♦r❡✱ ❡✈❡r② ❢✉♥❝t✐♦♥

z = f (t)

✸✷✶

✇✐t❤ ❛♥ ❛♣♣r♦♣r✐❛t❡ ❞♦♠❛✐♥ ❝r❡❛t❡s ❛ s❡q✉❡♥❝❡✿

zk = f (k) . ❆ ❢✉♥❝t✐♦♥ ✇✐t❤ ❝♦♠♣❧❡① ✈❛❧✉❡s ❞❡✜♥❡❞ ♦♥ ❛ r❛② ✐♥ t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs✱

s❡q✉❡♥❝❡✱ ♦r s✐♠♣❧② s❡q✉❡♥❝❡✳

{p, p + 1, ...}✱

✐s ❝❛❧❧❡❞ ❛♥

✐♥✜♥✐t❡

❊①❛♠♣❧❡ ✹✳✾✳✷✿ s♣✐r❛❧

❆ ❣♦♦❞ ❡①❛♠♣❧❡ ✐s t❤❛t ♦❢ t❤❡ s❡q✉❡♥❝❡ ♠❛❞❡ ♦❢ t❤❡ r❡❝✐♣r♦❝❛❧s✿

zk = ■t

cos k sin k + i. k k

t❡♥❞s t♦ 0 ✇❤✐❧❡ s♣✐r❛❧✐♥❣ ❛r♦✉♥❞ ✐t✳

❚❤❡ st❛rt✐♥❣ ♣♦✐♥t ♦❢ ❝❛❧❝✉❧✉s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛

♥✉♠❜❡rs

s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❧❡①

✐s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✐ts r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts ♦r✱ ✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♣♦✐♥ts

✭♦r ✈❡❝t♦rs✮ ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ s❡❡♥ ❛s ❛♥② ♣❧❛♥❡✿ ❚❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡

k t❤ ♣♦✐♥t t♦ t❤❡ ❧✐♠✐t ✐s ❣❡tt✐♥❣

s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r✳

❲❡ ✉s❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r ✈❡❝t♦rs ♦♥ t❤❡ ♣❧❛♥❡ ❜② s✐♠♣❧② r❡♣❧❛❝✐♥❣ ✈❡❝t♦rs ✇✐t❤ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛♥❞ ✏♠❛❣♥✐t✉❞❡✑ ✇✐t❤ ✏♠♦❞✉❧✉s✑✳

✹✳✾✳

❚❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ C ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R2

✸✷✷

❉❡✜♥✐t✐♦♥ ✹✳✾✳✸✿ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ❙✉♣♣♦s❡

C✳

{zk : k = 1, 2, 3, ...}

✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ✐✳❡✳✱ ♣♦✐♥ts ✐♥

❝♦♥✈❡r❣❡s

❲❡ s❛② t❤❛t t❤❡ s❡q✉❡♥❝❡

♣♦✐♥t ✐♥

C✱

❝❛❧❧❡❞ t❤❡

❧✐♠✐t

t♦ ❛♥♦t❤❡r ❝♦♠♣❧❡① ♥✉♠❜❡r

z✱

✐✳❡✳✱ ❛

♦❢ t❤❡ s❡q✉❡♥❝❡✱ ✐❢✿

||zk − z|| → 0

❛s

k → ∞,

❞❡♥♦t❡❞ ❛s ❢♦❧❧♦✇s✿

zk → z

❛s

k → ∞,

♦r

z = lim zk . k→∞

❝♦♥✈❡r❣❡♥t ❞✐✈❡r❣❡s✳

■❢ ❛ s❡q✉❡♥❝❡ ❤❛s ❛ ❧✐♠✐t✱ ✇❡ ❝❛❧❧ t❤❡ s❡q✉❡♥❝❡

✈❡r❣❡s ❀ ♦t❤❡r✇✐s❡ ✐t ✐s ❞✐✈❡r❣❡♥t

❛♥❞ ✇❡ s❛② ✐t

❛♥❞ s❛② t❤❛t ✐t

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♣♦✐♥ts st❛rt t♦ ❛❝❝✉♠✉❧❛t❡ ✐♥ s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r ❝✐r❝❧❡s ❛r♦✉♥❞ ❛ tr❡♥❞ ✐♥ ❛ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ✐s t♦ ❡♥❝❧♦s❡ t❤❡ t❛✐❧ ♦❢ t❤❡ s❡q✉❡♥❝❡ ✐♥ ❛

❞✐s❦ ✿

z✳

❝♦♥✲

❆ ✇❛② t♦ ✈✐s✉❛❧✐③❡

❚❤❡♦r❡♠ ✹✳✾✳✹✿ ❯♥✐q✉❡♥❡ss ♦❢ ▲✐♠✐t ❆ s❡q✉❡♥❝❡ ❝❛♥ ❤❛✈❡ ♦♥❧② ♦♥❡ ❧✐♠✐t ✭✜♥✐t❡ ♦r ✐♥✜♥✐t❡✮❀ ✐✳❡✳✱ ✐❢ a ❛♥❞ b ❛r❡ ❧✐♠✐ts ♦❢ t❤❡ s❛♠❡ s❡q✉❡♥❝❡✱ t❤❡♥ a = b✳

❉❡✜♥✐t✐♦♥ ✹✳✾✳✺✿ s❡q✉❡♥❝❡ t❡♥❞s t♦ ✐♥✜♥✐t② ❲❡ s❛② t❤❛t ❛ s❡q✉❡♥❝❡ ❡❛❝❤ r❡❛❧ ♥✉♠❜❡r ♥✉♠❜❡r

k > N✱

zk t❡♥❞s t♦ ✐♥✜♥✐t②

✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✿ ❋♦r

R✱ t❤❡r❡ ❡①✐sts s✉❝❤ ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r N

t❤❛t✱ ❢♦r ❡✈❡r② ♥❛t✉r❛❧

✇❡ ❤❛✈❡

||zk || > R . ❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥✿

zk → ∞

❛s

k → ∞.

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛♥♦t❤❡r ❛♥❛❧♦❣ ♦❢ ❛ ❢❛♠✐❧✐❛r t❤❡♦r❡♠ ❛❜♦✉t t❤❡ t♦♣♦❧♦❣② ♦❢ t❤❡ ♣❧❛♥❡✳

❚❤❡♦r❡♠ ✹✳✾✳✻✿ ❈♦♠♣♦♥❡♥t✇✐s❡ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❙❡q✉❡♥❝❡s ❆ s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs zk ✐♥ C ❝♦♥✈❡r❣❡s t♦ ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r z ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❜♦t❤ t❤❡ r❡❛❧ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ zk ❝♦♥✈❡r❣❡ t♦ t❤❡ r❡❛❧ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ z r❡s♣❡❝t✐✈❡❧②❀ ✐✳❡✳✱ zk → z ⇐⇒ Re(zk ) → Re(z) ❛♥❞ Im(zk ) → Im(z) .

✹✳✶✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿

C

✐s♥✬t ❥✉st

R2

✸✷✸

❚❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts ♦❢ s❡q✉❡♥❝❡s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✇✐❧❧ ❛❧s♦ ❧♦♦❦ ❢❛♠✐❧✐❛r✿

❚❤❡♦r❡♠ ✹✳✾✳✼✿ ❙✉♠ ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s ■❢ s❡q✉❡♥❝❡s

zk , uk

❝♦♥✈❡r❣❡✱ t❤❡♥ s♦ ❞♦❡s

z k + uk ✱

❛♥❞ ✇❡ ❤❛✈❡✿

lim (zk + uk ) = lim zk + lim uk .

k→∞

k→∞

k→∞

❚❤❡♦r❡♠ ✹✳✾✳✽✿ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s ■❢ s❡q✉❡♥❝❡

zk

❝♦♥✈❡r❣❡s✱ t❤❡♥ s♦ ❞♦❡s

czk

❢♦r ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r

c✱

❛♥❞ ✇❡

❤❛✈❡✿

lim c zk = c · lim zk .

k→∞

k→∞

❲♦✉❧❞♥✬t ❝❛❧❝✉❧✉s ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❜❡ ❥✉st ❛ ❝♦♣② ♦❢ ❝❛❧❝✉❧✉s ♦♥ t❤❡ ♣❧❛♥❡❄ ◆♦✱ ♥♦t ✇✐t❤ t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t✳

✹✳✶✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿

❙♦✱ t❤❡ ✈❡❝t♦r ❛❧❣❡❜r❛ ♦❢ ❏✉st ❧✐❦❡ ✐♥ ♣❧❛♥❡

C✳

R2 ✱

R2

✐s ✐♥❝❧✉❞❡❞ ✐♥ t❤❡ ❝♦♠♣❧❡① ❛❧❣❡❜r❛ ♦❢

♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ r❡❛❧ ♥✉♠❜❡r

r

▲❡t✬s st❛rt ✇✐t❤ ❞❡❣r❡❡s✿

1

1

❜❡❝♦♠❡s

i✱

✇❤✐❧❡

c

R2

❚❤❡r❡ ✐s ♠♦r❡ t♦ t❤❡ ❧❛tt❡r✳

✇✐❧❧ ❛❧s♦ r♦t❛t❡ ❡❛❝❤ ✈❡❝t♦r✳

i

i s❡✈❡r❛❧ t✐♠❡s✳ i ❜❡❝♦♠❡s −1✱ ❡t❝✳✿

❛♥❞ ♠✉❧t✐♣❧② ✐t ❜②

✐s♥✬t ❥✉st

✇✐❧❧ str❡t❝❤✴s❤r✐♥❦ ❛❧❧ ✈❡❝t♦rs ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❝♦♠♣❧❡①

❍♦✇❡✈❡r✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r

❊①❛♠♣❧❡ ✹✳✶✵✳✶✿ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜②

C✳

C

▼✉❧t✐♣❧✐❝❛t✐♦♥ ❜②

i

r♦t❛t❡s t❤❡ ♥✉♠❜❡r ❜②

90

✹✳✶✵✳

▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿

C

✐s♥✬t ❥✉st

✸✷✹

R2

r♦t❛t✐♦♥ ❢r♦♠ ✵ ❞❡❣r❡❡s t♦ ✾✵ i · i = i2 = −1 r♦t❛t✐♦♥ ❢r♦♠ ✾✵ ❞❡❣r❡❡s t♦ ✶✽✵ −1 · i = −i r♦t❛t✐♦♥ ❢r♦♠ ✶✽✵ ❞❡❣r❡❡s t♦ ✷✼✵ 2 −i · i = −i = 1 r♦t❛t✐♦♥ ❢r♦♠ ✷✼✵ ❞❡❣r❡❡s t♦ ✸✻✵ ❛♥❞ s♦ ♦♥✳ 1·i=i

❊①❛♠♣❧❡ ✹✳✶✵✳✷✿ ❝♦♠♣❧❡① ♠✉❧t✐♣❧✐❝❛t✐♦♥

❆ ♠♦r❡ ❝♦♠♣❧❡① ❡①❛♠♣❧❡✿ u

= 1 + 2i

v

=2+i

uv = 2 + 4i + i + 2i2 = (2 − 2) + (4 + 1)i = 0 + 5i

❚❤❡ r♦t❛t✐♦♥ ♦❢ v ✐s ✈✐s✐❜❧❡✿

■♥ ❝♦♥tr❛st✱ ✇❡ ❝❛♥ s❡❡ t❤❡ r❡s✉❧t ♦❢ ♠✉❧t✐♣❧②✐♥❣ v ❜② w = 2✿ ♥♦ r♦t❛t✐♦♥✳ ❙♦✱ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rt ♦❢ c ✐s r❡s♣♦♥s✐❜❧❡ ❢♦r r♦t❛t✐♦♥✳ ❍♦✇ ❞♦❡s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛✛❡❝t t♦♣♦❧♦❣②❄

❚❤❡♦r❡♠ ✹✳✶✵✳✸✿ Pr♦❞✉❝t ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s ■❢ s❡q✉❡♥❝❡s

zk , uk

❝♦♥✈❡r❣❡✱ t❤❡♥ s♦ ❞♦❡s

z k · uk ✱

❛♥❞

lim (zk · uk ) = lim zk · lim uk .

k→∞

k→∞

k→∞

✹✳✶✵✳

▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿

C

✸✷✺

R2

✐s♥✬t ❥✉st

Pr♦♦❢✳ ❙✉♣♣♦s❡

zk = ak + bk i → a + bi ❛♥❞ uk = pk + qk i = p + qi .

❚❤❡♥✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❈♦♠♣♦♥❡♥t✇✐s❡

❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠

❛❜♦✈❡✱ ✇❡ ❤❛✈❡✿

ak → a, bk → b ❛♥❞ pk → p, qk → q .

❚❤❡♥✱ ❜② t❤❡ Pr♦❞✉❝t

✱ ✇❡ ❤❛✈❡✿

❘✉❧❡ ❢♦r ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡s

ak pk → ap, ak qk → aq, bk pk → bp, bk qk → bq .

❚❤❡♥✱ ❛s ✇❡ ❦♥♦✇✱ zk · uk = (ak pk − bk qk ) + (ak qk + bk pk )i → (ap − bq) + (aq + bq)i = (a + bi)(p + qi) ,

❜② t❤❡ ❙✉♠

✳

❘✉❧❡ ❢♦r ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡s

❚❤❡♦r❡♠ ✹✳✶✵✳✹✿ ◗✉♦t✐❡♥t ❘✉❧❡ ❢♦r ❈♦♠♣❧❡① ❙❡q✉❡♥❝❡s ■❢ s❡q✉❡♥❝❡s

z k , uk

❝♦♥✈❡r❣❡ ✭✇✐t❤

uk 6= 0✮✱

t❤❡♥ s♦ ❞♦❡s

zk /uk ✱

❛♥❞

zk limk→∞ zk = , k→∞ uk limk→∞ uk lim

♣r♦✈✐❞❡❞

lim uk 6= 0 .

k→∞

❏✉st ❧✐❦❡ r❡❛❧ ♥✉♠❜❡rs✦

❊①❡r❝✐s❡ ✹✳✶✵✳✺ Pr♦✈❡ t❤❡ ❧❛st t❤❡♦r❡♠✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ st❛♥❞❛r❞✱ ❈❛rt❡s✐❛♥✱ r❡♣r❡s❡♥t❛t✐♦♥✱ ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r x = a + bi ❝❛♥ ❜❡ ❞❡✜♥❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s✳

❲❡ ❥✉st ❝♦♥t✐♥✉❡ ♦✉r ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇✐t❤ ❛ ♥❡✇ ♦♥❡✿ a + bi ←→ (a, b) ←→ (θ, r)

❚❤❡ t✇♦ q✉❛♥t✐t✐❡s θ ❛♥❞ r ❜❡❝♦♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❉❡✜♥✐t✐♦♥ ✹✳✶✵✳✻✿ ♠♦❞✉❧✉s ❛♥❞ ❛r❣✉♠❡♥t ❙✉♣♣♦s❡ z ✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r✳ ✶✳ ❚❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❧♦❝❛t✐♦♥ ♦❢ z ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ t♦ t❤❡ ♦r✐❣✐♥ O

✹✳✶✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿

C

✐s♥✬t ❥✉st

✐s ❝❛❧❧❡❞ t❤❡ ♠♦❞✉❧✉s ♦❢

R2

z

✸✷✻

❞❡♥♦t❡❞ ❜②✿

||z|| ✷✳ ❚❤❡ ❛♥❣❧❡ ♦❢ t❤❡ ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t ✐s ❝❛❧❧❡❞ t❤❡ ❛r❣✉♠❡♥t ♦❢

z

z

❢r♦♠ t❤❡ ♦r✐❣✐♥

O

✇✐t❤ t❤❡

x✲❛①✐s

❞❡♥♦t❡❞ ❜②✿

Arg(z)

❆ s✐♠♣❧❡ ❡①❛♠✐♥❛t✐♦♥ t❡❧❧s ✉s ❤♦✇ t♦ tr❛♥s✐t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❝♦♦r❞✐♥❛t❡ s②st❡♠s✿

❚❤❡♦r❡♠ ✹✳✶✵✳✼✿ ❈♦♥✈❡rs✐♦♥ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡rs ❙✉♣♣♦s❡

x = a + bi

✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡✿

✶✳ ❚❤❡ ♠♦❞✉❧✉s ♦❢

z

✐s ❢♦✉♥❞ ❜②✿

||z|| = ✷✳ ❚❤❡ ❛r❣✉♠❡♥t ♦❢

z

√

a2 + b2

✐s ❢♦✉♥❞ ❜②✿

Arg(z) = arctan

❆♥② t✇♦ r❡❛❧ ♥✉♠❜❡rs t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r✿

r≥0

❛♥❞

0 ≤ θ < 2π

b a

❝❛♥ s❡r✈❡ ❛s t❤♦s❡✳ ■t ✐s ❝❛❧❧❡❞ t❤❡ ❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢

  z = r cos θ + i sin θ ❚❤❡ ❛❧❣❡❜r❛ t❛❦❡s ❛ ♥❡✇ ❢♦r♠ t♦♦✳ ❲❡ ❞♦♥✬t ♥❡❡❞ t❤❡ ♥❡✇ r❡♣r❡s❡♥t❛t✐♦♥ t♦ ❝♦♠♣✉t❡ ❛❞❞✐t✐♦♥ ❛♥❞ ♠✉❧t✐✲ ♣❧✐❝❛t✐♦♥ ❜② r❡❛❧ ♥✉♠❜❡rs✱ ❜✉t ✇❡ ♥❡❡❞ ✐t ❢♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❲❤❛t ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿

❈♦♥s✐❞❡r✿

  z1 = r1 cos ϕ1 + i sin ϕ1

❛♥❞

  z2 = r2 cos ϕ2 + i sin ϕ2 ?

    z1 z2 = r1 cos ϕ1 + i sin ϕ1 ) · r2 cos ϕ2 + i sin ϕ2     = r1 r2 cos ϕ1 + i sin ϕ1 · cos ϕ2 + i sin ϕ2

 = r1 r2 cos ϕ1 cos ϕ2 + i sin ϕ1 cos ϕ2 + cos ϕ1 sin ϕ2 + i2 sin ϕ1 sin ϕ2 .

❲❡ ✉t✐❧✐③❡ t❤❡ ❢♦❧❧♦✇✐♥❣ tr✐❣♦♥♦♠❡tr✐❝ ✐❞❡♥t✐t✐❡s ✭❱♦❧✉♠❡ ✶✮✿

cos a cos b − sin a sin b = cos(a + b)

❛♥❞

cos a sin b + sin a cos b = sin(a + b) .

✹✳✶✵✳

▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿

❚❤❡♥✱

C

✐s♥✬t ❥✉st

R2

✸✷✼

  z1 z2 = r1 r2 cos(ϕ1 + ϕ2 ) + i sin(ϕ1 + ϕ2 ) .

❊①❛♠♣❧❡ ✹✳✶✵✳✽✿ ❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥

❲❡ ❝❛♥ s❡❡ t❤❡ ❛❜♦✈❡ ❝♦♠♣✉t❛t✐♦♥ ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✿

❲❡ ❤❛✈❡ ♣r♦✈❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✹✳✶✵✳✾✿ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❈♦♠♣❧❡① ◆✉♠❜❡rs ❲❤❡♥ t✇♦ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ♠✉❧t✐♣❧✐❡❞✱ t❤❡✐r ♠♦❞✉❧✐ ❛r❡ ♠✉❧t✐♣❧✐❡❞ ❛♥❞ t❤❡ ❛r❣✉♠❡♥ts ❛r❡ ❛❞❞❡❞✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

    r1 cos ϕ1 + i sin ϕ1 ) · r2 cos ϕ2 + i sin ϕ2   = r1 r2 cos(ϕ1 + ϕ2 ) + i sin(ϕ1 + ϕ2 ) ❊①❡r❝✐s❡ ✹✳✶✵✳✶✵

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❈♦♥s✐❞❡r t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ♠♦❞✉❧✐ ❛♥❞ t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ❛r❣✉♠❡♥ts ♦❢ t❤❡ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡ ❛♥❞ ♣r♦✈❡ t❤❛t t❤❡② ❝♦♥✈❡r❣❡✳ ❊①❡r❝✐s❡ ✹✳✶✵✳✶✶

✭❛✮ ❘❡♣r❡s❡♥t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣❧❡① ♥✉♠❜❡r ✐♥ t❤❡ st❛♥❞❛r❞ ❢♦r♠✿

(2 + 3i)(−1 + 2i)✳

■♥❞✐❝❛t❡ t❤❡ r❡❛❧

❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts✳ ✭❜✮ ❋✐♥❞ ✐ts ♠♦❞✉❧✉s ❛♥❞ ❛r❣✉♠❡♥t✳ ❊①❡r❝✐s❡ ✹✳✶✵✳✶✷

❙✐♠♣❧✐❢②

(1 + i)2 ✳

❊①❡r❝✐s❡ ✹✳✶✵✳✶✸

✭❛✮ ❋✐♥❞ t❤❡ r♦♦ts ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧

x2 + 2x + 2✳

✭❜✮ ❋✐♥❞ ✐ts

x✲✐♥t❡r❝❡♣ts✳

✭❝✮ ❋✐♥❞ ✐ts ❢❛❝t♦rs✳

❊①❡r❝✐s❡ ✹✳✶✵✳✶✹

❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ t❤❡ r♦♦ts ♦❢ t❤❡s❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s❄

✹✳✶✶✳

❉✐s❝r❡t❡ ❢♦r♠s

✸✷✽

✹✳✶✶✳ ❉✐s❝r❡t❡ ❢♦r♠s

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ♦✉t❧✐♥❡ ❛♥ ❛❧t❡r♥❛t✐✈❡ ✈✐❡✇ ♦♥ ❞✐s❝r❡t❡ ❝❛❧❝✉❧✉s✿ ❞✐✛❡r❡♥❝❡s✱ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts✱ s✉♠s✱ ❛♥❞ ❘✐❡♠❛♥♥ s✉♠s✳ ❍♦✇❡✈❡r✱ ✇❡ ✇✐❧❧ ❞❡❛❧ ✇✐t❤ s♦♠❡t❤✐♥❣ ❡✈❡♥ ♠♦r❡ ❢✉♥❞❛♠❡♥t❛❧ t❤❛♥ t❤♦s❡ ❢♦✉r✿ ❲❡ ♠♦✈❡ ❜❡②♦♥❞ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ ♥♦❞❡s ♦❢ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥s✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ y = f (x) ❛♥❞ ✇❡ ❛r❡ t♦ st✉❞② ✐ts ❜❡❤❛✈✐♦r ❛r♦✉♥❞ ❛ ♣♦✐♥t x = a✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ♦❢ f ❛t a ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∆f r✐s❡ ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡ = t❤❡ s❧♦♣❡ ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡ t❤r♦✉❣❤ (a, f (a)) = ∆x x=a r✉♥ ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡

❲❡ ❝❛♥ ❛❧✇❛②s s❡❡ ∆x✱ ∆y ♦♥ t❤❡ ❣r❛♣❤✿

❚❤✉s✱ ✇❡ ❤❛✈❡✿ • ∆x ✐s t❤❡ r✉♥ ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡✳

• ∆y ✐s t❤❡ r✐s❡ ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡✳

❚❤❡r❡ ✐s ❛ ❢✉♥❝t✐♦♥❛❧ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡♠✿ • y ❞❡♣❡♥❞s ♦♥ x ✈✐❛ • ∆y ❞❡♣❡♥❞s ♦♥ ∆x

y = f (x) .

❛♥❞ x ✈✐❛ ∆y =

∆f · ∆x . ∆x

❚❤❡ ❧❛tt❡r✱ tr✐✈✐❛❧✱ ❡q✉❛t✐♦♥ r❡❢❡rs t♦ ❛ s♣❡❝✐✜❝ ❧♦❝❛t✐♦♥✱ x = a ❛♥❞ y = f (a)✱ ♦♥ t❤❡ xy ✲♣❧❛♥❡✱ ❛♥❞ ✐t ✐s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♥❡✇ ✈❛r✐❛❜❧❡s ❛s t❤❡ ♦❧❞ ♦♥❡s ❤❛✈❡ ❜❡❡♥ s♣❡❝✐✜❡❞✳

✹✳✶✶✳

❉✐s❝r❡t❡ ❢♦r♠s

✸✷✾

❲❡ ❝❛♥✱ ❢✉rt❤❡r♠♦r❡✱ ♠❛❦❡ t❤❡s❡ ✈❛r✐❛❜❧❡s ❡①♣❧✐❝✐t✿

❇❡❧♦✇✱ ✇❡ ❛❞♦♣t ❛ s✐♠♣❧❡r✱ ✐❢ ❧❡ss ❡①♣❧✐❝✐t✱ ❛♣♣r♦❛❝❤✳ ❊①❛♠♣❧❡ ✹✳✶✶✳✶✿ ❜r♦❦❡♥ s♣❡❡❞♦♠❡t❡r

❘❡❝❛❧❧ ✇❤❛t ✇❡ st❛rt❡❞ ✇✐t❤ ✐♥ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣✳ ❙✉♣♣♦s❡ t❤❡ s♣❡❡❞♦♠❡t❡r ✐s ❜r♦❦❡♥ ❛♥❞ ✐♥ ♦r❞❡r t♦ ❡st✐♠❛t❡ ❤♦✇ ❢❛st ✇❡ ❛r❡ ❞r✐✈✐♥❣✱ ✇❡ ❧♦♦❦ ❛t t❤❡ ♦❞♦♠❡t❡r ❡✈❡r② ❤♦✉r✿

❚❤❛t✬s ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠✳ ❚♦ ✜♥❞ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❢♦r ❡✈❡r② ❤♦✉r✱ ✇❡ ❥✉st ❧♦♦❦ ❛t t❤❡ ❞✐✛❡r❡♥❝❡s✿

❚❤❛t✬s ❛ ❞✐s❝r❡t❡ 1✲❢♦r♠✳ ❆❧t❡r♥❛t✐✈❡❧②✱ t❤❡ ♦❞♦♠❡t❡r ✐s ❜r♦❦❡♥ ❛♥❞ ✇❡ ❧♦♦❦ ❛t t❤❡ s♣❡❡❞♦♠❡t❡r t♦ s❛♠♣❧❡ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡♥✱ ✈✐❛ t❤❡ ❘✐❡♠❛♥♥ s✉♠s✱ ✜♥❞ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✳ ▲❡t✬s st❛rt ♦✈❡r✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ s♦♠❡ ✐♥t❡r✈❛❧ [a, b] ♦♥ t❤❡ x✲❛①✐s✳

❚❤✐s t✐♠❡✱ ✇❡ ✇♦♥✬t ❛❞❞ s❡❝♦♥❞❛r② ♥♦❞❡s ❜✉t✱ ✐♥st❡❛❞✱ ❝♦♥s✐❞❡r ❛ ❝❡❧❧ ❚❤❡r❡ ❛r❡ t✇♦ t②♣❡s ♦❢ ♣✐❡❝❡s ✐♥ t❤❡ ✐♥t❡r✈❛❧✿ • t❤❡

♥♦❞❡s ✿

• t❤❡

❡❞❣❡s ✿

x = xk , k = 0, 1, ..., n ck = [xk−1 , xk ], k = 1, ..., n

❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡r✈❛❧✳

✹✳✶✶✳

❉✐s❝r❡t❡ ❢♦r♠s

✸✸✵

❚❤❡ ✐♥❝r❡♠❡♥ts ♦❢ x ❛r❡ st✐❧❧ ∆xk = xk − xk−1 ✳

❲❡ ✐♥tr♦❞✉❝❡ t❤❡s❡ ♥❛♠❡s✿

❉❡✜♥✐t✐♦♥ ✹✳✶✶✳✷✿ ❝❡❧❧s

• ❚❤❡ ♥♦❞❡s ❛r❡ ❝❛❧❧❡❞ 0✲❝❡❧❧s✳ • ❚❤❡ ❡❞❣❡s ❛r❡ ❝❛❧❧❡❞ 1✲❝❡❧❧s✳

❊①❛♠♣❧❡ ✹✳✶✶✳✸✿ ❝❡❧❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ❙♣❡❝✐✜❝ r❡♣r❡s❡♥t❛t✐♦♥s ❝❛♥ ❛❧s♦ ❜❡ ♣r♦✈✐❞❡❞ ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✱ ❝❤♦♦s✐♥❣✱ ❢♦r ❡①❛♠♣❧❡✱ ∆x = 1✿

❨♦✉ ❝❛♥ s❡❡ ❤♦✇ ❡✈❡r② ♦t❤❡r ❝❡❧❧ ✐s ❛ sq✉❛r❡ ❛♥❞ ❡✈❡r② ♦t❤❡r ✐s str❡t❝❤❡❞ ❤♦r✐③♦♥t❛❧❧② t♦ ❡♠♣❤❛s✐③❡ t❤❡ ❞✐✛❡r❡♥t ♥❛t✉r❡ ♦❢ t❤❡s❡ ❝❡❧❧s✿ ♥♦❞❡s ✈s✳ ❡❞❣❡s✳ ■♥ t❤❡ ♠♦t✐♦♥ ✐♥t❡r♣r❡t❛t✐♦♥✱ t❤❡r❡ ✐s ❛ ♥✉♠❜❡r ✭t❤❡ ❧♦❝❛t✐♦♥✮ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❡❛❝❤ ♥♦❞❡ ❛♥❞ ❛ ♥✉♠❜❡r ✭t❤❡ ❞✐s♣❧❛❝❡♠❡♥t✮ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❡❛❝❤ ❡❞❣❡✳

❉❡✜♥✐t✐♦♥ ✹✳✶✶✳✹✿ ❞✐s❝r❡t❡ ❢♦r♠ ❋♦r ❛ ❣✐✈❡♥ ♣❛rt✐t✐♦♥ ✭♦❢ ❛♥ ✐♥t❡r✈❛❧ ♦r t❤❡ ✇❤♦❧❡ r❡❛❧ ❧✐♥❡✮✱ ✇❡ ❞❡✜♥❡ t❤❡ ❢♦❧❧♦✇✲ ✐♥❣✿ • ❆ ❞✐s❝r❡t❡ ❢♦r♠ ♦❢ ❞❡❣r❡❡ 0 ✐s ❛ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ✇✐t❤ 0✲❝❡❧❧s ✭♥♦❞❡s✮ ❛s ✐♥♣✉ts✳ • ❆ ❞✐s❝r❡t❡ ❢♦r♠ ♦❢ ❞❡❣r❡❡ 1 ✐s ❛ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ✇✐t❤ 1✲❝❡❧❧s ✭❡❞❣❡s✮ ❛s ✐♥♣✉ts✳ ❲❡ ✉s❡ ❛rr♦✇s t♦ ♣✐❝t✉r❡ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛s ❝♦rr❡s♣♦♥❞❡♥❝❡s✿

❍❡r❡ ✇❡ ❤❛✈❡ t✇♦✿ • ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠ • ❛ ❞✐s❝r❡t❡ 1✲❢♦r♠

f : 0 7→ 2, 1 7→ 4, 2 7→ 3, ... s : [0, 1] 7→ 3, [1, 2] 7→ .5, [2, 3] 7→ 1, ...

✹✳✶✶✳

❉✐s❝r❡t❡ ❢♦r♠s

✸✸✶

❆ ♠♦r❡ ❝♦♠♣❛❝t ✇❛② t♦ ✈✐s✉❛❧✐③❡ ✐s t❤✐s✿

❲❡ ❝❛♥ ❛❧s♦

❧✐st t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ t✇♦ ❢✉♥❝t✐♦♥s✿

•

❛ ❞✐s❝r❡t❡

•

❛ ❞✐s❝r❡t❡

0✲❢♦r♠ f ✿ f (0) = 2, f (1) = 4, f (2) = 3, ... 1✲❢♦r♠ s✿













s [0, 1] = 3, s [1, 2] = .5, s [2, 3] = 1, ...

❊①❛♠♣❧❡ ✹✳✶✶✳✺✿ ❢♦r♠s ✇✐t❤ s♣r❡❛❞s❤❡❡t

❉✐s❝r❡t❡ ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② t❛❜❧❡s ✭s♣r❡❛❞s❤❡❡ts✮✿

❚❤❡ ♠♦st ❝♦♠♠♦♥ ✇❛② t♦ ✈✐s✉❛❧✐③❡ ❛ ❢✉♥❝t✐♦♥ ✐s ✇✐t❤ ✐ts ✇✐t❤

y = f (x)✿

•

❋♦r ❛ ❞✐s❝r❡t❡

•

❋♦r ❛ ❞✐s❝r❡t❡

(x, y)✱

❛

❣r❛♣❤✱ ✇❤✐❝❤ ❝♦♥s✐sts ♦❢ ♣♦✐♥ts ♦♥ t❤❡ xy✲♣❧❛♥❡

0✲❢♦r♠✱ x ✐s ❛ ♥♦❞❡✱ ❛ ♥✉♠❜❡r✱ ❛♥❞ y = f (x) ✐s ❛❧s♦ ❛ ♥✉♠❜❡r✳ ❚♦❣❡t❤❡r✱ ♣♦✐♥t ♦♥ t❤❡ xy ✲♣❧❛♥❡ ✭✇✐t❤ t❤❡ x✲❛①✐s s♣❧✐t ✐♥t♦ ❝❡❧❧s ❛s s❤♦✇♥ ❛❜♦✈❡✮✳ 1✲❢♦r♠✱ [A, B]

✐s ❛♥ ✐♥t❡r✈❛❧ ✐♥ t❤❡

t❤❡② ♣r♦❞✉❝❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♣♦✐♥ts ♦♥ t❤❡

x✲❛①✐s✱

xy ✲♣❧❛♥❡

t❤❡② ♣r♦❞✉❝❡

y = g([A, B]) ✐s ❛ ♥✉♠❜❡r✳ ❚♦❣❡t❤❡r✱ (x, y) ❢♦r ❡✈❡r② x ✐♥ [A, B]✳ ❚❤❡ r❡s✉❧t ✐s

❛♥❞

s✉❝❤ ❛s

❛ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥t✳ ❊✈❡♥ t❤♦✉❣❤ t❤❡s❡ ❢✉♥❝t✐♦♥s ♠❛② ❝♦♥s✐st ♦❢ ✉♥r❡❧❛t❡❞ ♣✐❡❝❡s✱ ✐t ✐s ♣♦ss✐❜❧❡ t❤❛t ✇❡ ❝❛♥ s❡❡ ❛

❝✉r✈❡ ✐❢ ✇❡ ③♦♦♠ ♦✉t✿

❝♦♥t✐♥✉♦✉s

✹✳✶✶✳

❉✐s❝r❡t❡ ❢♦r♠s

✸✸✷

❊①❛♠♣❧❡ ✹✳✶✶✳✻✿ ❣r❛♣❤s ♦❢ ❢♦r♠s ✇✐t❤ s♣r❡❛❞s❤❡❡t ❚♦ ✉♥❞❡rs❝♦r❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦✱ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠ ✐s s❤♦✇♥ ✇✐t❤ ❞♦ts ❛♥❞ t❤❛t ♦❢ ❛ ❞✐s❝r❡t❡ 1✲❢♦r♠ ✇✐t❤ ✈❡rt✐❝❛❧ ❜❛rs✿

◆❡①t✱ ✇❡ ❞✐s❝✉ss s♦♠❡ ♦❢ t❤❡ ❝❛❧❝✉❧✉s ✐ss✉❡s✳

❊①❛♠♣❧❡ ✹✳✶✶✳✼✿ ❞✐✛❡r❡♥❝❡ ▲❡t✬s ❝♦♥s✐❞❡r ❛♥ ❡①❛♠♣❧❡ ♦❢ ♠♦t✐♦♥✳ ❙✉♣♣♦s❡ ❛ 0✲❢♦r♠ p ❣✐✈❡s t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♣❡rs♦♥ ❛♥❞ s✉♣♣♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ❆t t✐♠❡ n ❤♦✉rs✱ ✇❡ ❛r❡ ❛t t❤❡ 5✲♠✐❧❡ ♠❛r❦✿ p(n) = 5✳ • ❆t t✐♠❡ n + 1 ❤♦✉rs✱ ✇❡ ❛r❡ ❛t t❤❡ 7✲♠✐❧❡ ♠❛r❦✿ p(n + 1) = 7✳ ❲❡ ❞♦♥✬t ❦♥♦✇ ✇❤❛t ❡①❛❝t❧② ❤❛s ❤❛♣♣❡♥❡❞ ❞✉r✐♥❣ t❤✐s ❤♦✉r ❜✉t t❤❡ s✐♠♣❧❡st ❛ss✉♠♣t✐♦♥ ✇♦✉❧❞ ❜❡ t❤❛t ✇❡ ❤❛✈❡ ❜❡❡♥ ✇❛❧❦✐♥❣ ❛t ❛ ❝♦♥st❛♥t s♣❡❡❞ ♦❢ 2 ♠✐❧❡s ♣❡r ❤♦✉r✳

◆♦✇✱ ✐♥st❡❛❞ ♦❢ ♦✉r ✈❡❧♦❝✐t② ❢✉♥❝t✐♦♥ v ❛ss✐❣♥✐♥❣ t❤✐s ✈❛❧✉❡ t♦ ❡❛❝❤ ✐♥st❛♥t ♦❢ t✐♠❡ ❞✉r✐♥❣ t❤✐s ♣❡r✐♦❞✱ ✐t ✐s ❛ss✐❣♥❡❞ t♦ t❤❡ ✇❤♦❧❡ ✐♥t❡r✈❛❧✿

♦r ❜❡tt❡r✿

v

= 2,

[n,n+1]

  v [n, n + 1] = 2 .

❚❤✐s ✇❛②✱ t❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ✈❡❧♦❝✐t② ❢✉♥❝t✐♦♥ ❛r❡ t❤❡ ❡❞❣❡s ❛♥❞ t❤❡ r❡s✉❧t✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❛ ❞✐s❝r❡t❡ 1✲❢♦r♠✦

✹✳✶✶✳ ❉✐s❝r❡t❡ ❢♦r♠s

✸✸✸

❚❤❡ ❢✉♥❝t✐♦♥s✱ ✇❤❡♥ ❞❡✜♥❡❞ ♦♥ t❤❡ ♥♦❞❡s✱ ❝❤❛♥❣❡ ❛❜r✉♣t❧② ❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ t❤❡ ❝❤❛♥❣❡ ♦✈❡r ❡✈❡r② ✐♥t❡r✈❛❧ [A, B] ✐s s✐♠♣❧② t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ✈❛❧✉❡s ❛t t❤❡ ♥♦❞❡s✱ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t✿ f (B) − f (A) .

❚❤❡ ♦✉t♣✉t ♦❢ t❤✐s s✐♠♣❧❡ ❝♦♠♣✉t❛t✐♦♥ ✐s t❤❡♥ ❛ss✐❣♥❡❞ t♦ t❤❡ ✐♥t❡r✈❛❧ [A, B]✿ [A, B] 7→ f (B) − f (A)

❏✉st ❛s ❜❡❢♦r❡✱ t❤❡ ❞✐✛❡r❡♥❝❡ st❛♥❞s ❢♦r t❤❡ ❝❤❛♥❣❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✹✳✶✶✳✽✿ ❞✐✛❡r❡♥❝❡ ♦❢ ❞✐s❝r❡t❡ 0✲❢♦r♠ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠ f ✐s ❛ ❞✐s❝r❡t❡ 1✲❢♦r♠ ❣✐✈❡♥ ❜② ✐ts ✈❛❧✉❡s ❛t ❡❛❝❤ ❡❞❣❡✿ ∆f (ck ) = f (xk ) − f (xk−1 )

❚❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❛ 0✲❢♦r♠ ❛♥❞ ✐ts ❞✐✛❡r❡♥❝❡ ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✿

❊①❛♠♣❧❡ ✹✳✶✶✳✾✿ ❞✐✛❡r❡♥❝❡ ✇✐t❤ s♣r❡❛❞s❤❡❡t ❚❤✐s ✐s ❤♦✇ ❛ s♣r❡❛❞s❤❡❡t ❝♦♠♣✉t❡s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② t❤❡ ❞❛t❛ ✐♥ t❤❡ ✜rst ❝♦❧✉♠♥✿

❊①❛♠♣❧❡ ✹✳✶✶✳✶✵✿ ❝♦♠♣✉t✐♥❣ ❞✐✛❡r❡♥❝❡s ❲❤❡♥ t❤❡ ❞✐s❝r❡t❡ 0✲❢♦r♠s ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② ❢♦r♠✉❧❛s✱ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ❛r❡ str❛✐❣❤t❢♦r✇❛r❞ ✭h = 1✮

✹✳✶✶✳

✸✸✹

❉✐s❝r❡t❡ ❢♦r♠s

✇✐t❤ ❛ ❝❤❛♥❝❡ ♦❢ s✐♠♣❧✐✜❝❛t✐♦♥✿ (1) f (n) = 3n2 + 1 =⇒ ∆f (cn ) = (3n2 + 1) − (3(n − 1)2 + 1) = 6n − 3 1 1 1 1 =⇒ ∆g (cn ) = − =− ❢♦r n 6= 0, 1 (2) g(n) = n n n−1 n(n − 1) (3) p(n) = 2n

=⇒ ∆p (cn ) = 2n − 2n−1 = 2n−1

❉❡✜♥✐t✐♦♥ ✹✳✶✶✳✶✶✿ s✉♠ ♦❢ ❞✐s❝r❡t❡ 1✲❢♦r♠ ❚❤❡ s✉♠ ♦❢ ❛ ❞✐s❝r❡t❡ 1✲❢♦r♠ g ✐s ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠ ❣✐✈❡♥ ❜② ✐ts ✈❛❧✉❡ ❛t ❡❛❝❤ ♥♦❞❡ xk , 1 ≤ k ≤ n, ♦❢ ❛ ♣❛rt✐t✐♦♥ ♦❢ [a, b] ❜②✿ X

g = g(c1 ) + g(c2 ) + ... + g(ck ) ,

[a,xk ]

✇❤❡r❡ c1 , c2 , ..., cn ❛r❡ t❤❡ ❡❞❣❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳ ❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ r❡❧❛t✐♦♥ ✐s ❜❡t✇❡❡♥ t❤❡ ❞✐✛❡r❡♥❝❡s ❛♥❞ s✉♠s✳ ❚❤❡ r❡s✉❧t ✐s ✈❡r② s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✶❀ t❤❡ t✇♦ ♦♣❡r❛t✐♦♥s ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r ✐♥ ❡✐t❤❡r ♦r❞❡r✿

❋✐rst✱ ✇❡ ❤❛✈❡ ❛ 0✲❢♦r♠ ❛♥❞ ❛ 1✲❢♦r♠✿ • ✐❢ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ♥♦❞❡s xk , k = 0, 1, 2, ..., n✱ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ t❤❡♥

• t❤❡ ❞✐✛❡r❡♥❝❡ g ♦❢ f ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❜②✿

g(ck ) = f (xk ) − f (xk−1 ) .

❚❤❡♦r❡♠ ✹✳✶✶✳✶✷✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ■ ❙✉♣♣♦s❡

f

✐s ❛ ❞✐s❝r❡t❡

0✲❢♦r♠✳ X [a,x]

❚❤❡♥✱ ❢♦r ❡❛❝❤ ♥♦❞❡

(∆f ) = f (x) − f (a) .

❙❡❝♦♥❞✱ ✇❡ ❤❛✈❡ ❛ 1✲❢♦r♠ ❛♥❞ ❛ 0✲❢♦r♠✿ • ✐❢ g ✐s ❞❡✜♥❡❞ ❛t t❤❡ ❡❞❣❡s ck , k = 1, 2, ..., n✱ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ t❤❡♥

• t❤❡ s✉♠ f ♦❢ g ✐s ❞❡✜♥❡❞ r❡❝✉rs✐✈❡❧② ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ❜②✿ f (xk ) = f (xk−1 ) + g(ck ) .

x

♦❢ t❤❡ ♣❛rt✐t✐♦♥✱ ✇❡ ❤❛✈❡✿

✹✳✶✶✳

❉✐s❝r❡t❡ ❢♦r♠s

✸✸✺ ❚❤❡♦r❡♠ ✹✳✶✶✳✶✸✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❉✐s❝r❡t❡ ❈❛❧❝✉❧✉s ■■ ❙✉♣♣♦s❡

g

✐s ❛ ❞✐s❝r❡t❡

1✲❢♦r♠✳

❚❤❡♥✱ ✇❡ ❤❛✈❡✿



∆

X [a,x]



g = g .

❏✉st ❛s ✐♥ ❈❤❛♣t❡r ✶✱ ✇❡ ❝❛rr② ♦✉t t❤❡ ❝♦♠♣✉t❛t✐♦♥s ✇✐t❤ ❛ s♣r❡❛❞s❤❡❡t✿

◆❡①t✱ ❝♦♠♣♦s✐t✐♦♥s✳ ◆❡①t✱ t❤❡r❡ ❛r❡ ♥♦ ❝♦♠♣♦s✐t✐♦♥s ♦❢ ❢♦r♠s✦ ❋♦r ❡①❛♠♣❧❡✱ t❤❡r❡ ✐s ♥♦ ✇❛② t♦ ❡①❡❝✉t❡ t❤❡s❡ ❝♦♥s❡❝✉t✐✈❡❧②✿ • 0✲❝❡❧❧ 7→ ♥✉♠❜❡r✱ ❢♦❧❧♦✇❡❞ ❜② • 0✲❝❡❧❧ 7→ ♥✉♠❜❡r

❚♦ ❜❡ ❛❜❧❡ t♦ ❢♦r♠ ❛ ❝♦♠♣♦s✐t✐♦♥✱ ♦♥❡ ♦❢ t❤❡s❡ ❤❛s t♦ ♠❛♣ ❝❡❧❧s t♦ ❝❡❧❧s✿ • 0✲❝❡❧❧ 7→ 0✲❝❡❧❧ 7→ ♥✉♠❜❡r

• 1✲❝❡❧❧ 7→ 1✲❝❡❧❧ 7→ ♥✉♠❜❡r

❲❡ ❝r❡❛t❡ ❛ ❝♦♠♣♦s✐t✐♦♥ q ◦ p ♦❢ ❛ 0✲ ♦r 1✲❢♦r♠ q ✇✐t❤ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥ ♦r ❢♦r♠ p ♦♥❧② ✇❤❡♥ t❤❡ ✈❛❧✉❡s ♦❢ p ❛r❡ 0✲ ❛♥❞ 1✲❝❡❧❧s r❡s♣❡❝t✐✈❡❧②✳ ❚♦ ❞❡✜♥❡ s✉❝❤ ❛ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥✱ ✇❡ ✇✐❧❧ r❡q✉✐r❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ p ❛ss✐❣♥s ❛ k ✲ ♦r (k − 1)✲❝❡❧❧ t♦ ❡❛❝❤ k ✲❝❡❧❧✿

✹✳✶✶✳

❉✐s❝r❡t❡ ❢♦r♠s

✸✸✻

❚❤❡r❡ ✐s ❛❧s♦ ❛ s♣❡❝✐❛❧ r❡q✉✐r❡♠❡♥t✿

❉❡✜♥✐t✐♦♥ ✹✳✶✶✳✶✹✿ ❝❡❧❧ ❢✉♥❝t✐♦♥ ❆ ❝❡❧❧ ❢✉♥❝t✐♦♥ y = p(x) ✐s ❛ ❢✉♥❝t✐♦♥ t❤❛t ❛ss✐❣♥s • •

❛ ♥♦❞❡ t♦ ❡❛❝❤ ♥♦❞❡✱ ❛♥❞ ❛♥ ❡❞❣❡ ♦r ❛ ♥♦❞❡ t♦ ❡❛❝❤ ❡❞❣❡✱

✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ ❡❛❝❤ ❡❞❣❡ r❡♠❛✐♥ ❡♥❞✲♣♦✐♥ts✿

   p [u, v] = p(u), p(v) ❚❤❡ r❡q✉✐r❡♠❡♥t ❣✉❛r❛♥t❡❡s ✏❝♦♥t✐♥✉✐t②✑✿

❇❡❝❛✉s❡ ♦❢ t❤❡ ♣r♦♣❡rt②✱ t❤❡ ✈❛❧✉❡s ♦❢ ❛ ❝❡❧❧ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❡❞❣❡s ❝❛♥ ❜❡ r❡❝♦♥str✉❝t❡❞ ❢r♦♠ ✐ts ✈❛❧✉❡s ♦♥ t❤❡ ♥♦❞❡s✳ ❚❤❡ ❢♦r♠❡r ✐s t❤❡♥ ❛♥❛❧♦❣♦✉s t♦ t❤❡ ❋♦r ❝♦♥✈❡♥✐❡♥❝❡✱ ✇❡ ❛ss✉♠❡ t❤❛t

∆

❞✐✛❡r❡♥❝❡

♦❢ t❤❡ ❝❡❧❧ ❢✉♥❝t✐♦♥✳

✐s ③❡r♦ ✇❤❡♥ ❝♦♠♣✉t❡❞ ♦✈❡r ❛♥② ♥♦❞❡

x✳

❇❡❧♦✇ ✐s ❛♥ ❛♥❛❧♦❣ ♦❢ t❤❡ ❝❤❛✐♥ r✉❧❡✿

❚❤❡♦r❡♠ ✹✳✶✶✳✶✺✿ ❈❤❛✐♥ ❘✉❧❡ ❚❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❧❛tt❡r ✇✐t❤ t❤❡ ❢♦r♠❡r❀ ✐✳❡✳✱ ❢♦r ❛♥② ❝❡❧❧ ❢✉♥❝t✐♦♥ x = p(t) ❢r♦♠ [a, b] t♦ [c, d] ❛♥❞ ❛♥② 0✲❢♦r♠ y = g(x) ♦♥ [c, d]✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐✛❡r❡♥❝❡s s❛t✐s❢②✿ ∆(g ◦ p) = ∆g ◦ p . ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ ❢♦r ❡❛❝❤ ❡❞❣❡

s✿ ∆(g ◦ p)(s) = ∆g (p(s)) .

❏✉st ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛ tr❛❞✐t✐♦♥❛❧ tr❡❛t♠❡♥t✱ ✇❡ ❛s❦✿ ■❢ ✇❡ s❛♠♣❧❡ ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ ❛♥ ✐♥t❡r✈❛❧✱ ✇❤❛t ❤❛♣♣❡♥s ✐❢ ✇❡ r❡✜♥❡ t❤❡ ♣❛rt✐t✐♦♥s❄ ❋✐rst✱ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡

❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t

❛s

∆x → 0❄

■t ❝♦♥✈❡r❣❡s t♦ t❤❡ ❞❡r✐✈❛t✐✈❡✿

✹✳✶✷✳ ❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s

✸✸✼

❇✉t ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❞✐✛❡r❡♥❝❡ ❛s ∆x → 0❄ ❆ ♥❡✇ ❝♦♥❝❡♣t ❡♠❡r❣❡s✿

■t ✐s ❛ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♦❢ ❞❡❣r❡❡ 1 ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ❙❡❝♦♥❞✱ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❛s ∆x → 0❄ ■t ❝♦♥✈❡r❣❡s t♦ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✿

❇✉t ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ s✉♠ ❛s ∆x → 0❄ ❚❤❡r❡ ✐s ♥♦ ❝♦♥✈❡r❣❡♥❝❡✦ ❯♥❧❡ss t❤❡ ❢✉♥❝t✐♦♥ ❜❡✐♥❣ s❛♠♣❧❡❞ ✐s ❛ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♦❢ ❞❡❣r❡❡ 1✳ ◆♦ ♥❡❡❞ ❢♦r ❛ ♥❡✇ ❝♦♥❝❡♣t✳

✹✳✶✷✳ ❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ♦✉t❧✐♥❡ ❛♥ ❛❧t❡r♥❛t✐✈❡ ✈✐❡✇ ♦♥ ❝❛❧❝✉❧✉s✿ ❞❡r✐✈❛t✐✈❡ ❛♥❞ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s✳ ❍♦✇❡✈❡r✱ ✇❡ ✇✐❧❧ ❞❡❛❧ ✇✐t❤ s♦♠❡t❤✐♥❣ ❡✈❡♥ ♠♦r❡ ❢✉♥❞❛♠❡♥t❛❧ t❤❛♥ t❤♦s❡ t✇♦✿ ❲❡ ♠♦✈❡ ❜❡②♦♥❞ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ ✐♥t❡r✈❛❧s✳ ◗✉❡st✐♦♥✿ ■s t❤❡ ❞❡r✐✈❛t✐✈❡

dy ❛ ❢r❛❝t✐♦♥❄ dx

❚❤❡ ❛♥s✇❡r t❤❛t ❢♦❧❧♦✇❡❞ t❤❡ ❞❡✜♥✐t✐♦♥ ✇❛s ❛♥ ❡♠♣❤❛t✐❝ ◆♦✦ ❆ ♠♦r❡ ❛❞✈❛♥❝❡❞ ❛♥s✇❡r ✇❡ ❣✐✈❡ ❤❡r❡ ✐s✿ ❨❡s✱ ❤❡r❡✬s ❤♦✇ ❛♥❞ ✇❤②✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥ y = f (x) ❛♥❞ ✇❡ ❛r❡ t♦ st✉❞② ✐ts ❜❡❤❛✈✐♦r ❛r♦✉♥❞ ❛ ♣♦✐♥t x = a✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡

✹✳✶✷✳ ❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s

✸✸✽

❛t a ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ r✐s❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ dy = t❤❡ s❧♦♣❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ t❤r♦✉❣❤ (a, f (a)) = dx x=a r✉♥ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡

❚❤✐s ✐s ❛ ❢r❛❝t✐♦♥ ❛❢t❡r ❛❧❧✦ ❊①❛♠♣❧❡ ✹✳✶✷✳✶✿

dx − dy ✲♣❧❛♥❡

❙♣❡❝✐✜❝❛❧❧②✱ s✉♣♣♦s❡ f (x) = x2 + 2x✳ ❆t a = 0✱ ✇❡ ❤❛✈❡ f (0) = 0✱ s♦ ♦✉r ✐♥t❡r❡st ✐s t❤❡ ♣♦✐♥t (0, 0)✳ ❚❤❡♥✱ dy = 2. = 2x + 2 dx x=0 x=0

■❢ t❤✐s ✐s ❛ ❢r❛❝t✐♦♥✱ ✇❤❛t ✇♦✉❧❞ ❜❡ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤✐s✿

dy = 2 · dx ?

■t ✐s t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✇r✐tt❡♥ ✇✐t❤ r❡s♣❡❝t t♦ dy ❛♥❞ dx✳ ❚❤✉s✱ t❤❡ ❡q✉❛t✐♦♥

dy = f ′ (a) · dx

r❡❢❡rs t♦ ❛ s♣❡❝✐✜❝ ❧♦❝❛t✐♦♥✱ x = a ❛♥❞ y = f (a)✱ ♦♥ t❤❡ xy ✲♣❧❛♥❡ ❛♥❞ ✐t ✐s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♥❡✇ ✈❛r✐❛❜❧❡s ❛s t❤❡ ♦❧❞ ♦♥❡s ❤❛✈❡ ❜❡❡♥ s♣❡❝✐✜❡❞✳ ❲❡ ❝❛♥ ❛❧✇❛②s s❡❡ dx✱ dy ♦♥ t❤❡ ❣r❛♣❤✿

❚❤✉s✱ ✇❡ ❤❛✈❡✿ • dx ✐s t❤❡ r✉♥ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡✳ • dy ✐s t❤❡ r✐s❡ ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡✳

❚❤❡② ❛r❡ ❝❛❧❧❡❞ t❤❡ ❞✐✛❡r❡♥t✐❛❧s ♦❢ x ❛♥❞ y r❡s♣❡❝t✐✈❡❧②✳

✹✳✶✷✳

❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s

✸✸✾

❲❛r♥✐♥❣✦ ❍❡r❡✱

X = dx ❛♥❞ Y = dy ❛r❡ ❥✉st ❝❡rt❛✐♥ ✈❛r✐❛❜❧❡s x ❛♥❞ y r❡s♣❡❝t✐✈❡❧②❀ t❤❡ ❧❛tt❡r ❞❡♣❡♥❞s

r❡❧❛t❡❞ t♦

♦♥ t❤❡ ❢♦r♠❡r ❧✐♥❡❛r❧②✿

Y =m·X.

❚❤❡ ❛❧❣❡❜r❛ ♠❛② ❝♦♠❡ ❢r♦♠ t❤❡ ❡①❛♠♣❧❡ ❛❜♦✈❡✿

• y ❞❡♣❡♥❞s ♦♥ x ✈✐❛ y = f (x)✳

• dy ❞❡♣❡♥❞s ♦♥ x ❛♥❞ dx ✈✐❛ dy = f ′ (x)dx✳

❊①❛♠♣❧❡ ✹✳✶✷✳✷✿ ❧✐♥❡❛r✐③❛t✐♦♥

●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ f (x) = x2 ✱ ✜♥❞ ✐ts ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❛t a = 1✳ ❙✐♥❝❡ f ′ (x) = 2x✱ ✇❡ s❡❡ t❤❛t f ′ (a) = f ′ (1) = 2 ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ f ❛t a = 1 ✐s

L(x) = f (a) + f ′ (a)(x − a) = 1 + 2(x − a) . ◆♦✇ ✇❡ r❡✲✐♥t❡r♣r❡t t❤❡s❡ q✉❛♥t✐t✐❡s✿ ✶✳ dx = x − a ✷✳ dy = L(x) − L(a) ❚❤❡♥✱ ✇❡ ❤❛✈❡✿

dy = 2 · dx .

❚❤❡ ❡q✉❛t✐♦♥ ❡①♣r❡ss❡s ♦✉r ❞❡r✐✈❛t✐✈❡ ✐♥ t❡r♠s ♦❢ t❤❡s❡ ♥❡✇ ✈❛r✐❛❜❧❡s✱ t❤❡ ❞✐✛❡r❡♥t✐❛❧s✳ ❲❡ ❝❛♣t✉r❡ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ x ❛♥❞ t❤❛t ♦❢ y ✕ ❝❧♦s❡ t♦ a✳ ■♥❞❡❡❞✱ y ❣r♦✇s t✇✐❝❡ ❛s ❢❛st ❛s x✳ ❲❡ ❛❝q✉✐r❡ t❤✐s ✐♥❢♦r♠❛t✐♦♥ ❜② ✐♥tr♦❞✉❝✐♥❣ ❛ ♥❡✇ ❝♦♦r❞✐♥❛t❡ s②st❡♠ (dy, dx)✳ ■♥ t❤✐s ❝♦♦r❞✐♥❛t❡ s②st❡♠✱ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✭❣✐✈❡♥ ❜② t❤❡ t❛♥❣❡♥t ❧✐♥❡✮ ❜❡❝♦♠❡s s✐♠♣❧② ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✳ ❚❤❡ ❛♥❛❧②s✐s ♣r❡s❡♥t❡❞ ❛❜♦✈❡ ❛♣♣❧✐❡s t♦ ❡✈❡r② ♣♦✐♥t ✕ ❛♥❞ t♦ ❛❧❧ ♣♦✐♥ts ❛t ♦♥❝❡✿

❘❡❝❛❧❧ ❛❧s♦ ❢r♦♠ ❈❤❛♣t❡r ✶ ❤♦✇ ✇❡ ❧❡❛r♥❡❞ t♦ ❧♦♦❦ ❛t t❤❡ ✐♥t❡❣r❛❧ ❞✐✛❡r❡♥t❧②✿ Z b k(x) dx . a

❲❡ ❝❤❛♥❣❡

✇❤❛t ✇❡ ✐♥t❡❣r❛t❡✳ ■♥st❡❛❞ ♦❢ ❛ ❢✉♥❝t✐♦♥✱ k(x)✱ ✐t ✐s ❛ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠✱ k(x) · dx✳

❉❡✜♥✐t✐♦♥ ✹✳✶✷✳✸✿ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♦❢ ❞❡❣r❡❡ 1 ❆ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ t✇♦ ✈❛r✐❛❜❧❡s✿

♦❢ ❞❡❣r❡❡ 1✱ ♦r s✐♠♣❧② ❛ 1✲❢♦r♠✱ ✐s ❞❡✜♥❡❞ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ϕ = ϕ(x, dx) = k(x) · dx ,

✇❤❡r❡ y = k(x) ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ x✳

✹✳✶✷✳

❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s

✸✹✵

❚❤❡ ❢✉♥❝t✐♦♥ ✐s s✐♠♣❧② ❧✐♥❡❛r ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s❡❝♦♥❞ ✈❛r✐❛❜❧❡✳

❲❛r♥✐♥❣✦ ❚❤❡ s②♠❜♦❧ ✏ ·✑ st❛♥❞s ❢♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ✐t ✐s

♦❢t❡♥ ♦♠✐tt❡❞✳

▲❡t✬s ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ❜❡❧♦✇✿

•

❋✐rst✱ ✇❡ ♣❧♦t t❤❡ ❣r❛♣❤ ♦❢

•

❚❤❡♥✱ ✇❡ ♦❜s❡r✈❡ t❤❛t

•

❋✐♥❛❧❧②✱ ✇❡ ❝♦♥♥❡❝t t❤❡s❡ ❞♦ts t♦ t❤❡ ❝✉r✈❡ ✇✐t❤

ϕ

✐s

k

✭❣r❡❡♥✮ ❛❜♦✈❡ t❤❡ ❧✐♥❡

0

✇❤❡♥

dx = 0

dx = 1✳

❛♥❞ ♣❧♦t ♣♦✐♥ts ♦♥ t❤❡

str❛✐❣❤t ❧✐♥❡s

x✲❛①✐s

✭r❡❞✮✳

✭♦r❛♥❣❡✮✳

❚❤❡ r❡s✉❧t ✐s t❤✐s s✉r❢❛❝❡✿

❆s ♣r❡s❡♥t❡❞ ❛❜♦✈❡✱ ❛ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♠❛② ❝♦♠❡ ❢r♦♠ t❤❡ ❢♦❧❧♦✇✐♥❣✿

y = f (x)

❛t

dy = f ′ (a) , dx

x = a =⇒

❛♥❞✱ ❢✉rt❤❡r♠♦r❡✱

=⇒ dy = f ′ (a) · dx . ❆ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ✐s ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❡①tr❛ ✈❛r✐❛❜❧❡s✱ ♦♥❝❡ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ♦❧❞ ♦♥❡s ❤❛s ❜❡❡♥ s♣❡❝✐✜❡❞✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐✛❡r❡♥t✐❛❧s ✈❛r✐❡s ❢r♦♠ ❧♦❝❛t✐♦♥ t♦ ❧♦❝❛t✐♦♥✳ ❙♦✱ ❞✐✛❡r❡♥t✐❛❧ ♦❢

x✱

✇❤✐❝❤ ✐s ❛ ✈❛r✐❛❜❧❡ s❡♣❛r❛t❡ ❢r♦♠✱ ❜✉t r❡❧❛t❡❞ t♦✱

❘❡❝❛❧❧ ❛❧s♦ ❤♦✇ t❤❡

dx

✐s t❤❡

x✳

❈❤❛✐♥ ❘✉❧❡✱ ✐♥ t❤❡ ▲❡✐❜♥✐③ ♥♦t❛t✐♦♥✱ ✇❛s ✐♥t❡r♣r❡t❡❞ ❛s ❛ ✏❝❛♥❝❡❧❧❛t✐♦♥✑

♦❢

du✿

dy 6 du dy = . dx 6 du dx ◆♦✇ ✇❡ ❝❛♥ s❡❡ t❤❛t ✐t ✐s ✐♥❞❡❡❞ ❛ ❝❛♥❝❡❧❧❛t✐♦♥✱ ✇❤❡♥

du

✐s ♥♦t ③❡r♦✳

❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s ❛r❡ ✇❤❛t ✇❡ ✐♥t❡❣r❛t❡✳ ❋♦r s✐♠♣❧✐❝✐t②✱ ✐♥st❡❛❞ ♦❢ ✉s✐♥❣ ♣❛rt✐t✐♦♥s ❛♥❞ ❞✐s❝r❡t❡ ❢♦r♠s t♦ ❞❡✜♥❡ t❤❡ ✐♥t❡❣r❛❧✱ ✇❡ ❥✉st r❡❢❡r t♦ t❤❡ ✏✉s✉❛❧✑ ✐♥t❡❣r❛❧✿

❉❡✜♥✐t✐♦♥ ✹✳✶✷✳✹✿ ✐♥t❡❣r❛❧ ♦❢ 1✲❢♦r♠ ❚❤❡

✐♥t❡❣r❛❧ ♦❢ ❛ 1✲❢♦r♠ ϕ = k dx ♦✈❡r ❛♥ ✐♥t❡r✈❛❧ [a, b] ✐s ❞❡✜♥❡❞ t♦ ❜❡ Z

❚❤❡♥ t❤❡ ❢♦r♠

k dx

✐s

f (u) ·

[a,b]

✐♥t❡❣r❛❜❧❡

▲❡t✬s t❛❦❡ ❛ ❧♦♦❦ ❛t ❤♦✇ ✇❡ r❡♣r❡s❡♥t❡❞ t❤❡ ❢♦r♠✉❧❛ ♦❢

Z

ϕ=

Z

b

k(x) dx . a

✇❤❡♥❡✈❡r

k

✐s ✐♥t❡❣r❛❜❧❡✳

✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥

du 6 dx = 6 dx

Z

f (u) du .

