高等数学 上 英文版【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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Chapter 0 Preliminary Knowledge1

0.1 Polar Coordinate System1

0.1.1 Plotting Points with Polar Coordinates2

0.1.2 Converting between Polar and Cartesian Coordinates2

0.2 Complex Numbers6

0.2.1 The Definition of the Complex Number6

0.2.2 The Complex Plane7

0.2.3 Absolute Value,Conjugation and Distance7

0.2.4 Polar Form of Complex Numbers8

Chapter 1 Theoretical Basis of Calculus9

1.1 Sets and Functions9

1.1.1 Sets and Their Operations10

1.1.2 Mappings and Functions15

1.1.3 The Primary Properties of Functions20

1.1.4 Composition of Functions22

1.1.5 Elementary Functions and Hyperbolic Functions23

1.1.6 Modeling Our Real World26

Exercises 1.132

1.2 Limits of Sequences of Numbers36

1.2.1 The Sequence37

1.2.2 Convergence of A Sequence38

1.2.3 Calculating Limits of Sequences48

Exercises 1.252

1.3 Limits of Functions55

1.3.1 Speed and Rates of Change55

1.3.2 The Concept of Limit of A Function59

1.3.3 Properties and Operation Rules of Functional Limits63

1.3.4 Two Important Limits66

Exercises 1.370

1.4 Infinitesimal and Infinite Quantities72

1.4.1 Infinitesimal Quantities and their Order72

1.4.2 Infinite Quantities76

Exercises 1.477

1.5 Continuous Functions78

1.5.1 Continuous Function and Discontinuous Points79

1.5.2 Operations on Continuous Functions and the Continuity of Elementary Functions83

1.5.3 Properties of Continuous Functions on a Closed Interval87

Exercises 1.591

Chapter 2 Derivative and Differential94

2.1 Concept of Derivatives94

2.1.1 Introductory Examples94

2.1.2 Definition of Derivatives95

2.1.3 Geometric Interpretation of Derivative98

2.1.4 Relationship between Derivability and Continuity100

Exercises 2.1102

2.2 Rules of Finding Derivatives104

2.2.1 Derivation Rules of Rational Operations104

2.2.2 Derivative of Inverse Functions108

2.2.3 Derivation Rules of Composite Functions109

2.2.4 Derivation Formulas of Fundamental Elementary Functions113

Exercises 2.2115

2.3 Higher-order Derivatives117

Exercises 2.3121

2.4 Derivation of Implicit Functions and Parametric Equations,Related Rates122

2.4.1 Derivation of Implicit Functions122

2.4.2 Derivation of Parametric Equations125

2.4.3 Related Rates128

Exercises 2.4131

2.5 Differential of the Function133

2.5.1 Concept of the Differential133

2.5.2 Geometric Meaning of the Differential135

2.5.3 Differential Rules of Elementary Functions137

Exercises 2.5139

2.6 Differential in Linear Approximate Computation140

Exercises 2.6141

Chapter 3 The Mean Value Theorem and Applications of Derivatives143

3.1 The Mean Value Theorem143

3.1.1 Rolle's Theorem143

3.1.2 Lagrange's Theorem146

3.1.3 Cauchy's Theorem151

Exercises 3.1152

3.2 L'Hospital's Rule154

Exercises 3.2162

3.3 Taylor's Theorem163

3.3.1 Taylor's Theorem163

3.3.2 Applications of Taylor's Theorem169

Exercises 3.3173

3.4 Monotonicity and Convexity of Functions174

3.4.1 Monotonicity of Functions174

3.4.2 Convexity of Functions,Inflections176

Exercises 3.4181

3.5 Local Extreme Values,Global Maxima and Minima183

3.5.1 Local Extreme Values183

3.5.2 Global Maxima and Minima187

Exercises 3.5192

3.6 Graphing Functions using Calculus194

Exercises 3.6197

Chapter 4 Indefinite Integrals198

4.1 Concepts and Properties of Indefinite Integrals198

4.1.1 Antiderivatives and Indefinite Integrals198

4.1.2 Properties of Indefinite Integrals199

Exercises 4.1201

4.2 Integration by Substitution202

4.2.1 Integration by the First Substitution202

4.2.2 Integration by the Second Substitution206

Exercises 4.2210

4.3 Integration by Parts213

Exercises 4.3220

4.4 Integration of Rational Fractions221

4.4.1 Integration of Rational Fractions221

4.4.2 Antiderivatives Not Expressed by Elementary Functions228

Exercises 4.4228

Chapter 5 Definite Integrals229

5.1 Concepts and Properties of Definite Integrals229

5.1.1 Instances of Definite Integral Problems229

5.1.2 The Definition of Definite Integral232

5.1.3 Properties of Definite Integrals234

Exercises 5.1239

5.2 The Fundamental Theorems of Calculus241

Exercises 5.2246

5.3 Integration by Substitution and by Parts in Definite Integrals249

5.3.1 Substitution in Definite Integrals249

5.3.2 Integration by Parts in Definite Integrals252

Exercises 5.3254

5.4 Improper Integral257

5.4.1 Integration on an Infinite Interval257

5.4.2 Improper Integrals with Infinite Discontinuities261

Exercises 5.4265

5.5 Applications of Definite Integrals266

5.5.1 Method of Setting up Elements of Integration266

5.5.2 The Area of a Plane Region268

5.5.3 The Arc Length of a Curve271

5.5.4 The Volume of a Solid275

5.5.5 Applications of Definite Integral in Physics278

Exercises 5.5282

Chapter 6 Infinite Series288

6.1 Concepts and Properties of Series with Constant Terms288

6.1.1 Examples of the Sum of an Infinite Sequence288

6.1.2 Concepts of Series with Constant Terms290

6.1.3 Properties of Series with Constant Terms294

Exercises 6.1297

6.2 Convergence Tests for Series with Constant Terms299

6.2.1 Convergence Tests of Series with Positive Terms299

6.2.2 Convergence Tests for Alternating Series306

6.2.3 Absolute and Conditional Convergence308

Exercises 6.2311

6.3 Power Series315

6.3.1 Functional Series315

6.3.2 Power Series and Their Convergence316

6.3.3 Operations of Power Series321

Exercises 6.3323

6.4 Expansion of Functions in Power Series326

6.4.1 Taylor and Maclaurin Series326

6.4.2 Expansion of Functions in Power Series328

6.4.3 Applications of Power Series Expansion of Functions332

Exercises 6.4335

6.5 Fourier Series336

6.5.1 Orthogonality of the System of Trigonometric Functions337

6.5.2 Fourier Series338

6.5.3 Convergence of Fourier Series340

6.5.4 Sine and Cosine Series344

Exercises 6.5345

6.6 Fourier Series of Other Forms347

6.6.1 Fourier Expansions of Periodic Functions with Period 2l347

6.6.2 Complex form of Fourier Series350

Exercises 6.6352

Bibliography353