Chapter 1 Fundamental Knowledge of Calculus 1
1.1 Mappings and Functions 1
1.1.1 Sets and Their Operations 1
1.1.2 Mappings and Functions 6
1.1.3 Elementary Properties of Functions 11
1.1.4 Composite Functions and Inverse Functions 14
1.1.5 Basic Elementary Functions and Elementary Functions 16
Exercises 1.1 A 23
Exercises 1.1 B 25
1.2 Limits of Sequences 26
1.2.1 The Definition of Limit of a Sequence 26
1.2.2 Properties of Limits of Sequences 31
1.2.3 Operations of Limits of Sequences 35
1.2.4 Some Criteria for Existence of the Limit of a Sequence 38
Exercises 1.2 A 44
Exercises 1.2 B 46
1.3 The Limit of a Function 46
1.3.1 Concept of the Limit of a Function 47
1.3.2 Properties and Operations of Limits for Functions 53
1.3.3 Two Important Limits of Functions 58
Exercises 1.3A 61
Exercises 1.3 B 63
1.4 Infinitesimal and Infinite Quantities 63
1.4.1 Infinitesimal Quantities 63
1.4.2 Infinite Quantities 65
1.4.3 The Order of Infinitesimals and Infinite Quantities 67
Exercises 1.4 A 71
Exercises 1.4 B 73
1.5 Continuous Functions 73
1.5.1 Continuity of Functions 74
1.5.2 Properties and Operations of Continuous Functions 76
1.5.3 Continuity of Elementary Functions 78
1.5.4 Discontinuous Points and Their Classification 80
1.5.5 Properties of Continuous Functions on a Closed Interval 83
Exercises 1.5 A 87
Exercises 1.5 B 89
Chapter 2 Derivative and Differential 91
2.1 Concept of Derivatives 91
2.1.1 Introductory Examples 91
2.1.2 Definition of Derivatives 92
2.1.3 Geometric Meaning of the Derivative 96
2.1.4 Relationship between Derivability and Continuity 96
Exercises 2.1 A 98
Exercises 2.1 B 99
2.2 Rules of Finding Derivatives 99
2.2.1 Derivation Rules of Rational（）perations 100
2.2.2 Derivation Rules of Composite Functions 101
2.2.3 Derivative of Inverse Functions 103
2.2.4 Derivation Formulas of Fundamental Elementary Functions 104
Exercises 2.2 A 105
Exereises 2.2 B 107
2.3 Higher Order Derivatives 107
Exercises 2.3 A 110
Exercises 2.3 B 111
2.4 Derivation of Implicit Functions and Parametric Equations,Related Rates 111
2.4.1 Derivation of Implicit Functions 111
2.4.2 Derivation of Parametric Equations 114
2.4.3 Related Rates 118
Exercises 2.4 A 120
Exercises 2.4 B 122
2.5 Differential of the Function 123
2.5.1 Concept of the Differential 123
2.5.2 Geometric Meaning of the Differential 125
2.5.3 Differential Rules of Elementary Functions 126
2.5.4 Differential in Linear Approximate Computation 127
Exercises 2.5 128
Chapter 3 The Mean Value Theorem and Applications of Derivatives 130
3.1 The Mean Value Theorem 130
3.1.1 Rolle's Theorem 130
3.1.2 Lagrange's Theorem 132
3.1.3 Cauchy's Theorem 135
Exercises 3.1 A 137
Exercises 3.1 B 138
3.2 L'Hospital's Rule 138
Exercises 3.2 A 144
Exercises 3.2 B 145
3.3 Taylor's Theorem 145
3.3.1 Taylor's Theorem 145
3.3.2 Applications of Taylor's Theorem 149
Exercises 3.3 A 150
Exercises 3.3 B 151
3.4 Monotonicity,Extreme Values,Global Maxima and Minima of Functions 151
3.4.1 Monotonicity of Functions 151
3.4.2 Extreme Values 153
3.4.3 Global Maxima and Minima and Its Application 156
Exercises 3.4 A 158
Exercises 3.4 B 160
3.5 Convexity of Functions,Inflections 160
Exercises 3.5 A 165