✐♥ ❈❤❛♣t❡r ✷✿

✹✳✶✷✳ ❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s

✸✹✶

■♥ ❧✐❣❤t ♦❢ t❤❡ ♥❡✇ ❞❡✜♥✐t✐♦♥✱ t❤✐s ✐s ❛ ❧✐t❡r❛❧ ❝❛♥❝❡❧❧❛t✐♦♥✳

❊①❡r❝✐s❡ ✹✳✶✷✳✺ ❙❤♦✇ t❤❛t t❤❡ s✉♠✱ ❜✉t ♥♦t t❤❡ ♣r♦❞✉❝t✱ ♦❢ t✇♦ 1✲❢♦r♠s ✐s ❛❧s♦ ❛ 1✲❢♦r♠✳ ❆♥❞ s♦ ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐s ❛❧s♦ tr❛♥s❧❛t❡❞ ✐♥t♦ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠s✳

❉❡✜♥✐t✐♦♥ ✹✳✶✷✳✻✿ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♦❢ ❞❡❣r❡❡ 0 ❆ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♦❢ ❞❡❣r❡❡ 0✱ ♦r s✐♠♣❧② ❛ 0✲❢♦r♠✱ ✐s ❛♥② ❢✉♥❝t✐♦♥ y = f (x) ♦❢ x✳ ❏✉st ❛s ❞✐s❝r❡t❡ ❢♦r♠s✱ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠s ♦❢ ❞✐✛❡r❡♥t ❞❡❣r❡❡s ❛r❡ ✐♥t❡r❝♦♥♥❡❝t❡❞✳

❉❡✜♥✐t✐♦♥ ✹✳✶✷✳✼✿ ❡①t❡r✐♦r ❞❡r✐✈❛t✐✈❡ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♦❢ ❞❡❣r❡❡ 0 ❚❤❡ ❡①t❡r✐♦r ❞❡r✐✈❛t✐✈❡ df ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ f ♦❢ ❞❡❣r❡❡ 0 ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ 1✲❢♦r♠ ❣✐✈❡♥ ❜②✿ df = f ′ (x) dx

❚❤✐s ♥♦t❛t✐♦♥ ✐s ✉s❡❞ ❛❧♦♥❣ ✇✐t❤ ♦t❤❡rs ✉s❡❞ ✇❤❡♥ t❤❡ ♥❛♠❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t❤❛t r❡❧❛t❡s x ❛♥❞ y ✐s ♥♦t ♣r♦✈✐❞❡❞✿

❉✐✛❡r❡♥t✐❛❧ df

❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❤❛✈❡ ❛s ❜❡❢♦r❡✿ ❲❡ ❝❛♥ ❛❧s♦ ❤❛✈❡✿

dy

d( )

y = x2 =⇒ dy = 2x dx . d(x2 ) = 2x dx .

❚❤✉s✱ t❤❡ ❡①t❡r✐♦r ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❝♦♥t❛✐♥s ❛❧❧ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ✐ts ❞❡r✐✈❛t✐✈❡✱ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❍♦✇❡✈❡r✱ t❤❡ ❢♦r♠❡r ♣r♦✈✐❞❡s ❛ ❞✐r❡❝t ❛♥s✇❡r t♦ t❤✐s q✉❡st✐♦♥✿ ◮ ■❢ ✇❡ ❛r❡ ❛t x = a ❛♥❞ ♠❛❦❡ ❛ st❡♣ dx✱ ✇❤❛t ✐s t❤❡ st❡♣ dy ♦❢ y ❄

❊①❛♠♣❧❡ ✹✳✶✷✳✽✿ ❞✐s♣❧❛❝❡♠❡♥t ❙✉♣♣♦s❡ x ✐s t✐♠❡ ❛♥❞ y = f (x) ✐s t❤❡ ❧♦❝❛t✐♦♥ ❛t t✐♠❡ x✳ ❚❤❡ ❡①t❡r✐♦r ❞❡r✐✈❛t✐✈❡ ♣r♦✈✐❞❡s ❛ ❞✐r❡❝t ❛♥s✇❡r t♦ t❤✐s q✉❡st✐♦♥✿ ◮ ❙✉♣♣♦s❡ x ✐s t✐♠❡ ❛♥❞ y = f (x) ✐s t❤❡ ❧♦❝❛t✐♦♥ ❛t t✐♠❡ x✳ ■❢ ❛t t✐♠❡ x = a ✇❡ ❛r❡ ❛t y = f (a) ❛♥❞ t❤❡♥ ✇❡ ♠♦✈❡ ❢♦r ❛ s❤♦rt s❤♦rt t✐♠❡ dx✱ ❤♦✇ ❢❛r ✇✐❧❧ ✇❡ ❣♦❄ ■t✬s t❤❡ ✈❡❧♦❝✐t② ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ t✐♠❡ ✐♥❝r❡♠❡♥t✿ ❉✐s♣❧❛❝❡♠❡♥t = f ′ (a) · dx , ❜✉t ♦♥❧② ✇❤❡♥ t❤❡ ✈❡❧♦❝✐t②✱ f ′ ✱ ✐s ❝♦♥st❛♥t✳ ■♥ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✱ t❤✐s ✐s ❛♥ ❡st✐♠❛t❡✳

❊①❛♠♣❧❡ ✹✳✶✷✳✾✿ ✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥ ❲❡ ❤❛✈❡ ✉s❡❞ t❤✐s ❛❧❣❡❜r❛ ❢♦r ✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥✳ ❈♦♥s✐❞❡r t❤❡ ✐♥t❡❣r❛❧✿ Z

2

2x sin x2 dx . 0

✹✳✶✷✳ ❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s

✸✹✷

❚❤❡ ✐❞❡❛ ✐s t♦ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇ ✈❛r✐❛❜❧❡

u = x2 .

❍❡r❡ ✐s ❛ ❢❛♠✐❧✐❛r ❝♦♠♣✉t❛t✐♦♥ ✐♥t❡r♣r❡t❡❞ ✐♥ ❛ ♥❡✇ ✇❛②✿ u = x2 =⇒ du = 2x dx Z x=2 2x sin x2 dx =⇒ x=0 Z u=22 sin u du = u=02

u=4 = − cos u u=0

❊①t❡r✐♦r ❞❡r✐✈❛t✐✈❡

du = 2x dx Z 2x sin x2 dx [0,2] Z sin u du =

[0, 2] → [02 , 22 ]

[02 ,22 ]

u=4 = − cos u u=0

= − cos 4 − (− cos 0) = − cos 4 − (− cos 0)

❖✉r ❞❡✜♥✐t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ tr❡❛ts t❤❡ ✐♥t❡❣r❛♥❞ ❛s ❛ s✐♠♣❧❡ ❝❛s❡ ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ t✇♦ ♥✉♠❜❡rs✳ ❚❤❛t ✐s ✇❤② ✇❡ ❛r❡ ❛t ❧✐❜❡rt② t♦ ❛❧❣❡❜r❛✐❝❛❧❧② ♠❛♥✐♣✉❧❛t❡ t❤❡s❡ ❡①♣r❡ss✐♦♥s t❤❡ ✇❛② ✇❡ ❤❛✈❡✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ s✐♠♣❧❡ r❡✲st❛t❡♠❡♥t ✇✐t❤ ♦✉r ♥❡✇ ♥♦t❛t✐♦♥ ♦❢ ❛ ❢❛♠✐❧✐❛r t❤❡♦r❡♠ ✭❈❤❛♣t❡r ✶✮✿

❚❤❡♦r❡♠ ✹✳✶✷✳✶✵✿ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ❙✉♣♣♦s❡ ϕ ✐s ❛ 1✲❢♦r♠ ✐♥t❡❣r❛❜❧❡ ♦♥ ✐♥t❡r✈❛❧ [a, b]✳ ❚❤❡♥✱ Z

[a,b]

ϕ = F (b) − F (a) ,

❢♦r ❛♥② 0✲❢♦r♠ F t❤❛t s❛t✐s✜❡s✿ dF = ϕ .

■♥ ♦r❞❡r t♦ st✉❞② ❛ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ y = f (x) ❛♥❞ ✐ts ❝❤❛♥❣❡✱ ✇❡ ♥♦✇ ❦❡❡♣ tr❛❝❦ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ✿ • t❤❡ ❧♦❝❛t✐♦♥s✱ x ✈s✳ y ✱ ❛♥❞ • t❤❡ ❞✐r❡❝t✐♦♥s✱ dx ✈s✳ dy ✳

❚❤❡ r❡❧❛t✐♦♥ ✐s ❛s ❢♦❧❧♦✇s✿

(x, dx) 7→ (y, dy) = f (x), f ′ (x)dx

❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ t♦ ❞✐s❝r❡t❡ ❢♦r♠s ❄



❲❡ ❦♥♦✇ t❤❛t ❞✐s❝r❡t❡ ❢♦r♠s ❝r❡❛t❡❞ ❜② s❛♠♣❧✐♥❣ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ r❡✜♥✐♥❣ ♣❛rt✐t✐♦♥s ❝♦♥✈❡r❣❡s t♦ ❛ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♦❢ t❤❡ s❛♠❡ ❞❡❣r❡❡✳ ❈❛♥ ✇❡ r❡✈❡rs❡ t❤✐s ❝♦rr❡s♣♦♥❞❡♥❝❡❄

❆♥② ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ s❛♠♣❧❡❞✳ ■t ♠❛tt❡rs ✇❤❡r❡✿ ✶✳ ❙❛♠♣❧✐♥❣ ❛t t❤❡ ♣r✐♠❛r② ♥♦❞❡s ♣r♦❞✉❝❡s ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠✳ ✷✳ ❙❛♠♣❧✐♥❣ ❛t t❤❡ s❡❝♦♥❞❛r② ♥♦❞❡s ♣r♦❞✉❝❡s ❛ ❞✐s❝r❡t❡ 1✲❢♦r♠✳ ❆ ❜❡tt❡r ✐❞❡❛✱ ❤♦✇❡✈❡r✱ ✐s t♦ s❛♠♣❧❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡❣r❡❡✳

❚❤❡♦r❡♠ ✹✳✶✷✳✶✶✿ ❉✐s❝r❡t❡ ❛♥❞ ❉✐✛❡r❡♥t✐❛❧ ❋♦r♠s ✶✳ ❆ ❞✐✛❡r❡♥t✐❛❧ 0✲❢♦r♠ ❡✈❛❧✉❛t❡❞ ❛t t❤❡ ♥♦❞❡s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ✐s ❛ ❞✐s❝r❡t❡ 0✲❢♦r♠✳

✹✳✶✷✳

❉✐✛❡r❡♥t✐❛❧ ❢♦r♠s

✸✹✸

✷✳ ❆ ❞✐✛❡r❡♥t✐❛❧ 1✲❢♦r♠ ❡✈❛❧✉❛t❡❞ ✕ ❜② ✐♥t❡❣r❛t✐♦♥ ✕ ❛t t❤❡ ✐♥t❡r✈❛❧s ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ✐s ❛ ❞✐s❝r❡t❡ 1✲❢♦r♠❀ ✐✳❡✳✱ ✐❢ ϕ ✐s ❛ ❞✐✛❡r❡♥t✐❛❧ 1✲❢♦r♠✱ t❤❡ ❝♦rr❡✲ s♣♦♥❞✐♥❣ ❞✐s❝r❡t❡ 1✲❢♦r♠ ✐s ❞❡✜♥❡❞ ❜②✿





s [A, B] =

❙♦✱ ❛ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♦❢ ❞❡❣r❡❡

1

✐s

Z

ϕ [A,B]

✐ts ✐♥t❡❣r❛❧s✿

■♥t❡❣r❛❧ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ✏❡✈❛❧✉❛t❡✑

✐♥♣✉t

→

↓ Z

[−1, 1]

 3x3 + sin x dx = 0 | {z } ↑

❞✐✛❡r❡♥t✐❛❧ ❢♦r♠

↑ ♦✉t♣✉t

■❢ ✇❡ s❡t t❤❡ ♠♦t✐♦♥ ✐♥t❡r♣r❡t❛t✐♦♥ ❛s✐❞❡✱ t❤❡ ♣✉r♣♦s❡ ♦❢ t❤❡ ♥❡✇ ❝♦♥❝❡♣t ✐s t♦ ♠❛❦❡ ❛ ❝❛r❡❢✉❧ ❞✐st✐♥❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❧♦❝❛t✐♦♥✱

◮

x✱

❛♥❞ t❤❡ ❞✐r❡❝t✐♦♥✱

dx✿

❍♦✇ ❢❛st ❛r❡ ✇❡ ❣♦✐♥❣ ❢r♦♠ t❤✐s ❧♦❝❛t✐♦♥ ✐♥ t❤❛t ❞✐r❡❝t✐♦♥❄

❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ ❞✐r❡❝t✐♦♥ ✭❛♥❞ ✐ts ♦♣♣♦s✐t❡✮ ❢♦r ❛ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ✈❛r✐❛❜❧❡✱ ❜✉t ✐♥✜♥✐t❡❧② ♠❛♥② ❢♦r ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❲❡ ✇✐❧❧ s❡❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠s ✐♥ t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ✐♥ ❱♦❧✉♠❡ ✹✳

❈❤❛♣t❡r ✺✿ ❙❡r✐❡s

❈♦♥t❡♥ts

✺✳✶ ❋r♦♠ ❧✐♥❡❛r t♦ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✳ ✺✳✷ ❚❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✸ ❙❡q✉❡♥❝❡s ♦❢ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✹ ■♥✜♥✐t❡ s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✺ ▼❛✐♥ ❝❧❛ss❡s ♦❢ s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✻ ❋r♦♠ ✜♥✐t❡ s✉♠s ✈✐❛ ❧✐♠✐ts t♦ s❡r✐❡s ✳ ✳ ✳ ✺✳✼ ❉✐✈❡r❣❡♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✽ ❙❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✾ ❈♦♠♣❛r✐s♦♥ ♦❢ s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✶✵ ❆❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✶✶ ❚❤❡ ❘❛t✐♦ ❚❡st ❛♥❞ t❤❡ ❘♦♦t ❚❡st ✳ ✳ ✳ ✺✳✶✷ P♦✇❡r s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✶✸ ❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✹✹ ✸✺✷ ✸✻✷ ✸✻✽ ✸✼✺ ✸✽✹ ✸✾✷ ✸✾✺ ✹✵✷ ✹✵✽ ✹✶✺ ✹✷✶ ✹✷✽

✺✳✶✳ ❋r♦♠ ❧✐♥❡❛r t♦ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s

❆♣♣r♦①✐♠❛t✐♥❣ ❢✉♥❝t✐♦♥s ✐s ❧✐❦❡ ❛♣♣r♦①✐♠❛t✐♥❣ ♥✉♠❜❡rs ✕ s✉❝❤ ❛s π ✱ e✱ ♦r t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ♦❢ ❛♥② ❢✉♥❝t✐♦♥ ✕ ❜✉t ❤❛r❞❡r✳ ❘❡❝❛❧❧ ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✻✮ t❤❡ ♠❡❛♥✐♥❣ ♦❢ ❧✐♥❡❛r✐③❛t✐♦♥ ✿ ❲❡ r❡♣❧❛❝❡ ❛ ❣✐✈❡♥ ❢✉♥❝t✐♦♥ y = f (x) ✇✐t❤ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ y = L(x) t❤❛t ❜❡st ❛♣♣r♦①✐♠❛t❡s ✐t ❛t ❛ ❣✐✈❡♥ ♣♦✐♥t✳ ❚❤✐s ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱ ❛♥❞ ✐t ❤❛♣♣❡♥s t♦ ❜❡ t❤❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ t❤❡ ❣r❛♣❤ ♦❢ ✇❤✐❝❤ ✐s t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛t t❤❡ ♣♦✐♥t✳ ❚❤❡ r❡♣❧❛❝❡♠❡♥t ✐s ❥✉st✐✜❡❞ ❜② t❤❡ ❢❛❝t t❤❛t ✇❤❡♥ ②♦✉ ③♦♦♠ ✐♥ ♦♥ t❤❡ ♣♦✐♥t✱ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✇✐❧❧ ♠❡r❣❡ ✇✐t❤ t❤❡ ❣r❛♣❤✿

✺✳✶✳

❋r♦♠ ❧✐♥❡❛r t♦ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s

✸✹✺

❍♦✇❡✈❡r✱ t❤❡r❡ ✐s ❛ ♠♦r❡ ❜❛s✐❝ ❛♣♣r♦①✐♠❛t✐♦♥✿ ❛

❝♦♥st❛♥t

❢✉♥❝t✐♦♥✱

y = C(x)✳

❊①❛♠♣❧❡ ✺✳✶✳✶✿ sq✉❛r❡ r♦♦t

▲❡t✬s r❡✈✐❡✇ t❤✐s ❡①❛♠♣❧❡ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✻✿

◮

❍♦✇ ❞♦ ✇❡ ❝♦♠♣✉t❡

❲❡ ❛♣♣r♦①✐♠❛t❡✳ ❢✉♥❝t✐♦♥

f (x) =

√

√

4.1

✇✐t❤♦✉t ❛❝t✉❛❧❧② ❡✈❛❧✉❛t✐♥❣

f (x) =

√

x❄ √

❙♣❡❝✐✜❝❛❧❧②✱ ✐♥ ♦r❞❡r t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ♥✉♠❜❡r ♦❢

x

✏❛r♦✉♥❞✑

4.1✱

✇❡ ❛♣♣r♦①✐♠❛t❡ t❤❡

a = 4✳

❲❡ ✜rst ❛♣♣r♦①✐♠❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛

❝♦♥st❛♥t

❢✉♥❝t✐♦♥✿

C(x) = 2 . ❚❤✐s ✈❛❧✉❡ ✐s ❝❤♦s❡♥ ❜❡❝❛✉s❡

f (a) =

√ √

4 = 2✳

❚❤❡♥ ✇❡ ❤❛✈❡✿

4.1 = f (4.1) ≈ C(4.1) = 2 .

■t ✐s ❛ ❝r✉❞❡ ❛♣♣r♦①✐♠❛t✐♦♥✿

❚❤❡ ♦t❤❡r✱ ❧✐♥❡❛r✱ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ✈✐s✐❜❧② ❜❡tt❡r✳ ❲❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛

❚❤✐s ✈❛❧✉❡ ✐s ❝❤♦s❡♥ ❜❡❝❛✉s❡

√

f (a) =

√

1 L(x) = 2 + (x − 4) . 4 4=2

❛♥❞

f ′ (a) =

1 ✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ 4

1 4.1 = f (4.1) ≈ L(4.1) = 2 + (4.1 − 4) = 2.025 . 4

❧✐♥❡❛r ❢✉♥❝t✐♦♥✿

✺✳✶✳

❋r♦♠ ❧✐♥❡❛r t♦ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s

✸✹✻

❲❡ ❤❛✈❡ ❢♦r ❛ ❢✉♥❝t✐♦♥ y = f (x) ❛♥❞ ❛ ♣♦✐♥t x = a✿ ❚❤❡ ❜❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥✿

C(x) = f (a) .

❚❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✿

L(x) = f (a)

+f ′ (a)(x − a) .

❲❡ s❤♦✉❧❞ ♥♦t✐❝❡ ❡❛r❧② ♦♥ t❤❛t t❤❡ ❧❛tt❡r ❥✉st ❛❞❞s ❛ ♥❡✇ ✭❧✐♥❡❛r✮ t❡r♠ t♦ t❤❡ ❢♦r♠❡r✦ ❲❛r♥✐♥❣✦

❚❤❡ ❧❛tt❡r ✐s ❜❡tt❡r t❤❛♥ t❤❡ ❢♦r♠❡r ✕ ❜✉t ♦♥❧② ✇❤❡♥ ✇❡ ♥❡❡❞ ♠♦r❡ ❛❝❝✉r❛❝②✳ ❖t❤❡r✇✐s❡✱ t❤❡ ❧❛tt❡r ✐s ✇♦rs❡ ❜❡❝❛✉s❡ ✐t r❡q✉✐r❡s ♠♦r❡ ❝♦♠♣✉t❛t✐♦♥✳

❈❛♥ ✇❡ ❞♦ ❜❡tt❡r t❤❛♥ t❤❡ ♠❛t✐♦♥❄ ❨❡s✳

❜❡st

❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥❄ ◆♦✳ ❈❛♥ ✇❡ ❞♦ ❜❡tt❡r t❤❛♥ t❤❡ ❜❡st

❧✐♥❡❛r

❛♣♣r♦①✐✲

❊①❛♠♣❧❡ ✺✳✶✳✷✿ r♦❛❞ ❝✉r✈❛t✉r❡

❖♥❡ ❝❛♥ ✉♥❞❡rst❛♥❞ ❧✐♥❡❛r✐③❛t✐♦♥ ❛s ✐❣♥♦r✐♥❣ t❤❡ s❤❛♣❡ ♦❢ t❤❡ r♦❛❞ ❛♥❞ ❝♦♥❝❡♥tr❛t✐♥❣ ♦♥ t❤❡ ❤❡❛❞❧✐❣❤ts ♦❢ t❤❡ ❝❛r ✭❈❤❛♣t❡r ✷❉❈✲✸✮ ♦♥❡ ❧♦❝❛t✐♦♥ ❛t ❛ t✐♠❡✿

■♥ ❈❤❛♣t❡r ✷❉❈✲✹✱ ✇❡ ❛❧s♦ ❧❡❛r♥❡❞ t❤❛t t❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ r♦❛❞ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❤♦✇ ❢❛st t❤❡ ❤❡❛❞❧✐❣❤ts t✉r♥ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❜② t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐t r❡♣r❡s❡♥ts✿

❆ ❢✉rt❤❡r ✐❞❡❛ ✐s t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❝✉r✈❡ ♦❢ t❤❡ r♦❛❞ ✇✐t❤ ❛ ❢♦r ❧♦✇❡r ❝✉r✈❛t✉r❡ ❛♥❞ s♠❛❧❧❡r ❢♦r ❤✐❣❤❡r ❝✉r✈❛t✉r❡✿

❝✐r❝❧❡

♦❢ ❛♥ ❛♣♣r♦♣r✐❛t❡ r❛❞✐✉s ✕ ❧❛r❣❡r

❊✈❡r② ❝✐r❝❧❡ ✐s ❛ q✉❛❞r❛t✐❝ ❝✉r✈❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐s s❡❡♥ ❛s ❛ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥✳ ❙✐♥❝❡ t❤❡ ♦♥❧② t❤✐♥❣ ✇❡ ✇❛♥t ❢r♦♠ ✐t ✐s ✐ts ❝✉r✈❛t✉r❡✱ ✇❡ ❝❛♥ r❡♣❧❛❝❡ t❤❡ ❝✐r❝❧❡ ✇✐t❤ ❛♥♦t❤❡r q✉❛❞r❛t✐❝ ❜✉t s✐♠♣❧❡r ✭❛♥❞ ❡①♣❧✐❝✐t❧② ❣✐✈❡♥✮ ❝✉r✈❡ ✕ t❤❡ ♣❛r❛❜♦❧❛✳ ❚❤❡ ❝✉r✈❛t✉r❡ ✐s ❢✉rt❤❡r st✉❞✐❡❞ ✐♥ ❱♦❧✉♠❡ ✹ ✭❈❤❛♣t❡r ✹❍❉✲✷✮✳ ❇❡❧♦✇ ✇❡ ✐❧❧✉str❛t❡ ❤♦✇ ✇❡ ❛tt❡♠♣t t♦ ❛♣♣r♦①✐♠❛t❡ ❛ ❢✉♥❝t✐♦♥ ❛r♦✉♥❞ t❤❡ ♣♦✐♥t (1, 1) ✇✐t❤ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s

✺✳✶✳

❋r♦♠ ❧✐♥❡❛r t♦ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s

✸✹✼

✜rst❀ ❢r♦♠ t❤♦s❡✱ ✇❡ ❝❤♦♦s❡ t❤❡ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t✳ ❚❤✐s ❧✐♥❡ t❤❡♥ ❜❡❝♦♠❡s ♦♥❡ ♦❢ t❤❡ ♠❛♥② ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ t❤❡ ❝✉r✈❡ t❤❛t ♣❛ss t❤r♦✉❣❤ t❤❡ ♣♦✐♥t❀ ❢r♦♠ t❤♦s❡✱ ✇❡ ❝❤♦♦s❡ t❤❡ t❛♥❣❡♥t ❧✐♥❡✿

s❡q✉❡♥❝❡ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s✦ ■♥❞❡❡❞✱ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✇✐❧❧ ❜❡❝♦♠❡ ♦♥❡ ♦❢ t❤❡ ♠❛♥② q✉❛❞r❛t✐❝ ❝✉r✈❡s ✕ ♣❛r❛❜♦❧❛s ✕ t❤❛t ♣❛ss t❤r♦✉❣❤ t❤❡ ♣♦✐♥t✳✳✳ ❛♥❞ ❛r❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡✳ ❲❤✐❝❤ ♦♥❡ ♦❢ t❤♦s❡ ❞♦ ✇❡ ❝❤♦♦s❡❄

◆♦✇✱ ✇❡ s❤❛❧❧ s❡❡ t❤❛t t❤❡s❡ ❛r❡ ❥✉st t❤❡ t✇♦ ✜rst st❡♣s ✐♥ ❛

■♥ ♦r❞❡r t♦ ❛♥s✇❡r t❤❛t✱ ✇❡ ♥❡❡❞ t♦ r❡✈✐❡✇ ❛♥❞ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤❡ ❜❡st ❝♦♥st❛♥t ❛♥❞ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ✇❡r❡ ❝❤♦s❡♥✳ ■♥ ✇❤❛t ✇❛② ❛r❡ t❤❡② t❤❡ ❜❡st❄ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❣✐✈❡♥ ❛♥❞ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❛♣♣r♦①✐♠❛t❡ ✐ts ❜❡❤❛✈✐♦r ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❛ ♣♦✐♥t✱ x = a✱ ✇✐t❤ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥ y = T (x)✳ ❚❤❡ ❧❛tt❡r ✐s t♦ ❜❡ t❛❦❡♥ ❢r♦♠ s♦♠❡ ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s t❤❛t ✇❡ ✜♥❞ s✉✐t❛❜❧❡✳ ❆ ❝❧❛ss ♦❢ r❡❧❛t✐✈❡❧② s✐♠♣❧❡ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ ❛❧s♦ q✉✐t❡ ✈❡rs❛t✐❧❡ ✐s t❤❡ ♣♦❧②♥♦♠✐❛❧s✳ ❲❤❛t ✇❡ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r ✐s t❤❡

❡rr♦r✱ ✐✳❡✳✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❢✉♥❝t✐♦♥ f

❚❤❡ t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ ✐s s❡❡♥ ❜❡❧♦✇✿

❛♥❞ ✐ts ❛♣♣r♦①✐♠❛t✐♦♥ T ✿

E(x) = |f (x) − T (x)|

❲❡ ❛r❡ s✉♣♣♦s❡❞ t♦ ♠✐♥✐♠✐③❡ t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ ✐♥ s♦♠❡ ✇❛②✳ ❖❢ ❝♦✉rs❡✱ t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ y = E(x) ✐s ❧✐❦❡❧② t♦ ❣r♦✇ ✇✐t❤ ♥♦ ❧✐♠✐t ❛s ✇❡ ♠♦✈❡ ❛✇❛② ❢r♦♠ ♦✉r ♣♦✐♥t ♦❢ ✐♥t❡r❡st✱ x = a✳✳✳ ❜✉t ✇❡ ❞♦♥✬t ❝❛r❡✳ ❲❡ ✇❛♥t t♦ ♠✐♥✐♠✐③❡ t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ a✱ ✇❤✐❝❤ ♠❡❛♥s ♠❛❦✐♥❣ s✉r❡ t❤❛t t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❡rr♦r ❛s x → a ❣♦❡s t♦ 0✦ ❚❤❡♦r❡♠ ✺✳✶✳✸✿ ❇❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥

❙✉♣♣♦s❡

f

✐s ❝♦♥t✐♥✉♦✉s ❛t

x=a

❛♥❞

C(x) = k ✐s ❛♥② ♦❢ ✐ts ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥s ✭✐✳❡✳✱ ❛r❜✐tr❛r② ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s✮✳ ❚❤❡♥✱

✺✳✶✳ ❋r♦♠ ❧✐♥❡❛r t♦ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s

✸✹✽

t❤❡ ❡rr♦r E ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♣♣r♦❛❝❤❡s 0 ❛t x = a ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦♥st❛♥t ✐s ❡q✉❛❧ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ❛t x = a✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ lim (f (x) − C(x)) = 0 ⇐⇒ k = f (a) .

x→a

Pr♦♦❢✳

❯s❡ t❤❡ ❙✉♠ ❘✉❧❡ ❢♦r ❧✐♠✐ts ❛♥❞ t❤❡♥ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ f ✿ 0 = lim (f (x) − C(x)) = lim f (x) − lim C(x) = f (a) − k . x→a

x→a

x→a

❚❤❛t✬s t❤❡ ❛♥❛❧♦❣ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠ ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✻✮✿ ❚❤❡♦r❡♠ ✺✳✶✳✹✿ ❇❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥

❙✉♣♣♦s❡ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a ❛♥❞ L(x) = f (a) + m(x − a)

✐s ❛♥② ♦❢ ✐ts ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❚❤❡♥✱ t❤❡ ❡rr♦r E ♦❢ t❤❡ ❛♣♣r♦①✐♠❛✲ t✐♦♥ ❛♣♣r♦❛❝❤❡s 0 ❛t x = a ❢❛st❡r t❤❛♥ x − a ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ ❧✐♥❡❛r t❡r♠ ✐s ❡q✉❛❧ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ❛t x = a✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f (x) − L(x) = 0 ⇐⇒ m = f ′ (a) . x→a x−a lim

Pr♦♦❢✳

❯s❡ t❤❡ ❙✉♠ ❘✉❧❡ ❢♦r ❧✐♠✐ts ❛♥❞ t❤❡♥ t❤❡ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ♦❢ f ✿ f (x) − f (a) f (x) − L(x) = lim − lim m = f ′ (a) − m . x→a x→a x→a x−a x−a

0 = lim

▲❡t✬s ❝♦♠♣❛r❡ t❤❡ ❝♦♥❞✐t✐♦♥s ✐♥ t❤❡ t✇♦ t❤❡♦r❡♠s✿ f (x) − C(x) → 0 ❛♥❞

f (x) − L(x) →0 x−a

❚❤❡ ❝♦♠♣❛r✐s♦♥ r❡✈❡❛❧s t❤❡ s✐♠✐❧❛r✐t② ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ ❤♦✇ ✇❡ ♠✐♥✐♠✐③❡ t❤❡ ❡rr♦r✦ ❚❤❡ ❞✐✛❡r❡♥❝❡ ✐s ✐♥ t❤❡ ❞❡❣r❡❡✿ ❤♦✇ ❢❛st t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ ❣♦❡s t♦ ③❡r♦✳ ■♥❞❡❡❞✱ ✇❡ ❧❡❛r♥❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✻ t❤❛t t❤❡ ❧❛tt❡r ❝♦♥❞✐t✐♦♥ ♠❡❛♥s t❤❛t f (x) − L(x) ❝♦♥✈❡r❣❡s t♦ 0 ❢❛st❡r t❤❛♥ x − a✱ ✐✳❡✳✱ f (x) − L(x) = o(x − a);

t❤❡r❡ ✐s ♥♦ s✉❝❤ r❡str✐❝t✐♦♥ ❢♦r t❤❡ ❢♦r♠❡r✳ ❙♦ ❢❛r✱ t❤✐s ✐s ✇❤❛t ✇❡ ❤❛✈❡ ❞✐s❝♦✈❡r❡❞✿

◮ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ❜✉✐❧t ❢r♦♠ t❤❡ ❜❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ❜② ❛❞❞✐♥❣ ❛ ❧✐♥❡❛r

t❡r♠✳

❚❤❡ ❜❡st ♦♥❡ ♦❢ t❤♦s❡ ❤❛s t❤❡ s❧♦♣❡ ✭✐ts ♦✇♥ ❞❡r✐✈❛t✐✈❡✮ ❡q✉❛❧ t♦ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f ❛t a✳ ❍♦✇ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s ✇✐❧❧ ♣r♦❣r❡ss ✐s ♥♦✇ ❝❧❡❛r❡r✿ ◮ ◗✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ❜✉✐❧t ❢r♦♠ t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❜② ❛❞❞✐♥❣ ❛ q✉❛❞r❛t✐❝

t❡r♠✳

✺✳✶✳ ❋r♦♠ ❧✐♥❡❛r t♦ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s

✸✹✾

❖♥❡ ♦❢ t❤❡♠ ♠✐❣❤t ❜❡ t❤❡ ❜❡st✿

❚♦ ❞❡❝✐❞❡ ✇❤✐❝❤ ♦♥❡ ♦❢ t❤♦s❡ ✐s t❤❡ ❜❡st✱ ✇❡ t❤✐♥❦ ❜② ❛♥❛❧♦❣② ❛♥❞ tr② t♦ ♠❛❦❡ t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥

E

❣♦ t♦

❜✉t ❡✈❡♥ ❢❛st❡r✿

❚❤❡♦r❡♠ ✺✳✶✳✺✿ ❇❡st q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥

❙✉♣♣♦s❡ f ✐s t✇✐❝❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a ❛♥❞ Q(x) = f (a) + f ′ (a)(x − a) + p(x − a)2

✐s ❛♥② ♦❢ ✐ts q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❚❤❡♥✱ t❤❡ ❡rr♦r E ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♣♣r♦❛❝❤❡s 0 ❛t x = a ❢❛st❡r t❤❛♥ (x − a)2 ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ q✉❛❞r❛t✐❝ t❡r♠ ✐s ❡q✉❛❧ t♦ ❤❛❧❢ ♦❢ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ❛t x = a✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿ f (x) − Q(x) 1 = 0 ⇐⇒ p = f ′′ (a) . 2 x→a (x − a) 2 lim

Pr♦♦❢✳ ❲❡ ❛♣♣❧② ▲✬❍♦♣✐t❛❧✬s r✉❧❡ t✇✐❝❡✿

f (x) − Q(x) x→a (x − a)2 f ′ (x) − f ′ (a) − 2p(x − a) = lim x→a 2(x − a) f ′′ (x) − 2p = lim x→a 2 f ′′ (x) = lim −p x→a 2 1 = f ′′ (a) − p . 2

0 = lim

❋✐rst✳

❆♥❞ s❡❝♦♥❞✳

0

✺✳✶✳

❋r♦♠ ❧✐♥❡❛r t♦ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s

✸✺✵

❖♥❝❡ ❛❣❛✐♥✱ t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ t❤❡♦r❡♠ ♠❡❛♥s t❤❛t

f (x) − Q(x)

❝♦♥✈❡r❣❡s t♦

0

❢❛st❡r t❤❛♥

(x − a)2 ✱

f (x) − Q(x) = o((x − a)2 ) . ❲❡ st❛rt t♦ s❡❡ ❛ ♣❛tt❡r♥✿

•

❚❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♥❣ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❣r♦✇✐♥❣✳

•

❚❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ❜❡✐♥❣ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ❛r❡ ❣r♦✇✐♥❣ t♦♦✳

❊①❛♠♣❧❡ ✺✳✶✳✻✿ sq✉❛r❡ r♦♦t ❋♦r t❤❡ ♦r✐❣✐♥❛❧ ❡①❛♠♣❧❡ ♦❢

f (x) =

f (x)

=

√

√

x

❛t

a = 4✱

✇❡ ❤❛✈❡✿

x

=⇒ f (4)

=2 1 = 4

1 =⇒ f ′ (4) = (x1/2 )′ = x−1/2 2 ′  1 1 1 −1/2 ′′ x = − x−3/2 =⇒ f ′′ (4) = − f (x) = 2 4 32 f ′ (x)

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

✇❤② ❜❡st✿

❛♣♣r♦①✐♠❛t✐♦♥s✿

C(x) =

2

❙❛♠❡ ✈❛❧✉❡ ❛s

1 2 + (x − 4) 4 1 1 2 + (x − 4) − (x − 4)2 2 · 32 4

L(x) = Q(x) =

f.

✳✳✳❆♥❞ s❛♠❡ s❧♦♣❡ ❛s

f.

✳✳✳❆♥❞ s❛♠❡ ❝♦♥❝❛✈✐t② ❛s

f.

❊①❛♠♣❧❡ ✺✳✶✳✼✿ s✐♥❡ ▲❡t✬s ❛♣♣r♦①✐♠❛t❡

f (x) = sin x

❛t

x = 0✳

❋✐rst✱ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡s✿

f (x)

= sin x

=⇒ f (0)

= 0 =⇒ C(x) = 0

f ′ (x)

= cos x

=⇒ f ′ (0)

= 1 =⇒ L(x) = x

′′

′′

f (x) = − sin x =⇒ f (0) = 0 =⇒ Q(x) = ?

❚❤❡r❡❢♦r❡✱ t❤❡ ❜❡st q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s✿

0 Q(x) = 0 + 1(x − 0) − (x − 0)2 = x . 2 ❙❛♠❡ ❛s t❤❡ ❧✐♥❡❛r✦ ❲❤②❄ ❇❡❝❛✉s❡ t❤❡ s✐♥❡ ✐s ♦❞❞✳

♦r

✺✳✶✳ ❋r♦♠ ❧✐♥❡❛r t♦ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s

✸✺✶

❊①❛♠♣❧❡ ✺✳✶✳✽✿ ❝♦s✐♥❡

▲❡t✬s ❛♣♣r♦①✐♠❛t❡ f (x) = cos x ❛t x = 0✳ ❋✐rst✱ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡s✿ f (x)

= cos x

=⇒ f (0)

=1

=⇒ C(x) = 1

f ′ (x)

= − sin x =⇒ f ′ (0)

=0

=⇒ L(x) = 1

f ′′ (x) = − cos x =⇒ f ′′ (0) = −1 =⇒ Q(x) = ?

❚❤❡r❡❢♦r❡✱ t❤❡ ❜❡st q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s✿ 1 1 Q(x) = 1 + 0(x − 0) − (x − 0)2 = 1 − x2 . 2 2

◆♦ ❧✐♥❡❛r t❡r♠✦ ❲❤②❄ ❇❡❝❛✉s❡ t❤❡ ❝♦s✐♥❡ ✐s ❡✈❡♥✳ ❊①❛♠♣❧❡ ✺✳✶✳✾✿ ❡❞❣❡ ❜❡❤❛✈✐♦r

❚❤❡ ❛♣♣❧✐❝❛❜✐❧✐t② ♦❢ t❤❡ t❤❡♦r❡♠s ❞❡♣❡♥❞s ♦♥ t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❈♦♥s✐❞❡r t❤❡s❡ t❤r❡❡ ❢❛♠✐❧✐❛r ❢✉♥❝t✐♦♥s ✭❱♦❧✉♠❡ ✷✮✿

❚❤❡s❡ ❛r❡ t❤❡ r❡s✉❧ts ♦❢ ♦✉r ❛♥❛❧②s✐s✿ ✶✳ ❋✉♥❝t✐♦♥ f (x) = sin x1 ✭✇✐t❤ f (0) = 0✮ ✐s ♥♦t ❝♦♥t✐♥✉♦✉s ❛t 0✱ ❛♥❞ ♥♦♥❡ ♦❢ t❤❡ t❤❡♦r❡♠s ❛♣♣❧②✳ ❚❤❡r❡ ✐s ♥♦ ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ❛t 0✱ ♦❢ ❛♥② ❦✐♥❞✳ ✷✳ ❋✉♥❝t✐♦♥ g(x) = x sin x1 ✭✇✐t❤ f (0) = 0✮ ✐s ❝♦♥t✐♥✉♦✉s ❛t 0✱ ❛♥❞ t❤❡ ✜rst t❤❡♦r❡♠ ❛♣♣❧✐❡s✳ ❇✉t ❛s ✐t✬s ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡✱ ❛♥❞ t❤❡ ♦t❤❡r t✇♦ t❤❡♦r❡♠s ❞♦ ♥♦t ❛♣♣❧②✳ ❚❤❡ ❜❡st ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ❛t 0 ✐s C(x) = 0✱ ❜✉t ✐t✬s ♥♦t✱ ❛♥❞ t❤❡r❡ ✐s ♥♦♥❡✱ ❛ ❣♦♦❞ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✳ ✸✳ ❋✉♥❝t✐♦♥ h(x) = x2 sin x1 ✭✇✐t❤ f (0) = 0✮ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t 0✱ ❛♥❞ t❤❡ ✜rst t✇♦ t❤❡♦r❡♠ ❛♣♣❧②✳ ❇✉t ✐t✬s ♥♦t t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡✱ ❛♥❞ t❤❡ ❧❛st t❤❡♦r❡♠ ❞♦❡s ♥♦t ❛♣♣❧②✳ ❚❤❡ ❜❡st ❧✐♥❡❛r ✭❛♥❞

✺✳✷✳ ❚❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s

✸✺✷

❝♦♥st❛♥t✮ ❛♣♣r♦①✐♠❛t✐♦♥ ❛t

0

L(x) = 0

✐s

❜✉t ✐t✬s ♥♦t✱ ❛♥❞ t❤❡r❡ ✐s ♥♦♥❡✱ ❛ ❣♦♦❞ q✉❛❞r❛t✐❝

❛♣♣r♦①✐♠❛t✐♦♥✳

❲❡ ✉s❡❞ s❡q✉❡♥❝❡s ♦❢ ♥✉♠❜❡rs t♦ ❛♣♣r♦①✐♠❛t❡ ♦t❤❡r ♥✉♠❜❡rs ✐♥ ❱♦❧✉♠❡ ✷❀ ♥♦✇ ✇❡ ✇✐❧❧ ✉s❡ s❡q✉❡♥❝❡s ♦❢

❢✉♥❝t✐♦♥s t♦ ❛♣♣r♦①✐♠❛t❡ ♦t❤❡r ❢✉♥❝t✐♦♥s✳ ■♥ ♦r❞❡r t♦ ❜❡ ❛❜❧❡ t♦ ❣♦ ❜❡②♦♥❞ q✉❛❞r❛t✐❝ ✐♥ ♦✉r s❡q✉❡♥❝❡ ♦❢ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s✱ ✇❡ r❡♥❛♠❡ t❤❡♠ ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❞❡❣r❡❡s ✿

T0 (x) = C(x) T1 (x) = L(x) T2 (x) = Q(x) ... ❊①❛♠♣❧❡ ✺✳✶✳✶✵✿ sq✉❛r❡ r♦♦t

❇❛❝❦ t♦ t❤❡ ♦r✐❣✐♥❛❧ ❡①❛♠♣❧❡ ♦❢

f (x) =

√

x

❛t

a = 4✳

❖♥❡ ❝❛♥ ❣✉❡ss ✇❤❡r❡ t❤✐s ✐s ❣♦✐♥❣✿

2 = T0 (x) f − T0 = o(1)

❝♦♥st❛♥t✿

1 (x − 4) + 2 = T1 (x) f − T1 = o(x − a) 4 1 (x − 4) + 2 = T2 (x) f − T2 = o((x − a)2 ) + 4 1 + (x − 4) + 2 = T3 (x) f − T3 = o((x − a)3 ) 4

❧✐♥❡❛r✿

❝✉❜✐❝✿

1 (x − 4)2 2 · 32 1 (x − 4)2 2 · 32

−

q✉❛❞r❛t✐❝✿

(?)(x − 4)3 − ✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

❲❡ ❛❞❞ ❛ t❡r♠ ❡✈❡r② t✐♠❡ ✇❡ ♠♦✈❡ ❞♦✇♥ t♦ t❤❡ ♥❡①t ❞❡❣r❡❡❀ ✐t✬s t❤❡ r❡❝✉rs✐✈❡ s✉♠ ✭❛s ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✶✮ ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ t❡r♠s t❤❛t ❦❡❡♣ ❛♣♣❡❛r✐♥❣✳ ❚❤❡ r❡s✉❧t✐♥❣ s❡q✉❡♥❝❡ ✐s ❝❛❧❧❡❞ ❛ ✏s❡r✐❡s✑✳

◆❡①t✱ t❤❡ ❣❡♥❡r❛❧ t❤❡♦r②✳

✺✳✷✳ ❚❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s

❆ ♣♦❧②♥♦♠✐❛❧ ✐s t②♣✐❝❛❧❧② ✇r✐tt❡♥ ✐♥ ✐ts st❛♥❞❛r❞ ❢♦r♠ ✿

P (x) = a0 + a1 x + a2 x2 + ... + an xn . ■t✬s t❤❡ s✉♠ ♦❢ ♠✉❧t✐♣❧❡s ✭❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥✮ ♦❢ t❤❡ ♣♦✇❡rs ♦❢

x✳

❍♦✇❡✈❡r✱ ✐♥ ♦r❞❡r t♦ ❜❡ ❛❜❧❡ t♦ ❝♦♥❝❡♥tr❛t❡ ♦♥ ❛ ♣❛rt✐❝✉❧❛r ✈❛❧✉❡ ♦❢ ♣♦❧②♥♦♠✐❛❧ ✐♥ t❡r♠s ♦❢ t❤❡ ❞❡✈✐❛t✐♦♥ ♦❢

x

❢r♦♠

a✱

✐✳❡✳✱

x − a✳

x✱

✐✳❡✳✱

x = a✱

✇❡ ✇❛♥t t♦ ❡①♣r❡ss t❤❡

❲❡ ✜♥❞ ❛ s♣❡❝✐❛❧ ❛♥❛❧♦❣ ♦❢ t❤❡ st❛♥❞❛r❞ ❢♦r♠

♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧❀ ❛ ♣♦❧②♥♦♠✐❛❧ ✐s st✐❧❧ t❤❡ s✉♠ ♦❢ ♣♦✇❡rs✱ ❥✉st ♥♦t ♦❢

x

❜✉t ♦❢

(x − a)✳

❆ ❢❛♠✐❧✐❛r ❡①❛♠♣❧❡ ♦❢ s✉❝❤ ❛ tr❛♥s✐t✐♦♥ ❝♦♠❡s ❢r♦♠ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧s ❛♥❞ t❤❡✐r ♣♦✐♥t✲s❧♦♣❡ ❢♦r♠ ✿

L(x) = mx + b = m(x − a) + d . ❚❤❡r❡ ✐s ❛❧s♦ t❤❡ ✈❡rt❡① ❢♦r♠ ♦❢ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ✭❈❤❛♣t❡r ✶P❈✲✹✮✿

Q(x) = a(x − h)2 + k .

✺✳✷✳

❚❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s

✸✺✸

❚❤✐s ✐s t❤❡ ❣❡♥❡r❛❧ r❡s✉❧t✿

❚❤❡♦r❡♠ ✺✳✷✳✶✿ ❈❡♥t❡r❡❞ ❋♦r♠ ♦❢ P♦❧②♥♦♠✐❛❧ a✱ ❡✈❡r② x = a✱ ✐✳❡✳✱

❋♦r ❡❛❝❤ r❡❛❧ ♥✉♠❜❡r ❢♦r♠

❝❡♥t❡r❡❞ ❛t

❞❡❣r❡❡

n

♣♦❧②♥♦♠✐❛❧

P

❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡

P (x) = c0 + c1 (x − a) + c2 (x − a)2 + ... + cn (x − a)n , ❢♦r s♦♠❡ r❡❛❧ ♥✉♠❜❡rs

Pr♦♦❢✳

❲❡ ✉s❡ t❤✐s ❝❤❛♥❣❡

♦❢ ✈❛r✐❛❜❧❡s ✿

c0 , ..., cn ✳

x 7→ x − a✱ ❛ s❤✐❢t t♦ t❤❡ r✐❣❤t ❜② a✳

❚❤❡ st❛♥❞❛r❞ ❢♦r♠ ✐s t❤❡♥ ❥✉st t❤❡ ❢♦r♠ ❝❡♥t❡r❡❞ ❛t x = 0✳ ■t ✐s ❢r♦♠ ❛♠♦♥❣ t❤❡s❡ ♣♦❧②♥♦♠✐❛❧s ✇❡ ✇✐❧❧ ❝❤♦♦s❡ t❤❡ ❜❡st ❛♣♣r♦①✐♠❛t✐♦♥ ❛t x = a✳ ❇❡❧♦✇ ✐s ❛ t❛❜❧❡ t❤❛t s❤♦✇s t❤❡ ♣r♦❣r❡ss ♦❢ ❜❡tt❡r ❛♥❞ ❜❡tt❡r ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦❧❧♦✇✐♥❣ t❤❡ ✐❞❡❛s ❞❡✈❡❧♦♣❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❚❤❡ ❞❡❣r❡❡s ♦❢ t❤❡s❡ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❧✐st❡❞ ✐♥ t❤❡ ✜rst ❝♦❧✉♠♥✱ ✇❤✐❧❡ t❤❡ ✜rst r♦✇ s❤♦✇s t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ t❡r♠s ♦❢ t❤❡s❡ ♣♦❧②♥♦♠✐❛❧s✿ ❞❡❣r❡❡s

n

...

3

2

1

0

0

c0 = T 0

1

c1 (x − a) + c0 = T1

c2 (x − a)2 + c1 (x − a) + c0 = T2

2 3

✳✳ ✳

c3 (x − a)3 + c2 (x − a)2 + c1 (x − a) + c0 = T3

✳✳ ✳

✳✳ ✳

✳✳ ✳

✳✳ ✳

✳✳ ✳

n cn (x − a)n + ... + c3 (x − a)3 + c2 (x − a)2 + c1 (x − a) + c0 = Tn

❖♥❝❡ ❛❣❛✐♥✱ t❤❡ s❡q✉❡♥❝❡ Tn ✐s t❤❡ s✉♠ ♦❢ t❤❡ s❡q✉❡♥❝❡ cn (x − a)n ✳

❍♦✇ ❞♦ ✇❡ ❝❤♦♦s❡ t❤❡ ❜❡st❄ ❲❡ r❡q✉✐r❡ t❤❛t t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ ❝♦♥✈❡r❣❡s t♦ ③❡r♦ ❢❛st❡r ❛♥❞ ❢❛st❡r✿

❉❡✜♥✐t✐♦♥ ✺✳✷✳✷✿ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ❙✉♣♣♦s❡ f ✐s n t✐♠❡s ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✳ ❚❤❡♥ t❤❡ nt❤ ♣♦❧②♥♦♠✐❛❧✱ n = 0, 1, 2, ...✱ ✐s ❞❡✜♥❡❞ r❡❝✉rs✐✈❡❧② ❜②✿ T0 = f (a),

Tn (x) = Tn−1 + cn (x − a)n ,

❚❛②❧♦r

✉♥❞❡r t❤❡ r❡q✉✐r❡♠❡♥t t❤❛t t❤❡ ❡rr♦r ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♣♣r♦❛❝❤❡s 0 ❛t x = a ❢❛st❡r t❤❛♥ (x − a)n ✿ f (x) − Tn (x) =0 x→a (x − a)n lim

✺✳✷✳

❚❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s

✸✺✹

❚❤❡ ❝♦❡✣❝✐❡♥ts c0 , c1 , ..., cn ❛r❡ ❝❛❧❧❡❞ t❤❡ ❚❛②❧♦r

❝♦❡✣❝✐❡♥ts ♦❢ f ✳

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❡rr♦r ✐s✿ E = f − Tn = o (x − a)n



❆♣♣❧②✐♥❣ t❤❡ P♦✇❡r ❋♦r♠✉❧❛ ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ r❡♣❡❛t❡❞❧② ♣r♦❞✉❝❡s ♠♦r❡ ❛♥❞ ♠♦r❡ ❜✉t s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r ❝♦❡✣❝✐❡♥ts ✉♥t✐❧ ✐t r❡❛❝❤❡s 1✿ x′ = 1, (x2 )′′ = (2x)′ = 2, (x3 )′′ = (3x2 )′′ = (6x)′ = 6, .. .

❚❤✐s ✐s t❤❡ ❣❡♥❡r❛❧ r❡s✉❧t✳

❚❤❡♦r❡♠ ✺✳✷✳✸✿ ♥t❤ ❞❡r✐✈❛t✐✈❡ ♦❢ ♥t❤ ♣♦✇❡r ❋♦r ❛♥② n = 1, 2, 3...✱ ✇❡ ❤❛✈❡✿ (xn )(n) = n!

Pr♦♦❢✳ (xn )′ =⇒ (xn ) ′′ =⇒ (xn ) ′′

✳✳ ✳

xn−1

= n· ′

= n · (n − 1)·

✳✳ ✳

✳✳ ✳

✳✳ ✳

=⇒ (xn )(n−1) = n · (n − 1) · ... · 3· n (n)

=⇒ (x )

x

 n−2 ′

x2

= n · (n − 1) · ... · 3 · 2· x

′

′

′

= n·

xn−1 =⇒

= n · (n − 1)·

xn−2 =⇒

= n · (n − 1) · (n − 2)·

✳✳ ✳

✳✳ ✳

xn−3 =⇒

✳✳ ✳

= n · (n − 1) · (n − 2) · ... · 3 · 2· x1 = n · (n − 1) · (n − 2) · ... · 3 · 2· 1

=⇒ = n!

❙❛♠❡ ❢♦r ♣♦✇❡rs ♦❢ (x − a)✿ (x − a)n

(n)

= n!

❙♦✱ t❤❛t✬s ✇❤② t❤❡ ❢❛❝t♦r✐❛❧ ❛♣♣❡❛rs ✐♥ t❤❡ ❢♦rt❤❝♦♠✐♥❣ ❢♦r♠✉❧❛s✳✳✳

❊①❡r❝✐s❡ ✺✳✷✳✹

Pr♦✈❡ t❤❡ ❧❛st ❢♦r♠✉❧❛✳ ❲❡ ❥✉♠♣ str❛✐❣❤t t♦ t❤❡ ❛♥s✇❡r✳ ❚❤✐s ✐s ♦✉r ♠❛✐♥ ✐♥t❡r❡st✳

❚❤❡♦r❡♠ ✺✳✷✳✺✿ ❚❛②❧♦r ❈♦❡✣❝✐❡♥ts ❙✉♣♣♦s❡ f ✐s n t✐♠❡s ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✳ ❚❤❡♥ t❤❡ ❚❛②❧♦r ❝♦❡✣❝✐❡♥ts ♦❢ f ♠✉st ❜❡✿ 1 1 c0 = f (a), c1 = f ′ (a), c2 = f (2) (a), ..., cn = f (n) (a) 2 n!

✺✳✷✳

✸✺✺

❚❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s

Pr♦♦❢✳

❋r♦♠ t❤❡ ❧❛st t❤❡♦r❡♠✱ ✐t ❢♦❧❧♦✇s t❤❛t✿ 1 (2) 1 c0 = T0 (a), c1 = T1′ (a), c2 = T2 (a), ..., cn = Tn(n) (a) . 2 n!

❲❡ ✉s❡ t❤❡ ❧✐♠✐t ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ❢♦r ❡❛❝❤ k = 0, 1, 2, ..., n✱ ❛s ❢♦❧❧♦✇s✳ ❙t❛rt ✇✐t❤✿ 0 : 0 = lim

x→a

f (x) − T0 (x) = lim (f (x) − T0 (x)) =⇒ T0 (a) = lim T0 (x) = lim f (x) = f (a) x→a x→a x→a (x − a)0

❲❡ ✉s❡ t❤❡ ❢❛❝t t❤❛t ❜♦t❤ T0 ❛♥❞ f ❛r❡ ❝♦♥t✐♥✉♦✉s ❛t x = a✳ ■t ❢♦❧❧♦✇s t❤❛t c0 = T0 (a) = f (a)✳ ◆❡①t✱ ✇❡ ✉s❡ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡ ✿ f ′ (x) − T1′ (x) f (x) − T1 (x) = lim =⇒ T1′ (a) = lim T1′ (x) = lim f ′ (x) = f ′ (a) 1 x→a x→a x→a x→a (x − a) 1

1 : 0 = lim

❲❡ ✉s❡ t❤❡ ❢❛❝t t❤❛t ❜♦t❤ T1′ ❛♥❞ f ′ ❛r❡ ❝♦♥t✐♥✉♦✉s ❛t x = a✳ ◆❡①t✱ ✇❡ ❛♣♣❧② ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡ t✇✐❝❡✿ f ′′ (x) − T ′′ 2 (x) f (x) − T2 (x) = lim =⇒ T ′′ 2 (a) = lim T ′′ 2 (x) = lim f ′′ (x) = f ′′ (a) x→a x→a x→a x→a (x − a)2 2

2 : 0 = lim

❲❡ ✉s❡ t❤❡ ❢❛❝t t❤❛t ❜♦t❤ T ′′ 2 ❛♥❞ f ′′ ❛r❡ ❝♦♥t✐♥✉♦✉s ❛t x = a✳ ❆♥❞ s♦ ♦♥✳ ❋♦r t❤❡ ❧❛st st❡♣✱ ✇❡ ❛♣♣❧② ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡ n t✐♠❡s✿ (n)

f (n) (x) − Tn (x) f (x) − Tn (x) = lim =⇒ Tn(n) (a) = lim Tn(n) (x) = lim f (n) (x) = f (n) (a) x→a x→a x→a x→a (x − a)n n!

n : 0 = lim

❲❡ ✉s❡ t❤❡ ❢❛❝t t❤❛t ❜♦t❤ Tn(n) ❛♥❞ f (n) ❛r❡ ❝♦♥t✐♥✉♦✉s ❛t x = a✳ ❊①❡r❝✐s❡ ✺✳✷✳✻

❋✐♥✐s❤ t❤❡ ♣r♦♦❢✳ ❊①❡r❝✐s❡ ✺✳✷✳✼

Pr♦✈❡ t❤❡ ❝♦♥✈❡rs❡✳ ❊①❡r❝✐s❡ ✺✳✷✳✽

Pr♦✈❡ t❤❛t t❤❡ nt❤ ❞❡❣r❡❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ♦❢ ❛♥ nt❤ ❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧ ✐s t❤❛t ♣♦❧②♥♦♠✐❛❧✳ ❲❛r♥✐♥❣✦ ❚❤❡s❡ ❛r❡ ♥✉♠❜❡rs✱ ♥♦t ❢✉♥❝t✐♦♥s❀ t❤❡ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❡✈❛❧✉❛t❡❞ ❛t

1 (k) . ck = f k! x=a

❙♦✱ ✇❡ ❤❛✈❡✿ T0 = f (a),

❛♥❞✱ ✐♥ s✐❣♠❛ ♥♦t❛t✐♦♥✿

x = a✿

Tn+1 (x) = Tn +

1 f (n+1) (a)(x − a)n+1 , (n + 1)!

✺✳✷✳

❚❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s

✸✺✻

❚❤❡♦r❡♠ ✺✳✷✳✾✿ ❚❛②❧♦r P♦❧②♥♦♠✐❛❧s ♦❢ ❋✉♥❝t✐♦♥s

❙✉♣♣♦s❡ f ✐s n t✐♠❡s ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✳ ❚❤❡♥ t❤❡ n✲t❤ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ♦❢ f ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ Tn (x) =

n X 1 (k) f (a)(x − a)k k! k=0

❚❤✐s ✐s ✐♥❞❡❡❞ ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ n✳ ■t ✐s ❝❡♥t❡r❡❞ ❛t a✳ ❊①❛♠♣❧❡ ✺✳✷✳✶✵✿ ❡①♣♦♥❡♥t ❛t ✵

❙♦♠❡ ❢✉♥❝t✐♦♥s ❛r❡ s♦ ❡❛s② t♦ ❞✐✛❡r❡♥t✐❛t❡ t❤❛t ✇❡ ❝❛♥ q✉✐❝❦❧② ✜♥❞ ❛❧❧ ♦❢ ✐ts ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s✳ ❋♦r ❡①❛♠♣❧❡✱ ❝♦♥s✐❞❡r f (x) = ex

❛t x = 0✳ ❚❤❡♥ f

(k)

❚❤❡r❡❢♦r❡✱

(0) = e

Tn (x) =

x

= 1. x=0

n X 1 k x . k! k=0

❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r♠✉❧❛✿ e≈1+

1 1 1 1 + + + ... + , 1! 2! 3! n!

✇❤✐❝❤ ❣✐✈❡s ❛ ❜❡tt❡r ❛❝❝✉r❛❝② ✇✐t❤ ❡❛❝❤ ♥❡✇ t❡r♠✳ ❚❤❡ ❡①❛❝t ♠❡❛♥✐♥❣ ♦❢ t❤✐s st❛t❡♠❡♥t ✐s ❡①♣❧❛✐♥❡❞ ❧❛t❡r✳ ❚❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ✐s ❡s♣❡❝✐❛❧❧② ❝♦♥✈❡♥✐❡♥t✿ Tn+1 (x) = Tn (x) +

1 (x − a)n+1 . (n + 1)!

❲❡ ✉s❡ ✐t ✐♥ ❛ s♣r❡❛❞s❤❡❡t ❛s ❢♦❧❧♦✇s✿ ❂❘❈❬✲✶❪✰❘✽❈✴❋❆❈❚✭❘✼❈✮✯✭❘❈✶✲❘✹❈✷✮✂❘✼❈

❚❤❡ ✜rst t❤r❡❡ ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡✿ 1 T0 (x) = 1, T1 (x) = x + 1, T2 (x) = x2 + x + 1 . 2

❲❡ ❝❛♥ ❝r❡❛t❡ ❛s ♠❛♥② ❛s ✇❡ ❧✐❦❡ ✐♥ ❡❛❝❤ ❝♦❧✉♠♥✿

✺✳✷✳

❚❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s

✸✺✼

❲❡ ❝❛♥ s❡❡ ❤♦✇ t❤❡ ❝✉r✈❡s st❛rt t♦ r❡s❡♠❜❧❡ t❤❡ ♦r✐❣✐♥❛❧ ❣r❛♣❤ ✕ ❜✉t ♦♥❧② ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ x = 0✳ ❊❧s❡✇❤❡r❡✱ ✇❡ ❦♥♦✇ t❤❛t ♣♦❧②♥♦♠✐❛❧s ❤❛✈❡ t❤❡ ♣r♦♣❡rt② t❤❛t

Tn (x) → ∞ ❛s x → ∞ . ❚❤❡r❡❢♦r❡✱ t❤❡② ❝❛♥ ♥❡✈❡r ❣❡t ❝❧♦s❡ t♦ t❤❡ ❤♦r✐③♦♥t❛❧ ❛s②♠♣t♦t❡ ❢♦r x → −∞ ✭❞❡❣r❡❡s 4 − 7✮✿

❲❡ ❝❛♥✬t s❡❡ ✐t ♦♥ t❤❡ ♦t❤❡r ❡♥❞✱ ❜✉t✱ ❢♦r x → +∞✱ ✇❡ ❦♥♦✇ ❢r♦♠ ▲✬❍♦♣✐t❛❧✬s ❘✉❧❡ t❤❛t ♣♦❧②♥♦♠✐❛❧s ❛r❡ t♦♦ s❧♦✇ t♦ ❝♦♠♣❡t❡ ✇✐t❤ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❣♦♦❞ ♥❡✇s ✐s t❤❛t✱ ✇✐t❤✐♥ ❛♥ ✐♥t❡r✈❛❧ ❝❡♥t❡r❡❞ ❛t 0✱ t❤❡r❡ ✐s ✈✐rt✉❛❧❧② ♥♦ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❣r❛♣❤s ✭❞❡❣r❡❡s 4 − 7✮✿

❲❡ ❤❛✈❡✱ ❜② ❢❛r✱ ♠♦r❡ t❤❛♥ ❥✉st ❛ s❡q✉❡♥❝❡ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❘❡❝❛❧❧ ❢r♦♠ ❱♦❧✉♠❡ ✶ t❤❛t ❛s ♥✉♠❜❡rs ❛r❡ r❡♣❧❛❝❡❞ ✇✐t❤ t❤❡✐r ❛♣♣r♦①✐♠❛t✐♦♥s✱ ✇❡ ❛r❡ st✐❧❧ ❛❜❧❡ t♦ ❞♦ ❛❧❣❡❜r❛ ✇✐t❤ t❤❡♠✳ ❙✐♠✐❧❛r❧②✱ ❛s ❢✉♥❝t✐♦♥s ❛r❡ r❡♣❧❛❝❡❞ ✇✐t❤ t❤❡✐r ❛♣♣r♦①✐♠❛t✐♦♥s✱ ✇❡ ❛r❡ st✐❧❧ ❛❜❧❡ t♦ ❞♦ ❝❛❧❝✉❧✉s ✇✐t❤ t❤❡♠✿

✺✳✷✳

❚❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s

✸✺✽

❚❤❡♦r❡♠ ✺✳✷✳✶✶✿ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ❚❛②❧♦r P♦❧②♥♦♠✐❛❧s ❙✉♣♣♦s❡ f ✐s n t✐♠❡s ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✳ ❚❤❡♥ t❤❡ ✜rst n ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ nt❤ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ♦❢ ❛ ❢✉♥❝t✐♦♥ f ❛❣r❡❡ ✇✐t❤ t❤♦s❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ✐ts❡❧❢❀ ✐✳❡✳✱ Tn(n) (a) = f (n) (a)

❈♦♥✈❡rs❡❧②✱ t❤✐s ♣♦❧②♥♦♠✐❛❧ ✐s t❤❡ ♦♥❧② ♦♥❡ t❤❛t s❛t✐s✜❡s t❤✐s ♣r♦♣❡rt②✳

Pr♦♦❢✳ ❙✐♥❝❡ Tn ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ n✱ t❤❡ ♦♥❧② t❡r♠ t❤❛t ♠❛tt❡rs ❢♦r ✐ts nt❤ ❞❡r✐✈❛t✐✈❡ ✐s t❤❡ ❧❛st ♦♥❡✱ cn (x − a)n ✳ ❇② t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡✱ ✇❡ ❤❛✈❡✿ Tn(n) (x) = cn (x − a)n

❊①❡r❝✐s❡ ✺✳✷✳✶✷

(n)

= cn · n! =

f (n) (a) · n! = f (n) (a) . n!

