The Calculus Tutoring Book

Front Cover......Page 1
Indefinite integral formulas 1......Page 3
Indefinite integral formulas 2......Page 4
Title Page......Page 5
Copyright......Page 6
CONTENTS......Page 7
Preface......Page 11
1.1 Introduction......Page 13
1.2 The Graph of a Function......Page 16
1.3 The Trigonometric Functions......Page 22
1.4 Inverse Functions and the Inverse Trigonometric Functions......Page 31
1.5 Exponential and Logarithm Functions......Page 35
1.6 Solving Inequalities Involving Elementary Functions......Page 41
1.7 Graphs of Translations, Reflections, Expansions and Sums......Page 44
Review Problems for Chapter 1......Page 45
2.1 Introduction......Page 53
2.2 Finding Limits of Combinations of Functions......Page 57
2.3 Indeterminate Limits......Page 60
Review Problems for Chapter 2......Page 63
3.1 Preview......Page 65
3.2 Definition and Some Applications of the Derivative......Page 68
3.3 Derivatives of the Basic Functions......Page 75
3.4 Nondifferentiable Functions......Page 82
3.5 Derivatives of Constant Multiples, Sums, Products and Quotients......Page 83
3.6 The Derivative of a Composition......Page 90
3.7 Implicit Differentiation and Logarithmic Differentiation......Page 93
3.8 Antidifferentiation......Page 96
Review Problems for Chapter 3......Page 104
4.1 Relative Maxima and Minima......Page 107
4.2 Absolute Maxima and Minima......Page 110
4.3 L'HBpital's Rule and Orders of Magnitude......Page 117
4.4 Indeterminate Products, Differences and Exponential Forms......Page 122
4.5 Drawing Graphs of Functions......Page 125
4.6 Related Rates......Page 128
4.7 Newton's Method......Page 132
4.8 Differentials......Page 134
4.9 Separable Differential Equations......Page 140
Review Problems for Chapter 4......Page 146
5.1 Preview......Page 149
5.2 Definition and Some Applications of the Integral......Page 151
5.3 The Fundamental Theorem of Calculus......Page 158
5.4 Numerical Integration......Page 163
5.5 Nonintegrable Functions......Page 167
5.6 Improper Integrals......Page 169
Review Problems for Chapter 5......Page 173
6.1 Further Applications of the Integral......Page 175
6.2 The Centroid of a Solid Hemisphere......Page 185
6.3 Area and Arc Length......Page 188
6.4 The Surface Area of a Cone and a Sphere......Page 193
6.5 Integrals with a Variable Upper Limit......Page 195
Review Problems for Chapter 6......Page 200
7.1 Introduction......Page 203
7.2 Substitution......Page 204
7.3 Pre-Table Algebra I......Page 207
7.4 Pre-Table Algebra II: Partial Fraction Decomposition......Page 210
7.5 Integration by Parts......Page 215
7.6 Recursion Formulas......Page 216
7.7 Trigonometric Substitution......Page 219
7.8 Choosing a Method......Page 221
7.9 Combining Techniques of Antidifferentiation with the Fundamental Theorem......Page 224
Review Problems for Chapter 7......Page 226
8.1 Introduction......Page 229
8.2 Geometric Series......Page 232
8.3 Convergence Tests for Positive Series I......Page 234
8.4 Convergence Tests for Positive Series II......Page 240
8.5 Alternating Series......Page 244
8.6 Power Series Functions......Page 250
8.7 Power Series Representations for Elementary Functions I......Page 253
8.8 Power Series Representations for Elementary Functions II (Maclaurin Series)......Page 260
8.9 The Taylor Remainder Formula and an Estimate for the Number e......Page 264
8.10 Power Series in Powers of x - b (Taylor Series)......Page 266
Review Problems for Chapter 8......Page 269
9.1 Introduction......Page 271
9.2 Vector Addition, Subtraction, Scalar Multiplication and Norms......Page 275
9.3 The Dot Product......Page 282
9.4 The Cross Product......Page 288
9.5 The Scalar Triple Product......Page 294
9.6 The Velocity Vector......Page 297
9.7 The Acceleration Vector......Page 302
Review Problems for Chapter 9......Page 306
10.1 Spheres......Page 309
10.2 Planes......Page 310
10.3 Lines......Page 314
10.4 Cylindrical and Quadric Surfaces......Page 319
10.5 Cylindrical and Spherical Coordinates......Page 324
Review Problems for Chapter 10......Page 329
11.1 Graphs and Level Sets......Page 331
11.2 Partial Derivatives......Page 337
11.3 Chain Rules for First-Order Partial Derivatives......Page 343
11.4 Chain Rules for Second-Order Partial Derivatives......Page 346
11.5 Maxima and Minima......Page 349
11.6 The Gradient......Page 358
11.7 Differentials and Exact Differential Equations......Page 367
Review Problems for Chapter 11......Page 373
12.1 Definition and Some Applications of the Double Integral......Page 375
12.2 Computing Double Integrals......Page 382
12.3 Double Integration in Polar Coordinates......Page 389
12.4 Area and Volume......Page 394
12.5 Further Applications of the Double Integral......Page 399
12.6 Triple Integrals......Page 403
12.7 Triple Integration in Spherical Coordinates......Page 410
12.8 Center of Mass......Page 416
Review Problems for Chapter 12......Page 420
Al Distance and Slope......Page 421
A2 Equations of Lines......Page 422
A3 Circles, Ellipses, Hyperbolas and Parabolas......Page 423
A4 The Binomial Theorem......Page 424
A5 Determinants......Page 425
A6 Polar Coordinates......Page 428
SOLUTIONS TO THE PROBLEMS......Page 431
LIST OF SYMBOLS......Page 539
INDEX......Page 542
AUTHORS' BIOGRAPHIES......Page 545
Indefinite integral formulas 3......Page 546
Indefinite integral formulas 4......Page 547
Back Cover......Page 548