Differential Calculus by SHANTI NARAYAN 10th Edition

THE

BOOK

WAS

DRENCHED

DIFFERENTIAL

CALCULUS

(Review published in

Gazette,

London,

December, 1953

THE

book

has reached Its 5th edition in _{9 years} and it

can be

assumed

that it

meets

all

demands.

Is it the

revie-wer's _{fancy to discern the influence of}G. H.

_Hardy

in the opening chapter

on

real

numbers,

which

are well and clearly dealt with ?

Or

is this _{only to} be _expected

from

an author of the race

which

_taught the rest of the world

how

to count ?

* .- "i

* *

The

course followed is

_{comprehensive}

and _thorough,

and there is a

_good

_chapter onvcurve _tracing.

The

author has a talent for clear

exposition, and is sympathetic to the

difficulties _{of the beginner.}

* * #

Answers

to _examples, of

which

there are

_good

and

_ample

selections, are given.

* # 3

Certaianly Mr. Narayan's

command

of English is

excellent

Our own

_young

scientific or mathematical

specialist,

grumbling

over French or

German

or Latin as additions to their studies,

would do

well to consider their Indian confreres, with English to master before their technical education can _begin.

DIFFERENTIAL

CALCULUS

FOR

B.

A.

&

B.

Sc.

STUDENTS

By

SHANTI

NARAYAN

Principal, and

Head

of the _{Department of} Mathematics

Hans

Raj College, Delhi University

TENTH

REVISED

EDITION

1962

S.

CHAND

&

CO.

DELHI

NEW

DELHI

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by Author

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TextBookofGeneral_Topology (Underpreparation)

Publishedby S.

CHAND

&

CO.

for

Shyam

Lai Charitable Trust,

16B/4, Asaf

AH

Road,

New

Delhi.

(All profitsfrom thisbook_{are spent}on charities.)

S.

CHAND

&

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Fountain DELHI

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First Edition 1942 Tenth Edition

1962

Price: Rs. 7-00

Published_by

G

S Shartna. for S. Chand

&

Co.,

Ram

Nagar,

New

Delhiand

Preface to the TenthEdition

The book

has beenrevised.

A

few

more

exercises

drawn

from

the _{recent university papers} have been _given.

30th April, 1962.

SHANTI

NARAYAN

PREFACE

This book is

meant

for students _preparing for the B.A.

and

B.Sc. examinations ofour universities.

Some

_topics of the

Honours

standard have also been included.

_They

are _given in the form of appendices to the relevant chapters.

The

treatment of the subject

is _rigorousbut no attempt has been

made

to state

and

_{prove the}

theorems _{in generalised} forms

and

under less restrictive conditions as

is the case with the

Modern

_Theory of Function. It has also been a constant endeavour ofthe author to seethat the _subject is not pre-sented just as a

body

of formulae. This is to see that the student doesnot form an_{unfortunate impression that the} study of Calculus consists _only in _acquiring a skill to _manipulate

some

formulae through 'constant drilling'.

The

_{book opens with} a brief 'outline of the _development of Real numbers, their_expression as infinite decimals

and

their repre-sentation

_by

_{points 'along} a line. This is followed

_by

a discussion of the graphs ofthe_{elementary functions x"> log} x, ex, sin x,

sin-1

*,

etc,

Some

of the difficulties attendant

_upon

the notion of inverse

functions have also been illustrated

_by

_precise formulation of Inverse _{trigonometrical} functions. It is _suggested that the teacher

in the classneedrefer to _{only a} fewsalient _{points of} this _{part of the}

book.

The

student _would,

on

his _{part, go through} the

same

in

complete details to acquire a sound grasp of the basis of the subject. This partis so _{presented that a student} would have no difficulty in

an independent studyofthesame.

The

first _partof the book is _analytical in character while the

later_part deals with the _{geometrical applications} of the subject.

But

this order of_{the subject} is

_by

no

means

_suggested to be _rigidly followed in the class.

A

different order

_may

_usefully be _adopted at tho discretionof theteacher.

An

_analysis of the _{'Layman's' concepts} has _frequently been

made

to serve as a basis for the _precise formulation of the

corres-ponding

'Scientist's' _concepts. This _specially relates to the

two

interpretation results analytically

been givento _bring

them

home

to the students.

A

_chapter

on 'Some

Important Curves' has been given before dealing with geometrical applications. Thiswillenable the student to get familiar with the

names and

_shapes of

some

of the _important curves. It is felt that a

student

would

have better _{understanding} of the _properties of a curve ifhe

knows

how

the curve lookslike. This _chapter will also serve as a useful introduction to the _subject of Double points of a curve.

Asymptote

of a curve has been defined as a line such that the

distance of

_any

_point

on

the curve

from

this line tends to zero as the point tends to infinity along the curve. It is believed that, ofall

the definitions of

an

_asymptote, this is the one which is

most

natural.

It embodies the idea to

which

the _concept of _{asymptotes owes} its

importance. Moreover, the definition _gives rise to a _simple

method

for _{determining the asymptotes.}

The

various _principles

and methods

have been _profusely

illus-trated

_by

means

ofa _large

number

_{of solved examples.}

I

am

indebtedto Prof. Sita

Ram

_{Gupta, M.A., P.E.S., formerly} of the

Government

College, Lahore

who

verykindly

went

through the manuscript

and

made

a

number

of suggestions.

My

thanks are also

due

to

_my

old _pupils

and

friends Professors

_Jagan

Nath

M.A.,

Vidya

Sagar

M

A.,

and

Om

Parkash

M

A., for the help which they

rendered

me

in _preparing this book.

Suggestions for

improvement

will _{be thankfully acknowledged.}

CHAPTER

1

Real _Numbers, Variables, Functions

ARTICLE

PAGE

Introduction ... 1

ri. Rational

Numbers

... 2

1'2. Irrational

Numbers.

Real

Numbers

... 5 1*3. Decimal _{representation of} real

numbers

... 6

1*4.

The modulus

of a real

number

... 9

1'5. Variables, Functions ... 11 1 6.

Some

_{important types of}

domains

of variation ... 13 1*7. _{Graphical representation of functions} ... 14

CHAPTER

II

Some

_{Important Classes} ofFunctionsand their _Graphs

2-1.

_Graphs

of

_y=x*

... 18

2*2. Monotoiiic Functions. Inverse Functions ... 20

2-3.

_Graph

of

y

=

x*in ... 22

2-4.

_Graph

of

_y=a

x ...

24

2-5.

_Graph

_{of >>^loga}

x

... 25

26.

_Graphs

ofsin x, cos x, tan x, cot x, sec x,

cosec x

...26

2-7.

