高等数学　英文　上 第2版PDF格式文档图书下载

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

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