3193 PROBLEMS IN MATHEMATICAL ANALYSIS

with Hints and Answers

G. Baranenkov, B. Demidovich, V. Efimenko, S. Kogan, G. Lunts, E. Porshneva,
E. Sycheva, S. Frolov, R. Shostak, A. Yanpolsky

edited by B. DEMIDOVICH, translated by G. YANKOVSKY

Chapter I
INTRODUCTION TO ANALYSIS

1.1 Functions   1.3.2 Limit of a function
1.1.1 Real numbers   1.3.3 One-sided limits
EXERCISES 166-180, 181-190, 191-198, 199-202,
203-215, 216-240, 241-252, 253-287
1.1.2 Definition of a function   1.4 Infinitely Small and Large Quantities
1.1.3 Domain of definition of a function   1.4.1 Infinitely small quantities (infinitesimals)
1.1.4 Inverse functions   1.4.2 Infinitely large quantities (infinitesimals)
EXERCISES 288 - 302
1.1.5 Compound and implicit functions   1.5 Continuity of Functions
1.1.6 Graph of a function
EXERCISES 1 - 43
  1.5.1 Definition of continuity

1.2 Graphs of Elementary Functions

EXERCISES 44 - 165

  1.5.2 Points of discontinuity of a function
1.3 Limits   1.5.3 Properties of continuous functions
EXERCISES 304 - 340
1.3.1 Limit of a sequence    

Chapter II
DIFFERENTIATION OF FUNCTIONS

2.1 Calculating Derivatives Directly

 

2.4.4 Segments associated with the tangent and the normal in a polar co-ordinate system, EXERCISES 621 - 666

2.1.1 Increment of the argument and increment of a function  

2. 5 Derivatives of Higher Orders

2.1.2 The derivative   2.5.1 Definition of higher derivatives
2.1.3 One-sided derivatives   2.5.2 Leibnitz's rule
2.1.4 Infinite derivative
EXERCISES 341 - 367
  2.5.3 Higher-order derivatives of parametrically represented functions,

2.2 Tables of Derivatives:

 

A. Higher-order derivatives of explicit functions, EXERCISES 667 - 691
B. Higher-Order Derivatives of Parametrically Represented Functions and of implicit Functions,
EXERCISES 692 - 712

2.2.1 Basic rules for derivatives  

2.6 Differentials of First and Higher Orders

2.2.2 Table of derivatives of basic functions   2.6.1 First-order differential
2.2.3 Rule for differentiating a compound function (chain rule)   2.6.2 Principal properties of differentials
A Algebraic Functions, EXERCISES 368 - 381,
B. Inverse Circular and Trigonometric Functions .
EXERCISES 382 - 389
C. Exponential and Logarithmic Functions,
EXERCISES 390 - 400
D. Hyperbolic and Inverse Hyperbolic Function
s, EXERCISES 401 - 408
  2.6.3. Applying differentials to approximate calculations

E. Compound Functions, EXERCISES 409 - 454
F. Miscellaneous Functions,
EXERCISES 455 - 563,
G. Logarithmic Derivative,
EXERCISES 564 - 580

 

2.6.4 Higher order differentials, EXERCISES 712 -755

2.3 Derivatives of Functions Not Represented Explicitly

 

2.7 Mean value theorems

2.3.1 Derivative of an inverse function   2.7.1 Rolle's theorem
2.3.2 The derivatives of parametrically represented functions   2.7.2 Lagrange's theorem
2.3.3 Derivative of an implicit function, EXERCISES 581 - 620  

2.7.3 Cauchy's theorem, EXERCISES 756 - 765

2.4 Geometry and Mechanics Applications of the Derivative

 

2.8. Taylor's Formula,
EXERCISES 766 - 775

2.4.1 Equations of the tangent and the normal  

2.9 L'Hospital - Bernoulli Rule for Evaluating Indeterminate Expressions

2.4.2 Angle between curves   2.9.l Evaluation of the indeterminate forms 0/0 and /
2.4.3 Segments associated with the tangent and the normal in an orthogonal co-ordinate system   2.9.2 Other Indeterminate forms EXERCISES 776 - 808

