Demidovich Problems in Mathematical Analysis
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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 ECKOMV AHAJ1H3V
I7od B.
H.
AE
rocydapcmeeHHoe a
M
G. Baranenkov* B. Drmidovich V. Efimenko, S. Kogan, G. Lunts>> E. Porshncva, E. bychfia, S. frolov, /?. bhostak, A. Yanpolsky
PROBLEMS IN
MATHEMATICAL ANALYSIS Under B.
the editorship of
DEMIDOVICH
Translated from the Russian by G.
YANKOVSKV
MIR PUBLISHERS Moscow
TO THE READER MIR opinion
of
Publishers would be the translation and
glad the
to
have your
design
of
this
book.
Please send your suggestions to 2, Pervy Rtzhtky Pereulok, Moscow, U. S. S. R.
Second Printing
Printed
in
the
Union
of
Soviet Socialist
Republic*
CONTENTS 9
Preface
Chapter Sec.
I.
INTRODUCTION TO ANALYSIS
1.
Functions
Sec. 2
Graphs
Sec. 3
Limits
of
11
Elementary Functions
16
22
Infinitely Small and Large Quantities Sec. 5. Continuity of Functions
33
Sec. 4
Chapter II Sec Sec
1.
2
36
DIFFERENTIATION OF FUNCTIONS Calculating Derivatives Directly Tabular Differentiation
42 46
Sec. 3 The Derivatwes of Functions Not Represented Explicitly Sec. 4. Geometrical and Mechanical Applications of the Derivative Sec 5 Derivatives of Higier Orders
Sec
6
Sec
7
Sec. 8
.
.
56
.
60
66
and Higher Orders Mean Value Theorems Taylor's Formula Differentials of First
9 The L'Hospital-Bernoulli Rule Forms
Sec
for
Evaluating
71
75
77
Indeterminate
78
THE EXTREMA OF A FUNCTION AND THE GEOMETRIC APPLICATIONS OF A DERIVATIVE
Chapter III
Sec.
1.
Sec. 2
Sec
3
Sec
4.
Sec. 5.
Chapter IV Sec.
1
Sec
2
Sec
3
Sec. 4
Sec. 5.
The Extrema of a Function The Direction of Concavity
of
One Argument
83
Points of Inflection
Asymptotes Graphing Functions by Characteristic Points Differential of an Arc Curvature
91
93
.
96 .
.
101
INDEFINITE INTEGRALS Direct Integration Integration by Substitution Integration by Parts
Standard Integrals Containing a Quadratic Trinomial Integration of Rational Functions
107
113 116
....
118 121
Contents
Sec. 6.
Integrating Certain Irrational Functions
125
Sec
Integrating Trigoncrretric Functions Integration of Hyperbolic Functions
128
7.
Sec. 8
Sec
9.
133
Using Ingonometric and Hyperbolic Substitutions
integrals of the
Form
f
R
(x,
^a^ + bx + c) dx,
for
Where R
Finding is
Ra-
a
tional Function
Sec
10
Sec
11
Sec.
12.
133
Integration of
Vanou* Transcendental Functions
135
Using Reduction Formulas Miscellaneous Examples on Integration
135 136
DEFINITE INTEGRALS
Chapter V Sec.
1.
Sec
2
The Definite Integral
as the Limit of a
Sum
138
Evaluating Ccfirite Integrals by Means of Indefinite Integrals 140
Sec. 3
Improper Integrals
Sec
Charge
4
of
143
Variable in a Definite Integral
146
Integration by Parts
Sec. 5.
149
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 Revolution
Sec
6
Sec
11
torrents
Sec
12.
Applying Definite Integrals
Centres
of
Gravity
150
153 158 161
166
Guldin's Theorems
to the Solution of Physical
168
Prob-
lems
Chapter VI. Sec.
1.
173
FUNCTIONS OF SEVERAL VARIABLES Basic Notions
180
Sec. 2. Continuity
184
Sec
3
Partial Derivatives
185
Sec
4
Total Differential of a Function
187
Sec
5
Differentiation of Composite Functions 190 Given Direction and the Gradient of a Function 193
Sec. 6. Derivative in a Sec. 7
Sec Sec Sec
HigKei -Order Derivatives and Differentials
197
8
Integration of Total Differentials 9 Differentiation of Implicit Functions 10 Change of Variables
202 205
.211 217
15
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 Curves
16
Envelope
232
Sec.
11.
Sec
12
Sec.
13
Sec
14
Sec
Sec
Taylor's Formula
.
.
.
.
.
Sec. 17. Arc Length o! a Space
Curve
.
220 222 227
230 234
Contents Sec.
18.
Sec.
19
The Vector Function of a Scalar Argument The Natural Trihedron of a Space Curve Curvature and Torsion of a Space Curve
Sec. 20.
Chapter VII. Sec.
The Double
1
Integral in Rectangular Coordinates
Sec. Sec.
7.
Triple Integrals
Sec.
8.
Improper Integrals
Sec. Sec.
Sec.
238
242
MULTIPLE AND LINE INTEGRALS
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 Mechanics
Sec.