_Graphs

of shr^x, cos^x,

tan^x,

cot~*x; see^x,

cosec^x ... 30

2*8. Function of a function ... 34

2-9. Classification offunctions ... 35

CHAPTER

III Continuity and Limit

3-1. _{Continuity of a function} ... 37

32. Limit ... 41

3*3.

Theorems

on limits .. 52

3-4. _Continuity ofsum, difference, product

and

quotient oftwo continuous functions.

Con-tinuity ofelementary functions ... 53

35.

Some

_{importantproperties} ofcontinuous functions 54

361.

Limit ofxn as -> oo ... 57

3'62. Limit ofxnjn I as n -> oo ... 58

3-63. Limit of(1

+

1//I)" as n ->oo ... 59

3-64. Limitof _[(a*~l)/Jt] as

x

-0

-... 63 3-65. Limit of[(x X

-^)/(x-o)]

as x->a ... 64 3-66. Limit of_{(sin .v/x)} as x->0 ... 64

3-7. _{Hyperbolic Functions}

and

their _graphs ... 66

)

CHAPTER

IV

Differentiation

4-1. Indroduction.

Rate

_{of change} ... 72

4*11. _{Derivability.} Derivative ... 72

4-12.

Derived

Function ... 74

4-14.

An

_{Important theorem}

... 74

4-15. Geometrical _{Interpretation} of a derivative ... 76

4*16. _Expressions for _velocity

and

acceleration ... 78

TO 4 22. Derivative of

x

a

4-3.

_Spme

_general theorems

on

differentiation ... 81

4-34. Derivative of a function ofa function ... 86

4*35. Differentiation ofinverse functions ... 88

4-36. Differentiation offunctions defined

_{by means}

ofa

_parameter

... 88

4*4. Derivatives_{of Trigonometrical Functions} ... 89

4*5. Derivative of_{Inverse Trigonometrical Functions} 93 4-61. Derivative of_loga

x

... 96

4-62. Derivative of (f ... 97

4-71. _{Derivatives of Hyperbolic Functions} ... 97

4-8. _{Derivatives of Inverse Hyperbolic Functions} ... 99

4-91. _Logarithmic differentiation ... 100

4*92. Transformationbefore differentiation ... 102

4-93. Differentiation 'ab initio* ... 104

Appendix

... 108

CHAPTER

V

Successive Differentiation

5-1. Notation ... 113

5*2. Calculation ofnth derivative.

Some

standard

results ... 116

5*3. Determinationof the th derivative of rational

functions ... 118

5-4. Determination of the nth derivative ofa

pro-duct of the

_powers

ofsines

and

cosines ... 120

5*5. Leibnitz's

theorem

... 121

5-6. Value of the nth derivative for

x0

... 123

CHAPTER

VI

General Theorems.

Mean

Value

Theorems

Introduction ... 130

tvl.

Rolled theorem

... ₁₃₀

6*2. _Lagrange's

mean

value

theorem

... 133

6*3.

Some

deductions

from

mean

value

theorem

... . 136

(

vn

)

6*6,6-7. Taylor's

development

ofa function in a finite

form ; Lagrange's

and

Cauchy's forms of

re-mainders ... 140

Appendix ... 145

CHAPTER

VII

Maxima

and

Minima,

Greatest and Least values

7-1. Definitions ... 148

7*2.

A

_{necessary condition} for extreme values ... 149

7-3,7-4. Criteria for extremevalues ... 151

7*6. _{Application to problems} ... 157

CHAPTER

VIII

Evaluation oflimits. Indeterminate forms

8-1. Introduction . ... 165

8-2. Limitof _[/(x)/ir

(x)]

when/(x)->0

and

F(x)->0 ... 166

8-4. Limit of_{[/(x)/F(x)] when/(;c)->oo}

and

_F(xj -*oo ... 170

8-5. Limit of _[/(x).F(x)]

_when/(x)->0

and

_F(x)->oo ... 173

8-6. Limitof_{[/(*) F(x)Jwhen/(x)->oo}

and

_F(x)->oc 174

F'(

\Y

8'7. Limit of _/(x) l" '

under various conditions ... 175

CHAPTER

IX

Taylor's Infinite Series

9*1. Definition of_convergence

and

of the

sum

ofan

infinite series ... 179

9-2,9*3. Taylor's

and

Maclaurin's infinite series ... 180

9-4.

Formal

_expansionsof functions ... 181

9-5.

Use

ofinfinite series for_evaluating limits ... 185

Appendix. Rigorous proofs of the expansions of ex, sin x, cos x, log (l-fx),

(l+x)

m ... 188

CHAPTER

X

FUNCTION

OF

WO

VARIABLES

Partial Differentiation 104 1. Introduction ... 193

10'2. Functions of

two

variables

and

their

domains

ofdefinition ... 193

10*4. _Continuity ofafunction of

two

variables ... 194

10*5. Limit ofa function of

two

variables ... 195

10'6. Partial Derivatives ... 196

10*7. Geometrical _{representation of a function} of

two

variables ... 197

10*71. Geometrical _{interpretation of}partialderivatives

(

10*81. Eider's

theorem

on

Homogeneous

functions ... 199

10*82. _{Choice of independent} variables.

A

new

Nota-tion ...

204

10*91.

Theorem

on

total Differentials.

_Approximate

Calculation ... 206

10*93* Differentiation of

_Composite

Functions ...

210

10*94. Differentiation of_Implicit Functions. ... 213

Appendix. Equalityof

Repeated

Derivatives.

Extreme

values of functions of

two

variables.

Lagrange's Multipliers ... 230

Miscellaneous Exercises I ... 232-237

CHAPTER

XI

Some

Important Curves

IT!. _{Explicit Cartesian} Equations.

Catenary

... 238

11*3. Parametric Cartesian Equations. Cycloid,

Hypocycloid. Epicycloid ... 239

11*4. _{Implicit Cartesian Equations.}

Branches

ofa curve. Cissoid, Strophoid, semi-Cubical

para-bola ... 243

11*5. Polar Co-ordinates ... 247

11-6. Polar _Equations. Cardioide, Leinniscate.

Curves

r

m

=a

m

cos _{w0, Spirals,}

Curves

r~a

sin 3$

and

r=a

sin U0 ...

248

CHAPTER

XII

Tangents and

Normals

12-1; Explicit, Implicit

and

Paramedic

Cartesian

equations ..

254

12*2.

Angle

ofintersection of

two

curves .. 262

12*3. Cartesian _sub-tangent

and

sub-normal ..

264

12*4. _{Pedal Equations. Cartesian equations} .. 266

12*5.