Chapter III
EXTREME VALUES OF A FUNCTION AND GEOMETRIC APPLICATIONS OF DERIVATIVES

3.1 Extreme values of a Function of One Argument

  3.3.2 Vertical asymptotes
3.1.1 Increase and decrease of functions   3.3.3 Inclined asymptotes
EXERCISES 901 - 915
3.1.2 Extreme values of a function   3.4 Graphing Functions by Characteristic Points
EXERCISES 916 - 992
3.1.3 Largest and smallest values
EXERCISES 811 - 890
 

3.5 Differential of an Arc. Curvature

3.2 The Direction of Concavity. Points of Inflection

 

3.5.1 Differential of an arc.

3.2.1 The concavity of the graph of a function   3.5.2 Curvature of a curve
3.2.2 Points of Inflection
EXERCISES 891 - 900
  3.5.3 Circle of curvature

3.3 Asymptotes

  3.5.4 Vertices of a curve
EXERCISES 993 - 1028
3.3.1. Definition    

Chapter IV
INDEFINITE INTEGRALS

4.1 Direct integration

 

4.5 Integration of Rational Functions

4.1.1 Basic rules of integration  

4.5.1 Method of undetermined coefficients

4.1.2 Table of standard Integrals
EXERCISES 1031 - 1050
  4.5.2 Ostrogradsky's method
EXERCISES 1280 - 1314
4.1.3 Integration under the sign of the differential
EXERCISES 1051 - 1144, 1145 - 1190
 

4.6 Integrating Certain Irrational Functions

4.2 Integration by Substitution

  4.6.1 Integrals of the form
EXERCISES 1315 - 1325
4.2.1 Change of variable in an indefinite Integral   4.6.2 Integrals of the form
4.2.2 Trigonometric substitutions
EXERCISES 1191 - 1210
 

4.6.3 Integrals of the form
EXERCISES 1326 - 1331

4.1.3 Integration under the sign of the differential
EXERCISES 1051 - 1144, EXERCISES 1145 - 1190
  4.6.4 Integrals of the binomial differentials
EXERCISES 1332 - 1337

4.2 Integration by Substitution

 

4.7 Integrating Trigonometric Functions

4.2.1 Change of variable in an indefinite Integral   4.7.1 Integrals of the form
EXERCISES 1338 - 1364
4.2.2 Trigonometric substitutions
EXERCISES 1191 - 1210
 

4.7.2 Integrals of the form

EXERCISES 1365 - 1372

4.3 Integration by Parts
EXERCISES 1211 - 1254

  4.7.3 Integrals of the form
EXERCISES 1373 - 1390

4.4 Standard Integrals Containing a second order polynomial

 

4.8 Integration of Hyperbolic Functions
EXERCISES 1391 - 1402

4.4.1 Integrals of the form

 

4.9 Using Trigonometric and Hyperbolic Substitutions for Integrals ol the Form
EXERCISES 1403 - 1414

4.4.2 Integrals of the form  

4.10 Integration of Various Transcendental Functions
EXERCISES 1415 - 1426

4.4.3 Integrals of the form  

4.11. Using Reduction Formulae
EXERCISES 1427 - 1430

4.4.4 Integrals of the form
EXERCISES 1259 - 1279
 

4.12 Miscellaneous Integration Examples
EXERCISES 1431 - 1500

Chapter V
DEFINITE INTEGRALS

5.1 The Definite Integral as the Limit of a Sum

 

5.8 Arc Length of a Curve

5.1.1 Integral sum

 

5.8.1. Arc length in rectangular coordinates

5.1.2 The definite Integral
EXERCISES 1501 - 1507
  5.8.2 Arc length of a curve represented parametrically
EXERCISES 1665 - 1684

5.2 Evaluating Definite Integrals byMeans of Indefinite Integrals

 

5.9 Volumes of Solids

5.2.1 A definite Integral with variable upper limit  

5.9.1 Volume of a solid of revolution

5.2.2 The Newton-Leibnitz formula
EXERCISES 1508 - 1545
  5.9.2 Computing volumes of solids from known cross-sections
EXERCISES 1685 - 1713

5.3 Improper Integrals

 