Pr♦✈❡ t❤❡ ❝♦♥✈❡rs❡✳

❊①❡r❝✐s❡ ✺✳✷✳✶✸ Pr♦✈❡ t❤❛t t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ♦❢ ❛♥ ❡✈❡♥ ✭♦❞❞✮ ❢✉♥❝t✐♦♥ ❤❛✈❡ ♦♥❧② ❡✈❡♥ ✭♦❞❞✮ t❡r♠s✳

❊①❛♠♣❧❡ ✺✳✷✳✶✹✿ ❡①♣♦♥❡♥t ❛t ✶ ▲❡t✬s ❛❣❛✐♥ ❝♦♥s✐❞❡r

f (x) = ex

❜✉t ❛t x = 1 t❤✐s t✐♠❡✳ ❲❡ st✐❧❧ ❤❛✈❡ ❛❧❧ t❤❡ ❞❡r✐✈❛t✐✈❡s r❡❛❞②✿ f

❚❤❡r❡❢♦r❡✱ t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ✐s✿

(k)

(1) = e x

= e. x=1

n X e (x − 1)k . Sn (x) = k! k=0

■t ✐s ♥❛♠❡❞ t❤✐s ✇❛② ✐♥ ♦r❞❡r t♦ ❝♦♠♣❛r❡ ✐t t♦ t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ♦❢ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ ❛t x = 0✿ Tn (x) =

n X 1 k x . k! k=0

❙♦✱ ✇❡ ❤❛✈❡ ❛ s✐♠♣❧❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡♠✿ Sn (x) = eTn (x − 1) .

❚❤❡ ♦r✐❣✐♥❛❧ ✐s s❤✐❢t❡❞ ♦♥❡ ✉♥✐t r✐❣❤t ❛♥❞ t❤❡♥ str❡t❝❤❡❞ ✈❡rt✐❝❛❧❧② ❜② ❛ ❢❛❝t♦r ♦❢ e✳✳✳ ❚❤❡ ✜rst t❤r❡❡ ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡✿ e S0 (x) = e, S1 (x) = e(x − 1) + e, S2 (x) = (x − 1)2 + e(x − 1) + e . 2

❲❡ s❡❡ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❞❡❣r❡❡s 0 − 7 ❜❡❧♦✇✿

✺✳✷✳

❚❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s

✸✺✾

❍♦✇ ✇❡❧❧ ❞♦ ♦✉r ❛♣♣r♦①✐♠❛t✐♦♥s ✇♦r❦❄ ❘❡❝❛❧❧ ❤♦✇ ❛ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥

f

❝♦♠❡s ✇✐t❤ ❛ ✏❢✉♥♥❡❧✑✿ ✐t ✐s ❝❡♥t❡r❡❞ ♦♥ t❤❡ t❛♥❣❡♥t ❧✐♥❡✱

❛♥❞ ✐ts ✉♣♣❡r ❛♥❞ ❧♦✇❡r ❜♦✉♥❞s ❛r❡ t❤❡ t✇♦ ♣❛r❛❜♦❧❛s ✭❥✉st ❧✐❦❡ q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s✮✳ ❚❤✐s ✐s ✇❤❡r❡ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥

f

♠✉st r❡s✐❞❡✿

❙✐♠✐❧❛r❧② t♦ t❤❡ ❡rr♦r ❜♦✉♥❞s ❢♦r ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s ✭❈❤❛♣t❡r ✸✮✱ ✇❡ ❤❛✈❡ ✐♥t❡r✈❛❧s t❤❛t ❝♦♥t❛✐♥ t❤❡ ✉♥❦♥♦✇♥ ✈❛❧✉❡ ✕ ✐❢ ✇❡ ✜① ❛ ✈❛❧✉❡ ♦❢

x✳

❋♦r ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❛ ❤✐❣❤❡r ❞❡❣r❡❡✱ t❤❡ ❜❧✉❡ ❧✐♥❡ ✐♥ t❤❡ ✐❧❧✉str❛t✐♦♥ ✐s t♦ ❜❡ r❡♣❧❛❝❡❞ ✇✐t❤ ❛ ♣❛r❛❜♦❧❛ ♦r ❛ ❤✐❣❤❡r ❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧✳ ❲❡ ❛❝❝❡♣t t❤❡ r❡s✉❧t ❜❡❧♦✇ ✇✐t❤♦✉t ♣r♦♦❢✿ ❚❤❡♦r❡♠ ✺✳✷✳✶✺✿ ❊rr♦r ❇♦✉♥❞ ❢♦r ❚❛②❧♦r ❆♣♣r♦①✐♠❛t✐♦♥

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ f ✐s (n + 1) t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ❢♦r ❡❛❝❤ i = 0, 1, 2, ..., n + 1✱ ✇❡ ❤❛✈❡ |f (i) (t)| < Ki ❢♦r ❡✈❡r② t ❜❡t✇❡❡♥ a ❛♥❞ x ,

❛♥❞ s♦♠❡ r❡❛❧ ♥✉♠❜❡r Ki ✳ ❚❤❡♥ En (x) = |f (x) − Tn (x)| ≤ Kn+1 ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❝♦♥s❡q✉❡♥❝❡ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

|x − a|n+1 . (n + 1)!

✺✳✷✳

❚❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s

✸✻✵

❈♦r♦❧❧❛r② ✺✳✷✳✶✻✿ ❚❛②❧♦r ❆♣♣r♦①✐♠❛t✐♦♥

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ f ✐s ✐♥✜♥✐t❡❧② ♠❛♥② t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ❢♦r ❡❛❝❤ i = 0, 1, 2, ...✱ ✇❡ ❤❛✈❡ |f (i) (t)| < Ki ❢♦r ❡✈❡r② t ❜❡t✇❡❡♥ a ❛♥❞ x ,

❛♥❞ s♦♠❡ r❡❛❧ ♥✉♠❜❡r Ki ✳ ❚❤❡♥ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ❛t x ❝♦♥✈❡r❣❡ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t x✿ Tn (x) → f (x) ❛s n → ∞ ❚❤❡ r❡s✉❧t ✇✐❧❧ s❡r✈❡ ❛s ❛ ❝♦r♥❡rst♦♥❡ ❢♦r t❤❡ r❡st ♦❢ t❤❡ t❤❡♦r②✳ ❊①❡r❝✐s❡ ✺✳✷✳✶✼

❉❡r✐✈❡ t❤❡ ❝♦r♦❧❧❛r② ❢r♦♠ t❤❡ t❤❡♦r❡♠✳

❏✉st ❛s ✇✐t❤ t❤❡ ❜♦✉♥❞s ❢♦r ✐♥t❡❣r❛❧s✱ ✇❡ ♥❡❡❞ t♦ ❦♥♦✇ ❜♦✉♥❞s ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥

f

✕ ❛ ♣r✐♦r✐ ✕ ✐♥ ♦r❞❡r t♦ ❦♥♦✇ ❤♦✇ ❢❛r ✐t ❝❛♥ ❣♦ ❢r♦♠ t❤❡ ✈❛❧✉❡ t❤❛t ✇❡ ❞♦ ❦♥♦✇✳ ◆♦t❡ ❛❧s♦ t❤❛t t❤❡

❡rr♦r ❜♦✉♥❞ r❡s❡♠❜❧❡s t❤❡

(n + 1)st

nt❤

❚❛②❧♦r t❡r♠✳

❚❤❡ ✐♥❡q✉❛❧✐t② ✐s ❛ ✏sq✉❡❡③❡✑ ✭❈❤❛♣t❡rs ✷❉❈✲✶ ❛♥❞ ✷❉❈✲✷✮ ❢♦r t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥

f✿

Tn (x) − En (x) < f (x) < Tn (x) + En (x) . ❚❤✐s ✐♥❡q✉❛❧✐t② ❣✉❛r❛♥t❡❡s t❤❛t✿

•

❚❤❡ ❡rr♦r ❛♣♣r♦❛❝❤❡s t♦

0

❛s

n→∞

•

❚❤❡ ❡rr♦r ❛♣♣r♦❛❝❤❡s t♦

0

❛s

x→a

■♥ ❢❛❝t✱ t❤❡ ❧❛tt❡r ❝♦♥✈❡r❣❡♥❝❡ ✐s

❢❛st ✿

❢♦r ❡❛❝❤

❢♦r ❡❛❝❤

x✳

n✳

q✉❛❞r❛t✐❝✱ ❝✉❜✐❝✱ ❡t❝✳

❊①❛♠♣❧❡ ✺✳✷✳✶✽✿ r♦♦t

▲❡t✬s s❡❡ ❤♦✇ ❝❧♦s❡ ✇❡ ❛r❡ t♦ t❤❡ tr✉t❤ ✇✐t❤ ♦✉r q✉❛❞r❛t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

f (x) =

√

x

❛r♦✉♥❞

a=4

❢r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❘❡❝❛❧❧ t❤❛t ✇❡ ❤❛✈❡✿

1 1 (x − 4)2 . T2 (x) = 2 + (x − 4) − 4 2 · 32

❚❤❡ r❡s✉❧t ❝♦♠❡s ❢r♦♠ ♦✉r ❝♦♠♣✉t❛t✐♦♥s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ✉♣ t♦ t❤❡ s❡❝♦♥❞ t♦ ✇❤✐❝❤ ✇❡ ❛❞❞ t❤❡ t❤✐r❞✿

f (x)

=

√

x

f ′ (x)

= (x1/2 )′ ′  1 −1/2 ′′ f (x) = x 2 ′ 1 −3/2 (3) f (x) = − x 4

◆❡①t✱ ✇❡ ♥♦t✐❝❡ t❤❛t

f (3)

=⇒ f (4) = 2 1 −1/2 1 = x =⇒ f ′ (4) = 2 4 1 −3/2 1 =− x =⇒ f ′′ (4) = − 4 32 3 1 −3 −5/2 x = x−5/2 =− 4 2 8

✐s ❞❡❝r❡❛s✐♥❣✳ ❚❤❡r❡❢♦r❡✱ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧

[4, +∞)✱ ♦✉r ❜❡st ✭s♠❛❧❧❡st✮ ✉♣♣❡r

❜♦✉♥❞ ❢♦r ✐t ✐s ✐ts ✐♥✐t✐❛❧ ✈❛❧✉❡✿

3 3 3 = . |f (3) (x)| ≤ |f (3) (4)| = 4−5/2 = 8 8 · 32 256 ❙♦✱ ♦✉r ❜❡st ✭s♠❛❧❧❡st✮ ❝❤♦✐❝❡ ♦❢ t❤❡ ✉♣♣❡r ❜♦✉♥❞ ✐s✿

K3 =

3 . 256

✺✳✷✳

❚❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s

✸✻✶

❚❤❡♥✱ E2 (x) = |f (x) − T2 (x)| ≤ K3 √ ❚❤✐s ✐s ✇❤❡r❡ t❤❡ ❣r❛♣❤ ♦❢ y = x ❧✐❡s✿

|x − 4|3 3 1 = |x − 4|3 = |x − 4|3 . 3! 256 · 3! 512

❲❡ ❝❛♥ ♥♦✇ ❛❞❞r❡ss t❤❡ ❛❝❝✉r❛❝② ♦❢ ♦✉r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

√

4.1✿

1 0.001 |4.1 − 4|3 = ≈ 0.000002 . 512 512 √ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❛♥ ✐♥t❡r✈❛❧ ✇❤❡r❡ t❤❡ tr✉❡ ✈❛❧✉❡ ♦❢ 4.1 ♠✉st ❧✐❡✿ √ T2 (4) − .000002 ≤ 4.1 ≤ T2 (4) + .000002 . E2 (4.1) =

❊①❛♠♣❧❡ ✺✳✷✳✶✾✿ ❡①♣♦♥❡♥t

▲❡t✬s ❡st✐♠❛t❡ e−0.01 ✇✐t❤✐♥ 6 ❞❡❝✐♠❛❧s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ s✉❝❤ ❛♥ n t❤❛t ✇❡ ❛r❡ ❣✉❛r❛♥t❡❡❞ t♦ ❤❛✈❡✿ e−0.01 − Tn (−.01) < 10−6 ,

✇❤❡r❡ Tn ✐s t❤❡ nt❤ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ♦❢ ex ❛r♦✉♥❞ x = 0✳ ❲❡ ❡st✐♠❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ✐♥t❡r✈❛❧ [−0.01, 0]✿ x (i) (e ) = ex ≤ 1 = Ki .

❚❤❡♥✱ ❤♦✇ ❞♦ ✇❡ ♠❛❦❡ t❤❡ ❡rr♦r ❜♦✉♥❞ s❛t✐s❢② t❤✐s✿

−0.01 |x − a|n+1 e − Tn (−.01) ≤ Kn+1 < 10−6 ? (n + 1)! ❲❡ r❡✲✇r✐t❡ ❛♥❞ s♦❧✈❡ t❤✐s ✐♥❡q✉❛❧✐t② ❢♦r n✿ 0.1n+1 < 10−6 . (n + 1)!

❆ ❧❛r❣❡ ❡♥♦✉❣❤ n ✇✐❧❧ ✇♦r❦✳ ❲❡ s✐♠♣❧② ❣♦ t❤r♦✉❣❤ ❛ ❢❡✇ ✈❛❧✉❡s ♦❢ n = 1, 2, 3, ... ✉♥t✐❧ t❤❡ ✐♥❡q✉❛❧✐t② ✐s s❛t✐s✜❡❞✿ n 3 4 5 n+1 0.1 0.000004167 0.000000083 0.000000001 (n + 1)!

■t✬s n = 4 ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ❛♥s✇❡r ✐s✿ 4 X 1 (−0.01)i . T4 (−0.01) = i! i=0

✺✳✸✳

❙❡q✉❡♥❝❡s ♦❢ ❢✉♥❝t✐♦♥s

✸✻✷

❲❡ ♥♦✇ ❝♦♠❡ ❜❛❝❦ t♦ t❤❡ r❡q✉✐r❡♠❡♥t ❢r♦♠ ❧❛st s❡❝t✐♦♥ t❤❛t t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s ❤❛✈❡ t♦ ♣r♦✈✐❞❡ ❢❛st❡r ❛♥❞ ❢❛st❡r ❝♦♥✈❡r❣❡♥❝❡ t♦ ③❡r♦ ♦❢ t❤❡ ❡rr♦r✳ ❈♦r♦❧❧❛r② ✺✳✷✳✷✵✿ ❊rr♦r ❈♦♥✈❡r❣❡♥❝❡

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ f ✐s (n + 1) t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ❢♦r ❡❛❝❤ i = 0, 1, 2, ..., n + 1✱ ✇❡ ❤❛✈❡ |f (i) (t)| < Ki ❢♦r ❡✈❡r② t ❜❡t✇❡❡♥ a ❛♥❞ x ,

❛♥❞ s♦♠❡ r❡❛❧ ♥✉♠❜❡r Ki ✳ ❚❤❡♥ En (x) |f (x) − Tn (x)| = → 0 ❛s x → a . n |x − a| |x − a|n

✺✳✸✳ ❙❡q✉❡♥❝❡s ♦❢ ❢✉♥❝t✐♦♥s

▲❡t✬s t❛❦❡ ❛ ❝❧♦s❡r ❧♦♦❦ ❛t t❤❡ t❤❡♦r❡♠s ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ • ❋✐rst✱ t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ♦❢ f ❢♦r♠ ❛

s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s

Tn : n = 0, 1, 2, 3, ... • ❙❡❝♦♥❞✱ s✉♣♣♦s❡ ✇❡ ✜① ❛ ✈❛❧✉❡ x ✇✐t❤✐♥ t❤❡ ✐♥t❡r✈❛❧✱ t❤❡♥ ✇❡ ❤❛✈❡ ❛

s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs

Tn (x) : n = 0, 1, 2, 3, ...

❚❤✐s s❡q✉❡♥❝❡ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt② ❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠ ❛❜♦✉t ❚❛②❧♦r s❡❝t✐♦♥✱ ❢♦r ❡❛❝❤ x✿

❆♣♣r♦①✐♠❛t✐♦♥ ✐♥ t❤❡ ❧❛st

Tn (x) → f (x) ❛s n → ∞ ❊①❛♠♣❧❡ ✺✳✸✳✶✿ ❡①♣♦♥❡♥t

■♥ ♦✉r ❛♣♣r♦①✐♠❛t✐♦♥s✱ ✇❡ ❝❤♦♦s❡ t♦ ❝♦♥❝❡♥tr❛t❡ ♦♥ ❛ s✐♥❣❧❡ ✈❛❧✉❡ ♦❢ x ❛t ❛ t✐♠❡✳ ❍♦✇ ❛❜♦✉t x = 2✿

❋r♦♠ t❤✐s s❡q✉❡♥❝❡ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ex ✱ ✇❡ t❛❦❡ t❤✐s✿ Tn (2) → e2 ❛s n → ∞ .

❆♥❞ s♦ ♦♥✳

✺✳✸✳

❙❡q✉❡♥❝❡s ♦❢ ❢✉♥❝t✐♦♥s

✸✻✸

❙✐♥❝❡ t❤✐s ❝♦♥✈❡r❣❡♥❝❡ ♦❝❝✉rs ❢♦r ❡❛❝❤ x✱ ✇❡ ❝❛♥ s♣❡❛❦ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✇❤♦❧❡ s❡q✉❡♥❝❡ ♦❢ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s✳ ❚❤❡ ❣❡♥❡r❛❧ ✐❞❡❛ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳

❉❡✜♥✐t✐♦♥ ✺✳✸✳✷✿ ♣♦✐♥t✇✐s❡ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s fn ❞❡✜♥❡❞ ♦♥ ✐♥t❡r✈❛❧ I ✳ ❲❡ s❛② t❤❛t t❤❡ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡s ♣♦✐♥t✇✐s❡ ♦♥ I t♦ ❛ ❢✉♥❝t✐♦♥ f ✐❢ ❢♦r ❡✈❡r② x✱ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛t x ❝♦♥✈❡r❣❡✱ ❛s ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs✱ t♦ t❤❡ ✈❛❧✉❡ ♦❢ f ❛t x✱ ✐✳❡✳✱ fn (x) → f (x) .

❖t❤❡r✇✐s❡✱ ✇❡ s❛② t❤❛t t❤❡ s❡q✉❡♥❝❡ ❞✐✈❡r❣❡s

♣♦✐♥t✇✐s❡✳

❲❛r♥✐♥❣✦ ■t ♦♥❧② t❛❦❡s ❞✐✈❡r❣❡♥❝❡ ❢♦r ❛ s✐♥❣❧❡ ✈❛❧✉❡ ♦❢

x

t♦

♠❛❦❡ t❤❡ ✇❤♦❧❡ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ❞✐✈❡r❣❡✳

❊①❛♠♣❧❡ ✺✳✸✳✸✿ s❤r✐♥❦✐♥❣ ❚❤✐s ✐s ❤♦✇✱ t②♣✐❝❛❧❧②✱ ❛ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡s✱ t♦ t❤❡ ③❡r♦ ❢✉♥❝t✐♦♥✱ f (x) = 0✱ ✐♥ t❤✐s ❝❛s❡✿

❚❤❡ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥str✉❝t❡❞ ❛s ❢♦❧❧♦✇s✿ g(x) = 3 − cos x,

❨♦✉ ❝❛♥ ❝❤♦♦s❡ ❛♥② ♦t❤❡r ❢✉♥❝t✐♦♥ g ✳

 n 2 fn (x) = g(x) . 3

❚❤❡ ♠✉❧t✐♣❧❡ ✐s ❣❡tt✐♥❣ s♠❛❧❧❡r ❛♥❞ s❤r✐♥❦s t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ g t♦✇❛r❞ t❤❡ x✲❛①✐s✳ ❚❤❡ ♣r♦♦❢ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✐s r♦✉t✐♥❡ ✭❛❢t❡r ❛❧❧✱ t❤❡ s❡q✉❡♥❝❡ ✐s ❣❡♦♠❡tr✐❝✮✿

❜② t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ t✐♦♥s ❝♦♥✈❡r❣❡s t♦ 0✿

 n  n 2 2 → 0, g(x) = |g(x)| |fn (x)| = 3 3

❘✉❧❡✳

❚❤✉s✱ ❡❛❝❤ ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡ ♣r♦❞✉❝❡❞ ❢r♦♠ t❤✐s s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝✲

❚❤❡② ❞♦ t❤✐s ✐♥❞❡♣❡♥❞❡♥t❧② ♦❢ ❡❛❝❤ ♦t❤❡r✳

✺✳✸✳ ❙❡q✉❡♥❝❡s ♦❢ ❢✉♥❝t✐♦♥s

✸✻✹

❊①❛♠♣❧❡ ✺✳✸✳✹✿ s❤✐❢t✐♥❣

❆♥♦t❤❡r s✐♠♣❧❡ ❝❤♦✐❝❡ ♦❢ ❛ s❡q✉❡♥❝❡ ✐s✿ fn (x) = f (x) +

1 → f (x) + 0 = f (x) , n

❢♦r ❡❛❝❤ x✱ ♥♦ ♠❛tt❡r t❤❛t f ✐s✳ ❊①❛♠♣❧❡ ✺✳✸✳✺✿ s❡❝❛♥ts ❛♥❞ t❛♥❣❡♥ts

❲❡ s❛✐❞✱ ✐♥❢♦r♠❛❧❧②✱ t❤❛t t❤❡ s❡❝❛♥t ❧✐♥❡s ❝♦♥✈❡r❣❡ t♦ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ✭❈❤❛♣t❡r ✷❉❈✲✸✮✿

▲❡t✬s ✐♥t❡r♣r❡t t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❡❝❛♥t ❧✐♥❡s t♦ t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛s ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s✳ ❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ f ✐s ❞❡✜♥❡❞ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝♦♥t❛✐♥s x = a✳ ❋♦r ❡❛❝❤ n = 1, 2, 3, ...✱ ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥ fn ❛s t❤❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ t❤❛t ♣❛ss❡s t❤r♦✉❣❤ t❤❡s❡ t✇♦ ♣♦✐♥ts✿ 

(a, f (a)) ❛♥❞

❚❤❡♥✱ fn (x) = f (a) +

1 a + ,f n



1 a+ n



.

f (a + 1/n) − f (a) (x − a) . 1/n

■❢ f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t x = a✱ t❤❡ ❢r❛❝t✐♦♥✱ ✐ts s❧♦♣❡✱ ❝♦♥✈❡r❣❡s t♦ f ′ (a) ❛s n → ∞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✿ f (a + 1/n) − f (a) → f ′ (a) . 1/n

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿ fn (x) = f (a) + ↓

f (x)

||

= f (a) +

f (a + 1/n) − f (a) (x − a) 1/n ↓ || f ′ (a)

(x − a)

❚❤✐s ♥❡✇ ❢✉♥❝t✐♦♥ ✐s t❤❡ ❜❡st ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ f ❛t a ❛♥❞ ✐ts ❣r❛♣❤ ✐s t❤❡ t❛♥❣❡♥t ❧✐♥❡ ♦❢ f ❛t a✳ ❲❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ f ✐s ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t❤✐s ♣♦✐♥t✱ ♦✉r s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ❞✐✈❡r❣❡s✳

✺✳✸✳

❙❡q✉❡♥❝❡s ♦❢ ❢✉♥❝t✐♦♥s

✸✻✺

❊①❛♠♣❧❡ ✺✳✸✳✻✿ ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s ❲❡ s❛✐❞ t❤❛t t❤❡ ❘✐❡♠❛♥♥ s✉♠s ✭s❛② ❧❡❢t ❡♥❞✮ ❝♦♥✈❡r❣❡ t♦ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✭❈❤❛♣t❡r ✶✮✿

Ln →

Z

b

f dx . a

❚❤❡r❡ ✐s ♥♦t❤✐♥❣ ✐♠♣r❡❝✐s❡ ❛❜♦✉t s❛②✐♥❣ t❤❛t✳

♥✉♠❜❡rs✳

❍♦✇❡✈❡r✱ t❤❛t ✇❛s ❛ s❡q✉❡♥❝❡ ♦❢

■s t❤❡r❡ ❛ s❡q✉❡♥❝❡ ♦❢

❢✉♥❝t✐♦♥s

❤❡r❡❄ ❨❡s✱ t❤❡ s❡q✉❡♥❝❡

♦❢ st❡♣✲❢✉♥❝t✐♦♥s t❤❛t r❡♣r❡s❡♥t t❤❡ ❧❡❢t✲❡♥❞ ❘✐❡♠❛♥♥ s✉♠s✳ ▲❡t✬s ♠❛❦❡ t❤✐s s♣❡❝✐✜❝✳

❙✉♣♣♦s❡

f

✐s ❞❡✜♥❡❞ ♦♥ ✐♥t❡r✈❛❧

[a, b]✳

❙✉♣♣♦s❡ ❛❧s♦ t❤❛t t❤❡ ✐♥t❡r✈❛❧ ✐s

❡q✉✐♣♣❡❞ ✇✐t❤ ❛ s♣❡❝✐✜❝ ❛✉❣♠❡♥t❡❞ ♣❛rt✐t✐♦♥✱ t❤❡ ❧❡❢t✲❡♥❞✱ ❢♦r ❡❛❝❤

(b − a)/n ❢♦❧❧♦✇s✿

❛♥❞

xi = a + ∆x · i✳

❚❤❡♥✱ ❢♦r ❡❛❝❤

fn (x) = F (xi )

n = 0, 1, 2, 3, ...✱

✇❤❡♥

n = 0, 1, 2, 3, ...✱

✇✐t❤

∆x =

❞❡✜♥❡ ❛ st❡♣✲❢✉♥❝t✐♦♥ ♣✐❡❝❡✇✐s❡ ❛s

xi ≤ x < xi+1 .

❚❤❡② ❛r❡ ♣❧♦tt❡❞ ✐♥ r❡❞✿

❲❤❡♥

f

✐s ❝♦♥t✐♥✉♦✉s✱ ✇❡ ❦♥♦✇ t❤❛t t❤✐s s❡q✉❡♥❝❡ ✇✐❧❧ ❝♦♥✈❡r❣❡ t♦

f

♣♦✐♥t✇✐s❡✳

❊①❡r❝✐s❡ ✺✳✸✳✼ Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t✳

❊①❡r❝✐s❡ ✺✳✸✳✽ ✭❛✮ ❙❤♦✇ t❤❛t ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♥♦t ❝♦♥t✐♥✉♦✉s ♦♥ t❤✐s ✐♥t❡r✈❛❧✱ ♦✉r s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ♠✐❣❤t ❞✐✈❡r❣❡✳ ✭❜✮ ❙❤♦✇ t❤❛t ✐t ❞♦❡s♥✬t

❤❛✈❡ t♦

❞✐✈❡r❣❡✱ ❤♦✇❡✈❡r✳

❊①❡r❝✐s❡ ✺✳✸✳✾ ❈♦♥s✐❞❡r ❛❧s♦ t❤❡ s❡q✉❡♥❝❡s ♦❢ st❡♣✲❢✉♥❝t✐♦♥s t❤❛t r❡♣r❡s❡♥t t❤❡ r✐❣❤t ❛♥❞ ♠✐❞❞❧❡ ♣♦✐♥t ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡ tr❛♣❡③♦✐❞ ❛♣♣r♦①✐♠❛t✐♦♥s✳

■♥ ❛❧❧ t❤❡s❡ ❡①❛♠♣❧❡s✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ♠❡❛♥s ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✭♠❛♥②✮ s❡q✉❡♥❝❡s ♦❢ ❍♦✇❡✈❡r✱ t❤❡ ✇❤♦❧❡ ❣r❛♣❤s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s

fn

♥✉♠❜❡rs✳

s❡❡♠ t♦ ❝♦♠♣❧❡t❡❧② ❛❝❝✉♠✉❧❛t❡ t♦✇❛r❞ t❤❡ ❣r❛♣❤ ♦❢

f✳

❚❤✐s ✐s ❛ ✏str♦♥❣❡r✑ ❦✐♥❞ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✿

❉❡✜♥✐t✐♦♥ ✺✳✸✳✶✵✿ ❝♦♥✈❡r❣❡s ✉♥✐❢♦r♠❧② ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s t❤❡ s❡q✉❡♥❝❡

❝♦♥✈❡r❣❡s ✉♥✐❢♦r♠❧② ♦♥ I

fn

❞❡✜♥❡❞ ♦♥ ✐♥t❡r✈❛❧

t♦ ❢✉♥❝t✐♦♥

f

I✳

❲❡ s❛② t❤❛t

✐❢ t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ♦❢

t❤❡ ❞✐✛❡r❡♥❝❡ ❝♦♥✈❡r❣❡s✱ ❛s ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs✱ t♦ ③❡r♦✿

max |fn (x) − f (x)| → 0 . I

❖t❤❡r✇✐s❡✱ ✇❡ s❛② t❤❛t t❤❡ s❡q✉❡♥❝❡

❞✐✈❡r❣❡s ✉♥✐❢♦r♠❧②✳

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ s❡q✉❡♥❝❡ ✇✐❧❧

f✱

♥♦ ♠❛tt❡r ❤♦✇ ♥❛rr♦✇✿

❡✈❡♥t✉❛❧❧②

✜t ❡♥t✐r❡❧② ✇✐t❤✐♥ ❛ str✐♣ ❛r♦✉♥❞

✺✳✸✳

❙❡q✉❡♥❝❡s ♦❢ ❢✉♥❝t✐♦♥s

✸✻✻

■t ✐s ❝❧❡❛r t❤❛t ❡✈❡r② ✉♥✐❢♦r♠❧② ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡s ♣♦✐♥t✇✐s❡ ✭t♦ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥✮✿

sup |fn (x) − f (x)| → 0 =⇒ |fn (x) − f (x)| → 0

❢♦r ❡❛❝❤

x

✐♥

I.

I

❊①❡r❝✐s❡ ✺✳✸✳✶✶

Pr♦✈❡ t❤❡ ❧❛st st❛t❡♠❡♥t✳

❚❤❡ ❝♦♥✈❡rs❡ ✐s♥✬t tr✉❡✳ ❊①❛♠♣❧❡ ✺✳✸✳✶✷✿ t♦♦t❤ s❡q✉❡♥❝❡

❲❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ I = [0, 1] t❤❛t s❛t✐s✜❡s✿ 1 ❢♦r x > ❛♥❞ t❤❡ r❡st ♦❢ t❤❡ ✈❛❧✉❡s ♣r♦❞✉❝❡ ❛ ✏t♦♦t❤✑ ♦❢ ❤❡✐❣❤t 1✿ n+1

fn (0) = 0 ❛♥❞ fn (x) = 0

❚❤❡♥✱ ✇❡ ❤❛✈❡✿

• fn • fn

❝♦♥✈❡r❣❡s t♦

0✱

t❤❡ ❢✉♥❝t✐♦♥✱ ♣♦✐♥t✇✐s❡ ❜❡❝❛✉s❡

❞♦❡s ♥♦t ❝♦♥✈❡r❣❡ t♦

str✐♣ ♥❛rr♦✇❡r t❤❛♥

0✱

fn (x)

❡✈❡♥t✉❛❧❧② ❜❡❝♦♠❡s ③❡r♦ ❢♦r ❡❛❝❤

x✳

t❤❡ ❢✉♥❝t✐♦♥✱ ✉♥✐❢♦r♠❧② ❜❡❝❛✉s❡ t❤❡ t♦♦t❤ ✇✐❧❧ ❛❧✇❛②s st✐❝❦ ♦✉t ♦❢ ❛♥②

1✳

❊①❛♠♣❧❡ ✺✳✸✳✶✸✿ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s

❋r♦♠ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t✱ ✇❤❡♥❡✈❡r t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧

I = [a, b]✱

f

❛r❡ ❦♥♦✇♥ t♦ ❜❡ ❜♦✉♥❞❡❞ ♦♥ ❛

t❤❡ s❡q✉❡♥❝❡ ♦❢ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ❝♦♥✈❡r❣❡s t♦

f

✉♥❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧✱ t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ ✐s♥✬t ❣✉❛r❛♥t❡❡❞❀ ❢♦r ❡①❛♠♣❧❡✱ ❝♦♥s✐❞❡r

(−∞, +∞)✿

I ✳ ❖♥ f (x) = cos x

✉♥✐❢♦r♠❧② ♦♥

❛♥ ♦♥

✺✳✸✳

❙❡q✉❡♥❝❡s ♦❢ ❢✉♥❝t✐♦♥s

✸✻✼

❚❤❡ r❡❛s♦♥ ✐s t❤❛t ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❛♥② ❞❡❣r❡❡ ❛❜♦✈❡

0

✇✐❧❧ ❡✈❡♥t✉❛❧❧② ❣♦ t♦ ✐♥✜♥✐t② ✭❱♦❧✉♠❡ ✷✮✳

P♦✐♥t✇✐s❡ ❝♦♥✈❡r❣❡♥❝❡ r❡♠❛✐♥s✳

▲❡t✬s ❝♦♠♣❛r❡ t❤❡ t✇♦ t②♣❡s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❛❣❛✐♥ ❜② r❡❢❡rr✐♥❣ t♦ t❤❡ ♦r✐❣✐♥❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✿

•

❙❡q✉❡♥❝❡

fn

❝♦♥✈❡r❣❡s t♦

◮ ❢♦r ❡❛❝❤ x✱

f

❢♦r ❛♥②

♣♦✐♥t✇✐s❡ ✐❢

ε>0

t❤❡r❡ ✐s s✉❝❤ ❛♥

N >0

t❤❛t

n > N =⇒ |fn (x) − f (x)| < ε . •

❙❡q✉❡♥❝❡

◮

fn

❝♦♥✈❡r❣❡s t♦

❢♦r ❛♥②

ε>0

f

✉♥✐❢♦r♠❧② ✐❢

t❤❡r❡ ✐s s✉❝❤ ❛♥

N >0

t❤❛t✱

❢♦r ❡❛❝❤ x✱

n > N =⇒ |fn (x) − f (x)| < ε . ❆s ②♦✉ s❡❡✱ ✇❡ ❥✉st ♠♦✈❡❞ ✏❢♦r ❡❛❝❤

x✑

✇✐t❤✐♥ t❤❡ s❡♥t❡♥❝❡✳

❍❡r❡ ✐s ❤♦✇ ❞✐✛❡r❡♥t t❤❡ ✐❞❡❛ ♦❢ ✏❝❧♦s❡✑ ✐s r❡❧❛t✐✈❡ t♦ t❤❡s❡ t✇♦ t②♣❡s ♦❢ ❝♦♥✈❡r❣❡♥❝❡✿

✺✳✹✳ ■♥✜♥✐t❡ s❡r✐❡s

✸✻✽

❊①❡r❝✐s❡ ✺✳✸✳✶✹ ■♥✈❡st✐❣❛t❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❡q✉❡♥❝❡ fn (x) =

1 ✳ nx

✺✳✹✳ ■♥✜♥✐t❡ s❡r✐❡s ❲❛r♥✐♥❣✦ ❙❡r✐❡s ❛r❡ s❡q✉❡♥❝❡s✳

❖✉r ❣♦❛❧ ✐s t♦ ❜❡ ❛❜❧❡ t♦ ✜♥❞ ♦✉t t❤❡ ❡①t❡♥t t♦ ✇❤✐❝❤ ♦✉r ❛♣♣r♦①✐♠❛t✐♦♥s ✕ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s ✕ ✇♦r❦✳ ❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ ♥❡❡❞ t♦ ❦♥♦✇ ❢♦r ✇❤❛t ✈❛❧✉❡s ♦❢ x t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s Tn (x) ♦❢ ❛ ❢✉♥❝t✐♦♥ f ❝♦♥✈❡r❣❡s t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x)✳ ❲❡ ✇✐❧❧ ❛❧s♦ ❡st✐♠❛t❡ t❤❡ ❡rr♦r✱ ❥✉st ❛s ✇❡ ❞✐❞ ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r✳ ❋♦❧❧♦✇✐♥❣ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✇❡ ❝❛♥ ❝♦♥s✐❞❡r s❡q✉❡♥❝❡s ♦❢ ❛♥② ❢✉♥❝t✐♦♥s ❛s ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❲❤❛t ♠❛❦❡s ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ❞✐✛❡r❡♥t❄ ■t✬s ✐♥ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s✱ ❢♦r ❛ ✜①❡❞ x✿ Tn+1 (x) = Tn (x) + cn+1 (x − a)n+1 .

❚❤❡ ❢♦r♠✉❧❛ ❝♦♥t✐♥✉❡s t♦ ❝♦♠♣✉t❡ t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ♦❢ ❤✐❣❤❡r ❛♥❞ ❤✐❣❤❡r ❞❡❣r❡❡s ❜② s✐♠♣❧② ❛❞❞✐♥❣ ♥❡✇ t❡r♠s t♦ t❤❡ ♣r❡✈✐♦✉s r❡s✉❧t✳ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❥✉st ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❉❡✜♥✐t✐♦♥ ✺✳✹✳✶✿ ♣♦✇❡r s❡r✐❡s ❝❡♥t❡r❡❞ ❛t ♣♦✐♥t ❆ s❡q✉❡♥❝❡ qn ♦❢ ♣♦❧②♥♦♠✐❛❧s ❣✐✈❡♥ ❜② ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿

qn+1 (x) = qn (x) + cn+1 (x − a)n+1 , n = 0, 1, 2, ... , ❢♦r s♦♠❡ ✜①❡❞ ♥✉♠❜❡r a ❛♥❞ ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦❡✣❝✐❡♥ts cn ✱ ✐s ❝❛❧❧❡❞ ❛ ♣♦✇❡r s❡r✐❡s ❝❡♥t❡r❡❞ ❛t a✳ P♦✇❡r s❡r✐❡s ♠❛② ❝♦♠❡ ❢✉❧❧② ❢♦r♠❡❞✳ ❍♦✇ ❞♦ ✇❡ ❛❞❞ t♦❣❡t❤❡r t❤❡ ✐♥✜♥✐t❡❧② ♠❛♥② t❡r♠s ♦❢ s✉❝❤ ❛ s❡q✉❡♥❝❡❄ ❱✐❛ ❧✐♠✐ts ✐s t❤❡ ♦♥❧② ❛♥s✇❡r✳ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❞❡✜♥✐t✐♦♥ ♦❢ t❤✐s ❝❤❛♣t❡r r❡♣❡❛ts t❤❡ ♦♥❡ ✐♥ ❈❤❛♣t❡r ✶✿

❉❡✜♥✐t✐♦♥ ✺✳✹✳✷✿ s❡q✉❡♥❝❡ ♦❢ s✉♠s✱ ♣❛rt✐❛❧ s✉♠s ❙✉♣♣♦s❡

an : n = s, s + 1, s + 2, ... ✐s ❛ s❡q✉❡♥❝❡✳ ■ts s❡q✉❡♥❝❡ ♦❢ s✉♠s

pn : n = s, s + 1, s + 2, ... ✐s t❤❡ s❡q✉❡♥❝❡ ❞❡✜♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿

ps = as ,

pn+1 = pn + an .

❚❤❡ s❡q✉❡♥❝❡ ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ♣❛rt✐❛❧ s✉♠s ♦❢ t❤❡ ♦r✐❣✐♥❛❧✳

✺✳✹✳

■♥✜♥✐t❡ s❡r✐❡s

✸✻✾

❚❤✐s ♣r♦❝❡ss ✐s ❛ ❢❛♠✐❧✐❛r ✇❛② ♦❢ ❝r❡❛t✐♥❣ ♥❡✇ s❡q✉❡♥❝❡s ❢r♦♠ ♦❧❞ ✭❱♦❧✉♠❡ ✷✮✳ ■♠❛❣✐♥❡ t❤❛t ✇❡ st❛❝❦ t❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡

an

✭❧❡❢t✮ ♦♥ t♦♣ ♦❢ ❡❛❝❤ ♦t❤❡r t♦ ♣r♦❞✉❝❡

■♥ t❤❡ ♥❡①t s❡✈❡r❛❧ s❡❝t✐♦♥s✱ ✇❡ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ s✉❝❤ s❡q✉❡♥❝❡s ♦❢

pn ✱

❛ ♥❡✇ s❡q✉❡♥❝❡ ✭r✐❣❤t✮✿

♥✉♠❜❡rs

✭r❛t❤❡r t❤❛♥ ❢✉♥❝t✐♦♥s✮ ❛♥❞

♦❝❝❛s✐♦♥❛❧❧② ❛♣♣❧② t❤❡ r❡s✉❧ts t♦ ♣♦✇❡r s❡r✐❡s✳ ❚❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡ ✐s ❧❡❢t ❜❡❤✐♥❞✱ ❛♥❞ ✐t ✐s

t❤❡ ❧✐♠✐t ♦❢ t❤✐s ♥❡✇ s❡q✉❡♥❝❡

t❤❛t ✇❡ ❛r❡ ❛❢t❡r✳

❊①❛♠♣❧❡ ✺✳✹✳✸✿ ❧✐♠✐t ♦❢ s✉♠s

■♥ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ t❛❜❧❡s ❜❡❧♦✇✱ ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡ ❣✐✈❡♥ ✐♥ t❤❡ ✜rst t✇♦ ❝♦❧✉♠♥s✳ ■ts ❢♦r♠✉❧❛ ✐s ❦♥♦✇♥ ❛♥❞✱ ❜❡❝❛✉s❡ ♦❢ t❤❛t✱ ✐ts ❧✐♠✐t ✐s ❛❧s♦ ❡❛s② t♦ ✜♥❞✳ s❡q✉❡♥❝❡ ♦❢ ✭♣❛rt✐❛❧✮ s✉♠s ♦❢ t❤❡ ✜rst✳ ■ts

nt❤

t❡r♠

❚❤❡ t❤✐r❞ ❝♦❧✉♠♥ s❤♦✇s t❤❡

nt❤ t❡r♠ ❢♦r♠✉❧❛ ✐s ✉♥❦♥♦✇♥ ❛♥❞✱

❜❡❝❛✉s❡ ♦❢ t❤❛t✱ ✐ts ❧✐♠✐t

✐s ♥♦t ❡❛s② t♦ ✜♥❞✳

n 1 2 3

an p n 1 1 1 1 1 1 1 + 2 1 2 1 1 1 1 + + 3 1 2 3

n 1 2 3

✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

n

1 n

1 1 1 1 + + + ... + , 1 2 3 n

✳ ✳ ✳

↓

∞

❢♦r♠✉❧❛❄

✳ ✳ ✳

✳ ✳ ✳

n

1 2n

✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

↓

↓

↓

0

an pn 1 1 1 1 1 1 1 + 2 1 2 1 1 1 1 + + 4 1 2 4

∞

?

✳ ✳ ✳

1 1 1 1 + + + ... + n−1 , 1 2 4 2

✳ ✳ ✳

✳ ✳ ✳

↓

↓

0

❢♦r♠✉❧❛❄

?

❊①❛♠♣❧❡ ✺✳✹✳✹✿ s♦♠❡t❤✐♥❣ ❢r♦♠ ♥♦t❤✐♥❣

❚❤✐s ❡①❛♠♣❧❡ ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✶✮ s❤♦✇s ✇❤❛t ❝❛♥ ❤❛♣♣❡♥ ✐❢ ✇❡ ✐❣♥♦r❡ t❤❡ ✐ss✉❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✿

❄

0 ==

0

❄

== (1 ❄

== ❄

== ❄

==

1 1 1

+0 −1) −1

+(1 +1

+0 −1) −1

+(1 +1

+0 −1) −1

+(1 +1

+... −1) −1

+... +...

+(−1 +1) +(−1 +1) +(−1 +1) +(−1 +1) +... +0

+0

+0

+0

+...

❄

== 1. ❘❡♠♦✈✐♥❣ t❤❡ ♣❛r❡♥t❤❡s❡s ✐♥ ❛♥ ✐♥✜♥✐t❡ ❝♦♠♣✉t❛t✐♦♥ ✐s ♥♦t ✇❤❛t ✇❡ ✇♦✉❧❞ ❞♦ ❛♥②♠♦r❡ ✕ ✐♥ ❧✐❣❤t ♦❢ ♦✉r ❞❡✜♥✐t✐♦♥✳ ❚❤❡ r❡s✉❧t q✉❛❧✐✜❡s ❛s ❛♥ ❡①❛♠♣❧❡ ♦❢ ✏s♦♠❡t❤✐♥❣ ❢r♦♠ ♥♦t❤✐♥❣✑✿

✺✳✹✳

■♥✜♥✐t❡ s❡r✐❡s

✸✼✵

■♥ t❤❡ ✏❝♦♠♣✉t❛t✐♦♥✑✱ ✇❡ ❣♦ ✕ ✐♠♣❧✐❝✐t❧② ✕ t❤r♦✉❣❤

t❤r❡❡

s❡q✉❡♥❝❡s✳ ❚❤❡ ❢❛❝t t❤❛t t❤❡ ✜rst ❛♥❞ t❤❡

t❤✐r❞ ❜♦t❤ ❝♦♥✈❡r❣❡ ♦❜s❝✉r❡s t❤❡ ❢❛❝t t❤❛t t❤❡ s❡❝♦♥❞ ❞✐✈❡r❣❡s✳ ❚♦ ❞❡t❡❝t t❤❡ s✇✐t❝❤❡s✱ ✇❡ ♦❜s❡r✈❡ t❤❛t t❤❡ s❡q✉❡♥❝❡s t❤❛t ♣r♦❞✉❝❡ t❤❡ t❤r❡❡ s✉♠s ❛r❡ ❞✐✛❡r❡♥t✦ ❲❡ ❧✐st t❤❡ ✜rst t✇♦ ❜❡❧♦✇✿

n 1 2 3 ✳ ✳ ✳

n ✳ ✳ ✳

an

pn

= pn

n

an

pn

=

pn

0=1−1 0

= 0

1

1

1

=

1

= 0

2

0

= 0

3

1−1

=

0=1−1 0+0+0

−1

=

1

0=1−1 0+0 ✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

0 = 1 − 1 0 + 0 + 0 + ... + 0 = 0

↓

∞

✳ ✳ ✳

✳ ✳ ✳

↓ 0

n

✳ ✳ ✳

✳ ✳ ✳

↓

↓

0

∞

1−1+1

1

✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

(−1)n 1 − 1 + 1 − ... + (−1)n = ✳ ✳ ✳

✳ ✳ ✳

↓

❉◆❊

❆s ❛ ❞✐✈❡r❣❡♥t s❡q✉❡♥❝❡✱ t❤❡ s❡❝♦♥❞ ♦♥❡ ❝❛♥♥♦t ❜❡ ❡q✉❛❧ t♦

❊①❡r❝✐s❡ ✺✳✹✳✺

❄ ❲❤✐❝❤ ♦❢ t❤❡ ✏ = =✑ s✐❣♥s ❛❜♦✈❡ ✐s ✐♥❝♦rr❡❝t❄

■♥ s✉♠♠❛r②✱

◮ ❆ s❡r✐❡s ✐s ❛ ♣❛✐r ♦❢ s❡q✉❡♥❝❡s✳ ❚❤❡ ✜rst ♣r♦❞✉❝❡s t❤❡ s❡❝♦♥❞ ✈✐❛ s✉♠♠❛t✐♦♥✱ ❢♦r ❡①❛♠♣❧❡✿

❛♥②t❤✐♥❣✳

✳ ✳ ✳

1, 0 ✳ ✳ ✳

↓

❉◆❊

✺✳✹✳

■♥✜♥✐t❡ s❡r✐❡s

✸✼✶

■s ✐t ❡✈❡♥ ♣♦ss✐❜❧❡ t♦ ✜♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s

pn

✇✐t❤♦✉t ✜rst ✜♥❞✐♥❣ t❤❡ ❢♦r♠✉❧❛s ❢♦r ✐ts

nt❤

t❡r♠❄ ❋♦r♠✉❧❛s t❤❛t ❤❛✈❡ ✏✳✳✳✑ ❞♦♥✬t ❝♦✉♥t✿ ❧✐st ♦❢ t❡r♠s ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡✿

1,

✭♣❛rt✐❛❧✮ s✉♠s s❡q✉❡♥❝❡✿

1,

❢♦r♠✉❧❛ ❢♦r

1 1 1 , , , ... 2 3 4

1 1+ , 2 1 1+ + 2 ... 1 1+ + 2

nt❤

t❡r♠

1 n

1 , 3 n 1 1 1 X1 + + ... + 3 4 n k=1 k

◆❡✐t❤❡r ❞♦ ❢♦r♠✉❧❛s t❤❛t r❡❧② ♦♥ t❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥✳ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♣♦s❡❞ ❜② s❡r✐❡s ✐s t❤❛t t❤❡② ♥❡✈❡r ❝♦♠❡ ✇✐t❤ ❛♥ ♣❛rt✐❛❧ s✉♠ ♦❢ ❛ ♣♦✇❡r s❡r✐❡s ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢

x

nt❤✲t❡r♠

❢♦r♠✉❧❛✦ ❆s ❛♥ ❡①❛♠♣❧❡✱ ❛♥

nt❤

❛♥❞✱ ❢r♦♠ ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ♣♦❧②♥♦♠✐❛❧s ✭❈❤❛♣t❡r

✶P❈✲✹✮✱ ✐t ❝❛♥ ♦♥❧② ❜❡ s✐♠♣❧✐✜❡❞ ✐♥ ❛ ❢❡✇ ✈❡r② s♣❡❝✐❛❧ ❝❛s❡s✱ s✉❝❤ ❛s t❤❡ ❇✐♥♦♠✐❛❧ ❋♦r♠✉❧❛ ✭❈❤❛♣t❡r ✶P❈✲✶✮✳

❉❡✜♥✐t✐♦♥ ✺✳✹✳✻✿ s✉♠ ♦❢ s❡q✉❡♥❝❡ ♦r s❡r✐❡s ❋♦r ❛ s❡q✉❡♥❝❡ t❤❡

an ✱

t❤❡ ❧✐♠✐t

s✉♠ ♦❢ t❤❡ s❡q✉❡♥❝❡

S

♦❢ ✐ts s❡q✉❡♥❝❡ ♦❢ ✭♣❛rt✐❛❧✮ s✉♠s

♦r✱ ♠♦r❡ ❝♦♠♠♦♥❧②✱ t❤❡

lim

n→∞

n X

✐s ❝❛❧❧❡❞ ❜②

ai = S .

i=s

❲❤❡♥ t❤❡ ❧✐♠✐t ♦❢ ♣❛rt✐❛❧ s✉♠s ❡①✐sts✱ ✇❡ s❛② t❤❛t t❤❡ ❧✐♠✐t ❞♦❡s ♥♦t ❡①✐st✱ ✇❡ s❛② t❤❛t t❤❡

pn

s✉♠ ♦❢ t❤❡ s❡r✐❡s ✿

s❡r✐❡s ❞✐✈❡r❣❡s✳

s❡r✐❡s ❝♦♥✈❡r❣❡s✳

❲❤❡♥ t❤❡

❆ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❞✐✈❡r❣❡♥❝❡

✐s t❤❡ ❞✐✈❡r❣❡♥❝❡ t♦ ✭♣❧✉s ♦r ♠✐♥✉s✮ ✐♥✜♥✐t② ❛♥❞ ✇❡ s❛② t❤❛t t❤❡ s✉♠ ✐s

lim

n→∞

n X i=s

✐♥✜♥✐t❡ ✿

ai = ∞ .

❖♥❝❡ ❛❣❛✐♥✱ ❛ s❡r✐❡s ✐s ❛ s❡q✉❡♥❝❡ ❜✉✐❧t ❢r♦♠ ❛♥♦t❤❡r ✈✐❛ r❡❝✉rs✐✈❡ ❛❞❞✐t✐♦♥✳

❲❛r♥✐♥❣✦ ❚❤❡ st❛rt✐♥❣ ♣♦✐♥t✱ s✱ ♦❢ s✉♠♠❛t✐♦♥ ❞♦❡s♥✬t ❛✛❡❝t ❝♦♥✈❡r❣❡♥❝❡ ❜✉t ❞♦❡s ❛✛❡❝t t❤❡ s✉♠ ✇❤❡♥ t❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s✳

✺✳✹✳

■♥✜♥✐t❡ s❡r✐❡s

✸✼✷

❊①❛♠♣❧❡ ✺✳✹✳✼✿ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♣♦✇❡r s❡r✐❡s x

◆♦t❡ t❤❛t ❛ ♣♦✇❡r s❡r✐❡s ♠❛② ❝♦♥✈❡r❣❡ ❢♦r s♦♠❡ ✈❛❧✉❡s ♦❢

❛♥❞ ❞✐✈❡r❣❡ ❢♦r ♦t❤❡rs✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡

s❡r✐❡s

1 + x + x2 + .. . • •

x=0 x = 1✳

❝♦♥✈❡r❣❡s ❢♦r ❞✐✈❡r❣❡s ❢♦r

❜✉t

❲❡ ❛❧s♦ ❞❡♠♦♥str❛t❡ ❜❡❧♦✇ t❤❛t ✐t ❝♦♥✈❡r❣❡s ❢♦r

x = 1/2✳

❚❤✐s ✐s ❛♥ ❛❜❜r❡✈✐❛t❡❞ ♥♦t❛t✐♦♥ t♦ ✇r✐t❡ t❤❡ ❧✐♠✐t ♦❢ ♣❛rt✐❛❧ s✉♠s✿

❙✉♠ ♦❢ s❡r✐❡s ∞ X

ai = lim

i=s

n→∞

n X

ai

i=s

❲❛r♥✐♥❣✦ ❚❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s♥✬t ❛ s❡r✐❡s✳

❘❡❝❛❧❧ t❤❛t ❤❡r❡

Σ

st❛♥❞s ❢♦r ✏❙✑ ♠❡❛♥✐♥❣ ✏s✉♠✑✳ ❚❤✐s ✐s ❤♦✇ t❤❡ ♥♦t❛t✐♦♥ ✐s ❞❡❝♦♥str✉❝t❡❞✿

❙✐❣♠❛ ♥♦t❛t✐♦♥ ❢♦r s❡r✐❡s ❜❡❣✐♥♥✐♥❣

❛♥❞ ❡♥❞ ✈❛❧✉❡s ❢♦r

k

↓

∞ k=0

X

1 1 + k k 2 3



↑

=

❛ s♣❡❝✐✜❝ s❡q✉❡♥❝❡

7 2 ↑

❛ s♣❡❝✐✜❝ ♥✉♠❜❡r✱

± ∞,

♦r ✏❉◆❊✑

❲❛r♥✐♥❣✦ ■♥ s♦♠❡ s♦✉r❝❡s✱ t❤❡ ✇♦r❞ ✏s❡r✐❡s✑ ♠✐❣❤t r❡❢❡r t♦ ♦♥❡✱ ♦r ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❛❧❧✱ ♦❢ t❤❡s❡ t❤r❡❡✿

•

t❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡

•

✐ts s❡q✉❡♥❝❡ ♦❢ s✉♠s

•

t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❧❛tt❡r✳

an ✱ pn ✱

❛♥❞

❊①❛♠♣❧❡ ✺✳✹✳✽✿ ✜♥❡t❡♥❡ss ❲❤❡♥ t❤❡r❡ ❛r❡ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② ♥♦♥✲③❡r♦ ❡❧❡♠❡♥ts ✐♥ t❤❡ s❡q✉❡♥❝❡✱ ✐ts s✉♠ ✐s s✐♠♣❧❡✿

ai = 0

❢♦r ❡❛❝❤

i > N =⇒

∞ X i=1

❚❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s t♦ t❤✐s ♥✉♠❜❡r✳

ai =

N X i=1

ai .

✺✳✹✳ ■♥✜♥✐t❡ s❡r✐❡s

✸✼✸

❊①❛♠♣❧❡ ✺✳✹✳✾✿ ❞✐✈❡r❣❡♥t

◆♦♥✲③❡r♦ ❝♦♥st❛♥t ✐s t❤❡ s✐♠♣❧❡st ❡①❛♠♣❧❡ ♦❢ ❛ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ❜✉t ❛ ❞✐✈❡r❣❡♥t s❡r✐❡s✿

ai = 1 ❢♦r ❡❛❝❤ i =⇒

∞ X

ai = lim

i=1

❊①❛♠♣❧❡ ✺✳✹✳✶✵✿

sin n

n→∞

n X i=1

1 = lim n = ∞ . n→∞

❛♥❞ r❛♥❞♦♠ s❡q✉❡♥❝❡

❈♦♥s✐❞❡r t❤❡ s❡q✉❡♥❝❡

an = sin n . ■t ❧♦♦❦s ✉♥✐❢♦r♠❧② s♣r❡❛❞ ❜❡t✇❡❡♥ −1 ❛♥❞ 1 ✭❣r❡❡♥✮✿

■ts s❡q✉❡♥❝❡ ♦❢ s✉♠s ❛❧s♦ ❧♦♦❦s ✉♥✐❢♦r♠❧② s♣r❡❛❞ ❜❡t✇❡❡♥✱ 0 ❛♥❞ 2 ✭♦r❛♥❣❡✮✳ ■t✱ t❤❡♥✱ ❛♣♣❡❛rs t♦ ❞✐✈❡r❣❡✳ ❆ s❡r✐❡s ♦❢ ❛ tr✉❧② r❛♥❞♦♠ s❡q✉❡♥❝❡ ❞✐✈❡r❣❡s ✐♥ ❛ ❞✐✛❡r❡♥t ✇❛②✿

❲❡ s❛✇ t❤❛t✱ ✇❤❡♥ ❢❛❝✐♥❣ ✐♥✜♥✐t②✱ ❛❧❣❡❜r❛ ♠❛② ❢❛✐❧✳ ❇✉t ✐t ✇♦♥✬t ✐❢ t❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡✦ ■♥ t❤❛t ❝❛s❡✱ t❤❡ s❡r✐❡s ❝❛♥ ❜❡ s✉❜❥❡❝t❡❞ t♦ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ♣♦✇❡r s❡r✐❡s t♦♦ ❝❛♥ ❜❡ s✉❜❥❡❝t❡❞ t♦ ❝❛❧❝✉❧✉s ♦♣❡r❛t✐♦♥s✳✳✳ ❊①❛♠♣❧❡ ✺✳✹✳✶✶✿ ❞❡❝✐♠❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s

❋♦r ❛ ❣✐✈❡♥ r❡❛❧ ♥✉♠❜❡r✱ ✇❡ ❝❛♥ ❝♦♥str✉❝t ❛ s❡r✐❡s t❤❛t t❡♥❞s t♦ t❤❛t ♥✉♠❜❡r ✕ ✈✐❛ tr✉♥❝❛t✐♦♥s ♦❢ ✐ts ❞❡❝✐♠❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❚❤❡ s❡q✉❡♥❝❡

an = 0.9, 0.09, 0.009, 0.0009, ... t❡♥❞s t♦ 0 . ❇✉t ✐ts s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐❛❧ s✉♠s

pn = 0.9, 0.99, 0.999, 0.9999, ... t❡♥❞s t♦ 1 .

✺✳✹✳ ■♥✜♥✐t❡ s❡r✐❡s

✸✼✹

❚❤❡ s❡q✉❡♥❝❡

an = 0.3, 0.03, 0.003, 0.0003, ... t❡♥❞s t♦ 0 .

❇✉t ✐ts s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐❛❧ s✉♠s pn = 0.3, 0.33, 0.333, 0.3333, ... t❡♥❞s t♦ 1/3 .

❚❤❡ ✐❞❡❛ ♦❢ s❡r✐❡s t❤❡♥ ❤❡❧♣s ✉s ✉♥❞❡rst❛♥❞ ✐♥✜♥✐t❡ ❞❡❝✐♠❛❧s✳ • ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ 0.9999...❄ ■t ✐s t❤❡ s✉♠ ♦❢ t❤❡ s❡r✐❡s✿ ∞ X i=1

9 · 10−i .

• ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ 0.3333...❄ ■t ✐s t❤❡ s✉♠ ♦❢ t❤❡ s❡r✐❡s✿ ∞ X i=1

3 · 10−i .

❊①❡r❝✐s❡ ✺✳✹✳✶✷

❋✐♥❞ s✉❝❤ ❛ s❡r✐❡s ❢♦r 1/6✳ ❲❡ ❦♥♦✇ t❤❛t ❛ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ❝❛♥ ❤❛✈❡ ♦♥❧② ♦♥❡ ❧✐♠✐t✳ ❚❤❡♦r❡♠ ✺✳✹✳✶✸✿ ❯♥✐q✉❡♥❡ss ♦❢ ❙✉♠ ♦❢ ❙❡r✐❡s

❆ s❡r✐❡s ❝❛♥ ❤❛✈❡ ♦♥❧② ♦♥❡ s✉♠ ✭✜♥✐t❡ ♦r ✐♥✜♥✐t❡✮✳ ❚❤✉s✱ ✇❡ ❛r❡ ❥✉st✐✜❡❞ t♦ s♣❡❛❦ ♦❢ t❤❡ s✉♠✳ ❚❤✐s ❝♦♥❝❧✉s✐♦♥ ♠❛❦❡s ♣♦ss✐❜❧❡ t❤❡ t❤❡♦r② ✇❡ s❤❛❧❧ ❞❡✈❡❧♦♣✿ ◮ ❆♥② ♣♦✇❡r s❡r✐❡s ❞❡✜♥❡s ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ✐ts ❞♦♠❛✐♥ ❝♦♥s✐st✐♥❣ ♦❢ t❤♦s❡ ✈❛❧✉❡s ♦❢ x ❢♦r ✇❤✐❝❤

t❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s✳

❊①❛♠♣❧❡ ✺✳✹✳✶✹✿ s❡r✐❡s ✈s✳ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s

❚❤❡ ✇❛② ✇❡ ✉s❡ t❤❡ ❧✐♠✐ts t♦ tr❛♥s✐t✐♦♥ ❢r♦♠ s❡q✉❡♥❝❡s t♦ s❡r✐❡s ✐s ❢❛♠✐❧✐❛r✳ ■t ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ tr❛♥s✐t✐♦♥ ❢r♦♠ ❛ ❢✉♥❝t✐♦♥ f t♦ ✐ts ✐♥t❡❣r❛❧ ♦✈❡r [0, ∞)✿

■♥❞❡❡❞✱ ❝♦♠♣❛r❡✿

Z

∞

1 ∞ X i=1

f (x) dx = lim

b→∞

ai

= lim

n→∞

Z

b

f (x) dx

1 n X

ai

i=1

❋✉rt❤❡r♠♦r❡✱ t❤❡ ❧❛tt❡r ✇✐❧❧ ❢❛❧❧ ✉♥❞❡r t❤❡ s❝♦♣❡ ♦❢ t❤❡ ❢♦r♠❡r ✐❢ ✇❡ ❝❤♦♦s❡ f t♦ ❜❡ t❤❡ st❡♣✲❢✉♥❝t✐♦♥

✺✳✺✳

▼❛✐♥ ❝❧❛ss❡s ♦❢ s❡r✐❡s

✸✼✺

♣r♦❞✉❝❡❞ ❜② t❤❡ s❡q✉❡♥❝❡ an ✿ f (x) = a[x] . ❲❛r♥✐♥❣✦ ❚❤❡ ✇♦rst ♠✐st❛❦❡ ♦♥❡ ❝❛♥ ♠❛❦❡ ✐s t♦ ❝♦♥❢✉s❡ t❤❡ ❧✐♠✐t ♦❢

an

✇✐t❤ t❤❡ ❧✐♠✐t ♦❢

pn ✳

✺✳✺✳ ▼❛✐♥ ❝❧❛ss❡s ♦❢ s❡r✐❡s

❚❤❡ ❦❡② t♦ ❡✈❛❧✉❛t✐♥❣ s✉♠s ♦❢ ❛ s❡r✐❡s✱ ♦r ❞✐s❝♦✈❡r✐♥❣ t❤❛t ✐t ❞✐✈❡r❣❡s✱ ✐s t♦ ✜♥❞ ❛♥ n pn ♦❢ an ✳

t❤ ♣❛rt✐❛❧ s✉♠

❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r t❤❡

❲❡ ❤❛✈❡ t♦ ❜❡ ❝❛r❡❢✉❧ ❛♥❞ ♥♦t t❛❦❡ t❤❡ ❢♦r♠❡r ♦✈❡r t❤❡ ❧❛tt❡r✿ lim an ✈s✳

n→∞

lim pn

n→∞

❊①❛♠♣❧❡ ✺✳✺✳✶✿ ❝♦♥st❛♥t

❘❡❝❛❧❧ ❛❜♦✉t t❤❡

❝♦♥st❛♥t s❡q✉❡♥❝❡ t❤❛t✱ ❢♦r ❛♥② r❡❛❧ c✱ ✇❡ ❤❛✈❡ lim c = c .

n→∞

❚❤❡ r❡s✉❧t t❡❧❧s ✉s ♥♦t❤✐♥❣ ❛❜♦✉t t❤❡ s❡r✐❡s ♣r♦❞✉❝❡❞ ❜② t❤❡ s❡q✉❡♥❝❡✦ ❈♦♥s✐❞❡r ✐♥st❡❛❞✿ c + c + c + ... = lim

n→∞

n X k=1

c = lim nc . n→∞

❚❤❡r❡❢♦r❡✱ s✉❝❤ ❛ ❝♦♥st❛♥t s❡r✐❡s ❞✐✈❡r❣❡s ✉♥❧❡ss c = 0✳ ❚❤❡ r❡s✉❧t ✐s ✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿ ∞ X k=1

❊①❛♠♣❧❡ ✺✳✺✳✷✿ ❛r✐t❤♠❡t✐❝

▼♦r❡ ❣❡♥❡r❛❧❧②✱ ❝♦♥s✐❞❡r ✇❤❛t ✇❡ ❦♥♦✇ ❛❜♦✉t ✇❡ ❤❛✈❡✿

c = ∞.

❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s ❀ ❢♦r ❛♥② r❡❛❧ ♥✉♠❜❡rs m, b > 0✱

  −∞ lim (b + nm) = b n→∞   +∞

✐❢ m < 0 , ✐❢ m = 0 , ✐❢ m > 0 .

