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OSMANIA UNIVERSITY LIBRARYCall No.

g>^^

Accession No.

Author

This book should be returned on or before the date

last

marked below

M

I

RP.

PUBLISH KS

C.

T. C. BapaneHKoe. B. 77 AeMudoeuH, B. A M. Koean, r Jl JJyHit, E noptuneea, C. B. P. fl. UlocmaK, A. P.

E

H. Ctweea,

SAflAMM H VnPA)KHEHHfl

no

MATEMATM H ECKOMVAHAJ1H3V

I7odB.

H.

AE

rocydapcmeeHHoea

M

G. Baranenkov* B. Drmidovich V. Efimenko, S. Kogan, G. Lunts>> E. Porshncva, E. bychfia, S. frolov, /?. bhostak, A. Yanpolsky

PROBLEMSIN

MATHEMATICALANALYSISUnderB.the editorship of

DEMIDOVICH

Translated from the Russian byG.

YANKOVSKV

MIR PUBLISHERSMoscow

TO THE READERMIRopinionbook.of

Publishers would be the translation and

gladthe

to

have yourof

design

this

Please send your suggestions to 2, Pervy Rtzhtky Pereulok, Moscow, U. S. S. R.

Second Printing

Printed

in

the

Union

of

Soviet Socialist

Republic*

CONTENTSPreface9I.1.

ChapterSec.

INTRODUCTION TO ANALYSISFunctions11

Sec. 2 Sec. 3 Sec. 4

GraphsLimits

of

Elementary Functions

16

22

Infinitely Small and Large Quantities Sec. 5. Continuity of Functions

3336

Chapter IISec Sec1.

DIFFERENTIATION OF FUNCTIONSCalculating Derivatives Directly Tabular Differentiation.

4246.

2

Sec. 3 The Derivatwes of Functions Not Represented Explicitly Sec. 4. Geometrical and Mechanical Applications of the Derivative Sec 5 Derivatives of Higier Orders

56

.

60

6671

SecSec Sec

67

Sec. 8

and Higher Orders Mean Value Theorems Taylor's FormulaDifferentials of Firstfor

75

77

9 The L'Hospital-Bernoulli Rule Forms

Evaluating

Indeterminate

78

Chapter III

THE EXTREMA OF A FUNCTION AND THE GEOMETRIC APPLICATIONS OF A DERIVATIVE1.

Sec.

Sec. 2

The Extrema of a Function The Direction of Concavity.

of

One Argument

8391

Points of Inflection

Sec Sec

34.

Sec. 5.

Asymptotes Graphing Functions by Characteristic Points Differential of an Arc Curvature

9396..

101

Chapter IVSec.1

INDEFINITE INTEGRALSDirect Integration Integration by Substitution Integration by Parts

107

Sec Sec

2 3

113116

Sec. 4

Sec. 5.

Standard Integrals Containing a Quadratic Trinomial Integration of Rational Functions

....

118121

ContentsIntegrating Certain Irrational Functions Integrating Trigoncrretric Functions Integration of Hyperbolic Functions

Sec. 6.

125 128 133for

Sec Sec

7.

Sec. 89.

Using Ingonometric and Hyperbolic Substitutions

Findingis

integrals of thetional Function

Form

f

R

(x,

^a^ + bx + c) dx,

Where R

a

Ra133

Sec

101112.

Integration of

Vanou* Transcendental Functions

135

SecSec.

Using Reduction Formulas Miscellaneous Examples on Integration

135136

Chapter VSec.1.

DEFINITE INTEGRALSThe Definite IntegralImproper Integralsas the Limit of a

Sum

138

Sec Sec

2

Evaluating Ccfirite Integrals by Means of Indefinite Integrals 140143 146

Sec. 3

4

Charge

of

Variable in a Definite Integral

Sec. 5.

