Problems in mathematical analysis

Title......Page 3
Contents......Page 5
Preface......Page 9
1. Functions......Page 11
ANSWERS......Page 396
2. Graphs of Elementary Functions......Page 16
=Answers(44-165)......Page 397
3. Limits......Page 22
=Answers(166-287)......Page 398
4. Infinitely Small and Large Quantities......Page 33
=Answers(304-340)......Page 400
5. Continuity of Functions......Page 36
1. Calculating Derivatives Directly......Page 42
=Answers(341-367)......Page 401
2. Tabular Differentiation......Page 46
=Answers(368-580)......Page 402
3. The Derivatives of Functions Not Represented Explicitly......Page 56
=Answers(621-666)......Page 405
4. Geometrical and Mechanical Applications of the Derivative......Page 60
5. Derivatives of Higher Orders......Page 66
=Answers(667-711)......Page 406
6. Differentials of First and Higher Orders......Page 71
=Answers(712-755)......Page 407
7. Mean Value Theorems......Page 75
=Answers(811-890)......Page 408
8. Taylor's Formula......Page 77
9. The L'Hospital-Bernoulli Rule for Evaluating Indeterminate Forms......Page 78
1. The Extrema of a Function of One Argument......Page 83
2. The Direction of Concavity Points of Inflection......Page 91
=Answers(891-900)......Page 410
3. Asymptotes......Page 93
=Answers(916-992)......Page 411
4. Graphing Functions by Characteristic Points......Page 96
5. Differential of an Arc Curvature......Page 101
=Answers(1031-1190)......Page 415
1. Direct Integration......Page 107
2. Integration by Substitution......Page 113
=Answers(1191-1210)......Page 418
3. Integration by Parts......Page 116
=Answers(1211-1254)......Page 419
4. Standard Integrals Containing a Quadratic Trinomial......Page 118
=Answers(1280-1314)......Page 421
5. Integration of Rational Functions......Page 121
6. Integrating Certain Irrational Functions......Page 125
=Answers(1315-1337)......Page 422
7. Integrating Trigonometric Functions......Page 128
=Answers(1338-1390)......Page 423
9. Using Trigonometric and Hyperbolic Substitutions for Finding Integrals of the Form R(x,sqrt(ax2+bx+c))......Page 133
=Answers(1391-1402)......Page 424
=Answers(1431-1500)......Page 425
11. Using Reduction Formulas......Page 135
12. Miscellaneous Examples on Integration......Page 136
1. The Definite Integral as the Limit of a Sum......Page 138
=Answers(1508-1545)......Page 427
2. Evaluating Definite Integrals by Means of Indefinite Integrals......Page 140
3. Improper Integrals......Page 143
=Answers(1576-1598)......Page 428
4. Charge of Variable in a Definite Integral......Page 146
5. Integration by Parts......Page 149
=Answers(1623-1664)......Page 429
6. Mean-Value Theorem......Page 150
7. The Areas of Plane Figures......Page 153
8. The Arc Length of a Curve......Page 158
=Answers(1714-1726)......Page 430
9. Volumes of Solids......Page 161
10. The Area of a Surface of Revolution......Page 166
11. Moments, Centres of Gravity, Guldin's Theorems......Page 168
=Answers(1727-1750)......Page 431
12. Applying Definite Integrals to the Solution of Physical Problems......Page 173
=Answers(1751-1781)......Page 432
1. Basic Notions......Page 180
=Answers(1782-1796)......Page 434
2. Continuity......Page 184
=Answers(1801-1830)......Page 435
3. Partial Derivatives......Page 185
4. Total Differential of a Function......Page 187
=Answers(1856-1875)......Page 436
5. Differentiation of Composite Functions......Page 190
6. Derivative in a Given Direction and the Gradient of a Function......Page 193
=Answers(1896-1940)......Page 437
7. Higher-Order Derivatives and Differentials......Page 197
8. Integration of Total Differentials......Page 202
9. Differentiation of Implicit Functions......Page 205
=Answers(1941-1968)......Page 438
10. Change of Variables......Page 211
=Answers(1981-1995)......Page 439
11. The Tangent Plane and the Normal to a Surface......Page 217