❚❤❡ r❡s✉❧t t❡❧❧s ✉s ♥♦t❤✐♥❣ ❛❜♦✉t t❤❡ s❡r✐❡s✦ ■♥st❡❛❞✱ ❧❡t✬s ❡①❛♠✐♥❡ t❤❡ ♣❛rt✐❛❧ s✉♠s ❛♥❞✱ ❜❡❝❛✉s❡ ❡❛❝❤ ✐s ❝♦♠♣r✐s❡❞ ♦❢ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② t❡r♠s✱ ✇❡ ❝❛♥ ♠❛♥✐♣✉❧❛t❡ t❤❡♠ ❛❧❣❡❜r❛✐❝❛❧❧② ❜❡❢♦r❡ ✜♥❞✐♥❣ t❤❡ ❧✐♠✐t✱ ❛s ❢♦❧❧♦✇s✿ b + (b + m) + (b + 2m) + (b + 3m) + ... = lim

n→∞

n X k=0

(b + km) .

✺✳✺✳

▼❛✐♥ ❝❧❛ss❡s ♦❢ s❡r✐❡s

✸✼✻

❊①❡r❝✐s❡ ✺✳✺✳✸ ❙❤♦✇ t❤❛t s✉❝❤ ❛ s❡r✐❡s ❞✐✈❡r❣❡s ✉♥❧❡ss b = m = 0✿ ∞ X k=0

(b + km) = ∞ .

❚❤❡r❡ ❛r❡ ♠♦r❡ ✐♥t❡r❡st✐♥❣ s❡r✐❡s✳

❉❡✜♥✐t✐♦♥ ✺✳✺✳✹✿ ❤❛r♠♦♥✐❝ s❡r✐❡s ❚❤❡ s❡r✐❡s ♣r♦❞✉❝❡❞ ❜② t❤❡ s❡q✉❡♥❝❡ ♦❢ r❡❝✐♣r♦❝❛❧s an = 1/n✱ ∞ X 1 k=1

✐s ❝❛❧❧❡❞ t❤❡

k

=

1 1 1 1 1 + + + + ... + + ... , 1 2 3 4 k

❤❛r♠♦♥✐❝ s❡r✐❡s✳

❊①❛♠♣❧❡ ✺✳✺✳✺✿ ❤❛r♠♦♥✐❝ s❡r✐❡s ❇❡❧♦✇ ✇❡ s❤♦✇ t❤❡ ✉♥❞❡r❧②✐♥❣ s❡q✉❡♥❝❡✱ an = 1/n✱ t❤❛t ✐s ❦♥♦✇♥ t♦ ❝♦♥✈❡r❣❡ t♦ ③❡r♦ ✭❧❡❢t✮✱ ❛♥❞ ✐ts s❡q✉❡♥❝❡ ♦❢ t❤❡ ♣❛rt✐❛❧ s✉♠s ✭r✐❣❤t✮✿

P❧♦tt✐♥❣ t❤❡ ✜rst 3000 t❡r♠s ♦❢ t❤❡ ❧❛tt❡r s❡❡♠s t♦ s✉❣❣❡sts t❤❛t ✐t ❛❧s♦ ❝♦♥✈❡r❣❡s✳ ❊①❛♠✐♥✐♥❣ t❤❡ ❞❛t❛✱ ✇❡ ✜♥❞ t❤❛t t❤❡ s✉♠ ✐s♥✬t ❧❛r❣❡✱ s♦ ❢❛r✿ 1 1 1 1 1 + + + + ... + ≈ 8.59 . 1 2 3 4 3000

❲❡ ❦♥♦✇ ❜❡tt❡r t❤❛♥ t♦ t❤✐♥❦ t❤❛t t❤✐s t❡❧❧s ✉s ❛♥②t❤✐♥❣❀ t❤❡ s❡r✐❡s ❝❤❛♣t❡r✳

❞✐✈❡r❣❡s ❛s ✇❡ ✇✐❧❧ s❤♦✇ ✐♥ t❤✐s

❊①❛♠♣❧❡ ✺✳✺✳✻✿ ❤✐❣❤❡r ♣♦✇❡rs ■❢ ✇❡ r❡♣❧❛❝❡ t❤❡ ♣♦✇❡r ♦❢ k ✐♥ t❤❡ ❤❛r♠♦♥✐❝ s❡r✐❡s✱ ✇❤✐❝❤ ✐s −1✱ ✇✐t❤ −1.1✱ t❤❡ s❡r✐❡s✱ ∞ X 1 , 1.1 k k=1

t❤❡ ❣r❛♣❤ ♦❢ ✐ts s✉♠s ❧♦♦❦s ❛❧♠♦st ❡①❛❝t❧② t❤❡ s❛♠❡✿

✺✳✺✳

▼❛✐♥ ❝❧❛ss❡s ♦❢ s❡r✐❡s

✸✼✼

❍♦✇❡✈❡r✱ ✇❡ s❤♦✇ ❜❡❧♦✇ t❤❛t ✐t ✐s ❝♦♥✈❡r❣❡♥t✦

❊①❛♠♣❧❡ ✺✳✺✳✼✿ ❢❛❝t♦r✐❛❧s ■♥ ❝♦♥tr❛st✱ t❤✐s ✐s ❤♦✇ ❢❛st t❤❡ s❡r✐❡s ♦❢ t❤❡

r❡❝✐♣r♦❝❛❧s ♦❢ t❤❡ ❢❛❝t♦r✐❛❧s

✱

∞ X 1 , k! k=1 ❝♦♥✈❡r❣❡s✿

❆s s❤♦✇♥ ✐♥ ❡❛r❧✐❡r ✐♥ t❤✐s ❝❤❛♣t❡r✱ ✐t ❝♦♥✈❡r❣❡s t♦

e✳

❖♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t s❡r✐❡s ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳

❉❡✜♥✐t✐♦♥ ✺✳✺✳✽✿ ❣❡♦♠❡tr✐❝ s❡r✐❡s ❚❤❡ s❡r✐❡s ♣r♦❞✉❝❡❞ ❜② t❤❡ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥

∞ X

an = a · rn

ark = a + ar1 + ar2 + ar3 + ar4 + ... + ark + ... ,

k=0

✐s ❝❛❧❧❡❞ t❤❡

❘❡❝❛❧❧ t❤❡ ❢❛❝t ❛❜♦✉t ❣❡♦♠❡tr✐❝

✇✐t❤ r❛t✐♦

❣❡♦♠❡tr✐❝ s❡r✐❡s

♣r♦❣r❡ss✐♦♥s

✇✐t❤ r❛t✐♦

r✳

❢r♦♠ ❱♦❧✉♠❡ ✶ ✭❈❤❛♣t❡r ✶❉❈✲✶✮✳

r✱

✺✳✺✳

▼❛✐♥ ❝❧❛ss❡s ♦❢ s❡r✐❡s

✸✼✽

❚❤❡♦r❡♠ ✺✳✺✳✾✿ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ●❡♦♠❡tr✐❝ Pr♦❣r❡ss✐♦♥ ❚❤❡ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ ✈❛❧✉❡ ♦❢

r✿

r

❝♦♥✈❡r❣❡s ❛♥❞ ❞✐✈❡r❣❡s ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡

 ❞✐✈❡r❣❡s    0 lim rn = n→∞  1    +∞

✐❢ ✐❢ ✐❢ ✐❢

r ≤ −1, |r| < 1, r = 1, r > 1.

❲❛r♥✐♥❣✦ ❚❤❡ r❡s✉❧t t❡❧❧s ✉s ♥♦t❤✐♥❣ ❛❜♦✉t t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❡r✐❡s✳

❊①❛♠♣❧❡ ✺✳✺✳✶✵✿ ❣❡♦♠❡tr✐❝

❚❤❡ ❝♦♥str✉❝t✐♦♥ s❤♦✇♥ ❜❡❧♦✇ ✐s r❡❝✉rs✐✈❡❀ ✐t ❝✉ts ❛ sq✉❛r❡ ✐♥t♦ ❢♦✉r✱ t❛❦❡s t❤❡ ✜rst t❤r❡❡✱ ❛♥❞ t❤❡♥ r❡♣❡❛ts t❤❡ ♣r♦❝❡❞✉r❡ ✇✐t❤ t❤❡ ❧❛st ♦♥❡✿

❚❤❡s❡ ❛r❡ t❤❡ ❛r❡❛s✿ ✜rst t✇♦ sq✉❛r❡s t❤✐r❞ sq✉❛r❡ ✜rst t✇♦ sq✉❛r❡s t❤✐r❞ sq✉❛r❡ 1 1 + 4 4 1 2 1 2

1 4 1 4 1 + 4

1 16 1 16 1 + 16

1 1 + 16 16 1 8 1 + 8

... ... ... +... = 1 ?

❇❡❝❛✉s❡ t❤❡② ❛r❡ ❛❧❧ ❝✉t ♦✉t ♦❢ t❤❡ ❜✐❣ sq✉❛r❡ ❛♥❞ ❛❧❧ ♣❛rts ♦❢ t❤❡ sq✉❛r❡ ❛r❡ ❝♦✈❡r❡❞✱ t❤❡ t♦t❛❧ s✉♠ ♠✉st ❜❡ 1✳ Pr♦✈✐❞❡❞ s✉❝❤ ❛ s✉♠ ♠❛❦❡s s❡♥s❡ ✐♥ t❤❡ ✜rst ♣❧❛❝❡✦ ❉♦❡s ✐t❄ ❊❛❝❤ st❡♣ ❝r❡❛t❡s t✇♦ t❡r♠s ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡r✐❡s✿     n X 1 = 2k k=1

1 1 + 2 4

+

1 1 + 8 16

❄

+ ... == 1

❚❤❡ s✉♠ ♦❢ t❤❡ s❡r✐❡s t❤❡♥ ♠✉st ❜❡ 1✱ ✐❢ ✐t ❝♦♥✈❡r❣❡s✳ ❚❤✐s ✐s ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s ❛♥❞ t❤❡ ❝♦♥str✉❝t✐♦♥ s✉❣❣❡sts t❤❛t ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ r❛t✐♦ r = 1/2 ❝♦♥✈❡r❣❡s✳ ▲❡t✬s ✐♥✈❡st✐❣❛t❡ t❤❡ ♣❛rt✐❛❧ s✉♠s ♦❢ t❤❡ ❣❡♥❡r❛❧ ❣❡♦♠❡tr✐❝ s❡r✐❡s✱ ar0 , ar1 , ar2 , ...arn−1 , arn , ... ,

✺✳✺✳

▼❛✐♥ ❝❧❛ss❡s ♦❢ s❡r✐❡s

✸✼✾

✉♥❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ r❡str✐❝t✐♦♥s✿

❲❡ ✇✐❧❧ ♥❡❡❞ ❛♥

❡①♣❧✐❝✐t

r 6= 0, a 6= 0 . ❢♦r♠✉❧❛ ❢♦r t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s ♦❢ t❤✐s s❡q✉❡♥❝❡✿

pn = ar0 + ar1 + ar2 + ... + arn−1 + arn . ❚❤❡ ✏✳✳✳✑ ♣❛rt ✐s ✇❤❛t st❛♥❞s ✐♥ t❤❡ ✇❛②✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❡✈❡r tr✐❝❦ ✭♣r❡s❡♥t❡❞ ✜rst ✐♥ ❈❤❛♣t❡r ✶P❈✲✶✮ s♦❧✈❡s t❤❡ ♣r♦❜❧❡♠✳ ❇❡❧♦✇✱ ✇❡ ✇r✐t❡ t❤❡ ♣❛rt✐❛❧ s✉♠

pn

✐♥ t❤❡ ✜rst r♦✇✱ ✐ts ♠✉❧t✐♣❧❡

rpn

✭❛❧❧ t❡r♠s ❛r❡ ♠✉❧t✐♣❧✐❡❞ ❜②

r✮ ✐♥ t❤❡ s❡❝♦♥❞✱

nt❤

s✉❜tr❛❝t t❤❡♠✱

❛♥❞ t❤❡♥ ❝❛♥❝❡❧ t❤❡ t❡r♠s t❤❛t ❛♣♣❡❛r t✇✐❝❡✿

= ar0

pn rpn

=

ar1

+ar1

+ar2

+... +arn−1

+arn

+ ar2

+ ar3

+... + arn

+ arn+1

❙✉❜tr❛❝t

pn − rpn = ar0 − ar1 +ar1 − ar2 +ar2 − ar3 +... +arn−1 − arn +arn − arn+1 = ar0

−arn+1 .

❚❤❡ ✏✳✳✳✑ ♣❛rt ✐s ❣♦♥❡✦ ❚❤❡r❡❢♦r❡✱

pn (1 − r) = a − arn+1 . ❚❤✉s✱ ✇❡ ❤❛✈❡ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r t❤❡

nt❤

t❡r♠ ♦❢ t❤❡ ♣❛rt✐❛❧ s✉♠✿

pn = a

1 − rn+1 . 1−r

▲❡t✬s ♥♦t❡ t❤❛t t❤✐s ✐s t❤❡ s✉♠ ♦❢ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ❛♥❞ ❛ ❝♦♥st❛♥t s❡q✉❡♥❝❡✿

pn =

−ra n a r + . 1−r 1−r

❲❡ t❤❡♥ ❤❛✈❡ ❛♥ ✐♥t❡r❡st✐♥❣ ♠❛t❝❤✿

❚❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s

❚❤❡ ✐♥t❡❣r❛❧

♦❢ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥

♦❢ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥

✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥

✐s ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥

✇✐t❤ t❤❡ s❛♠❡ r❛t✐♦ ♣❧✉s ❛ ❝♦♥st❛♥t✳ ■t✬s ❡❛s② t♦ ✜♥❞ t❤❡ ❧✐♠✐t ♦❢ t❤✐s s❡q✉❡♥❝❡ ❜❡❝❛✉s❡

rn+1

✇✐t❤ t❤❡ s❛♠❡ ❜❛s❡ ♣❧✉s ❛ ❝♦♥st❛♥t✳ ✐s t❤❡ ♦♥❧② t❡r♠ t❤❛t ♠❛tt❡rs✳ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ✐s t❤❡

❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✺✳✺✳✶✶✿ ❙✉♠ ♦❢ ●❡♦♠❡tr✐❝ ❙❡r✐❡s ❚❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ r❛t✐♦

r

❝♦♥✈❡r❣❡s ✐❢ ❛♥❞ ♦♥❧② ✐❢

|r| < 1 ; ✐♥ t❤❛t ❝❛s❡✱ t❤❡ s✉♠ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

∞ X k=0

ark =

a 1−r

✺✳✺✳

▼❛✐♥ ❝❧❛ss❡s ♦❢ s❡r✐❡s

✸✽✵

Pr♦♦❢✳

❲❡ ✉s❡ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛ ❛♥❞ t❤❡ ❢❛♠✐❧✐❛r ♣r♦♣❡rt✐❡s ♦❢ ❧✐♠✐ts✿ n X

ark = lim pn

k=0

n→∞

1 − rn+1 = lim a n→∞ 1−r a = lim (1 − rn+1 ) n→∞ 1−r   a 1 − lim rn+1 . = n→∞ 1−r

❚♦ ✜♥✐s❤✱ ✇❡ ✐♥✈♦❦❡ t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡ ❛❜♦✉t t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❣❡♦♠❡tr✐❝

♣r♦❣r❡ss✐♦♥s✳

❚❤❡ t❤❡♦r❡♠ ✐s ❝♦♥✜r♠❡❞ ♥✉♠❡r✐❝❛❧❧② ❜❡❧♦✇✿

❊①❡r❝✐s❡ ✺✳✺✳✶✷

❊①♣❧❛✐♥ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡s❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s ❛♥❞ t❤❡ s❡r✐❡s ✇✐t❤ r = 1✳ ❊①❛♠♣❧❡ ✺✳✺✳✶✸✿ ❣❡♦♠❡tr✐❝ s❡r✐❡s

❲❡ ❛♣♣❧② t❤❡ t❤❡♦r❡♠ ❛s ❢♦❧❧♦✇s✿ ❣❡♦♠❡tr✐❝ s❡r✐❡s 1 1 1 + + + ... 2 4 8 1 1 1 1+ + + + ... 3 9 27 1 1 1 1 − + − + ... 2 4 8 1+

✜rst t❡r♠ a r❛t✐♦ r s✉♠ 1 1 1

1 2 1 3 1 − 2

1 =2 1 − 1/2 3 1 = 1 − 1/3 2 2 1 = 1 + 1/2 3

1 + 1.1 + 1.12 + 1.13 + ...

1

1.1

❞✐✈❡r❣❡s

1 − 1 + 1 − 1 + ...

1

−1

❞✐✈❡r❣❡s

❚❤✐s ♣♦✇❡r❢✉❧ t❤❡♦r❡♠ ✇✐❧❧ ❛❧s♦ ❛❧❧♦✇ ✉s t♦ st✉❞② ♦t❤❡r s❡r✐❡s ❜② ❝♦♠♣❛r✐♥❣ t❤❡♠ t♦ ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s✳

✺✳✺✳

▼❛✐♥ ❝❧❛ss❡s ♦❢ s❡r✐❡s

✸✽✶

❊①❛♠♣❧❡ ✺✳✺✳✶✹✿ ❩❡♥♦✬s ♣❛r❛❞♦① ❘❡❝❛❧❧ ❛ s✐♠♣❧❡ s❝❡♥❛r✐♦ ✭❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✶✮✿ ❆s ②♦✉ ✇❛❧❦ t♦✇❛r❞ ❛ ✇❛❧❧✱ ②♦✉ ❝❛♥ ♥❡✈❡r r❡❛❝❤ ✐t ❜❡❝❛✉s❡ ♦♥❝❡ ②♦✉✬✈❡ ❝♦✈❡r❡❞ ❤❛❧❢ t❤❡ ❞✐st❛♥❝❡✱ t❤❡r❡ ✐s st✐❧❧ ❞✐st❛♥❝❡ ❧❡❢t✱ ❡t❝✳ ■t ✇✐❧❧ t❛❦❡ ✐♥✜♥✐t❡❧② ♠❛♥② st❡♣s t♦ r❡❛❝❤ t❤❡ ✇❛❧❧✿

❲❡ ♥♦✇ ❦♥♦✇ t❤❛t t❤❡ s✉♠ ♦❢ t❤❡ ❞✐st❛♥❝❡s ✐s 1 ❛s ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s✿ 1 1 1 1 + + + + ... = 1 . 1 2 4 8

t✐♠❡

❇✉t ✇❡ ❦♥❡✇ t❤❛t✦ ❲❤❛t r❡s♦❧✈❡s t❤❡ ♣❛r❛❞♦① ✐s t❤❡ ❢❛❝t t❤❛t t❤❡ ♣❡r✐♦❞s ❛❧s♦ ❢♦r♠ ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s✳ ❙✉♣♣♦s❡ v ✐s t❤❡ s♣❡❡❞ ♦❢ t❤❡ ♣❡rs♦♥✳ ❚❤❡♥✱ t✐♠❡ ♣❡r✐♦❞s ❛r❡ ❝♦♠♣✉t❡❞ ❛s t❤❡ ❞✐st❛♥❝❡ ♦✈❡r t❤❡ s♣❡❡❞✿ 1 1 1 /v, /v, /v, ... 2 4 8

❚❤✐s ✐s ❛ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡ ✇✐t❤✿ a=

1 , r = 1/2 . 2v

■ts s✉♠ ✐s ❢♦✉♥❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠✿ 1 11 11 11 + + + ... = . 2v 4v 8v v

❚❤✐s ✐s ❤♦✇ ❧♦♥❣ ✐t t❛❦❡s✳ ❙♦✱ ❡✈❡♥ t❤♦✉❣❤ ✐t t❛❦❡s ✐♥✜♥✐t❡❧② ♠❛♥② t❛❦❡s ✐s♥✬t ✐♥✜♥✐t❡✦

st❡♣s t♦ r❡❛❝❤ t❤❡ ✇❛❧❧✱ t❤❡ t✐♠❡ ✐t

❊①❛♠♣❧❡ ✺✳✺✳✶✺✿ ✐♥✜♥✐t❡ ❞❡❝✐♠❛❧s ▲❡t✬s r❡♣r❡s❡♥t t❤❡ ♥✉♠❜❡r 0.44444... ❛s ❛ s❡r✐❡s✳ ■❢ ✇❡ ❝❛♥ ❞❡♠♦♥str❛t❡ ❝♦♥✈❡r❣❡♥❝❡✱ t❤✐s ✐s ✇❤❛t ✇❡ ❤❛✈❡✿ 0.4444... = 0.4 = 4 · 0.1

+0.04

+0.004

+4 · 0.04

+4 · 0.004 +4 · 0.0004 +...

= 4 · 10−1 +4 · 10−2 +4 · 10−3

+0.0004 +4 · 10−4

+... +...

❚❤✐s ✐s ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ t❤❡ ✜rst t❡r♠ a = .4 ❛♥❞ t❤❡ r❛t✐♦ r = 0.1 < 1✳ ❚❤❡r❡❢♦r❡✱ ✐t ❝♦♥✈❡r❣❡s ❜② t❤❡ t❤❡♦r❡♠ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿ n X i=0

n

0.4 · 0.1 =

n X i=0

arn =

a 0.4 4 = = . 1−r 1 − 0.1 9

✺✳✺✳ ▼❛✐♥ ❝❧❛ss❡s ♦❢ s❡r✐❡s

✸✽✷

❊①❡r❝✐s❡ ✺✳✺✳✶✻

❯s❡ t❤❡ ❧❛st ❡①❛♠♣❧❡ t♦ s❤♦✇ t❤❛t ❢♦r ❛♥② ❞✐❣✐t d ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥✿ .dddd... =

d . 9

❊①❛♠♣❧❡ ✺✳✺✳✶✼✿ ❣❡♦♠❡tr✐❝

❍❡r❡ ✐s ❛♥♦t❤❡r ❣❡♦♠❡tr✐❝ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s✿

■♥st❡❛❞ ♦❢ ✜♥❞✐♥❣ t❤❡ s✉♠ ♦❢ ❛ s❡r✐❡s ❛s ❛ ♥✉♠❜❡r✱ ✇❡ ❛r❡ ♣r♦❞✉❝✐♥❣ ❛ s❡r✐❡s ❢r♦♠ ❛ ♥✉♠❜❡r✳ ❲❡ st❛rt ✇✐t❤ ❛ s✐♠♣❧❡ ♦❜s❡r✈❛t✐♦♥✿ 1=

3 1 1 + = (3 + 1) . 4 4 4

❲❡ r❡♣❧❛❝❡ 1 ✇✐t❤ t❤✐s ❡①♣r❡ss✐♦♥✱ ✇❤✐❝❤ ❛❧s♦ ❝♦♥t❛✐♥s 1✱ t♦ ❜❡ r❡♣❧❛❝❡❞ ✇✐t❤ t❤✐s ❡①♣r❡ss✐♦♥ ❛❣❛✐♥✱ ❡t❝✳ ❚❤✐s ✇❛②✱ ✐♥ ❝♦♥tr❛st t♦ ❛❧❧ ♦t❤❡r ❡①❛♠♣❧❡s✱ ✇❡ st❛rt ✇✐t❤ t❤❡ s✉♠ ❛♥❞ t❤❡♥ ❛❝q✉✐r❡ ❛ s❡r✐❡s ❢♦r ✐t✿ 1 1 = (3 + 1) 4 1 = (3 + 1) 4  1 1 = 3 + (3 + 1) 4 4  1 1 = 3 + (3 + 1) 4 4 !  1 1 1 = 3 + 3 + (3 + 1) 4 4 4

= = = = =

3 4 3 4 3 4 3 4 3 4

1 4 1 + ·1 4   1 3 1 + + 4 4 4 1 3 + 2 ·1 + 2 4 4   1 3 3 1 + 2 + 2 + 4 4 4 4 +

= ... ∞ X 3 . = 4n n=1

❚❤✐s ✐♥✜♥✐t❡ ❝♦♠♣✉t❛t✐♦♥ ♠❛❦❡s s❡♥s❡✱ t❤❛♥❦s t♦ t❤❡ ❧❛st t❤❡♦r❡♠✳ ❊①❛♠♣❧❡ ✺✳✺✳✶✽✿ ♣♦✇❡r s❡r✐❡s

▲❡t✬s ♥♦t ❢♦r❣❡t ✇❤② ✇❡ ❛r❡ ❞♦✐♥❣ t❤✐s✳ ❈♦♥s✐❞❡r t❤❡ ❢❛♠✐❧✐❛r ✭❛♥❞ t❤❡ s✐♠♣❧❡st✮ ♣♦✇❡r s❡r✐❡s✿ 1 + x + x2 + x3 + ...

■t ❝♦♥✈❡r❣❡s ❢♦r x = 0 ❛♥❞ ❞✐✈❡r❣❡s ❢♦r x = 2✳ ❲❡ ❦♥♦✇ ♠♦r❡ ♥♦✇✳ ❋♦r ❡❛❝❤ x✱ t❤✐s ✐s ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ r❛t✐♦ r = x✳ ❚❤❡r❡❢♦r❡✱ ✐t ❝♦♥✈❡r❣❡s ❢♦r ❡✈❡r② x t❤❛t s❛t✐s✜❡s |x| < 1✳ ❚❤❡ s✉♠ ✐s ❛ ♥✉♠❜❡r✱ ❛♥❞ t❤✐s ♥✉♠❜❡r ✐s t❤❡ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛t x✳ ❚❤❡ ✐♥t❡r✈❛❧ (−1, 1) ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ t❤✐s ✇❛②✳ ❚❤❡ t❤❡♦r❡♠ ❡✈❡♥ ♣r♦✈✐❞❡s ❛ ❢♦r♠✉❧❛ ❢♦r t❤✐s ❢✉♥❝t✐♦♥✿ 1 + x + x2 + x3 + ... =

1 . 1−x

✺✳✺✳

▼❛✐♥ ❝❧❛ss❡s ♦❢ s❡r✐❡s

✸✽✸

❚❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s ❣✐✈❡♥ ❤❡r❡ ✐s ❝❧❡❛r❀ ✐t✬s t❤❡✐r ❞♦♠❛✐♥s✳ ❚❤❡ ♣❛rt✐❛❧ s✉♠s ♦❢ t❤❡ s❡r✐❡s ❛r❡ ♣♦❧②♥♦♠✐❛❧s t❤❛t ❛♣♣r♦①✐♠❛t❡ t❤❡ ❢✉♥❝t✐♦♥✿

❚❤❡ ♠✐s♠❛t❝❤ ✐s ✈✐s✐❜❧❡ ✐♥ t❤❡ ❣r❛♣❤ ❛s t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❢❛✐❧s t♦ ✐♠♣r♦✈❡ ♦✉ts✐❞❡ t❤❡ ✐♥t❡r✈❛❧ (−1, 1)✳ ❊①❡r❝✐s❡ ✺✳✺✳✶✾

❈♦♥✜r♠ t❤❛t t❤❡s❡ ❛r❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❊①❡r❝✐s❡ ✺✳✺✳✷✵

❋✐♥❞ t❤❡ s✉♠s ♦❢ t❤❡ s❡r✐❡s ❣❡♥❡r❛t❡❞ ❜② ❡❛❝❤ ♦❢ t❤❡s❡ s❡q✉❡♥❝❡s ♦r s❤♦✇ t❤❛t ✐t ❞♦❡s♥✬t ❡①✐st✿ ✶✳ 1/1, 1/3, 1/5, 1/7, 1/9, 1/11, 1/13, ... ✷✳ 1/0.9, 1/0.99, 1/0.999, 1/0.9999, ... ✸✳ 1, −1, 1, −1, ... ✹✳ 1, −1/2, 1/4, −1/8, ... ✺✳ 1, 1/4, 1/16 , 1/64, ... ❊①❛♠♣❧❡ ✺✳✺✳✷✶✿ t❡❧❡s❝♦♣✐♥❣ s❡r✐❡s

❲❤❡♥ ❛ s❡r✐❡s ✐s♥✬t ❣❡♦♠❡tr✐❝✱ ✐t ♠✐❣❤t st✐❧❧ ❜❡ ♣♦ss✐❜❧❡ t♦ s✐♠♣❧✐❢② t❤❡ ♣❛rt✐❛❧ s✉♠s ✈✐❛ ❛❧❣❡❜r❛✐❝ tr✐❝❦s ❛♥❞ ✜♥❞ ✐ts s✉♠✿ ∞ X k=1

∞

 1 1 − k k+1  n  X 1 1 = lim − n→∞ k k+1 k=1       1 1 1 1 1 1− = lim + − + ... + − n→∞ 2  3 +1   n n    2 1 1 1 1 1 1 1 + − + + ... + − + − = lim 1 + − + n→∞ 2 2 3 3 n n n+1  1 = lim 1 − n→∞ n+1

X 1 = k(k + 1) k=1

= 1.



✺✳✻✳ ❋r♦♠ ✜♥✐t❡ s✉♠s ✈✐❛ ❧✐♠✐ts t♦ s❡r✐❡s

✸✽✹

❊①❡r❝✐s❡ ✺✳✺✳✷✷

❊①♣❧❛✐♥ ❤♦✇ t❤❡ ❛❜♦✈❡ ❝♦♠♣✉t❛t✐♦♥ ✐s ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♥❞ ✇❤② ✐t ♠❛tt❡rs✿ ∞ X k=1

 ∞  X 1 1 1 = − k(k + 1) k k+1 k=1       1 1 1 1 1 + + ... + + ... − − = 1− 2  3 +1 2   n n  1 1 1 1 1 1 =1+ − + + − + + ... + − + + ... 2 2 3 3 n n = 1.

❊①❡r❝✐s❡ ✺✳✺✳✷✸

❯s❡ t❤❡ ❛❧❣❡❜r❛✐❝ tr✐❝❦ ❢r♦♠ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡ t♦ ❡✈❛❧✉❛t❡ t❤✐s ✐♥t❡❣r❛❧✿ Z

1 dx . x(x + 1)

❚❤❡ ●❡♦♠❡tr✐❝ ❙❡r✐❡s ❚❤❡♦r❡♠✱ ❛s ✇❡❧❧ ❛s ♦t❤❡r t❤❡♦r❡♠s ❛❜♦✉t ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s❡r✐❡s ♣r❡s❡♥t❡❞ ❜❡❧♦✇✱ ❛❧❧♦✇s ✉s t♦ ❜❡ ❜♦❧❞❡r ✇✐t❤ ❝♦♠♣✉t❛t✐♦♥s t❤❛t ✐♥✈♦❧✈❡ ✐♥✜♥✐t❡❧② ♠❛♥② st❡♣s✳

✺✳✻✳ ❋r♦♠ ✜♥✐t❡ s✉♠s ✈✐❛ ❧✐♠✐ts t♦ s❡r✐❡s ▲❡t✬s r❡♠❡♠❜❡r t❤❛t✱ ✐♥✐t✐❛❧❧②✱ ◮ t❤❡r❡ ❛r❡ ♥♦ s❡r✐❡s✳

❆ s❡r✐❡s ✐s s❤♦rt❤❛♥❞ ❢♦r ✇❤❛t ✇❡ ❞♦ ✇✐t❤ ❛ s❡q✉❡♥❝❡✳ ❆ s❡r✐❡s ✐s✱ t❤❡r❡❢♦r❡✱ ❛❧✇❛②s ❛ ♣❛✐r ♦❢ s❡q✉❡♥❝❡s✿

❲❤❛t ✇❡ ❞♦ ✐s r❡❝✉rs✐✈❡ ❛♥❞✱ ✇❤❡♥ t❤❡ s❡q✉❡♥❝❡ ✐s ✐♥✜♥✐t❡✱ ✐♥✈♦❧✈❡s ✐♥✜♥✐t❡❧② ♠❛♥② st❡♣s✳ ❆t t❤❡ ♥❡①t st❛❣❡✱ ♥♦t❛t✐♦♥ t❛❦❡s ♦✈❡r ❛♥❞ t❤❡ s❡r✐❡s st❛rt t♦ ❜❡ tr❡❛t❡❞ ❛s ❡♥t✐t✐❡s✿ ❙❡r✐❡s

∞ X

ak ✐s t❤❡ ❧✐♠✐t ♦❢ ♣❛rt✐❛❧ s✉♠s ♦❢ ak .

k=m

❊①❡r❝✐s❡ ✺✳✻✳✶

❙❤♦✇ t❤❛t ❡✈❡r② s❡q✉❡♥❝❡ ✐s t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s ♦❢ s♦♠❡ s❡q✉❡♥❝❡✳ ❍✐♥t✿ ❯s❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❝♦♥✲ str✉❝t✐♦♥✿ an+1 = ∆pn = pn+1 − pn .

✺✳✻✳ ❋r♦♠ ✜♥✐t❡ s✉♠s ✈✐❛ ❧✐♠✐ts t♦ s❡r✐❡s

✸✽✺

■♥ ❱♦❧✉♠❡ ✶ ✭❈❤❛♣t❡r ✶P❈✲✶✮ ✇❡ ✐♥tr♦❞✉❝❡❞ ✜♥✐t❡ s✉♠s✱ ✐✳❡✳✱ s✉♠s ♦❢ t❤❡ t❡r♠s ♦❢ ❛ ❣✐✈❡♥ s❡q✉❡♥❝❡ an ♦✈❡r ❛♥ ✐♥t❡r✈❛❧ [a, b] ♦❢ ✈❛❧✉❡s ♦❢ k✿ q X

ak = ap + ap+1 + ... + aq ,

k=p

❛♥❞ t❤❡✐r ♣r♦♣❡rt✐❡s✳ ❲❡ ✉s❡❞ t❤♦s❡ ♣r♦♣❡rt✐❡s t♦ st✉❞② t❤❡ ❘✐❡♠❛♥♥ s✉♠s ✐♥ ❈❤❛♣t❡r ✶❀ t❤❡s❡ s✉♠s ❛r❡ t❤❡ ❛r❡❛s ✉♥❞❡r t❤❡ ❣r❛♣❤s ♦❢ t❤❡ st❡♣✲❢✉♥❝t✐♦♥s ♣r♦❞✉❝❡❞ ❜② t❤❡ s❡q✉❡♥❝❡s✳ ◆♦✇ ✇❡ ❥✉st ♥❡❡❞ t♦ tr❛♥s✐t✐♦♥ t♦ ✐♥✜♥✐t❡ s✉♠s ✈✐❛ ❧✐♠✐ts✳ ❆s ❛ ♠❛tt❡r ♦❢ ♥♦t❛t✐♦♥✱ ✇❡ ✇✐❧❧ ♦❢t❡♥ ♦♠✐t t❤❡ ❜♦✉♥❞s ✐♥ t❤❡ s✐❣♠❛ ♥♦t❛t✐♦♥ ❢♦r s❡r✐❡s✿

❙✐❣♠❛ ♥♦t❛t✐♦♥ ∞ X

ak ✐s r❡♣❧❛❝❡❞ ✇✐t❤

k=m

X

ak

❚❤❡ r❡❛s♦♥ ✐s ✇❤② t❤✐s ✐s ❛❝❝❡♣t❛❜❧❡ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❈♦♥s✐❞❡r t❤❡s❡ t✇♦ ✜♥✐t❡ s✉♠s✿ n X k=a

uk ❛♥❞

n X

uk .

k=b

❚❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡♠ ✐s ❥✉st ❛ ✜♥✐t❡❧② ♠❛♥② t❡r♠s ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡② ❡✐t❤❡r ❜♦t❤ ❝♦♥✈❡r❣❡ ♦r ❜♦t❤ ❞✐✈❡r❣❡✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❚r✉♥❝❛t✐♦♥ Pr✐♥❝✐♣❧❡ ❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✶✮✳ ❚❤❡ ♥♦t❛t✐♦♥ ✐s ❡s♣❡❝✐❛❧❧② ❛♣♣r♦♣r✐❛t❡ ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ❤♦♣❡ ♦❢ ✜♥❞✐♥❣ t❤❡ s✉♠ ♦❢ t❤❡ s❡r✐❡s✳ ■♥ t❤❛t ❝❛s❡✱ ✇❡ ❢❛❝❡ ❛ s✐♠♣❧❡ ❞✐❝❤♦t♦♠②✿ • ✐t ❝♦♥✈❡r❣❡s✱ ♦r • ✐t ❞✐✈❡r❣❡s✳

◆❡①t✱ ✇❡ ♣r♦✈❡ s❡✈❡r❛❧ t❤❡♦r❡♠s ❛❜♦✉t ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s❡r✐❡s✳ ❚❤❡② ❛❧❧ ❢♦❧❧♦✇ t❤❡ s❛♠❡ ✐❞❡❛✿ ◮ ■❢ ❛♥ ❛❧❣❡❜r❛✐❝ r❡❧❛t✐♦♥ ❡①✐sts ❢♦r ✜♥✐t❡ s✉♠s✱ t❤❡♥ t❤✐s r❡❧❛t✐♦♥ r❡♠❛✐♥s ✈❛❧✐❞ ❢♦r ✐♥✜♥✐t❡ s✉♠s✱

✐✳❡✳✱ s❡r✐❡s✱ ♣r♦✈✐❞❡❞ t❤❡② ❝♦♥✈❡r❣❡✳

❋✐rst✱ t❤❡ ❝♦♠♣❛r✐s♦♥ ♣r♦♣❡rt✐❡s✳ ■❢ t✇♦ s❡q✉❡♥❝❡s ❛r❡ ❝♦♠♣❛r❛❜❧❡✱ t❤❡♥ s♦ ❛r❡ t❤❡✐r s✉♠s✿

✺✳✻✳ ❋r♦♠ ✜♥✐t❡ s✉♠s ✈✐❛ ❧✐♠✐ts t♦ s❡r✐❡s

✸✽✻

■♥ ❢❛❝t✱ t❤✐s s✐♠♣❧❡ ❛❧❣❡❜r❛ t❡❧❧s t❤❡ ✇❤♦❧❡ st♦r②✿ u ≤ U

❆❞❞ t❤❡ t✇♦ ✐♥❡q✉❛❧✐t✐❡s ♣r♦❞✉❝✐♥❣ t❤❡ t❤✐r❞✿

v ≤ V

u+v ≤ U +V

❚❤❡ ♦♥❧② ❞✐✛❡r❡♥❝❡ ✐s t❤❛t ✇❡ ❤❛✈❡ ♠♦r❡ t❤❛♥ ❥✉st t✇♦ t❡r♠s✿ up ≤ Up

up+1 ≤ Up+1 ... ... ...

❆❞❞ t❤❡ s❡q✉❡♥❝❡ ♦❢ ✐♥❡q✉❛❧✐t✐❡s ♣r♦❞✉❝✐♥❣ t❤❡ ♥❡✇ ♦♥❡✿

uq ≤ Uq

up + ... + uq ≤ Up + ... + Uq q q X X un ≤ Un n=p

n=p

❚❤✐s ✐s t❤❡ ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ❙✉♠s ♣r❡s❡♥t❡❞ ✐♥ ❱♦❧✉♠❡ ✶ ✭❈❤❛♣t❡r ✶P❈✲✶✮ t❤❛t ✇❛s ✉s❡❞ t♦ st✉❞② ❘✐❡♠❛♥♥ s✉♠s ❛♥❞ t❤❡♥ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s ✐♥ ❈❤❛♣t❡r ✶✳ ❩♦♦♠✐♥❣ ♦✉t ❤❡❧♣s ✉s s❡❡ t❤❛t t❤❡ ❧❛r❣❡r ❢✉♥❝t✐♦♥✱ ♦r ❛ s❡q✉❡♥❝❡✱ ❛❧✇❛②s ❝♦♥t❛✐♥s ❛ ❧❛r❣❡r ❛r❡❛ ✉♥❞❡r ✐ts ❣r❛♣❤✿

❚❛❦✐♥❣ t❤❡ ❧✐♠✐t q → ∞ ❛❧❧♦✇s ✉s t♦ ♠❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❧✉s✐♦♥ ❜❛s❡❞ ♦♥ t❤❡ ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ▲✐♠✐ts ♦❢ ❙❡q✉❡♥❝❡s✳ ❚❤❡♦r❡♠ ✺✳✻✳✷✿ ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ❙❡r✐❡s ❆♥ ✐♥❡q✉❛❧✐t② ❜❡t✇❡❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❡r♠s ♦❢ t✇♦ s❡r✐❡s ❤♦❧❞s ❢♦r t❤❡✐r s✉♠s t♦♦✳

■♥ ♦t❤❡r ✇♦r❞s✱ s✉♣♣♦s❡

un

❛♥❞

Un

❛r❡ s❡q✉❡♥❝❡s✳

❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ❡✐t❤❡r ♦❢

t❤❡ t✇♦ s❡r✐❡s t❤❡② ♣r♦❞✉❝❡ ❝♦♥✈❡r❣❡s✳ ❚❤❡♥✿

un ≤ Un =⇒

X

un ≤

X

Un .

✺✳✻✳

❋r♦♠ ✜♥✐t❡ s✉♠s ✈✐❛ ❧✐♠✐ts t♦ s❡r✐❡s

✸✽✼

❊①❛♠♣❧❡ ✺✳✻✳✸✿ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ ❣❡♦♠❡tr✐❝ ❚❤❡ s❡r✐❡s

∞ X n=1

2n

1 −1

✐s♥✬t ❣❡♦♠❡tr✐❝✳✳✳ ❜✉t ❝❧♦s❡✳ ❈♦♥s✐❞❡r t❤✐s ♦❜✈✐♦✉s ❢❛❝t✿

2n − 1 ≤ 2n =⇒

2n

1 1 ≥ n. −1 2

❚❤❡ s❡r✐❡s ♣r♦❞✉❝❡❞ ❜② t❤❡ s❡q✉❡♥❝❡ ♦♥ t❤❡ r✐❣❤t ❝♦♥✈❡r❣❡s t♦ t❤❛t

∞ X n=1

2n

1✳

■t ❢♦❧❧♦✇s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠

1 ≥ 1, −1

✐❢ ✇❡ ❝❛♥ ❡st❛❜❧✐s❤ t❤❛t t❤✐s s❡r✐❡s ❝♦♥✈❡r❣❡s✳

❆ r❡❧❛t❡❞ r❡s✉❧t ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳

❚❤❡♦r❡♠ ✺✳✻✳✹✿ ❙tr✐❝t ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ❙❡r✐❡s ❙✉♣♣♦s❡

un

❛♥❞

Un

❛r❡ s❡q✉❡♥❝❡s✳

❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ❡✐t❤❡r ♦❢ t❤❡ t✇♦ s❡r✐❡s

t❤❡② ♣r♦❞✉❝❡ ❝♦♥✈❡r❣❡s✳ ❚❤❡♥✿

un < Un =⇒

X

un
pn . ■t ✐s ❛❧s♦

❜♦✉♥❞❡❞ ✿ pn = d1 · 0.1 + d2 · 0.01 + d3 · 0.001 + ... + dn · 0.1n

< 10 · 0.1 + 10 · 0.01 + 10 · 0.001 + ... + 10 · 0.1n

< 10 · 0.1 + 10 · 0.01 + 10 · 0.001 + ... + 10 · 0.1n + ... 1 = ❇❡❝❛✉s❡ ✐t✬s ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ r = 0.1 . 1 − 0.1 = 9. ❚❤❡r❡❢♦r❡✱ t❤❡ s❡q✉❡♥❝❡ ✐s ❝♦♥✈❡r❣❡♥t ❜② t❤❡

▼♦♥♦t♦♥❡ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠

✭❱♦❧✉♠❡ ✷✮✳

❚❤❡ r❡s✉❧t ❡①♣❧❛✐♥s ✇❤② t❤❡ ▼♦♥♦t♦♥❡ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠ ✐s ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡

♦❢ ❘❡❛❧ ◆✉♠❜❡rs ❀ ❛♥ ✏✐♥❝♦♠♣❧❡t❡✑

❈♦♠♣❧❡t❡♥❡ss Pr♦♣❡rt②

r♦♣❡ ✇♦♥✬t ❤❛♥❣✿

❊①❡r❝✐s❡ ✺✳✻✳✶✶

Pr♦✈❡ t❤❡ t❤❡♦r❡♠ ❜② ✉s✐♥❣ t❤❡ ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ❙❡r✐❡s ✐♥st❡❛❞✳

❊①❡r❝✐s❡ ✺✳✻✳✶✷

❙t❛t❡ ❛♥❞ ♣r♦✈❡ ❛♥ ❛♥❛❧♦❣♦✉s t❤❡♦r❡♠ ❢♦r

❜✐♥❛r②

❛r✐t❤♠❡t✐❝✳

▼♦r❡♦✈❡r✱ t❤❡ ❞❡❝✐♠❛❧ ♥✉♠❜❡rs ❛r❡ ❛❧s♦ s✉❜❥❡❝t t♦ t❤❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❛❧❣❡❜r❛✐❝ t❤❡♦r❡♠s ❛❜♦✈❡✳ ❊①❛♠♣❧❡ ✺✳✻✳✶✸✿ ❛❧❣❡❜r❛ ♦❢ ❞❡❝✐♠❛❧s

❲❡ ❝❛♥ ❛❞❞ ✐♥✜♥✐t❡ ❞❡❝✐♠❛❧s✿

u

= .u1 u2 ...un ... =

∞ X k=1

v

= .v1 v2 ...vn ...

=

∞ X k=1

u+v

=

∞ X k=1

uk · (0.1)k vk · (0.1)k (uk + vk ) · (0.1)k

✺✳✼✳ ❉✐✈❡r❣❡♥❝❡

✸✾✷

❲❡ ❝❛♥ ❛❧s♦ ♠✉❧t✐♣❧② ❛♥ ✐♥✜♥✐t❡ ❞❡❝✐♠❛❧ ❜② ❛♥♦t❤❡r r❡❛❧ ♥✉♠❜❡r✿ = .u1 u2 ...un ... =

u

∞ X k=1

c·u

=

∞ X k=1

uk · (0.1)k c · uk · (0.1)k

❚❤❡ ❢♦r♠✉❧❛s ❞♦♥✬t t❡❧❧ ✉s t❤❡ ❞❡❝✐♠❛❧ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ u + v ✭✉♥❧❡ss un + vn < 10✮ ♦r c · u ✳

✺✳✼✳ ❉✐✈❡r❣❡♥❝❡

❲❤❛t ✐❢ ✇❡ ❢❛❝❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ✇✐t❤ s❡r✐❡s t❤❛t ❞✐✈❡r❣❡ ❄ ❚❤❡ ❧❛✇s ❛❜♦✈❡ t❡❧❧ ✉s ♥♦t❤✐♥❣✦ ❋♦r ❡①❛♠♣❧❡✱ t❤✐s ✐s t❤❡ ❙✉♠ ❘✉❧❡ ✿ X X X (an + bn ) ❝♦♥✈❡r❣❡s❀ an , bn ❝♦♥✈❡r❣❡ =⇒

❜✉t t❤✐s ✐s♥✬t✿

❊①❡r❝✐s❡ ✺✳✼✳✶

X

an ,

X

bn ❞✐✈❡r❣❡ 6=⇒

X

(an + bn ) ❞✐✈❡r❣❡s✳

❙❤♦✇ t❤❛t t❤✐s st❛t❡♠❡♥t ✇♦✉❧❞ ✐♥❞❡❡❞ ❜❡ ✉♥tr✉❡✳

❊①❛♠♣❧❡ ✺✳✼✳✷✿ ❝♦♥✈❡r❣❡♥t ♣❧✉s ❞✐✈❡r❣❡♥t ❈♦♠♣✉t❡ t❤❡ s✉♠✿

 ∞  X 1 en + . 2n 3 n=1

❚❤✐s s✉♠ ✐s ❛ ❧✐♠✐t✱ ❛♥❞ ✇❡ ❝❛♥✬t ❛ss✉♠❡ t❤❛t t❤❡ ❛♥s✇❡r ✇✐❧❧ ❜❡ ❛ ♥✉♠❜❡r✳ ▲❡t✬s tr② t♦ ❛♣♣❧② t❤❡ ❙✉♠ ❘✉❧❡✿  ∞  X en 1 + n 2 3 n=1

∞ ∞ X X en 1 + == ❨❡s✱ ❜✉t ♦♥❧② ✐❢ t❤❡ t✇♦ s❡r✐❡s ❝♦♥✈❡r❣❡✦ n 2 3 n=1 n=1 ∞  n ∞ X X 1 1 n = e . ❉♦ t❤❡②❄ + 2 3 n=1 n=1

❄

❚❤❡s❡ ❛r❡ t✇♦ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ t❤❡ r❛t✐♦s ❡q✉❛❧ t♦✱ r❡s♣❡❝t✐✈❡❧②✱ 1/2 ❛♥❞ e✳ ❚❤❡ ✜rst ♦♥❡ ✐s s♠❛❧❧❡r t❤❛♥ 1 ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s✱ ❜✉t t❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ❧❛r❣❡r t❤❛♥ 1 ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ s❡r✐❡s ❞✐✈❡r❣❡s✦ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t ✐t ✐s ✉♥❥✉st✐✜❡❞ t♦ ✉s❡ t❤❡ ❙✉♠ ❘✉❧❡✳ ❲❡ ♣❛✉s❡ ❛t t❤✐s ♣♦✐♥t✳✳✳ ❛♥❞ t❤❡♥ tr② t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ❜② ♦t❤❡r ♠❡❛♥s✳ ❲❡ r❡❝❛❧❧ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✶ t❤❛t✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ ✐♥✜♥✐t❡ ❧✐♠✐ts✱ ✇❡ ❛❞❤❡r❡ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ r✉❧❡s ✭k 6= 0✮✿

❆❧❣❡❜r❛ ♦❢ ■♥✜♥✐t✐❡s

♥✉♠❜❡r ±∞

k

+ (±∞) = ±∞

+ (±∞) = ±∞

·

(±∞) = ± sign(k)∞

✺✳✼✳

❉✐✈❡r❣❡♥❝❡

✸✾✸

■t ❢♦❧❧♦✇s t❤❛t t❤❡ s❡r✐❡s ✐♥ t❤❡ ❧❛st ❡①❛♠♣❧❡ ❞✐✈❡r❣❡s t♦ ✐♥✜♥✐t②✳ ❚❤❡s❡ ❢♦r♠✉❧❛s s✉❣❣❡st t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐✈❡r❣❡♥❝❡ r❡s✉❧t ❢♦r s❡r✐❡s✳ ❚❤❡♦r❡♠ ✺✳✼✳✸✿ P✉s❤ ❖✉t ❚❤❡♦r❡♠ ❢♦r ❙❡r✐❡s

✶✳ ■❢ t❤❡ ✈❛❧✉❡s ♦❢ ❛ s❡r✐❡s ❧✐❡ ❛❜♦✈❡ t❤♦s❡ ♦❢ ❛ s❡r✐❡s t❤❛t ❞✐✈❡r❣❡s t♦ ♣♦s✐t✐✈❡ ✐♥✜♥✐t②✱ t❤❡♥ s♦ ❞♦❡s t❤✐s s❡r✐❡s✳ ✷✳ ■❢ t❤❡ ✈❛❧✉❡s ♦❢ ❛ s❡r✐❡s ❧✐❡ ❜❡❧♦✇ t❤♦s❡ ♦❢ ❛ s❡r✐❡s t❤❛t ❞✐✈❡r❣❡s t♦ ♥❡❣❛t✐✈❡ ✐♥✜♥✐t②✱ t❤❡♥ s♦ ❞♦❡s t❤✐s s❡r✐❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ s✉♣♣♦s❡ an ❛♥❞ bn ❛r❡ s❡q✉❡♥❝❡s✳ ❙✉♣♣♦s❡ t❤❛t✱ ❢♦r ❛♥ ✐♥t❡❣❡r p✱ ✇❡ ❤❛✈❡✿ an ≥ bn ❢♦r n ≥ p .

❚❤❡♥✿

X

X

an = +∞ ⇐=

an = −∞ =⇒

X

X

bn = +∞ bn = −∞

Pr♦♦❢✳

■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ P✉s❤

❖✉t ❚❤❡♦r❡♠ ❢♦r ❙❡q✉❡♥❝❡s ✭❱♦❧✉♠❡ ✷✮✳

❙♦✱ t❤❡ s♠❛❧❧❡r s❡r✐❡s✱ ✐❢ ✐t ❣♦❡s t♦ +∞✱ ♣✉s❤❡s t❤❡ ❧❛r❣❡r ♦♥❡ ✉♣✱ t♦ +∞✳ ❆♥❞ t❤❡ ❧❛r❣❡r s❡r✐❡s✱ ✐❢ ✐t ❣♦❡s t♦ −∞✱ ♣✉s❤❡s t❤❡ ❧❛r❣❡r ♦♥❡ ❞♦✇♥✱ t♦ −∞✳ ❊①❛♠♣❧❡ ✺✳✼✳✹✿ ❝♦♠♣❛r✐s♦♥

❈♦♥s✐❞❡r t❤✐s ♦❜✈✐♦✉s ❢❛❝t✿

1 1 ≥ . 2 − 1/n 2

■t ❢♦❧❧♦✇s t❤❛t

∞ X n=1

1 = ∞. 2 − 1/n

❊①❡r❝✐s❡ ✺✳✼✳✺

●✐✈❡ ❡①❛♠♣❧❡s ♦❢ s❡r✐❡s t❤❛t s❤♦✇ t❤❛t t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ t❤❡♦r❡♠ ✐s ✉♥tr✉❡✳ ❲❛r♥✐♥❣✦

◆♦t ❛❧❧ ❞✐✈❡r❣❡♥t s❡r✐❡s ❞✐✈❡r❣❡ t♦ ✐♥✜♥✐t②✳

❲❡ t✉r♥ t♦ ❛❧❣❡❜r❛✿ ❚❤❡♦r❡♠ ✺✳✼✳✻✿ ❉✐✈❡r❣❡♥❝❡ ♦❢ ❙✉♠ ♦❢ ❙❡r✐❡s

❙✉♣♣♦s❡ an ❛♥❞ bn ❛r❡ s❡q✉❡♥❝❡s✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡✿ • ❚❤❡ s✉♠ ♦❢ ❛ ❞✐✈❡r❣❡♥t s❡r✐❡s ❛♥❞ ❛ ❝♦♥✈❡r❣❡♥t s❡r✐❡s ❞✐✈❡r❣❡s✿ X

an ❞✐✈❡r❣❡s✱

X

bn ❝♦♥✈❡r❣❡s

=⇒

X

(an + bn ) ❞✐✈❡r❣❡s✳

• ❋♦r s❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s✱ t❤❡ s✉♠ ♦❢ t✇♦ ❞✐✈❡r❣❡♥t s❡r✐❡s ❞✐✈❡r❣❡s✿ X X X an ❞✐✈❡r❣❡s✱ bn ❞✐✈❡r❣❡s =⇒ (an + bn ) ❞✐✈❡r❣❡s✳

✺✳✼✳

❉✐✈❡r❣❡♥❝❡

✸✾✹ ❚❤❡♦r❡♠ ✺✳✼✳✼✿ ❉✐✈❡r❣❡♥❝❡ ♦❢ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ♦❢ ❙❡r✐❡s ❙✉♣♣♦s❡

an

✐s ❛ s❡q✉❡♥❝❡✳

❚❤❡♥✱ ❛ ❝♦♥st❛♥t✱ ♥♦♥✲③❡r♦ ♠✉❧t✐♣❧❡ ♦❢ ❛ ❞✐✈❡r❣❡♥t

s❡r✐❡s ❞✐✈❡r❣❡s✿

X

an

❞✐✈❡r❣❡s

=⇒

X

kan

, k 6= 0 .

❞✐✈❡r❣❡s

❊①❡r❝✐s❡ ✺✳✼✳✽

Pr♦✈❡ t❤❡s❡ t❤❡♦r❡♠s✳ ❚❤❡ ♣❛tt❡r♥ t❤❛t ✇❡ ♠❛② ❤❛✈❡ ♥♦t✐❝❡❞ ✐s t❤❛t ❝♦♥st❛♥t❧② ❛❞❞✐♥❣ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs t❤❛t ❣r♦✇ ✇✐❧❧ ❣✐✈❡ ②♦✉ ✐♥✜♥✐t② ❛t t❤❡ ❧✐♠✐t✳ ❚❤❡ s❛♠❡ ❝♦♥❝❧✉s✐♦♥ ✐s✱ ♦❢ ❝♦✉rs❡✱ tr✉❡ ❢♦r ❛ ❝♦♥st❛♥t s❡q✉❡♥❝❡✳ ❲❤❛t ✐❢ t❤❡ s❡q✉❡♥❝❡ ❞❡❝r❡❛s❡s❄ ❚❤❡♥ ✐t ❞❡♣❡♥❞s✳ ❋♦r ❡①❛♠♣❧❡✱ an = 1 + 1/n ❞❡❝r❡❛s❡s ❜✉t t❤❡ s❡r✐❡s st✐❧❧ ❞✐✈❡r❣❡s✳ ■t ❛♣♣❡❛rs t❤❛t t❤❡ s❡q✉❡♥❝❡ s❤♦✉❧❞ ❛t ❧❡❛st ❞❡❝r❡❛s❡ t♦ ③❡r♦✳ ❚❤❡ ❛❝t✉❛❧ r❡s✉❧t ✐s ❝r✉❝✐❛❧✳ ❚❤❡♦r❡♠ ✺✳✼✳✾✿ ❉✐✈❡r❣❡♥❝❡ ❚❡st ❢♦r ❙❡r✐❡s ■❢ ❛ s❡q✉❡♥❝❡ ❞♦❡s♥✬t ❝♦♥✈❡r❣❡ t♦ ③❡r♦✱ ✐ts s✉♠ ❞✐✈❡r❣❡s✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

X

an

❝♦♥✈❡r❣❡s

=⇒ an → 0

Pr♦♦❢✳

❲❡ ❤❛✈❡ t♦ ✐♥✈♦❦❡ t❤❡ ❞❡✜♥✐t✐♦♥✳ ❇✉t ❧❡t✬s t✉r♥ t♦ t❤❡ ❝♦♥tr❛♣♦s✐t✐✈❡ ❢♦r♠ ♦❢ t❤❡ t❤❡♦r❡♠✿ ◮ ■❢ ❛ s❡r✐❡s ❝♦♥✈❡r❣❡s✱ t❤❡♥ t❤❡ ✉♥❞❡r❧②✐♥❣ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡s t♦ 0✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ♥❡❡❞ t♦ ♣r♦✈❡ t❤✐s✿ lim

n→∞

n X k=1

ak = P =⇒ lim an = 0 , n→∞

✇❤❡r❡ P ✐s s♦♠❡ ♥✉♠❜❡r✳ ❖r✱ ❢♦r t❤❡ s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐❛❧ s✉♠s pn ✱ ✇❡ ❤❛✈❡ t♦ ❞❡♠♦♥str❛t❡ t❤✐s✿ lim pn = P =⇒ lim an = 0 .

n→∞

n→∞

❘❡❝❛❧❧ t❤❡ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s ♦❢ an ✿ pn+1 = pn + an ,

❛♥❞✱ ❛❝❝♦r❞✐♥❣❧②✱ ✇❡ ❤❛✈❡ t❤❡ ♦r✐❣✐♥❛❧ ❛s t❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐✛❡r❡♥❝❡s ♦❢ pn ✿ an = pn − pn−1 .

❚❤❡r❡❢♦r❡✱ lim an = lim (pn − pn−1 )

n→∞

n→∞

❇♦t❤ ❝♦♥✈❡r❣❡✱ ❛♥❞ ✇❡ ❛♣♣❧②✳✳✳

= lim pn − lim pn−1 ✳✳✳t❤❡ ❙✉♠ ❘✉❧❡ ❢♦r ❙❡q✉❡♥❝❡s✳ n→∞

=P −P = 0.

n→∞

❚❤❡ ❧✐♠✐t ✐s t❤❡ s❛♠❡ ❜❡❝❛✉s❡ ✐t✬s t❤❡ s❛♠❡ s❡q✉❡♥❝❡✳

✺✳✽✳

✸✾✺

❙❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s

❚❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ t❤❡♦r❡♠ ✐s ✉♥tr✉❡ ❛s ✇✐❧❧ ❜❡ s❡❡♥ ❢r♦♠ t❤❡ ❡①❛♠♣❧❡ ♦❢ t❤❡ ❤❛r♠♦♥✐❝ s❡r✐❡s✿ X

X

an ❝♦♥✈❡r❣❡s =⇒ an → 0

an ❝♦♥✈❡r❣❡s 6⇐= an → 0

❊①❛♠♣❧❡ ✺✳✼✳✶✵✿ t❡st ❞✐✈❡r❣❡♥❝❡

❚❤❡ t❡st ✐s ❢♦r ❞✐✈❡r❣❡♥❝❡ ❛♥❞ ♥♦t❤✐♥❣ ❡❧s❡✿ 

1 ✶✳ lim 1 + n



6= 0 =⇒

X

X

1 1+ n

✷✳ lim sin n 6= 0

=⇒

✸✳ lim

=⇒ t❡st ❢❛✐❧s✳

1 =0 n 1 ✹✳ lim 2 = 0 n



❞✐✈❡r❣❡s✳

sin n ❞✐✈❡r❣❡s✳

=⇒ t❡st ❢❛✐❧s✳ ❲❛r♥✐♥❣✦ ❚❤❡ ❢❛✐❧✉r❡ ♦❢ t❤❡ t❡st ❞♦❡s♥✬t ♣r♦✈❡ ❛♥②t❤✐♥❣✦

✺✳✽✳ ❙❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s

❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s✉❝❤ s❡r✐❡s ✐s ❡❛s✐❡r t♦ ❞❡t❡r♠✐♥❡✳ ❆❧❧ ✇❡ ♥❡❡❞ ✐s t❤❡ ▼♦♥♦t♦♥❡

❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠

❢r♦♠ ❱♦❧✉♠❡ ✷ ✭❈❤❛♣t❡r ✷❉❈✲✶✮✿

◮ ❊✈❡r② ♠♦♥♦t♦♥❡ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡s✳

❖❢ ❝♦✉rs❡✱ ✇❡ ✇✐❧❧ ♥♦t ❛♣♣❧② t❤❡ t❤❡♦r❡♠ t♦ t❤❡ s❡q✉❡♥❝❡ ❜✉t r❛t❤❡r t♦ ✐ts s❡q✉❡♥❝❡ ♦❢ s✉♠s✦ ■♥❞❡❡❞✱ ✐❢ t❤❡ ♦r✐❣✐♥❛❧ s❡q✉❡♥❝❡ an ❤❛s ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s✿ an > 0✱ t❤❡♥ t❤❡ ♥❡✇ s❡q✉❡♥❝❡ ♦❢ s✉♠s ✐s ✐♥❝r❡❛s✐♥❣ ✿ pn+1 = pn + an+1 ≥ pn .

■❢ t❤❡ ❉✐✈❡r❣❡♥❝❡ ❚❡st ✐s ❛❧s♦ s❛t✐s✜❡❞✱ t❤❡ ♣❛✐r ♦❢ t❤❡ s❡q✉❡♥❝❡ ❛♥❞ ✐ts s✉♠s ✇✐❧❧ ❤❛✈❡ t♦ ❧♦♦❦ ❥✉st ❧✐❦❡ t❤❡ ❣❡♥❡r✐❝ ✐❧❧✉str❛t✐♦♥ ✇❡ ❤❛✈❡ ❜❡❡♥ ✉s✐♥❣✿

❆ ❣♦♦❞ ❡①❛♠♣❧❡ ✐s t❤❡ ❤❛r♠♦♥✐❝ s❡r✐❡s ❜❡❧♦✇✿

✺✳✽✳ ❙❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s

✸✾✻

❙♦✱ ✐❢ s✉❝❤ ❛ s❡q✉❡♥❝❡ ✐s ❛❧s♦ ❜♦✉♥❞❡❞✱ ✐t ✐s ❝♦♥✈❡r❣❡♥t✳ ❚❤❡r❡❢♦r❡✱ s✉❝❤ ❛ s❡r✐❡s ❝❛♥✬t ❥✉st ❞✐✈❡r❣❡✱ ❛s ❞♦❡s❀ ✐t ❤❛s t♦ ❞✐✈❡r❣❡ t♦ ✐♥✜♥✐t②✳

X

sin n

❚❤❡♦r❡♠ ✺✳✽✳✶✿ ◆♦♥✲♥❡❣❛t✐✈❡ ❙❡r✐❡s

X

❋♦r ❛ s❡r✐❡s an ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s✱ an ≥ 0✱ t❤❡r❡ ❝❛♥ ❜❡ ♦♥❧② ❤❛✈❡ t✇♦ ♦✉t❝♦♠❡s✿ • ✐t ❝♦♥✈❡r❣❡s✱ ♦r • ✐t ❞✐✈❡r❣❡s t♦ ✐♥✜♥✐t②✳ ❚❤✐s ♦❜s❡r✈❛t✐♦♥ s✐❣♥✐✜❝❛♥t❧② s✐♠♣❧✐✜❡s t❤✐♥❣s❀ t❤❡ s✉♠ ♦❢ t❤❡ s❡r✐❡s ✐s ❡✐t❤❡r✿ • ❛ ♥✉♠❜❡r✱ ♦r • t❤❡ ✐♥✜♥✐t②✳

❚❤❡ ❧❛tt❡r ♦♣t✐♦♥ ❛♥❞ t❤❡ ❢♦r♠❡r ♦♣t✐♦♥ ❛r❡ ♦❢t❡♥ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ✇❤❡♥❡✈❡r t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ s✉♠ ✐s ♥♦t ❜❡✐♥❣ ❝♦♥s✐❞❡r❡❞✿ ❙✉♠ ♦❢ s❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s

X

X

an = ∞ ♦r

an < ∞

▼❛♥② t❤❡♦r❡♠s ✐♥ t❤❡ r❡st ♦❢ t❤❡ ❝❤❛♣t❡r ✇✐❧❧ ♦♥❧② t❡❧❧ t❤❡ ❢♦r♠❡r ❢r♦♠ t❤❡ ❧❛tt❡r✳✳✳ ❊①❛♠♣❧❡ ✺✳✽✳✷✿ ❞✐✈❡r❣❡♥❝❡

❙✐♥❝❡ t❤❡ s❡r✐❡s



(−1)n lim 1 + n X



= 1 6= 0 ,

(−1)n 1+ n



,

❢❛✐❧s t❤❡ ❉✐✈❡r❣❡♥❝❡ ❚❡st ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❞✐✈❡r❣❡s✳ ▼♦r❡♦✈❡r✱ X

(−1)n 1+ n



= ∞.