235
246 252
2
256 258
259 230 262
Dependent
on
a
Parameter.
Improper 269
Multifle Integrals 9 Line Integrals
273
Sec.
Sec.
10.
Surface Integrals
284
Sec.
11.
Sec.
12.
The Ostrogradsky-Gauss Formula Fundamentals of Field Theory
288
286
Chapter VIII. SERIES Sec.
Number
1.
293
Series
Sec. 2. Functional Series
304
Sec. 3. Taylor's Series Sec. 4. Fourier's Series
318
Chapter Sec.
311
IX DIFFERENTIAL EQUATIONS 1.
Verifying Solutions. Forming Differential Equations Curves. Initial Conditions
of
Fami-
lies of
Sec. 2
Sec.
3.
327
Orthogonal Trajectories Sec. 4
322
324 First-Order Differential Equations First-Order Diflerential Equations with Variables Separable. First-Order
Sec. 5. First-Order
Homogeneous Linear
Differential
Differential
330
Equations Equations.
Bernoulli's
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 Equations Sec.
12.
Linear Differential Equations of Second Order with Constant
Coefficients
351
Contents
8 Sec. 13. Linear
Differential
Equations of Order
Higher
than
Two 356
with Constant Coefficients Sec
14.
Euler's Equations
Sec
15.
Systems
of
Differential
357
Integration of Differential Equations by
Sec. 16.
359
Equations
Means
of
Power
Se-
361
ries
Sec
17.
Chapter X.
Problems on Fourier's Method
363
APPROXIMATE CALCULATIONS 367
Sec. 2.
Operations on Approximate Numbers Interpolation of Functions
Sec.
Computing the^Rcal Roots
376
Sec.
1
3.
Sec. 4
Numerical, Integration of
Nun
Equations Functions
Integration of Ordinary DilUrtntial Equations Sec. 6. Approximating Ftuncr's Coefficients Sec. 5.
372
of
er:ca1
382 .
.
384
3>3
ANSWERS
396
APPENDIX
475
I.
II.
Greek Alphabet Some Constants
Inverse Quantities, Powers, Roots, Logarithms Trigonometric Functions V. Exponential, Hyperbolic and Trigonometric Functions VI. Some Curves
475
475
III.
476
IV
478 479 480
PREFACE This collection of problems and exercises in mathematical analcovers the maximum requirements of general courses in ysis higher mathematics for higher technical schools. It contains over 3,000 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 applications all
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,
method,
and
the
Fourier
approximate calculaiions. Experience shows that problems given in this book not only fully satisfies
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 section to 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.
and
Chapter I
INTRODUCTION TO ANALYSIS
Sec.
1.
Functions
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 |aj as real
ana
b:
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 is either 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,
< <
lem 21)
Example
1.
Determine the domain
of definition of the function 1
Solution.
The function
is
defined
x that
is,
if
|x|>
1.
oo0,
domain
/-
134*.
in the polar
functions
(spiral of Archimedes).
(logarithmic spiral).
(hyperbolic spiral).
= 2cosip (circle). ' = -^- (straight line). = sec*y (parabola).
135. r 136. 137.
/-
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 parametri-
= =
cally:
=
=
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
'
^3,
= rTT'
^0//wm ^
Descartes).
/==
2- (branch of y=2 = # = 2 sin (segment of *-/t\ y=t x^a (2 cos/ cos2/), = a(2sin/
147. xasfc'-t^143. jc 2cos f f
149. 150.
J/
t
t
,
2
/
f
a hyperbola). a straight line).
2
2
/
1
,
*/
t
sin 2/) (cardioid).
Cjnstruct 'the graphs of the following functions defined implicitly:
151*.x 152.
2
+
2
=
25 (circle). xy--= 12 (hyperbola). */
= 2jc (parabola). 154. ^1 + ^! = = jc'(10 155. 156*. x T + y T =;aT (astroid). 153*.
2
i/
j/*
t
157*. x 158. *'
=
2
Graphs
Sec. 2]
2
of
Elementary Functions
21
a"
=e =
* 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
+
+
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.
Fig. 5
6
Fig
Construct the graph of this function and value. 163. Given a triangle ACB with BC a, AC
=
ACB = x (Fig. 6). angle area ABC as a function of Express # of this function and find its greatest value.
$
=
Give
164.
a) 2x'
b) x* c)
+
A
find
=b
x.
a graphic solution of the equations:
5x + 2 = x 1=0;
0;
d) e)
= 0.1jc; logJt
f)
I0'
x
x=l cot
= x\ 4
x^x
xy=10, x
b)
xr/-6,
c)
x
d)
*
e)
#=sinx,
f
y
5sin;c;
(0-\ = N Thus, for
every
number
positive
(e).
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), or lim f(x) x -a if
for
every 8
>
\f(x)A
such that
f(jO
= 4,
a| 00
of the sequences:
0.23, 0.233, 0.2333,
Find the
172.
1
1/2
1/2;
d) 0.2,
171.
'
liai/(x) X
00
= 0.001.
following
-oo
Hm
*
C+D
(
"*
+
)(>.