_{Angle between}

radius vector

and

_tangent .. 267

12-6. Perpendicular

from

thepole to a tangent ..

270

12-7. _{Polar sub-tangent}

and

sub-normal

.. 271

12-8. _{Pedal Equations. Polar equations} .. 272

CHAPTER

XIII

Derivative of

Arcs

13'1.

On

the

_meaning

of lengths ofarcs.

An

axiom

... 274

13>2.

_Length

of

an

arc as a function .,. 275

13-3.

To

determine ds/dx for the curve

_y=f(x)

...

275

13-4.

To

determinedsjdt for the curve

*=/(0,

?=/(')

-

276

CHAPTER

XIV

Concavity, Convexity, Inflexion

14-). Definitions ... 281

14*2. _{Investigation of the conditions} for a curve to be concave

_upwards

or

downwards

or to

have

in-flexion at a_point ... 282

1-t-3.

Another

criterion for _point of inflexion ...

284

14-4. _Concavity

and

convexity w.r. to line ...

285

CHAPTER

XV

Curvature, Evolutes

15-1. Introduction, Definition ofCurvature ...

290

15-2. Curvature ofa circle ... 291

15vJ.

Radius

ofCurvature ... 291

15-4.

Radius

ofCurvature for Cartesian curves. Explicit Equations. Implicit Equations. Parametric _Equations.

Newton's

for-mulae

forradius ofcurvature. Generalised

Newtonian

Formula. ... 292

lo'-l'O.

Radius

ofCurvature for _{polar curves} ...

298

15-47.

Radius

ofCurvature for _pedal curves ...

299

15*48.

Radius

ofCurvature for _{tangential polar curves} ... 300

15-51. Centre Curvature. Evolute. Involute. Circle

of curvature.

Chord

ofcurvature ...

304

15*5").

Two

_{properties of evolutes} ... 310

CHAPTER

XVI

Asymptotes

16-1. Definition ... 313

16'2. Determination of_Asj^mptotes ... 313

IO31. Determination of

_Asymptotes

_{parallel to}

co-ordinate axes ,.:_< ...

315

lfi-32.

_Asymptotes

_{of general rational Alge^raic'Equations} 317

1(5-4.

_Asymptotes

_by

_inspection

"

...

324

lfi-5. Intersection of a curve

and

its _asymptotes ...

325

](>(>.

_Asymptotes

_by

_expansion ...

328

I(r7. Position ofa curve relative toits

_asymptotes

...

329

1C-8.

_Asymptotes

in _{polar co-ordinates} ... 331

CHAPTER

XVII

Singular Points, Multiple Points

17*1. Introduction ...

335

17-2. Definitions ...

336

17-31.

_Tangents

at the _origin ... 336

Types

of cusps

17-6. Radii of curvature_{at multiple points} ...

345

CHAPTER

XVIII

Curve Tracing

18*2. Procedure for _tracing curves ...

348

18-3.

_Equations

ofthe

form

}> 2

=/(x) ...

349

18-4.

_Equations

ofthe

form

_{y*+yf(x)+F(x)-^0}

...

357

18-6. Polar Curves ...

359

18-6. Parametric

_Equations

...

364

CHAPTER

XIX

Envelopes 19f

l.

One

_parameter

_family ofcurves ... 369

19*2, 19'3. Definitions.

Envelopes

ofy=MX-t-ajm

obtained 'ab initio* ...

369

19*4.

Determination

of

_Envelope

...

270

19*5. Evolute of curve as the

_Envelope

of its

normals

... 37:2

19*6. Geometrical relations

between

a family of curves

and

its _envelope ... 372

Miscellaneous Exercises II ... 378-82

CHAPTER

I

REAL

NUMBERS

FUNCTIONS

Introduction.

The

_{subject of} Differential Calculus takes its

stand

_upon

the _aggregate of

numbers

and

it iswith

numbers and

with the various operations with

them

that it _{primarily concerns}

itself. _{It specially} introduces

and

deals with

what

is called _Limiting

operation in addition to _{being concerned} with _{the Algebraic} opera-tionsof Addition

and

_{Multiplication}

and

their inverses, Subtraction

and

Division,

and

is a developmentof the _important notion of

Instan-taneous rate _{of change} which is itselfalimited _{idea and, as such,}it

finds application to all those branches of

human

_{knowledge which}

dealwith the same.

Thus

it is _appliedto _Geometry, Mechanics

and

other branches ofTheoretical _Physics

and

also toSocial Sciences such as Economics

and

_Psychology.

It

_may

be noted here thatthis_application is _essentiallybased

on

the notion of _measurement,

_whereby

we

_{employe-numbers} to measure the _{particular quantity}or _{magnitude which}is the object of investigation in

any

department of _knowledge. In Mechanics, for

instance,

we

are concerned with the notion oftime and, therefore, in the application of Calculus to Mechanics, the first _step istd~correlate the

two

notions of

_{Time and Number,}

i.e., to measure timeinterms

of numbers. Similar is the case with other notions such as _Heat, Intensity of Light, Force,

Demand,

Intelligence, etc.

The formula^

tionof

an

entity in terms of _numbers, i.e., measurement, must, of

course, take note of theproperties which

we

intuitively associate with the same. This

remark

will later

on

be illustrated with reference to the concepts ofVelocity, Acceleration Gwry^fr&e, etc.

The

_importance of_{numlTe^l^/Bfe-sradj^bf}

_i^subject

in

hand

being thusclear,

we

will in some*.ofthe;:f#ltowig.articles, see

how we

were first introduced to the notion of

number and

how, in courseof

time,thisnotion

came

to*be _subjectedto a series of_{generalisations.} It is, however, notintendedto give here

any

logically

connect-ed amount

of the _development of tl>e systemofreal numbers, also

known

as Arithmetic Continuum

and

_{only a very} brief reference to

some

well

known

salient facts will suffice for our purpose.

An

excellent account of the. _-Development of

numbers

is _given in

'FundamentalsofAnalysis'

by

Landau.

It

_may

also be mentioned here.that_{even though} itsatisfies a

deep

philosophical need to base the _{theory part of Calculus}

on

the notion *>f

number

ialohe, to the entire exclusion of every physical basis, butarigid insistence

on

the

same

is not within the scope of

this

book

and

intuitive _geometrical notion ofPoint, Distance, etc., will sometimes be_appealed to for _{securing simplicity.}

1-1. Rational numbers and their _{representation}by pointsalong

a

straightline.

I'll. Positive _Integers. It

was

to thenumbers, 1, 2, 3, 4, etc.r

that

we

werefirst introduced_through the _processof counting certain

objects.

The

totalityof these

numbers

is

known

as_{the aggregate} of naturalnumbers, wholenumbersorpositive integers.