5.10 Area of a Surface of Revolution
EXERCISES 1714 -1726

5.3.1 Integrals of unbounded functions  

5.11 Moments. Centres of Gravity. Guldin's Theorems

5.3.2. Integrals with Infinite limits
EXERCISES 1546 - 1575
 

5.11.1 Static moment

5.4 Change of Variable in a Definite Integral
Exercises 1576 - 1598

  5.11.2 Moment of inertia

5.5 Integration by Parts
EXERCISES 1599 - 1609

  5.11.3 Centre of gravity

5.6 Mean-Value Theorem

  5.11.4 Guldin's theorems
EXERCISES 1727 -1750
5.6.1 Evaluation of Integrals  

5.12 Applying Definite Integrals to the Solution of Physical Problems

5.6.2 The mean value of a function
EXERCISES 1610 - 1622
 

5.12.1 Path traversed by a point

5.7 The Areas of Plane Figures

  5.12.2 Work of a force
5.7.1 Area in rectangular coordinates   5.12.3 Kinetic energy
5.7.2 Area in polar coordinates
EXERCISES 1623 - 1664
  5.12.4 Pressure of a liquid
EXERCISES 1751 - 1771, MISCELLANEOUS EXERCISES

CHAPTER VI
FUNCTIONS OF SEVERAL VARIABLES

6.1 Basic Concepts

 

6.10 Change of Variables

6.1.1 The concept of a function of several variables: Functional notation   6.10.1. Change of variables in expressions containing ordinary derivatives
6.1.2 Domain of definition of a function   6.10.2 Change of variables in expressions containing partial derivatives
EXERCISES 1969 - 1980
6.1.3 Level lines and level surfaces of a function
EXERCISES 1782 - 1796
 

6.11 The Tangent Plane and the Normal to a Surface

6.2 Continuity

  6.11.1 The equations of a tangent plane and a normal for the case of explicit representation of a surface
6.2.1 The limit of a function   6.11.2 Equations of the tangent plane and the normal for an implicitly represented surface
EXERCISES 1981 - 1995
6.2.2 Continuity and points of discontinuity
EXERCISES 1797 - 1800
 

6.12 Taylor's Formula for a Function of Several Variables
EXERCISES 1996 - 2006

6.3 Partial Derivatives

 

6.13 The Extreme Value of a Function of Several Variables

6.3.1 Definition of a partial derivative   6.13.1 Definition of an extreme value of a function
6.3.2 Euler's theorem
EXERCISES 1801 - 1830
  6.13.2. Necessary conditions for an extreme value

6.4 Total Differential of a Function

  6.13.3 Sufficient conditions for an extreme value
6.4.1 Total Increment of a function   6.13.4. The case of a function of many variables
6.4.2 The total differential of a function   6.13.5 Conditional extremum
6.4.3 Applying the total differential of a function to approximate calculations
EXERCISES 1831 - 1855
  6.13.6 Largest and smallest values of a function
EXERCISES 2008 - 2033

6.5 Differentiation of Composite Functions

 

6.14 Finding the Greatest and Smallest Values of Functions
EXERCISES 2034 - 2052

6.5.2 The case of several independent variables
EXERCISES 1856 - 1875
 

6.15. Singular Points of Plane Curves

6.6 Derivative in a Given Direction and the Gradient of a Function

  6.15.1 Definition of a singular point
6.6.1 The derivative of a function In a given direction   6.15.2 Basic types of singular points
EXERCISES 2053 - 2062
6.6.2 The gradient of a function
EXERCISES 1876 - 1890
 

6.16 Envelopes

6.7 Higher-Order Derivatives and Differentials

  6.16.1. Definition of an envelope
6.7.1 Higher order partial derivatives   6.16.2 Equations of an envelope
EXERCISES 2063 - 2070
6.7.2 Higher-order differentials
EXERCISES 1891 - 1925
 

6.17 Arc Length of a Space Curve
EXERCISES 2071 - 2077

6.8 Integration of Total Differentials

 

6.18 The Vector Function of a Scalar Argument

6.8.1 Condition for a total differential   6.18.1 The derivative of the vector function of a scalar argument:
6.8.2 The case of three variables
EXERCISES 1926 - 1940
  6.18.2 Basic rules for differentiating the vector function of a scalar argument
EXERCISES 2078 - 2089

6.9 Differentiation of Implicit Functions

 