❊①❡r❝✐s❡ ✺✳✽✳✸

Pr♦✈❡ t❤❛t t❤❡ ❤❛r♠♦♥✐❝ s❡r✐❡s ❞✐✈❡r❣❡s ❜② ❢♦❧❧♦✇✐♥❣ t❤✐s ♦❜s❡r✈❛t✐♦♥✿ ❢♦r ❡❛❝❤ k ❝♦♥s❡❝✉t✐✈❡ t❡r♠s✱ 1 1 1 1 k 1 , , ..., ✱ t❤❡② ❛r❡ ❛❧❧ ≥ ✱ s♦ t❤❡✐r s✉♠ ✐s ≥ = ✳ k+1 k+2 2k 2k 2k 2

❲❡ ♥♦✇ ❛❞❞r❡ss t❤❡ ✐ss✉❡ ♦❢ s❡r✐❡s ✈s✳ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s ♦✈❡r ✐♥✜♥✐t❡ ❞♦♠❛✐♥s✳

✺✳✽✳

✸✾✼

❙❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s

❇♦t❤ ❛r❡ ❧✐♠✐ts✿ Z

∞

f (x) dx = lim

b→∞

1 ∞ X

ai

= lim

i=1

n→∞

Z

b

f (x) dx

1 n X

ai

i=1

❆♥❞ t❤❡ ♥♦t❛t✐♦♥ ♠❛t❝❤❡s t♦♦✳ ◆♦t ♦♥❧② t❤❡ ✐♥t❡❣r❛❧ ❜✉t ❛❧s♦ t❤❡ s✉♠ ♦♥ t❤❡ r✐❣❤t r❡♣r❡s❡♥t ❛r❡❛s ✉♥❞❡r ❣r❛♣❤s ♦❢ ❝❡rt❛✐♥ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ ✜♥✐t❡ ✐♥t❡r✈❛❧s✳ ❇♦t❤ t❤❡ ✐♥t❡❣r❛❧ ❛♥❞ t❤❡ s✉♠ ♦♥ t❤❡ ❧❡❢t ❛r❡ ❝♦♠♣✉t❡❞ ❛s ❧✐♠✐ts ♦❢ t❤❡ ♦♥❡s ♦♥ t❤❡ r✐❣❤t✿

❲❡ ❝❛♥ ❝♦♥❥❡❝t✉r❡ ♥♦✇ t❤❛t ✐❢ f ❛♥❞ an ❛r❡ r❡❧❛t❡❞✱ t❤❡♥ t❤❡s❡ ❧✐♠✐ts✱ t❤♦✉❣❤ ♥♦t ❡q✉❛❧✱ ♠❛② ❜❡ r❡❧❛t❡❞ t♦♦✳ ❚❤❡ s❡q✉❡♥❝❡ ♠❛② ❝♦♠❡ ❢r♦♠ s❛♠♣❧✐♥❣ t❤❡ ❢✉♥❝t✐♦♥✿ an = f (n), n = 1, 2, 3, ...

▲✐❦❡ t❤✐s✿

❊①❛♠♣❧❡ ✺✳✽✳✹✿ ✐♥t❡❣r❛❧s ✈s✳ s✉♠s

❈♦♥s✐❞❡r t❤✐s ♣❛✐r✿

f (x) = e−x ❛♥❞ an = e−n .

❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤❡ ❢♦r♠❡r ❛♥❞ t❤❡ s✉♠ ♦❢ ❝❡rt❛✐♥ r❡❣✐♦♥s ❛♥❞ ✇❡ ❝❛♥ ♣❧❛❝❡ ♦♥❡ ❜❡❧♦✇ ♦r ❛❜♦✈❡ t❤❡ ♦t❤❡r✿

♦❢ t❤❡ ❧❛tt❡r ❄

❇♦t❤ ❛r❡ t❤❡ ❛r❡❛s

✺✳✽✳ ❙❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s

✸✾✽

❚❤❡ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧ ✐s ❡❛s② t♦ ❝♦♠♣✉t❡✿

Z

∞

e

−x

dx = lim

b→∞

1

▼❡❛♥✇❤✐❧❡✱ t❤❡ s❡r✐❡s ✐s ❣❡♦♠❡tr✐❝ ✇✐t❤

Z

b 1

e−x dx = lim −(e−b − e1 ) = e . b→∞

r = 1/e✿ ∞ X

e−n =

n=1

1/e . 1 − 1/e

❇♦t❤ ❝♦♥✈❡r❣❡✦

❆ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡② ❡✐t❤❡r ❜♦t❤ ❝♦♥✈❡r❣❡ ♦r ❜♦t❤ ❞✐✈❡r❣❡ ✭t♦ ✐♥✜♥✐t②✮ ❢♦r ❛♥②

•

t❤❡ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ✇✐t❤ ❜❛s❡

•

t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ r❛t✐♦

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡✿

Z

1/a

♦✈❡r

[1, ∞)

a > 0✿

❛♥❞

1/a✳ ∞

−x

a

1

dx < ∞ ⇐⇒

∞ X n=1

a−n < ∞ .

❊①❡r❝✐s❡ ✺✳✽✳✺

Pr♦✈❡ t❤❡ st❛t❡♠❡♥t✳

❲❡ ✇✐❧❧ ♥♦✇ tr② t♦ ❛♣♣❧② t❤❡ ✐❞❡❛ t♦ ❛ ❣❡♥❡r❛❧ s❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s ✐❢ ✇❡ ❝❛♥ ♠❛t❝❤ ✐t ✇✐t❤ ❛♥ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧✳ ❚❤❡ ❤♦♣❡ ✐s t❤❛t ✇❡ ❝❛♥ ❤❛♥❞❧❡ ✐♥t❡❣r❛❧s ❜❡tt❡r ✕ ✇✐t❤ ❛❧❧ t❤❡ t♦♦❧s ✐♥ ❈❤❛♣t❡r ✷ ✕ t❤❛♥ t❤❡ s❡r✐❡s✳ ❋✐rst✱ ✇❡ ♠✐❣❤t ❞✐s❝♦✈❡r t❤❛t ♦✉r s❡r✐❡s ✐s ✏❞♦♠✐♥❛t❡❞✑ ❜② ❛ ❝♦♥✈❡r❣❡♥t ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧✿

∞ X k=1

an ≤

Z

∞ 1

f (x) dx < ∞ .

❚❤❡ ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞ ✇❤❡♥✿

an ≤ f (x)

❢♦r ❡✈❡r②

x

✐♥

[n, n + 1] .

✺✳✽✳ ❙❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s

✸✾✾

❖r✱ ✇❡ ♠✐❣❤t ❞✐s❝♦✈❡r t❤❛t ♦✉r s❡r✐❡s ✏❞♦♠✐♥❛t❡s✑ ❛ ❞✐✈❡r❣❡♥t ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧✿

∞ X k=1

an ≥

Z

∞

f (x) dx = ∞ .

1

❚❤❡ ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞ ✇❤❡♥✿

an ≥ f (x)

❢♦r ❡✈❡r②

x

✐♥

[n, n + 1] .

❚❤❡r❡ ✐s ❛ ✇❛② t♦ ❝♦♠❜✐♥❡ t❤❡s❡ t✇♦ ❝♦♥❞✐t✐♦♥s ✐♥t♦ ♦♥❡✳ ❚❤❡ ✐❞❡❛ ✐s t♦ ✏sq✉❡❡③❡✑ t❤❡ s❡q✉❡♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❢✉♥❝t✐♦♥s✳ ❇✉t ✇❤❡r❡ ❞♦❡s t❤❡ ♦t❤❡r ❢✉♥❝t✐♦♥ ❝♦♠❡ ❢r♦♠❄ ❲❡ s❤✐❢t t❤❡ ❣r❛♣❤ ♦❢

f

t♦ t❤❡ r✐❣❤t ❜②

1

✉♥✐t ✐♥

♦r❞❡r t♦ ♣✉t ✐t ❛❜♦✈❡ t❤❡ s❡q✉❡♥❝❡✿

f (x) ≤ an ≤ f (x − 1) . ❚❤✐s ✐s t❤❡ ♠❛✐♥ ✐❞❡❛✿

❚❤❡ ❝♦♥❞✐t✐♦♥s t❤❛t ♠❛❦❡ s✉r❡ t❤❛t t❤❡ ♣✐❝t✉r❡ ✐s ❥✉st✐✜❡❞ ❛r❡ ❧✐st❡❞ ❜❡❧♦✇✳ ❚❤❡♦r❡♠ ✺✳✽✳✻✿ ■♥t❡❣r❛❧ ❈♦♠♣❛r✐s♦♥ ❚❡st

❙✉♣♣♦s❡ t❤❛t ♦♥ [1, ∞)✱ ✶✳ f ✐s ❝♦♥t✐♥✉♦✉s❀ ✷✳ f ✐s ❞❡❝r❡❛s✐♥❣❀ ✸✳ f ✐s ♥♦♥✲♥❡❣❛t✐✈❡✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ✇❡ ❤❛✈❡ ❛ s❡q✉❡♥❝❡✿ an = f (n), n = 1, 2, ...

❚❤❡♥ t❤❡ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧ ❛♥❞ t❤❡ s❡r✐❡s ❜❡❧♦✇✱ Z

∞

f (x) dx ❛♥❞

1

∞ X

an ,

n=1

❡✐t❤❡r ❜♦t❤ ❝♦♥✈❡r❣❡ ♦r ❜♦t❤ ❞✐✈❡r❣❡ ✭t♦ ✐♥✜♥✐t②✮❀ ✐✳❡✳✱ Z

∞ 1

f (x) dx < ∞ ⇐⇒

∞ X n=1

an < ∞ .

Pr♦♦❢✳

❋r♦♠ t❤❡ s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ❢♦r ❡✈❡r②

n = 2, 3, ...

❛♥❞ ❛❧❧

f (x) ≤ an ≤ f (x − 1) . ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡✿

Z

n+1 n

f (x) dx ≤ an ≤

Z

n+1 n

f (x − 1) dx ,

n ≤ x ≤ n + 1✱

✇❡ ❤❛✈❡

✺✳✽✳

❙❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s

✹✵✵

♦r✱ ❛❢t❡r ❛ s✉❜st✐t✉t✐♦♥ ♦♥ t❤❡ r✐❣❤t✱

Z ❆❞❞✐♥❣ ❛❧❧ t❤❡s❡ ❢♦r

n = 2, 3, 4, ...✱ Z

n+1

f (x) dx ≤ an ≤

n

Z

n

f (x) dx . n−1

✇❡ ♦❜t❛✐♥✿

∞ 2

f (x) dx ≤

∞ X n=2

an ≤

Z

∞

f (x) dx .

1

◆♦✇✱ ❡✐t❤❡r ♦❢ t❤❡ t✇♦ ❝♦♥❝❧✉s✐♦♥s ♦❢ t❤✐s t❤❡♦r❡♠ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣❛rt ♦❢ t❤❡

♥❡❣❛t✐✈❡ ❙❡r✐❡s ❚❤❡♦r❡♠

◆♦♥✲

❛❜♦✈❡✳ ❚❤✐s ✐s t❤❡ ♠❛✐♥ st❡♣ ♦❢ t❤❡ ♣r♦♦❢✿

❊①❡r❝✐s❡ ✺✳✽✳✼ ❉❡♠♦♥str❛t❡ t❤❛t ♥♦♥❡ ♦❢ t❤❡ t❤r❡❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ❝❛♥ ❜❡ ❞r♦♣♣❡❞✳

❲❛r♥✐♥❣✦ ❚❤❡ s✉♠ ♦❢ t❤❡ s❡r✐❡s ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❜② t❤❡ ✐♥✲ t❡❣r❛❧s ❜✉t r❡♠❛✐♥s ✉♥❦♥♦✇♥✳

❊①❛♠♣❧❡ ✺✳✽✳✽✿ ❤❛r♠♦♥✐❝ s❡r✐❡s ❞✐✈❡r❣❡s ❯♥❧✐❦❡ ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s✱ t❤❡ s✉♠s ♦❢ t❤❡ ❤❛r♠♦♥✐❝ s❡r✐❡s ❞♦❡s♥✬t ❤❛✈❡ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛✳ ❆ ❝♦♠✲ ♣❛r✐s♦♥ ✐s✱ t❤❡r❡❢♦r❡✱ ♥❡❝❡ss❛r②✳ ❚❤❡ ❤❛r♠♦♥✐❝ s❡r✐❡s ❞✐✈❡r❣❡s ❜❡❝❛✉s❡

Z

∞

1 dx = lim ln x = +∞ . x→+∞ x

1

▼♦r❡ ❣❡♥❡r❛❧ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳

❈♦r♦❧❧❛r② ✺✳✽✳✾✿ ❆

p✲s❡r✐❡s✱

• •

p✲s❡r✐❡s

✐✳❡✳✱

X 1 , np p > 1 ❛♥❞ 0 < p ≤ 1✳

❝♦♥✈❡r❣❡s ✇❤❡♥ ❞✐✈❡r❣❡s ✇❤❡♥

Pr♦♦❢✳ ❖♥❝❡ t❤❡ ❢✉♥❝t✐♦♥

f

✐s ❝❤♦s❡♥✿

f (x) =

1 , xp

✺✳✽✳

❙❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s

✹✵✶

t❤❡ r❡st ♦❢ t❤❡ ♣r♦♦❢ ✐s ♣✉r❡❧② ❝♦♠♣✉t❛t✐♦♥❛❧✳ ■♥❞❡❡❞✱

Z

∞

f (x) dx =

1

Z

∞

x−p dx

1

     lim

b 1 −p+1 x ✐❢ p 6= 1, b→∞ −p + 1 1 = b    ✐❢ p = 1, lim ln x b→∞ 1  1   (b−p+1 − 1−p+1 ) ✐❢ p 6= 1, lim b→∞ −p + 1 b =   ✐❢ p = 1,  lim (ln b − ln 1) b→∞

   

1 −p + 1 = ∞    ∞

1

✐❢

p > 1,

✐❢

p = 1, p < 1.

✐❢

❚❤❡ ❤❛r♠♦♥✐❝ s❡r✐❡s✳

❚❤✉s✱ ♥♦t ♦♥❧② t❤❡ ❤❛r♠♦♥✐❝ s❡r✐❡s ❞✐✈❡r❣❡ ❜✉t ✐t ❛❧s♦ s❡♣❛r❛t❡s t❤❡ ❞✐✈❡r❣❡♥t

p✲s❡r✐❡s

❢r♦♠ t❤❡ ❝♦♥✈❡r❣❡♥t

♦♥❡s✿

❊①❡r❝✐s❡ ✺✳✽✳✶✵ ❙❤♦✇ t❤❛t t❤❡ t❤❡♦r❡♠ ❢❛✐❧s ✐❢ ✇❡ ❞r♦♣ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞❡❝r❡❛s✐♥❣✳

❈❛♥ t❤✐s

❛ss✉♠♣t✐♦♥ ❜❡ ✇❡❛❦❡♥❡❞❄

❊①❡r❝✐s❡ ✺✳✽✳✶✶ ❙❤♦✇ t❤❛t t❤❡ t❤❡♦r❡♠ ❢❛✐❧s ✐❢ ✇❡ ❞r♦♣ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ♥♦♥✲♥❡❣❛t✐✈❡✳ ❛ss✉♠♣t✐♦♥ ❜❡ ✇❡❛❦❡♥❡❞❄

❈❛♥ t❤✐s

✺✳✾✳

❈♦♠♣❛r✐s♦♥ ♦❢ s❡r✐❡s

✹✵✷

✺✳✾✳ ❈♦♠♣❛r✐s♦♥ ♦❢ s❡r✐❡s

■♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✇❡ ♠❛t❝❤❡❞ s❡r✐❡s ✇✐t❤ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s ✐♥ ♦r❞❡r t♦ ❞❡r✐✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦r ❞✐✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❢♦r♠❡r ❢r♦♠ t❤❛t ♦❢ t❤❡ ❧❛tt❡r✳ ◆♦✇ ✇❡ ❢♦❧❧♦✇ t❤✐s ✐❞❡❛ ❜✉t✱ ✐♥st❡❛❞✱ ❝♦♠♣❛r❡ s❡r✐❡s t♦ ♦t❤❡r s❡r✐❡s✳

❲❛r♥✐♥❣✦ ❚❤❡ ❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ❙❡r✐❡s ❞♦❡s♥✬t ❤❡❧♣ ❤❡r❡ ❜❡❝❛✉s❡ ✐t ❛ss✉♠❡s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❜♦t❤ s❡r✐❡s✳

❚❤❡ ♣❧❛♥ ✐s ❛s ❢♦❧❧♦✇s✿

◮

❈♦♠♣❛r❡ ❛ ♥❡✇ s❡r✐❡s ✭✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s✮ t♦ ❛♥ ♦❧❞ ♦♥❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡✴❞✐✈❡r❣❡♥❝❡ ♦❢

✇❤✐❝❤ ✐s ❦♥♦✇♥✳ ❚❤❡ st❛rt✐♥❣ ♣♦✐♥t ✐s t❤❡

❈♦♠♣❛r✐s♦♥ ❘✉❧❡ ❢♦r ❙✉♠s

❡❧❡♠❡♥ts ✐s s♠❛❧❧❡r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢

an ≤ bn ✱

✭❈❤❛♣t❡r ✶P❈✲✶✮✿ ❚❤❡ s✉♠ ♦❢ ❛ s❡q✉❡♥❝❡ ✇✐t❤ s♠❛❧❧❡r

t❤❡♥ ✇❡ ❤❛✈❡ ❢♦r ❛♥②

q X n=p

an ≤

q X

p, q

✇✐t❤

p ≤ q✿

bn .

n=p

❚❤❡r❡❢♦r❡✱ t❤❡ s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐❛❧ s✉♠s ♦❢ t❤❡ ✜rst s❡r✐❡s ✐s ✏❞♦♠✐♥❛t❡❞✑ ❜② t❤❛t ♦❢ t❤❡ s❡❝♦♥❞✳ ■❢ t❤❡ s❡❝♦♥❞ ❝♦♥✈❡r❣❡s✱ ✐ts s✉♠ ✐s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ❢♦r t❤❡ ♣❛rt✐❛❧ s✉♠s ♦❢ t❤❡ ✜rst✳ ❚❤❡♥✱ t❤❡ ✜rst s❡r✐❡s ❝♦♥✈❡r❣❡s t♦♦ ❜② t❤❡ t❤❡♦r❡♠ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳ ❚❤✐s ✐s t❤❡ ♠❛✐♥ r❡s✉❧t✳

❚❤❡♦r❡♠ ✺✳✾✳✶✿ ❉✐r❡❝t ❈♦♠♣❛r✐s♦♥ ❚❡st ❢♦r ❙❡r✐❡s ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ s❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s t❤❛t s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣✿

0 ≤ an ≤ bn , ❢♦r ❛❧❧

n✳

❚❤❡♥✱ t❤❡♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡✴❞✐✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❧❛r❣❡r✴s♠❛❧❧❡r ✐♠♣❧✐❡s

t❤❡ ❝♦♥✈❡r❣❡♥❝❡✴❞✐✈❡r❣❡♥❝❡ ♦❢ t❤❡ s♠❛❧❧❡r✴❧❛r❣❡r❀ ✐✳❡✳✱

X

X

an < ∞ ⇐=

an = ∞ =⇒

X

X

bn < ∞

bn = ∞

❊①❡r❝✐s❡ ✺✳✾✳✷ Pr♦✈❡ t❤❡ s❡❝♦♥❞ ♣❛rt✳

■♥ ♦r❞❡r t♦ ❛♣♣❧② t❤❡ t❤❡♦r❡♠ t♦ ❛ ♣❛rt✐❝✉❧❛r ✕ ♥❡✇ ✕ s❡r✐❡s✱ ✇❡ s❤♦✉❧❞ tr② t♦ ♠♦❞✐❢② ✐ts ❢♦r♠✉❧❛ ✇❤✐❧❡ ♣❛②✐♥❣ ❛tt❡♥t✐♦♥ t♦ ✇❤❡t❤❡r ✐t ✐s ❣❡tt✐♥❣ s♠❛❧❧❡r ♦r ❧❛r❣❡r✳

❊①❛♠♣❧❡ ✺✳✾✳✸✿

p✲s❡r✐❡s

❛s ❛ st❛rt

▲❡t✬s ❣♦ ❜❛❝❦✇❛r❞s ❛t ✜rst✳

❲❡ ❝♦♥s✐❞❡r t❤❡

p✲s❡r✐❡s

❛♥❞ s❡❡ ✇❤❡t❤❡r ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❦♥♦✇♥

❝♦♥✈❡r❣❡♥❝❡ ❢❛❝ts ❛❜♦✉t t❤❡♠✳ ❙♦♠❡ s❡r✐❡s ❝❛♥ ❜❡

♠♦❞✐✜❡❞ ✐♥t♦

s✉❝❤ ❛ s❡r✐❡s✳ ❚❤❡ ♦♥❡ ♦♥ t❤❡ ❧❡❢t ✐s ✉♥❢❛♠✐❧✐❛r ❜✉t ❝❛♥ ❜❡ ♠❛t❝❤❡❞

✺✳✾✳

❈♦♠♣❛r✐s♦♥ ♦❢ s❡r✐❡s

✹✵✸

✇✐t❤ ❛ ❢❛♠✐❧✐❛r ♦♥❡✿

n2

1 1 ≤ 2. +1 n

❲❡ r❡♠♦✈❡ ✏ +1✑ ❛♥❞ t❤❡ ♥❡✇ s❡r✐❡s ✐s ✏❧❛r❣❡r✑✳ ❚❤❡♥✱ ❜② t❤❡ t❤❡♦r❡♠✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s❡r✐❡s ✭♦♥ t❤❡ ❧❡❢t✮ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤✐s

X

p✲s❡r✐❡s

✇✐t❤

p = 2 > 1✿

X 1 1 < ∞. ≤ n2 + 1 n2

❙✐♠✐❧❛r❧②✱ ✇❡ ❝❛♥ ♠♦❞✐❢② t❤✐s s❡r✐❡s✿

1 1 ≥ 1/2 . −1 n

n1/2

❲❡ r❡♠♦✈❡ ✏ −1✑ ❛♥❞ t❤❡ ♥❡✇ s❡r✐❡s ✐s ✏s♠❛❧❧❡r✑✳ ❚❤❡♥✱ t❤❡ ❞✐✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s❡r✐❡s ✭♦♥ t❤❡

p✲s❡r✐❡s

❧❡❢t✮ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❞✐✈❡r❣❡♥❝❡ ♦❢ t❤✐s

X

✇✐t❤

p = 1/2 < 1✿

X 1 1 = ∞. ≥ n1/2 + 1 n1/2

◆♦✇ ❧❡t✬s tr② t♦ ♠♦❞✐❢② t❤✐s s❡r✐❡s✿

n2

1 1 ≥ 2. −1 n

❲❡ r❡♠♦✈❡ ✏ −1✑ ❛♥❞ t❤❡ ♥❡✇ s❡r✐❡s ✐s ✏s♠❛❧❧❡r✑✳ ❚❤❡♥✱ t❤❡ ❞✐✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s❡r✐❡s ✭♦♥ t❤❡ ❧❡❢t✮ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❞✐✈❡r❣❡♥❝❡✳✳✳ ✇❛✐t✱ t❤✐s

X

p✲s❡r✐❡s

✇✐t❤

p=2>1

❝♦♥✈❡r❣❡s✦ ❙♦✱ ✇❡ ❤❛✈❡✿

X 1 1 < ∞. ≥ n2 − 1 n2

❚❤❡r❡ ✐s ♥♦t❤✐♥❣ ✇❡ ❝❛♥ ❝♦♥❝❧✉❞❡ ❢r♦♠ t❤✐s ♦❜s❡r✈❛t✐♦♥✳ ❙✐♠✐❧❛r❧②✱ ❧❡t✬s tr② t♦ ♠♦❞✐❢② t❤✐s s❡r✐❡s✿

1 1 ≤ 1/2 . +1 n

n1/2

❲❡ r❡♠♦✈❡ ✏ +1✑ ❛♥❞ t❤❡ ♥❡✇ s❡r✐❡s ✐s ✏❧❛r❣❡r✑✳ ❚❤❡♥✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s❡r✐❡s ✭♦♥ t❤❡ ❧❡❢t✮ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❝♦♥✈❡r❣❡♥❝❡✳✳✳ ❜✉t t❤✐s

X

p✲s❡r✐❡s

✇✐t❤

p = 1/2 < 1

❞✐✈❡r❣❡s✦ ❙♦✱ ✇❡ ❤❛✈❡✿

X 1 1 = ∞. ≤ n1/2 − 1 n1/2

❚❤❡r❡ ✐s ♥♦t❤✐♥❣ ✇❡ ❝❛♥ ❝♦♥❝❧✉❞❡ ❢r♦♠ t❤✐s ♦❜s❡r✈❛t✐♦♥✳

❊①❛♠♣❧❡ ✺✳✾✳✹✿

p✲s❡r✐❡s

❛s ❛ ❣♦❛❧

❘❡♠♦✈✐♥❣ ✏ −1✑ ❛♥❞ ✏ +1✑ ❢❛✐❧❡❞ t♦ ♣r♦❞✉❝❡

n2

✉s❡❢✉❧

1 −1

s❡r✐❡s ❢♦r t❤❡s❡ t✇♦✿

❛♥❞

1 . +1

n1/2

❲❡ ✇✐❧❧ ❤❛✈❡ t♦ ❜❡ s✉❜t❧❡r ✐♥ ✜♥❞✐♥❣ ❝♦♠♣❛r✐s♦♥s✳ ▲❡t✬s tr② ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❲❡ ❛❞❞ ✏ 2✑ ✐♥ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ ❞✐s❝♦✈❡r t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s tr✉❡ ❢♦r ❛❧❧

n = 2, 3, ...✿ n2

2 1 ≤ 2. −1 n

❚❤❡ ♥❡✇ s❡r✐❡s ✐s ✏❧❛r❣❡r✑✳ ❚❤❡♥✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s❡r✐❡s ✭♦♥ t❤❡ ❧❡❢t✮ ❢♦❧❧♦✇s ❢r♦♠ t❤❡

✺✳✾✳

❈♦♠♣❛r✐s♦♥ ♦❢ s❡r✐❡s

✹✵✹

❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤✐s ✭♠✉❧t✐♣❧❡ ♦❢ ❛✮

p✲s❡r✐❡s X

✇✐t❤

p = 2 > 1✿

X 1 1 < ∞. ≤ n2 − 1 n2

◆❡①t✱ ✇❡ ❢♦❧❧♦✇ t❤✐s ✐❞❡❛ ❢♦r t❤❡ ♦t❤❡r s❡r✐❡s✳ ❲❡ ❛❞❞ ✏ 2✑ ✐♥ t❤❡ ❞❡♥♦♠✐♥❛t♦r ❛♥❞ ❞✐s❝♦✈❡r t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s tr✉❡ ❢♦r ❛❧❧

n = 1, 2, ...✿

1 1 ≥ 1/2 . +1 2n

n1/2

❚❤❡ ♥❡✇ s❡r✐❡s ✐s ✏s♠❛❧❧❡r✑✳ ❚❤❡♥✱ t❤❡ ❞✐✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s❡r✐❡s ✭♦♥ t❤❡ ❧❡❢t✮ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❞✐✈❡r❣❡♥❝❡ ♦❢ t❤✐s ✭♠✉❧t✐♣❧❡ ♦❢ ❛✮

p✲s❡r✐❡s X

p = 1/2 < 1✿

✇✐t❤

X 1 1 = ∞. ≥ n1/2 + 1 2n1/2

❲❤❛t ✐s t❤❡ ❧❡ss♦♥❄ ■t ✐s ❤❛r❞ t♦ ❝♦♠♣❛r❡ s❡q✉❡♥❝❡s ✐♥ t❤✐s ♠❛♥♥❡r✳

❢✉♥❝t✐♦♥s ♣r❡✈✐♦✉s❧②❄ ❆ ✇❛② t♦ ❝♦♠♣❛r❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s f, g ❛t ✐♥✜♥✐t② ✐s t♦ ❝♦♥s✐❞❡r t❤❡✐r r❡❧❛t✐✈❡ ♦r❞❡r ♦❢ ♠❛❣♥✐t✉❞❡ ❛s ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✻✳ ■t ✐s ❞❡✜♥❡❞ ✈✐❛ t❤❡ ❧✐♠✐t ♦❢

❇✉t ❞✐❞ ✇❡ ❝♦♠♣❛r❡ t❤❡✐r r❛t✐♦✿

f (x) = L. x→+∞ g(x) lim

■❢

L

✐s ✐♥✜♥✐t❡✱ ✇❡ s❛② t❤❛t

f

❤❛s ❛ ❣r❡❛t❡r ♠❛❣♥✐t✉❞❡✿

f >> g . ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤✐❡r❛r❝❤②✿

... >> ex >> ... >> xn >> ... >> x2 >> x >>

√

x >> ... >> 1 >> ... >>

1 1 >> 2 >> ... >> e−x >> ... x x

❲❡ ❛♣♣❧② t❤✐s ✐❞❡❛ t♦ s❡r✐❡s✳ ❊①❛♠♣❧❡ ✺✳✾✳✺✿ ❝♦♠♣❛r✐s♦♥

❚❤❡ ✐❞❡❛ ✐s t❤❛t s♦♠❡ ♣❛✐rs ♦❢ s❡r✐❡s ❝♦♥✈❡r❣❡ ♦r ❞✐✈❡r❣❡ t♦❣❡t❤❡r ✇❤❡♥ t❤❡② ❛r❡ ❝♦♠♣❛r❛❜❧❡ ✐♥ s♦♠❡ ✇❛②✳ ■❢ ♦♥❡ ♦❢ t❤❡♠ ✐s ❢❛♠✐❧✐❛r✱ ✇❡ ♠❛② ❤❛✈❡ ❞✐s❝♦✈❡r❡❞ ✇❤❛t t❤❡ ♦t❤❡rs ❞♦✳ ❍❡r❡ ❛r❡ t❤❡ ✜rst t❤r❡❡ t♦ ❜❡ ❝♦♠♣❛r❡❞✿

n2

1 −1

✈s✳

1 n2

✈s✳

n2

1 . +1

✺✳✾✳

❈♦♠♣❛r✐s♦♥ ♦❢ s❡r✐❡s

✹✵✺

❚❤✐s ✐s t❤❡ s❡❝♦♥❞ tr✐♣❧❡✿

1 1 1 ✈s✳ 1/2 ✈s✳ 1/2 . −1 n n +1

n1/2

❲❡ ❝♦♥s✐❞❡r t❤❡s❡ r❛t✐♦s✿ n2

1 1 1 1 ÷ 2 → 1 ❛♥❞ 1/2 ÷ 1/2 → 1 . ±1 n n ±1 n

❚❤❡ s❡❝♦♥❞ s❡r✐❡s ❝♦♥✈❡r❣❡s ✐♥ t❤❡ ❢♦r♠❡r ❝❛s❡ ❛♥❞ ❞✐✈❡r❣❡s ✐♥ t❤❡ ❧❛tt❡r✳ ❚❤❡♥ s♦ ❞♦❡s t❤❡ ✜rst ♦♥❡✳ ❲❡ s✉♠♠❛r✐③❡ t❤✐s ✐❞❡❛ ❜❡❧♦✇✳

❉❡✜♥✐t✐♦♥ ✺✳✾✳✻✿ ♦r❞❡r ♦❢ ♠❛❣♥✐t✉❞❡ ❢♦r s❡q✉❡♥❝❡s ■❢ t❤❡ ❧✐♠✐t ❜❡❧♦✇ ✐s ✐♥✜♥✐t❡✱ ♦r ✐ts r❡❝✐♣r♦❝❛❧ ✐s ③❡r♦✱ lim

n→∞

an = ∞ ♦r bn

lim

n→∞

bn = 0, an

❛♥❞ ❜♦t❤ s❡q✉❡♥❝❡s ❤❛✈❡ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s✱ ✇❡ s❛② t❤❛t an ✐s ♦❢

❤✐❣❤❡r ♦r❞❡r

✺✳✾✳

❈♦♠♣❛r✐s♦♥ ♦❢ s❡r✐❡s

✹✵✻

t❤❛♥ bn ✱ ❛♥❞ ✇❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥✿ an >> bn

❛♥❞

bn = o(an )

❛♥❞

bn >> an ,

❚❤❡ ❧❛tt❡r r❡❛❞s ✏❧✐tt❧❡ ♦✑✳ ❲❤❡♥

an >> bn ✇❡ s❛② t❤❛t t❤❡② ❤❛✈❡

t❤❡ s❛♠❡ ♠❛❣♥✐t✉❞❡✱ ❛♥❞ ✇❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥✿ an ∼ bn

❲❡ ❛♣♣❧② t❤✐s t❡r♠✐♥♦❧♦❣② t♦ ❜♦t❤ s❡q✉❡♥❝❡s ❛♥❞ s❡r✐❡s✳ ❚❤❡♦r❡♠ ✺✳✾✳✼✿ ▲✐♠✐t ❈♦♠♣❛r✐s♦♥ ❚❡st ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ s❡q✉❡♥❝❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s✳

❚❤❡♥✱ t❤❡ ❝♦♥✈❡r✲

❣❡♥❝❡✴❞✐✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❧❛r❣❡r✴s♠❛❧❧❡r ✐♥ ♠❛❣♥✐t✉❞❡ s❡r✐❡s ✐♠♣❧✐❡s t❤❡ ❝♦♥✲ ✈❡r❣❡♥❝❡✴❞✐✈❡r❣❡♥❝❡ ♦❢ t❤❡ s♠❛❧❧❡r✴❧❛r❣❡r s❡r✐❡s❀ ✐✳❡✳✱ t❤❡r❡ ❛r❡ t❤r❡❡ ❝❛s❡s ❢♦r

an ≥ 0, bn ≥ 0✿ an ∼ b n

❈❛s❡ ✶✱

an > bn

❈❛s❡ ✸✱

X

X

X

an < ∞ ⇐⇒

bn < ∞ =⇒

bn = ∞ =⇒

X

X

X

bn < ∞ .

an < ∞ .

an = ∞ .

Pr♦♦❢✳

❙✉♣♣♦s❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡✐r r❛t✐♦ ❜❡❧♦✇ ❡①✐sts✱ ❛s ❛ ♥✉♠❜❡r ♦r ❛s ✐♥✜♥✐t②✿

an = L. n→∞ bn lim

■♥ ❈❛s❡s ✶ ❛♥❞ ✷✱

L

✐s ❛ ♥✉♠❜❡r✳

❚❤❡♥ t❤❡

❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡

✭❈❤❛♣t❡r ✷❉❈✲✶✮

st❛t❡s✿

◮

❋♦r ❡❛❝❤

▲❡t✬s ❝❤♦♦s❡

ε>0

ε = 1✳

t❤❡r❡ ✐s s✉❝❤ ❛♥

❚❤❡♥✱ ❢♦r t❤❡ ❢♦✉♥❞

N

t❤❛t ❢♦r ❡✈❡r②

N✱

n>N

✇❡ ❤❛✈❡✿

an − L < ε . bn ✇❡ ❤❛✈❡

an < L + ε = L + 1, bn ❢♦r ❡✈❡r②

n < N✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ ❛ ❝♦♠♣❛r✐s♦♥ ♦❢ ✭t❤❡ t❛✐❧s ♦❢ ✮ t✇♦ s❡q✉❡♥❝❡s✿

an < (L + 1)bn . ◆♦✇✱ ❜② t❤❡

❚❤❡♥ ❜② t❤❡

❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❙❡r✐❡s✱ ✇❡ ❤❛✈❡✿ X

bn < ∞ =⇒

❉✐r❡❝t ❈♦♠♣❛r✐s♦♥ ❚❡st✱ ✇❡ ❤❛✈❡✿ X

X

(L + 1)bn < ∞ .

an < ∞ .

✺✳✾✳

❈♦♠♣❛r✐s♦♥ ♦❢ s❡r✐❡s

✹✵✼

❊①❡r❝✐s❡ ✺✳✾✳✽ Pr♦✈❡ ❈❛s❡ ✸✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢

an =L n→∞ bn lim

✐s ❛ ♥✉♠❜❡r ♦r ✐♥✜♥✐t②✱ t❤❡♥ t❤❡s❡ ❛r❡ t❤❡ t❤r❡❡ ❝❛s❡s✿ ❈❛s❡ ✶✱

L>0:

❆ ✏♣❡r❢❡❝t✑ ♠❛t❝❤✿ ❜♦t❤ ❝♦♥✈❡r❣❡ ♦r ❜♦t❤ ❞✐✈❡r❣❡✳

❈❛s❡ ✷✱

L=0:

❚❤❡ ❞❡♥♦♠✐♥❛t♦r ✏❞♦♠✐♥❛t❡s✑ t❤❡ ♥✉♠❡r❛t♦r✳

❈❛s❡ ✸✱

L=∞:

❚❤❡ ♥✉♠❡r❛t♦r ✏❞♦♠✐♥❛t❡s✑ t❤❡ ❞❡♥♦♠✐♥❛t♦r✳

❊①❡r❝✐s❡ ✺✳✾✳✾ ❆♣♣❧② t❤❡ t❤❡♦r❡♠ t♦ t❤❡ t✇♦ tr✐♣❧❡s ♦❢ s❡r✐❡s ✐♥ t❤❡ ❡①❛♠♣❧❡s ❛❜♦✈❡✳

❊①❛♠♣❧❡ ✺✳✾✳✶✵✿ ✜♥❞ ❛ ❝♦♠♣❛r✐s♦♥ s❡r✐❡s ❈♦♥s✐❞❡r t❤❡ s❡r✐❡s✿

X

√

1 . n2 + n + 1

❲❡ ♥❡❡❞ t♦ ❞❡t❡r♠✐♥❡ t♦ ✇❤❛t s✐♠♣❧❡r s❡r✐❡s t❤✐s s❡r✐❡s ✐s ✏s✐♠✐❧❛r✑✳ ❚❤❡ ❧❡❛❞✐♥❣ t❡r♠ ♦❢ t❤❡ ❡①♣r❡ss✐♦♥ 2 ✐♥s✐❞❡ t❤❡ r❛❞✐❝❛❧ ✐s n ✳ ❚❤❡r❡❢♦r❡✱ ✇❡ s❤♦✉❧❞ ❝♦♠♣❛r❡ ♦✉r s❡r✐❡s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿

X 1 X1 √ = , n n2 t❤❡ ❞✐✈❡r❣❡♥t ❤❛r♠♦♥✐❝ s❡r✐❡s✳ ❲❡ ❡✈❛❧✉❛t❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ r❛t✐♦ ♥♦✇✿

√

n2

1 1 n ÷ =√ 2 +n+1 n n +n+1 1 =√ n2 + n + 1/n 1 =p (n2 + n + 1)/n2 1 =p 1 + 1/n + 1/n2 1 →√ 1+0+0

❛s

n→∞

= 1. ❙♦✱ ❈❛s❡ ✶ ♦❢ t❤❡

▲✐♠✐t ❈♦♠♣❛r✐s♦♥ ❚❤❡♦r❡♠ ❛♣♣❧✐❡s ❛♥❞ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ♦✉r s❡r✐❡s ❞✐✈❡r❣❡s✳

❊①❡r❝✐s❡ ✺✳✾✳✶✶ ❏✉st✐❢② t❤❡ ✐♥t❡r♠❡❞✐❛t❡ st❡♣s ✐♥ t❤❡ ❛❜♦✈❡ ❝♦♠♣✉t❛t✐♦♥✳

❊①❡r❝✐s❡ ✺✳✾✳✶✷ ❍♦✇ ❞♦❡s t❤❡ t❤❡♦r❡♠ ❛♣♣❧② ✐❢ ✇❡ r❡♠♦✈❡

+n

❢r♦♠ t❤❡ ❛❜♦✈❡ s❡r✐❡s❄

❊①❡r❝✐s❡ ✺✳✾✳✶✸ ❍♦✇ ❞♦❡s t❤❡ t❤❡♦r❡♠ ❛♣♣❧② ✐❢ ✇❡ r❡♣❧❛❝❡

n2

✇✐t❤

n3

✐♥ t❤❡ ❛❜♦✈❡ s❡r✐❡s❄

✺✳✶✵✳

❆❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡

✹✵✽

❊①❛♠♣❧❡ ✺✳✾✳✶✹✿ ♣♦✇❡r s❡r✐❡s

❚❤❡ ✐❞❡❛ ♦❢ t❤❡ t❤❡♦r❡♠ ❛♣♣❧✐❡s ✇❡❧❧ t♦

X

♣♦✇❡r s❡r✐❡s✳ ■❢ ✇❡ ❤❛✈❡ t❤❡s❡ t✇♦✱

cn (x − a)n

X

❛♥❞

t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ t❤❡♦r❡♠ ❜❡❝♦♠❡s✿

dn (x − a)n ,

cn cn (x − a)n = → L, dn (x − a)n dn ✇❤❡♥

cn , dn > 0 ❛♥❞ x > a✳ ❍♦✇❡✈❡r✱ ❛r❡ t❤❡s❡ s❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s❄ x = a ✇✐❧❧ ❝❤❛♥❣❡ t❤❡ s✐❣♥ ♦❢ t❤❡ t❡r♠ ❢♦r ❡❛❝❤ ♦❞❞ n✦ ❚❤❡ ❛♣♣❧✐❝❛❜✐❧✐t② ♦❢

t❤r♦✉❣❤

◆♦❀ ❥✉st ♣❛ss✐♥❣ t❤❡s❡ t❤❡♦r❡♠s ✐s

✈❡r② ❧✐♠✐t❡❞✳

✺✳✶✵✳ ❆❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡

❋r♦♠ t❤✐s ♣♦✐♥t✱ ✇❡ ❛❜❛♥❞♦♥ t❤❡ s❡✈❡r❡ r❡str✐❝t✐♦♥ t❤❛t t❤❡ t❡r♠s ♦❢ t❤❡ s❡r✐❡s ❝❛♥✬t ❜❡ ♥❡❣❛t✐✈❡✳ ❚❤❡♥✱ ♥♦♥❡ ♦❢ t❤❡ r❡s✉❧ts ✐♥ t❤❡ ❧❛st t✇♦ s❡❝t✐♦♥s ❛♣♣❧✐❡s✦ ❚❤❡ ♣❧❛♥ ✐s t♦ ♠❛❦❡ ❢r♦♠ ♦✉r s❡r✐❡s ❛ ♥♦♥✲♥❡❣❛t✐✈❡ s❡r✐❡s ❛♥❞ s❡❡ ✐❢ t❤✐s ♥❡✇ s❡r✐❡s ❝♦♥✈❡r❣❡s ♦r ❞✐✈❡r❣❡s ✐♥ ❤♦♣❡ t❤❛t t❤✐s ✇✐❧❧ t❡❧❧ ✉s s♦♠❡t❤✐♥❣ ❛❜♦✉t t❤❡ ♦r✐❣✐♥❛❧ s❡r✐❡s✳ ❖❢ ❝♦✉rs❡✱ t❤❡ ❡❛s✐❡st ✇❛② t♦ ❞♦ t❤✐s ✐s t♦ t❛❦❡ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡✿

X

an

❝r❡❛t❡s

X

|an | .

❲❡ ❝❛♥ ❛❧s♦ ✇❛❧❦ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥ ❛♥❞ t❛❦❡ ❛♥② s❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s✱ ✐t ✏❛❧t❡r♥❛t❡✑✿

◆♦✇✱ ✇❤✐❝❤ ✐s

X

bn

❝r❡❛t❡s

X

bn ≥ 0 ✱

❛♥❞ ♠❛❦❡

(−1)n bn , bn ≥ 0 .

♠♦r❡ ❧✐❦❡❧② ❧✐❦❡❧② t♦ ❝♦♥✈❡r❣❡✱ t❤❡ ❢♦r♠❡r ♦r t❤❡ ❧❛tt❡r❄

❊①❛♠♣❧❡ ✺✳✶✵✳✶✿ ❢❛♠✐❧✐❛r s❡r✐❡s

❙♦♠❡ ♦❢ t❤♦s❡ ❛r❡ ❡❛s② t♦ ❛♥❛❧②③❡✿ X • (−1)n ❞✐✈❡r❣❡s ✕ ❛❝❝♦r❞✐♥❣ t♦ t❤❡

•

X

1

❞✐✈❡r❣❡s t♦♦✳

1 (−1)n n ❝♦♥✈❡r❣❡s 2 X 1 ❝♦♥✈❡r❣❡s t♦♦✳ • 2n

•

❉✐✈❡r❣❡♥❝❡ ❚❡st✳

X

✕ ❛❝❝♦r❞✐♥❣ t♦ t❤❡

●❡♦♠❡tr✐❝ ❙❡r✐❡s ❚❤❡♦r❡♠✳

■♥ ❣❡♥❡r❛❧✱ t❤✐s ✐s ✇❤❛t s✉❝❤ ❛ ♣❛✐r ♦❢ s❡r✐❡s ❧♦♦❦s ❧✐❦❡ ✭t❤❡ s❡q✉❡♥❝❡s ❛r❡ ❛❜♦✈❡ ❛♥❞ t❤❡✐r s❡q✉❡♥❝❡s ♦❢ ♣❛rt✐❛❧ s✉♠s ❛r❡ ❜❡❧♦✇✮✿

✺✳✶✵✳ ❆❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡

✹✵✾

❆s ✇❡ ❦♥♦✇ ❛♥❞ ❝❛♥ s❡❡ ❤❡r❡✱ t❤❡ ♥♦♥✲♥❡❣❛t✐✈❡ ♦♥❡ ♣r♦❞✉❝❡s t❤❡ s❡q✉❡♥❝❡ ♦❢ s✉♠s t❤❛t ✐s ✐♥❝r❡❛s✐♥❣✳ ■t ♠❛② ♦r ♠❛② ♥♦t ❝♦♥✈❡r❣❡ ❞❡♣❡♥❞✐♥❣ ♦♥ ❤♦✇ ♠✉❝❤ ✇❡ ❛❞❞ ❛t ❡✈❡r② st❡♣✳ ❇✉t ❢♦r t❤❡ ❧❛tt❡r✱ ❤❛❧❢ ♦❢ t❤❡s❡ ✉♣✲st❡♣s ❛r❡ ❝❛♥❝❡❧❡❞ ❜② t❤❡ ❞♦✇♥✲st❡♣s✦ ❚❤✐s s✉❣❣❡sts t❤❛t ✐❢ t❤❡ ❢♦r♠❡r ✐s s❧♦✇✐♥❣ ❞♦✇♥✱ t❤❡♥ s♦ ✐s t❤❡ ❧❛tt❡r✳ ❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❢♦r♠❡r t❤❡♥ ✐♠♣❧✐❡s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❧❛tt❡r✳✳✳

❊①❛♠♣❧❡ ✺✳✶✵✳✷✿

p✲s❡r✐❡s

❈♦♥s✐❞❡r t❤❡ ❢❛♠✐❧✐❛r t❤❡

p✲s❡r✐❡s

✇✐t❤

p = 2✿ X 1 . n2

■t ✐s ❝♦♥✈❡r❣❡♥t✳ ■ts ✏❛❧t❡r♥❛t✐♥❣✑ ✈❡rs✐♦♥ ✐s

X

(−1)n

1 . n2

❚❤❡② ❛r❡ s❤♦✇♥ ❛❜♦✈❡✳ ❍♦✇ ❞♦ t❤❡s❡ t✇♦ ❝♦♠♣❛r❡ ❄ ❚❤❡ ❢♦r♠❡r ❛♣♣❡❛rs ✏s♠❛❧❧❡r✑ t❤❛♥ t❤❡ ❧❛tt❡r✿

(−1)n

1 1 ≤ 2. 2 n n

❉♦❡s ✐t ♠❡❛♥ t❤❛t ✐t ♠✉st ❝♦♥✈❡r❣❡❄ ◆♦✱ ✐t ♠✐❣❤t st✐❧❧ ❞✐✈❡r❣❡ ❜❡❝❛✉s❡ t❤❡ ◆♦♥✲♥❡❣❛t✐✈❡ ❙❡r✐❡s ❚❤❡♦r❡♠ ❞♦❡s♥✬t ❛♣♣❧②✳ ❍♦✇❡✈❡r✱ ❛ ❝❧❡✈❡r tr✐❝❦ ✐s t♦ ✉s❡ ❛ ❤✐❞❞❡♥ ♥♦♥✲♥❡❣❛t✐✈❡ s❡r✐❡s❀ ✐t ✐s t❤❡ s✉♠ ♦❢ t❤❡ t✇♦✿

(−1)n

1 1 + 2 ≥ 0. 2 n n

❲❤❛t ❞♦ ✇❡ ❦♥♦✇ ❛❜♦✉t ✐t❄ ❆♥ ✐♥s✐❣❤t❢✉❧ ♦❜s❡r✈❛t✐♦♥ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❡q✉❛❧✐t②✿

(−1)n

1 1 2 + 2 ≤ 2, 2 n n n

❲❡ ❤❛✈❡ t✇♦ ♥♦♥✲♥❡❣❛t✐✈❡✲t❡r♠ s❡r✐❡s ❛♥❞ t❤❡ ❜✐❣❣❡r ♦♥❡ ✐s ❝♦♥✈❡r❣❡♥t✦

✺✳✶✵✳

❆❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡

✹✶✵

❚❤❡r❡❢♦r❡✱ t❤❡ s♠❛❧❧❡r s❡r✐❡s ✐s ❝♦♥✈❡r❣❡♥t t♦♦✱

X

1 1 (−1) 2 + 2 n n n

❚❤❡r❡❢♦r❡✱ s♦ ✐s t❤❡ ♦r✐❣✐♥❛❧ s❡r✐❡s✱

❜② t❤❡

X

❙✉♠ ❘✉❧❡s ❢♦r ❙❡r✐❡s

✳

(−1)n



< ∞.

1 , n2

▲❡t✬s ❣❡♥❡r❛❧✐③❡ t❤✐s ❡①❛♠♣❧❡✳ ❲❡ ❛r❡ ❛❢t❡r ❛ ♣❛rt✐❝✉❧❛r ❦✐♥❞ ♦❢ sq✉❡❡③❡✿

❚❤❡♦r❡♠ ✺✳✶✵✳✸✿ ❙q✉❡❡③❡ ❚❤❡♦r❡♠ ❢♦r ❙❡r✐❡s ❙✉♣♣♦s❡ s❡q✉❡♥❝❡s

❢♦r ❛❧❧

n✳

❚❤❡♥✱ ✐❢

an , bn

X

✇✐t❤

bn ≥ 0

s❛t✐s❢②✿

−bn ≤ an ≤ bn bn

❝♦♥✈❡r❣❡s✱ t❤❡♥ s♦ ❞♦❡s

X

an ✳

✺✳✶✵✳ ❆❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡

✹✶✶

Pr♦♦❢✳ ❋✐rst✱ ♥♦t❡ t❤❛t t❤❡ sq✉❡❡③❡ t❤❛t ✇❡ ❤❛✈❡ ♣r♦✈❡s ♥♦t❤✐♥❣ ❛❜♦✉t ❛❧❧ ✇❡ ❞❡r✐✈❡ ✐s t❤❛t

bn

an → 0

an

✉♥❧❡ss ✇❡ ❤❛✈❡

bn → 0 ✳

❊✈❡♥ t❤❡♥✱

t♦♦✳ ❲❤❛t ❛❜♦✉t t❤❡ s❡r✐❡s ❄ ❲❡ t❛❦❡ ❛ ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤✳ ❋✐rst✱ ✇❡ ❛❞❞

t♦ t❤❡ t❤r❡❡ ♣❛rts ♦❢ t❤❡ ❛❜♦✈❡ ✐♥❡q✉❛❧✐t②✿

0 ≤ an + bn ≤ bn + bn = 2bn . ▲❡t✬s ❞❡✜♥❡ ❛ ♥❡✇ s❡q✉❡♥❝❡✿

cn = an + bn ❢♦r ❛❧❧

n✳

❚❤❡♥✱ ✇❡ ❤❛✈❡ ❛ ♥❡✇ sq✉❡❡③❡ ✿

❚❤❡ ❧❛st s❡q✉❡♥❝❡ ♣r♦❞✉❝❡s ❛ s❡r✐❡s✱ ❚❤❡♥

X

cn

X

0 ≤ cn ≤ 2bn . 2bn ✱

t❤❛t ❝♦♥✈❡r❣❡s ❜② t❤❡ ❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❙❡r✐❡s✳

❛❧s♦ ❝♦♥✈❡r❣❡s ❜② t❤❡ ◆♦♥✲♥❡❣❛t✐✈❡ ❙❡r✐❡s ❚❤❡♦r❡♠ ❛♥❞ t❤❡ ❉✐r❡❝t ❈♦♠♣❛r✐s♦♥ ❚❤❡♦r❡♠✳

❋✐♥❛❧❧②✱ t❤❡ ♦r✐❣✐♥❛❧ s❡r✐❡s✱

X

an =

X

(cn − bn ) ,

❝♦♥✈❡r❣❡s ❛s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t✇♦ ❝♦♥✈❡r❣❡♥t s❡r✐❡s ❜② t❤❡ ❙✉♠ ❘✉❧❡s ❢♦r ❙❡r✐❡s✳

❙♦✱ ✇❡ ❝❛♥ ✉s❡ t❤❡ ❢❛❝t t❤❛t

bn

❝♦♥✈❡r❣❡s t♦ ♣r♦✈❡ t❤❛t

an

❝♦♥✈❡r❣❡s t♦♦✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ✜♥❞ t❤✐s ❝♦♥✈❡♥✐❡♥t

bn ❄ ❚❤❡r❡ ✐s ♦♥❡ ♥❛t✉r❛❧ ❝❤♦✐❝❡✿

bn = |an | . ❚❤✐s t✐♠❡ t❤❡ sq✉❡❡③❡ ✐s ✏♣❡r❢❡❝t✑✿ ◆♦t ♦♥❧② t❤❡ s❡q✉❡♥❝❡ ✐s ❜♦✉♥❞❡❞ ❜② t❤♦s❡ t✇♦❀ ✐t ✐s✱ ✐♥ ❢❛❝t✱ ❛❧✇❛②s ❡q✉❛❧ t♦ ♦♥❡ ♦r t❤❡ ♦t❤❡r✦ ❍❡r❡ ✐s ❛♥ ✐❧❧✉str❛t✐♦♥✿

■t✬s ❛s ✐❢ ❛ ❜❛❧❧ ✐s ❝♦♥t✐♥✉♦✉s❧② ❜♦✉♥❝✐♥❣ ♦✛ t❤❡ ❝❡✐❧✐♥❣ ❛♥❞ t❤❡ ✢♦♦r ♦❢ ❛ ❝♦rr✐❞♦r✳✳✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❡♣t ✐s ❝r✉❝✐❛❧✿

❉❡✜♥✐t✐♦♥ ✺✳✶✵✳✹✿ s❡r✐❡s ❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧② ❲❡ s❛② t❤❛t ❛ s❡r✐❡s

X

|an |✱

❝♦♥✈❡r❣❡s✳

X

an

❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧② ✐❢ ✐ts s❡r✐❡s ♦❢ ❛❜s♦❧✉t❡ ✈❛❧✉❡s✱

❲❛r♥✐♥❣✦ ❚❤❡ ✇♦r❞ ✏❛❜s♦❧✉t❡✑ r❡❢❡rs t♦ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡✳

✺✳✶✵✳

❆❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡

✹✶✷

❚❤❡ ❧❛st t❤❡♦r❡♠ ✐♠♣❧✐❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t r❡s✉❧t✳

❚❤❡♦r❡♠ ✺✳✶✵✳✺✿ ❆❜s♦❧✉t❡ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠ ■❢ ❛ s❡r✐❡s ❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧②✱ t❤❡♥ ✐t ❝♦♥✈❡r❣❡s❀ ✐✳❡✳✱

X

|an | < ∞ =⇒

X

an

❝♦♥✈❡r❣❡s

❊①❡r❝✐s❡ ✺✳✶✵✳✻ ✭❛✮ ❙❤♦✇ t❤❛t ✐❢ ❛ s❡r✐❡s ❤❛s ♦♥❧② ♣♦s✐t✐✈❡ t❡r♠s✱ t❤❡♥ ❛❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡ ♠❡❛♥s ❝♦♥✈❡r❣❡♥❝❡✳ ✭❜✮ ❙❤♦✇ t❤❛t ✐❢ ❛ s❡r✐❡s ❤❛s ♦♥❧② ♥❡❣❛t✐✈❡ t❡r♠s✱ t❤❡♥ ❛❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡ ♠❡❛♥s ❝♦♥✈❡r❣❡♥❝❡✳

❊①❛♠♣❧❡ ✺✳✶✵✳✼✿ ♣♦✇❡r s❡r✐❡s ❋♦r ♣♦✇❡r s❡r✐❡s✱ t❤❡ t❤❡♦r❡♠ ❜❡❝♦♠❡s✿

❢♦r ❡❛❝❤

x✳

X

|cn ||x − a|n < ∞ =⇒

X

cn (x − a)n

❝♦♥✈❡r❣❡s

,

❚❤❡ s❡r✐❡s ♦❢ ❛❜s♦❧✉t❡ ✈❛❧✉❡s ♦❢ t❤❡ ❚❛②❧♦r s❡r✐❡s ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ✐s ✐❧❧✉str❛t❡❞

❜❡❧♦✇✿

▲♦♦❦ ❛t t❤❡ ❝✉s♣s❀ t❤❡s❡ ❛r❡♥✬t ♣♦✇❡rs ❛♥❞ t❤❡ ♥❡✇ s❡r✐❡s ✐s

♥♦t

❛ ♣♦✇❡r s❡r✐❡s✳

❚❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ t❤❡♦r❡♠ ✐s ❢❛❧s❡✿

X

X

|an | < ∞ =⇒ |an | < ∞ 6⇐=

X

X

an

❝♦♥✈❡r❣❡s

an

❝♦♥✈❡r❣❡s

❉❡✜♥✐t✐♦♥ ✺✳✶✵✳✽✿ s❡r✐❡s ❝♦♥✈❡r❣❡s ❝♦♥❞✐t✐♦♥❛❧❧② ❲❡ s❛② t❤❛t ❛ s❡r✐❡s ♦❢ ❛❜s♦❧✉t❡ ✈❛❧✉❡s✱

X

X

an

|an |✱

❝♦♥✈❡r❣❡s ❝♦♥❞✐t✐♦♥❛❧❧② ❞♦❡s ♥♦t✳

❚❤❡♥✱ ❡✈❡r② ✭♥✉♠❡r✐❝❛❧✮ s❡r✐❡s ❝♦♥✈❡r❣❡s ❡✐t❤❡r ❛❜s♦❧✉t❡❧② ♦r ❝♦♥❞✐t✐♦♥❛❧❧②✳

✐❢ ✐t ❝♦♥✈❡r❣❡s ❜✉t ✐ts s❡r✐❡s

✺✳✶✵✳

❆❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡

✹✶✸

❊①❛♠♣❧❡ ✺✳✶✵✳✾✿ ♥♦♥✲♥❡❣❛t✐✈❡

❖❢ ❝♦✉rs❡✱ ❛❧❧ ❝♦♥✈❡r❣❡♥t ♥♦♥✲♥❡❣❛t✐✈❡✲t❡r♠ s❡r✐❡s ❝♦♥✈❡r❣❡ ❛❜s♦❧✉t❡❧②✳ ❋♦r ❡①❛♠♣❧❡✱ ❛❧❧

p > 1✱

❝♦♥✈❡r❣❡ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❝♦♥✈❡r❣❡ ❛❜s♦❧✉t❡❧②✳

p✲s❡r✐❡s

✇✐t❤

X 1 , np

❈♦♥✈❡rs❡❧②✱ ❢♦r ❡✈❡r② ❝♦♥✈❡r❣❡♥t s❡r✐❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ t❡r♠s✱ ✇❡ ♥♦✇ ❦♥♦✇ ♦t❤❡r ✭❛❜s♦❧✉t❡❧②✮ ❝♦♥✈❡r❣❡♥t s❡r✐❡s✳ ❋♦r ❡①❛♠♣❧❡✱ s✐♥❝❡ ❛

p✲s❡r✐❡s

✇✐t❤

p > 1✱

X 1 , np

❝♦♥✈❡r❣❡s✱ t❤❡♥ ✐ts ❛❧t❡r♥❛t✐♥❣ ✈❡rs✐♦♥✱

X (−1)n np

✐s ❛❧s♦ ❝♦♥✈❡r❣❡♥t✱ ❛❜s♦❧✉t❡❧②✱ ❜❡❝❛✉s❡✿

❊①❛♠♣❧❡ ✺✳✶✵✳✶✵✿

p✲s❡r✐❡s

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❛s ❛

p✲s❡r✐❡s

✇✐t❤

,

(−1)n 1 np = np .

p ≤ 1✱ X 1 , np

❞✐✈❡r❣❡s✱ ❞♦❡s ✐t ♠❡❛♥ t❤❛t ✐ts ❛❧t❡r♥❛t✐♥❣ ✈❡rs✐♦♥✱

X (−1)n np

❛❧s♦ ❞✐✈❡r❣❡s❄ ◆♦✳

,

❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s♦❧✈❡s t❤❡ ✐ss✉❡✳ ❚❤❡♦r❡♠ ✺✳✶✵✳✶✶✿ ▲❡✐❜♥✐③ ❆❧t❡r♥❛t✐♥❣ ❙❡r✐❡s ❚❡st

❙✉♣♣♦s❡ ❛ s❡q✉❡♥❝❡ bn s❛t✐s✜❡s✿ ✶✳ bn > 0 ❢♦r ❛❧❧ n❀ ✷✳ bn > bn+1 ❢♦r ❛❧❧ n❀ ❛♥❞ ✸✳ bn → 0 ❛s n → ∞✳ X X ❚❤❡♥ t❤❡ ❛❧t❡r♥❛t✐♥❣ ✈❡rs✐♦♥ ♦❢ t❤❡ s❡r✐❡s bn ✱ t❤❡ s❡r✐❡s (−1)n bn ✱ ❝♦♥✈❡r❣❡s✳

✺✳✶✵✳

❆❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡

✹✶✹

Pr♦♦❢✳ ❚❤❡

✐❞❡❛

♦❢ t❤❡ ♣r♦♦❢ ✐s ❛s ❢♦❧❧♦✇s✳ ❋✐rst✱ t❤❡ s❡q✉❡♥❝❡ ❛❧t❡r♥❛t❡s ❜❡t✇❡❡♥ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡✳ ❆s

❛ r❡s✉❧t✱ t❤❡ s❡q✉❡♥❝❡ ♦❢ ✐ts s✉♠s ❣♦❡s ✉♣ ❛♥❞ ❞♦✇♥ ❛t ❡✈❡r② st❡♣✳ ❋✉rt❤❡r♠♦r❡✱ ❡❛❝❤ st❡♣ ✐s s♠❛❧❧❡r t❤❛♥ t❤❡ ❧❛st ❛♥❞ t❤❡ s✇✐♥❣ ✐s ❞✐♠✐♥✐s❤✐♥❣✳ ▼♦r❡♦✈❡r✱ ✐t ✐s ❞✐♠✐♥✐s❤✐♥❣ t♦ ③❡r♦✳ ❚❤❛t✬s ❝♦♥✈❡r❣❡♥❝❡✦

▲❡t✬s ❝♦♥s✐❞❡r t❤❡ s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐❛❧ s✉♠s ♦❢ ♦✉r s❡r✐❡s✿

pn =

n X

(−1)k bk .

k=1

❲❡ ❡①❛♠✐♥❡ t❤❡ ❜❡❤❛✈✐♦r ✐♥ t❤❡ s✉❜s❡q✉❡♥❝❡s ♦❢ ♦❞❞✲ ❛♥❞ ❡✈❡♥✲♥✉♠❜❡r❡❞ ❡❧❡♠❡♥ts✳ ❋♦r t❤❡ ♦❞❞✿

p2k+1 − p2k−1 = (−1)2k b2k + (−1)2k+1 b2k+1 = b2k − b2k+1 > 0.

❆❝❝♦r❞✐♥❣ t♦ ❝♦♥❞✐t✐♦♥ ✷✳

❚❤❡r❡❢♦r❡✱

p2k+1 ր ❙✐♠✐❧❛r❧②✱ ❢♦r t❤❡ ❡✈❡♥

p2k+2 − p2k = (−1)2k+1 b2k+1 + (−1)2k+2 b2k+2 = −b2k+1 + b2k+2

< 0.

❆❝❝♦r❞✐♥❣ t♦ ❝♦♥❞✐t✐♦♥ ✷✳

❚❤❡r❡❢♦r❡✱

p2k ց ❲❡ ❤❛✈❡ t✇♦ ♠♦♥♦t♦♥❡ s❡q✉❡♥❝❡s t❤❛t ❛r❡ ❛❧s♦ ❜♦✉♥❞❡❞✿

p1 ≤ pn ≤ p2 .