Integration by Parts

149150

Sec

Mean-Value Theorem Sec. 7. The Areas of Plane Figures Sec 8. The Arc Length of a Curve Sec 9 Volumes of Solids Sec 10 The Area of a Surface of Revolution6

153158161

166168

SecSec

1112.

torrents

Centres

of

Gravity

Guldin's Theorems

Applying Definite Integrals

to the Solution of Physical

Prob173

lems

Chapter VI.Sec.1.

FUNCTIONS OF SEVERAL VARIABLESBasic NotionsPartial Derivatives

180184

Sec. 2. Continuity

Sec

34

185187

SecSec

Total Differential of a Function

5

Sec. 6. Derivative in aSec. 7

Differentiation of Composite Functions 190 Given Direction and the Gradient of a Function 193

HigKei -Order Derivatives and Differentials

197

SecSecSecSec.

Integration of Total Differentials 9 Differentiation of Implicit Functions 10 Change of Variables11.

8

202205

.211217. .

SecSec.

121314

Sec

Sec

15 16

The Tangent Plane and the Normal to a Surface for a Function of Several Variables . The Extremum of a Function of Several Variables .... * Firdirg the Greatest and tallest Values of Functions Smcular Points of Plane CurvesTaylor's Formula.

220222227

.

230 232234

Sec

Envelope

.

.

Sec. 17. Arc Length o! a Space

Curve

ContentsSec.18.

Sec.

19

Sec. 20.

The Vector Function of a Scalar Argument The Natural Trihedron of a Space Curve Curvature and Torsion of a Space Curve

235238

242

Chapter VII.Sec.Sec. Sec.Sec.1

MULTIPLE AND LINE INTEGRALSThe DoubleIntegral in Rectangular Coordinates

Sec.Sec. Sec.

Change of Variables in a Double Integral 3. Computing Areas 4. Computing Volumes 5. Computing the Areas of Surfaces 6 Applications of the Double Integral in Mechanics27.

246 252 256258

259 230262

Triple Integrals

Sec.

8.

Improper Integrals

Dependent

on

a

Parameter.

Improper269

Sec.

Multifle Integrals 9 Line Integrals10.11.

273284 286

Sec.

Surface Integrals

Sec.Sec.

12.

The Ostrogradsky-Gauss Formula Fundamentals of Field Theory

288

Chapter VIII. SERIESSec.1.

Number

Series

293304311

Sec. 2. Functional Series

Sec. 3. Taylor's Series Sec. 4. Fourier's Series

318

ChapterSec.

IX DIFFERENTIAL EQUATIONS1.

lies of

Verifying Solutions. Forming Differential Equations Curves. Initial Conditions

of

Fami322

Sec. 2

Sec.

3.

324 First-Order Differential Equations First-Order Diflerential Equations with Variables Separable. 327Differential

Orthogonal TrajectoriesSec. 4

First-Order

HomogeneousLinear

EquationsEquations.Bernoulli's

330

Sec. 5. First-Order

Differential

332 Equation 335 Sec. 6 Exact Differential Equations. Integrating Factor Sec 7 First-Order Differential Equations not Solved for the Derivative 337 339 Sec. 8. The Lagrange and Clairaut Equations Sec. 9. Miscellaneous Exercises on First-Order Differential Equations 340 345 Sec. 10. Higher-Order Differential Equations 349 Sec. 11. Linear Differential EquationsSec.12.

Linear Differential Equations of Second Order with Constant351

Coefficients

8Sec. 13. Linear

ContentsDifferential

Equations of Order

Higher

than

Two356357

with Constant CoefficientsSecSec14.15.

Euler's Equations

Systems

of

Differential

Equations

359

Sec. 16.ries

Integration of Differential Equations by

Means

of

Power

Se-

36117.

Sec

Problems on Fourier's Method

363

Chapter X.Sec.1

APPROXIMATE CALCULATIONSOperations on Approximate Numbers Interpolation of Functions

367372376 382.

Sec. 2.

Sec.

3.

Computing the^Rcal RootsNumerical, Integration of

Sec. 4Sec. 5.

