IB Diploma Program Mathematics Course Companion Higher Level Option: Calculus [1 ed.] 0198304846, 9780198304845

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O X

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M AT H E M AT I C S

P

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M

E

HI GH E R

LE V E L

C A LC U LU S

Josip Harcet

Lorraine Heinrichs

Palmira Mariz Seiler

Marlene Torres-Skoumal

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v

Contents

Chapter

1.1

1

Patterns

From

limits

of

to

inni ty

sequences

to

2

limits

of

functions

1.2

Squeeze

limits

1.3

From

of

of

and

convergent

evaluation

limits

of

algebra

indeterminate

of

limits

sequences

to

10

limits

2

Smoothness

Continuity

on

13

an

and

in

mathematics

22

dierentiability

inter val

Theorems

2.3

Dierentiable

2.4

Limits

2.5

What

2.6

Limits

Theorem

and

at

Introduction

and

of

continuous

functions:

Mean

point,

L’Hopital’s

are

functions

3

Value

Theorem

indeterminate

functions

of

and

functions?

limits

49

Improper

Integral

4.5

The

4.6

Comparison

98

convergence

tests

Integrals

test

p-series

for

test

114

for

Limit

comparison

Ratio

test

for

4.9

Absolute

Conditional

112

test

4.7

4.10

110

convergence

4.8

convergence

test

for

convergence

convergence

convergence

of

convergence

Representing

5.2

Representing

5.3

Representing

5.4

Taylor

Power

115

118

119

series

of

series

3.2

Dierential

3.3

Separable

variables,

equations

and

3.4

Modeling

of

3.5

First

3.6

Homogeneous

3.7

Euler

and

dierential

their

solutions

Equations

with

polynomic

Series

Functions

120

122

130

by

1

132

Power

Series

as

Functions

135

Series

Functions

by

2

138

Polynomials

5.5

Taylor

5.6

Using

5.7

Useful

54

of

Everything

Power

dynamic

Classications

5

5.1

of

50

3.1

and

143

Maclaurin

Taylor

Series

to

Series

146

approximate

functions

applications

156

of

power

series

161

56

separated

variables

Answers

168

Index

185

61

dierential

graphs

growth

of

and

their

solutions

order

integrating

equations

and

factors

and

dierential

69

exact

Method

63

decay

phenomena

vi

to

96

104

4.3

42

graphs

phenomena

equations

inni te

series

4.4

33

forms,

r ule

smooth

Modeling

equations

the

28

Rolle’s

sequences

Chapter

in

convergence

24

about

a

and

ni te

4.2

Chapter

2.2

The

Series

for

7

4

4.1

of

sequences

sequences:

and

the

functions

Chapter

2.1

theorem

Divergent

forms

1.4

3

Chapter

73

dierential

substitution

for

rst

equations

methods

80

order

85

Patterns

1

CHAPTER

sequences

Informal

Before

1

Simplify

2

1

x

3

n

−

n

3




log

(L

+

5),

3

for

p

=

int

(log

(L

+

5))

+

,

it

is

tr ue

that

n

≥

p

⇒

u

3




L

⇒

2

>

L

⇒

n

>

log

(L),

for

2

p

=

int

(log

(L))

+

,

it

is

tr ue

that

n

≥

p

⇒

|u

2

Therefore

|

>

L.

n

lim

u

=

+∞

n n → ∞

The

following

have

limit

Theorem

impor tant

innity

3:

Let

with

}

{u

and

useful

convergent

be

a

theorem

sequences

relates

that

sequences

have

limit

that

zero.

sequence.

n

1 i

lim

u

=

+∞

⇒

lim

=

0

n n → ∞

n → ∞

u n

1 ii

lim u

=

0

⇒

lim

=

+∞

n n → ∞

n → ∞

u n

This

the

theorem

small,

and

innitely

the

formalizes

reciprocal

of

the

reciprocal

large.

algebra

of

results

something

This

is

innity .

of

that

when

example,

have

large

something

impor tant

For

you

innitely

is

been

innitely

it

comes

using

using

something

small

to

is

intuitively:

innitely

something

setting

Theorem

3,

⎛ u

it

r ules

is

⎞

easy

for

to

a

n

prove

that

if

lim u

=

a



and

0

= 0,

lim v

then

lim

⎜

n

n

n → ∞

x → ∞

x → ∞

⎟

=

v ⎝

=

+∞.

