Joseph Edwards Differential Calculus with Applications and Numerous Examples an elementary treatise

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V

\

\ \

DIFFEKENTIAL CALCULUS.

"'

xi \

^.^ 2\\ Ar\

DIFFERENTIAL CALCULUS WITH

APPLICATIONS AND NUMEROUS EXAMPLES:

AN ELEMENTARY

TREATISE.

BY

JOSEPH EDWAEDS,

M.A.,

FORMERLY FELLOW OF SIDNEY SUSSEX COLLEGE, CAMBRIDGE.

'onbon:

MACMILLAN AND AND NEW YORK. 1886. [All Rights Reserved.]

CO.,

$rtut*l> at the

ntb*rsttj)

BY ROBERT MACLEHOSE, WEST NILE STREET, GLASGOW.

PREFACE. THE

object

of the

present volume

is

to

offer to

the

student a fairly complete account of the elementary portions of the Differential Calculus, unencumbered,

by

such parts of\ the subject as are not usually read in colleges

and

Where and

schools.

a choice of method exists, geometrical proofs

illustrations

have been in most cases adopted

in

preference to purely analytical processes. It has

been the constant endeavour of the author

to impress

upon the mind of the student the geometrical

meaning of

differentiation

measurement of

its

rates of growth.

character of the operator of combination

and

which

--

as a

it satisfies

aspect as a

means of

The purely

analytical

symbol and the laws have also been fully

considered.

The applications of the Calculus i

to the treatment of

PREFACE.

vi

at an earlier stage plane curves have been introduced

than usual, from the interesting and important nature

At the same

of the problems to be discussed.

the chapters on Undetermined

time,

Forms and Maxima and

Minima, which have been thereby postponed,

may be

read

in their ordinary place if thought desirable.

The

and

direct

have

hyperbolic functions

inverse

~\

been freely used, and the convenient notation

-,

to

'

denote partial differentiation, has been adopted. It is

sets of easy illustrative

hoped that the frequent

examples introduced throughout the text will be found useful before attacking the

the copious

Many

selections

at

more

difficult

ends

the

of

problems in

the

chapters.

of these examples have been selected from various

university and college examination papers, others from

Home

papers set in the India and

Civil Service

and

Woolwich examinations, and many are new. I

have to thank the Rev. H.

P.

Gurney, M.A., formerly

Senior Fellow of Clare College, Cambridge, for the kind interest he has taken in the preparation of this work,

and

for

much

many

assisted

useful

suggestions.

I

have also been

in the revision of proof sheets and in

the verification of examples

by

J.

Wilson, Esq.,

M.A V

PR ErA CE. formerly Fellow of Christ's College, one of H.M. Inspectors of Schools,

and

also

by H. G. Edwards,

B.A., late Scholar of Queen's College.

that the

book

I

Esq.,

hope therefore

will not be found to contain

many

serious

A errors. '

JOSEPH EDWARDS. 80 CAMBRIDGE GARDENS.

NORTH KENSIN GTON, November, 1886.

VV.

,

EEKATA. Page

82, line

258, Ex.

l.-For 1

(a).

b" +1

For

read b n read

.

+

CONTENTS. PRINCIPLES AND PROCESSES OF THE DIFFERENTIAL CALCULUS.

CHAPTER DEFINITIONS. Object of the Calculus,

3-9

Definitions,

LIMITS.

.......

ARTS.

1-2

I.

and Fundamental

PAGES. 1

:

2-5

.

5-9

10-12

Limits.

13-17

18-24

Undetermined Forms, Four Important Undetermined Forms,

25

Hyperbolic Functions,

16-17

26-36

Infinitesimals,

17-24

Illustrations

CHAPTER

Principles,

....

9-11

12-15

II.

FUNDAMENTAL PROPOSITIONS. 26-30

37-38

Tangent to a Curve,

39-41

Differential Coefficients,

42-44

45-54

Examples, Notation, a as Rate-Measurer, Aspect

.

.

.

30-34 34-36 37-41

55-57

Constant, Sum, Product, Quotient, Function of a Function,

58-62

Inverse Functions,

44-48

CHAPTER

42-44

III.

STANDARD FORMS. xn

........

63-67

Differentiation of

68-73 74-81

The Circular Functions, The Inverse Trigonometrical Functions,

82

Interrogative Character of the Integral Calculus,

,

a*,

log x,

.