Equations Functions.

of

Integration of Ordinary DilUrtntial Equations Sec. 6. Approximating Ftuncr's Coefficientser:ca1

Nun

384

3>3396

ANSWERSAPPENDIXI.

475475

II.

Greek Alphabet Some Constants

475

Inverse Quantities, Powers, Roots, Logarithms Trigonometric Functions V. Exponential, Hyperbolic and Trigonometric Functions VI. Some CurvesIII.

476 478479 480

IV

PREFACEThis collection of problems and exercises in mathematical analcovers the maximum requirements of general courses in ysis higher mathematics for higher technical schools. It contains over3,000 problems sequentially arranged in Chapters I to X covering branches of higher mathematics (with the exception of analytical geometry) given in college courses. Particular attention is given to the most important sections of the course that require established skills (the finding of limits, differentiation techniques, the graphing of functions, integration techniques, the applicationsall

of definite integrals, series, the solution of differential equations). Since some institutes have extended courses of mathematics,

the authors have included problems on field theory,

the

Fourier

method,

and

the number of the requireiren s of the student, as far as practical mas!ering of the various sections of the course goes, but also enables the instructor to supply a varied choice of problems in each sectionto select problems for tests and examinations. Each chap.er begins with a brief theoretical introduction that covers the basic definitions and formulas of that section of the course. Here the most important typical problems are worked out in full. We believe that this will greatly simplify the work of the student. Answers are given to all computational problems; one asterisk indicates that hints to the solution are given in the answers, two asterisks, that the solution is given. The are frequently illustrated by drawings. problems This collection of problems is the result of many years of teaching higher mathematics in the technical schools of the Soviet Union. It includes, in addition to original problems and examples, a large number of commonly used problems.

approximate calculaiions. Experience shows that problems given in this book not only fully satisfies

and

Chapter I

INTRODUCTION TO ANALYSIS

Sec.

1.

Functions

ana

1. Real nurrbers. Rational and irrational numbers are collectively known numbers The absolute value of a real number a is understood to be the nonnegative number \a\ defined by the conditions' \a\=a if a^O, and = a if a < 0. The following inequality holds for all real numbers a |ajas realb:

2. Definition of a function. If to every value*) of a variable x, which belongs to son.e collection (set) E, there corresponds one and only one finite value of the quantity /, then y is said to be a function (single-valued) of x or a dependent tariable defined on the set E. x is the a r gument or independent variable The fact that y is a Junction of x is expressed in brief form by the notation y~l(x) or y = F (A), and the 1'ke If to every value of x belonging to some set E there corresponds one or several values of the variable /y, then y is called a multiple- valued function of x defined on E. From now on we shall use the word "function" only in the meaning of a single-valued function, if not otherwise stated 3 The domain of definition of a function. The collection of values of x for which the given function is defined is called the domain of definition (or the domain) of this function. In the simplest cases, the domain of a function iseither a closed interval [a.b\, which is the set of real numbers x that satisfy the inequalities or an open intenal (a.b), which :s the set of real numbers that satisfy the inequalities a a more comx b. Also possible is plex structure of the domain of definition of a function (see, for instance, Prob-

a^^^b,

< 0,of the function is a setof

oo

1.1

Thus, the

domain-\-

two

inter-

and

1

(spiral of Archimedes).

(logarithmic spiral).

(hyperbolic spiral).

/-

= 2cosip (circle). ' = -^- (straight line). = sec*y (parabola).= ==

138*. r=10sin3(p (three-leafed rose) 139*. r a(l fcoscp) (a>0) (cardioid). 2 I 143*. r a cos2(p (a>0) (lemniscate). Cjnstruct the graphs of the functions represented parametrically:t* (semicubical parabola). 141*. x t\ y 142*. *=10 cos/, y=sin/ (ellipse). 3 1 143*. *=10cos /, 10 sin / (astroid). 144*. jc a(cos/-f / sin/), t/ a(sm / /cos/)

=

=

y=

=

(involute of a

circle).