0 n

⎠

a

The

notation

needs

to

be

inter preted

in

the

context

of

limits

and

0

seen

as

a

simplication

simplied

language,

summarized

by

the

the

of

mathematical

algebra

following

of

limits

language.

involving

Using

this

innity

can

be

table:

Chapter

1

11

(±∞)

+

(±∞)

=

±∞

(±∞)

×

(±∞)

(±∞)

×

(

=

± (± ∞)

−

(± ∞)

=

indeterminate

+∞

a

+

(

± ∞ )

=

± ∞ ,

a

−

(

± ∞)

=

∓ ∞, a ∈ R

± ∞)

=

∞

± a

×

(± ∞)

=

±∞,

a

>

0

a

a

×

(±∞)

=

n

∞,

a


0

the

following

n

e

e

=

n !

n

limits:

3

3

n

+

n

5n

3

+ 6n

− 1

n

6n

n

+

2n

+

3

1

n

⎛

n

(

+

5

−

n

+

2

)

lim

f

n →∞

n →∞

n

+

⎜ +

+

n →∞

3)

( h

+

)

n

n →∞

5 ⋅ 3

n

+

7

lim

n

4

2

n

n +1

2

n

⎟ n

(n

n →∞

lim

g

⎟

lim

n

3

5

⎜

1 d

n →∞

+

−

lim

c

n →∞

+

3

lim

2

2n

2

2

n b

lim

= n

1C

5n

e

d

d

n

lim n →∞

> 1

b

Evaluate

a

b

n

n

n

,

n

2

Exercise

+

where

=

1

4

n

+

e

⎝

n

n

k

3

3

n

⎛

n

∑

3

n

n

+

( 2k

1)

∑(

lim

lim

j

n

EXTENSION

lim

k

⎜ n + 1

⎝

)

lim

l

n

2

n →∞

n →∞

1

k =0

k =0

i

2

1

n →∞

n

2n

n →∞

1

+

3

QUESTION

n

2

Given

lim

that

n



1,

n →∞

ln

ln n

Show

a

lim

that



n →∞

0.

Hence

nd

the

value

of

(n

+

k

) ,

lim n → +∞

n

where

k

>

0.

n

n

5

⎛

Find

b

the

value

of

n

lim

⎞

n ⎜

⎟

n →∞

n

⎝

1.4

From

During

in

an

of

he

ver y

to

make

intuition

clear

was

sequences

several

this

forced

often

to

rst

the

to

area

limits

of

mathematics

mathematicians

to

Cauchy

dene

ones

to

mathematicians

leading

denitions.

the

similar

of

centur y ,

functions

with

develop

and

9th

attempt

limits

and

the

limits

⎠

we

the

use

to

incorrect

was

limit

today .

a

He

functions

worked

more

deal

this

it

the

was

group

function

on

concepts

rigorous.

with

results,

among

of

of

ver y

of

lim

f

the

concept

of

calculus

study

of

innity ,

impor tant

to

mathematicians,

precisely ,

inter preted

As

of

(

x

using

)

=

b

terms

as

a

x →a

relation

between

innitely

Over

the

small,

course

functions

in

order

Among

were

to

denition,

the

of

the

with

there

which

is

when

dierence

the

is

∆y

following

proposed

deal

them,

variables:

a

by

the

=

50

various

pitfalls

of

denition

based

f

on

the

dierence

(x)

−

b

years,

Δx

also

other

=

Cauchy’s

limit

study

of

of

a

becomes

innitely

denitions

These

denition

function

limits

−

becomes

mathematicians.

of

x

of

of

limits

were

which

known

numerical

small.

explored

were

as

of

exposed.