While _{the operations of addition} and _{multiplications}

are^,un-restrictedly possible in relationto the aggregate of positive integers, this isnot the _{case in respect} of the inverse _{operations of subtraction}

and

'division. _Thus, for_example, the _symbols

are_{meaningless in respect} of_{the aggregate ofpositive integers.}

1*12. Fractional numbers.

At

a later _stage, anotherclass of

numbers

like_{p\q (e.g., |, |)} where

_{p and}

_qare natural numbers,

was

added

to the former class. This is

known

as the class offractions

and

it_obviously includes natural

numbers

as a sub-class ; q being

equal to 1 in this case.

The

introduction of Fractional

numbers

is _motivated, from

an

abstract point of view, to render Division _{unrestrictedly possible}and,

from

concrete _point of view, to render

numbers

serviceable for

measurement

also in addition _{to counting.}

1-13. Rational numbers. Still later, the class of

numbers was

enlarged

by

incorporating in it the class _{of negative} fractions includ-ing negative integers

and

zero.

The

entireaggregate ofthese

numbers

is

known

as the _aggregate of rational numbers.

_Every

rational

number

is _{expressible as p\q,} where

_p

and

_q are

_{any two}

_integers, positive

and

negative

and

qis not zero.

The

introduction of _Negative

numbers

is _motivated, from an

abstract _point of view/ to render Subtraction _{always possible and,} from concrete _point of view^, to facilitate a unified treatment oi

oppositely directed pairs ofentities such as, gain

and

loss, rise

and

fall, etc.

1-14. Fundamental _operationson rational numbers.

An

impor-tant _property of the _aggregate of rational

numbers

is that the operations ofaddition, multiplication, subtraction

and

division can be performed

upon any two

such numbers, (with one exception which

is considered below in

_{T15) and}

the

number

obtained as the result of these operations is _{again a} rational number.

This _property is

expressed

by

saying tli^t the

ag^egate

of

rational

numbers

is closed with _respect to the four fundamental operations.

1-15. _Meaningless operationofdivision _by zero. It is _important

REAL

NUMBERS

3 by zero' which is a _{meaningless operation.} This

_may

be seen as follows :

To

divide a

_by

b

amounts

to _{determining a}

number

c such that

bc=a,

and

the division willbe _{intelligible only,} if

and

_only if, the

determi-nation ofc is _{uniquely possible.}

Now,

there is no

number

which

when

_multiplied

_by

zero pro-duces a

number

other than zero so that a_JO is no

number

_{when 0^0.}

Also any

number

when

multiplied

by

zero _{produces zero} so _{that 0/0}

may

be any number.

On

account of_{this impossibility} in one case

and

indefiniteness in

the other, the operation ofdivision

_by

zero must be _alwaysavoided.

A

_{disregard of}this _{exception often} leads to absurd results as is

illustrated below in (/).

(i) Let jc-6.

Then

;c2

_-36:=.x-6,

or _{(x--6)(jc-f.6)=Jt} 8.

Dividing both sides by jc 6,

we

get

jc+6=l.

6+6=1,

i.e., 12

=

1.

which is _clearly absurd.

Division

_by

jc 6, which is zero here, is _responsible for this absurd conclusion.

(ii)

We

may

also remark inthis connection that

X

^lf

=

(

_*~^~

6)

=x+6,

only

when

*j66. ... ₍₁₎

For

*=6,

the left

hand

_expression, (je 2

36)/(x 6), is

meaning-less whereas the _right

hand

_expression, x-f6, is _equal to 12 so that the equalityceases to hold for Jt=6.

The

_{equality (1)} above is _proved

_by

_dividing the numerator

and

denominator ofthe fraction (x2 36)/(x 6)

by

(.x 6)

and

this

operation ofdivision is _{possible only}

when

the divisor _{(jc -6)}

_^0,

i.e*9

when x^6.

This explains the restrictedcharacter_{of the equality (1).} Ex. 1. Showthat theaggregate ofnaturalnumbers is not closed with

respect to the operations of subtraction and division. Also show that the

aggregateof positivefractions isnot closedwith respect to the operations of subtraction.

Ex. 2. Show that every rational number is_expressible asa _terminating

orarecurring decimal.

To

decimalise_plq, wehavefirst todivide/?by qand theneach remainder,

after multiplication with 10, is to be divided by qto obtain the successive

figures in the decimal expression of p/q. The decimal expression will be

terminatingif,atsomestage,theremainder vanishes. Otherwise, the process

will be unending. In the latter case,theremainderwillalways beoneof the

From

this _stage onward, the quotients will also _repeat themselvesandthe

decimal_{expressionwill, therefore,}berecurring.

Thestudentwillunderstand the_argumentbetterif he actually expresses

somefractionalnumbers, say3/7, 3/13,31/123, indecimalnotation.

Ex. 3. For what valuesofxare_{the following equalities not valid} :

(0 -!.

W

--*+.

-

tan

-1*16. _{Representation of}rational numbers bypoints along a line

or _{by segments of a} line.

The

mode

of representing rational

numbers

by

points along aline or

_by

_segmentsofaline, which

may

be

known

as _{the number-axis,} will

now

be _explained.

We

start with

_{marking an}

_{arbitrary point}

O

on

the number-axis

and

_calling itthe _origin or zero _point.

The

number

zero will

be represented

by

the point O.

The

_point

O

divides the

number

axis into

two

_parts or sides.

Any

one of these

_may

be _{called positive}

and

the other, then negative.

o

i Usually, the number-axis is "

Q

-

"ft

-

drawn

parallel

to the _printed

lines of the _page

and

the _right

Fi - !

hand

_side of

O

_is termed

posi-tive

and

the left

hand

side of

O

_negative.

On

the _positive side,

we

take

an

_{arbitrary length}

OA,

and

call itthe unit _length.

We

say, then, that the

number

1 isrepresented

by

the point A. After _having fixed

an

_{origin,positive sense}

and

a unit _length on the

number

axis inthe

manner

indicated above,

we

are in a _position to determine a _{point representing}

_any

_given rational

number

as explainedbelow :

Positive integers. Firstly,

we

consider

any

positive integer, m.

We

take a _point

on

the _positive side of the line such thatits

distance from

O

is

m

times the unit length

OA.

This _point will be reached

_by

_measuring successively

m

steps each equal to

OA

starting from O. This _point, then, is said to _{represent the positive} inte-.

ger, m.

Negative integers.

To

represent a negative integer, m, w< take a_point

on

_{the negative}side of

O

such that its distance from

is

m

_{times the unit length}

OA.

This _{point represents the negative integer,}

m.