+ 3)
n l)
2n+11 2
178*. lim n
-* CD
J'
notations:
= oo.
6
Limits
Sec. 3]
Hm (Vn +
179.
n -+ -o/%
1
\f~n).
<
/
i
180. lim
When x -+
the
seeking is
it
oo,
useful
of a ratio of two integral polynomials in n to divide both terms of the ratio by x , where
limit
first
the highest decree of these polynomials. A similar procedure is also possible in ing irrational terms.
Example lim
_
2.
* lim
lim
181. r
182.
cases for fractions contain-
1.
J2^-3)(3t-f^)(4A'-6)
Example
many
-. or
lim
=.
.
^rrr. *
*86.
^^.
187.
~'
=
J
lim
1
lim *
+
.,
3* O ^2
184.
185.
lim
V
J"
.
__
+5*
8v
lim
1/
-11-
.
jc
lim ->*
h
3
-\- \
10-j-
A:
^
Y L
4-
188.
.
+7
X*
lim *
lim
1.
^~~^=J.
00
183.
* as n is
189.
lirn
-r-r~c
*
-
^5
190.
lim
Vx + Vx If
P(A-)
and Q
(x)
are integral
polynomials and P
(u)
+
or
Q
(a)
then the limit of the rational fraction lim
is
obtained directly. But if P(a) Q(a)=0, then
=
out of the fraction
Example
P Q
(x)
it
is
advisable to camel the binomial *
once or several times.
3.
lim
/'T
4
^
lim !*""!!)
f
xf ??
Hm
^^4.
a
Introduction to Analysis
26
101.
^{.
lim
*-.
192. lim * _.
*-+>
The expressions containing irrational ized by introducing a new variable. Example
I
196. lim
^
*
|
\Ch.
Um ^
198.
*
_
terms are
fl
in
cases rational-
many
Find
4.
lim
Solution. Putting
we
!+* =
have
E=1
lim
199. lim X -
Mm
^
* -4^-. *~
200. lim
T
=
lim
"
2
,'~ 201. ,,. lim
l
1
*/',
3/ t/x
~
,
]
x .
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.
Example
5.
=
=
lim
lim
x -+a(X
a)(Vx
_^ + V a)
!
lim
*-> a
203.
204.
-49Q-.
lim
li.n
*-*
j-^=
/
x
.
^
206.
207.
2
jc
-f
lim
lim *-+<
1
205.
lim *-+'
^L""
/
*
.
1
208.
lim ^-*o
!
V
a
2\f~i
-=f. __
Limits
Sec. #]
_ 209.
27
212.
K
lim
213
[/*(*
^i 214.
Hm(]/xfa
+ a)
xj.
-6* 4 6-*).
'
210. lim
211.
lim X-
li.
|
Jf--fCO
215.
The formula llm X-
X
-i
r frequently used when solving the following examples. cos a. sin a and lim cos granted that lim sin *
=
Example
a)
!!!
lim;
lim
227. a)
lim
X-*
-> CO sill
,.
3x
228.
xsinl;
lim x sin
b)
b)li.n^. X 217.
lim
00
.
*
x) tan
(1
-~-
lim X -0
sin
5*
sin
2*
229.
'
.
^
Jt-M
218.
taken
is
6. lira
216.
It
*=
lim cot 2x cot f-^ *
* -+0
x). /
\
sin JTX
219.
lim M
^
=
.
sin BJIJC
230.
l
lim *-
220.
lim ( n n-*cc
221.
lim
222.
lim
223.
lim
224.
lim
225.
lim
226.
lim
\
sin-). n I
Jt
231.
lim
232.
lim
cosmx
-
arc sin
crs^ tan*
236. '
sui
A:
lim JC
I
- cosn V*
tan
233.
*
ji
1-2
lim
^
'
sin six
\
*
for
"28
_ _ Introduction to Analysis
m.
24
.=T.
ta
-
!!?.
one should bear
mind
in
C=4"; lim (p(x)
if
= ^ /l
1
cp
A and
(x)
and
if
(x)
where
e
\|?
lim ty(x)^=
is
= B,
(x)
oo,
then the problem of finding
=
we
Hm a
(x)
^
Hm
(x)
a
= 2.718
Example
7.
...
is
Napier's number.
Find lim
Solution. Here, lim
(5111=2 X
and
lim
Jf-^O \
hence, lim x-*o
Example
Solution.
put
q>(x)=
1
+a(x),
as x -+ a and, lien^e,
-*
1
-
lim
solved in straightforward fashion; lim \|) (x) then co, lini(pU)=l and
the limit of (3)
x
(3)
that:
lim
where a
-=C
there are final limits
if
3)
'"""
taking limits of the form lim loo
*
272*.
y=lim n-*c
= \im
273. y
*
i
(x^O).
xn
J/V-t-a
2 .
a->o
274.
t/
275.