.

.

.

.

49-51

51-53 54-57

58

CONTENTS.

x

PA3ES.

ARTS.

84

Table of Results to be remembered, Cases of the Form w,

85

Hyperbolic Functions.

86-87

Illustrations of Differentiation,

83

Results,

CHAPTER

59-60

.

60-61 .

.

.

.

61-62

.

.

62-65

.

.

.

.

IV.

SUCCESSIVE DIFFEKENTIATION. 88-89

Repeated Differentiations,

90-94

d dx

95-97

Standard Results and Processes,

98-100 101

as

.

.

an Operative Symbol,

72-73 74-78

.

....

78-82 82-85

Leibnitz's Theorem,

Note on Partial Fractions,

CHAPTER

85-88

.

.

V.

EXPANSIONS. 102

Enumeration

103

Method Method Method

-yl04-lll 112

of

92

Methods,

Algebraical and Trigonometrical Methods, 92-94 94-99 Taylor's and Maclaurin's Theorems,

I.

II.

.

Differentiation or Integration of

III.

known 99-101

Series,

113

Method IV.

114-119

Continuity and Discontinuity,

120

Lagrange-Formula for Remainder after n Terms

By

a Differential Equation,

.

.

101-103 104-107

of

'

Taylor's Series,

121-122 123-124

.

.

.

126-128

.

.

107-109

Cauchy and Schlomilch and Roche, Application to Maclaurin's Theorem and Special Cases of Taylor's Theorem, .

109-110

.

110

Geometrical Illustration of Lagrange-Formula, Failure of Taylor's and Maclaurin's Theorems,

.

110-111

.

111-114

Formulae

of

.

125

.

129

Examples

130

Bernoulli's

of Application of

.

.

Lagrange-Formula,

.

Numbers,

CHAPTER

114-115 115-117

VI.

PARTIAL DIFFERENTIATION.

....

132-134

Meaning

135-136

Geometrical Illustrations,

137-139

Differentials,

129-132

140-144

Total Differential and Total Differential Coefficient,

132-134

of Partial Differentiation, .

.

126-127

127-129

CONTENTS.

xi PACKS.

ARTS.

145

146-150

151-152

153 154-155

156-160 161-168

an Implicit Function, 135 135-138 Order of Partial Differentiations Commutative, Second Differential Coefficient of an Implicit Function, 138-139 Differentiation of

.

.

.

.

An An

Illustrative Process,

Important Theorem,

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

Extensions of Taylor's and Maclaurin's Theorems, Homogeneous Functions. Euler's Theorems, .

141

141-143

.

143-145

.

145-152

APPLICATIONS TO PLANE CURVES.

CHAPTER

VII.

TANGENTS AND NORMALS. 169-171

172 173 174-178 179-181

182-183 184-186

....

Equation of Tangent in various Forms, Equation of Normal, Tangents at the Origin, Geometrical Results. Cartesians and Polars, Polar Subtangent, Subnormal, etc., Polar Equations of Tangent and Normal, Number of Tangents and Normals from a given ih point to a Curve of the n degree. .

.

.

.

.

187

188-190

Polar Line, Conic, Cubic, etc., Pedal Equation of a Curve,

.

.

159-161

161-163 164-165

.

.

165-169

.

.

169-171

.

.

171-172

.

.

172-174 174-175

.

.

.

175-177

195-199

Pedal Curves, , Tangential- Polar Equation, Important Geometrical Results,

200

Tangential Equation,

201-204

Inversion,

187-190

205-207

Polar Reciprocals,

190-192

191-193

194

177-181 .

....

211-213

To find the Oblique Asymptotes, Number of Asymptotes to a Curve

214

Asymptotes

215

Method

216

Particular Cases of the General Theorem,

217-218

Limiting

219-220

Asymptotes by Inspection,

n ih

of the

parallel to the Co-ordinate Axes, of Partial Fractions for Asymptotes, of

Curve at

181

181-186 186-187

VIII.

ASYMPTOTES.

Form

.

.

CHAPTER 208-210

...

.... .... .

Infinity, .

.

degree,

203-205

206

.

.

206-208

.

.

208-209

.

.

209-211

.

.

211-213

.

213-214

CONTENTS.

XI 1

PAGES.

ARTS.

21

Curve through points with

222 223-225

226-229

its

of intersection of a given curve

..... ....