145*. ^

146

'

147. xasfc'-t^143. jc 2cos f f

149. 150.

2- (branch of y=2 = # = 2 sin (segment of *-/t\ y=t x^a (2 cos/ cos2/), = a(2sin/1tt

^'^3,f

=

J/

=rTT'

^0//wm ^

Descartes).

/==

,

2

/

a hyperbola). a straight line).sin 2/) (cardioid).

2

2

/

,

t

*/

Cjnstruct 'the graphs of the following functions defined implicitly:

151*.x152.

= 2jc (parabola). 154. ^1 + ^! = = jc'(10 155. 156*. x T + y T =;aT (astroid).153*.2 i/

25 (circle). xy--= 12 (hyperbola).*/

2

+

2

=

j/*

t

2

157*. x 158. *'

=

Sec. 2]

Graphs

of

Elementary Functions

21

* 159*. |/V y (logarithmic spiral). 8 160*. x* 3x// (folium of Descartes). y 161. Derive the conversion formula Irom the Celsius scale (Q to the Fahrenheit scale (F) if it is known that corresponds

+

2

+

=e =

a"

0C

to

32F

and 100C corresponds to 212F. Construct the graph of the function obtained. 162. Inscribed in a triangle (base 6^=10, altitude h

rectangle (Fig. 5). tion of the base x.

6) is a the area of the rectangle y as a funcExpress

=

Fig. 5

Fig

6

ACB = x (Fig. 6). angle area ABC as a function of Express # of this function and find its greatest value.

Construct the graph of this function and value. 163. Given a triangle ACB with BC a, AC

find

its

greatest

=

=b

and a variable

$

=

A

x.

Plot the graph

164.

Give

a graphic solution of the equations:0;

a) 2x'

b) x*c)

+

5x + 2 = x 1=0;

d)e)f)

I0'

x

= x\45sin;c;

x=lcot

= 0.1jc; logJt

x^x

(0-\ = NThus, for

every

positive

number

there will

be a number

Af=

1

such

N we will have inequality (2) Consequently, the number 2 is that for n the limit of the sequence x n (2n-\- l)/(n-fl), hence, formula (1) is true. 2. The limit of a function. We say that a function / (x) -*- A as x -+ a (A and a are numbers), orlim f(x) x -aif

>

= A,

for

every 8|

>

we havefor

6

=6

()

>f(jO

such that

\f(x)A *10-jA:

Y Lh

^

184.

limlim-

43

8v

+5*

189.

lirn

185.

-r-r~c

*

^5are integral

190.

lim

Vx + VxIf

P(A-)

and Q

(x)

polynomials and P

(u)

+

or

Q

(a)

then the limit of the rational fractionlim

is

obtained directly. But if P(a) Q(a)=0, then

=

it

is

advisable to camel the binomial *

a

out of the fraction

P Q

(x)

once or several times.

Example

3.

lim

/'T

4

^

lim !*""!!)

f

xf ??

Hm

^^4.

26

Introduction to Analysis

101.

lim

^{.*

198.

Um ^*-+>fl

_in

\Ch.

I

192. lim* _.|

*-.*

196. lim

^

The expressions containing irrational ized by introducing a new variable.Example4.

terms are

many

cases rational-

Findlim

Solution. Putting

we

!+* =Mm

havelim

E=11

^

*/',

=

lim

"

2

199. limX -

200. lim

* -4^-. *~l

201. ,'~ ,,. limx.

3/ t/x

~

,

]

T

Another way of finding the limit of an irrational expression is to transfer the irrational term from the numerator to the denominator, or vice versa, from the denominator to the numerator.

Examplelim

5.

=

=

lim

x -+a(X

a)(Vx

_^+ V a)lim!!

*-> a

^

jc

-f

V

a

2\f~i

203.

lim

-. Q -49j-^=.