❚❤❡r❡❢♦r❡✱ ❜♦t❤ ❝♦♥✈❡r❣❡ ❜② t❤❡

▼♦♥♦t♦♥❡ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠

◆❡①t✱ ❝♦♥s✐❞❡r t❤❡s❡ t✇♦ ❧✐♠✐ts✳ ❇② t❤❡

✳

❙q✉❡❡③❡ ❚❤❡♦r❡♠

✇❡ ❤❛✈❡✿

lim (−1)n bn = 0 ,

n→∞

✺✳✶✶✳

❚❤❡ ❘❛t✐♦ ❚❡st ❛♥❞ t❤❡ ❘♦♦t ❚❡st

❢r♦♠ ❝♦♥❞✐t✐♦♥ ✸✳ ❚❤❡♥✱ ❜② t❤❡ ❙✉♠

✹✶✺

❘✉❧❡ ✇❡ ❤❛✈❡✿

lim p2k+1 − lim p2k = lim (p2k+1 − p2k ) = lim (−1)2k+1 b2k+1 = 0 .

n→∞

n→∞

n→∞

n→∞

❚❤❡♥✱ t❤❡ ❧✐♠✐ts ♦❢ t❤❡ ♦❞❞ ❛♥❞ t❤❡ ❡✈❡♥ ♣❛rt✐❛❧ s✉♠s ❛r❡ ❡q✉❛❧✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✇❤♦❧❡ s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐❛❧ s✉♠s pn ❝♦♥✈❡r❣❡s t♦ t❤❡ s❛♠❡ ❧✐♠✐t✳ ❊①❡r❝✐s❡ ✺✳✶✵✳✶✷

Pr♦✈✐❞❡ ❛ ♣r♦♦❢ ❢♦r t❤❡ ❧❛st st❡♣✳ ❈♦r♦❧❧❛r② ✺✳✶✵✳✶✸ ❆❧❧ ❛❧t❡r♥❛t✐♥❣

p✲s❡r✐❡s✱

✐✳❡✳✱

X (−1)n np

❝♦♥✈❡r❣❡✳

,

■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❛❧t❡r♥❛t✐♥❣ p✲s❡r✐❡s✱ X (−1)n np

❝♦♥✈❡r❣❡ ❝♦♥❞✐t✐♦♥❛❧❧②✳

, ✇✐t❤ 0 < p < 1 ,

✺✳✶✶✳ ❚❤❡ ❘❛t✐♦ ❚❡st ❛♥❞ t❤❡ ❘♦♦t ❚❡st

❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ t❤❡ ❈♦♠♣❛r✐s♦♥

❚❡st

✐s ♦❢t❡♥ ❤♦✇ t♦ ✜♥❞ ❛ s❡r✐❡s ❣♦♦❞ ❢♦r ❝♦♠♣❛r✐s♦♥✳

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❝❤♦♦s❡ ❛ s✐♥❣❧❡ t②♣❡ ♦❢ s❡r✐❡s ❛♥❞ ❞❡r✐✈❡ ❛❧❧ t❤❡ ❝♦♥❝❧✉s✐♦♥s ✇❡ ❝❛♥ ❛❜♦✉t t❤❡ s❡r✐❡s t❤❛t ❝♦♠♣❛r❡ ✇❡❧❧ ✇✐t❤ ✐t✳ ❚❤✐s ❝❤♦✐❝❡ ✐s✱ ♦❢ ❝♦✉rs❡✱ t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s✳ ❚❤❡ ♠♦st ✇❡❧❧✲✉♥❞❡rst♦♦❞ s❡r✐❡s ✐s t❤❡ st❛♥❞❛r❞ ❣❡♦♠❡tr✐❝ s❡r✐❡s ❜② ✐ts r❛t✐♦ r ≥ 0✿ • ■❢ r < 1✱ t❤❡♥

• ■❢ r > 1✱ t❤❡♥

X

X

X

rn ✳ ■ts ❝♦♥✈❡r❣❡♥❝❡ ✐s ❢✉❧❧② ❞❡t❡r♠✐♥❡❞

rn ❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧②✳

rn ❞✐✈❡r❣❡s✳

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ s❡q✉❡♥❝❡ ❤❛s t♦ ❣♦ t♦ 0 ❢❛st ❡♥♦✉❣❤ ❢♦r t❤❡ s❡r✐❡s t♦ ❝♦♥✈❡r❣❡✳ ❚❤✐s ✐❞❡❛ ♦❢ t❤❡ r❛t✐♦ ❛♥❞ t❤❡s❡ t✇♦ ❝♦♥❞✐t✐♦♥s r❡❛♣♣❡❛r ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛ ❣❡♥❡r✐❝ s❡r✐❡s✳ ■♥❞❡❡❞✱ ❡✈❡r② s❡r✐❡s

X

an ❤❛s t❤❡

r❛t✐♦✱ ❛ s❡q✉❡♥❝❡✿ rn =

an+1 . an

■♥ ❝♦♥tr❛st t♦ t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s✱ t❤❡ r❛t✐♦ ❞❡♣❡♥❞s ♦♥ n✳ ❇✉t ✐ts ❧✐♠✐t ❞♦❡s ♥♦t✦ ❆s ✐t t✉r♥s ♦✉t✱ t❤❡ s❡r✐❡s ❡①❤✐❜✐ts t❤❡ s❛♠❡ ❝♦♥✈❡r❣❡♥❝❡ ♣❛tt❡r♥ ❛s t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ t❤❡ r❛t✐♦ ❡q✉❛❧ t♦ t❤✐s ❧✐♠✐t✿

✺✳✶✶✳

❚❤❡ ❘❛t✐♦ ❚❡st ❛♥❞ t❤❡ ❘♦♦t ❚❡st

✹✶✻

❚❤❡♦r❡♠ ✺✳✶✶✳✶✿ ❘❛t✐♦ ❚❡st ❢♦r ❙❡r✐❡s

❙✉♣♣♦s❡ an ✐s ❛ s❡q✉❡♥❝❡ ✇✐t❤ ♥♦♥✲③❡r♦ t❡r♠s✳ ❙✉♣♣♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t ❡①✐sts✱ ❛s ❛ ♥✉♠❜❡r ♦r ❛s ✐♥✜♥✐t②✿ an+1 r = lim n→∞ an

❚❤❡♥ ✇❡ ❤❛✈❡✿ X ✶✳ ■❢ r < 1✱ t❤❡♥ an ❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧②✳

✳

✷✳ ■❢ r > 1✱ t❤❡♥

X

an ❞✐✈❡r❣❡s✳

■❢ r = 1 ♦r t❤❡ ❧✐♠✐t ❞♦❡s♥✬t ❡①✐st✱ ✇❡ s❛② t❤❛t t❤❡ t❡st ❢❛✐❧s✳ Pr♦♦❢✳

❙✉♣♣♦s❡✱ ❢♦r ❛ s❡q✉❡♥❝❡ ✇✐t❤ ♣♦s✐t✐✈❡ t❡r♠s✱ ✇❡ ❤❛✈❡✿ r = lim

n→∞

an+1 . an

❚❤❡♥✱ ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t✱ ✇❡ ❝♦♥❝❧✉❞❡✿ an+1 < s, ❢♦r ❛❧❧ n ≥ N , an

❢♦r s♦♠❡ N ❛♥❞ ❛♥② s > r✳ ❚❤❡r❡❢♦r❡✱ an+1 < san , ❢♦r ❛❧❧ n ≥ N .

❚❤✐s ✐♥❡q✉❛❧✐t② ✐s ♥♦✇ ❛♣♣❧✐❡❞ ♠✉❧t✐♣❧❡ t✐♠❡s✱ st❛rt✐♥❣ ❛t ❛♥② t❡r♠ m > N ✿ am < sam−1 < s(sam−2 ) = s2 am−2 < s2 (sam−3 ) = s3 am−3 < ... < sm−N aN .

❆t t❤❡ ❡♥❞✱ ✇❡ ❤❛✈❡ ❛ ❝♦♠♣❛r✐s♦♥ ♦❢ ♦✉r s❡r✐❡s ❛♥❞ ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ r❛t✐♦ s✿ am
1✳ ■♥ t❤❡ s✐♠♣❧❡st ❝❛s❡✱ ✇❡ ❤❛✈❡ ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✿ R=

❚❤❡♥✱ t❤❡ ❝♦♥❞✐t✐♦♥ r(x) < 1 ❜❡❝♦♠❡s✿ |x − a|

1

cn+1 . limn→∞ cn

1 < 1 ✱ ♦r |x − a| < R . R

❚❤❡s❡ x✬s ❢♦r♠ ❛♥ ✐♥t❡r✈❛❧ ❝❡♥t❡r❡❞ ❛t a✿ {x : |x − a| < R} = {x : a − R < x < a + R} .

✺✳✶✶✳

✹✶✽

❚❤❡ ❘❛t✐♦ ❚❡st ❛♥❞ t❤❡ ❘♦♦t ❚❡st

❲❤❡♥ ♦✉r ❧✐♠✐t ✐s ③❡r♦✱ ✇❡ ❤❛✈❡ R = ∞ ❛♥❞ ✇❤❡♥ ✐t ✐s ✐♥✜♥✐t②✱ ✇❡ ❤❛✈❡ R = 0✳ ❈♦r♦❧❧❛r② ✺✳✶✶✳✺✿ ❘❛t✐♦ ❚❡st ❢♦r P♦✇❡r ❙❡r✐❡s

❙✉♣♣♦s❡ cn ✐s ❛ s❡q✉❡♥❝❡ ✇✐t❤ ♥♦♥✲③❡r♦ t❡r♠s✳ ❙✉♣♣♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t ❡①✐sts✱ ❛s ❛ ♥✉♠❜❡r ♦r ❛s ✐♥✜♥✐t②✿

❚❤❡♥✱ t❤❡r❡ ❛r❡ t❤r❡❡ ❝❛s❡s✿

cn+1 1 = lim R n→∞ cn

❚❤❡ s❡r✐❡s

X

cn (x − a)n ...

❈❛s❡ ✶✿ R = ∞ ❝♦♥✈❡r❣❡s ❢♦r ❡❛❝❤ x; ❈❛s❡ ✷✿ 0 < R < ∞ ❝♦♥✈❡r❣❡s ❢♦r ❡❛❝❤ x ✐♥ t❤❡ ✐♥t❡r✈❛❧✿ (a − R, a + R),

❛♥❞ ❞✐✈❡r❣❡s ❢♦r ❡❛❝❤ x ✐♥ t❤❡ r❛②s✿

❈❛s❡ ✸✿ R = 0

(−∞, a − R), (a + R, +∞);

❞✐✈❡r❣❡s ❢♦r ❡❛❝❤ x 6= a.

❚❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤✐s ✐♥t❡r✈❛❧✱ a − R ❛♥❞ a + R, ✇✐❧❧ ❤❛✈❡ t♦ ❜❡ tr❡❛t❡❞ s❡♣❛r❛t❡❧②✳ ❊①❛♠♣❧❡ ✺✳✶✶✳✻✿ s❡r✐❡s ♦❢ ♣♦✇❡rs

❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ 1 + x + x2 + ... ✐s ✐❧❧✉str❛t❡❞ ❜❡❧♦✇✳ ❙✐♥❝❡ cn = 1 ❛♥❞ R = 1✱ ✇❡ ✐♥❢❡r ❢r♦♠ t❤❡ t❤❡♦r❡♠ t❤❡ ❞✐✈❡r❣❡♥❝❡ ❢♦r x ♦✉ts✐❞❡ t❤❡ ✐♥t❡r✈❛❧ [−1, 1] ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ✐♥s✐❞❡ t❤❡ ✐♥t❡r✈❛❧ (−1, 1) ✭t❤❡ ❧❛tt❡r ✐s t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ ❢♦r♠❡r✮✳ ❚❤❡ t✇♦ ♣❛rt✐❛❧ s✉♠s✱ 1 + x + x2 + ... + x9 ❛♥❞ 1 + x + x2 + ... + x10 ,

❛r❡ s❤♦✇♥✿

✺✳✶✶✳

❚❤❡ ❘❛t✐♦ ❚❡st ❛♥❞ t❤❡ ❘♦♦t ❚❡st

✹✶✾

❊✈❡♥ t❤♦✉❣❤ t❤❡ ♣❛rt✐❛❧ s✉♠s ❛r❡ ❞❡✜♥❡❞ ❢♦r ❛❧❧ ♣♦ss✐❜❧❡ r❡❛❧ ✈❛❧✉❡s ♦❢ x✱ t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② t❤❡ s❡r✐❡s✿ f (x) = 1 + x + x2 + ... ,

✐s ✉♥❞❡✜♥❡❞ ♦✉ts✐❞❡ t❤❡ ✐♥t❡r✈❛❧ (−1, 1)✳ ❊①❛♠♣❧❡ ✺✳✶✶✳✼✿ ❡①♣♦♥❡♥t✐❛❧

▲❡t✬s ❝♦♥✜r♠ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❚❛②❧♦r s❡r✐❡s ❢♦r f (x) = ex ❝❡♥t❡r❡❞ ❛t a = 0✳ ❲❡ ❦♥♦✇ t❤❛t t❤✐s ✐s ❛ ♣♦✇❡r s❡r✐❡s ✇✐t❤ cn =

❚❤❡♥✱

1 . n!

cn+1 n! 1 = lim 1/(n + 1)! = lim = lim = 0 . c = lim n→∞ (n + 1)! n→∞ n n→∞ n→∞ cn 1/n!

❚❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s ❢♦r ❛❧❧ x✳ ❙♦✱ (−∞, +∞) ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜② t❤❡ s❡r✐❡s✳ ◆❡①t✱ t❤❡r❡ ✐s ❛♥♦t❤❡r ✇❛② t♦ ❡①tr❛❝t t❤❡ r❛t✐♦ r ❢r♦♠ ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s an = rn ✿ r=

√ n

an .

■♥ ❝♦♥tr❛st t♦ t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s✱ t❤✐s ♥✉♠❜❡r ❞❡♣❡♥❞s ♦♥ n✳ ❇✉t ✐ts ❧✐♠✐t ❞♦❡s ♥♦t✦ ❆♥❛❧♦❣♦✉s❧② t♦ t❤❡ ❘❛t✐♦ ❚❡st✱ t❤❡ s❡r✐❡s ❡①❤✐❜✐ts t❤❡ s❛♠❡ ❝♦♥✈❡r❣❡♥❝❡ ♣❛tt❡r♥ ❛s t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ t❤❡ r❛t✐♦ ❡q✉❛❧ t♦ t❤✐s ❧✐♠✐t✿ ❚❤❡♦r❡♠ ✺✳✶✶✳✽✿ ❘♦♦t ❚❡st

❙✉♣♣♦s❡ an ✐s ❛ s❡q✉❡♥❝❡✳ ❙✉♣♣♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t ❡①✐sts✱ ❛s ❛ ♥✉♠❜❡r ♦r ❛s ✐♥✜♥✐t②✿ r = lim

n→∞

p n

|an |

❚❤❡♥✱ ✇❡ ❤❛✈❡✿ X ✶✳ ■❢ r < 1✱ t❤❡♥ an ❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧②✳ ✷✳ ■❢ r > 1✱ t❤❡♥

X

an ❞✐✈❡r❣❡s✳

■❢ r = 1 ♦r t❤❡ ❧✐♠✐t ❞♦❡s♥✬t ❡①✐st✱ t❤❡ t❡st ❢❛✐❧s✳ Pr♦♦❢✳

❙✉♣♣♦s❡✱ ❢♦r ❛ s❡q✉❡♥❝❡ ✇✐t❤ ♣♦s✐t✐✈❡ t❡r♠s✱ ✇❡ ❤❛✈❡✿ r = lim

n→∞

r n

an . a0

✺✳✶✶✳

❚❤❡ ❘❛t✐♦ ❚❡st ❛♥❞ t❤❡ ❘♦♦t ❚❡st

✹✷✵

❚❤❡♥✱ ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t✱ ✇❡ ❝♦♥❝❧✉❞❡✿ r n

an < R ❢♦r ❛❧❧ n ≥ N , a0

❢♦r s♦♠❡ N ❛♥❞ ❛♥② R > r✳ ❚❤❡r❡❢♦r❡✱

an < Rn a0 ❢♦r ❛❧❧ n ≥ N .

❲❡ t❤✉s ❤❛✈❡ ❛ ❝♦♠♣❛r✐s♦♥ ♦❢ ♦✉r s❡r✐❡s ❛♥❞ t❤❡ ❣❡♦♠❡tr✐❝ X s❡r✐❡s ✇✐t❤ r❛t✐♦ R✳ ❚❤❡ ❧❛tt❡r ❝♦♥✈❡r❣❡s ✇❤❡♥ R < 1 ❛♥❞ t❤❡♥✱ ❜② t❤❡ ❈♦♠♣❛r✐s♦♥ ❚❡st✱ s♦ ❞♦❡s an ✳ ❙✐♥❝❡ R ✐s ❛♥② ♥✉♠❜❡r ❛❜♦✈❡ R✱ t❤✐s ❝♦♥❞✐t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦ r < 1✳

❊①❡r❝✐s❡ ✺✳✶✶✳✾ ❈♦♠♣❧❡t❡ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠✳

❊①❡r❝✐s❡ ✺✳✶✶✳✶✵ X n ✳ 2n

❊①❛♠✐♥❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❡r✐❡s

❊①❡r❝✐s❡ ✺✳✶✶✳✶✶

❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❧✐♠✐ts ✐♥ t❤❡ t✇♦ t❤❡♦r❡♠s❄

❊①❛♠♣❧❡ ✺✳✶✶✳✶✷✿ r❡❝✉rs✐✈❡❧② ❞❡✜♥❡❞ s❡q✉❡♥❝❡s ❚❤❡ ❘♦♦t ❚❡st r❡q✉✐r❡s t❤❡ nt❤ t❡r♠ s❡q✉❡♥❝❡ t♦ ❜❡ ❦♥♦✇♥✦ ■♥ ❝♦♥tr❛st✱ t❤❡ ❘❛t✐♦ ❚❡st ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ s❡q✉❡♥❝❡s ❞❡✜♥❡❞ r❡❝✉rs✐✈❡❧②✳ ❋♦r ❡①❛♠♣❧❡✱ ❧❡t an+1 = an ·

2n + 1 . 3n + 1

❚❤❡r❡ ✐s ♥♦ ❞✐r❡❝t ❢♦r♠✉❧❛ ❜✉t ✐ts ❝♦♥✈❡r❣❡♥❝❡ ✐s ♣r♦✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣✉t❛t✐♦♥✿ 2n + 1 2 an+1 = → < 1. an 3n + 1 3

❚❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s✳ ▲❡t✬s ❝♦♥s✐❞❡r ❛ ♣♦✇❡r

s❡r✐❡s ❛❣❛✐♥✿

X

cn (x − a)n .

❚❤❡♥ t❤❡ t❤❡♦r❡♠ ❛♣♣❧✐❡s t♦ t❤✐s ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡✿

an = cn (x − a)n ,

❛s ❢♦❧❧♦✇s✿ r(x) = lim

n→∞

p p n |cn (x − a)n | = |x − a| lim n |cn | . n→∞

❚❤❡♥ ✐t ❝♦♥✈❡r❣❡s ❢♦r t❤♦s❡ ✈❛❧✉❡s ♦❢ x ❢♦r ✇❤✐❝❤ ✇❡ ❤❛✈❡✿ |x − a|
1✳

✺✳✶✷✳

P♦✇❡r s❡r✐❡s

✹✷✸

❊①❡r❝✐s❡ ✺✳✶✷✳✹ Pr♦✈❡ t❤❡ st❛t❡♠❡♥ts ✐♥ t❤❡ ❧❛st s❡♥t❡♥❝❡✳

❊①❡r❝✐s❡ ✺✳✶✷✳✺ ❙❦❡t❝❤ t❤❡ ❞✐✈❡r❣❡♥❝❡ ❛t x = −1 ❛♥❞ x = 1✳

❲❛r♥✐♥❣✦ ❚❤❡s❡ ❛r❡ t✇♦ ❞✐✛❡r❡♥t ❢✉♥❝t✐♦♥s✿ 1 + x + x2 + x3 + ... ❛♥❞

1 . 1−x

❚❤❡ r❡❣✐♦♥s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❤❛✈❡ ❜❡❡♥ ✐♥t❡r✈❛❧s ♦❢ ❛ s♣❡❝✐❛❧ ❦✐♥❞✿

❚❤❡② ❛r❡ ❝❡♥t❡r❡❞ ♦♥ a✳ ❚❤❡ ✇❤♦❧❡ ❧✐♥❡ ✐s ❛ ♣♦ss✐❜✐❧✐t②✳ ❚❤❡ r❡st ❛r❡ ✜♥✐t❡✳ ❚❤♦s❡ ✐♥t❡r✈❛❧s ❤❛✈❡ ❡♥❞✲♣♦✐♥ts✳ ❇♦t❤✱ ♦♥❡✱ ♦r ♥❡✐t❤❡r ♠❛② ❜❡❧♦♥❣ t♦ t❤❡ ❞♦♠❛✐♥✳ ❆ s✐♥❣❧❡ ♣♦✐♥t ✐s ❛❧s♦ ❛ ♣♦ss✐❜✐❧✐t②✳ ❲❡ ✇✐❧❧ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ t❡r♠✐♥♦❧♦❣②✿

❉❡✜♥✐t✐♦♥ ✺✳✶✷✳✻✿ ✐♥t❡r✐♦r ♦❢ ✐♥t❡r✈❛❧ [a, b]

❚❤❡

✐♥t❡r✐♦r ♦❢

[a, b) (a, b]

✇✐t❤ a < b ✐s (a, b)✳

(a, b)

❚❤❡

✐♥t❡r✐♦r ♦❢ (−∞, +∞) ✐s (−∞, +∞)✳

❇❡❧♦✇ ✐s ❛ ❝♦♠♣❛❝t s✉♠♠❛r② ♦❢ t❤❡ r❡s✉❧t ❛❜♦✈❡✿

❚❤❡♦r❡♠ ✺✳✶✷✳✼✿ ■♥t❡r✈❛❧ ♦❢ ❈♦♥✈❡r❣❡♥❝❡ ✶✳ ❚❤❡ ❞♦♠❛✐♥ ♦❢ ❛ ♣♦✇❡r s❡r✐❡s ❝❡♥t❡r❡❞ ❛t

x

✐s ❛♥ ✐♥t❡r✈❛❧ ❝❡♥t❡r❡❞ ❛t

a✳

✷✳ ❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐s ❛❜s♦❧✉t❡ ✐♥ t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ ✐♥t❡r✈❛❧ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✳ ✸✳ ❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐s ✉♥✐❢♦r♠ ♦♥ ❛♥② ❝❧♦s❡❞ ✐♥t❡r✈❛❧ ✐♥ t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ ✐♥t❡r✈❛❧ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✳

Pr♦♦❢✳ ❋♦r s✐♠♣❧✐❝✐t②✱ ✇❡ ❛ss✉♠❡ t❤❛t a = 0✳ ■❢ t❤❡ ♦♥❧② ❝♦♥✈❡r❣❡♥t ✈❛❧✉❡ ✐s x = 0✱ ✇❡ ❛r❡ ❞♦♥❡❀ t❤❛t✬s t❤❡ ✐♥t❡r✈❛❧✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t t❤❡r❡ ✐s ❛♥♦t❤❡r ✈❛❧✉❡✱ x = b 6= 0✱ ✐✳❡✳✱ ∞ X n=0

cn bn ❝♦♥✈❡r❣❡s✳

✺✳✶✷✳

P♦✇❡r s❡r✐❡s

✹✷✹

❚❤❡♥✱ ❜② t❤❡ ❉✐✈❡r❣❡♥❝❡

❚❡st✱ ✇❡ ❤❛✈❡

❛♥❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱

cn bn → 0 ,

|cn bn | ≤ M ❢♦r s♦♠❡ M .

❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ❡st✐♠❛t❡ ♦✉r s❡r✐❡s ❛s ❢♦❧❧♦✇s✿

x n x n |cn x | ≤ |cn b | ≤ M . b b n

n

❚❤❡ ❧❛st s❡q✉❡♥❝❡ ✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ❛♥❞ ✐ts s❡r✐❡s ❝♦♥✈❡r❣❡s ✇❤❡♥❡✈❡r |x/b| < 1✱ ♦r |x| < |b|✳ ❚❤❡r❡❢♦r❡✱ ❜② t❤❡ ❉✐r❡❝t ❈♦♠♣❛r✐s♦♥ ❚❡st✱ ♦✉r s❡r✐❡s ❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧② ❢♦r ❡✈❡r② x ✐♥ t❤❡ ✐♥t❡r✈❛❧ (−|b|, |b|)✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ❛♥② t✇♦ ♣♦✐♥ts ✐♥ t❤❡ ❞♦♠❛✐♥ ♣r♦❞✉❝❡ ❛♥ ✐♥t❡r✈❛❧ t❤❛t ❧✐❡s ❡♥t✐r❡❧② ✐♥s✐❞❡ t❤❡ ❞♦♠❛✐♥✳ ❲❡ ♣r♦✈❡❞ ✐♥ ❈❤❛♣t❡r ✷❉❈✲✶ t❤❛t t❤✐s ♣r♦♣❡rt② ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❛t ♦❢ ❜❡✐♥❣ ❛♥ ✐♥t❡r✈❛❧✳ ■♥ t❤❡ ♣r♦♦❢✱ ✇❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐s ❛❧s♦ ❛❜s♦❧✉t❡✳ ❋♦r t❤❡ ❧❛st ♣❛rt✱ ✇❡ ♠♦❞✐❢② t❤❡ ❛❜♦✈❡ ♣r♦♦❢ s❧✐❣❤t❧②✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♥✉♠❜❡r p ✇✐t❤ 1 < p < 1✳ ❚❤❡♥✱ ❢♦r ❛♥② x ✐♥ t❤❡ ✐♥t❡r✈❛❧ [−p|b|, p|b|]✱ ✇❡ ❤❛✈❡ ❛ ❝♦♠♣❛r✐s♦♥ ♦❢ ♦✉r s❡r✐❡s ✇✐t❤ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✐♥❞❡♣❡♥❞❡♥t ♦❢ x✿ x n p|b| n |x| n ≤M ≤ M pn . |cn xn | ≤ M ≤ M b |b| |b|

❉❡✜♥✐t✐♦♥ ✺✳✶✷✳✽✿ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡

❚❤❡ ♠✐♥✐♠❛❧ ❞✐st❛♥❝❡ R ✭t❤❛t ❝♦✉❧❞ ❜❡ ✐♥✜♥✐t❡✮ ❢r♦♠ a t♦ t❤❡ ♣♦✐♥t ❢♦r ✇❤✐❝❤ t❤❡ s❡r✐❡s ❞✐✈❡r❣❡s ✐s ❝❛❧❧❡❞ t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♣♦✇❡r s❡r✐❡s ❝❡♥t❡r❡❞ ❛t a✳ ❚❤✐s ❞❡✜♥✐t✐♦♥ ✐s ❧❡❣✐t✐♠❛t❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❊①✐st❡♥❝❡

♦❢ sup ❚❤❡♦r❡♠ ❢r♦♠ ❈❤❛♣t❡r ✷❉❈✲✶✳

■❢ R ✐s t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡✱ t❤❡♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞♦♠❛✐♥ ✐♥t❡r✈❛❧ ✐s 2R ✭✇❡ ✇✐❧❧ s❡❡ ✐♥ ❱♦❧✉♠❡ ✺ t❤❛t t❤❡r❡ ✐s✱ ✐♥ ❢❛❝t✱ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s R✮✳ ❚❤❡s❡ ❛r❡ t❤❡ t❤r❡❡ ♣♦ss✐❜✐❧✐t✐❡s✿

❲❡ ♠❛❦❡ t❤❡ ❧❛st t❤❡♦r❡♠ ♠♦r❡ s♣❡❝✐✜❝ ❜❡❧♦✇✿

❚❤❡♦r❡♠ ✺✳✶✷✳✾✿ ❘❛❞✐✉s ♦❢ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ P♦✇❡r ❙❡r✐❡s ❙✉♣♣♦s❡

R

✐s t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ ♣♦✇❡r s❡r✐❡s✿

∞ X n=0

cn (x − a)n .

❚❤❡♥✱ ✇❡ ❤❛✈❡✿ ✶✳ ❲❤❡♥

R = ∞✱

t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ s❡r✐❡s ✐s

(−∞, +∞)✳

✺✳✶✷✳

P♦✇❡r s❡r✐❡s

✹✷✺ R < ∞✱ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ s❡r✐❡s ✐s ❛♥ ✐♥t❡r✈❛❧ a − R ❛♥❞ a + R ✭♣♦ss✐❜❧② ✐♥❝❧✉❞❡❞ ♦r ❡①❝❧✉❞❡❞✮✳ ❲❤❡♥ R = 0✱ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ s❡r✐❡s ✐s {a}✳

✷✳ ❲❤❡♥ ✸✳

✇✐t❤ t❤❡ ❡♥❞✲♣♦✐♥ts

❊①❡r❝✐s❡ ✺✳✶✷✳✶✵

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ❢♦r ❝❛s❡ ✸✳ ▲❡t✬s ❛ss✐❣♥ ❞♦♠❛✐♥s t♦ ♣♦✇❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❡st❛❜❧✐s❤❡❞ ♣r❡✈✐♦✉s❧②✿ s❡r✐❡s ∞ X

✐ts s✉♠

xn

=

n=0

∞ X 1 n x n! n=0 ∞ X (−1)k k=0

(2k + 1)!

∞ X (−1)k k=0

(2k)!

✐ts ❞♦♠❛✐♥

1 (−1, 1) 1−x

= ex

(−∞, +∞)

x2k+1 = sin x

(−∞, +∞)

= cos x

(−∞, +∞)

x2k

❊①❡r❝✐s❡ ✺✳✶✷✳✶✶

Pr♦✈❡ t❤❡ ❧❛st t✇♦✳ ❍✐♥t✿ ❙t❛rt ✇✐t❤ t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s✳ ❚❤❡ ♠❛✐♥ ♠❡t❤♦❞ ♦❢ ✜♥❞✐♥❣ ❛ ♣♦✇❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s ✐ts ❚❛②❧♦r ❛r❡ ✐♥❞✐r❡❝t✳

s❡r✐❡s✳ ❖t❤❡r ♠❡t❤♦❞s

❊①❛♠♣❧❡ ✺✳✶✷✳✶✷✿ ♣♦✇❡r s❡r✐❡s ✈✐❛ s✉❜st✐t✉t✐♦♥

❙♦♠❡t✐♠❡s ✇❡ ❝❛♥ ✜♥❞ t❤❡ ♣♦✇❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❜② ✐♥❣❡♥✐♦✉s❧② ❛♣♣❧②✐♥❣ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s✱ ❜❛❝❦✇❛r❞✿ ∞

X 1 = rn , ❢♦r |r| < 1 . 1 − r n=0

❲❡ ❤❛✈❡ ✉s❡❞ t❤✐s ✐❞❡❛ t♦ ✜♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥✿ ∞

X 1 = xn . 1 − x n=0

◆♦✇✱ t❤❡ ❢✉♥❝t✐♦♥ f (x) =

1 1 − x2

✐s r❡❝♦❣♥✐③❡❞ ❛s t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ✜rst ❢♦r♠✉❧❛ ✇✐t❤ r = x2 ✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ s✐♠♣❧② ✇r✐t❡ ✐t ❛s ❛ ♣♦✇❡r s❡r✐❡s ✭✇✐t❤ ♦♥❧② ❡✈❡♥ t❡r♠s✮✿ ∞

∞

X X 1 2 n = (x ) = x2n . 1 − x2 n=0 n=0

✺✳✶✷✳

P♦✇❡r s❡r✐❡s

✹✷✻

❙✐♠✐❧❛r❧②✱ ✇❡ ❝❤♦♦s❡ r = x3 ✱ ❛❢t❡r ❢❛❝t♦r✐♥❣✱ ❜❡❧♦✇✿ ∞ ∞ X X 1 x 3 n =x =x (x ) = x3n+1 . 1 − x3 1 − x3 n=0 n=0

❖♥❡ ♠♦r❡ ✭r = 2x✮✿

∞

∞

X X 1 = (2x)n = 2 n · xn . 1 − 2x n=0 n=0

❚❤❡ ❝❡♥t❡r ♦❢ ❛ s❡r✐❡s ❝♦♥str✉❝t❡❞ t❤✐s ✇❛② ♠❛② ❜❡ ❡❧s❡✇❤❡r❡✿ ∞

∞

X X 1 1 = = (1 − x)n = (−1)(x − 1)n . x 1 − (1 − x) n=0 n=0

❚❤✐s ♠❡t❤♦❞ ❛♠♦✉♥ts t♦ ❛ ❝❤❛♥❣❡

♦❢ ✈❛r✐❛❜❧❡s✳

❊①❡r❝✐s❡ ✺✳✶✷✳✶✸

❋✐♥❞ t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r ❡❛❝❤ ♦❢ t❤❡s❡ ♣♦✇❡r s❡r✐❡s✳ ❚❤❡ ❛❜♦✈❡ t❡r♠✐♥♦❧♦❣② s✐♠♣❧✐✜❡s t❤❡ t✇♦ r❡s✉❧ts ❡st❛❜❧✐s❤❡❞ ♣r❡✈✐♦✉s❧②✳ ❚❤❡♦r❡♠ ✺✳✶✷✳✶✹✿ ❘❛t✐♦ ❛♥❞ ❘♦♦t ❚❡sts ❢♦r P♦✇❡r ❙❡r✐❡s

❙✉♣♣♦s❡ cn ✐s ❛ s❡q✉❡♥❝❡ ✇✐t❤ ♥♦♥✲③❡r♦ t❡r♠s✳ ❙✉♣♣♦s❡ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❧✐♠✐ts ❡①✐sts✱ ❛s ❛ ♥✉♠❜❡r ♦r ❛s ✐♥✜♥✐t②✿ cn 1 ♦r R = p R = lim . n→∞ cn+1 limn→∞ n |cn |

❚❤❡♥✱ R ✐s t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♣♦✇❡r s❡r✐❡s ∞ X n=0

cn (x − a)n .

❊①❛♠♣❧❡ ✺✳✶✷✳✶✺✿ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡

▲❡t✬s ❝♦♥s✐❞❡r ❋♦❧❧♦✇✐♥❣ t❤❡ ❘❛t✐♦

X xn n

.

❚❡st✱ ✇❡ ♥❡❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t✿ 1/n = lim n + 1 = 1 . R = lim n→∞ 1/(n + 1) n→∞ n

❚❤❡r❡❢♦r❡✱ t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ ❞♦♠❛✐♥ ✐s (−1, 1)✳ ◆♦✇ t❤❡ ❡♥❞✲♣♦✐♥ts✿ x=1

=⇒

x = −1 =⇒

X xn

n X xn n

X1 ✐s t❤❡ ❞✐✈❡r❣❡♥t ❤❛r♠♦♥✐❝ s❡r✐❡s. n X (−1)n = ✐s t❤❡ ❝♦♥✈❡r❣❡♥t ❛❧t❡r♥❛t✐♥❣ ❤❛r♠♦♥✐❝ s❡r✐❡s. n X =

❚❤❡r❡❢♦r❡✱ t❤❡ ❞♦♠❛✐♥ ✐s [−1, 1)✳ ❲❡ ❤❛✈❡ ♦♥❡ ♠♦r❡ ♣♦✐♥t ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ ❆ ❣✐✈❡♥ ♣♦✇❡r s❡r✐❡s ♣r♦❞✉❝❡s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ ✐ts ❞♦♠❛✐♥✳ ❇❡❝❛✉s❡ ♦❢ t❤❡

xn ✳

❯♥✐q✉❡♥❡ss ♦❢ t❤❡ ❙✉♠ ♦❢

✺✳✶✷✳ P♦✇❡r s❡r✐❡s

✹✷✼

❙❡r✐❡s✱ ✐t ❝❛♥✬t ♣r♦❞✉❝❡ t✇♦✦ ▲❡t✬s ❧♦♦❦ ❛t t❤✐s ✐❞❡❛ ❢r♦♠ t❤❡ ♦t❤❡r ❞✐r❡❝t✐♦♥✿ ◮ ❈❛♥ ❛ ❢✉♥❝t✐♦♥ ❤❛✈❡ t✇♦ ❞✐✛❡r❡♥t r❡♣r❡s❡♥t❛t✐♦♥s ❜② ❛ ♣♦✇❡r s❡r✐❡s❄

❚❤❡ ✐ss✉❡ ✐s ❤❛♥❞❧❡❞ ✐♥ t❤❡ ♠❛♥♥❡r t❤❛t ✇♦✉❧❞ ❛♣♣❧② t♦ ♣♦❧②♥♦♠✐❛❧s✳ ❊①❛♠♣❧❡ ✺✳✶✷✳✶✻✿ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ♣♦❧②♥♦♠✐❛❧s

❚❤❡r❡ ✐s t❤❡ st❛♥❞❛r❞ ✇❛② t♦ r❡♣r❡s❡♥t t❤❡♠✳ ❋♦r ❡①❛♠♣❧❡✱ x + x = 2x ✐s ❞✐s❝♦✈❡r❡❞ t♦ ❜❡ ❛ s✐♥❣❧❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦♥❝❡ ✇❡ ❝❤♦♦s❡ t♦ ❝♦♠❜✐♥❡ t❤❡ s✐♠✐❧❛r t❡r♠s✳ ▲❡t✬s ❝♦♥s✐❞❡r ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥❀ s✉♣♣♦s❡ l(x) = mx + b = nx + c ❢♦r ❡❛❝❤ x .

❙✐♥❝❡ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ❝❛♥ ♦♥❧② ❤❛✈❡ ♦♥❡ s❧♦♣❡ ❛♥❞ ♦♥❡ y ✲✐♥t❡r❝❡♣t✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t m = n ❛♥❞ b = c✳ ❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ r❡♣r❡s❡♥t❛t✐♦♥✦ ▲❡t✬s ❝♦♥s✐❞❡r ❛ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧❀ s✉♣♣♦s❡ p(x) = ax2 + bx + c = dx2 + ex + f ❢♦r ❡❛❝❤ x .

❉♦ t❤❡s❡ ❝♦❡✣❝✐❡♥ts ❤❛✈❡ t♦ ❜❡ t❤❡ s❛♠❡❄ ❆❣❛✐♥✱ ✇❡ ♥♦t✐❝❡ t❤❛t c ✐s t❤❡ y ✲✐♥t❡r❝❡♣t✱ ❛♥❞ s♦ ✐s f ✳ ❚❤❡② ♠✉st ❜❡ ❡q✉❛❧ ✐❢ t❤✐s ✐s t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥✦ ❲❡ ❝❛♥ ✐♥t❡r♣r❡t t❤✐s ❣❡♦♠❡tr✐❝ ♦❜s❡r✈❛t✐♦♥ ❛❧❣❡❜r❛✐❝❛❧❧②✳ ❲❡ ❥✉st ♣❧✉❣ ✐♥ x = 0 ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥✿ p(0) = a02 + b0 + c = d02 + e0 + f =⇒ c = f .

❲❡ ❝❛♥ ♥♦✇ ❝❛♥❝❡❧ t❤❡s❡ ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥ ❢♦r p✿ p(x) = ax2 + bx + c = dx2 + ex + f =⇒ ax2 + bx = dx2 + ex =⇒ x(ax + b) = x(dx + e) .

❙✐♥❝❡ ✐t ❤♦❧❞s ❢♦r ❛❧❧ x✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ✇❡ ❤❛✈❡ t✇♦ ❡q✉❛❧ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s✿ ax + b = dx + e .

■t ❢♦❧❧♦✇s t❤❛t a = d ❛♥❞ b = e ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❛❜♦✈❡ ❛♥❛❧②s✐s✳ ❲❡ ❝❛♥ ❝♦♥t✐♥✉❡ ♦♥ ✇✐t❤ ❤✐❣❤❡r ❛♥❞ ❤✐❣❤❡r ❞❡❣r❡❡s✳ ❚❤❡ ❣❡♥❡r❛❧ r❡s✉❧t ✐s ❜❡❧♦✇✳ ❚❤❡♦r❡♠ ✺✳✶✷✳✶✼✿ ❯♥✐q✉❡♥❡ss ♦❢ P♦❧②♥♦♠✐❛❧ ❘❡♣r❡s❡♥t❛t✐♦♥

■❢ t✇♦ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❡q✉❛❧✱ t❤❡♥ t❤❡✐r ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦❡✣❝✐❡♥ts ✐♥ t❤❡ st❛♥❞❛r❞ ❝❡♥t❡r❡❞ ❢♦r♠ ❛r❡ ❡q✉❛❧ t♦♦✱ ✐✳❡✳✱ N X n=0

=⇒

cn (x − a) cn

N

=

N X n=0

=

dn (x − a)n dn

❢♦r ❛❧❧ x ❢♦r ❛❧❧ n = 0, 1, 2, 3, ..., N

Pr♦♦❢✳

❚❤❡ ♣r♦♦❢ ✐s ❜② ✐♥❞✉❝t✐♦♥ ♦✈❡r t❤❡ ❞❡❣r❡❡ N ✳ ❚❤❡ tr✐❝❦ ❝♦♥s✐sts ♦❢ ❛ s✉❜st✐t✉t✐♦♥ x = a ✐♥t♦ t❤✐s ❢♦r♠✉❧❛❀ ❛❧❧ t❡r♠s ✇✐t❤ (x − a) ❞✐s❛♣♣❡❛r✿ c0 = d 0 .

✺✳✶✸✳ ❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s

✹✷✽

◆♦✇ t❤❡s❡ t✇♦ t❡r♠s ❛r❡ ❝❛♥❝❡❧❡❞ ❢r♦♠ ♦✉r ❡q✉❛t✐♦♥✱ ♣r♦❞✉❝✐♥❣✿ N X n=1

n

cn (x − a) =

N X n=1

dn (x − a)n .

❆s ②♦✉ ❝❛♥ s❡❡✱ t❤❡ s✉♠♠❛t✐♦♥ st❛rts ✇✐t❤ n = 1 ♥♦✇✳ ❚❤❡ ♣♦✇❡r ♦❢ (x − a) ✐♥ ❡✈❡r② t❡r♠ ✐s t❤❡♥ ❛t ❧❡❛st 1✳ ❲❡ ❝❛♥ ♥♦✇ ❢❛❝t♦r ♦✉t (x − a)✱ ♣r♦❞✉❝✐♥❣✿ (x − a)

N X

(x − a)

N −1 X

♦r

n=1

k=0

cn (x − a)n−1 = (x − a)

k

ck+1 (x − a) = (x − a)

N X

dn (x − a)n−1 ,

N −1 X

dk+1 (x − a)k .

n=1

k=0

❙✐♥❝❡ t❤✐s ❤♦❧❞s ❢♦r ❛❧❧ x✱ t❤❡ ♣♦❧②♥♦♠✐❛❧s ✐♥ ♣❛r❡♥t❤❡s❡s ✭♦❢ ❞❡❣r❡❡ N − 1✮ ❛r❡ ❡q✉❛❧✳ ❚❤❡② ♠✉st ❤❛✈❡ ❡q✉❛❧ ❝♦❡✣❝✐❡♥ts ❜② t❤❡ ✐♥❞✉❝t✐✈❡ ❛ss✉♠♣t✐♦♥✳ ❚❤❡ ✐❞❡❛ ❛♣♣❧✐❡s ✭❜✉t ♥♦t t❤❡ ♣r♦♦❢✱ t♦ ❜❡ ♣r♦✈✐❞❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✮ t♦ ♣♦✇❡r s❡r✐❡s✿ ❚❤❡♦r❡♠ ✺✳✶✷✳✶✽✿ ❯♥✐q✉❡♥❡ss ♦❢ P♦✇❡r ❙❡r✐❡s ❘❡♣r❡s❡♥t❛t✐♦♥

■❢ t✇♦ ♣♦✇❡r s❡r✐❡s ❛r❡ ❡q✉❛❧✱ ❛s ❢✉♥❝t✐♦♥s✱ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧✱ t❤❡♥ t❤❡✐r ❝♦r✲ r❡s♣♦♥❞✐♥❣ ❝♦❡✣❝✐❡♥ts ❛r❡ ❡q✉❛❧ t♦♦✱ ✐✳❡✳✱ ∞ X n=0

=⇒

cn (x − a) cn

n

∞ X

=

n=0

=

dn (x − a)n dn

❢♦r ❛❧❧ a − r < x < a + r, ❢♦r ❛❧❧ n = 0, 1, 2, 3, ...

r>0

❲❛r♥✐♥❣✦ ❚❤❡ t❤❡♦r❡♠ ❞♦❡s♥✬t ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❧❛st ❜❡❝❛✉s❡ t❤❡ ❢❛❝t t❤❛t t✇♦ ❧✐♠✐ts ❛r❡ ❡q✉❛❧ ❞♦❡s♥✬t ✐♠♣❧② t❤❛t s♦ ❛r❡ t❤❡ s❡q✉❡♥❝❡s✳

✺✳✶✸✳ ❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s

■s t❤❡r❡ ❛ r❡❛s♦♥ t♦ st✉❞② ♣♦✇❡r s❡r✐❡s ❜❡s✐❞❡s ❛s ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s❄ ❙✉♣♣♦s❡ t❤❡ ❝❡♥t❡r a ✐s ❣✐✈❡♥✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐t❡♠s ❝r❡❛t❡ ❛ ❢✉♥❝t✐♦♥✱ f ✿ ✶✳ ❚❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs✱ c0 , c1 , c2 , ...

✷✳ ❋♦r ❡❛❝❤ ✐♥♣✉t x✱ ✐ts ✈❛❧✉❡ ✉♥❞❡r t❤✐s ❢✉♥❝t✐♦♥ ✐s ❝♦♠♣✉t❡❞ ❜② s✉❜st✐t✉t✐♥❣ ✐t ✐♥t♦ ❛ ❢♦r♠✉❧❛✱ ❛ ♣♦✇❡r s❡r✐❡s ✇✐t❤ t❤❡ s❡q✉❡♥❝❡ s❡r✈✐♥❣ ❛s ✐ts ❝♦❡✣❝✐❡♥ts✿ f (x) =

∞ X n=0

cn (x − a)n .

✸✳ ❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s t❤❡ r❡❣✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❡r✐❡s✳

✺✳✶✸✳

❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s

✹✷✾

P♦✇❡r s❡r✐❡s ❛r❡ ❢✉♥❝t✐♦♥s✦ ❚❤❡r❡ ❛r❡ ♠❛② ❜❡ ♣♦✇❡r s❡r✐❡s ❢♦r ❡✈❡r② ♦❝❝❛s✐♦♥✱ ❜✉t ❛r❡ t❤❡② ❛s ❣♦♦❞ ❛s ❢✉♥❝t✐♦♥s❄ ❲❡ st❛rt ✇✐t❤

❛❧❣❡❜r❛✳

❏✉st ❛s ✇✐t❤ ❢✉♥❝t✐♦♥s ✐♥ ❣❡♥❡r❛❧✱ ✇❡ ❝❛♥ ❝❛rr② ♦✉t ✭s♦♠❡✮ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ♦♥ ♣♦✇❡r s❡r✐❡s✱ ♣r♦❞✉❝✐♥❣ ♥❡✇ ♣♦✇❡r s❡r✐❡s✳ ❍♦✇❡✈❡r✱ ✇❤❛t ✐s tr✉❧② ✐♠♣♦rt❛♥t ✐s t❤❛t ✇❡ ❝❛♥ ❞♦ t❤❡s❡ ♦♣❡r❛t✐♦♥s ✐❞❡❛ ❝♦♠❡s ❢r♦♠ ♦✉r ❡①♣❡r✐❡♥❝❡ ✇✐t❤

t❡r♠ ❜② t❡r♠✳

❚❤❡

♣♦❧②♥♦♠✐❛❧s ❀ ❛❢t❡r ❛❧❧ t❤❡ ♦♣❡r❛t✐♦♥s✱ ✇❡ ✇❛♥t t♦ ♣✉t t❤❡ r❡s✉❧t ✐♥ t❤❡

st❛♥❞❛r❞ ❢♦r♠✱ ✐✳❡✳✱ ✇✐t❤ ❛❧❧ t❡r♠s ❛rr❛♥❣❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♣♦✇❡rs✳

❊①❛♠♣❧❡ ✺✳✶✸✳✶✿ ❛❧❣❡❜r❛ ♦❢ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ ♣♦✇❡r s❡r✐❡s ❋✐rst✱ ✇❡ ❝❛♥ ❛❞❞ t✇♦ ♣♦❧②♥♦♠✐❛❧s ♦♥❡ t❡r♠ ❛t ❛ t✐♠❡✿

p(x)

=1

+2x

+3x2

q(x)

=7

+5x

−2x2

p(x) + q(x) = (1 + 7) +(2 + 5)x +(3 − 2)x2 ❲❡ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ❛❧s♦ ❛❞❞ t✇♦ s❡r✐❡s ♦♥❡ t❡r♠ ❛t ❛ t✐♠❡ ✭✇❤❡♥ t❤❡② ❝♦♥✈❡r❣❡✮✿

p(x)

= c0

+c1 x

+c2 x2

+...

q(x)

= d0

+d1 x

+d2 x2

+...

p(x) + q(x) = (c0 + d0 ) +(c1 + d1 )x +(c2 + d2 )x2 +... ❙❡❝♦♥❞✱ ✇❡ ❝❛♥ ♠✉❧t✐♣❧② ❛ ♣♦❧②♥♦♠✐❛❧ ❜② ❛ ♥✉♠❜❡r ♦♥❡ t❡r♠ ❛t ❛ t✐♠❡✿

p(x)

=1

+2x

+3x2

2p(x) = (2 · 1) +(2 · 2)x +(2 · 3)x2 ❲❡ ❛❧s♦ ♠✉❧t✐♣❧② ❛ s❡r✐❡s ❜② ❛ ♥✉♠❜❡r ♦♥❡ t❡r♠ ❛t ❛ t✐♠❡ ✭✇❤❡♥ ✐t ❝♦♥✈❡r❣❡s✮✿

p(x)

= c0

+c1 x

+c2 x2

+...

kp(x) = (kc0 ) +(kc1 )x +(kc2 )x

2

+...

❚❤❡ ❝❛✈❡❛t ✏✇❤❡♥ t❤❡② ❝♦♥✈❡r❣❡✑ ❞✐s❛♣♣❡❛rs ✐❢ t❤❡ st❛rt✐♥❣ ♣♦✐♥t ✐s ❛ ❢✉♥❝t✐♦♥✱ ♥♦t ❛ s❡r✐❡s✳

❚❤❡ ❣❡♥❡r❛❧ r❡s✉❧t ✐s ❜❡❧♦✇✿

❚❤❡♦r❡♠ ✺✳✶✸✳✷✿ ❚❡r♠✲❜②✲❚❡r♠ ❆❧❣❡❜r❛ ♦❢ P♦✇❡r ❙❡r✐❡s

❙✉♣♣♦s❡ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② ♣♦✇❡r s❡r✐❡s✿ f (x) =

∞ X

cn (x − a)n ❛♥❞ g(x) =

∞ X

cn (x − a)n +

n=0

∞ X n=0

dn (x − a)n .

❚❤❡♥ ✇❡ ❤❛✈❡✿ ✶✳ ❚❤❡ ❢✉♥❝t✐♦♥ f + g ✐s r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ♣♦✇❡r s❡r✐❡s t❤❛t ✐s t❤❡ t❡r♠✲❜②✲ t❡r♠ s✉♠ ♦❢ t❤♦s❡ ♦❢ f ❛♥❞ g ✱ ❞❡✜♥❡❞ ♦♥ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡✐r ❞♦♠❛✐♥s✿ (f + g)(x) =

n=0

∞ X n=0

dn (x − a)n =

∞ X n=0

(cn + dn )(x − a)n

✷✳ ❚❤❡ ❢✉♥❝t✐♦♥ kf ✱ ❢♦r ❛♥② ❝♦♥st❛♥t k ✱ ✐s r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ♣♦✇❡r s❡r✐❡s t❤❛t

✺✳✶✸✳

❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s

✹✸✵

✐s t❤❡ t❡r♠✲❜②✲t❡r♠ ♣r♦❞✉❝t ♦❢ t❤❛t ♦❢ f ✱ ❞❡✜♥❡❞ ♦♥ t❤❡ s❛♠❡ ❞♦♠❛✐♥✿ (kf )(x) = k ·

∞ X n=0

cn (x − a)n =

∞ X n=0

(kcn )(x − a)n

Pr♦♦❢✳

❚❤❡ ✜rst ❝♦♥❝❧✉s✐♦♥ ✐s ❥✉st✐✜❡❞ ❜② t❤❡

❈♦♥st❛♥t ▼✉❧t✐♣❧❡ ❘✉❧❡ ❢♦r ❙❡r✐❡s✳

❙✉♠ ❘✉❧❡ ❢♦r ❙❡r✐❡s✳

❚❤❡ s❡❝♦♥❞ ❝♦♥❝❧✉s✐♦♥ ✐s ❥✉st✐✜❡❞ ❜② t❤❡

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡s❡ ✏✐♥✜♥✐t❡✑ ♣♦❧②♥♦♠✐❛❧s ❜❡❤❛✈❡ ❥✉st ❧✐❦❡ ♦r❞✐♥❛r② ♣♦❧②♥♦♠✐❛❧s✱ ✇❤❡r❡✈❡r t❤❡② ❝♦♥✈❡r❣❡✳ ◆❡①t ✐s ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ✐♥t❡❣r❛t✐♦♥✱ ✐✳❡✳✱

❝❛❧❝✉❧✉s✱ ♦❢ ♣♦✇❡r s❡r✐❡s✳

❲❡ ✇✐❧❧ s❡❡ t❤❛t✱ ❥✉st ❛s ✇✐t❤ ❢✉♥❝t✐♦♥s ✐♥ ❣❡♥❡r❛❧✱ ✇❡ ❝❛♥ ❝❛rr② ♦✉t t❤❡ ❝❛❧❝✉❧✉s ♦♣❡r❛t✐♦♥s ♦♥ ♣♦✇❡r s❡r✐❡s✱ ♣r♦❞✉❝✐♥❣ ♥❡✇ ♣♦✇❡r s❡r✐❡s✳ ❍♦✇❡✈❡r✱ ✇❤❛t ✐s tr✉❧② ✐♠♣♦rt❛♥t ✐s t❤❛t ✇❡ ❝❛♥ ❞♦ t❤❡s❡ ♦♣❡r❛t✐♦♥s

t❡r♠✳

❊①❛♠♣❧❡ ✺✳✶✸✳✸✿ ❝❛❧❝✉❧✉s ♦❢ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ ♣♦✇❡r s❡r✐❡s

▲❡t✬s ❞✐✛❡r❡♥t✐❛t❡ ❛♥❞ ✐♥t❡❣r❛t❡ ✕ ♦♥❧② t❤❡

P♦✇❡r ❋♦r♠✉❧❛ r❡q✉✐r❡❞ ✕ ♣♦❧②♥♦♠✐❛❧s✳

❋✐rst✱ ✇❡ ❝❛♥ ❞✐✛❡r❡♥t✐❛t❡ ❛ ♣♦❧②♥♦♠✐❛❧ ♦♥❡ t❡r♠ ❛t ❛ t✐♠❡✿

p(x)

= 1 +2x +3x2

p′ (x) = 0 +2

+3 · 2x

▼❛②❜❡ t❤✐s ✐s ❥✉st t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❛ ♣♦✇❡r s❡r✐❡s✿

p(x)

= 1 +2x +3x2

p′ (x) = 0 +2

+...

+3 · 2x +...

❙❡❝♦♥❞✱ ✇❡ ❝❛♥ ✐♥t❡❣r❛t❡ ❛ ♣♦❧②♥♦♠✐❛❧ ♦♥❡ t❡r♠ ❛t ❛ t✐♠❡✿

p(x) = 1 +2x +3x2 Z p(x) dx = C +x +x2 +x3 ❲❤❛t ✐❢ t❤✐s ✇❛s ❥✉st t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❛ ❝♦♥✈❡r❣❡♥t ♣♦✇❡r s❡r✐❡s❄

t❡r♠ ❜②

✺✳✶✸✳

❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s

✹✸✶

❊①❛♠♣❧❡ ✺✳✶✸✳✹✿ ❞✐✛❡r❡♥t✐❛t✐♦♥ ▲❡t✬s ❞✐✛❡r❡♥t✐❛t❡ t❤❡ t❡r♠s ♦❢ t❤❡ ♣♦✇❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✿ ex   d y dx

=

❄

1   d y dx

(ex )′ == 0 = =

+ x   d y dx

+ 1

1

+

1 2 1 3 x + x 2! 3!   d d ↓ dx y dx

+ ... +

1 n x n!  d y dx

+

1 xn+1 (n +1)!  d y dx

+...

1 1 2 1 n−1 1 2x + 3x + ... + nx + (n + 1)xn +... 2! 3! n! (n + 1)! 1 2 1 1 n x xn−1 + x + x + + ... + +... 2! (n − 1)! n!

+

ex

■t ✇♦r❦s✦

❊①❛♠♣❧❡ ✺✳✶✸✳✺✿ ✐♥t❡❣r❛t✐♦♥ ▲❡t✬s ✐♥t❡❣r❛t❡ t❤❡ t❡r♠s ♦❢ t❤❡ ♣♦✇❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥✿ ex  R  y Z

=

1  R  y

❄

ex dx == C+ x =

C+ x

=

K+ ex

+ x  R  y

+

1 2 x 2!  R  y

+ ... +

1 n x n! R  y

+

1 xn+1 (n+ 1)! R  y

+...

1 2 1 11 3 1 1 1 x + x + ... + xn+1 + xn+2 +... 2 2! 3 n! n + 1 (n + 1)! n + 2 1 2 1 3 1 1 x + x xn+1 + xn+2 +... + + ... + 2! 3! (n + 1)! (n + 2)!

+

■t ✇♦r❦s✦

❊①❡r❝✐s❡ ✺✳✶✸✳✻ Pr♦✈✐❞❡ ❞❡t❛✐❧s ♦❢ t❤❡ ❛❜♦✈❡ ❝♦♠♣✉t❛t✐♦♥s✳ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ t❡r♠s ✐s ❡❛s②✿  d cn (x − a)n = ncn (x − a)n−1 , dx

Z

 cn (x − a)n dx =

cn (x − a)n+1 . n+1

❚❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠ ✐s ❝❡♥tr❛❧✿

❚❤❡♦r❡♠ ✺✳✶✸✳✼✿ ❚❡r♠✲❜②✲❚❡r♠ ❈❛❧❝✉❧✉s ♦❢ P♦✇❡r ❙❡r✐❡s ❙✉♣♣♦s❡ t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ ♣♦✇❡r s❡r✐❡s✱ f (x) =

∞ X n=0

an (x − a)n ,

✐s ♣♦s✐t✐✈❡ ♦r ✐♥✜♥✐t❡✳ ❚❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ f r❡♣r❡s❡♥t❡❞ ❜② t❤✐s ♣♦✇❡r s❡r✐❡s ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ✭❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐♥t❡❣r❛❜❧❡✮ ♦♥ t❤❡ ✐♥t❡r✈❛❧ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✱ ❛♥❞ t❤❡ ♣♦✇❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ✐ts ❞❡r✐✈❛t✐✈❡ ❛♥❞ ✐ts ❛♥t✐❞❡r✐✈❛t✐✈❡ ❝♦♥✈❡r❣❡ ✐♥ t❤❡ ✐♥t❡r✐♦r ♦❢ t❤✐s ✐♥t❡r✈❛❧ ❛♥❞ ❛r❡ ❢♦✉♥❞ ❜② t❡r♠✲❜②✲t❡r♠ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞

✺✳✶✸✳

❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s

✹✸✷

✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ ♣♦✇❡r s❡r✐❡s ♦❢

∞ X

′

f (x) =

n=0

f

r❡s♣❡❝t✐✈❡❧②✱ ✐✳❡✳✱

cn (x − a)

n

!′

=

∞ X n=0

cn (x − a)n

′

❛♥❞

Z

f (x) dx =

∞ X

Z

n=0

cn (x − a)

n

!

dx =

∞ Z X n=0

cn (x − a)n dx

❲✐t❤ t❤✐s t❤❡♦r❡♠✱ t❤❡r❡ ✐s ♥♦ ♥❡❡❞ ❢♦r t❤❡ r✉❧❡s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦r ✐♥t❡❣r❛t✐♦♥ ❡①❝❡♣t ❢♦r t❤❡

❋♦r♠✉❧❛ ✦

P♦✇❡r

❲❛r♥✐♥❣✦ ❚❤❡ t❤❡♦r❡♠ s♣❡❛❦s ♦❢ t❤❡

✐♥t❡r✐♦r

♦❢ t❤❡ ✐♥t❡r✈❛❧❀

t❤❡r❡ ✐s ♥♦ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ✐ts ❡♥❞✲♣♦✐♥ts✳

▲❡t✬s ✜♥❞ t❤❡ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s ❢♦r t❤❡ ♥❡✇ s❡r✐❡s ✐♥ t❤❡ st❛♥❞❛r❞ ❢♦r♠✿

❈♦r♦❧❧❛r② ✺✳✶✸✳✽✿ ❚❡r♠✲❜②✲❚❡r♠ ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ P♦✇❡r ❙❡r✐❡s ■♥ t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ ✐♥t❡r✈❛❧ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ ♣♦✇❡r s❡r✐❡s✱ ✇❡ ❤❛✈❡✿

f ′ (x) =

∞ X n=0

◆♦t❡ t❤❡ ✐♥✐t✐❛❧ ✐♥❞❡① ♦❢

1

✐♥st❡❛❞ ♦❢

0

(cn (x − a)n )′ =

∞ X n=1

ncn (x − a)n−1 =

∞ X k=0

(k + 1)ck+1 (x − a)k .

✐♥ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡✳

❈♦r♦❧❧❛r② ✺✳✶✸✳✾✿ ❚❡r♠✲❜②✲❚❡r♠ ■♥t❡❣r❛t✐♦♥ ♦❢ P♦✇❡r ❙❡r✐❡s ■♥ t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ ✐♥t❡r✈❛❧ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ ♣♦✇❡r s❡r✐❡s✱ ✇❡ ❤❛✈❡✿

Z

f (x) dx =

∞ Z X n=0

∞ ∞ X X cn ck−1 n+1 (x−a) = C+ (x−a)k . cn (x−a) dx = C+ n+1 k n=0 k=1 n

❊①❛♠♣❧❡ ✺✳✶✸✳✶✵✿ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ■♥ ❝♦♥tr❛st t♦ t❤❡ ❧❛st ❡①❛♠♣❧❡✱ ❧❡t✬s ✉s❡ t❤❡ t❤❡♦r❡♠ t♦ ✏❞✐s❝♦✈❡r✑ ❛ s♦❧✉t✐♦♥ r❛t❤❡r t❤❛♥ ❝♦♥✜r♠ ✐t✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ♠♦❞❡❧ t❤❛t st❛t❡s t❤❛t ❛ q✉❛♥t✐t②✬s r❛t❡ ♦❢ ❝❤❛♥❣❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ ✭♦r ❡q✉❛❧ ❛t ✐ts s✐♠♣❧❡st✮ t♦ t❤❡ ❝✉rr❡♥t ✈❛❧✉❡s ♦❢ t❤❡ q✉❛♥t✐t②✳ ❲❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤✐s ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿

f′ = f . ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥

y = f (x)

✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ ♣♦✇❡r s❡r✐❡s✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡

✺✳✶✸✳

❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s

✹✸✸

s❡r✐❡s ❛♥❞ t❤❡♥ ♠❛t❝❤ t❤❡ t❡r♠s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❡q✉❛t✐♦♥✿ f f

′

= c0 + =

c1 ւ

=⇒ f′ ||

f

c1 x

c2 x2

+ +

2c2 x

ւ

c3 x3

+

+ 3c3 x ւ

= c0 +

||

c1 x

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❯♥✐q✉❡♥❡ss

||

+

c2 x

2

2

...

+

+

...

+

...

= c1 + 2c2 x + 3c3 x2 + ||

+

...

+

ncn xn−1

+

||

...

+ cn−1 x

n−1

cn xn ncn x

n−1

+... +...

ւ

+ (n + 1)cn+1 xn +...

+

||

cn xn

+...

♦❢ P♦✇❡r ❙❡r✐❡s✱ t❤❡ ❝♦❡✣❝✐❡♥ts ❤❛✈❡ t♦ ❜❡ ❡q✉❛❧✦

❲❡ ❤❛✈❡ ❝r❡❛t❡❞ ❛ s❡q✉❡♥❝❡ ♦❢ ❡q✉❛t✐♦♥s✿ c1 2c2 3c3 ... (n + 1)cn+1 ... ||

c0

||

c1

||

c2

||

...

cn

...

❲❡ ❤❛✈❡ ❝♦♥✈❡rt❡❞ ❛ ❝❛❧❝✉❧✉s ♣r♦❜❧❡♠ ✐♥t♦ ❛♥ ❛❧❣❡❜r❛ ♣r♦❜❧❡♠✦ ❲❡ ❢❛❝❡ ✐♥✜♥✐t❡❧② ♠❛♥② ❡q✉❛t✐♦♥s ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ✉♥❦♥♦✇♥s✳ ❲❡ st❛rt s♦❧✈✐♥❣ ♦♥❡ ❡q✉❛t✐♦♥ ❛t ❛ t✐♠❡ ❛♥❞ s✉❜st✐t✉t❡ t❤❡ r❡s✉❧t ✐♥t♦ t❤❡ ♥❡①t ❡q✉❛t✐♦♥✱ ♠♦✈✐♥❣ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✿ c1 =⇒ c1 = c0

2c2

||

||

c0

=⇒ c2 = c0 /2

3c3 ||

=⇒ c1 = c0

=⇒ c3 = c0 /(2 · 3)

=⇒ c2 = c0 /2

❚❤❡ ♣❛tt❡r♥ ❜❡❝♦♠❡s ❝❧❡❛r✿ cn+1 =

❚❤❡r❡❢♦r❡✱ cn =

4c4 ||

=⇒ c3 = ...

cn . n

c0 . n!