Heine’s

sequences:

Chapter

1

13

Denition:

Let

I

be

an

open

inter val

of

real

numbers.

Let

f

: I

→

R This

a  I ,

and

lim

then

f

x

(

)

exists

and

lim

x →a

f

(

x

)

=

b

if

and

only

if

for

be

sequence

{a

}

such

a

that

∈ I

for

all

studying

and

n ∈ Z

lim a

n

n

f

(

(a

=

))

n

will

for

results

in

a,

=

n

Chapter

n →∞

lim

useful

x →a

+

any

denition

ver y

2

b

n →∞

This

of

a

denition

function

need

to

nd

is

at

ver y

a

two

useful

point

x

=

when

a

sequences

does

{u

}

we

not

and

want

to

exist.

{v

n

}

In

such

show

this

that

case,

lim

that

the

you

u

lim

f

(

(u

≠

))

n

lim

f

(

(v

just

=

lim

v

n

n

=

a

but

n

n → +∞

n → +∞

limit

n → +∞

))

n

n → +∞

Example



⎧3 x Show

that

the

function

f

dened

by

f

(

x

)

=

⎩

u

=

2

v

and

−

=

2

x


2

–6

–8

Example

the

could

this

illustrates

reach

method

proof

14

9

intuitive

way

the

of

(shown

Patterns

the

relation

learned

same

example

innity

to

between

nd

conclusion

substitution

in

to

you

9)

does

limits

by

proof

give

a

although

demonstrating

functions

substituting x

not

does,

a

of

=

rigorous

it

is

2

as

in

par t

both

result

sufcient

of

limit

core

branches

about

for

that

the

the

of

limit

examination

of

a

function

course.

the

of

a

In

does

practical

piecewise

function

purposes.

in

not

function.

the

exist,

terms,

way

and

you

However ,

that

the

Example

x

=

2

is

When

just

9

shows

dierent

we

one

want

side

a

from

to

of

the

of

Let

inter val

be

an

lim

Then:

(

f

)

x

limit

over

exists

whose

behavior

the

point

right

x

its

study

Denition

I

function

and

2

a

the

we

the

of

use

to

right

the

the

of

x

left

=

function

lateral

of

2.

on

limits:

point:

real

lim

f

numbers.

(

x

)

=

b

Let

if

and

f

: I

→

only

if

R

and

a  I .

for

+

+

x →a

x →a

any

to

behavior

=

at

behavior

sequence

{a

}

such

that

n

+

a

●

 I

for

all

n ∈ Z

n

+

a

●

>

a

for

all

n ∈ Z

n

lim

●

a

=

a

n n →∞

we

have

lim

f

(

(a

n

))

=

b

n →∞

The

lim

(

f

)

x

is

called

right

limit

of

f

at

x

=

a.

+

x →a

The

denition

of

left

Denition

of

Let

inter val

I

be

an

left

limit

limit

of

at

is

a

real

similar:

point:

numbers.

Let

f

: I

→

R

and

a  I Note

lim

(

f

x

in

denitions

)

exists

and

x →a

lim

f

(

x

)

=

b

if

and

only

if

for

any

these

of

right

and

sequence

x →a

left

{a

that

}

such

that

limits,

type

of

I

can

inter val:

be

any

open,

n

closed,

semi-open,…,

+

●

a

 I

for

n ∈ Z

all

n

etc.

+

●

a

>

a

for

all

n ∈ Z

n

lim

●

a

=

a

n n →∞

we

have

lim

f

(

(a

n

))

=

b

n →∞

When

the

of

a

is

an

function

I,

the

lim

endpoint

at

f

that

(

x

)

of

point

exists

the

as

a

inter val

lateral

exactly

I,

we

limit.

when

dene

If

both

a

is

the

not

lateral

limit

an

limits

of

endpoint

exist

x →a

and

lim −

x →a

f

(

x

)

=

lim

f

(

x

)

+

x →a

Chapter

1

15

Example

0

1

⎧

,

⎪

Consider

the

function

f

dened

by

(

f

x

)

=

⎨ x

⎪ ⎩

−2