6.19 The Natural Trihedron of a Space Curve
EXERCISES 2090- 2103

6.9.1 Case of one independent variable  

6.20. Curvature and Torsion of a Space Curve

6.9.2 Case of several independent variables   6.20.1. Curvature
6.9.3 A system of implicit functions   6.20.2 Torsion
6.9.4 Parametric representation of a function
EXERCISES 1941 - 1968
  6.20.3. Frenet formulae
EXERCISES 2104 - 2112

Chapter VII
MULTIPLE AND LINE INTEGRALS

7.1 The Double Integral in Rectangular Co-ordinates

  7.8.2 Improper double and triple integrals
EXERCISES 2273 - 2292
7.1.1 Direct computation of double integrals  

7.9 Line Integrals

7.1.2. Setting up the limits of integration in a double integral
EXERCISES 2113 - 2159
 

7.9.1 Line Integrals of the first type

7.2 Change of Variables in a Double Integral

  7.9.2 Line integrals of the second type
7.2.1 Double Integral in polar co-ordinates   7.9.3. The case of a total differential
7.2.2. Double Integral in curvilinear co-ordinates
EXERCISES 2160 - 2174
  7.9.4 Green's formula for a plane

7.3 Computing Areas

  7.9.5 Applications of line integrals
EXERCISES 2293 - 2309/Integrals of the First Type ,
EXERCISES 2310 -2326/Integrals of the Second Type ,
EXERCISES 2327 - 2335/ Green's formula ,
EXERCISES 2336 - 2346/ Applications of Line Integral
7.3.1 Area in rectangular co-ordinates  

7.10 Surface Integrals

7.3.2 Area in polar coordinates
EXERCISES 2175 - 2187
 

7.10.1 Surface Integral of the first type

7.4 Computing Volumes
EXERCISES 2188 - 2212

  7.10.2 Surface Integral of the second type

7.5 Computing the Areas of Surfaces
EXERCISES 2213 - 2224

  7.10.3 Stokes' formula
EXERCISES 2347 - 2360

7.6 Applications of the Double Integral in Mechanics

 

7.11 Ostrogradsky-Gauss Formula
EXERCISES 2361 -2370

7.6.1 Mass and static moments of a lamina  

7.12 Fundamentals of Field Theory

7.6.2 The co-ordinates of the centre of gravity of a lamina
7.6.3 Moments of inertia of a lamina
EXERCISES 2225 - 2239
 

7.12.1 Scalar and vector fields

7.7 Triple Integrals

  7.12.2 Gradient
7.7.1 Triple integrals in rectangular co-ordinates   7.12.3 Divergence and rotation
7.7.2 Change of variables in a triple integral   7.12.4 Flux of a vector
7.7.3 Applications of triple Integrals
EXERCISES 2240 - 2272
  7.12.5 Circulation of a vector, the work of a field

7.8 Improper Integrals Depending on a Parameter. Improper Multiple Integrals

  7.12.6. Potential and solenoidal fields
EXERCISES 2371 - 2400
7.8.1 Differentiation with respect to a parameter    

Chapter VIII
SERIES

8.1 Number Series

  8.2.3 Uniform convergence
EXERCISES 2510 - 2586
8.1.1 Fundamental concepts  

8.3 Taylor's Series

8.1.2 Tests of convergence and divergence of positive series   8.3.1 Expanding a function in a power series
8.1.3 Tests for convergence of alternating series   8.3.2 Techniques employed for expanding in power series
8.1.4 Series with complex terms   8.3.3 Taylor's series for a function of two variables
EXERCISES 2587 - 2670
8.1.5 Operations on series
EXERCISES 2401 - 2509
 

8.4 Fourier Series

8.2 Functional Series

  8.4.1 Dirichlet's theorem
8.2.1 Region of convergence   8.4.2 Incomplete Fourier series
8.2.2 Power series   8.4.3 Fourier series of a period 2l
EXERCISES 2671 - 2703

Chapter IX
DIFFERENTIAL EQUATIONS

9.1 Verifying Solutions. Forming Differential Equations of Families of Curves. Initial Conditions

  9.7.1 First-order differential equations of higher powers
9.1.1 Basic concepts   9.7.2 Solving a differential equation by introduction of a parameter
EXERCISES 2812 - 1819
9.1.2. Initial conditions
EXERCISES 2704 -2732
 