12. Taylor's Formula for a Function of Several Variables......Page 220
=Answers(2008-2033)......Page 440
13. The Extremum of a Function of Several Variables......Page 222
14. Fir ding the Create st and Smallest Values of Functions......Page 227
=Answers(2034-2052)......Page 441
15. Singular Points of Plane Curves......Page 230
=Answers(2078-2089)......Page 442
16. Envelope......Page 232
17. Arc Length of a Space Curve......Page 234
18. The Vector Function of a Scalar Argument......Page 235
19. The Natural Trihedron of a Space Curve......Page 238
=Answers(2090-2103)......Page 443
20. Curvature and Torsion of a Space Curve......Page 242
=Answers(2113-2159)......Page 444
1. The Double Integral in Rectangular Coordinates......Page 246
2. Change of Variables in a Double Integral......Page 252
=Answers(2160-2174)......Page 446
3. Computing Areas......Page 256
=Answers(2175-2187)......Page 445
4. Computing Volumes......Page 258
=Answers(2213-2224)......Page 448
5. Computing the Areas of Surfaces......Page 259
6. Applications of the Double Integral in Mechanics......Page 260
=Answers(2240-2272)......Page 449
7. Triple Integrals......Page 262
8. Improper Integrals Dependent on a Parameter. Improper Multiple Integrals......Page 269
=Answers(2273-2292)......Page 451
9. Line Integrals......Page 273
=Answers(2293-2346)......Page 452
10. Surface Integrals......Page 284
=Answers(2371-2400)......Page 453
11. The Ostrogradsky-Gauss Formula......Page 286
12. Fundamentals of Field Theory......Page 288
1. Number Series......Page 293
=Answers(2401-2509)......Page 454
2. Functional Series......Page 304
=Answers(2510-2586)......Page 455
3. Taylor's Series......Page 311
=Answers(2587-2670)......Page 456
4. Fourier's Series......Page 318
=Answers(2671-2703)......Page 459
1. Verifying Solutions. Forming Differential Equations of Families of Curves. Initial Conditions......Page 322
=Answers(2768-2784)......Page 462
2. First-Order Differential Equations......Page 324
3. First-Order Differential Equations with Variables Separable. Orthogonal Trajectories......Page 327
4. First-Order Homogeneous Differential Equations......Page 330
5. First-Order Linear Differential Equations. Bernoulli's Equation......Page 332
=Answers(2812-2821)......Page 463
6. Exact Differential Equations. Integrating Factor......Page 335
9. Miscellaneous Exercises on First-Order Differential Equations 340......Page 
8. The Lagrange and Clairaut Equations......Page 339
=Answers(2833-2910)......Page 464
10. Higher-Order Differential Equations......Page 345
=Answers(2911-2967)......Page 466
11. Linear Differential Equations......Page 349
=Answers(2976-3044)......Page 467
12. Linear Differential Equations of Second Order with Constant Coefficients......Page 351
13. Linear Differential Equations of Order Higher than Two with Constant Coefficients......Page 356
=Answers(3078-3092)......Page 469
14. Euler's Equations......Page 357
15. Systems of Differential Equations......Page 359
16. Integration of Differential Equations by Means of Power Series......Page 361
=Answers(3108-3127)......Page 471
17. Problems on Fourier's Method......Page 363
1. Operations on Approximate Numbers......Page 367
2. Interpolation of Functions......Page 372
=Answers(3128-3137)......Page 472
3. Computing the Real Roots of Equations......Page 376
=Answers(3190-3193)......Page 474
4. Numerical, Integration of Functions......Page 382
5. Numerical Integration of Contrary Differential Equations......Page 384
6. Approximating Fourier Coefficients......Page 393
II. Some Constants......Page 475
III. Inverse Quantities, Powers, Roots, Logarithms......Page 476
IV. Trigonometric Functions......Page 478
V. Exponential, Hyperbolic and Trigonometric Functions......Page 479
VI. Some Curves......Page 480
INDEX......Page 488