Fractions. _Finally, let_p\q be

_any

fraction ; q being a _positive

integer. Let

OA

be divided into q equal parts ;

OB

being one

of them.

We

take _{$ point}

on

the _{positiveor negative} side of

O

accordingas

p

is_positive or _{negative such that} its distance

from

O

is

p

times _(or,

_p

times_jf

_p

is _{negative) the distance}

OB.

REAL

NUMBERS

6 Ifa_point

P

_represents a rational

number

_pjq, thenthe measure of the _length

OP

is _{clearly p\q}or _p\qaccording as the

number

is

positive or negative.

Sometimes

we

_saythat the number, p/q, is represented

by

the

segment

OP.

1-2. Irrational numbers. Real numbers.

We

have seen inthe

last article that _every rational

number

can be _represented

_by

a point ofaline. _Also, it is_easy, toseethat

we

can cover the line with such points as closely as

we

like.

The

natural _question

now

arises,

"Is the converse true ?" Is it _{possible to assign} a rational

number

to_{every point of the} number-axis ?

A

_{simple consideration,} as

de-tailed _{below,will clearly}

show

thatit is not so.

Construct a square each of

whose

sidesis _equalto_{the unit length}

OA

and

_{take a point}

P

on

the nutnber-axis such that

OP

is _equal in

the lengthto the _diagonal of the _square.

It will

now

be

shown

+hat the _point

P

cannot _correspond to a rational

number

i.e., the _length of

OP

cannot have a

rational

number

as its measure.

u

7i

r

Fig. 2.

If _possible, let its measure be a rational

number

_pjq so that,

by

Pythagoras's theorem,

we

have

=

_{2, i.e.,p* 2q*. (0}

We

_may

_suppose that

p

and

_qhave

no

common

factor, for, such factors, ifany, can becancelled to begin with.

Firstly

we

notice that

so_{that the square of} an even

number

is even

and

that of

an odd

number

isodd.

From

thHtion

_(/),

we

see, that

p

2 is an even number.

Therefore,

p

itlHpinst

be even. Let, the

Thus, #2 is alsc

equalto 2n where nis

an

_integer.

Hence

_p

and

_{#mfcrommo*factor}

2

and

this conclusion

con-traflicts the hypothesis

thaPthej^fce

no

common

factor.

Thus

the

measure _-y/2 of

OP

isnot a rationaffiumber. There exists,therefore^ a _point

on

the number-axis not_{corresponding to}

_any

rational number, Again,

we

take a point

L

on

the line_{such that the length}

QL

The

_length

OL

cannot have arational measure. For, if

possi-ble, let

_m\n

be the measureof

OL.

p /0

m

mq

-\/2=-- or \/2

=

>

q v n

v

_p '

which statesthat _<\/2 is a rational _{number, being equal} to _mqjnp. This is a contradiction.

Hence

L

cannot _correspond to a rational number.

Thus

we

see thatthere exist anunlimited

number

of _points on the number-axis whichdo_{not correspond} to rational numbers.

If

_{we now}

_{require that our aggregate of}

_numbers

_{should be such} that afterthe choice ofunit _length

on

the line, every point of the

line shouldhave a

number

_{corresponding} to it _(or that _{every length}

should be _{capable of measurement),}

we

are forced to extendour

sys-tem

of

numbers

further

_by

the introduction of

what

are called

irra-tionalnumbers.

We

will thus associate an irrational

number

to _{every point} of the line _{which does not correspond} to a rational number.

A

method

of _representing irrational

numbers

in thedecimal

notation is _givenin the next article 1-3.

Def. Real number.

A

number, rational or irrational, is calleda realnumber.

Theaggregate of rational and irrational

number

is, thus, the aggregate ofreal numbers.

Each

real

number

is _represented

_by

some _{point of the}

number-axis and each _point of the number-axis has

some

real number, rational or_{irrational, corresponding}to it.

Or,

we

might say, that each real

number

is the measureofsome length

OP

and

that the _aggregate of real

numbers

is _enough to measure_everylength.

1-21.

Number

andPoint. If

_any

_{number, say} x, is

represent-ed

_by

a _point P,then

we

usuallysay that the point

P

is x.

Thus

the terms,

number

and _{point, are generally} used in

an

indistinguishable manner.

1-22. Closed _{and open} intervals, seta,b betwo given

numbers

such that

#<6.

Then

the set of

numbers

x

such

_thata^x^fe

is

calleda closed intervaldenoted

_by

the _symbol [a, b].

Also the set of

numbers

x

such that

a<x<b

is called an openintervaldenoted _bythe _{symbol (a,b).}

The number

b ais referred to as _{the length of}[a, b]as also of

<.

*).

1-3. Decimal _{representation} of real numbers. Let

P

be

_any

given point ofthe number-axis.

We

now

seek to obtain the decimal representationof the

number

associated with the point P.

REAL

NUMBERS

7

To

start with,

we

suppose that the point

P

lies on the _positive side ofO.

Let the _{points corresponding} to _integers be

marked

on

the number-axis sothat the wholeaxis isdivided into intervals _{of length} one each.

Now,

if

P

coincides _{with a point ofdivision,} it _correspondsto

an

integer

and

we

need proceed no further. Incase

P

falls between

two

points of division, say a, a

+

l,

we

sub-divide the interval (a,

a+l)

into _{10 equal parts} so_{that the length of each part} is _T1_T.

The

_points

ofdivision, now, are,

#, tf-fTIP

If

P

coincides with any ofthese points ofdivision, then it corres-ponds to a rational number. In the alternative_case, it fallsbetween

two

_{points of}division, say

i.e.,

o.a

_v

_tf.K+1),

where, al9 is

any

one of the integers 0, 1, 2, 3, , 9.

We

again sub-divide the interval

r

, a, , <Ji+i

n

L*

+

io'

a+

10

_J

into _{10 equal parts} so _{that the length of each part} is _1/102.

The

_points ofdivision, now,are

fll _!_" 4_ l - i a_> . 2 //-u'7! . 9 io fl

+io

+

io'

The

_point

P

will cither coincide with one of the above _points of division (in which case it _corresponds to arational _number) or will

lie between

two

_points ofdivision _say 10*

i.e.,

where

a₂isone of the_integers 0, 1, 2,..., 9.

We

_{again sub-divide} this last interval

and

continue to _repeat the _process. After a

number

of_{steps, say}n, the point

P

will either

be

found to coincide with

some

_point of division _(in this case it

corresponds toa rational number) or lie between

two

_points of the

form

the distancebetween which is _1/10*

and

which _{clearly gets} smaller

and

smaller as nincreases.

