3193 PROBLEMS IN MATHEMATICAL ANALYSIS

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

 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
 2.1 Calculating Derivatives Directly 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 Functions, 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 2.3 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 2.4 Geometry and Mechanics Applications of the Derivative 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 3.2.1 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
 4.1 Direct integration 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 4.1.3 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 4.7.3 Integrals of the form EXERCISES 1373 - 1390 4.4 Standard Integrals Containing a second order polynomial 4.4.1 Integrals of the form 4.4.2 Integrals of the form 4.4.3 Integrals of the form 4.4.4 Integrals of the form EXERCISES 1259 - 1279

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 5.3.1 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 5.11.2 Moment of inertia 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
 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 6.3 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 6.5.2 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 6.8 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 6.9.1 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
 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 7.10.2 Surface Integral of the second type 7.10.3 Stokes' formula EXERCISES 2347 - 2360 7.6 Applications of the Double Integral in Mechanics 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
 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
 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 9.2.3 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 9.6.1 Exact differential equations 9.6.2 Integrating factor EXERCISES 2802 - 2811 9.7 First-Order Differential Equations not Solved for the Derivative
 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 10.3.1 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