❚❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞✦ ■♥❞❡❡❞✱ ✇❡ ❤❛✈❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ✇❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r✿ f (x) =

∞ X 1 n x = c0 x , n! n! n=0

∞ X c0 n=0

n

❛♥❞ ✐t ✇✐❧❧ ❣✐✈❡ ✉s t❤❡ ✈❛❧✉❡s ♦❢ f ✇✐t❤ ❛♥② ❛❝❝✉r❛❝② ✇❡ ✇❛♥t✳ ❚❤❡ ♦♥❧② ♠✐ss✐♥❣ ♣❛rt ✐♥ t❤✐s ♣r♦❣r❛♠ ✐s t❤❡ ♣r♦♦❢ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❀ ✐t ✐s ❞♦♥❡ ✇✐t❤ t❤❡ ❘❛t✐♦ ❚❡st ✭s❡❡♥ ♣r❡✈✐♦✉s❧②✮✿ R = ∞✳ ❆s ❛ ❜♦♥✉s ✭❥✉st ❛ ❜♦♥✉s✦✮✱ ✇❡ r❡❝♦❣♥✐③❡ t❤❡ r❡s✉❧t✐♥❣ s❡r✐❡s✿ f (x) = c0 ex ✳ ❊①❡r❝✐s❡ ✺✳✶✸✳✶✶

❆♣♣❧② t❤❡ ♠❡t❤♦❞ t♦ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✿ ✭❛✮ f ′ = 2f ❀ ✭❜✮ f ′ = f + 1❀ ✭❝✮ f ′′ = f ❀ ✭❞✮ f ′′ = −f ✳ ❊①❡r❝✐s❡ ✺✳✶✸✳✶✷

❙❤♦✇ t❤❛t t❤❡ ♠❡t❤♦❞ ❞♦❡s♥✬t ✇♦r❦ ❢♦r f ′ = f 2 ✳ ❚❤✐s ♠❡t❤♦❞ ♦❢ s♦❧✈✐♥❣ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✐s ❢✉rt❤❡r ❞✐s❝✉ss❡❞ ✐♥ ❈❤❛♣t❡r ✺❉❊✲✷✳ ❚❤❡ ♥❡✇ ❛♣♣r♦❛❝❤ t♦ ❝❛❧❝✉❧✉s ✐s ❜❛s❡❞ ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❡♣t✿

✺✳✶✸✳

❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s

✹✸✹

❉❡✜♥✐t✐♦♥ ✺✳✶✸✳✶✸✿ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥ ❆ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧ t❤❛t ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ ♣♦✇❡r s❡r✐❡s ✐s ❝❛❧❧❡❞ ❛♥❛❧②t✐❝ ♦♥ t❤✐s ✐♥t❡r✈❛❧✳ ❲❡ ❝❛♥ ✜♥❛❧❧② ♣r♦✈❡ t❤❡ ✐♠♣♦rt❛♥t r❡s✉❧t ✇❡ ♣✉t ❢♦r✇❛r❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✿

❈♦r♦❧❧❛r② ✺✳✶✸✳✶✹✿ ❯♥✐q✉❡♥❡ss ♦❢ P♦✇❡r ❙❡r✐❡s ❘❡♣r❡s❡♥t❛t✐♦♥ ❆♥ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥ ❤❛s ❛ ✉♥✐q✉❡ ♣♦✇❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥✱ ✐✳❡✳✱ ✐❢ t✇♦ ♣♦✇❡r s❡r✐❡s ❛r❡ ❡q✉❛❧✱ ❛s ❢✉♥❝t✐♦♥s✱ ♦♥ ❛♥ ♦♣❡♥ ✐♥t❡r✈❛❧✱ t❤❡♥ t❤❡✐r ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦❡✣❝✐❡♥ts ❛r❡ ❡q✉❛❧ t♦♦✱ ✐✳❡✳✱ f (x) =

∞ X

cn (x − a)

n=0

=⇒

n

=

∞ X n=0

cn =

dn (x − a)n

❢♦r ❛❧❧ a − r < x < a + r, r > 0 ❢♦r ❛❧❧ n = 0, 1, 2, 3, ...

dn

Pr♦♦❢✳ ■t✬s t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥❀ t❤❡r❡❢♦r❡✱ t❤❡ ✈❛❧✉❡ ❛t x = a ✐s t❤❡ s❛♠❡✿ f (a) =

∞ X n=0

cn (a − a)

n

f (a) = c0

=

∞ X n=0

dn (a − a)n =⇒

= d0

■t✬s t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥❀ t❤❡r❡❢♦r❡✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛t x = a ✐s t❤❡ s❛♠❡✿ ′

f (a) =

∞ X n=1

cn n(a − a)

n−1

f ′ (a) = c1

=

∞ X n=1

dn n(a − a)n−1 =⇒

= d1

❚❤❡♥ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✱ s♦ ♦♥✳

❈♦r♦❧❧❛r② ✺✳✶✸✳✶✺ ❊✈❡r② ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥ ✐s ✐♥✜♥✐t❡❧② ♠❛♥② t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡✳ ❲❤❛t ❛❜♦✉t t❤❡ ❝♦♥✈❡rs❡❄ ❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ♣♦✇❡r s❡r✐❡s❄

❊①❛♠♣❧❡ ✺✳✶✸✳✶✻✿ ❞♦♠❛✐♥ ♦❢ ♣♦✇❡r s❡r✐❡s ❈♦♥s✐❞❡r ❛❣❛✐♥✿ f (x) =

1 . x−1

❚❤❡ ♣✐❝t✉r❡ ✐❧❧✉str❛t❡s t❤❡ ❧✐♠✐t❛t✐♦♥s ♦❢ r❡♣r❡s❡♥t✐♥❣ ❢✉♥❝t✐♦♥s ❜② ♣♦✇❡r s❡r✐❡s✳ ❚❤❡ ❝❡♥t❡rs ✇❡ tr② ❛r❡ a = 0, −.5, .5✿

✺✳✶✸✳

❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s

✹✸✺

▲❛r❣❡ ♣❛rts ♦❢ t❤❡ ❣r❛♣❤s ❧✐❡ ♦✉ts✐❞❡ t❤❡ str✐♣ ❡✈❡r② t✐♠❡✳ ❲❤②❄ ❇❡❝❛✉s❡ t❤❡ ❞♦♠❛✐♥ ♦❢ ❛♥② ♦❢ t❤❡s❡ ♣♦✇❡r s❡r✐❡s ✐s ❛♥ ✐♥t❡r✈❛❧ ❝❡♥t❡r❡❞ ❛t a t❤❛t ❝❛♥♥♦t ❝♦♥t❛✐♥ 1✦ ❚❤❡ ❣♦♦❞ ♥❡✇s ✐s t❤❛t t♦❣❡t❤❡r t❤❡② ❝♦✈❡r t❤❡ ✇❤♦❧❡ ❣r❛♣❤✳ ❚❤❡ t❡r♠✲❜②✲t❡r♠ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛❧❧♦✇s ✉s t♦ r❡❞✐s❝♦✈❡r t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s✳ ❙✉♣♣♦s❡ ✇❡ ❛❧r❡❛❞② ❤❛✈❡ ❛ ♣♦✇❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ R > 0✿ f (x) = c0 + c1 (x − a) + c2 (x − a)2 + c3 (x − a)3 + ... + cn (x − a)n + ...

▲❡t✬s ❡①♣r❡ss t❤❡ ❝♦❡✣❝✐❡♥ts ✐♥ t❡r♠s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐ts❡❧❢✳ ❚❤❡ tr✐❝❦ ✐s ♣❛rt❧② t❤❡ s❛♠❡ ❛s ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❧❛st t❤❡♦r❡♠✿ ❲❡ s✉❜st✐t✉t❡ x = a ✐♥t♦ t❤✐s ❛♥❞ t❤❡ ❞❡r✐✈❡❞ ❢♦r♠✉❧❛s✱ ❡①❝❡♣t t❤✐s t✐♠❡ ✇❡ ❞♦♥✬t ❞✐✈✐❞❡ ❜✉t r❛t❤❡r ❞✐✛❡r❡♥t✐❛t❡✳ ❋✐rst s✉❜st✐t✉t✐♦♥ ❣✐✈❡s ✉s ✇❤❛t ✇❡ ❛❧r❡❛❞② ❦♥♦✇✿ f (a) = c0 .

❲❡ ❤❛✈❡ ❢♦✉♥❞ c0 ✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡✱ f ′ (x) = c1 + 2c2 (x − a) + 3c3 (x − a)2 + ... + ncn (x − a)n−1 + ... ,

❛♥❞ s✉❜st✐t✉t❡ x = a✱ ❣✐✈✐♥❣ ✉s✿

f ′ (a) = c1 .

❲❡ ❤❛✈❡ ❢♦✉♥❞ c1 ✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡ ♦♥❡ ♠♦r❡ t✐♠❡✱ f ′′ (x) = 2c2 + 3c3 (x − a) + ... + n(n − 1)cn (x − a)n−2 + ... ,

❛♥❞ s✉❜st✐t✉t❡ x = a✱ ❣✐✈✐♥❣ ✉s✿

f ′′ (a) = 2c2 .

❆❢t❡r m st❡♣s✱ ✇❡ ❤❛✈❡✿ f (m) (x) = m(m− 1)...3 · 2 · 1 · cm +m(m− 1)...3· 2· cm−1 (x− a)+ ... +n· (n− 1) · · · (n− m) · cn (x− a)n−m + ... ,

❛♥❞ s✉❜st✐t✉t✐♥❣ x = a ❣✐✈❡s ✉s✿

f (m) (a) = m!cm .

❲❡ ❤❛✈❡ ❢♦✉♥❞ cm ✳ ❚❤❡ r❡s✉❧t ♦❢ t❤✐s ❝♦♠♣✉t❛t✐♦♥ ✐s ♣r❡s❡♥t❡❞ ❜❡❧♦✇✿

✺✳✶✸✳

❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s

✹✸✻

❚❤❡♦r❡♠ ✺✳✶✸✳✶✼✿ ❚❛②❧♦r ❈♦❡✣❝✐❡♥ts

■❢ ❛ ❢✉♥❝t✐♦♥ ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ ♣♦✇❡r s❡r✐❡s ✇✐t❤ ❛ ♣♦s✐t✐✈❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r✲ ❣❡♥❝❡✱ f (x) = c0 + c1 (x − a) + c2 (x − a)2 + c3 (x − a)3 + ... + cn (x − a)n + ... ,

t❤❡♥ ✐ts ❝♦❡✣❝✐❡♥ts ❛r❡ t❤❡ ❚❛②❧♦r ❝♦❡✣❝✐❡♥ts✿ cn =

f (n) (a) . n!

❚❤✉s✱ t❤❡ nt❤ ♣❛rt✐❛❧ s✉♠ ♦❢ t❤✐s s❡r✐❡s ✐s t❤❡ nt❤ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ♦❢ f ✳

❉❡✜♥✐t✐♦♥ ✺✳✶✸✳✶✽✿ ❚❛②❧♦r s❡r✐❡s ❙✉♣♣♦s❡ f ✐s ❛♥ ✐♥✜♥✐t❡❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥✳ ❚❤❡♥ t❤❡ ♣♦✇❡r s❡r✐❡s c0 + c1 (x − a) + c2 (x − a)2 + c3 (x − a)3 + ... + cn (x − a)n + ... ,

✇✐t❤ ❝♦❡✣❝✐❡♥ts✿

f (n) (a) n! ♦❢ f ❛t x = a✳ cn =

✐s ❝❛❧❧❡❞ t❤❡ ❚❛②❧♦r

s❡r✐❡s

❊①❛♠♣❧❡ ✺✳✶✸✳✶✾✿ s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ▲❡t✬s ✜♥❞ t❤❡ ♣♦✇❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r f (x) = sin x ❛t x = 0✳ ❲❡ st❛rt ✇✐t❤ ✇❤❛t ✇❡ ❛❧r❡❛❞② ❦♥♦✇✿

❲❡ ♥❡❡❞ t❤❡♠ ❛❧❧✿ f (x) ′

= sin x

=⇒ f (0) ′

=0

=⇒ T0 (x) = 0

f (x)

= cos x

=⇒ f (0)

=1

=⇒ T1 (x) = x

f ′′ (x)

= − sin x =⇒ f ′′ (0)

=0

=⇒ T2 (x) = x

1 ′ ′ f ′′ (x) = − cos x =⇒ f ′′ (0) = −1 =⇒ T3 (x) = 1 − x3 6 ...

❚❤❡ s❡q✉❡♥❝❡ st❛rts t♦ r❡♣❡❛t ✐ts❡❧❢✱ ❡✈❡r② ❢♦✉r st❡♣s✳ ❖❢ ❝♦✉rs❡✱ ❡✈❡r② ♣♦❧②♥♦♠✐❛❧ ❧❡❛✈❡s ❢♦r ✐♥✜♥✐t② ❡✈❡♥t✉❛❧❧②✱ ❜✉t t❤❡ r❡s❡♠❜❧❛♥❝❡ ❡①t❡♥❞s ❢✉rt❤❡r ❛♥❞ ❢✉rt❤❡r ❢r♦♠ t❤❡ ❝❡♥t❡r✿

✺✳✶✸✳

❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s

✹✸✼

❚❤❡r❡ ❛r❡ ♥♦ ❡✈❡♥ ♣♦✇❡rs ♣r❡s❡♥t ❜❡❝❛✉s❡ t❤❡ s✐♥❡ ✐s ♦❞❞✳ ❚❤❡r❡❢♦r❡✱ f (2m−1) (0) = (−1)m .

❲❡ ❤❛✈❡ t❤❡ ❚❛②❧♦r ❝♦❡✣❝✐❡♥ts✿ c2m−1 =

(−1)m . (2m − 1)!

▲❡t✬s ❛♣♣r♦①✐♠❛t❡ f (x) = cos x ❛t x = 0✳ ❲❡ st❛rt ✇✐t❤ ✇❤❛t ✇❡ ❛❧r❡❛❞② ❦♥♦✇✿

❲❡ ♥❡❡❞ t❤❡♠ ❛❧❧✿ f (x) ′

f (x) f ′′ (x) ′

f ′′ (x)

= cos x

=⇒ f (0) ′

= − sin x =⇒ f (0)

=1

=⇒ T0 (x) = 1

=0

=⇒ T1 (x) = 1

1 = −1 =⇒ T2 (x) = 1 − x2 2 1 ′ =⇒ T3 (x) = 1 − x2 =⇒ f ′′ (0) = 0 2 (4) =⇒ f (0) = 1

= − cos x =⇒ f ′′ (0)

= sin x

f (4) (x) = cos x ...

❚❤❡ s❡q✉❡♥❝❡ st❛rts t♦ r❡♣❡❛t ✐ts❡❧❢✱ ❡✈❡r② ❢♦✉r st❡♣s✳ ❖♥❝❡ ❛❣❛✐♥✱ ❡✈❡r② ♣♦❧②♥♦♠✐❛❧ ❧❡❛✈❡s ❢♦r ✐♥✜♥✐t② ❡✈❡♥t✉❛❧❧②✱ ❜✉t t❤❡ r❡s❡♠❜❧❛♥❝❡ ❡①t❡♥❞s ❢✉rt❤❡r ❛♥❞ ❢✉rt❤❡r ❢r♦♠ t❤❡ ❝❡♥t❡r✿

✺✳✶✸✳ ❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s

✹✸✽

❚❤❡r❡ ❛r❡ ♥♦ ♦❞❞ ♣♦✇❡rs ♣r❡s❡♥t ❜❡❝❛✉s❡ t❤❡ ❝♦s✐♥❡ ✐s ❡✈❡♥✳ ❚❤❡r❡❢♦r❡✱ f (2m) (0) = (−1)m .

❲❡ ❤❛✈❡ t❤❡ ❚❛②❧♦r ❝♦❡✣❝✐❡♥ts✿ c2m =

(−1)m . (2m)!

❚❤✐s ✐s ✇❤❛t t❤❡ ❣r❛♣❤s ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ t❤❡s❡ ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥s ✇♦✉❧❞ ❧♦♦❦ ❧✐❦❡✿

❚❤❡② ❡✈❡♥t✉❛❧❧② ❜❡❝♦♠❡ ❜❛❞✱ ❜✉t t❤❡ ✐♥t❡r✈❛❧ ✇❤❡r❡ t❤✐♥❣s ❛r❡ ❣♦♦❞ ✐s ❡①♣❛♥❞✐♥❣✳ ❚❤❡ s✉r♣r✐s✐♥❣ ❜②♣r♦❞✉❝t ♦❢ t❤❡ t❤❡♦r❡♠ ✐s t❤❡ ❝♦♥❝❧✉s✐♦♥ t❤❛t t❤❡ ✇❤♦❧❡ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ✈❛❧✉❡s ♦❢ ✐ts ❞❡r✐✈❛t✐✈❡s ❛t ❛ s✐♥❣❧❡ ♣♦✐♥t✳ ❈♦r♦❧❧❛r② ✺✳✶✸✳✷✵

f, g ❛r❡ ❛♥❛❧②t✐❝ ♦♥ ✐♥t❡r✈❛❧ (a−R, a+R), R > 0✳ ■❢ t❤❡② ❤❛✈❡ ♠❛t❝❤✐♥❣ ❞❡r✐✈❛t✐✈❡s ♦❢ ❛❧❧ ♦r❞❡rs ❛t a✱ t❤❡② ❛r❡ ❡q✉❛❧ ♦♥ t❤✐s ✐♥t❡r✈❛❧❀ ✐✳❡✳✱ ❙✉♣♣♦s❡

f (n) (a) = g (n) (a)

❢♦r ❛❧❧

n = 0, 1, 2, ... =⇒ f (x) = g(x)

❢♦r ❛❧❧

x

✐♥

(a−R, a+R) .

❊①❡r❝✐s❡ ✺✳✶✸✳✷✶

Pr♦✈❡ t❤❡ ❝♦r♦❧❧❛r②✳ ❍✐♥t✿ ❈♦♥s✐❞❡r f − g✳ ❙✐♥❝❡ t❤❡s❡ ❞❡r✐✈❛t✐✈❡s ❛r❡✱ ✐♥ t✉r♥✱ ❢✉❧❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♦♥ ❛ s♠❛❧❧ ✭♥♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧✮ ✐♥t❡r✈❛❧ ❛r♦✉♥❞ t❤✐s ♣♦✐♥t✱ ✇❡ ❝♦♥❝❧✉❞❡✿ ◮ ❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✇❛② t♦ ❡①t❡♥❞ t❤❡ ❢✉♥❝t✐♦♥ ❜❡②♦♥❞ t❤✐s ✐♥t❡r✈❛❧✳

✺✳✶✸✳

❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s

✹✸✾

❋♦r ❡①❛♠♣❧❡✱ ❛ ❢✉♥❝t✐♦♥ ❝♦♥st❛♥t ❛r♦✉♥❞ t❤❡ ♣♦✐♥t ✇✐❧❧ ❤❛✈❡ t♦ ❜❡ ❝♦♥st❛♥t ❡✈❡r②✇❤❡r❡ ❡❧s❡✳ ❆ ❢✉♥❝t✐♦♥ ❧✐♥❡❛r ❛r♦✉♥❞ ❛ ♣♦✐♥t ✇✐❧❧ ❤❛✈❡ t♦ ❜❡ ❧✐♥❡❛r ❡✈❡r②✇❤❡r❡ ❡❧s❡✳ ❆♥❞ t❤❡ s❛♠❡ ❢♦r t❤❡ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s t♦♦✳ ❇❡❧♦✇ ✐s t❤❡ ❣❡♥❡r❛❧ s✐t✉❛t✐♦♥✿

■♥ ♦t❤❡r ✇♦r❞s✱ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s ❛r❡ ❡①tr❡♠❡❧② ✏♣r❡❞✐❝t❛❜❧❡✑✿ ❖♥❝❡ ✇❡ ❤❛✈❡ ❞r❛✇♥ ❛ t✐♥② ♣✐❡❝❡ ♦❢ t❤❡ ❣r❛♣❤✱ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✇❛② t♦ ❝♦♥t✐♥✉❡ t♦ ❞r❛✇ ✐t✳ ❲❡ ❝❛♥ ✐♥❢♦r♠❛❧❧② ✐♥t❡r♣r❡t t❤✐s ✐❞❡❛ ❛s ❢♦❧❧♦✇s✿

◮

❉r❛✇✐♥❣ ❛ ❝✉r✈❡ ✇✐t❤ ❛ s✐♥❣❧❡ str♦❦❡ ♦❢ t❤❡ ♣❡♥ ♣r♦❞✉❝❡s ❛♥ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥✱ ❜✉t st♦♣♣✐♥❣

✐♥ t❤❡ ♠✐❞❞❧❡ t♦ ❞❡❝✐❞❡ ❤♦✇ t♦ ♣r♦❝❡❡❞ ✐s ❧✐❦❡❧② t♦ ♣r❡✈❡♥t t❤✐s ❢r♦♠ ❤❛♣♣❡♥✐♥❣✳ ❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ♣♦✇❡r s❡r✐❡s❄ ❚❤❡♦r❡♠ ✺✳✶✸✳✷✷✿ ❘❡♣r❡s❡♥t❛t✐♦♥ ❜② P♦✇❡r ❙❡r✐❡s

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ f ✐s ✐♥✜♥✐t❡❧② ♠❛♥② t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ♦♥ ✐♥t❡r✈❛❧ (a−R, a+ R)✱ ❛♥❞ t❤❡s❡ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❜♦✉♥❞❡❞ ❜② t❤❡ s❛♠❡ ❝♦♥st❛♥t M ✿ ❢♦r ❛❧❧ x ✐♥ (a − R, a + R) .

|f (n) (x)| ≤ M

❚❤❡♥ f ✐s ❛♥❛❧②t✐❝✳ Pr♦♦❢✳

❏✉st ❝♦♠♣❛r❡ t❤❡ ❚❛②❧♦r s❡r✐❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✇✐t❤ t❤✐s ❝♦♥✈❡r❣❡♥t s❡r✐❡s✿

XM

n!

❊①❡r❝✐s❡ ✺✳✶✸✳✷✸

❲❡❛❦❡♥ t❤❡ ❜♦✉♥❞❡❞♥❡ss ❝♦♥❞✐t✐♦♥✳

.

✺✳✶✸✳ ❈❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s

✹✹✵

❊①❛♠♣❧❡ ✺✳✶✸✳✷✹✿ ❚❛②❧♦r s❡r✐❡s ✈✐❛ s✉❜st✐t✉t✐♦♥s

❋✐♥❞ t❤❡ ♣♦✇❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ♦❢

f (x) = x−3

❛r♦✉♥❞ a = 1✳ ❲❡ ❞✐✛❡r❡♥t✐❛t❡ ❛♥❞ s✉❜st✐t✉t❡✿ f (x)

= x−3

f ′ (x)

= (−3)x−4

f ′′ (x)

= (−3)(−4)x−5

f ..

(3)

(x) = (−3)(−4)(−5)x

⇒ f (1) = 1

⇒ c0 = 1

⇒ f ′ (1) = −3

⇒ c1 = −3

⇒ f ′′ (1) = 12

−6

⇒ f

...

...

(3)

...

⇒ c2 = 12/2 = 6

(1) = −60

f (n) (x) = (−3)(−4) · · · (−2 − n)x−3−n ⇒ f (n) (1) =

⇒ c3 = −60/6 = −10 ...

...

n

(−1) (2 + n)! (n + 1)(n + 2) ⇒ cn = (−1)n 2 2

❚❤❛t✬s t❤❡ nt❤ ❚❛②❧♦r ❝♦❡✣❝✐❡♥t✳ ❍❡♥❝❡✱ f (x) =

∞ X

(−1)n

n=1

(n + 1)(n + 2) (x − a)n . 2

❇✉t ✇❤❛t ✐s t❤❡ ✐♥t❡r✈❛❧ ♦❢ ❝♦♥✈❡r❣❡♥❝❡❄ ❚❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✐s t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ♥❡❛r❡st ♣♦✐♥t ❢♦r ✇❤✐❝❤ t❤❡r❡ ✐s ♥♦ ❝♦♥✈❡r❣❡♥❝❡✱ ✇❤✐❝❤ ❛♣♣❡❛rs t♦ ❜❡ 0✱ s♦ R = 1✳ ❲❡ ❝♦♥✜r♠ t❤✐s ✇✐t❤ t❤❡ ❘❛t✐♦ ❚❡st ✿ (n + 1)(n + 2) (n + 2)(n + 3) (n + 1)(n + 2) lim (−1)n = lim ÷ (−1)n+1 = 1. n→∞ n→∞ (n + 2)(n + 3) 2 2

❚❤❡♥ t❤❡ ❡♥❞✲♣♦✐♥ts ♦❢ t❤❡ ✐♥t❡r✈❛❧ ❛r❡ 0 ❛♥❞ 2✳ ❚❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s ✐♥ t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ ✐♥t❡r✈❛❧✱ ❛♥❞ ✇❤❛t✬s ❧❡❢t ✐s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ❛t t❤❡ ❡♥❞✲♣♦✐♥ts✿ x = 0, x = 2,

X

X

(n + 1)(n + 2) (n + 1)(n + 2)(−1)n

❞✐✈❡r❣❡s✳ ❞✐✈❡r❣❡s✳

❚❤✉s✱ (0, 2) ✐s t❤❡ ✐♥t❡r✈❛❧ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❡r✐❡s✳ ■t ✐s ❛❧s♦ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ s❡r✐❡s ❡✈❡♥ t❤♦✉❣❤ ✐t✬s s♠❛❧❧❡r t❤❛♥ t❤❛t ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥✳ ❚❤❡ ♠✐s♠❛t❝❤ ❜❡t✇❡❡♥ t❤❡ ❞♦♠❛✐♥s ♦❢ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡ ✐♥t❡r✈❛❧s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡✐r ♣♦✇❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ✐s t❤❡ ✭♠❛✐♥✮ r❡❛s♦♥ ✇❤② t❤❡ ♠❛t❝❤ ❜❡t✇❡❡♥ ❝❛❧❝✉❧✉s ♦❢ ❢✉♥❝t✐♦♥s ❛♥❞ ❝❛❧❝✉❧✉s ♦❢ ♣♦✇❡r s❡r✐❡s ✐s ✐♠♣❡r❢❡❝t✳ ❆t ❧❡❛st✱ t❤❡ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ✐♥t❡❣r❛t✐♦♥✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❚❡r♠✲❜②✲❚❡r♠ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ■♥t❡❣r❛t✐♦♥✱ ❛r❡ ♣❡r❢❡❝t❧② r❡✢❡❝t❡❞ ✐♥ t❤✐s ♠✐rr♦r✿ f   d y dx f

′

❚❛②❧♦r −−−−−−−→

X

cn (x − a)n   d y dx

X ❚❛②❧♦r −−−−−−−→ (cn (x − a)n )′

Z

f  R  y

❚❛②❧♦r −−−−−−−→

X

cn (x − a)n  R  y

XZ ❚❛②❧♦r (cn (x − a)n ) dx f dx −−−−−−−→

■♥ t❤❡ ✜rst ❞✐❛❣r❛♠✱ ✇❡ st❛rt ✇✐t❤ ❛ ❢✉♥❝t✐♦♥ ❛t t❤❡ t♦♣ ❧❡❢t ❛♥❞ t❤❡♥ ✇❡ ♣r♦❝❡❡❞ ✐♥ t✇♦ ✇❛②s✿ • ❘✐❣❤t✿ ✜♥❞ ✐ts ❚❛②❧♦r s❡r✐❡s✳ ❚❤❡♥ ❞♦✇♥✿ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ r❡s✉❧t t❡r♠ ❜② t❡r♠✳ • ❉♦✇♥✿ ❞✐✛❡r❡♥t✐❛t❡ ✐t✳ ❚❤❡♥ r✐❣❤t✿ ✜♥❞ ✐ts ❚❛②❧♦r s❡r✐❡s✳

❚❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡✦

❚❤✐s st✉❞② ✇✐❧❧ ❝♦♥t✐♥✉❡ ✇✐t❤ ❢✉rt❤❡r ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❈❤❛♣t❡r ✺❉❊✲✷✳

❊①❡r❝✐s❡s

❈♦♥t❡♥ts ✶ ❊①❡r❝✐s❡s✿ ❇❛❝❦❣r♦✉♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✹✶

✷ ❊①❡r❝✐s❡s✿ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✹✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✹✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺✶

✸ ❊①❡r❝✐s❡s✿ ❉❡r✐✈❛t✐✈❡s ✹ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s

✺ ❊①❡r❝✐s❡s✿ ■♥❢♦r♠❛t✐♦♥ ❢r♦♠ t❤❡ ❞❡r✐✈❛t✐✈❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺✸

✻ ❊①❡r❝✐s❡s✿ ❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺✹

✼ ❊①❡r❝✐s❡s✿ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺✺

✽ ❊①❡r❝✐s❡s✿ ■♥t❡❣r❛t✐♦♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺✼

✾ ❊①❡r❝✐s❡s✿ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺✾

✶✵ ❊①❡r❝✐s❡s✿ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ r❡❧❛t❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✻✶

✶✶ ❊①❡r❝✐s❡s✿ ❙❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✹✻✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷ ❊①❡r❝✐s❡s✿ ❙❡q✉❡♥❝❡s ❛♥❞ s❡r✐❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸ ❊①❡r❝✐s❡s✿ P♦✇❡r s❡r✐❡s ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✻✻ ✹✻✽

✶✳ ❊①❡r❝✐s❡s✿ ❇❛❝❦❣r♦✉♥❞

❊①❡r❝✐s❡ ✶✳✶

❊①❡r❝✐s❡ ✶✳✹

❲❤❛t ❛r❡ t❤❡ ♠❛①✱ ♠✐♥✱ ❛♥❞ ❛♥② ❜♦✉♥❞s ♦❢ t❤❡ s❡t

❙t❛t❡ t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤✐s st❛t❡♠❡♥t✿ ✏❚❤❡ ❝♦♥✈❡rs❡

♦❢ ✐♥t❡❣❡rs❄ ❲❤❛t ❛❜♦✉t

R❄

♦❢ t❤❡ ❝♦♥✈❡rs❡ ♦❢ ❛ tr✉❡ st❛t❡♠❡♥t ✐s tr✉❡✑✳

❊①❡r❝✐s❡ ✶✳✺ ❊①❡r❝✐s❡ ✶✳✷

■s t❤❡ ❝♦♥✈❡rs❡ ♦❢ ❛ tr✉❡ st❛t❡♠❡♥t tr✉❡❄

❘❡♣r❡s❡♥t t❤❡s❡ s❡ts ❛s ✐♥t❡rs❡❝t✐♦♥s ❛♥❞ ✉♥✐♦♥s✿ ✶✳

(0, 5)

✷✳

{3}

✸✳

∅

✹✳

{x : x > 0 ❖❘ x

❊①❡r❝✐s❡ ✶✳✸

■s t❤❡ ❝♦♥✈❡rs❡ ♦❢ t❤❡ ❝♦♥✈❡rs❡ ♦❢ ❛ tr✉❡ st❛t❡♠❡♥t tr✉❡❄

✐s ❛♥ ✐♥t❡❣❡r}

✶✳ ❊①❡r❝✐s❡s✿ ❇❛❝❦❣r♦✉♥❞

✺✳

{x : x

✹✹✷

✐s ❞✐✈✐s✐❜❧❡ ❜②

6}

❞r✐✈❡r ❣♦t ❧♦st ❛♥❞ ❞r♦✈❡ ✜✈❡ ❡①tr❛ ♠✐❧❡s✳ ✺✳ ❋r✐❞❛②✱ ■ ❤❛✈❡ ❜❡❡♥ t❛❦✐♥❣ ❛ t❛①✐ t♦ t❤❡ st❛✲

❊①❡r❝✐s❡ ✶✳✻

❚r✉❡ ♦r ❢❛❧s❡✿

t✐♦♥ ❛❧❧ ✇❡❡❦ ♦♥ ❝r❡❞✐t❀ ■ ♣❛② ✇❤❛t ■ ♦✇❡

0 = 1 =⇒ 0 = 1❄

t♦❞❛②✳ ❲❤❛t ✐❢ t❤❡r❡ ✐s ❛♥ ❡①tr❛ ❝❤❛r❣❡ ♣❡r r✐❞❡ ♦❢

❊①❡r❝✐s❡ ✶✳✼

m

❞♦❧✲

❧❛rs❄

Pr♦✈❡ ♦r ❞✐s♣r♦✈❡✿ ❊①❡r❝✐s❡ ✶✳✶✹

max{max A, max B} = max(A ∪ B) .

▲❡t

✭❛✮ ■❢✱ st❛rt✐♥❣ ✇✐t❤ ❛ st❛t❡♠❡♥t ❝♦♥❝❧✉s✐♦♥s ②♦✉ ❛rr✐✈❡ t♦

A✱

A❄

❛♥❞

g : C → D

❜❡ t✇♦ ♣♦ss✐✲

❜❧❡ ❢✉♥❝t✐♦♥s✳ ❋♦r ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s✱

❊①❡r❝✐s❡ ✶✳✽

❝❧✉❞❡ ❛❜♦✉t

f : A → B

0 = 1✱

A✱

❛❢t❡r ❛ s❡r✐❡s ♦❢

st❛t❡ ✇❤❡t❤❡r ♦r ♥♦t ②♦✉ ❝❛♥ ❝♦♠♣✉t❡

• D⊂B

✇❤❛t ❝❛♥ ②♦✉ ❝♦♥✲

✭❜✮ ■❢✱ st❛rt✐♥❣ ✇✐t❤ ❛ st❛t❡♠❡♥t

❛❢t❡r ❛ s❡r✐❡s ♦❢ ❝♦♥❝❧✉s✐♦♥s ②♦✉ ❛rr✐✈❡ t♦

✇❤❛t ❝❛♥ ②♦✉ ❝♦♥❝❧✉❞❡ ❛❜♦✉t

f ◦ g✿

• C⊂A

0 = 0✱

A❄

• B⊂D • B=C

❊①❡r❝✐s❡ ✶✳✾

❲❡ ❦♥♦✇ t❤❛t ✏■❢ ✐t r❛✐♥s✱ t❤❡ r♦❛❞ ❣❡ts ✇❡t✑✳ ❉♦❡s ✐t ♠❡❛♥ t❤❛t ✐❢ t❤❡ r♦❛❞ ✐s ✇❡t✱ ✐t ❤❛s r❛✐♥❡❞❄

❊①❡r❝✐s❡ ✶✳✶✺

❊①❡r❝✐s❡ ✶✳✶✵

♦❢ ✐ts ✈❛❧✉❡s✳ ▼❛❦❡ s✉r❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♦♥t♦✳

❋✉♥❝t✐♦♥

y = f (x)

❆ ❣❛r❛❣❡ ❧✐❣❤t ✐s ❝♦♥tr♦❧❧❡❞ ❜② ❛ s✇✐t❝❤ ❛♥❞✱ ❛❧s♦✱ ✐t

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ♦❢ s♦♠❡

x −1 0 1 2 3 4 5

♠❛② ❛✉t♦♠❛t✐❝❛❧❧② t✉r♥ ♦♥ ✇❤❡♥ ✐t s❡♥s❡s ♠♦t✐♦♥

y = f (x) −1

❞✉r✐♥❣ ♥✐❣❤tt✐♠❡✳ ■❢ t❤❡ ❧✐❣❤t ✐s ❖❋❋✱ ✇❤❛t ❞♦ ②♦✉ ❝♦♥❝❧✉❞❡❄

4 5

2

❊①❡r❝✐s❡ ✶✳✶✻

❊①❡r❝✐s❡ ✶✳✶✶

■❢ ❛♥ ❛❞✈❡rt✐s❡♠❡♥t ❝❧❛✐♠s t❤❛t ✏❆❧❧ ♦✉r s❡❝♦♥❞✲ ❤❛♥❞ ❝❛rs ❝♦♠❡ ✇✐t❤ ✇♦r❦✐♥❣ ❆❈✑✱ ✇❤❛t ✐s t❤❡ ❡❛s✲ ✐❡st ✇❛② t♦ ❞✐s♣r♦✈❡ t❤❡ s❡♥t❡♥❝❡❄

❋✉♥❝t✐♦♥

y = f (x)

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ♦❢ s♦♠❡

♦❢ ✐ts ✈❛❧✉❡s✳ ❆❞❞ ♠✐ss✐♥❣ ✈❛❧✉❡s ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ♦♥❡✲t♦✲♦♥❡✳

x −1 0 1 2 3 4 5

❊①❡r❝✐s❡ ✶✳✶✷

y = f (x) −1

❚❡❛❝❤❡rs ♦❢t❡♥ s❛② t♦ t❤❡ st✉❞❡♥t✬s ♣❛r❡♥ts✿ ✏■❢ ②♦✉r st✉❞❡♥t ✇♦r❦s ❤❛r❞❡r✱ ❤❡✬❧❧ ✐♠♣r♦✈❡✑✳

0 5

0

❲❤❡♥ ❤❡

✇♦♥✬t ✐♠♣r♦✈❡ ❛♥❞ t❤❡ ♣❛r❡♥ts ❝♦♠❡ ❜❛❝❦ t♦ t❤❡ t❡❛❝❤❡r✱ ❤❡ ✇✐❧❧ ❛♥s✇❡r✿ ✏❍❡ ❞✐❞♥✬t ✐♠♣r♦✈❡✱ t❤❛t ♠❡❛♥s ❤❡ ❞✐❞♥✬t ✇♦r❦ ❤❛r❞❡r✑✳ ❆♥❛❧②③❡✳

❊①❡r❝✐s❡ ✶✳✶✼

h(x) = sin2 x + sin3 x ❛s g ◦ f ♦❢ t✇♦ ❢✉♥❝t✐♦♥s y = f (x)

❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞

z = g(y)✳

❊①❡r❝✐s❡ ✶✳✶✸

❙✉♣♣♦s❡ t❤❡ ❝♦st ✐s

f (x)

❞♦❧❧❛rs ❢♦r ❛ t❛①✐ tr✐♣ ♦❢

♠✐❧❡s✳ ■♥t❡r♣r❡t t❤❡ ❢♦❧❧♦✇✐♥❣ st♦r✐❡s ✐♥ t❡r♠s ♦❢ ✶✳ ▼♦♥❞❛②✱ ■ t♦♦❦ ❛ t❛①✐ t♦ t❤❡ st❛t✐♦♥

5

x f✳

❊①❡r❝✐s❡ ✶✳✶✽

❋✉♥❝t✐♦♥ ♠✐❧❡s

❛✇❛②✳

y = f (x)

✉❡s✳ ❋✐♥❞ ✐ts ✐♥✈❡rs❡ ❛♥❞ r❡♣r❡s❡♥t ✐t ❜② ❛ s✐♠✐❧❛r t❛❜❧❡✳

x 0 1 2 3 4

✷✳ ❚✉❡s❞❛②✱ ■ t♦♦❦ ❛ t❛①✐ t♦ t❤❡ st❛t✐♦♥ ❜✉t t❤❡♥ r❡❛❧✐③❡❞ t❤❛t ■ ❧❡❢t s♦♠❡t❤✐♥❣ ❛t ❤♦♠❡ ❛♥❞

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ✐ts ✈❛❧✲

y = f (x) 0 1 2 4 3

❤❛❞ t♦ ❝♦♠❡ ❜❛❝❦✳ ✸✳ ❲❡❞♥❡s❞❛②✱ ■ t♦♦❦ ❛ t❛①✐ t♦ t❤❡ st❛t✐♦♥ ❛♥❞ ■ ❣❛✈❡ ♠② ❞r✐✈❡r ❛ ✜✈❡ ❞♦❧❧❛r t✐♣✳ ✹✳ ❚❤✉rs❞❛②✱ ■ t♦♦❦ ❛ t❛①✐ t♦ t❤❡ st❛t✐♦♥ ❜✉t t❤❡

❊①❡r❝✐s❡ ✶✳✶✾

❋✐♥❞ t❤❡ ❢♦r♠✉❧❛s ♦❢ t❤❡ ✐♥✈❡rs❡s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s✿ ✭❛✮

f (x) = (x + 1)3 ❀

✭❜✮

g(x) = ln(x3 )✳

✶✳ ❊①❡r❝✐s❡s✿ ❇❛❝❦❣r♦✉♥❞

✹✹✸

❊①❡r❝✐s❡ ✶✳✷✵

❊①❡r❝✐s❡ ✶✳✷✽

f, g ✱

●✐✈❡♥ t❤❡ t❛❜❧❡s ♦❢ ✈❛❧✉❡s ♦❢ ✈❛❧✉❡s ♦❢

✜♥❞ t❤❡ t❛❜❧❡ ♦❢

f ◦ g ❢♦r t❤❡ ❢✉♥❝t✐♦♥s ❣✐✈❡♥ ❜② f (u) = u + u ❛♥❞ g(x) = 3❄ ✭❛✮ ❲❤❛t ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ f ◦ g ❢♦r t❤❡ ❢✉♥❝t✐♦♥s ❣✐✈❡♥ ❜② √ f (u) = 2 ❛♥❞ g(x) = x❄ ✭❛✮ ❲❤❛t ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥

2

f ◦ g✿ x y = g(x)

y z = f (y)

0

0

0 4

1

4

1 4

❊①❡r❝✐s❡ ✶✳✷✾

2

3

2 0

❋✉♥❝t✐♦♥

3

0

3 1

✉❡s✳ ❋✐♥❞ ✐ts ✐♥✈❡rs❡ ❛♥❞ r❡♣r❡s❡♥t ✐t ❜② ❛ s✐♠✐❧❛r

4

1

4 2

y = f (x)

t❛❜❧❡✳

x 0 1 2 3 4 y = f (x) 1 2 0 4 3

❲❤❛t ✐❢ t❤❡ ❧❛st r♦✇s ✇❡r❡ ♠✐ss✐♥❣❄

❊①❡r❝✐s❡ ✶✳✷✶

❊①❡r❝✐s❡ ✶✳✸✵

❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥ ❜❡❧♦✇ ❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥

f ◦g

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ✐ts ✈❛❧✲

♦❢ t✇♦ ❢✉♥❝t✐♦♥s✿

●✐✈❡ ❡①❛♠♣❧❡s ♦❢ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ t❤❡✐r ♦✇♥ ✐♥✲ ✈❡rs❡s✳

p h(x) = 2x3 + x .

❊①❡r❝✐s❡ ✶✳✸✶

❊①❡r❝✐s❡ ✶✳✷✷

❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥

h(x) = 2 sin3 x + sin x + 5

h(x) = tan(2x) ❢✉♥❝t✐♦♥s ♦❢ ✈❛r✐❛❜❧❡s x

❘❡♣r❡s❡♥t t❤✐s ❢✉♥❝t✐♦♥✿

❛s t❤❡

❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦

❛♥❞

y✳

❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ♦♥❡ ♦❢ ✇❤✐❝❤ ✐s tr✐❣♦♥♦♠❡tr✐❝✳

❊①❡r❝✐s❡ ✶✳✸✷

❘❡♣r❡s❡♥t t❤✐s ❢✉♥❝t✐♦♥✿

❊①❡r❝✐s❡ ✶✳✷✸

f ♣❡r❢♦r♠s t❤❡ ♦♣❡r❛t✐♦♥✿ ✏t❛❦❡ ❜❛s❡ 2 ♦❢ ✑✱ ❛♥❞ ❢✉♥❝t✐♦♥ g ♣❡r❢♦r♠s✿

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ t❤❡ ❧♦❣❛r✐t❤♠

✏t❛❦❡ t❤❡ sq✉❛r❡ r♦♦t ♦❢ ✑✳ t❤❡ ✐♥✈❡rs❡s ♦❢

f

❛♥❞

g✳

t❤❡s❡ ❢♦✉r ❢✉♥❝t✐♦♥s✳

✭❛✮ ❱❡r❜❛❧❧② ❞❡s❝r✐❜❡

✭❜✮ ❋✐♥❞ t❤❡ ❢♦r♠✉❧❛s ❢♦r

h(x) =

x3 + 1 , x3 − 1

❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ♦❢ ✈❛r✐❛❜❧❡s ❛♥❞

x

y✳

✭❝✮ ●✐✈❡ t❤❡♠ ❞♦♠❛✐♥s ❛♥❞

❝♦❞♦♠❛✐♥s✳ ❊①❡r❝✐s❡ ✶✳✸✸ ❊①❡r❝✐s❡ ✶✳✷✹

y = f (x)

❋✉♥❝t✐♦♥

✶✳ ❘❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥

h(x) =

p

t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s

x2 − 1 ❛s f ❛♥❞ g ✳

✈❛❧✉❡s✳ ■s t❤❡ ❢✉♥❝t✐♦♥ ♦♥❡✲t♦ ♦♥❡❄ ❲❤❛t ❛❜♦✉t ✐ts ✐♥✈❡rs❡❄

x 0 1 2 3 4

y = g(x) = 2x − 1✳

✷✳ Pr♦✈✐❞❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝♦♠♣♦s✐t✐♦♥

f (g(x))

♦❢

f (u) = u2 + u

❛♥❞

Pr♦✈✐❞❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝♦♠♣♦s✐t✐♦♥

f (u) = sin u

y = f (x) 0 1 2 1 2

❊①❡r❝✐s❡ ✶✳✸✹

❊①❡r❝✐s❡ ✶✳✷✺

♦❢

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ♦❢ ✐ts

❛♥❞

g(x) =

√

y = f (g(x))

x✳

y = f (x)

❋✉♥❝t✐♦♥

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❜② ❛ ❧✐st ♦❢ ✐ts

✈❛❧✉❡s✳ ■s t❤❡ ❢✉♥❝t✐♦♥ ♦♥❡✲t♦ ♦♥❡❄ ❲❤❛t ❛❜♦✉t ✐ts ✐♥✈❡rs❡❄

❊①❡r❝✐s❡ ✶✳✷✻

x 0 1 2 3 4

Pr♦✈✐❞❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢

f (u) = u2 − 3u + 2

❛♥❞

y = f (g(x))

❊①❡r❝✐s❡ ✶✳✸✺

❊①❡r❝✐s❡ ✶✳✷✼

❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❢✉♥❝✲ t✐♦♥

2

f (x) = 3x + 1❄

❞♦♠❛✐♥s✳

y = f (x) 7 5 3 4 6

g(x) = x✳

❍✐♥t✿

❈❤♦♦s❡ ❛♣♣r♦♣r✐❛t❡

❋✉♥❝t✐♦♥s

y = f (x)

❛♥❞

u = g(y)

❛r❡ ❣✐✈❡♥ ❜❡❧♦✇

❜② t❛❜❧❡s ♦❢ s♦♠❡ ♦❢ t❤❡✐r ✈❛❧✉❡s✳ Pr❡s❡♥t t❤❡ ❝♦♠✲ ♣♦s✐t✐♦♥

u = h(x)

♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❜② ❛ s✐♠✐❧❛r

✶✳ ❊①❡r❝✐s❡s✿ ❇❛❝❦❣r♦✉♥❞

✹✹✹

❊①❡r❝✐s❡ ✶✳✹✹

t❛❜❧❡✿

x 0 1 2 3 4 y = f (x) 1 1 2 0 2 y

0 1 2 3 4

❚❤❡ t❛①✐ ❝❤❛r❣❡s $1.75 ❢♦r t❤❡ ✜rst q✉❛rt❡r ♦❢ ❛ ♠✐❧❡ ❛♥❞ $0.35 ❢♦r ❡❛❝❤ ❛❞❞✐t✐♦♥❛❧ ✜❢t❤ ♦❢ ❛ ♠✐❧❡✳ ❋✐♥❞ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ♠♦❞❡❧s t❤❡ t❛①✐ ❢❛r❡ f ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ♠✐❧❡s ❞r✐✈❡♥✱ x✳

u = g(y) 3 1 2 1 0 ❊①❡r❝✐s❡ ✶✳✹✺ ❊①❡r❝✐s❡ ✶✳✸✻

✭❛✮ ❆❧❣❡❜r❛✐❝❛❧❧②✱ s❤♦✇ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ f (x) = x2 ✐s ♥♦t ♦♥❡✲t♦✲♦♥❡✳ ✭❜✮ ●r❛♣❤✐❝❛❧❧②✱ s❤♦✇ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ g(x) = 2x+1 ✐s ♦♥❡✲t♦✲♦♥❡✳ ✭❝✮ ❋✐♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ g ✳ ❊①❡r❝✐s❡ ✶✳✸✼

❉❡s❝r✐❜❡ ✕ ❜♦t❤ ❣❡♦♠❡tr✐❝❛❧❧② ❛♥❞ ❛❧❣❡❜r❛✐❝❛❧❧② ✕ t✇♦ ❞✐✛❡r❡♥t tr❛♥s❢♦r♠❛t✐♦♥s t❤❛t ♠❛❦❡ ❛ 1 × 1 sq✉❛r❡ ✐♥t♦ ❛ 2 × 3 r❡❝t❛♥❣❧❡✳ ❊①❡r❝✐s❡ ✶✳✸✽

❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥ts (−1, 1) ❛♥❞ (−1, 5)✳

❊①❡r❝✐s❡ ✶✳✸✾

❋✐♥❞ t❤❡ ✈❛❧✉❡ ♦❢ k s♦ t❤❛t t❤❡ ❧✐♥❡ ❝♦♥t❛✐♥✐♥❣ t❤❡ ♣♦✐♥ts (−6, 0) ❛♥❞ (k, −5) ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡ ❧✐♥❡ ❝♦♥t❛✐♥✐♥❣ t❤❡ ♣♦✐♥ts (4, 3) ❛♥❞ (1, 7)✳

❊①❡r❝✐s❡ ✶✳✹✻

❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ s❛t✐s✜❡❞ ❜② ❛❧❧ ♣♦✐♥ts t❤❛t ❧✐❡ 2 ✉♥✐ts ❛✇❛② ❢r♦♠ t❤❡ ♣♦✐♥t (−1, −2) ❛♥❞ ❜② ♥♦ ♦t❤❡r ♣♦✐♥ts✳

❊①❡r❝✐s❡ ✶✳✹✼

❋♦r t❤❡ ♣♦❧②♥♦♠✐❛❧s ❣r❛♣❤❡❞ ❜❡❧♦✇✱ ✜♥❞ t❤❡ ❢♦❧✲ ❧♦✇✐♥❣✿

1 2 3 s♠❛❧❧❡st ♣♦ss✐❜❧❡ ❞❡❣r❡❡ s✐❣♥ ♦❢ t❤❡ ❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥t ❞❡❣r❡❡ ✐s ♦❞❞✴❡✈❡♥

❈♦♥s✐❞❡r tr✐❛♥❣❧❡ ABC ✐♥ t❤❡ ♣❧❛♥❡ ✇❤❡r❡ A = (3, 2)✱ B = (3, −3)✱ C = (−2, −2)✳ ❋✐♥❞ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ s✐❞❡s ♦❢ t❤❡ tr✐❛♥❣❧❡✳ ❊①❡r❝✐s❡ ✶✳✹✵

❋✐♥❞ ❛❧❧ x s✉❝❤ t❤❛t t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts (3, −8) ❛♥❞ (x, −6) ✐s 5✳ ❊①❡r❝✐s❡ ✶✳✹✽ ❊①❡r❝✐s❡ ✶✳✹✶

❋✐♥❞ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts ♦❢ ✐♥t❡rs❡❝✲ t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ (x − 1)2 + (y − 2)2 = 6 ✇✐t❤ t❤❡ ❛①❡s✳

❋✐♥❞ ❛ ♣♦ss✐❜❧❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ♣❧♦tt❡❞ ❜❡✲ ❧♦✇✿

❊①❡r❝✐s❡ ✶✳✹✷

❙♦❧✈❡ t❤❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ ( x − y = −1, 2x + y = 0. ❊①❡r❝✐s❡ ✶✳✹✾ ❊①❡r❝✐s❡ ✶✳✹✸

❆ ♠♦✈✐❡ t❤❡❛t❡r ❝❤❛r❣❡s $10 ❢♦r ❛❞✉❧ts ❛♥❞ $6 ❢♦r ❝❤✐❧❞r❡♥✳ ❖♥ ❛ ♣❛rt✐❝✉❧❛r ❞❛② ✇❤❡♥ 320 ♣❡♦♣❧❡ ♣❛✐❞ ❛♥ ❛❞♠✐ss✐♦♥✱ t❤❡ t♦t❛❧ r❡❝❡✐♣ts ✇❡r❡ $3120✳ ❍♦✇ ♠❛♥② ✇❡r❡ ❛❞✉❧ts ❛♥❞ ❤♦✇ ♠❛♥② ✇❡r❡ ❝❤✐❧❞r❡♥❄

❆ ❢❛❝t♦r② ✐s t♦ ❜❡ ❜✉✐❧t ♦♥ ❛ ❧♦t ♠❡❛s✉r✐♥❣ 240 ❢t ❜② 320 ❢t✳ ❆ ❜✉✐❧❞✐♥❣ ❝♦❞❡ r❡q✉✐r❡s t❤❛t ❛ ❧❛✇♥ ♦❢ ✉♥✐❢♦r♠ ✇✐❞t❤ ❛♥❞ ❡q✉❛❧ ✐♥ ❛r❡❛ t♦ t❤❡ ❢❛❝t♦r② ♠✉st s✉rr♦✉♥❞ t❤❡ ❢❛❝t♦r②✳ ❲❤❛t ♠✉st t❤❡ ✇✐❞t❤ ♦❢ t❤❡ ❧❛✇♥ ❜❡❄

✶✳ ❊①❡r❝✐s❡s✿ ❇❛❝❦❣r♦✉♥❞

❊①❡r❝✐s❡ ✶✳✺✵

❆ ❢❛❝t♦r② ♦❝❝✉♣✐❡s ❛ ❧♦t ♠❡❛s✉r✐♥❣ 240 ❢t ❜② 320 ❢t✳ ❆ ❜✉✐❧❞✐♥❣ ❝♦❞❡ r❡q✉✐r❡s t❤❛t ❛ ❧❛✇♥ ♦❢ ✉♥✐✲ ❢♦r♠ ✇✐❞t❤ ❛♥❞ ❡q✉❛❧ ✐♥ ❛r❡❛ t♦ t❤❡ ❢❛❝t♦r② ♠✉st s✉rr♦✉♥❞ t❤❡ ❢❛❝t♦r②✳ ❲❤❛t ♠✉st t❤❡ ✇✐❞t❤ ♦❢ t❤❡ ❧❛✇♥ ❜❡❄

✹✹✺

y = f (−x) − 1✳

❊①❡r❝✐s❡ ✶✳✺✶

▼❛❦❡ ❛ ✢♦✇❝❤❛rt ❛♥❞ t❤❡♥ ♣r♦✈✐❞❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ y = f (x) t❤❛t r❡♣r❡s❡♥ts ❛ ♣❛r❦✐♥❣ ❢❡❡ ❢♦r ❛ st❛② ♦❢ x ❤♦✉rs✳ ■t ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ ❢r❡❡ ❢♦r t❤❡ ✜rst ❤♦✉r ❛♥❞ $1 ♣❡r ❤♦✉r ❜❡②♦♥❞✳

❊①❡r❝✐s❡ ✶✳✺✽

❚❤❡ ❣r❛♣❤ ❞r❛✇♥ ✇✐t❤ ❛ s♦❧✐❞ ❧✐♥❡ ✐s y = x3 ✳ ❲❤❛t ❛r❡ t❤❡ ♦t❤❡r t✇♦❄

❊①❡r❝✐s❡ ✶✳✺✷

❊①♣❧❛✐♥ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s✿ r

√ x−1 x−1 . ❛♥❞ √ x+1 x+1

❊①❡r❝✐s❡ ✶✳✺✸

❈❧❛ss✐❢② t❤❡s❡ ❢✉♥❝t✐♦♥s✿ ❢✉♥❝t✐♦♥

♦❞❞ ❡✈❡♥ ♦♥t♦ ♦♥❡✲t♦✲♦♥❡

f (x) = 2x − 1

❊①❡r❝✐s❡ ✶✳✺✾

❚❤❡ ❣r❛♣❤ ❜❡❧♦✇ ✐s t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x) = A sin x + B ❢♦r s♦♠❡ A ❛♥❞ B ✳ ❋✐♥❞ t❤❡s❡ ♥✉♠❜❡rs✳

g(x) = −x + 2 h(x) = 3

❊①❡r❝✐s❡ ✶✳✺✹

❚❤❡ ❣r❛♣❤ ♦❢ y = f (x) ✐s ♣❧♦tt❡❞ ❜❡❧♦✇✳ ❙❦❡t❝❤ y = −f (x + 5) − 6✳

❊①❡r❝✐s❡ ✶✳✻✵

❍❛❧❢ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛♥ ❡✈❡♥ ❢✉♥❝t✐♦♥ ✐s s❤♦✇♥ ❜❡✲ ❧♦✇❀ ♣r♦✈✐❞❡ t❤❡ ♦t❤❡r ❤❛❧❢✿

❊①❡r❝✐s❡ ✶✳✺✺

■s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ ♦❞❞✲ ✴❡✈❡♥ ♦❞❞✴❡✈❡♥❄ ❊①❡r❝✐s❡ ✶✳✺✻

❋✐♥❞ t❤❡ ❢♦r♠✉❧❛s ♦❢ t❤❡ ✐♥✈❡rs❡s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s✿ ✭❛✮ f (x) = (x + 1)3 ❀ ✭❜✮ g(x) = ln(x3 )✳ ❊①❡r❝✐s❡ ✶✳✺✼

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ y = f (x) ✐s ❣✐✈❡♥ ❜❡✲ ❧♦✇✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ y = 2f (3x) ❛♥❞ t❤❡♥

❊①❡r❝✐s❡ ✶✳✻✶

❍❛❧❢ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ❛♥ ♦❞❞ ❢✉♥❝t✐♦♥ ✐s s❤♦✇♥ ❛❜♦✈❡❀ ♣r♦✈✐❞❡ t❤❡ ♦t❤❡r ❤❛❧❢✳

✷✳ ❊①❡r❝✐s❡s✿ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②

✹✹✻

✷✳ ❊①❡r❝✐s❡s✿ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②

❊①❡r❝✐s❡ ✷✳✶

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

y = f (x)✱ ✇❤❡r❡ x ✐s f (x) ✐s t❤❡

t❤❡ ✐♥❝♦♠❡ ✭✐♥ t❤♦✉s❛♥❞s ♦❢ ❞♦❧❧❛rs✮ ❛♥❞

t❛① ❜✐❧❧ ✭✐♥ t❤♦✉s❛♥❞s ♦❢ ❞♦❧❧❛rs✮ ❢♦r t❤❡ ✐♥❝♦♠❡ ♦❢

x✱ ✇❤✐❝❤ ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿ ♥♦ t❛① ♦♥ t❤❡ ✜rst $10, 000✱ t❤❡♥ 5% ❢♦r t❤❡ ♥❡①t $10, 000✱ ❛♥❞ 10% ❢♦r t❤❡ r❡st ♦❢ t❤❡ ✐♥❝♦♠❡✳ ■♥✈❡st✐❣❛t❡ ✐ts ❧✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②✳

❊①❡r❝✐s❡ ✷✳✷

❊①♣❧❛✐♥ ✇❤② t❤❡ ❧✐♠✐t

lim sin

x→0

1 x

❞♦❡s ♥♦t ❡①✐st✳

❊①❡r❝✐s❡ ✷✳✾

❊①♣r❡ss t❤❡ ❛s②♠♣t♦t❡s ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❛s ❧✐♠✐ts ❛♥❞ ✐❞❡♥t✐❢② ♦t❤❡r ♦❢ ✐ts ✐♠♣♦rt❛♥t ❢❡❛t✉r❡s✿ ❊①❡r❝✐s❡ ✷✳✸

✭❛✮ ❙t❛t❡ t❤❡

ε✲ δ

❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t✳

❞❡✜♥✐t✐♦♥ t♦ ♣r♦✈❡ t❤❛t

✭❜✮ ❯s❡ t❤❡

2

lim x = 0✳

x→0

❊①❡r❝✐s❡ ✷✳✹

✭❛✮ ❙t❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♠✐t✳ ✭❜✮ ❯s❡ t❤❡ ❞❡✜♥✐✲ t✐♦♥ t♦ ♣r♦✈❡ t❤❛t

lim x3 6= 3✳

x→0

❊①❡r❝✐s❡ ✷✳✶✵

f ✐s (a, b)✑❄

❚r✉❡ ♦r ❢❛❧s❡✿ ✏■❢ ✐s ❜♦✉♥❞❡❞ ♦♥

❊①❡r❝✐s❡ ✷✳✺

(a, b)✱

t❤❡♥

f

f

✐s

[a, b)✱

t❤❡♥

f

[a, ∞)✱

t❤❡♥

f

❝♦♥t✐♥✉♦✉s ♦♥

✭❛✮ ❙t❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛♥ ✐♥✜♥✐t❡ ❧✐♠✐t✳ ✭❜✮ ❯s❡ t❤❡ ❞❡✜♥✐t✐♦♥ t♦ ♣r♦✈❡ t❤❛t

lim x3 = +∞✳

❊①❡r❝✐s❡ ✷✳✶✶

x→+∞

❚r✉❡ ♦r ❢❛❧s❡✿ ✏■❢ ❜♦✉♥❞❡❞ ♦♥

f

✐s ❝♦♥t✐♥✉♦✉s ♦♥

[a, b]✱

t❤❡♥

[a, b]✑❄

❊①❡r❝✐s❡ ✷✳✻

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ t✇♦ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡s✿

x=0

❛♥❞

x = 2✳

❊①❡r❝✐s❡ ✷✳✶✷

✐s ❜♦✉♥❞❡❞ ♦♥

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛ ❤♦r✐③♦♥t❛❧

x = 2✳

y = −1✱

❝♦♥t✐♥✉♦✉s ♦♥

❊①❡r❝✐s❡ ✷✳✶✸

❊①❡r❝✐s❡ ✷✳✼

❛s②♠♣t♦t❡✿

f ✐s [a, b)✑❄

❚r✉❡ ♦r ❢❛❧s❡✿ ✏■❢

❛♥❞ ❛ ✈❡rt✐❝❛❧ ❛s②♠♣t♦t❡✿

f ✐s ❝♦♥t✐♥✉♦✉s [a, ∞)✑❄

❚r✉❡ ♦r ❢❛❧s❡✿ ✏■❢ ✐s ❜♦✉♥❞❡❞ ♦♥

♦♥

❊①❡r❝✐s❡ ✷✳✶✹ ❊①❡r❝✐s❡ ✷✳✽

■❞❡♥t✐❢② ❛❧❧ ✐♠♣♦rt❛♥t ❢❡❛t✉r❡s ♦❢ t❤✐s ❣r❛♣❤✿

❚r✉❡ ♦r ❢❛❧s❡✿

✏❊✈❡r② ❢✉♥❝t✐♦♥ ✐s ❜♦✉♥❞❡❞ ♦♥ ❛

❝❧♦s❡❞ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧✑❄

✷✳ ❊①❡r❝✐s❡s✿ ▲✐♠✐ts ❛♥❞ ❝♦♥t✐♥✉✐t②

❊①❡r❝✐s❡ ✷✳✶✺

❚❤❡ ❣r❛♣❤ ♦❢ f ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ■t ❤❛s ❛s②♠♣t♦t❡s✳ ❉❡s❝r✐❜❡ t❤❡♠ ❛s ❧✐♠✐ts✳ ❍✐♥t✿ ✉s❡ ❜♦t❤ +∞ ❛♥❞ −∞✳

❊①❡r❝✐s❡ ✷✳✶✻

❆ ❤♦✉s❡ ❤❛s 4 ✢♦♦rs ❛♥❞ ❡❛❝❤ ✢♦♦r ❤❛s 7 ✇✐♥❞♦✇s✳ ❲❤❛t ✇❛s t❤❡ ②❡❛r ✇❤❡♥ t❤❡ ❞♦♦r♠❛♥✬s ❣r❛♥❞✲ ♠♦t❤❡r ❞✐❡❞❄ ❊①❡r❝✐s❡ ✷✳✶✼

■❧❧✉str❛t❡ ✇✐t❤ ♣❧♦ts ✭s❡♣❛r❛t❡❧②✮ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❜❡❤❛✈✐♦r✿ ✭❛✮ f (x) → +∞ ❛s x → 1❀ ✭❜✮ f (x) → −∞ ❛s x → 2+ ❀ ✭❝✮ f (x) → 3 ❛s x → −∞✳ ❊①❡r❝✐s❡ ✷✳✶✽

●✐✈❡♥ f (x) = −(x − 3)4 (x + 1)3 ✳ ❋✐♥❞ t❤❡ ❧❡❛❞✐♥❣ t❡r♠ ❛♥❞ ✉s❡ ✐t t♦ ❞❡s❝r✐❜❡ t❤❡ ❧♦♥❣ t❡r♠ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❊①❡r❝✐s❡ ✷✳✶✾

✭❛✮ ❙t❛t❡ t❤❡ ■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠✳ ✭❜✮ ●✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ✐ts ❛♣♣❧✐❝❛t✐♦♥✳

✹✹✼

✸✳ ❊①❡r❝✐s❡s✿ ❉❡r✐✈❛t✐✈❡s

✹✹✽

✸✳ ❊①❡r❝✐s❡s✿ ❉❡r✐✈❛t✐✈❡s

❊①❡r❝✐s❡ ✸✳✶

❚❤r❡❡ str❛✐❣❤t ❧✐♥❡s ❛r❡ s❤♦✇♥ ❜❡❧♦✇✳ ❲❤❛t ✐s s♦ s♣❡❝✐❛❧ ❛❜♦✉t t❤❡♠❄ ❋✐♥❞ t❤❡✐r s❧♦♣❡s✳

