دانلود کتاب A course in calculus and real analysis

فهرست مطالب :

Preface......Page 6
Contents......Page 8
1 Numbers and Functions......Page 11
1.1 Properties of Real Numbers......Page 12
1.2 Inequalities......Page 20
1.3 Functions and Their Geometric Properties......Page 23
Exercises......Page 42
2.1 Convergence of Sequences......Page 51
2.2 Subsequences and Cauchy Sequences......Page 64
2.3 Cluster Points of Sequences......Page 69
Exercises......Page 72
3.1 Continuity of Functions......Page 77
3.2 Basic Properties of Continuous Functions......Page 83
3.3 Limits of Functions of a Real Variable......Page 93
Exercises......Page 108
4 Differentiation......Page 115
4.1 Derivative and Its Basic Properties......Page 116
4.2 Mean Value Theorem and Taylor Theorem......Page 129
4.3 Monotonicity, Convexity, and Concavity......Page 137
4.4 L\'Hôpital\'s Rule......Page 143
Exercises......Page 151
5.1 Absolute Minimum and Maximum......Page 159
5.2 Local Extrema and Points of Inflection......Page 162
5.3 Linear and Quadratic Approximations.......Page 169
5.4 Picard and Newton Methods......Page 173
Exercises......Page 185
6.1 Riemann Integral......Page 191
6.2 Integrable Functions......Page 201
6.3 Fundamental Theorem of Calculus......Page 212
6.4 Riemann Sums......Page 220
6.5 Riemann Integrals over Bounded Sets......Page 226
Exercises......Page 235
7 Elementary Transcendental Functions......Page 243
7.1 Logarithmic and Exponential Functions......Page 244
7.2 Trigonometric Functions......Page 256
7.3 Sine of the Reciprocal......Page 268
7.4 Polar Coordinates......Page 275
7.5 Transcendence......Page 284
Exercises......Page 289
Revision Exercises......Page 299
8.1 Area of a Region Between Curves......Page 305
8.2 Volume of a Solid......Page 312
8.3 Arc Length of a Curve......Page 324
8.4 Area of a Surface of Revolution......Page 332
8.5 Centroids......Page 338
8.6 Quadrature Rules......Page 350
Exercises......Page 367
9.1 Convergence of Series......Page 375
9.2 Convergence Tests for Series......Page 382
9.3 Power Series......Page 391
9.4 Convergence of Improper Integrals......Page 401
9.5 Convergence Tests for Improper Integrals......Page 409
9.6 Related Improper Integrals......Page 416
Exercises......Page 427
10 Sequences and Series of Functions, Integrals Depending on a Parameter......Page 435
10.1 Pointwise Convergence of Sequences......Page 436
10.2 Uniform Convergence of Sequences......Page 439
10.3 Uniform Convergence of Series......Page 448
10.4 Weierstrass Approximation Theorems......Page 458
10.5 Bounded Convergence......Page 468
10.6 Riemann Integrals Depending on a Parameter......Page 476
10.7 Improper Integrals Depending on a Parameter......Page 481
Exercises......Page 502
A.1 Equivalence Relations......Page 513
A.2 Cauchy Sequences of Rational Numbers......Page 515
A.3 Uniqueness of a Complete Ordered Field......Page 522
B.1 Complex Numbers and Complex Functions......Page 527
B.2 Polynomials and Their Roots......Page 529
References......Page 533
List of Symbols and Abbreviations......Page 537
Index......Page 542