206.

lim

-=f. __

204.

li.n

*-*

/

207.

lim*-+ COsill

227. a)b)

lim

xsinl;.

b)li.n^. X217.218.219. 220.221.,.

lim x sinX-*00

*-~-

3x

228.229.

limJt-M

(1

x) tan

.

^

sin

5*'

lim X -0

sin

2*=.

* -+0

lim cot 2x cot f-^ *\

x)./

sin JTX

limM

^

sin BJIJCl

230.

lim*Jt

ji

*

lim ( nn-*cc\

sin-). n I

231.

lim

1-2

lim

222.

lim lim lim

232. 233.

lim

cosmxtanA:

- cosnV*

\

*

sui

223.224. 225.226.

limJC

-

arc sin

^

limlimI

crs^ tan*

236.'

limsin six

'

"28

_ _Introduction to Analysis

[Ch. 1

m.

nx

ta-* 1

.=T.I

24

-

n!!?.

*""r

'"""

f

p

Jt

When

taking limits of the formlim l\

M

3x4-2/i

].

251. lirn(l-.o

+ sinjc) *.J_ *;

/^i

2 \*a

245>

Jill ( 2?+T )/1

252**. a) lim (cos x)X ~*.

\

246.

... V

HmflIim(l

-) /

.

b)

H

247

f I)*.

30

When solving the problems that follow, limit lim/(x) exists and is positive, thenlim [In /(*)] x-+a

_ _Introduction to Analysisit

[Ch.if

!

is

useful to

know

that

the

= In

[HmX-+Q

f (x)].

Example

tO.

Prove that

Solution.

We

have

lim X-*0

ln

Xis

X-+Q

Formula

(*)

frequently used in the solution of problems.

253. lim [In*-

(2*+!)X.

254.

li

-

255.

,_* \

" limfjlnl/J-i^). lX/

260*.

n

llmn(^/a ^V)

pCLX

256. lim *[ln(jt+l)0).

ptX.

257.

lim.Hm=.ital!the*

-*

.

-^o

258*.

263. a)(a

limlim

259*.

>0).(see

b)

x*

Problems 103 and

104).

Find

following limits that occur on one side:

264. a)

lira

*_^

.

fa

Hm*" +

i

b)Jirn*265.*-*-*

p===.267-

1+ ' T

a/lLutanh*;b)

a ) lim

limtanh*,*->+

*-b) Hm*-*+

where tanh^ =266. a)lira

^^~.

268. a) limb) |im

V

;

Sec. 31

Limits

31

269. a)

lim-^4i;'

270. a)

Hm-^-; x~*

Construct the graphs of the following functions: 2 \im (cos "*). 271**. y

=

n->oo

*

272*.

y=limn-*c

*

i

xn2.

(x^O).

273. y274.t/

= \ima->o

J/V-t-a

= li;n|=li

275.

t/

-* oo. as n 279. Find the limit of the perimeters of regular n-gons inscribed oo. in a circle of radius R and circumscribed about it as n 20. Find the limit of the sum of the lengths of the ordinates

of the

curve

y

= e~*cos nx,

drawn

x 0, 1, 2, ..., n, as n *oo. 281. Find the limit of the sum of the areas of the squares constructed on the ordinates of the curveat the points

=

as

on bases, where x=^l, 2, 3, ..., n, provided that n 282. Find the limit of the perimeter of a broken line

*oo.

M^.. .Mn

inscribed in a logarithmic spiral

Introduction to Analysis

[Ch.

I

oo), if the vertices of this broken line have, respectively, (as n the polar anglesai.e.,if

|a(x)|-

100* -1,000;

c)

b)

7+2-

Sec. 5. Continuity of Functions

1. Definition of continuity.

A

function

/ (x)

is

continuous

when x =

(or "at the point g"), if: 1) this function is defined at the point g, that is, there exists a number / (g); 2) there exists a finite limit lim f (x); 3) this lim-

x-4it

is

equal to the value of the function at the point

g,

i.e.,

llmf*-*fc

(*)

= /().