= li;n|
t/
=
li
-*<
276. Transform a
common
the
following
fraction:
a
Regard 277.
it
What
mixed
periodic
fraction
into
= 0.13555...
as the limit of the corresponding finite fraction. will happen to the roots of the quadratic equation
if the coefficient a approaches zero while the coefficients b and c are constant, and fc^=0? 278. Find the limit of the interior angle of a regular n-gon > 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,
=
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 curve
drawn
as
at the points
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 angles a i.e.,
<
|a(x)|3.
P'ot the graph of this function.
331. Prove that the Dirichlet function %(x) which is zero for and unity for rational x, is discontinuous for every t
irrational x value of x.
Investigate the following functions for continuity and construct their graphs:
= \in\ y = lim (x arc tan nx).
332. y 333.
Continuity of Functions
Sec. 5]
334. a) y
= sgnx,
function sgn x
is
b)
y
=x
sgnx,
c)
i/
= sgn(sinjt),
where the
defined by the formulas:
+
I
sgn x
=
1,
if
*>0,
0,
if
x=
-1,
if
*.
usually called differentiation of the function. The derivative y'=f' (x) is the rate of change of the function at the point x. Example 3. Find the derivative of the function
Finding the derivative
/'
is
y Solution.
From formula
= x*.
we have
(1)
Ay = (*+
xi
A*)*
2*Ax+ (Ax) 1
and
Hence,
'=
L^
lim
Ax 5*.
lim AJC->O
One-sided derivatives. The expressions
/'_(*)= lim AJ:-*--O
f
(*+**)-/(*) Ax
and
/(x)=
lim
Ax
'
44
[Ch. 2
Differentiation of Functions
are called, respectively, the left-hand or right-hand derivative of the function f(x) at the point x. For /' (x) to exist, it is necessary and sufficient that /'.(*)
Example 4 Find
By the
Solution.
and
/'_ (0)
f^
4.
Infinite derivative.
(0)
the function
we have
=
L
lim
=
lim A*--t-o
Ax
some point we have
at
If
(0) of
+
/'
definition /'_ (0)
= /+(*).
/(*+**)-/(*)_. Uoo,
lim
we say that the continuous function / (x) has an infinite derivative at x. In this case, the tangent to the graph of the function y f(x) is perpendicular to the x-axis. Example 5. Find /' (0) of the function
then
=
Solution.
We
V=V*
have
/'0)=llm *~ Ax
=
]im
-=-=0?
What
347. in the
is
mean
the
l^x^4?
interval
rate of change of the
=
2
function y
+
= x*
+
2/ 348. The law of motion of a point is s 3/ 5, where the distance s is given in centimetres and the time t is in seconds. What is the average velocity of the point over the interval of time from t~\ to ^ 5? 2* in the interval 349. Find the mean rise of the curve y
=
=
mean
350. Find the [x,
x+Ax]. 351. What
rise of the
curve
j/
= /(x)
in the interval
=
f(x) by the rise of the curve y given point x? 352. Define: a) the mean rate of rotation; b) the instantaneous rate of rotation. 353. A hot body placed in a medium of lower temperature cools off. What is to be understood by: a) the mean rate of cooling; b) the rate of cooling at a given instant? 354. What is to be understood by the rate of reaction of a substance in a chemical reaction? 355. Let /(X) be the mass of a non- homogeneous rod over the interval [0, x]. What is to be understood by: a) the mean linear density of the rod on the interval [x, x+Ax]; b) the linear density of the rod at a point x? is
to be understood
at a
M=
356.
x=
2,
if:
ative y'
Find
the
a)
Ax-1;
ratio
when x^2?
b)
of
the
Ax = 0.1;
c)
function
*/
=
Ax -0.01. What
at the point is
the deriv-
_ _ Differentiation of Functions
46
357**. Find the derivative of the function y 358. Find {/'=
a)
=x
t/
f
c)
;
^
lirn
y
[C/t.
2
= ianx.
of the functions:
=
359. Calculate f'(8), if 1 360. Find /'(0), /'(I), /'(2), if /(*) *(*- 1) (x-2)V 361. At what points does the derivative of the function #* coincide numerically with the value of the function itself, /(#) that is, /(*) /'(*)? 362. The law of motion of a point is s 5/*, where the distance s is in metres and the time t is in seconds. Find the speed
=
=
=
at
*
=
= 3.
Find
363.
the
of
slope
the
tangent
=
drawn
at a point with abscissa x 2. 364. Find the slope of the tangent
the point
the
to
to the
curve y = Q.lx*
curve y=sinjt at
0).
(ji,
f (*) \ /
= -i
are the slopes of the tangents to the curves y
=~
365. Find the value of the derivative of the function at the point
366*.
x = X Q (x
What
of their intersection?
Find the
367**. Show that the following functions derivatives at the indicated points:
y=^?_
b) y c)
y
=l/xl
at
x x
= |cosx|
at
*=
Sec. 2. Tabular
1. Basic v
ty(x) are
at
5)
(*)'=,;
6)
3)
(
)'-' r
4)
fc
= 0,
1,
(cu)'=cu
have
finite
2,
rules for finding a derivative. If c is a constant functions that have derivatives, then
2)
(c)'
jt,
do not
angle be-
Differentiation
= 0;
1)
x
+ 0).
and y = x* at the point tween these tangents.
a)
i
;
t;';
7)-
==
(v
* 0).
and
w
=
o>(jc) '
Sec. 2]
__
2. Table I.
of derivatives of basic functions n n- 1
(x )'
III.