The

_process can _clearly be continued _{indefinitely.}

The

successive intervals inwhich

P

lies_go on _shrinking

and

will

clearly close

up

to the point P.

This point

P

isthen _represented

_by

theinfinitedecimal Conversely, consider

any

infinite decimal

and

construct the series ofintervals

[a, 0+1], [a.a& a.a^+l], [

Each

of these intervals lies _{within the preceding one}

; their

lengths go

on

diminishing and

by

taking n sufficientlylarge

we

can

make

the_{length as}nearto zero as

we

like.

We

thus see thatthese

in-tervals shrink to a _point. This fact is related to the _intuitively perceived aspectof the continuity of a straightline.

Thus

thereisoneand _{only one point}

common

to this series of intervals

and

thisis the _{pointrepresentedby}the decimal

Combining the results of this article with that of Ex. 2, !!,

p. 3,

we

seethat every decimal,finite or infinite, denotes a numberwhich isrationalifthe decimal is _{terminating or recurringand}irrational in the contrary case.

Let, now,

P

lie on the _negative side ofO.

Then

the

number

representingit is

#-#i#2**#i...

where

a.a^

...a_n...

is the

number

representingthe pointP'on the positive sideoft?such

that

PP'

is bisected at O.

Ex. 1. Calculate thecuberoot_of2to threedecimalplaces.

We

have

I=*l<2and2

3

_=8>2.

l<3/2<2.

We

consider thenumbers

1,M,

1'2,... 1-9,2,

whichdivide the interval[1, 2]into 10 equal partsandfindtwosuccessivenunv

bers such that the cubeof thefirst is

<2

andof thesecondis >2.

We

findthat

Againconsider thenumbers

1-2, 1*21, 1-22 ..., 1-29, l'3r

REAL NUMBERS

We

findthat

(l'25)8=l-953125<2and(l-26)8-2'000376>2.

l-25<3/2<l-26. Again,thenumbers

1-25, 1-251, 1-752...,1*259,1-26

divide the interval[1-25, 1-26] into 10 equal parts

We

find that (1-259)=1-995616979<2 and(1'26) 8 =2'000376>2 !-259<3/2<l-26. Hence

Thusto threedecimalplaces,wehave

?/2=l-259.

Ex. 2. Calculatethecube rootof5 to2 decimalplaces.

Note. The method describedaboveinEx. 1 whichisindeed very cum-bersome, has only been given to illustrate the basicand elementary natureor

the_problem. In actual _practice, however, other methods involving infinite

seriesorotherlimitingprocessesare employed.

1*4.

The

modulusofa real number.

Def.

_By

the_{modulus of a}realnumber, x, is meant the number

x,

x

or _{according as}

x

is_{positive, negative,}orzero.

Notation* _{The smybol}

\

x

\

isusedto denote themodulus ofx.

Thus

the

modulus

of a

number

means

the

same

_thing as ite

numericalor absolute value. _{For example,}

we

have

|3

|

=3;|

-3

|

=-(-3)=3;

[ |

=0

; |

5-7

|

=

|

7-5

|

=2.

The modulus

of the difference between

two numbers

is the

measure of the distance between the _{corresponding} points on the number-axis.

Some

results _involving moduli.

We

now

state

some

_simple

and

useful results _{involving the} moduli of numbers.

1-41.

|

a+b

|<

|

a

|

+

| b | ,

i.e., themodulus ofthe

sum

of two numbers is less than or equal to the

sum

_oftheirmoduli.

The

result is almost self-evident.

To

enable the reader to see

its truth

more

_clearly,

we

_split it

_up

into

two

cases _giving exampfes-ofeach.

Case 1. Let a,

b

have the

same

sign.

In this case,

we

clearly have

I

a+6

I

=

| a |

+

|

b

| .

e.g., |

7+3

|

=

| 7 |

+

| 3 | ,

DIFFERENTIAL CALCULUS

Case II. Let a, b have oppositesigns.

Inthiscase,

we

clearly have

I

a+b

|

<

| a |

+

| b \ , e.g.,

4=

|

7-3

|

<

| 7 I

+

| 3 |

=10.

Thus

in either case,

we

have

I

a+b

\

<

\a\

+

\b\

. 1-42. | ab |

=

| a | . | b | ,

e.g., themodulus oftheproduct of two numbers is_equaltothe

pro-duct _oftheirmoduli,

e.g-, | 4-3 |

=12=

| 4 | . | 3 | ; |

(-4)(-3)

|

=12=

|

-4

| . |.

-3

| ;

1-43. Ifx, a, /, be three numberssuch that

I x I <*> (^)

then

i.e.,

x

lies between a Iand_{a-\-lor that}

x

_belongs to the _open interval

The

_inequality (A) implies thaithe numericaldifference between

a

and x must

be less than /, so that the point

x

_(which

may

He tothe

right or totheleft ofa) can, atthe most, be at a distance / from the

pointa.

a,

d+t

Now,

from the figure,

we

clearly

seethat this is _possible, if

and

_only

Fig 3 iff

x

lies between a I

and

a+l.

It

_may

also be at once seen that

I

*-!<'

is _equivalent to _sayingthat, x, belongs to the closed interval [a-l, a+l]. Ex. 1. //1

a-b

| </, 1

b-c

\

<m,

show that \

a-c

| </+/w.

We

have |

a-c

\

=

1

a-b+b

c\

<

1

a-b

\

+

\

b-c

\

<l+m.

2. Give the equivalents of the following in terms of the modulus notation :

(0

_-1<*<3.

() 2<x<5.

(m)

-3<;t<7.

(iv)

/~s<x</+s.

3. Givethe equivalentsofthe followingby doingawaywith the

modu-lusnotation:

(/) | x

_

2

| <3. (//)

\x+\\ <2.

(7)

0<

|

x-\

| <2.

4. If

_y=

\ x\

+

1jc 1 | , thenshowthat

l 2x,fora;<0 forO<A:<l.

FUNCTIONS

11

1*5. Variables, Functions.

We

give below

some

examples to enable the reader to understand

and

formulate the notion of a vari-able

and

a function.

Ex. 1. Consider two numbers

_{x and y}

connectedby the relation,

"where

we

take _{only the positive value of}the_square root.

Before _considering this relation,

we

observe that there is

no

real

number

whose

_square is _negative

and

hence, so far as real

num-bers are _concerned, the _square root _{of a negative}

number

does not

exist.

Now,

1 jc2, is positive or zero so long as

x

2

is less than or

equal to 1. This is the case if

and

_only if

x

is

_any

number

satisfying the relation

i.e.,

when

x

belongs to the interval

[1,

I].

If,

now,

we

assign

any

value to

x

_belonging to the interval

[ 1, 1], then the given equation determines a unique corresponding value ofy.