Asymptotes, Newton's Theorem, Other Definitions of Asymptotes, Curve in general on opposite sides of the Asymptote

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

at opposite extremities.

230 231-233 234-235

236

Curvilinear Asymptotes, Linear Asymptote obtained

Exceptions,

by Expansion,

215-216

.

.

216-218

.

.

219-221

.

221-224

Polar Equation to Asymptote, Circular Asymptotes,

CHAPTER

215

224

IX.

SINGULAR POINTS.

..... ....

238-240

Concavity and Convexity, Points of Inflexion and Undulation,

229-231

241-248

Analytical Conditions,

231-238

249-250

Multiple Points,

251-253 254-257

Double Points, To examine the Nature

258-259

To

260-261

262-263

Singularities of Transcendental Curves, Maclaurin's Theorem with regard to Cubics,

264

Points of Inflexion on a Cubic are Collinear,

237

238-240

.

.

240-242

.

on a Curve, 242-248

of a specified point discriminate the Species of a Cusp,

.

.

CHAPTER

.

.

248-253

.

.

254-256

.

.

256-257

.

.

257-258

X.

CURVATURE. 265-266 267-268 269-271

272-275 276-279

280 281-282

283

284 285

Angle of Contingence. Average Curvature, Curvature of a Circle. Radius of Curvature, Formula for Intrinsic Equations, Formulae for Cartesian Equations Curvature at the Origin, Formula for Pedal Equations, Formulae for Polar Curves,

.

.

....

Tangential -Polar Formula, Conditions for a Point of Inflexion,

.

.

286-290

291-294

Intrinsic Equations,

.

266-268

268-269 269-272

273-275 276-277

277-278

....

Co-ordinates of Centre of Curvature, Involutes and Evolutes,

265-266

.

.

278 278-279 279-281

281-285 285-288

CONTENTS.

xiii

ARTS.

PACKS.

295-297

Contact.

298 299-300

Osculating Circle, Conic of Closest Contact,

301

Tangent and Normal as Axes; x and y

Analytical Conditions,

.

.

.

288-293

.

293-294

.

294-297

CHAPTER

in terms of

297-298

s,

XI.

ENVELOPES.

Parameter

302-303

Families of Curves

304

The Envelope touches each

;

;

Envelope,

.

of the Intersecting

31 1

.

Mem-

bers of the Family,

311-312

.

305

General Investigation of Equation to Envelope,

306-307

313-314 Envelope of A\* + 2B\ + C=0, Indeterminate Multipliers, 315-318 Several Parameters. Converse Problem. Given the Family and the Envelope to tind the Relation between the Parameters, 318-320 320 Evolutes, Pedal Curves, 321-322

308-311

312 313 314

312-313

.

.

CHAPTER

XII.

CURVE TRACING. 315-317

Nature of the Problem

Order

;

of

Procedure in

....

Cartesians,

318-319 320-321

322 323-325

326

Examples, Newton's Parallelogram, Order of Procedure for Polar Curves, Curves of the Classes r = a sin nd, r sin nd .

Curves of the

Class rn

.

= a* cos nd.

330-333

...

.

.

333-340 340-344

.

.

.

.

.

.

344-345

.

.

345-347

.

.

347-352

a,

Spirals,

APPLICATION TO THE EVALUATION OF SINGULAR FORMS AND MAXIMA

AND MINIMA CHAPTER

VALUES. XIII.

UNDETERMINED FORMS. 327-329

Forms

330

Algebraical Treatment,

331-334

Form

335

Form

to be discussed,

I**

-,

x

oo

,

.

.

.

.

... ... .

361-362 362-365

365-369

369

CONTENTS.

XIV

PAGES.

ARTS.

836-338

Form

00

369-373

00

339

Form

340

Forms 0, 00,

341

A Useful

342

dy

dx

co

oo

,

373

.

1*,

374

Example,

374

of Doubtful

Value at a Multiple Point,

CHAPTER

375

XIV.

MAXIMA AND MINIMA ONE INDEPENDENT VARIABLE.

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

343-344

Elementary Methods,

345-347

The General Problem.

348-349

Properties of

Definition,

Maxima and Minima

Values. for Discovery and Discrimination,

381-383 384-386

Criteria

.