❊①❡r❝✐s❡ ✸✳✻

❚❤❡ ♣♦s✐t✐♦♥ ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ❛r❡ ♣❧♦tt❡❞ ❜❡❧♦✇✳ P❧♦t t❤❡ ❛❝❝❡❧❡r❛t✐♦♥✳

❊①❡r❝✐s❡ ✸✳✷

✭❛✮ ❙✉♣♣♦s❡ ❞✉r✐♥❣ t❤❡ ✜rst 2 s❡❝♦♥❞s ♦❢ ✐ts ✢✐❣❤t ❛♥ ♦❜❥❡❝t ♣r♦❣r❡ss❡❞ ❢r♦♠ ♣♦✐♥t (0, 0) t♦ (1, 0) t♦ (2, 0)✳ ❲❤❛t ✇❛s ✐ts ❛✈❡r❛❣❡ ✈❡❧♦❝✐t② ❛♥❞ ❛✈❡r❛❣❡ ❛❝❝❡❧❡r❛t✐♦♥❄ ✭❜✮ ❲❤❛t ✐❢ t❤❡ ❧❛st ♣♦✐♥t ✐s (1, 1) ✐♥st❡❛❞❄ ❊①❡r❝✐s❡ ✸✳✸

❙✉♣♣♦s❡ t ✐s t✐♠❡ ❛♥❞ x ✐s t❤❡ ♣r✐❝❡ ♦❢ ❜r❡❛❞✳ ❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t ✐ts ❞②♥❛♠✐❝s❄ ❇❡ ❛s s♣❡❝✐✜❝ ❛s ♣♦ss✐❜❧❡✳

❊①❡r❝✐s❡ ✸✳✼

❊❛❝❤ ♦❢ t❤❡s❡ str❛✐❣❤t ❧✐♥❡s ❛r❡ ❞r♦✇♥ t❤r♦✉❣❤ t✇♦ ♣♦✐♥t ♦❢ t❤❡ ❣r❛♣❤✳ ❲❤❛t ❞♦ t❤❡② t❡❧❧ ✉s ❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥❄ ❊①❡r❝✐s❡ ✸✳✹

❋✐♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥ts ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❛t❛✿ x

y = f (x)

−1

2

1

2

3

3

5

3

7

−2

9

5

❊①❡r❝✐s❡ ✸✳✽ ❊①❡r❝✐s❡ ✸✳✺

P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t② ❢♦r t❤❡ ❢♦❧✲ ❧♦✇✐♥❣ ♣♦s✐t✐♦♥ ❢✉♥❝t✐♦♥✿

❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✱ ❝♦♠♣✉t❡ t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ f (x) = x2 + 1 ❛t a = 2 ✇✐t❤ h = 0.2 ❛♥❞ h = 0.1✳ ❊①♣❧❛✐♥ t❤❡ ❞✐✛❡r❡♥❝❡✳

✸✳ ❊①❡r❝✐s❡s✿ ❉❡r✐✈❛t✐✈❡s

✹✹✾

❊①❡r❝✐s❡ ✸✳✾

✭❛✮ ❈♦♠♣✉t❡ t❤❡ ❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ❢♦r t❤❡ ❢✉♥❝t✐♦♥

f (x) = 3x2 − x

❛t

a = 1

❛♥❞

h = .5✳

✭❜✮ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ s❡❝❛♥t t♦ t❤❡ ❣r❛♣❤ ♦❢

y = f (x)

❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤✐s ❛✈❡r❛❣❡ r❛t❡ ♦❢

❝❤❛♥❣❡✳ ❊①❡r❝✐s❡ ✸✳✶✹

❨♦✉ ❤❛✈❡ r❡❝❡✐✈❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡♠❛✐❧ ❢r♦♠ ②♦✉r ❊①❡r❝✐s❡ ✸✳✶✵

❜♦ss✿ ✏❚✐♠✱ ▲♦♦❦ ❛t t❤❡ ♥✉♠❜❡rs ✐♥ t❤✐s s♣r❡❛❞✲ ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊s✲

s❤❡❡t✳ ❚❤✐s st♦❝❦ s❡❡♠s t♦ ❜❡ ✐♥❝❤✐♥❣ ✉♣✳✳✳ ❉♦❡s

∆f t✐♠❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ∆x ❢♦r

✐t❄ ■❢ ✐t ❞♦❡s✱ ❤♦✇ ❢❛st❄ ❚❤❛♥❦s✳ ✕ ❚♦♠✑✳ ❉❡s❝r✐❜❡

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

x = 0, 4,

❛♥❞

6

❛♥❞

f (x)

∆x = 0.5✳

②♦✉r ❛❝t✐♦♥s✳

❊①❡r❝✐s❡ ✸✳✶✺

■❢ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❡q✉❛❧✱ ❞♦ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ❤❛✈❡ t♦ ❜❡ ❡q✉❛❧ t♦♦❄

❊①❡r❝✐s❡ ✸✳✶✻

❋✐♥❞ t❤❡ t❛♥❣❡♥t ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t

(2, 1)

t♦

t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✇❤✐❝❤ ✐s

ex

2

✳

❊①❡r❝✐s❡ ✸✳✶✶

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

f (x)

✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊s✲

❊①❡r❝✐s❡ ✸✳✶✼

t✐♠❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❢♦r

❲❤❛t ❞♦ t❤❡s❡ str❛✐❣❤t ❧✐♥❡s t❡❧❧ ✉s ❛❜♦✉t t❤❡ ❢✉♥❝✲

x = 2, 4, 9

t✐♦♥❄

❛♥❞

∆x = 1✳

❊①❡r❝✐s❡ ✸✳✶✷

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

f

✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊st✐♠❛t❡

t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ q✉♦t✐❡♥t ❛♥❞

∆x = 2, 1, 0.5✳

∆f ∆x

❢♦r

x=1 ❊①❡r❝✐s❡ ✸✳✶✽

✭❛✮ ❙t❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝✲ t✐♦♥ ❛t ♣♦✐♥t

a✳

✭❜✮ Pr♦✈✐❞❡ ❛ ❣r❛♣❤✐❝❛❧ ✐♥t❡r♣r❡t❛✲

t✐♦♥ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥✳

❊①❡r❝✐s❡ ✸✳✶✾

❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✱

f (x) = x2 + 1

❛t

❝♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢

a = 2✳

❊①❡r❝✐s❡ ✸✳✶✸

❊①❡r❝✐s❡ ✸✳✷✵

❚❤❡ s❡❝❛♥t ❧✐♥❡ ♦❢ t❤❡ s✐❣♥ ❢✉♥❝t✐♦♥ ❛r❡ s❤♦✇♥ ❜❡✲

❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

❧♦✇✳ ❲❤❛t ❞♦ t❤❡② t❡❧❧ ②♦✉ ❛❜♦✉t t❤❡ ❞✐✛❡r❡♥t✐❛✲

♠❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡

❜✐❧✐t② ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t

x = 0❄

❛♥❞

6.

f (x)

✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊st✐✲

f ′ (x) ❢♦r x = 0, 4,

✸✳ ❊①❡r❝✐s❡s✿ ❉❡r✐✈❛t✐✈❡s

✹✺✵

❊①❡r❝✐s❡ ✸✳✷✶ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

f (x)

✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊st✐✲

♠❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛♥❞

f ′ (x) ❢♦r x = 2, 4,

9✳

❊①❡r❝✐s❡ ✸✳✷✷ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

f

✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊st✐♠❛t❡

t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡

f′

❢♦r

x=0

❛♥❞

x = 4✳

❙❤♦✇ ②♦✉r ❝♦♠♣✉t❛t✐♦♥s✳

❊①❡r❝✐s❡ ✸✳✷✸ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥

f (x)

✐s ❣✐✈❡♥ ❜❡❧♦✇✳ ❊st✐✲

♠❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛♥❞

6✳

f ′ (x) ❢♦r x = 1, 3✱

✹✳ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s

✹✺✶

✹✳ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s

❊①❡r❝✐s❡ ✹✳✶

❚❤❡ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❛ ❝✐t② ❞❡❝❧✐♥❡s ❜② 10% ❡✈❡r② ②❡❛r✳ ❍♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ t♦ ❞r♦♣ t♦ 50% ♦❢ t❤❡ ❝✉rr❡♥t ♣♦♣✉❧❛t✐♦♥❄ ❊①❡r❝✐s❡ ✹✳✷

❚❤❡ ❢✉♥❝t✐♦♥ y = f (x) s❤♦✇♥ ❜❡❧♦✇ r❡♣r❡s❡♥ts t❤❡ ❧♦❝❛t✐♦♥ ✭✐♥ ♠✐❧❡s✮ ♦❢ ❛ ❤✐❦❡r ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✭✐♥ ❤♦✉rs✮✳ ❙❦❡t❝❤ t❤❡ ❤✐❦❡r✬s ✈❡❧♦❝✐t② ❛s t❤❡ ❞✐✛❡r✲ ❡♥❝❡ q✉♦t✐❡♥t✳

❊①❡r❝✐s❡ ✹✳✻

❙✉♣♣♦s❡ t❤❡ ❛❧t✐t✉❞❡✱ ✐♥ ♠❡t❡rs✱ ♦❢ ❛♥ ♦❜❥❡❝t ✐s ❣✐✈❡♥ ❜② t❤❡ ❢✉♥❝t✐♦♥ t2 + t✱ ✇❤❡r❡ t ✐s t✐♠❡✱ ✐♥ s❡❝♦♥❞s✳ ❲❤❛t ✐s t❤❡ ✈❡❧♦❝✐t② ✇❤❡♥ t❤❡ ❛❧t✐t✉❞❡ ✐s 12 ♠❡t❡rs❄ ❊①❡r❝✐s❡ ✹✳✼

❊①❡r❝✐s❡ ✹✳✸

❚❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ♦❜❥❡❝t ❛t t✐♠❡ t ✐s ❣✐✈❡♥ ❜② v(t) = 1 + 3t2 . ■❢ ❛t t✐♠❡ t = 1 t❤❡ ♦❜❥❡❝t ✐s ❛t ♣♦s✐t✐♦♥ x = 4, ✇❤❡r❡ ✐s ✐t ❛t t✐♠❡ t = 0❄ ❊①❡r❝✐s❡ ✹✳✹

❚❤❡ ❣r❛♣❤s ♦❢ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♠♦✈✐♥❣ ♦❜❥❡❝t ❛r❡ s❤♦✇♥ ❜❡❧♦✇✳ ❉❡s❝r✐❜❡ ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣✳

❚❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ♦❜❥❡❝t ❛t t✐♠❡ t ✐s ❣✐✈❡♥ ❜② v(t) = 1 + et . ■❢ ❛t t✐♠❡ t = 0 t❤❡ ♦❜❥❡❝t ✐s ❛t x = 2, ✇❤❡r❡ ✐s ✐t ❛t t✐♠❡ t = 1❄ ❊①❡r❝✐s❡ ✹✳✽

❚❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ❛t t✐♠❡ t ✐s ❣✐✈❡♥ ❜② a(t) = 3t. ■❢ ❛t t✐♠❡ t = 1 t❤❡ ✈❡❧♦❝✐t② ♦❢ ♦❜❥❡❝t ✐s ❛t v(1) = −1, ✇❤❛t ✐s ✐t ❛t t✐♠❡ t = 0❄ ❊①❡r❝✐s❡ ✹✳✾

❙✉♣♣♦s❡ s(t) r❡♣r❡s❡♥ts t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ❛t t✐♠❡ t ❛♥❞ v(t) ✐ts ✈❡❧♦❝✐t②✳ ■❢ v(t) = sin t − cos t ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥ ✐s s(0) = 0, ✜♥❞ t❤❡ ♣♦s✐t✐♦♥ s(1). ❊①❡r❝✐s❡ ✹✳✶✵

❙✉♣♣♦s❡ t❤❡ s♣❡❡❞ ♦❢ ❛ ❝❛r ✇❛s ❣r♦✇✐♥❣ ❝♦♥t✐♥✉✲ ♦✉s❧② ❢♦❧❧♦✇✐♥❣ t❤❡ r✉❧❡ 55 + 5t ♣❡r ❤♦✉r✱ ✇❤❡r❡ t ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❤♦✉rs ♣❛ss❡❞ s✐♥❝❡ ✐t ✇❛s 250 ♠✐❧❡s ❛✇❛② ❢r♦♠ ❛ ❝✐t②✳ ❍♦✇ ❢❛r ✐s ✐t ❢r♦♠ t❤❡ ❝✐t② ❛❢t❡r 3 ❤♦✉rs ♦❢ ❞r✐✈✐♥❣ t♦✇❛r❞s ✐t❄ ❊①❡r❝✐s❡ ✹✳✶✶ ❊①❡r❝✐s❡ ✹✳✺

❚❤❡ ❣r❛♣❤s ♦❢ t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♠♦✈✐♥❣ ♦❜❥❡❝t ❛r❡ s❤♦✇♥ ❜❡❧♦✇✳ ❉❡s❝r✐❜❡ ✇❤❛t ✐s ❤❛♣♣❡♥✐♥❣✳

▲❡t x r❡♣r❡s❡♥t t❤❡ t✐♠❡ ♣❛ss❡❞ s✐♥❝❡ t❤❡ ❝❛r ❧❡❢t t❤❡ ❝✐t②✳ ❚❤❡ t❛❜❧❡ ❜❡❧♦✇ t❡❧❧s ❢♦r ✇❤❛t ✈❛❧✉❡s ♦❢ x t❤❡ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ t❤❡ ❝❛r ❛r❡ ♣♦s✐t✐✈❡✱ ♥❡❣❛t✐✈❡✱ ♦r ③❡r♦✳ ▲❡t f (x) r❡♣r❡s❡♥t t❤❡

✹✳ ❊①❡r❝✐s❡s✿ ▼♦❞❡❧s

✹✺✷

❞✐st❛♥❝❡ ♦❢ t❤❡ ❝❛r ❢r♦♠ t❤❡ ❝✐t②✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢

f✳ x

✈❡❧♦❝✐t②

❛❝❝❡❧❡r❛t✐♦♥

0

0

+

1

+

2

0

−

3

−

−

−

❊①❡r❝✐s❡ ✹✳✶✷

t s❡❝♦♥❞s ❛❢t❡r ✐t ✐s f (t) = −16t2 + 8t + 6✳ ❊①♣❧❛✐♥ t❤❡ ♥✉♠❜❡rs −16, 8, 6✳

❚❤❡ ❤❡✐❣❤t ♦❢ t❤❡ ❜❛❧❧ ✭✐♥ ❢❡❡t✮ t❤r♦✇♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ♠❡❛♥✐♥❣ ♦❢

❊①❡r❝✐s❡ ✹✳✶✸

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦❢ ❛ ❝❛r ✐s ❣✐✈❡♥ ❜❡❧♦✇✳ P❧♦t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥t✐♥❣ t❤❡ ❞✐s✲ t❛♥❝❡ ♦❢ t❤❡ ❝❛r ❢r♦♠ t❤❡ st❛rt✐♥❣ ♣♦✐♥t✳

❊①❡r❝✐s❡ ✹✳✶✹

❙✉♣♣♦s❡ t❤❡ s♣❡❡❞ ♦❢ ❛ ❝❛r ✇❛s ❝❤❛♥❣✐♥❣ ❝♦♥t✐♥✉✲ ♦✉s❧② ❢♦❧❧♦✇✐♥❣ t❤❡ r✉❧❡

60 − t2

♣❡r ❤♦✉r✱ ✇❤❡r❡

t ✐s

t❤❡ ♥✉♠❜❡r ♦❢ ❤♦✉rs ♣❛ss❡❞ s✐♥❝❡ ♥♦♦♥✳ ❋✐♥❞ t❤❡ ❛✈❡r❛❣❡ s♣❡❡❞ ♦❢ t❤❡ ❝❛r ❜❡t✇❡❡♥

1

♣♠ ❛♥❞

3

♣♠✳

❊①❡r❝✐s❡ ✹✳✶✺

❙✉♣♣♦s❡ t❤❡ ❛❧t✐t✉❞❡✱ ✐♥ ♠❡t❡rs✱ ♦❢ ❛♥ ♦❜❥❡❝t ✐s ❣✐✈❡♥ ❜② t❤❡ ❢✉♥❝t✐♦♥

y = t2 + t, t ≥ 0, ✇❤❡r❡

t

✐s t✐♠❡✱ ✐♥ s❡❝✳ ❲❤❛t ✐s t❤❡ ✈❡❧♦❝✐t② ✇❤❡♥

t❤❡ ❛❧t✐t✉❞❡ ✐s

12

♠❡t❡rs❄

❊①❡r❝✐s❡ ✹✳✶✻

❋✐♥❞ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ♦❢ ❛ ❢r❡❡ ❢❛❧❧✐♥❣ ♦❜❥❡❝t ❢r♦♠ t❤✐s ❞❛t❛✿

✺✳ ❊①❡r❝✐s❡s✿ ■♥❢♦r♠❛t✐♦♥ ❢r♦♠ t❤❡ ❞❡r✐✈❛t✐✈❡s

✹✺✸

✺✳ ❊①❡r❝✐s❡s✿ ■♥❢♦r♠❛t✐♦♥ ❢r♦♠ t❤❡ ❞❡r✐✈❛t✐✈❡s

❊①❡r❝✐s❡ ✺✳✶

❊①❡r❝✐s❡ ✺✳✶✵

❋✐♥❞ ❛❧❧ ❧♦❝❛❧ ♠❛①✐♠❛ ❛♥❞ ♠✐♥✐♠❛ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

❋✐♥❞ t❤❡ ❧♦❝❛❧ ♠❛①✐♠✉♠ ❛♥❞ ♠✐♥✐♠✉♠ ♣♦✐♥ts ♦❢

3

f (x) = x3 − 3x✳

f (x) = x − 3x − 1✳

t❤❡ ❢✉♥❝t✐♦♥

❊①❡r❝✐s❡ ✺✳✷

❊①❡r❝✐s❡ ✺✳✶✶

✭❛✮ ❆♥❛❧②③❡ t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥

4

f (x) = x −2x f✳

2

✳ ✭❜✮ ❯s❡ ♣❛rt ✭❛✮ t♦ s❦❡t❝❤

❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥ ✇❤❛t ♣♦✐♥ts ✐s

f

❞♦❡s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢

✐ts ❣r❛♣❤ ♦❢

f

✐s ❣✐✈❡♥ ❜❡❧♦✇✳

❝♦♥t✐♥✉♦✉s❄

f

✭❛✮ ❆t

✭❜✮ ❆t ✇❤❛t ♣♦✐♥ts

❡①✐st❄

❊①❡r❝✐s❡ ✺✳✸

❙✉♣♣♦s❡ t❤❡ ❢✉♥❝t✐♦♥s t❤❛t ❢♦❧❧♦✇ ❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡✳ ✭❛✮ ❋✐♥✐s❤ t❤❡ st❛t❡♠❡♥t ✏■❢

h′ (x) = 0

❢♦r ❛❧❧

(a, b)✱ t❤❡♥✳✳✳✑✳ ✭❜✮ ❋✐♥✐s❤ t❤❡ st❛t❡♠❡♥t ′ f (x) = g ′ (x) ❢♦r ❛❧❧ x ✐♥ (a, b)✱ t❤❡♥✳✳✳✑✳

✐♥

x ✏■❢

❊①❡r❝✐s❡ ✺✳✹

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❏✉st✐❢② t❤❡ ❣r❛♣❤ ❜② st✉❞②✐♥❣ t❤❡

√

xe−x ✳ ❞❡r✐✈❛t✐✈❡s ♦❢ f ✳ f (x) =

❊①❡r❝✐s❡ ✺✳✶✷ ❊①❡r❝✐s❡ ✺✳✺

■♥❞✐❝❛t❡ ✇❤✐❝❤ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❜❡❧♦✇ ✐s ✭✶✮ ❙t❛t❡ ❘♦❧❧❡✬s ❚❤❡♦r❡♠ ❛♥❞ ✐❧❧✉str❛t❡ ✐t ✇✐t❤ ❛

tr✉❡ ♦r ❢❛❧s❡ ✭♥♦ ♣r♦♦❢ ♥❡❝❡ss❛r②✮✿

s❦❡t❝❤✳ ✭❜✮ ◗✉♦t❡ ❛♥❞ st❛t❡ t❤❡ t❤❡♦r❡♠✭s✮ ♥❡❝❡s✲ s❛r② t♦ ♣r♦✈❡ ✐t✳

✭❝✮ ❲❤❛t t❤❡♦r❡♠ ❢♦❧❧♦✇s ❢r♦♠

✐t❄

✶✳ ■❢ t❤❡ ❢✉♥❝t✐♦♥

f

✐s ✐♥❝r❡❛s✐♥❣✱ t❤❡♥ s♦ ✐s

f −1 .

✷✳ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❤❛s ❛♥ ❛s②♠♣t♦t❡✳ ✸✳ ■❢

❊①❡r❝✐s❡ ✺✳✻

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜❡❧♦✇✳ Pr♦✲ ✈✐❞❡ ❥✉st✐✜❝❛t✐♦♥ ❢♦r ❡❛❝❤ ❢❡❛t✉r❡ ♦❢ t❤❡ ❣r❛♣❤✿

x2 + 7x + 3 . f (x) = x

f ′ (c) = 0✱

t❤❡♥

❧♦❝❛❧ ♠✐♥✐♠✉♠ ♦❢

c ✐s f✳

❛ ❧♦❝❛❧ ♠❛①✐♠✉♠ ♦r ❛

✹✳ ■❢ ❛ ❢✉♥❝t✐♦♥ ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ t❤❡♥ ✐t ✐s ❝♦♥✲ t✐♥✉♦✉s✳ ✺✳ ■❢ t✇♦ ❢✉♥❝t✐♦♥s ❛r❡ ❡q✉❛❧✱ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ❛r❡ ❛❧s♦ ❡q✉❛❧✳

❊①❡r❝✐s❡ ✺✳✼

✻✳ ■❢

✭❛✮ ❙t❛t❡ t❤❡ ▼❡❛♥ ❱❛❧✉❡ ❚❤❡♦r❡♠✳ ✭❜✮ ❱❡r✐❢② t❤❛t t❤❡ ❢✉♥❝t✐♦♥

f (x) =

x x+2 s❛t✐s✜❡s t❤❡ ❤②♣♦t❤❡s❡s ♦❢

t❤❡ t❤❡♦r❡♠ ♦♥ t❤❡ ✐♥t❡r✈❛❧

[1, 4]✳

❊①❡r❝✐s❡ ✺✳✽

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x) = x4 − x2 ✳

Pr♦✈✐❞❡ ❥✉st✐✜❝❛t✐♦♥ ❢♦r ❡❛❝❤ ❢❡❛t✉r❡ ♦❢ t❤❡ ❣r❛♣❤✳

❊①❡r❝✐s❡ ✺✳✾

❋✐♥❞ ❣❧♦❜❛❧ ♠❛①✐♠❛ ❛♥❞ ♠✐♥✐♠❛ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱

f (x) = x3 − 3x

♦♥ t❤❡ ✐♥t❡r✈❛❧

[−2, 10]✳

t✇♦

❢✉♥❝t✐♦♥s

❛r❡

❡q✉❛❧✱

❞❡r✐✈❛t✐✈❡s ❛r❡ ❛❧s♦ ❡q✉❛❧✳

t❤❡✐r

❛♥t✐✲

✻✳ ❊①❡r❝✐s❡s✿ ❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s

✹✺✹

✻✳ ❊①❡r❝✐s❡s✿ ❈♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s

❊①❡r❝✐s❡ ✻✳✶

❊①❡r❝✐s❡ ✻✳✶✷

❙✉♣♣♦s❡ f (1) = 3 ❛♥❞ f ′ (1) = 2✳ ❯s❡ t❤✐s ✐♥❢♦r♠❛✲ t✐♦♥ t♦ ✜❧❧ ✐♥ t❤❡ ❜❧❛♥❦s✿

■s ✐t ♣♦ss✐❜❧❡ t❤❛t ❜♦t❤ F (x) ❛♥❞ F (2x) ❛r❡ ❜♦t❤ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ s♦♠❡ ♥♦♥✲③❡r♦ ❢✉♥❝t✐♦♥ f ❄

′ f −1 ( ) =

❊①❡r❝✐s❡ ✻✳✶✸

❊✈❛❧✉❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = x2 ex ✳

❊①❡r❝✐s❡ ✻✳✷

❉✐✛❡r❡♥t✐❛t❡ t❤✐s✿

❊①❡r❝✐s❡ ✻✳✶✹

❋✐♥❞ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ h(x) = x2 + x + 1. ❲❤❛t ❞♦❡s ✐t t❡❧❧ ②♦✉ ❛❜♦✉t t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f ?

g(t) = t cos t sin t . ❊①❡r❝✐s❡ ✻✳✸

❉✐✛❡r❡♥t✐❛t❡✿

❊①❡r❝✐s❡ ✻✳✶✺

ln(sin x) . x

❋✐♥❞ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ h(x) = 2xπ ✳ ❊①❡r❝✐s❡ ✻✳✶✻

❊①❡r❝✐s❡ ✻✳✹

❈♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = ex

2 +3x

✳

❊①❡r❝✐s❡ ✻✳✶✼

❊①❡r❝✐s❡ ✻✳✺

❊✈❛❧✉❛t❡

d dx

 sin x · ex+1 ✳

❊①❡r❝✐s❡ ✻✳✻

❊✈❛❧✉❛t❡

d dx

 t

cos t + e ✳ ❍✐♥t✿ ✇❛t❝❤ t❤❡ ✈❛r✐❛❜❧❡s✳

❊①❡r❝✐s❡ ✻✳✼

❊✈❛❧✉❛t❡

dy dx

❈♦♠♣✉t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = ln(3x + 2)✳

❢♦r y = sin e2x .

❋✐♥❞ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ h(x) = xex ✳ ❊①❡r❝✐s❡ ✻✳✶✽

❋✐♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s✿ ✭❛✮ 3xe + eπ ✱ ✭❜✮ 7 ln x + (1/x) − ln 2. ❊①❡r❝✐s❡ ✻✳✶✾

❊✈❛❧✉❛t❡

dy dx

❢♦r y=

√

ex .

❊①❡r❝✐s❡ ✻✳✽

❊①❡r❝✐s❡ ✻✳✷✵

❊✈❛❧✉❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ♦❢ f (x) = xesin x .

❋✐♥❞ t❤❡ s❧♦♣❡s ♦❢ t❤❡ t❛♥❣❡♥t ❧✐♥❡s t♦ t❤❡ ❡❧❧✐♣s❡ x2 + 2y 2 = 1 ❛t t❤❡ ♣♦✐♥ts ✇❤❡r❡ ✐t ❝r♦ss❡s t❤❡ ❞✐❛❣♦♥❛❧ ❧✐♥❡ y = x✳

❊①❡r❝✐s❡ ✻✳✾

❙✉♣♣♦s❡ f ′ (1) = 2✱ g ′ (2) = 3✱ ❛♥❞ h′ (1) = 6✱ ✇❤❡r❡ h = g ◦ f ✳ ❲❤❛t ✐s f (1)❄

❊①❡r❝✐s❡ ✻✳✷✶

❊✈❛❧✉❛t❡

dy dx

❢♦r y = sin cos(−x)✳

❊①❡r❝✐s❡ ✻✳✶✵

■s ✐t ♣♦ss✐❜❧❡ t❤❛t ❜♦t❤ F (x) ❛♥❞ F (2x) ❛r❡ ❜♦t❤ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ s♦♠❡ ❢✉♥❝t✐♦♥ f ❄ ❊①❡r❝✐s❡ ✻✳✶✶

■s sin x + 3x ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ cos x2 ❄

❊①❡r❝✐s❡ ✻✳✷✷

❙✉♣♣♦s❡ x sin y + y 2 = x✳ ❋✐♥❞

dy dx ✳

✼✳ ❊①❡r❝✐s❡s✿ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

✹✺✺

✼✳ ❊①❡r❝✐s❡s✿ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

❊①❡r❝✐s❡ ✼✳✶

❊①❡r❝✐s❡ ✼✳✽

✭❛✮ ❙t❛t❡ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✳ ✭❜✮ ❯s❡ ♣❛rt ✭❛✮ t♦ ❡✈❛❧✉❛t❡

❙✉♣♣♦s❡

Z

Z

1

x sin dx . 3 −1

x

0

f dx = 0, 0

Z

2

f dx = 2 . 1

3

f dx, 1

Z

Z

1

(f (x) + 3) dx, 0

4

f dx . 2

2

0

❊①❡r❝✐s❡ ✼✳✾

❙✉♣♣♦s❡ ❛ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❜②✿

✭❛✮ Z▼❛❦❡ ❛ s❦❡t❝❤ ♦❢ t❤❡ ❧❡❢t✲❡♥❞ ❘✐❡♠❛♥♥ s✉♠s ❢♦r

4

et dt .

❊①❡r❝✐s❡ ✼✳✸

1√

0

Z

✭❛✮ ❙t❛t❡ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s✳ ✭❜✮ ❯s❡ ♣❛rt ✭❛✮ t♦ ❡✈❛❧✉❛t❡ Z

f dx = 2,

Z

❋✐♥❞

❊①❡r❝✐s❡ ✼✳✷

d dx

1

x dx ✇✐t❤ n = 4 ✐♥t❡r✈❛❧s✳ ✭❜✮ ❙t❛t❡ t❤❡

❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✳

F (x) =

●✐✈❡♥ f (x) = x2 + 1, ✇r✐t❡ ✭❜✉t ❞♦ ♥♦t ❡✈❛❧✉❛t❡✮ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❢♦r t❤❡ ✐♥t❡❣r❛❧ ♦❢ f ❢r♦♠ −1 t♦ 2 ✇✐t❤ n = 6 ❛♥❞ ❧❡❢t ❡♥❞s ❛s s❛♠♣❧❡ ♣♦✐♥ts✳ ▼❛❦❡ ❛ s❦❡t❝❤✳ ❊①❡r❝✐s❡ ✼✳✺

Pr♦✈✐❞❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ ✈✐❛ ✐ts ❘✐❡♠❛♥♥ s✉♠s✳ ▼❛❦❡ ❛ s❦❡t❝❤✳

x

f dx. 2

❋✐♥❞✱ ✐♥ t❡r♠s ♦❢ F ✱ t❤❡ ❢♦❧❧♦✇✐♥❣✿ Z

❊①❡r❝✐s❡ ✼✳✹

Z

4

f dx, 0

Z

2

f dx, 1

Z

−1 0

f dx,

Z

2 1

(f (x) − 1) dx .

❊①❡r❝✐s❡ ✼✳✶✵

❊✈❛❧✉❛t❡ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ♦❢ f ❜❡❧♦✇ ♦♥ t❤❡ ✐♥✲ t❡r✈❛❧ [−1, 1.5] ✇✐t❤ n = 5✳ ❲❤❛t ❛r❡ ✐ts s❛♠♣❧❡ ♣♦✐♥ts❄ ❲❤❛t ❞♦❡s ✐t ❡st✐♠❛t❡❄

❊①❡r❝✐s❡ ✼✳✻

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s ✐♥❝❧✉❞❡s Z b

t❤❡ ❢♦r♠✉❧❛

a

f (x) dx = F (b) − F (a)✳ ✭❛✮ ❙t❛t❡

t❤❡ ✇❤♦❧❡ t❤❡♦r❡♠✳ ✭❜✮ Pr♦✈✐❞❡ ❞❡✜♥✐t✐♦♥s ♦❢ t❤❡ ✐t❡♠s ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ ❢♦r♠✉❧❛✳ ❊①❡r❝✐s❡ ✼✳✼

✭❛✮ ❙t❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧ Z

b

a

f (x) dx ❛♥❞ ✐❧❧✉str❛t❡ t❤❡ ❝♦♥str✉❝t✐♦♥ ✇✐t❤ ❛

s❦❡t❝❤✳ ✭❜✮ ❯s❡ t❤❡ ❞❡✜♥✐t✐♦♥ t♦ ❥✉st✐❢② t❤❛t Z

b

cf (x) dx = c a

❢♦r ❛ ❝♦♥st❛♥t c✳

Z

b

f (x) dx a

❊①❡r❝✐s❡ ✼✳✶✶

❲r✐t❡ ✭❞♦♥✬tZ❡✈❛❧✉❛t❡✮ t❤❡ ❧❡❢t✲❡♥❞ ❘✐❡♠❛♥♥ s✉♠ ♦❢ t❤❡ ✐♥t❡❣r❛❧

5

0

f (x) dx ❢♦r ❢✉♥❝t✐♦♥ f s❤♦✇♥ ❜❡❧♦✇

✇✐t❤ n = 5 ✐♥t❡r✈❛❧s✳

✼✳ ❊①❡r❝✐s❡s✿ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧

✹✺✻

•

Z

u dv = uv...

• u = cos t =⇒ du = ... ❊①❡r❝✐s❡ ✼✳✶✼

❙✉♣♣♦s❡ t❤❛t F ✐s ❛♥ ❛♥t✐❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❞✐✛❡r❡♥✲ t✐❛❜❧❡ ❢✉♥❝t✐♦♥ f ✳ ■❢ F ✐s ✐♥❝r❡❛s✐♥❣ ♦♥ [a, b]✱ ✇❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t f ❄

❊①❡r❝✐s❡ ✼✳✶✷

❲r✐t❡ t❤❡ ❢♦r♠✉❧❛ ❛♥❞ ✐❧❧✉str❛t❡ ✇✐t❤ ❛Zs❦❡t❝❤ t❤❡ 3 f (x) dx ❧❡❢t✲❡♥❞ ❘✐❡♠❛♥♥ s✉♠ L4 ♦❢ t❤❡ ✐♥t❡❣r❛❧ 1 ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ♣❧♦tt❡❞ ❛❜♦✈❡✳ ❊①❡r❝✐s❡ ✼✳✶✸

❲r✐t❡ t❤❡ ♠✐❞✲♣♦✐♥t Z ❘✐❡♠❛♥♥ s✉♠ t❤❛t ❛♣♣r♦①✐✲ ♠❛t❡s t❤❡ ✐♥t❡❣r❛❧

1

0

sin x dx ✇✐t❤✐♥ .01✳

❊①❡r❝✐s❡ ✼✳✶✹

❙❡t ✉♣ t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❢♦r t❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s R ❛s t❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t✇♦ ❝✉r✈❡s✱ ♣r♦✈✐❞❡ ❛♥ ✐❧❧✉str❛t✐♦♥ ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ❢♦r♠✉❧❛✳ ❊✈❛❧✉❛t❡ ❢♦r ❡①tr❛ ✺ ♣♦✐♥ts✳ ❊①❡r❝✐s❡ ✼✳✶✺

▲❡t I =

Z

8 2

f dx✳ ✭❛✮ ❯s❡ t❤❡ ❣r❛♣❤ ♦❢ y = f (x)

❜❡❧♦✇ t♦ ❡st✐♠❛t❡ L4 , M4 , R4 ✳ ✭❜✮ ❈♦♠♣❛r❡ t❤❡♠ t♦ I ✳

❊①❡r❝✐s❡ ✼✳✶✻

❈♦♠♣❧❡t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts✿ • (f (x) · x2 )′ = f ′ (x) · x2 + ... Z • x−1 dx = ... •

Z

f ′ (x) dx = ...

✽✳ ❊①❡r❝✐s❡s✿ ■♥t❡❣r❛t✐♦♥

✹✺✼

✽✳ ❊①❡r❝✐s❡s✿ ■♥t❡❣r❛t✐♦♥

❊①❡r❝✐s❡ ✽✳✶

❊①❡r❝✐s❡ ✽✳✾

❊①❡❝✉t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜st✐t✉t✐♦♥ ✐♥ t❤❡ ✐♥t❡❣r❛❧ ✭❞♦♥✬t ❡✈❛❧✉❛t❡ t❤❡ r❡s✉❧t✐♥❣ ✐♥t❡❣r❛❧✮✿

❊✈❛❧✉❛t❡ t❤❡ ✐♥t❡❣r❛❧ ❜② s✉❜st✐t✉t✐♦♥

Z

√

cos x + sin x dx,

Z

u = sin x .

2

xex dx .

❊①❡r❝✐s❡ ✽✳✶✵ ❊①❡r❝✐s❡ ✽✳✷

❋✐♥❞ ❛❧❧ ❛♥t✐❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥✿

❙✉♣♣♦s❡ s(t) r❡♣r❡s❡♥ts t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ❛t t✐♠❡ t ❛♥❞ v(t) ✐ts ✈❡❧♦❝✐t②✳ ■❢ v(t) = sin t − cos t ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥ ✐s s(0) = 0, ✜♥❞ t❤❡ ♣♦s✐t✐♦♥

❊①❡r❝✐s❡ ✽✳✶✶

s(1).

f (x) = e−x .

❋✐♥❞ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡ F ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x) = 3x2 − 1 s❛t✐s❢②✐♥❣ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ F (1) = 0✳

❊①❡r❝✐s❡ ✽✳✸ ❊①❡r❝✐s❡ ✽✳✶✷

❊✈❛❧✉❛t❡

Z

❊✈❛❧✉❛t❡ t❤❡ ✐♥t❡❣r❛❧

e3x dx .

Z

❊①❡r❝✐s❡ ✽✳✹

❊✈❛❧✉❛t❡

Z

2

x

(e +

√

1

x3 dx .

0

❊①❡r❝✐s❡ ✽✳✶✸

x+x

−1

) dx .

1

❊✈❛❧✉❛t❡✿

Z

x dx −

Z

x2 dx .

−2

Z

x−2 dx .

2

❊①❡r❝✐s❡ ✽✳✺ ❊①❡r❝✐s❡ ✽✳✶✹

❊✈❛❧✉❛t❡

Z

2

ex 2x dx .

❊①❡r❝✐s❡ ✽✳✻

❊✈❛❧✉❛t❡

❊✈❛❧✉❛t❡✿

Z

x

dx −

❊①❡r❝✐s❡ ✽✳✶✺

■♥t❡❣r❛t❡ ❜② ♣❛rts✿

Z

2x sin 5x dx .

Z

3xe−x dx .

❊①❡r❝✐s❡ ✽✳✼

❊✈❛❧✉❛t❡

❊①❡r❝✐s❡ ✽✳✶✻

Z

3

❯s❡ t❤❡ t❛❜❧❡ ♦❢ ✐♥t❡❣r❛❧s t♦ ❡✈❛❧✉❛t❡✿

et+1 dx .

Z

1

❍✐♥t✿ ✇❛t❝❤ t❤❡ ✈❛r✐❛❜❧❡s✳ ❊①❡r❝✐s❡ ✽✳✶✼

❊①❡r❝✐s❡ ✽✳✽

❈❛❧❝✉❧❛t❡✿

sin−1 2x dx .

Z 

e

sin x2 +77

′

❊✈❛❧✉❛t❡✿ dx .

Z

1 0

1 dx . 2x

✽✳ ❊①❡r❝✐s❡s✿ ■♥t❡❣r❛t✐♦♥

✹✺✽

❊①❡r❝✐s❡ ✽✳✶✽ ❯s❡ s✉❜st✐t✉t✐♦♥ t♦ ❡✈❛❧✉❛t❡ t❤❡ ✐♥t❡❣r❛❧✿

Z

π

sin x cos2 x dx .

0

❊①❡r❝✐s❡ ✽✳✶✾ ■♥t❡❣r❛t❡ ❜② ♣❛rts✿

Z

x(ln x)2 dx .

❊①❡r❝✐s❡ ✽✳✷✵ ❯s❡ s✉❜st✐t✉t✐♦♥ t♦ ❡✈❛❧✉❛t❡ t❤❡ ✐♥t❡❣r❛❧✿

Z

1

√

0

1 dx . 4 − x2

❊①❡r❝✐s❡ ✽✳✷✶ ❯s❡ t❤❡ t❛❜❧❡ ♦❢ ✐♥t❡❣r❛❧s t♦ ❡✈❛❧✉❛t❡✿

Z

x2 (

p

x2 − 4 −

p x2 + 9) dx .

❊①❡r❝✐s❡ ✽✳✷✷ ❊✈❛❧✉❛t❡

Z

x sin x dx .

❊①❡r❝✐s❡ ✽✳✷✸ ❯s❡ s✉❜st✐t✉t✐♦♥

u = 1 + x2 t♦ ❡✈❛❧✉❛t❡ t❤❡ ✐♥t❡❣r❛❧ Z p 1 + x2 x5 dx .

❊①❡r❝✐s❡ ✽✳✷✹ ❯s❡ s✉❜st✐t✉t✐♦♥ t♦ ❡✈❛❧✉❛t❡ t❤❡ ✐♥t❡❣r❛❧✿

Z

π

sin x cos2 x dx .

0

❊①❡r❝✐s❡ ✽✳✷✺ ❊✈❛❧✉❛t❡ t❤❡ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧✿

Z

∞ 1

1 dx . 2x

❊①❡r❝✐s❡ ✽✳✷✻ ❋✐♥❞ t❤❡ ❛♥t✐❞❡r✐✈❛t✐✈❡

x

e +x

F

f (x) = F (0) = 1✳

♦❢ t❤❡ ❢✉♥❝t✐♦♥

s❛t✐s❢②✐♥❣ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥

✾✳ ❊①❡r❝✐s❡s✿ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ✐♥t❡❣r❛❧s

✹✺✾

✾✳ ❊①❡r❝✐s❡s✿ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ✐♥t❡❣r❛❧s

❊①❡r❝✐s❡ ✾✳✶

❊①❡r❝✐s❡ ✾✳✾

❚❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ ❣r❛♣❤s ♦❢

y=

0, ❛♥❞ x = 1 ✐s r❡✈♦❧✈❡❞ ❛❜♦✉t t❤❡ x✲❛①✐s✳

√

x, y =

❋✐♥❞ t❤❡

x✲❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❝❡♥t❡r 2 3 ❜❡t✇❡❡♥ y = x ❛♥❞ y = x ✳

❋✐♥❞ t❤❡ r❡❣✐♦♥

♦❢ ♠❛ss ♦❢ t❤❡

s✉r❢❛❝❡ ❛r❡❛ ♦❢ t❤❡ s♦❧✐❞ ❣❡♥❡r❛t❡❞✳ ❊①❡r❝✐s❡ ✾✳✶✵

❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ r✐❣❤t ❝✐r❝✉❧❛r ❝♦♥❡ ♦❢ r❛❞✐✉s

❊①❡r❝✐s❡ ✾✳✷

❆ ❝❤♦r❞ ♦❢ ❛ ❝✐r❝❧❡ ✐s ❛ str❛✐❣❤t ❧✐♥❡ s❡❣♠❡♥t ✇❤♦s❡

R

❛♥❞ ❤❡✐❣❤t

h

❜② ❛♥② ♠❡t❤♦❞ ②♦✉ ❧✐❦❡✳

❡♥❞✲♣♦✐♥ts ❧✐❡ ♦♥ t❤❡ ❝✐r❝❧❡✳ ❋✐♥❞ t❤❡ ❛✈❡r❛❣❡ ❧❡♥❣t❤ ♦❢ ❛ ❝❤♦r❞ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❞✐❛♠❡t❡r✳

❲❤❛t

❛❜♦✉t ♣❛r❛❧❧❡❧❄

❊①❡r❝✐s❡ ✾✳✶✶

❈♦♠♣✉t❡ t❤❡ ❛✈❡r❛❣❡ ❛r❡❛ ♦❢ t❤❡ ❝r♦ss s❡❝t✐♦♥ ♦❢ t❤❡ s♣❤❡r❡ ♦❢ r❛❞✐✉s

1✳

❊①❡r❝✐s❡ ✾✳✸

❋✐♥❞ t❤❡ ❛✈❡r❛❣❡ ❧❡♥❣t❤ ♦❢ ❛ s❡❣♠❡♥t ✐♥ ❛ sq✉❛r❡ ♣❛r❛❧❧❡❧ t♦ ✭❛✮ t❤❡ ❜❛s❡✱ ✭❜✮ t❤❡ ❞✐❛❣♦♥❛❧✳

❊①❡r❝✐s❡ ✾✳✶✷

❋✐♥❞ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ t❤❡ r❡❣✐♦♥ ❜❡❧♦✇ ❢♦r

y = 2x

0 ≤ x ≤ 1✳

❊①❡r❝✐s❡ ✾✳✹

❋✐♥❞ ✭❜② ✐♥t❡❣r❛t✐♦♥✮ t❤❡ ❧❡♥❣t❤ ♦❢ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s

❊①❡r❝✐s❡ ✾✳✶✸

❚❤❡ ✈♦❧✉♠❡ ♦❢ ❛ s♦❧✐❞ ✐s t❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤❡ ❛r❡❛s

r✳

♦❢ ✐ts ❝r♦ss✲s❡❝t✐♦♥s✳ ❊①♣❧❛✐♥ ❛♥❞ ❥✉st✐❢② ✉s✐♥❣ ❘✐❡✲ ♠❛♥♥ s✉♠s✳

❊①❡r❝✐s❡ ✾✳✺

❋✐♥❞ t❤❡ ❛r❡❛ ❡♥❝❧♦s❡❞ ❜② t❤❡ ❝✉r✈❡s ❜❡❧♦✇✿

❊①❡r❝✐s❡ ✾✳✶✹

❚❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ ❣r❛♣❤s ♦❢

y = x2 +

1, y = 0, x = 0 ❛♥❞ x = 1 ✐s r❡✈♦❧✈❡❞ ❛❜♦✉t t❤❡ x✲❛①✐s✳ ❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ❛r❡❛ ♦❢ t❤❡ s♦❧✐❞ ❣❡♥❡r❛t❡❞✳ ❊①❡r❝✐s❡ ✾✳✶✺

❚❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ ❣r❛♣❤s ♦❢

y = x2 +

1, y = 0, x = 0, ❛♥❞ x = 1 ✐s r❡✈♦❧✈❡❞ ❛❜♦✉t t❤❡ y ✲❛①✐s✳ ❋✐♥❞ t❤❡ ✈♦❧✉♠❡ ❛r❡❛ ♦❢ t❤❡ s♦❧✐❞ ❣❡♥❡r❛t❡❞✳ ❊①❡r❝✐s❡ ✾✳✻

❊①❡r❝✐s❡ ✾✳✶✻

❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② ❛♥❞

y = 3✳

2

y = x −1

❆♥ ❛q✉❛r✐✉♠

2

♠ ❧♦♥❣✱

1

♠ ✇✐❞❡✱ ❛♥❞

1

♠ ❞❡❡♣ ✐s

❢✉❧❧ ♦❢ ✇❛t❡r✳ ❋✐♥❞ t❤❡ ✇♦r❦ ♥❡❡❞❡❞ t♦ ♣✉♠♣ ❤❛❧❢ ♦❢ t❤❡ ✇❛t❡r ♦✉t ♦❢ t❤❡ ❛q✉❛r✐✉♠ ✭t❤❡ ❞❡♥s✐t② ♦❢ ✇❛t❡r ✐s

❊①❡r❝✐s❡ ✾✳✼

❙✉♣♣♦s❡

f

f

✐s ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥✳ ✭❛✮ ❙❤♦✇ t❤❛t Z a

f dx = 0✳

✐s ❛❧s♦ ♦❞❞ t❤❡♥

−a

r❡❧❛t❡❞ ❢♦r♠✉❧❛ ❢♦r ❛♥ ❡✈❡♥

✭❜✮ ❙✉❣❣❡st ❛

f✳

1000

3

❦❣✴♠ ✮✳

❊①❡r❝✐s❡ ✾✳✶✼

❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ s✉r❢❛❝❡ ♦❢ r❡✈♦❧✉t✐♦♥ ❛r♦✉♥❞ t❤❡

x✲❛①✐s

♦❜t❛✐♥❡❞ ❢r♦♠

y=

√

x, 4 ≤ x ≤ 9✳

❊①❡r❝✐s❡ ✾✳✽

❊①❡r❝✐s❡ ✾✳✶✽

❋✐♥❞ t❤❡ ❝❡♥tr♦✐❞ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡

❋✐♥❞ t❤❡ ❝❡♥tr♦✐❞ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡

❝✉r✈❡s

2

y = x , y = 1✳

❝✉r✈❡s

y = 4 − x 2 , y = x + 2✳

✾✳ ❊①❡r❝✐s❡s✿ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ✐♥t❡❣r❛❧s

✹✻✵

❊①❡r❝✐s❡ ✾✳✶✾ ❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② ❛♥❞

y = 3✳

y = x2 − 1

❊①❡r❝✐s❡ ✾✳✷✵ ❋✐♥❞

t❤❡

f (x) = e

x

❛r❡❛

✉♥❞❡r

t❤❡

❢r♦♠

x = −1

t♦

❣r❛♣❤

♦❢

t❤❡

❢✉♥❝t✐♦♥

x = 1✳

❊①❡r❝✐s❡ ✾✳✷✶ ❋✐♥❞

2

t❤❡

2x − 3

❛✈❡r❛❣❡

✈❛❧✉❡

♦♥ t❤❡ ✐♥t❡r✈❛❧

♦❢

t❤❡

f (x) =

❢✉♥❝t✐♦♥

[1, 3]✳

❊①❡r❝✐s❡ ✾✳✷✷ ❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡

x✲❛①✐s✱

❛♥❞ t❤❡ ❧✐♥❡s

x=1

❛♥❞

y =

x = 4✳

√

x✱

✶✵✳ ❊①❡r❝✐s❡s✿ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ r❡❧❛t❡❞

✹✻✶

✶✵✳ ❊①❡r❝✐s❡s✿ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ r❡❧❛t❡❞

❊①❡r❝✐s❡ ✶✵✳✶

❊①❡r❝✐s❡ ✶✵✳✼

❉❡s❝r✐❜❡ t❤❡ ♠♦t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ✇✐t❤ ♣♦s✐t✐♦♥

✭✶✮ ❙❦❡t❝❤ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

(x, y)✱

sin 2t✳

✇❤❡r❡

❣❧❡ ♦❢ t❤✐s ✐♥t❡rs❡❝t✐♦♥✳

x = 2 + t cos t, y = 1 + t sin t, ❛s

t

✈❛r✐❡s ✇✐t❤✐♥

x = cos t, y =

✭✷✮ ❚❤❡ ❝✉r✈❡ ✐♥t❡rs❡❝ts ✐ts❡❧❢✳ ❋✐♥❞ t❤❡ ❛♥✲

[0, ∞)✳

❊①❡r❝✐s❡ ✶✵✳✽

❋✐♥❞ ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ s♣✐r❛❧ ❝♦♥✈❡r❣✐♥❣ t♦ t❤❡ ♦r✐❣✐♥ ❛s ❜❡❧♦✇✿

❊①❡r❝✐s❡ ✶✵✳✷

❙✉♣♣♦s❡ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐s ❣✐✈❡♥ ❜②

x = cos 3t, y = 2 sin t. ❙❡t ✉♣✱ ❜✉t ❞♦ ♥♦t ❡✈❛❧✉❛t❡✱ t❤❡ ✐♥t❡❣r❛❧s t❤❛t r❡♣✲ r❡s❡♥t ✭❛✮ t❤❡ ❛r❝✲❧❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡✱ ✭❜✮ t❤❡ ❛r❡❛ ♦❢ t❤❡ s✉r❢❛❝❡ ♦❜t❛✐♥❡❞ ❜② r♦t❛t✐♥❣ t❤❡ ❝✉r✈❡ ❛❜♦✉t t❤❡

x✲❛①✐s✳

❊①❡r❝✐s❡ ✶✵✳✸

❙✉♣♣♦s❡ ❝✉r✈❡

C

✐s t❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥

y = f (x)✳

✭❛✮ ❋✐♥❞ ❛ ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ C ✳ ✭❜✮ ❋✐♥❞ ❛ ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢

C

t❤❛t ❣♦❡s ❢r♦♠

r✐❣❤t t♦ ❧❡❢t✳

❊①❡r❝✐s❡ ✶✵✳✾

P❧♦t t❤✐s ❡♥t✐r❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿

x = sin t, y =

cos 2t✳ ❊①❡r❝✐s❡ ✶✵✳✹

❋✐♥❞ ❛❧❧ ♣♦✐♥ts ♦♥ t❤❡ ❝✉r✈❡

x = cos 3t, y = 2 sin t ✇❤❡r❡ t❤❡ t❛♥❣❡♥t ✐s ❡✐t❤❡r ❤♦r✐③♦♥t❛❧ ♦r ✈❡rt✐❝❛❧✳

❊①❡r❝✐s❡ ✶✵✳✶✵

❋✐♥❞ ❛ ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ❜❡❧♦✇✱ ❛ s♣✐r❛❧ ✇r❛♣♣✐♥❣ ❛r♦✉♥❞ ❛ ❝✐r❝❧❡✳ ❲❤❛t ❛❜♦✉t ♦♥❡ t❤❛t ✐s ✇r❛♣♣✐♥❣ ❢r♦♠ t❤❡ ✐♥s✐❞❡❄ ✭♥♦ ♣r♦♦❢ ♥❡❝❡ss❛r②✮✿

❊①❡r❝✐s❡ ✶✵✳✺

❙❦❡t❝❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿

x = | cos t|, y = | sin t|, −∞ < t < +∞. ❉❡s❝r✐❜❡ t❤❡ ❝✉r✈❡ ❛♥❞ t❤❡ ♠♦t✐♦♥✳

❊①❡r❝✐s❡ ✶✵✳✻

❙❦❡t❝❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✿

1 • x(t) = , y(t) = sin t, t > 0 t • x = cos t, y = 2

❊①❡r❝✐s❡ ✶✵✳✶✶

●✐✈❡♥ ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ 2

• x = 1/t, y = 1/t , t > 0

x = sin t, y = t2 ✳

t❤❡ ❧✐♥❡✭s✮ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ ❛t t❤❡ ♦r✐❣✐♥✳

❋✐♥❞

✶✵✳ ❊①❡r❝✐s❡s✿ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ r❡❧❛t❡❞

✹✻✷

❊①❡r❝✐s❡ ✶✵✳✶✷

❊①❡r❝✐s❡ ✶✵✳✷✷

❋✐♥❞ ❛ ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ t❤❛t ❧♦♦❦s ❧✐❦❡ t❤❡ ✜❣✉r❡ ❡✐❣❤t ♦r ❛ ✢♦✇❡r ✭♥♦ ♣r♦♦❢ ♥❡❝✲ ❡ss❛r②✮✳

❲❤❛t ❝❛♥ ②♦✉ s❛② ❛❜♦✉t t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ t❤❡ r♦♦ts ♦❢ t❤❡s❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s❄

❊①❡r❝✐s❡ ✶✵✳✶✸

❙❦❡t❝❤ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡✿ x = t2 − 1, y = 2t2 + 3 . ❊①❡r❝✐s❡ ✶✵✳✶✹

❙✉♣♣♦s❡ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ r❡♣r❡✲ s❡♥ts t❤❡ ♠♦t✐♦♥ ♦❢ ❛♥ ♦❜❥❡❝t ♦♥ t❤❡ ♣❧❛♥❡✿ x = 3t − 1, y = t2 − 1 .

✭❛✮ ❲❤❡♥ ❞♦❡s t❤❡ ♦❜❥❡❝t ❝r♦ss t❤❡ x✲❛①✐s❄ ✭❜✮ ❲❤❡♥ ❞♦❡s t❤❡ ♦❜❥❡❝t ❝r♦ss t❤❡ y ✲❛①✐s❄ ❊①❡r❝✐s❡ ✶✵✳✶✺

❘❡♣r❡s❡♥t ❛s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ t❤❡ r♦t❛t✐♦♥ ♦❢ ❛ r♦❞ ♦❢ ❧❡♥❣t❤ 2 t❤❛t ♠❛❦❡s ♦♥❡ ❢✉❧❧ t✉r♥ ❡✈❡r② 3 s❡❝♦♥❞s✳ ❊①❡r❝✐s❡ ✶✵✳✶✻

❖♥❡ ❝✐r❝❧❡ ✐s ❝❡♥t❡r❡❞ ❛t (0, 0) ❛♥❞ ❤❛s r❛❞✐✉s 1✳ ❚❤❡ s❡❝♦♥❞ ✐s ❝❡♥t❡r❡❞ ❛t (3, 3)✳ ❲❤❛t ✐s t❤❡ r❛✲ ❞✐✉s ♦❢ t❤❡ s❡❝♦♥❞ ✐❢ t❤❡ t✇♦ ❝✐r❝❧❡s t♦✉❝❤❄

❊①❡r❝✐s❡ ✶✵✳✷✸

P❧♦t t❤❡ ❝✉r✈❡ r = 2 cos(3θ)✳ ❋♦r ✺ ❡①tr❛ ♣♦✐♥ts✱ ✜♥❞ t❤❡ ❧✐♥❡✭s✮ t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡✳ ❊①❡r❝✐s❡ ✶✵✳✷✹

✭❛✮ P❧♦t t❤❡s❡ ♣♦✐♥ts ✐♥ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s✿ (r, θ) = (0, 1), (1, 0), (1, π), (2, 3π)✳ ✭❜✮ ❙❦❡t❝❤ t❤❡s❡ t❤r❡❡ ♣♦❧❛r ❝✉r✈❡s✿ r = 1, θ = 0, r = θ✳ ❊①❡r❝✐s❡ ✶✵✳✷✺

❋✐♥❞ ❛ ♣♦❧❛r r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝✉r✈❡ s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ❜❡❧♦✇✱ ❛ s♣✐r❛❧ ✇r❛♣♣✐♥❣ ❛r♦✉♥❞ ❛ ❝✐r❝❧❡✳ ❲❤❛t ❛❜♦✉t ♦♥❡ t❤❛t ✐s ✇r❛♣♣✐♥❣ ❢r♦♠ t❤❡ ✐♥s✐❞❡❄ ✭♥♦ ♣r♦♦❢ ♥❡❝❡ss❛r②✮✿

❊①❡r❝✐s❡ ✶✵✳✶✼

❘❡♣r❡s❡♥t ✐♥ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s t❤❡s❡ ♣♦✐♥ts ❣✐✈❡♥ ❜② t❤❡✐r ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s✿ ✭❛✮ (1, 2)❀ ✭❜✮ (−1, −1)❀ ✭❝✮ (0, 0)✳ ❊①❡r❝✐s❡ ✶✵✳✶✽

❘❡♣r❡s❡♥t ✐♥ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s t❤❡s❡ ♣♦✐♥ts ❣✐✈❡♥ ❜② t❤❡✐r ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s✿ ✭❛✮ θ = 0, r = −1❀ ✭❜✮ θ = π/4, r = 2❀ ✭❝✮ θ = 1, r = 0✳ ❊①❡r❝✐s❡ ✶✵✳✶✾

✭❛✮ ❘❡♣r❡s❡♥t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣❧❡① ♥✉♠❜❡r ✐♥ t❤❡ st❛♥❞❛r❞ ❢♦r♠✿ (2 + 3i)(−1 + 2i)✳ ■♥❞✐❝❛t❡ t❤❡ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts✳ ✭❜✮ ❋✐♥❞ ✐ts ♠♦❞✉❧❡ ❛♥❞ ❛r✲ ❣✉♠❡♥t✳ ❊①❡r❝✐s❡ ✶✵✳✷✵

❙✐♠♣❧✐❢② (1 + i)2 ✳

❊①❡r❝✐s❡ ✶✵✳✷✻

■♥❞✐❝❛t❡ ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛r❡ tr✉❡ ♦r ❢❛❧s❡✿ ✶✳ ■♥

♣♦❧❛r

❝♦♦r❞✐♥❛t❡s✱ (1, π/2) (−1, −π/2) r❡♣r❡s❡♥t t❤❡ s❛♠❡ ♣♦✐♥t✳

❛♥❞

✷✳ ❚❤❡ ❝✉r✈❡ r = 3 + cos θ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♦r✐❣✐♥✳ ✸✳ ❚❤❡ ❝✉r✈❡ r = cos 2θ ✐s ❝❧♦s❡❞✳

❊①❡r❝✐s❡ ✶✵✳✷✶

✹✳ ❚❤❡ ❝✉r✈❡ r = 1 + cos θ ✐s ❜♦✉♥❞❡❞✳

✭❛✮ ❋✐♥❞ t❤❡ r♦♦ts ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ x2 + 2x + 2✳ ✭❜✮ ❋✐♥❞ ✐ts x✲✐♥t❡r❝❡♣ts✳ ✭❝✮ ❋✐♥❞ ✐ts ❢❛❝t♦rs✳

✺✳ ❚❤❡ ❣r❛♣❤ ♦❢ r = θ2 ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐♥ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s✳

✶✵✳ ❊①❡r❝✐s❡s✿ P❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ r❡❧❛t❡❞

✻✳ ■♥ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s✱

(−1, −π/2)

A = (1, π/2)

B=

r❡♣r❡s❡♥t t❤❡ s❛♠❡ ♣♦✐♥t✳

✼✳ ❚❤❡ s❧♦♣❡ ♦❢ t❤❡ ♣♦❧❛r ❝✉r✈❡ t♦

❛♥❞

✹✻✸

r =0

✐s ❡q✉❛❧

0✳

✽✳ ❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❝✉r✈❡

r = cos 2θ

✾✳ ❚❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡

x = t2 , y = sin t

✐s ❛ s♣✐r❛❧✳ ✐s

❜♦✉♥❞❡❞✳ ✶✵✳ ❚❤❡ ❣r❛♣❤ ♦❢

r = θ2

❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛

♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ✐♥ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s✳

✶✶✳ ❊①❡r❝✐s❡s✿ ❙❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

✹✻✹

✶✶✳ ❊①❡r❝✐s❡s✿ ❙❡✈❡r❛❧ ✈❛r✐❛❜❧❡s

❊①❡r❝✐s❡ ✶✶✳✶

❊①❡r❝✐s❡ ✶✶✳✻

❉r❛✇ ❛ ❢❡✇ ❧❡✈❡❧ ❝✉r✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x, y) =

❙❤♦✇ t❤❛t t❤❡ ❧✐♠✐t ❞♦❡s♥✬t ❡①✐st✿

2

x + y✳

xy . 2 x + y2 (x,y)→(0,0) lim

❊①❡r❝✐s❡ ✶✶✳✷ ❚❤❡ ❣r❛♣❤ ♦❢ ❢✉♥❝t✐♦♥

y = g(x)

s❤♦✇♥ ❜❡❧♦✇✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ ♦♥

x✱

t✇♦

♦❢

♦♥❡

✈❛r✐❛❜❧❡ ✐s

z = f (x, y) = g(x)

✈❛r✐❛❜❧❡s✱ ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥❧②

❣✐✈❡♥ ❜② t❤❡ s❛♠❡ ❢♦r♠✉❧❛✳

✇❤❡r❡ t❤❡ ❣r❛❞✐❡♥t ♦❢

f

✐s ❡q✉❛❧ t♦

❋✐♥❞ ❛❧❧ ♣♦✐♥ts

0✳

❊①❡r❝✐s❡ ✶✶✳✼ ❉r❛✇ t❤❡ ❝♦♥t♦✉r ♠❛♣ ✭❧❡✈❡❧ ❝✉r✈❡s✮ ♦❢ t❤❡ ❢✉♥❝✲ t✐♦♥

f (x, y) = ey/x ✳

❊①♣❧❛✐♥ ✇❤❛t t❤❡ ❧❡✈❡❧ ❝✉r✈❡s

❛r❡✳

❊①❡r❝✐s❡ ✶✶✳✽ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

z = f (x, y)

t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ ✇❤✐❝❤ ❤❛✈❡ t❤❡ ❢♦❧✲

❧♦✇✐♥❣ s✐❣♥s✿

fx > 0, fxx > 0, fy < 0, fyy < 0 . ❊①❡r❝✐s❡ ✶✶✳✾ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

z = f (x, y)

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❛❧♦♥❣ ✇✐t❤ ❢♦✉r ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳ ❙❦❡t❝❤ t❤❡ ❣r❛❞✐❡♥t ❢♦r ❡❛❝❤ ♦♥ ❛ s❡♣❛r❛t❡

xy ✲♣❧❛♥❡✿

❊①❡r❝✐s❡ ✶✶✳✸ ❋✐♥❞ ❛❧❧ ❝r✐t✐❝❛❧ ♣♦✐♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x, y) =

2x3 − 6x + y 2 − 2y + 7✳ ❊①❡r❝✐s❡ ✶✶✳✹ ❙❦❡t❝❤ t❤❡ ❝♦♥t♦✉r ✭❧❡✈❡❧✮ ❝✉r✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ s❤♦✇♥ ❜❡❧♦✇✱ ❛❧♦♥❣ ✇✐t❤ ♣♦✐♥ts

A, B, C, D✱

♦♥ t❤❡

xy ✲♣❧❛♥❡✿

❊①❡r❝✐s❡ ✶✶✳✶✵ ❋✐♥❞ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❢✉♥❝t✐♦♥

f (x, y) = x2 y −3

(1, 1)✳ ❯s❡ t❤✐s ✐♥❢♦r♠❛t✐♦♥ t♦ s❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ f ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤✐s ♣♦✐♥t✳ ❊①♣❧❛✐♥✳ ❛t t❤❡ ♣♦✐♥t

❊①❡r❝✐s❡ ✶✶✳✶✶ ❚❤❡ ❣r❛♣❤ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s

z = f (x, y)

✐s ❣✐✈❡♥ ❜❡❧♦✇ ❛❧♦♥❣ ✇✐t❤ ❢♦✉r ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤✳ Pr♦✈✐❞❡ t❤❡ s✐❣♥s ✭♣♦s✐t✐✈❡ ♦r ♥❡❣❛t✐✈❡✮ ♦❢ t❤❡ ♣❛r✲

❊①❡r❝✐s❡ ✶✶✳✺

f (x, y) = z = −1, 0, 1, 2 .

❙❦❡t❝❤ t❤❡ ❧❡✈❡❧ ❝✉r✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥

2xy + 1

❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❛❧✉❡s ♦❢

t✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢

∂f

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