The

_symbol

x

which, in the _present case, can take

up

as its

value

_any

number

_belonging to the interval [ 1, 1], is called the

independent variable

and

the interval

_[1,

_{1] is} called its domain

o?

variation.

The

_{symbol y}

which has a _{value corresponding} to each value of

x in the interval

_[1,

_{1] is}calledthe dependent variable or &_function of

x

defined in the interval [" 1, 1].

2. Consider the two numbers

_{x and y}

connectedby the relation,

Here, the determination of y for

x

~2

involves the meaningless operation of division

by

zero _and, therefore, the relation doesnot assign

any

value to

_y

_{corresponding} to jc=2.

But

for _every other value of

x

therelationdoes _assigna value to_y.

Here,

x

isthe _independentvariable

whose domain

of variation consists of the entire_{aggregate of}real

numbers

_{excluding the}

number

2

_{and y}

is a function of

x

defined for this

domain

of variation ofx.

3. Considerthe two numbers

x and y

with their _relationship

definedby the equations

y=^x*

when

x<0,

...(/)

y=x

when

_0<x<l,

_(")

y=ljx

when

x>l.

...(iii)

Theserelations _assign a definite value to

_y

_{corresponding} to every volue ofx, although the value of

y

is notdetermined

by

asingle

formula as in

_Examples

1

and

2. In orderto determine a value of

y

corresponding to a given value ofx,

we

have

to select one of the three equations depending

upon

the value of

x

in question.

For

instance,

forx=

2,

y=(

2)2

=4,

[Equation (/)

v

2<0

forx=,

j=J

[Equation (i7)

v 0<J<1

for

*=3,

_^=|

[Equation (in)

v

3>1.

Here

_again,

_y

is a functionof

x

9 defined fortheentire aggregate

ofreal numbers.

This _exampleillustrates

an

_{important point that} itis not neces-sary that only one formula should be usedto determine

_y

as a func-tion of x.

What

is _required is _simplytheexistence ofalaw or laws which _assign a valueto

_y

_{corresponding} to each value of

x

in its

domain

ofvariation.

4. Let

Here y

is a function of

x

defined for _{the aggregate of positive}

integers only. 5. Let

Here

we

have a function of

x

defined for the entire _{aggregate of} real numbers.

It

_may

benoticed that the

same

functioncan also be defined as follows :

ys=

x

when

y=

x

when

x<0.

6. Let

yt=z\lqf when

x

is a rational

number

plq in its lowest terms r

y=0

t

when x

is irrational.

Hence

_again

_y

is a function of

x

defined for the entire

aggre-gate ofrealnumbers.

1*51. Independent variable and its domain of variation.

The

above examples lead usto _{the following precise definitions}ofvariable

and

function.

_Ifxis

_{a symbol} which doesnot denote _anyfixed

number

but is _{capable of assuming} as itsvalue _any one _{of a}set _ofnumbers,, then

x

is calleda variable

and

this set _{of numbers} is said to be its

domain

ofvariation.

1*52. Functionand itsdomain ofdefinition. _// toeach value _of anindependent variablex, belonging to its domain of variation, there

FUNCTIONS

13

then

_y

is saidto beafunction of

x

definedin the domain

of

variation _of

K, which isthen called the_{domain of}definition ofthefunction.

1*53. Notationfor afunction.

The

fact that

_y

is a function of

a

variable

x

is _{expressed symbolically}as

is readas the _'/' ofx.

Ifx, be

_any

_particular value of

x

_belonging to the

domain

of variationof

X

9then the corresponding value of the function is

denot-ed _{by/(x). Thus, iff(x)} be the function considered in Ex. 3, 1*5, p. 11, then

If functional _{symbols be required} for

two

or

more

_functions,

thenit is usual_{to replace} the latter

_/in

the

_symbol

_f(x)

_by

other

letters suchas F,G, etc.

1*6.

Some

_{importanttypes} ofdomainsof variation.

Usually the

domain

ofvariation ofa variable

x

is an interval

a, b], i.e.,

x

can

assume

as its value

any

number

greater than or

equalto a

and

less than _{or equal}to _{6, i.e.,}

Sometimes itbecomes _necessary to _distinguish between closed

and

_open intervals.

If

x

can take

_up

as its value

_any

number

_greater than aor less

than b but neithera nor b i.e., if

a<x<b

y then

we

say that its

domain

of variation is

an

_open interval denoted

_by

(a, b)to

distin-guish it from _{[a, b]} which denotes a closed interval where

x

can take

up

the values a

and

b also.

We

_may

similarly have semi-clos3(l or semi-opened intervals

[a, b),

a<x<6

; (a, b],

a<x^b

as

domains

ofvariation.

We

may

also have

domains

of variation _extending without

bound

in one or the other directions i.e., the intervals

(00,

b] or

x<6

; [a, GO) or

x>a

;

(00

, oo) or

any

x.

Here

it should be noted that the _symbols 00,00 are

no

numbers in

_any

sense whatsoever. Yet, in the _{following pages they}

will be used in various

_ways

(but, of.course never as numbers)

and

in each case itwill be_explicitlymentionedas to

what

_{they stand} for.

Here, for_example, the

_symbol

( 00, b) denotes the

domain

of

vari-ation ofa variable which can take

_up

as its value

_any

number

less

thanor _equalto b.

Similar _{meanings have been assigned}to the _symbols

(0, GO),

(00

, oo).

Constants.

A

_symbolwhich denotes a certain _fixed

number

is

It has

become

_customaryto use earlier letters of the alp}ifl,betr

like a, h, c

; a, (J,

y

_vassymbols for constants

and

the latter letters

like JC, y, z ; w, v, w*as symbols for variables.

Note. Thefollowing points aboutthe definitionofa function should be

carefully noted :

1.

A

functionneednot benecessarily definedbya formula or formulae so

that thevalue ofthe functioncorrespondingto anygiven valueof the indepen-dentvariableis_{given by}substitution. All that is necessary isthatsome ruleor

setofrulesbe given whichprescribe avalueof the function for every valueof theindependent variable_{which belongs} to thedomainofdefinitionof the

func-tion. _{[Refer Ex.} 6,page12.]

2. It is_{not necessary}that there should beasingleformulaor rule forthe

wholedomainofdefinition ofthe function. _[ReferEx.3,page 11.]

Ex. 1*

Show

that the _{domain of}definition ofthefunction

is the open interval _{(I, 2).}

For

x=l

and

2 the denominator becomes zero. Also for

x<l

and

x>2

the expression (1 -Jt) (x 2) under the radical sign becomes

negative.