386-392

350

Analytical Investigation,

393-395

351

396-398

352-353

Implicit Functions, Several Dependent Variables,

354

Function of a Function,

400-404

355

Singularities,

404-405

.

.

.

.

.

398-400

APPENDIX. 356-366

On

the Properties of the Cycloid,

ANSWERS TO THE EXAMPLES,

.

417-424 427-439

PEINCIPLES

AND PEOCESSES OF THE

DIFFEEENTIAL CALCULUS.

CHAPTER DEFINITIONS.

I.

LIMITS.

Primary Object of the Differential Calculus. In Nature we frequently meet with quantities 1.

observed for some period of time, are found undergo increase or decrease ; for instance, the

which, to

if

distance of a

moving

particle

from a

known

fixed point ordinate of a given

in its path, the length of a moving curve, the force exerted upon a piece of soft iron is

gradually

made

to

which

approach one of the poles of a

magnet. When such quantities are made the subject of mathematical investigation, it often becomes necessary to estimate

their rates

This

of growth.

is

the

primary

object of the Differential Calculus. 2.

In the

first

six chapters

we

shall

be concerned with

the description of an instrument for the measurement of such rates, and in framing rules for its formation and use,

and the student must make himself as proficient as possible in its manipulation. These chapters contain the whole machinery of the Differential Calculus. The remaining chapters simply consist of various applications of the methods and formulae here established.

A

DEFINITIONS.

-2

We

3.

LIMITS.

commence with an explanation

of

several

technical terms which are of frequent occurrence in this subject, and with the meanings of which the student

should be familiar from the outset. 4.

Constants and Variables.

A

CONSTANT is a quantity which, during any set of mathematical operations, retains the same value. A VARIABLE is .a quantity which, during any set of mathematical operations, does not retain the same value, but

is

capable of assuming different values.

Ex. The area of any triangle on a given base and between given parallels is a constant quantity so also the base, the distance ;

between the parallel

lines,

the

sum

of the angles of the triangle are

constant quantities. But the separate angles, the sides, the position of the vertex are variables.

It has

become conventional

to

make

a,b,C,...,a,/3,y,..., from

use of the letters

the beginning of the

alphabet to denote constants and to retain later letters, for such as u, v, iv, x, y, 2, and the Greek letters tj, f, ;

variables.

Dependent and Independent Variables, An INDEPENDENT VARIABLE is one which 5.

may

any arbitrary value that may be assigned to it* A DEPENDENT VARIABLE is owwhich assumes

take

its

in consequence of some second variable or system of variables taking up any set of arbitrary values that in >/ be assigned to them. 6.

Functions.

When one quantity depends upon another system of

others, so

that

it

or

upon a

assumes a definite

valn(x) == sec #,

= sec ,

and

53

r

(05

+ h),

SGc(x-\-h)

,

h =Q-j=Lt dx

h

cosx

cos

-

h

.

sin

-

r-

(x+h)

.

h

/

oj

9

COS^C 73. Differential Coefficient of cosec x.

If

u = (p(x) (x) = sin ~

l

x,

x = smu, and x+h = siuU h = sinU siuu,

du_ dx

T4

Uu

T^

;

Uu

STANDARD FORMS.

55

77. Differential Coefficient of

u = (f>(x) = tan~ lx,

If

# = tan h = tan

Hence therefore

u,

and

U

x-\-h = ta n

tan

u,

Uu = Ltu ax h = Lt u=u Uu

du =

and

-j

T

JLtho T

U

l

T

T

-Uu

=u tan

sec%

^ cos

+ ta,n

1

jj U

2

u

1

+x

78. Differential Coefficient

u

If

x = cotu, and x+h = cot U\ h = cot U cot u,

Hence therefore

Uu

=

-7

dx

=

J-

Uu

h

U --

U

.

Ltrr= u ^- TTT sin (

U

TTT

v

sin

.

U sin u

u) 1

1

l+cot2u 79. Differential Coefficient of sec- l x.

^

If

Hence

x = secu, and

therefore

fi

A and

= sQC Usecu,

du "= -7

T

jJth

Uu Uu =u T

o

=jL/6rr_ 9/ w

5

/i

T = Ltrr ,

-

COS

U

COS

fr

U

u

tan

TT cos u U

,

T-PP

\

-Uu y^p

sec

U-secu

COS

U

COS

TT

U

2

'

56

STANDARD FORMS.