Exercises 3.5 B 166
3.6 Asymptotes and Graphing Functions 166
Exercises 3.6 170
Chapter 4 Indefinite Integrals 172
4.1 Concepts and Properties of Indefinite Integrals 172
4.1.1 Antiderivatives and Indefinite Integrals 172
4.1.2 Formulas for Indefinite Integrals 174
4.1.3Operation Rules of Indefinite Integrals 175
Exercises 4.1 A 176
Exercises 4.1 B 177
4.2 Integration by Substitution 177
4.2.1 Integration by the First Substitution 177
4.2.2 Integration by the Second Substitution 181
Exercises 4.2 A 184
Exercises 4.2 B 186
4.3 Integration by Parts 186
Exercises 4.3 A 193
Exercises 4.3 B 194
4.4 Integration of Rational Functions 194
4.4.1 Rational Functions and Partial Fractions 194
4.4.2 Integration of Rational Fractions 197
4.4.3 Antiderivatives Not Expressed by Elementary Functions 201
Exercises 4.4 201
Chapter 5 Definite Integrals 202
5.1 Concepts and Properties of Definite Integrals 202
5.1.1 Instances of Definite Integral Problems 202
5.1.2 The Definition of the Definite Integral 205
5.1.3 Properties of Definite Integrals 208
Exercises 5.1 A 213
Exercises 5.1 B 214
5.2 The Fundamental Theorems of Calculus 215
5.2.1 Fundamental Theorems of Calculus 215
5.2.2 Newton-Leibniz Formula for Evaluation of Definite Integrals 217
Exercises 5.2 A 219
Exercises 5.2 B 221
5.3 Integration by Substitution and by Parts in Definite Integrals 222
5.3.1 Substitution in Definite Integrals 222
5.3.2 Integration by Parts in Definite Integrals 225
Exercises 5.3 A 226
Exercises 5.3 B 228
5.4 Improper Integral 229
5.4.1 Integration on an Infinite Interval 229
5.4.2 Improper Integrals with Infinite Discontinuities 232
Exercises 5.4 A 235
Exercises 5.4 B 236
5.5 Applications of Definite Integrals 237
5.5.1 Method of Setting up Elements of Integration 237
5.5.2 The Area of a Plane Region 239
5.5.3 The Arc Length of Plane Curves 243
5.5.4 The Volume of a Solid by Slicing and Rotation about an Axis 247
5.5.5 Applications of Definite Integral in Physics 249
Exercises 5.5 A 252
Exercises 5.5 B 254
Chapter 6 Differential Equations 256
6.1 Basic Concepts of Differential Equations 256
6.1.1 Examples of Differential Equations 256
6.1.2 Basic Concepts 258
Exercises 6.1 259
6.2 First-Order Differential Equations 260
6.2.1 First-Order Separable Differential Equation 260
6.2.2 Equations can be Reduced to Equations with Variables Separable 262
6.2.3 First-Order Linear Equations 266
6.2.4 Bernoulli's Equation 269
6.2.5 Some Examples that can be Reduced to First-Order Linear Equations 270
Exereises 6.2 272
6.3 Reducible Second Order Differential Equations 273
Exercises 6.3 276
6.4 Higher-Order Linear Differential Equations 277
6.4.1 Some Examples of Linear Differential Equation of Higher-Order 277
6.4.2 Structure of Solutions of Linear Differential Equations 279
Exercises 6.4 282
6.5 Linear Equations with Constant Coefficients 283
6.5.1 Higher Order Linear Homogeneous Equations with Constant Coefficients 283
6.5.2 Higher-Order Linear Nonhomogeneous Equations with Constant Coefficients 287
Exercises 6.5 294
6.6Euler's Differential Equation 295
Exercises 6.6 296
6.7 Applications of Differential Equations 296
Exercises 6.7 301
Bibliography 303