高等数学 上 英文版

Chapter 0 Preliminary Knowledge 1

0.1 Polar Coordinate System 1

0.1.1 Plotting Points with Polar Coordinates 2

0.1.2 Converting between Polar and Cartesian Coordinates 2

0.2 Complex Numbers 6

0.2.1 The Definition of the Complex Number 6

0.2.2 The Complex Plane 7

0.2.3 Absolute Value,Conjugation and Distance 7

0.2.4 Polar Form of Complex Numbers 8

Chapter 1 Theoretical Basis of Calculus 9

1.1 Sets and Functions 9

1.1.1 Sets and Their Operations 10

1.1.2 Mappings and Functions 15

1.1.3 The Primary Properties of Functions 20

1.1.4 Composition of Functions 22

1.1.5 Elementary Functions and Hyperbolic Functions 23

1.1.6 Modeling Our Real World 26

Exercises 1.1 32

1.2 Limits of Sequences of Numbers 36

1.2.1 The Sequence 37

1.2.2 Convergence of A Sequence 38

1.2.3 Calculating Limits of Sequences 48

Exercises 1.2 52

1.3 Limits of Functions 55

1.3.1 Speed and Rates of Change 55

1.3.2 The Concept of Limit of A Function 59

1.3.3 Properties and Operation Rules of Functional Limits 63

1.3.4 Two Important Limits 66

Exercises 1.3 70

1.4 Infinitesimal and Infinite Quantities 72

1.4.1 Infinitesimal Quantities and their Order 72

1.4.2 Infinite Quantities 76

Exercises 1.4 77

1.5 Continuous Functions 78

1.5.1 Continuous Function and Discontinuous Points 79

1.5.2 Operations on Continuous Functions and the Continuity of Elementary Functions 83

1.5.3 Properties of Continuous Functions on a Closed Interval 87

Exercises 1.5 91

Chapter 2 Derivative and Differential 94

2.1 Concept of Derivatives 94

2.1.1 Introductory Examples 94

2.1.2 Definition of Derivatives 95

2.1.3 Geometric Interpretation of Derivative 98

2.1.4 Relationship between Derivability and Continuity 100

Exercises 2.1 102

2.2 Rules of Finding Derivatives 104

2.2.1 Derivation Rules of Rational Operations 104

2.2.2 Derivative of Inverse Functions 108

2.2.3 Derivation Rules of Composite Functions 109

2.2.4 Derivation Formulas of Fundamental Elementary Functions 113

Exercises 2.2 115

2.3 Higher-order Derivatives 117

Exercises 2.3 121

2.4 Derivation of Implicit Functions and Parametric Equations,Related Rates 122

2.4.1 Derivation of Implicit Functions 122

2.4.2 Derivation of Parametric Equations 125

2.4.3 Related Rates 128

Exercises 2.4 131

2.5 Differential of the Function 133

2.5.1 Concept of the Differential 133

2.5.2 Geometric Meaning of the Differential 135

2.5.3 Differential Rules of Elementary Functions 137

Exercises 2.5 139

2.6 Differential in Linear Approximate Computation 140

Exercises 2.6 141

Chapter 3 The Mean Value Theorem and Applications of Derivatives 143

3.1 The Mean Value Theorem 143

3.1.1 Rolle's Theorem 143

3.1.2 Lagrange's Theorem 146

3.1.3 Cauchy's Theorem 151

Exercises 3.1 152

3.2 L'Hospital's Rule 154

Exercises 3.2 162

3.3 Taylor's Theorem 163

3.3.1 Taylor's Theorem 163

3.3.2 Applications of Taylor's Theorem 169

Exercises 3.3 173

3.4 Monotonicity and Convexity of Functions 174

3.4.1 Monotonicity of Functions 174

3.4.2 Convexity of Functions,Inflections 176

Exercises 3.4 181

3.5 Local Extreme Values,Global Maxima and Minima 183

3.5.1 Local Extreme Values 183

3.5.2 Global Maxima and Minima 187

Exercises 3.5 192

3.6 Graphing Functions using Calculus 194

Exercises 3.6 197

Chapter 4 Indefinite Integrals 198

4.1 Concepts and Properties of Indefinite Integrals 198

4.1.1 Antiderivatives and Indefinite Integrals 198

4.1.2 Properties of Indefinite Integrals 199

Exercises 4.1 201

4.2 Integration by Substitution 202

4.2.1 Integration by the First Substitution 202

4.2.2 Integration by the Second Substitution 206

Exercises 4.2 210

4.3 Integration by Parts 213

Exercises 4.3 220

4.4 Integration of Rational Fractions 221

4.4.1 Integration of Rational Fractions 221

4.4.2 Antiderivatives Not Expressed by Elementary Functions 228

Exercises 4.4 228

Chapter 5 Definite Integrals 229

5.1 Concepts and Properties of Definite Integrals 229

5.1.1 Instances of Definite Integral Problems 229

5.1.2 The Definition of Definite Integral 232

5.1.3 Properties of Definite Integrals 234

Exercises 5.1 239

5.2 The Fundamental Theorems of Calculus 241

Exercises 5.2 246

5.3 Integration by Substitution and by Parts in Definite Integrals 249

5.3.1 Substitution in Definite Integrals 249

5.3.2 Integration by Parts in Definite Integrals 252

Exercises 5.3 254

5.4 Improper Integral 257

5.4.1 Integration on an Infinite Interval 257

5.4.2 Improper Integrals with Infinite Discontinuities 261

Exercises 5.4 265

5.5 Applications of Definite Integrals 266

5.5.1 Method of Setting up Elements of Integration 266

5.5.2 The Area of a Plane Region 268

5.5.3 The Arc Length of a Curve 271

5.5.4 The Volume of a Solid 275

5.5.5 Applications of Definite Integral in Physics 278

Exercises 5.5 282

Chapter 6 Infinite Series 288

6.1 Concepts and Properties of Series with Constant Terms 288

6.1.1 Examples of the Sum of an Infinite Sequence 288

6.1.2 Concepts of Series with Constant Terms 290

6.1.3 Properties of Series with Constant Terms 294

Exercises 6.1 297

6.2 Convergence Tests for Series with Constant Terms 299

6.2.1 Convergence Tests of Series with Positive Terms 299

6.2.2 Convergence Tests for Alternating Series 306

6.2.3 Absolute and Conditional Convergence 308

Exercises 6.2 311

6.3 Power Series 315

6.3.1 Functional Series 315

6.3.2 Power Series and Their Convergence 316

6.3.3 Operations of Power Series 321

Exercises 6.3 323

6.4 Expansion of Functions in Power Series 326

6.4.1 Taylor and Maclaurin Series 326

6.4.2 Expansion of Functions in Power Series 328

6.4.3 Applications of Power Series Expansion of Functions 332

Exercises 6.4 335

6.5 Fourier Series 336

6.5.1 Orthogonality of the System of Trigonometric Functions 337

6.5.2 Fourier Series 338

6.5.3 Convergence of Fourier Series 340

6.5.4 Sine and Cosine Series 344

Exercises 6.5 345

6.6 Fourier Series of Other Forms 347

6.6.1 Fourier Expansions of Periodic Functions with Period 2l 347

6.6.2 Complex form of Fourier Series 350

Exercises 6.6 352

Bibliography 353