= nx
(sinx)'
IV. (cosx)'
V1T
VIII.
.
47
= cos*. = sin*.
(arrdn*)':=
(
ZL.
(arccos*)'=
IX. (arc
_
Tabular Differentiation
|
*
|
<
1).
-
"~56(2*
1)'
414. t/=J/T^J?". 415.
y=^/
416. w=(a''.
24(2^1)'
'
40(2x
I)''
have:
Differentiation of Functions
50
417.
t/
= (3
2 sin*)
Solution.
2 sin x) 4
4 Jt)
2 sin x)'
-(3
419. y r/=J/coU
/coU.
423. y j/
427.
428
y
426.
-
429.
430. 431.
(
2
3
= |/ 3sin*--2cos* y=
f
/.
9
.
t/
y=/2ex + tart y = sin 3* + cos -|-
Solution
^ ^
1
sin Sm
5
f^
,
= cos 3^3*)' -sin 4 f 4V + 5
\
5 /
]
,
cos 2
x Y/-
x 5 "^2
432. t/=sin(x
f
=
433. /(x) cos(ct; 434. /(0=sin/si ,._
"
l+cos2*
= acot~ = ~ y = arc sin 2x.
436. /(x) 437.
438.
t/
Solution, y'
439. y
=
= arcsin^.
440. /(x)
2 cos x)
-- CGSJC = 30-^-3 cos *
424. y
= cosec ^+sec 425. f f(x) = 6(1 _ 3cosx) {/= 1/1 + arc sin x. (arc sin x) y = J/arc tan *
422.
2 sin x)*
t
= 2x + 5 cos' *.
421*. x
= 5 (3
.
-
418. j/=tanjc
[Ch. 2
.
= 5 (32 sin y'
- 10 cos x (3
420. y
5
_
= arccosJ/7.
441. y
= arc tan.
442. y
=
=
_
Tabular Differentiation
Sec. 2]
443.
AAA 444.
= 5e~*. 447. y = arc cose*. 448 0=1 = X = logsinjc. 5 449. = x 10'*. 450. y= ln(l *'). = = \n* * 451. ts'm2 y In(lnjc). f(t) 5 4 arc x sin sinx). y== \n(e* + = arctan (lnA:) + ln(arctan^). y = /In x+l + In (1/7+1). t/
>
-
.
t/
j/
445.
446. 452.
453. 454.
2
j/
t
.
t/
F. Miscellaneous Functions
455**. y=sin'5jccos*y.
15
10 3)'
458.
j/=
460.
az
461. y 462.
463.
f/
^-i-jc 2 x*
= :
3
= |-
y=44
465.
t/
=x
4
(a
__J "2 |/
=
470. z
=
468.
|
469.
471.
/(0=(2/-
(Jt-i-2)
1 '
51
52
Differentiation of Functions
[C/t.
2
--
= ln(]/l+e*-l)-ln(/l # = ^ cos'x (3 cos * 5).
473. y
2
474. ...
475
= (tan
-
2
4
*
l)(tan
x-HOtan 2 *-fl)
476. y=-ian*5x.
= ^ sin (x
477. y
478. j/=sin 479.
=
493. 494.
2
A;+sin'x.
ianx + x.
tan *
486. y
= arc sin
487. y
=
488. y
= 4~- af c sin fx
2
491.
jc
* .
cos *
^ V
.
\
^
= K^
2
490. t/=jt/a
y=/a sin + p cos x. y = arc sinjc + arccosA; 2
= arc sin
489. y
^f +cotx.
=
^-T +a 8
y=arcsin(l
.
= -^ (arc sin*)
2
arc cos jt.
=jc-I y = ln(arcsin5x). y = arc sin (Inx). =
495.
496.
-7-. x-\- 1
2. The derivatives of functions represented parametrically. related to an argument x by means of a parameter t
If
a function
t
t
*-
I
=
then
*t or, in other notation, *JL
t^dx' dt
Example
2.
Find
^, dx
if
x
a cos
y = a sin Solution.
We
=
find
t, /
a sin/ and -r-
d\
dt
_
_ dx
3. The derivative and y
is
of an
= acosf.
a
implicit
sin
Whence
>
/
function.
If
the
relationship between
x
given in implicit form,
F(x,y) = Q,
(I)
then to find the derivative
y'x y' in the simplest cases it is sufficient: 1) to 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/)
and
= 0,
3) to solve the resulting equation for
Example
3.
Find the derivative
yx
(2)
/'.
if
0.
Solution. ito
zero,
we
Forming the derivative
get 3*'
of the left
(3)
side
+ 3y V -3a (y + xy') = 0,
of (3)
and equating
it
58
Differentiation of Functions
[Ch. 2
whence 2
y
581. Find the derivative
,_* ay ~~axy*' xy
a)
f/
c)
y
if
=
= 0. .dy
In the following problems, find the derivative
#'=^
*
the
functions y represented parametrically: 582.