U-u

STANDARD FORMS. definition;

chapter

Ex.

we

(i.)

but by aid of Prop. VI. of the preceding can simplify the proofs considerably.

If

x = sin

we have

dx

whence

-=-

du

and therefore

and

57

u

= cos u

;

:

-=-=- =

dx

dx

cos

u

cos~ 1 o;=

since

~2

,

we have Ex.

(ii.)

dcos~ l x= dx

u

If

x = tan

we have

dx

,

whence

-,

du

c and therefore

1

dx

,,2>

cot" 1 ^ =

and since

-

A ,

we have 1

Ex.

(iii.)

If

+a

5

u x = vers u = l

we have

dx

= sin u

whence

-y

and therefore

-r-=

;

=

sinu

cos

STANDARD FORMS.

58

whence

82.

also

d covers" 1 ^ dx

1

The Integral Calculus.

Suppose any expression in terms of x given can we find a function of which that expression is the differential ;

The problem here suggested

coefficient?

is

inverse to

The dissuch functions is the fundamental aim of the The function whose differential Calculus.

that considered in the Differential Calculus.

covery of Integral

the given expression is said to be the For example, if (x) is (x) C, C being any

+

arbitrary constant. The notation by which this "

f(j>(x)dx being read

is

expressed

integral of

Thus we have seen d/

\

-T-T-(sm

x)

'(x)

= cos x, 1+a2

'

etc.,

whence

it

follows immediately that cos xdx = sin x,

f

I

1+x* etc.,

is

with respect to x"

STANDARD FORMS. where the arbitrary constant may bemadded in each case if desired.

We

do not propose to enter upon any description of the various operations of the Integral Calculus, but it will be found that for integration we shall require to 83.

remember the same

list

of standard forms that

is

estab-

lished in the present chapter and tabulated below, and it is advantageous to learn each formula here in its double

We

have therefore ventured to tabulate the standard forms for Differentiation and Integration toaspect.

Moreover, we shall find it convenient to be gether. able to use the standard forms of integration in several of our subsequent articles.

TABLE OF RESULTS TO BE COMMITTED TO MEMORY.

= xn = ax

= ex

du

-_n

-^-

.

dx

du = -^-

.

dx

ci

K

-Y-

.

dx

a

e

u = sin x. u = cosx. u = tan x.

loge a.

= ex

x

=,

= ex

dx

dx

or

x

= cos x.

-= dx

s\u x.

= sec Tdx

*/cos

xdx

o?.

.

=T

-. a log e

= sin x.

fsinxdx

=cos x.

fsocPxdx

= tan x.

39.

/

45.

a '

/

'i

I

40.

y=

41.

y=

v=

y= a 1 y = e *cos(&tan~ o;) y=

o

t

X'-X+l

51.

'

53.

?/

=

= 55.

2/

= 1

+

y=

56.

57.

/

= logn (#),

where log w means log log log

(repeated

=

58.

>Jb

1

n

+ a+ *Jb-a tan

log a;

tan

=

59. 60. 1

-

x p O

fi5 \J f

11 W

A A V/VJ

xjC rt/ C/ ^^~ C/

61.

67. 68.

62. 69.

64.

...

times).

a

63.

+ a cos a;

b

52.

44.

'

-

48.

50. 42.

a

'

'

y= y= y=

9

c08a:

(sina;) cota:

(cota)

+(cos,

+

coth

(cothic)

1

*.

STANDARD FORMS. 71.

78.

j^L

96. Differentiate

97. Differentiate sec' 1

^

T L

*j3L>

with regard to

_ with

-

98. Differentiate tan' 1

x/l-ce Differentiate tan" 1

99.

100. Differentiate

w cc

-^-

with regard to

2

regard to sec'

1

_ 2x*

2x

with regard to sin" 1

1 log tan" ^ with regard to

Sln s

x? ..

1

X

A1 V/

X

.

102.

TIt"}/ / J. L

to oo

y*

/

(/

i>ty

.

ax

Ify =

x

-

y log aj

prove

+ fu a! 1+

1

I

= dx

1-+

lH-...tooo,

1

03. If

v = a; + - 1 x +-

x+-X

i rv A

1U4.

Tf

Liy =

+

...

tO

00,

,

cosa;

sin

j.

x

cos

105. If

106.

'

'

*7 /

_

r/c

prove

~

\

T/

c?v -

1

+ 2y + cos

=

V

=

cos

sin

a?