(1)

Putting

where Ag

^0, conditionlim

(1)

may belim

rewritten as(g)]

A/(g) =

l/(g+ Ag)-f

= 0.

(2)

or the function / (x) is continuous at the point g if (and only if) at this point to an infinitesimal increment in the argument there corresponds an infinitesimal increment in the function. If a function is continuous at every point of some region (interval, etc.), then it is said to be continuous in this region. Example 1. Prove that the function

yfs

= sin xx.

continuousSolution.

for every

We

value of the argument

havecos

Ay = sinSince

.

_

51

j/

2

j/

t

.

t/

F. Miscellaneous Functions

455**. y=sin'5jccos*y.

15

103)'

458.

j/=

460.461. y 462.

az

^-i-jc 2x*

=:

3

f/

= |4

463.

y=4-

465.

t/

=x

4

(a

__J"2(Jt-i-2)

1 '

468. 469.

|/

=

|

470. z

=

471.

/(0=(2/-

52

Differentiation of Functions

[C/t.

2

473. y474.

= ln(]/l+e*-l)-ln(/l # = ^ cos'x (3 cos * 5).2

475

...

-

= (tan

-2

*

4

l)(tan

x-HOtan 2 *-fl)485. #

476. y=-ian*5x. 477. y

= arc sin = arc sin =cos *

= ^ sin (x2

2

).

486. y

*.

478. j/=sin 479.

(O2

487. y488. y

^V^2

.

*/

= 3sinA:cos =-o-

A;+sin'x.

= 4~- af c sin fx\

I/*

-) a /. CL

480. w481. y

O

tan *

5

ianx + x.

489. y

= K^

x*8

+ a arc sin2

=

^f +cotx.2

490. t/=jt/a491.

^-T +a

arc

sin-.

482. 483.

y=/a sin + p cos x. y = arc sinjc + arccosA;jc2

2

y=arcsin(l

a

.

484. y

= -^ (arc sin*)=

2

arc cos jt.

492.493. 494.

=jc-I y = ln(arcsin5x). y = arc sin (Inx).5tan-i-

495.

496.497. 498. 499. 500.

l

find tind

n0)q/(or

558.

Given the functions /(x)=l.

x

and

cp(jc)

=

sin^

r

nna find

2^ff

(1)

559. Prove that the derivative of an even function is an odd function, and the derivative of an odd function is an even function.

560. Prove that thea periodic function.

derivative of a periodic

functionthe

is

also

xy' = d-x)y-

561.

ShowShow

that

the

function

y = xe~*

satisfies

equatione

3. The derivativeand yis

of an

implicit

the

relationship between

x

given in implicit form,

F(x,y) = Q,then to find the derivative

(I)

y' y' in the simplest cases it is sufficient: 1) to x calculate the derivative, with respect to x, of the left side of equation (1), taking y as a function of x\ 2) to equate this derivative to zero, that is, to put

~F(A:,f/)

= 0,/'.

(2)

and

3) to solve the resulting equation for

Example

3.

Find the derivative

yx

if

0.

(3)

Solution.ito

Forming the derivative3*'

of the left

side

of (3)

zero,

we

and equating

it

get

+ 3y V -3a (y + xy') = 0,

58

Differentiation of Functions

[Ch. 2

whencey

,_* ay ~~axy*'xyf/

2

581. Find the derivativea)

if

=

c)

y

= 0.#'=^.dy*

In the following problems, find the derivative

the

functions y represented parametrically:582.589.(

x = acos*f,

\ y

583.

590.

= b sin* x = acos* y=b sin8

t.

t,1.

cos

3

/

T^nr584.591.sin?=

8

/

V coslr

x585.592.

= arc cosarc sin__*~ ^

y(

586.

593.'

{

y~=e:^

587.

= a( In tan + cos = a(sin + cosO.-2-

sin

^)

>

t

588.t

/

cos/).^

595. Calculate

~

when:

f/t cn a sin

= 4= a(t = a(lsin/

if

sin

/),

cos/).