589.
x = acos*f,
(
= b sin* x = acos* y=b sin
\ y
590.
583.
t.
t,
8
cos
3
1.
/
T^nr 591.
584.
sin
8
/
?=
V coslr
585.
x
= arc cos
y
arc sin
592.
__
(
586.
'
y~=e:
{
= a( In tan + cos = a(sin + cosO. -2-
587.
t
588. /
t
~
595. Calculate
cos/).
when
f
///i
/t cn a sin
= 4= a(t = a(l
^
:
Solution.
*~ ^
593.
y
sin/
-r-~
a(l
cosO
1
cos/
if
sin
/),
cos/).
^
sin
^)
>
The Derivatives
Sec. 3]
Functions Not Represented Explicitly
of
59
and S1
fdy\
=
^ **
when
"T I
596.
Find
=
/
if<
!
i
= tlnt, in/ y=^ x = e cosf, = ^ sm^./ \ x
'
/
~.
597.
dv dx
^
i
Find
i
when
,
f
= 44 ji
-r
f
if
<
<
.
f/
698.
Prove that a function # represented parametrically by the
equations
satisfies
599.
the equation
When x = 2
the following equation jc
Does
when
it
_
x = 2?
= Va*
x*.
(x*)'
Is
x
= :r
5//+10 = 0.
2x
602.
5 + ? =1 s
8
8
x
604.
x'-
605.
l/^ +
606.
l/S + /~* =
607.
/'= y
--=a
f
possible to perform term-by-term
+ y*^0'? it
required to find the deriva-
is
609.
a cos
6I0
'
tan//
611.
xy-
613.
^=
-
603.
-t-y
= (2x)
implicit functions y.
of the
601.
608.
*
it
examples that follow
In the y'
true:
follow from this that
600. Let y differentiation of
tive
is
= 2x.
2
.
K^ = /"a.
0.3 sin y
'/a*.
= *.
2
614.
616.
arctan
^-
= -^
l
60
Differentiation of Functions
2
+ y = care tan 2
617.
1/x
619.
Find
Solution.
#=1, we
at the point
y'
Differentiating,
obtain
2*/'
= l+3f/',
620. Find the indicated points: a)
b)
for for
2
c)
we get whence
for
if
==y* + 3xy*y'.
x
Putting
2y' 0'
derivatives y' of
y)
x+l ye = e # = y
A! (1,1),
=
x*
618.
.
(Ch.
=
l
and
1.
functions y at the
specified
x=
*= x=
and and
y=l;
and
r/=l.
*/=!;
Sec. 4. Geometrical and Mechanical Applications of the Derivative
1. Equations of the tangent and the normal. From the geometric significance of a derivative it follows that the equation of the tangent to a curve will be t/ ) (* y = f(x) or F(x,y)=Q at a point
M
where
,
M
value of the derivative y' at the point (X Q y Q ). The straight through the point of tangency perpendicularly to the tangent is called the normal to the curve. For the normal we have the equation
y'Q is the
line passing
,
Y\ 2. The angle between angle between the curves
curves.
The
d and
at their
10 {Z
the
is
M
Q
common
A and
M
M
point
B
point Using a familiar formula of analytic geometry,
M
to
these
y Q ) (Fig. 12) tangents curves at the
(*
between
co
angle
,
the
.
we
get
3. Segments associated with the tangent and the normal in a rectangular coordinate system. The tangent and the normal determine the following four
Sec
Geometrical and Mechanical Applications of the Deriiative
4]
segments (Fig.
13): t
= TM
is
NM
n
the so-called
is
S t = TK
is
S n = KN
is
segment
and tan
\
y = y'Q
Sn N
/f
Fig.
KM = \y
of the tangent,
the subtangent, the segment of the normal, the subnormal.
St
Since
61
,
X
13
it
follows that
j/o
4. Segments
associated with the tangent
tern of coordinates. If a en in polar coordinates
tion
r
= /(q>),
then
formed by the tangent
OM
the
MT
u.
angle
and the
radius vector r (Fig. 14), defined by the following formula:
The tangent
and the normal
curve is givby the equa-
is
.
MN
MT
and the normal together with the radithe point of tangency and with the perpendicular to the radius vector drawn through the pole determine the following four segat the point us vector of
ments
M
(see Fig.
\Af
Fig.
