demidovich problems in analysis
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OU_1 58871 >CD
CO
OUP-43
30-1.71
5,000
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
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