+

*

*

+

...

cc

+ Jetc.

tO

00.

t/

a? -

sin x'

v sin x + v sin

to

oo,

x .

dx

2y-l If #n = the sum

common

"T"

x

1 *J

tC

1

ft/

1

1)T*OV*

-,

-i

--

Sin

= - as dx 2-x+-JC + -

prove -^

,

ratio,

of a G. P. to

prove that

n terms

of which r

is

the

STANDARD FORMS. a+

1 ch

+

mo 108.

nGiven

/^

'

%+...+ i

i

,

r2cos20

/a

1

Oa = rsm0n +

,

3

~

-i

and

dx\Q

i

""

r cos 30 +rcos + -

CHAPTER

V.

EXPANSIONS. The student

102.

will

have already met with several

expansions of given explicit functions in ascending in T tegral powers of the independent variable for example, ;

those for (x+a) n , e x log(l-f#), ,

tan^x,

sin x, cosx,

which

occur in ordinary Algebra and Trigonometry.

The

principal

methods of development in common use

may be briefly classified as follows I. By purely Algebraical or Trigonometrical processes. II. By Taylor's or Maclaurin's Theorems. III. By Differentiation or Integration of a known :

series, or

IV.

By

equivalent process. the use of a differential equation.

These methods we proceed to explain and exemplify. 103.

Ex.

1.

METHOD

Algebraic and Trigonometrical Methods. Find the first three terms of the expansion of log sec x in I.

ascending powers of

x.

By Trigonometry oos x

=

1

1

.V

2

-

2!

Hence where

log sec

x

+ ,

X4

X*

4!

6!

-4-

log cos x x 1 xt 3? .

+ .

log

^_-_+_-...s

(1

z\

EXPANSIONS. and expanding log(l

by the logarithmic theorem we obtain

z)

9J

=

^T O

r -__ + :L_ '" 6

,r.4

,2

|

93

41^6!

.2!

__

.

"

24

720

# _ "~ 4

h ence

Ex.

log sec

x

Expand

2.

A-

6

=+"+ 2 12 45 cos 3.^ in

......

powers of

x.

4 cos% = cos 3x + 3 cos #

Since

_

,

'

r

"4f'

2!

',

_ lV r

weobtain

sin%=i

Similarly

j

(3

3

-

-3)^3 1

5

(3

-

O2n - 1

Ex.

3.

volving

Expand tan #

in

-O

:

7

;

'-

_o

powers of x as far as the term

x*. /vW>

~ ^ __

Since

tan.v=

.

-)

~"

_j_

3

5

1

'

1

/y4

/jii

l_^+^_... we may by

actual division

...

.

show that .3

tan # =# 4-

3

+

g

15

x* +

.

.

.

in-

EXPANSIONS. Ex.

Expand |{log(l + #)}

4.

in

powers of

(l+x)y=ev los(l+x

Since

we

2

x.

),

have, by expanding each side of this identity, )g8

Hence, equating

+ y(y

-

coefficients of

y

2 ,

-a

series

-

*- etc.,

which may be written in the form

EXAMPLES. 1.

Prove

2.

Prove cosher = 1

/n

+

"

(3n

-

.

Prove that

riogd+a)]^ /!

where

r

p

r\

Pk denotes

the

sum

of all products

k at a time of the

first

>

natural numbers.

104.

METHOD

II. Taylor's

and Maclaurin's Theorems.

been discovered that the Binomial, Exponential, and other well-known expansions are all particular cases It has

of one general theorem known as Taylor's Theorem, which has for its object the expansion of f(x h} in ascending integral positive powers of h, f(x) being a function of a.

+

1

of

any form

whatever.

It will

be found that such an

not always possible, but we reserve for later articles [120 to 128] a rigorous discussion of the

expansion

is

limitations of the theorem.

EXPANSIONS.

95 *

105.

The theorem

referred to is that

under certain

circumstances

'

+!/(*) +

to infinity,

...

."

expansion off(x-\-h) in powers of h. This result was first published by Taylor in 1715, in " his Methodus Incrementorum Directa et Iriversa." In

O

,+

............ (3)

etc.

Hence putting x

()

in (1),

(2), (3), ...,

4

and

we have

=/(0), A=/'(0), A, =/"(()), substituting these values in (1)

f(x) =/(())

+ xf(0) +

O

etc., ...