14
14): t
= MT
n=
MN
S t = OT S n = ON
is is is
is
the segment of the polar tangent, the segment of the polar normal, the polar subtangent,
the polar subnormal.
in a polar sys-
[Ch. 2
Differentiation of Functions
62
These segments are expressed by the following formulas:
to
621. What angles cp are formed with the x-axis by the tangents 2 x x at points with abscissas: the curve y
=
x = 0;
a)
x=l/2;
b)
x=l?
c)
2x. Whence Solution. We have y'^\ 45; b) tan 9 0, q>=0; l, q>
=
= =
622. At
N
2
the
sect
=
=
what angles do the sine and y= s'm2x inter-
axis
of
abscissas
at
the
origin? 623. At what angle does the tanianx intersect the gent curve y
Fig. 15
=
axis of abscissas at the origin? tx 624. At what angle does the curve y e*' intersect the line x 2? straight 625. Find the points at which the tangents to the curve 4 20 are parallel to the jc-axis. 12x* y =z 3* -f. 4x* 626. At what point is the tangent to the parabola
=
=
+
=
+ =
3 0? 5x y 627. Find the equation of the parabola y~x*-}-bx-\-c that is tangent to the straight line x y at the point (1,1). 628. Determine the slope of the tangent to the curve x*+y* Q at the point (1,2). 2 629. At what point of the curve y 2x* is the tangent perto 2 the line 0? straight pendicular 630. Write the equation of the tangent and the normal to the parallel to the straight line
xy7 =
= + 4x3y =
parabola
y=
at the point Solution.
We
= [y'] x= i = -T-
K
x
with abscissa x = 4. have y
f
2
1
k
,/-
whence
7=;
V
the
x
slope
the
of
Since the point of tangency has coordinates
*
#2 =
= 4,
follows that the equation of the tangent is 1/4 (* 4) or x Since the slope of the normal must be perpendicular, *,
whence the equation
of the
normal:
tangent
y = 2,
4 (x
4) or
4x
+y
It
4# + 4 = 0.
= -4; t/2 =
is
18
0.
_
Geometrical and Mechanical Applications of the Derivative
Sec. 4\
63
631. Write the equations of the tangent and the normal to the 4# 3 at the point (2,5). 2x* curve y = x' 632. Find the equations of the tangent and the normal to the curve
+
at the point (1,0).
633. Form the equations of the tangent and the normal to the curves at the indicated points: tan2x at the origin; a) y b)
= y = arc
sin
^^
at
the
point
of
intersection
with
the
A:-axis; c)
d)
y = arc cos 3x at the point of intersection with the y-axis; at the point of intersection with the #-axis; y = ln* ~ x* at the points of intersection with the straight y=e }
e)
line
y=
1.
634. Write the equations of the tangent point (2,2) to the curve t*
and the normal
at the
'
635. Write the equations of the tangent to the curve
at the origin
and
at the
point
^
= j-
636. Write the equations of the tangent and the normal to the x* 6=0 at the point with ordinate y 3. 2x y* 637. Write the equation of the tangent to the curve x* y*
curve
=
+ +
+
2xy = Q at the point (1,1). 638. Write the equations of the tangents and the normals to the curve y = (x 3) at the points of its intersection 2)(x l)(jt with the #-axis. 639. Write the equations of the tangent and the normal to the 4 curve y* = 4x 6xy at the point (1,2). 640*. Show that the segment of the tangent to the hyperbola xy = a* (the segment lies between the coordinate axes) is divided in two at the point of tangency. 641. Show that in the case of the astroid x 2 8 + y*t* = a*/ J the segment of the tangent between the coordinate axes has a constant value equal to a.
+
/
_
___
_
[Ch. 2
Differentiation of Functions
64
642.
Show
that the normals to the involute of the circle
x = a(cost
+
t
y = a(sinf
sin/),
t
cost)
are tangents to the circle 643. Find the angle at which the parabolas y (x 2 _. x i ntersect. 4 _|_ 6* y 2 x and y 644. At what angle do the parabolas y
=
=
Sect? 645.
=
Show
2
+
that the curves y 4x 2x 8 and y are tangent to each other at the point (3,34). Will
same thing 646.
at
Show
2
2)
and
= x*
inter-
= x*
x
we have
-\-
10
the
(2,4)? that the hyperbolas
intersect at a right angle. 647. Given a parabola y* 4x. At the point (1,2) evaluate the lengths of the segments of the subtangent, subnormal, tangent,
=
and normal. 648. Find the length of the segment 2* at any point of it. curve y 649.
Show
that in the
equilateral
of the
subtangent of the 2
2
=
2
a the y hyperbola x to the radius vector equal
length of the normal at any point is of this point. 650. Show that the length of the segment of the subnormal 2 2 2 a at any point is equal to the abscissa in the hyperbola x y
=
of this point.
651. x*
y
Show
that the segments of the subtangents of the ellipse
2
jjr+frl
an d the circle
2
x -+y
2
= a*
at
points
with the same
abscissas are equal. What procedure of construction of the tangent to the ellipse follows from this? 652. Find the length of the segment ol the tangent, the normal, the subtangent, and the subnormal of the cycloid ( \
x = a(t a(ts'mt), y = a(l
an arbitrary point t~t 653. Find the angle between the tangent and the radius vector of the point of tangency in the case of the logarithmic spiral at
.
Find the angle between the tangent and the radius vecthe point of tangency in the case of the lemniscate 1 a cos 2q>.