///i

y

Solution.

-r-~

a(l

cosO

1

cos/

Sec. 3]

The Derivatives

of

Functions Not Represented Explicitly

59

andS1

fdy\

=

"TI

596.

Find~.i

^ **dv dx

when

/

=,

!

ifl

\nxX

-1

*^

]

A

h~? 2 A'

795.796.

lim lim

L__lyA)

_^/

AO J

797

limf-^^'*

*

^Vc2

^}2cosx/have**

798.

lim

A;*.

Solution.

We

=

r/;

In

y=?x

In

A".

lim In

t/

= limjtln x

=

s

lim

p

= limj~

0,

whence lim//=l,

that

is,

ImiA^

l.

82

Differentiation of Functions

(Ch. 2

799. limx*.a

804.

li

V-Htan

800. limx4 * "*.1

805. Hmftan^f) 4 /X-+l\1

\*.

801. linue s/n*->0

*.

806. lim (cot x) lnX-H)costa

802.

lim(l-*)

807.

lta(I) x / x-*o \

".*.

803.

lim(l+xX-+0

2

)*-

808. lim (cot x)* in

809. Prove that the limits of

Xa)sin*

cannot

be found

by the L'Hospital-Bernoulli

rule.

Find these

limits directly.

810*. Show that the area of a circular segment with minor and central angle a, which has a chord (Fig. 20), is

AB=b

CD=A

approximately

with an arbitrarily small relative error when a

->0.

Chapter III

THE EXTREMA OF A FUNCTION AND THE GEOMETRIC APPLICATIONS OF A DERIVATIVE

Sec.

1.

The Extrema

of

a Function of One Argument

of tunctions. Tlu Junction y f(x) is called on some interval if, fo. any points x and x 2 which belong to this interval, from the inequality A',f(x Q ) then the point x is called theJ

Figfor

23

minimum

point

of

the)

functionis

y

f(x),

while the number

x lf the inequality any point xj^x l of some f(*)0 for X Q d/ (0)and so e*as

> +x1

when x when x

^ 0,

we

set

out to prove.

Prove the inequalities: x sin x 858. x

^lA:

^

whenwhen

860.

~--10 graph.c)

We

seek inclined asymptotes, and find

,=bl

limX -> +oo

= 0,oo,

lim#->-t-oo

y

thus, there is no right asymptote. From the symmetry of the curve it follows that there is no left-hand asymptote either. d) We find the critical points of the first and second kinds, that is, points at which the first (or, respectively, the second) derivative of the given function vanishes or does not exist.

We

have:

,

derivatives y' and \f are nonexistent only at that is, only at points where the function y itself does not exist; and so the critical points are only those at which y' and y" vanish. From (1) and (2) it follows that

The

x=l,

y'=Qr/"

=

when x= when x =

V$\and

x=and(1,

3.

Thus,

y' retains al),(

in each of the intervals constant_ sign(l,3,

(

00,

J/T),of

(-V3,intervals

(1,

1),(

V$)

and (V~3

t

+00),(0,1),

00,

3),

1),

To determine the signs of y' (or, respectively, y") in each of the indicated intervals, it is sufficient to determine the sign of y' (or y") at some one point of each of these intervals.

(1,

/3)

in

each(3,

the

0),

and

+00).

Sec

4]

Graphing Functions by Characteristic Points

97

It is convenient to tabulate the results of such an investigation (Table I), calculating also the ordinates of the characteristic points of the graph of the function. It will be noted that due to the oddness of the function r/, it is enough to calculate only for Jc^O; the left-hand half of the graph is constructed by the principle of odd symmetry.

Table I

e)

Usin^ the results33).

of

the investigation,

we

construct

the

graph

of

the

function (Fig

-/

Fig. 33

4-1900

Extrema and the Geometric Applications

of

a Derivative

[Ch. 3]

Example

2.

Graph the functionIn x

xSolution, a) The domain of definition of the function is 0