Thus

the function is not defined for _*<; 1

and

x^2.

For

x>l

and

<2

the expression under the radical _sign is _positive so thata value ofthe function is determinable.

Hence

the function is defined

in^heopen

interval (1, 2).

2. Showthat thedomainofdefinitionof the "function ^[(1 x)(A* 2)] is

theclosed interval [1,2].

3. Show that thedomainofdefinitionofthe functions are(0, oo)and ( oo,0)respectively.

4. Obtainthedomainsofdefinition of the functions

(0 ^(2x+l) (//) 1/U

+

cosA-) (iii) V(l-f-2 sin x).

1-7. _{Graphical representation of}functions.

Letus consider a function

y=f(x)

defined in an interval [a, b]. ... _(/)

To

_represent it _graphically,

we

take

two

_straight lines

X'OX

fwid

Y'OY

at right angles to each other as in Plane Analytical Geometry* These are the

two

co-ordinate axes.

We

take

O

as _origin for both the axes

and

select unit intervals

on

OX,

OF

(usuallyof the

same

lengths). Also as usual,

OX,

OY

arqtaken as positive directionsonthe

two

axes.

FUNCTIONS

15

To

_{the corresponding}

number

y, avS determined

from

(/), there

corresponds a point

N

on

K-axis such that

N

y

o

Completingthe rectangle

OMPN,

we

obtain _{a point}

P

which is

said to _correspond to the _pair of

y*

numbers

x, y.

Thus

to _every

number

x

be-longingto the interval [<?, b], there

corresponds a

number

y

determined

by

the functional _{equation yc=f(x)}

and

to this _pair of _{numbers, x,}

_y

corresponds a point

P

as obtained

above. Fig%4

The

totality of these points, obtained

by

givingdifferent values, to x, issaid to bethe _graph ofthe function_/(;c)

_{and y=f(x)}

is saidto"

be the _{equation of the graph.}

Examples

1.

The

_graph of the function considered in Ex. 3, 1'5,

page 11, is

Fig. 5 2.

The

_graph of_>>

=

(jc 2

excluding the point P(l, 2).

1) isthe straight line

y=x-\-l

o

Fig. 6

3.

The

_graphof

_y=*x

\ consistsof a discrete setof

4.

The

_graph

ofthe function x,

when

1,

when

I

x when

is as_given. Fig. 7

5.

Draw

the graph of the _function which denotes the_positive square root

o/x

2.

As

_*v/xa_{=jc or}

_x

according as

x

is _positive or _negative, the graph(Fig. 8) of

V*

2 *s the _graph of the function_{/(x) where,}

x,

when

xi f

=H

x,

when x<0.

o

Fig. 8 Fig.9

The

student should

_compare

the_{graph (Fig. 8)} of_\/*2_{with the}

graph (Fig. 9) ofx.

The

reader

_may

seethat

6.

Draw

the_{graph of}

>-

*

Fig.10

We

have

{

x+lx=l

2x

_{when x<0,}

x+lx=l

when

_0<x

x+x

^l=2x

1

when

Thus

we

have the _graph as

drawn

:

The

_graph consists _{of parts of} 3 _straight lines

,,

^

corresponding to the intervals ^-oo,0]. [0, !],[!, x).

FUNCTIONS

17

7.

*

Dawn

the graph of

[*],

where [x] denotes the greatest integer not greater than x.

We

have _f0, for

<*<!,

y=4

1, for 1

<x<2,

1^2, for 2

<ix<3

and

so on.

The

value of

_y

for _{negative values of}

x

can also be _similarly given.

The

_{right-hand end-point of each segment} of the lineis not a

point of the graph.

Yk

~

O

Fig. 11.

Exercises

1.

Draw

the_graphs of the following functions :

1, whenA-<CO ( -v

w

*

-l,when.v>0

<)/(*)

=

_{!-*,

x, whenOjcCjc^i ,.^ f,^ ! . , 2-*,

when^v^i

(")/(*)

=

^ 1, whenx

=

4 ; <0/(*)

=

,when ,when 1 A,when f A-2 whenA<TO f I/*.when

^<

_

_^ A , wften

x^u

,

(v/)/(A-)

-

^ 0, whenA

=

:

\

^v, when

A>0;

^ -l/jc.whe

2.

Draw

the_graphs of the following functions : x.x ^*

2

,.^ , ^(x-\) 2

(/)

00

x

+

Thepositive value of the squareroot istobe takenineachcase. 3.

Draw

thegraphs of the functions :

(0 |*|. (//) |*|

+

|x

+

l|. (///)

2|*-1

|

+3|

4.

Draw

thegraphs ofthe functions:

W

MV

()

M

+

[*fl]

CHAPTER

II

SOME

IMPORTANT

CLASSES

OF

FUNCTIONS

AND

THEIR

GRAPHS

Introduction. _{This chapter}will_{deal with the graphs}

and

some

simple propertiesof the elementary functions

x

n, a

x

, _loga

x

;

sinx, cos x, tan x, cot x, sec x, cosec

x

;

sin~1

x, cc5s~1_x, tan"~3

x, cot~1x, sec^x, cosec-3_*.

The

_logarithmic function is _{inverse of the exponential just as} the inverse _{trigonometric} functions are inverses of the _{corresponding} trigonometric functions.

The

trigonometric functions being periodic, the inverse trigonometric are _{multiple-valued}

and

_special care _has, therefore, to be taken to define

them

so asto introduce

them

as

single-valued.

2-1. _{Graphicalrepresentation of}the_function

y=x*

;

n beingany integer,positive ornegative.

We

have, here, really to discuss a class of functions obtained

_by

giving different integral values to n.

It will be seen that, from the point ofviewof graphs, the whole of this class of functions divides itself into four sub-classes a& follows :

(i)

when

n is a positive even integer ; (//) when n is a positive odd

integer ; (Hi) when n is a negativeeven integer (iv) when n is a negative

odd integer.

The

_{functions belonging} to the

same

sub-class will be seen to

have

_graphs similar in _general outlines

and

_{differing only} in details.

Each

of these four cases will

now

be taken

_up

one

_by

one. 2-11. Let n be a _positive even _integer.

following are, obviously, the properties of the graph of

_y=x

n whatever_positive even _integral value, n

may

have.

(i)

^=0, when

x=0

;

y=l, when

x=l

;

>>=!,

when

x=

1.

The

_{graph, therefore,}

through the points,

O

_(0,0),

A

_(1,1), A'

_(-1,

_1).

(ii)

y

is positive

when

x

is

posi-tiveor_negative.

Thus

no _point

on

the graph lies in the third or the fourth quadrant.

18

The

passes

(it,apositive even integer)