654. tor r*
=
of
Sec. 4]
Geometrical and Mechanical Applications of the Derivative
65
655. Find the lengths of the segments of the polar subtangent, subnormal, tangent and normal, and also the angle between the tangent and the radius vector of the point of tangency in the case of the spiral of Archimedes
at a
point with polar angle
/sin a
^-,
point
g is the acceleration of gravity. Find the and the distance covered. Also determine the
is in
its
direction.
motion along a hyperbola
r/
=
so that its
x increases uniformly at a rate of 1 unit per second. the rate of change of its ordinate when the point passes
abscissa
What
=
the time and
is
/
*/
is
through (5,2)? 662. At what point of the parabola
y*=\8x does the ordinate increase at twice the rate of the abscissa? 663. One side of a rectangle, a 10 cm, is of constant length, while the other side, b, increases at a constant rate of 4 cm,'sec. At what rate are the diagonal of the rectangle and its area increas30 cm? ing when 6 664. The radius of a sphere is increasing at a uniform rate of 5 cm/sec. At what rate are the area of the surface of the sphere and the volume of the sphere increasing when the radius
=
=
becomes 50 cm? 665.
=10
A
point
is
in
motion along the spiral
of
Archimedes
so that the angular velocity of rotation of its radius constant and equal to 6 per second. Determine the rate of elongation of the radius vector r when r 25 cm. 666. A nonhomogeneous rod AB is 12 cm long. The mass of a part of it, AM, increases with the square of the distance of the from the end A and is 10 gm when 2 cm. moving point, Find the mass of the entire rod AB and the linear density at any point M. What is the linear density of the rod at A and S?
(a
vector
cm)
is
=
M
AM =
Sec. 5. Derivatives of Higher Orders
1. Definition of higher derivatives. A derivative of the second order, or Ihe second derivative, of the function y=f(x) is the derivative of its derivative; that is,
= u,
*'
Solution. lim
" ; X->1
777.
lim
xcosx v
m
r
^r 2 "~~ 7/
OJi
~~
9
*
sinx ,
779.
lir
*7on
is
I-*
tj 11IU
'
ji
x
*->il_si n :r
780.
lim
tan*
sin*
exam-
sec
,.
781.
2tan* 1+COS4* 2
'
x
785.
lim-
*-cot^ 2
lim^. tan5x
782.
S
^Ji
786.
lim
y
s
*-"
787.
.
i .
1.
lim
[,
cos x) cot
(1
x
x->o
.
n x lim -r-?=r
,
)
lim 1
783.
81
n
A-
x +
1
lim
Solution,
(1
.
^cos
x) cot
^
= Hm
,
cos A
'^- lim X->0
Sill AT
;
Forms
L' Hospital-Bernoulli Rule for Indeterminate
Sec. 9]
Sill
A
linrZi!iJ^(
788.
lim(l
A*)
Ian
~
a
792.
.
^
,V-^l
X^X
lim arc sin x cot x.
789.
-, V
lim x" sin
793.
'
hnilnxln
(x
1).
X-+Q ri
790.
n>0.
lim(jc e?"*), v^o
791. lim x sin
794.
lim ^
*i
f^ \
A
n"^ lr %
]
.
Solution.
A
=
[-
A
n
1
A'
1
,
= lim
lim j
^^MiiA-l
A
795.
lim
796.
lim
(A
1)
x
*
11
A
= lim
j
\nx
[
-1
*^
A -7
T-
A
h~? 2 A'
]
L__l
limf-^^' *
1
X
y 797
~ >l
^Vc
_^/
A)
AO J
^} 2cosx/
2
798.
lim
A;*.
Solution.
s
lim
p
We
= lim j~
have
0,
**
=
r/;
whence lim//=l,
In
y=?x
that
In
is,
A".
ImiA^
lim In
l.
t/
= limjtln x
=
82
(Ch. 2
Differentiation of Functions
799. limx*.
804.
li
V-H tan
a
800. limx4 * "*.
\
805. Hmftan^f) 4 /
1
X-+l\
1
801. linue s/n
806. lim (cot x) ln
*.
*->0
*.
X-H) cos
802.
lim(l-*)
803.
lim(l+x
ta
807.
2
)*-
lta(I) x / x-*o \
".
808. lim (cot x)* in
*.
X-+0
809. Prove that the limits of
X a)
cannot
sin*
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
1. Increase and
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',Q in the interval (00, increases Similar1) and the function in this interval we find that y'--10 graph. c)
We
seek inclined asymptotes, and find
,=
lim X -> +
bl
oo
lim
-x
= 0,
y
oo,
#->-t-oo
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:
,
x=l,
The
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
when x= when x =
y'=Q r/"
y' retains a
Thus,
(-V3, intervals
l), (
(1,
00,
V$\ and
x=
3.
in each of the intervals constant_ sign
1),
3),
=
(
(l, 3,
V$)
and (V~3
1),
(1,
t
0),
+00), (0,
1),
and (1,
/ 3)
(
00,
in
each
and
(3,
J/T), of
+00).
the
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.
Sec
Graphing Functions by Characteristic Points
4]
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 results
function (Fig
of
the investigation,
33).
-/
Fig. 33
4-1900
we
construct
the
graph
of
the
Extrema and the Geometric Applications
Example
2.
of
a Derivative
[Ch. 3]
Graph the function In x
x Solution, a) The domain of definition of the function is 0
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