9.8 Lagrange and Clairaut Equations

9.2. First-Order Differential Equations

   9.8.1 Lagrange equation
9.2.1 Types of first-order differential equations   9.8.2 Clairaut equation
EXERCISES 2822 - 2832
9.2.2 Direction field
EXERCISES 2733 - 2737
 

9.9 Miscellaneous Exercises on-First-Order Differential Equations
EXERCISES 2833 - 2910

9.2.3 Cauchy's theorem  

9.10 Higher-Order Differential Equations

9.2.4 Euler's broken line method
EXERCISES 2738 - 2741
  9.10.1 Case of direct integration

9.3 First-Order Differential Equations with Separable Variables . Orthogonal Trajectories

  10.2 Cases of reduction of order
EXERCISES 2911 - 2967
9.3.1 First-order equations with separable variables  

9.11 Linear Differential Equations

9.3.2 Certain differential equations which reduce to equations with separable variables   9.11.1 Homogeneous equations
9.3.3 Orthogonal trajectorles   9.11.2 Non-homogeneous equations
EXERCISES 2968 - 2975
9.3.4 Forming differential equations
EXERCISES 2742 - 2767
  9.12 Linear Differential Equations of Second Order with Constant Coefficients

9.4 First-Order Homogeneous Differential Equations

  9.12.1 Homogeneous equations
9.4.1 Homogeneous equations   9.12.2 Non-homogeneous equations
9.4.2 Equations which reduce to homogeneous equations
EXERCISES 2768 - 2764
  9.12.3 The principle of superposition of solutions
EXERCISES 2976 - 3044

9.5 First-Order Linear Differential Equations. Bernoulli's Equation

 

9.13. Linear Differential Equations of Order Higher than Two with Constant Coefficients

9.5.1 Linear equations   9.13.1 Homogeneous equations
9.5.2 Bernoulli's equation
EXERCISES 2785 - 2801
  9.13.2 Non-homogeneous equations
EXERCISES 3045 - 3066

9.6 Exact Differential Equations. Integrating Factor

 

9.14 Euler's Equations
EXERCISES 3068 - 3077

9.6.1 Exact differential equations  

9.15 Systems of Differential Equations
EXERCISES 3078 - 3092

9.6.2 Integrating factor
EXERCISES 2802 - 2811
 

9.16 Integration of Differential Equations by Means of Power Series
EXERCISES 3093 - 3102

9.7 First-Order Differential Equations not Solved for the Derivative

 

9.17 Problems on Fourier's Method
EXERCISES 3103 - 3107

CHAPTER X
APPROXIMATE CALCULATIONS

10.1 Operations on Approximate Numbers

  10.3.2 The rule of proportionate parts (chord method)
10.1.1 Absolute error   10.3.3 Newton's method (tangent method)
10.1.2. Relative error   10.3.4 Iterative method
10.1.3 Number of correct decimals   10.3.5 The case of a system of two equations
EXERCISES 3138 - 3159
10.1.4 Addition and subtraction of approximate numbers  

10. 4 Numerical Integration of Functions

10.1.5 Multiplication and division of approximate numbers  

10.4.1 Trapezoidal formula

10.1.6 Powers and roots of approximate numbers   10.4.2 Simpson's rule (parabolic formula)
EXERCISES 3160 - 3175
10.1.7 Calculating the error of the result of various operations on approximate numbers  

10.5 Numerical Integration of Ordinary Differential Equations

10.1.8 Establishing admissible errors of approximate numbers for a given error in the result of operations on them. EXERCISES 3108 - 3127  

10.5.1. A method of successive approximation (Picard's method)

10.2 Interpolation of Functions

  10.5.2 Runge-Kutta method
10.2.1. Newton's interpolation formula   10.5.3 Milne's method
10.2.2 Lagrange's Interpolation formula
EXERCISES 3128 - 3137
  10.5.4 Adams' method:
EXERCISES 3176 - 3189

10. 3. Computing Real Roots of Equations

 

10.6 Approximating Fourier Coefficients
EXERCISES 3190 - 3193

10.3.1 Establishing initial approximations of roots    

APPENDIX

I. GREK Alphabet

 

IV Trigonometric Functions

II Some Constants

 

V Exponential, Hyperbolic, Trigonometric Functions

III Inverse Quantities, Powers, Roots, Logarithms

 

VI Curves (for reference)

INDEX