;

.

.

109. It will be noticed that in the above proofs there is nothing to indicate in what cases the expansions assumed in the equations

numbered

(1) in

G

each of the

last

two

EXPANSIONS.

98

articles are illegitimate,

and we

shall

student to Arts. 120 to 128 for a fuller

have to refer the and more rigorous

discussion.

proceeding farther, that the student should satisfy himself that the well known n x e sinx, etc., expansions of such functions as (x + h) 110.

It is important, before

,

,

are really all included in the general results of Arts. 107, 108.

For example, if f(x) = xn ,f(x + h) = (x + h)", f(x) = nxn = n(n I)xu ~ 2 etc. Hence Taylor's Theorem,

-

,

gives the binomial expansion (x

+ h)" = a" + nhxn

~l

+

-^hV- + 2

. . .

i

= e? then f(x) ex f"(x) ex = 1,/(0) = 1, /'(O) = 1, etc. /(O)

Again, suppose f(x) therefore

Hence

Maclaurin's. f(x)

t

,

,

etc.,

Theorem,

=/(0) + xf(0) +

/"

nrfL

x

e

gives

the result

= l+a; +

known

+ +

...,

as the Exponential Theorem.

We

append a few examples which admit of expansion, and to which therefore the results of Arts. 107, 111.

108 apply. EXAMPLES. Prove the following results 5

= o:- x? 4- x 3

!

5

-.... !

:

EXPANSIONS.

/vi

>

,

99

3.

tan- 1 * =#->-+ 5 3

4.

e*cos

x = I + 2^cos

?r .

a?

4

+ 2*cos

+ 2W-

2ir

4

- + 2*cos -21

-+

43!

.

,

+....

4 n\ ,1

5.

6.

u

**

"7

GT"*"*

(

*

^^

f^tr^

I

Z.

^

~

. ..

'^

't

^ -

(

^

8.

S in-

^ 2 )l 9.

log sin (# + K)

= log sin ^ + h cot ^ -

2

cosec 2^ +

3!

1+

3 sm'%

.10.

METHOD 112.

III.

Expansion by Differentiation or Integration of

known

series or equivalent process.

The method Ex.

1.

To

Suppose then

of treatment is indicated in the following examples l expand tan~ x in powers of x. Gregory's Series.

= tan- = a + a^x + a.^ 2 + a$P + / (x) = ---- = ai + 2^^ + Sa^e2 + 4 4^3 + 1

.*?

f(x)

.

X ~p ^/

Hence, comparing these expansions, we have

i

a

Also,

therefore

= -1, = tan -1 = nir 1)

3

3

tan \v = nir + x- ^--+ -

3

5a 5 =l,

etc.

-'+..

. .

,

. .

.

:

:

EXPANSIONS.

100 This result

be obtained immediately by integration of the

may

1

series for 1

the constant a being determined as before.

To expand sin~ l x. = sin~ J .r = Suppose f(x) Ex.

2.

therefore

+ a ^K + a.^x* =a

=

1

2

4-

4

4

AJ 4-

.

.

.

,

But

2.4 Hence, comparing these

we have

series,

a.2 = a^ = a6 =

and

!

= !,

...

3a 3 =i,

=0, = 1.3

5%

2.4'

Also

.3.5 2.4.6'

1.3.r'

Hence

1

'

2.4

3

5

.

7

and, as before, this might have been obtained immediately bv integration of the expansion of

Ex.

3.

Again,

if

a

known

series

from it by differentiation. For example, borrowing the Ex. 2 of the next Art.,

1 2 series for (sin" ^) established in

viz.

#*

we

be given, we can obtain others

2#*

.

2.4 x6

obtain at once by differentiation 2 sin2.4

.

1

.*-

.

,

,

2.4.6

a*

2.4.6

EXAMPLES. 3

= smhlog(.>; + V 1 + ^

1.

Prove

2.

~ Prove tanh \v = x + '+--+....

>2

)

3

5

-^

=

.r

3

,1.3 ~~ >

ji> '

5

EXPANSIONS. .

And

Deduce from Ex.

3,

/i

-i

z\\

101

Art, 112,

2.4 x 1 "3 ~3~5~ "~3T5'T" % %A

fi

*

hence by putting .r=shi0, prove 2

= 1 - sin2 *

6 cot

sin4