Contents

Chapter 1 Fundamental Knowledge of Calculus1

1.1 Mappings and Functions1

1.1.1 Sets and Their Operations1

1.1.2 Mappings and Functions6

1.1.3 Elementary Properties of Functions11

1.1.4 Composite Functions and Inverse Functions14

1.1.5 Basic Elementary Functions and Elementary Functions16

Exercises 1.1 A23

Exercises 1.1 B25

1.2 Limits of Sequences26

1.2.1 The Definition of Limit of a Sequence26

1.2.2 Properties of Limits of Sequences31

1.2.3 Operations of Limits of Sequences35

1.2.4 Some Criteria for Existence of the Limit of a Sequence38

Exercises 1.2 A44

Exercises 1.2 B46

1.3 The Limit of a Function46

1.3.1 Concept of the Limit of a Function47

1.3.2 Properties and Operations of Limits for Functions53

1.3.3 Two Important Limits of Functions58

Exercises 1.3 A61

Exercises 1.3 B63

1.4 Infinitesimal and Infinite Quantities63

1.4.1 Infinitesimal Quantities63

1.4.2 Infinite Quantities65

1.4.3 The Order of Infinitesimals and Infinite Quantities67

Exercises 1.4 A71

Exercises 1.4 B73

1.5 Continuous Functions73

1.5.1 Continuity of Functions74

1.5.2 Properties and Operations of Continuous Functions76

1.5.3 Continuity of Elementary Functions78

1.5.4 Discontinuous Points and Their Classification80

1.5.5 Properties of Continuous Functions on a Closed Interval83

Exercises 1.5 A87

Exercises 1.5 B89

Chapter 2 Derivative and Differential91

2.1 Concept of Derivatives91

2.1.1 Introductory Examples 91

2.1.2 Definition of Derivatives92

2.1.3 Geometric Meaning of the Derivative96

2.1.4 Relationship between Derivability and Continuity96

Exercises 2.1 A98

Exercises 2.1 B99

2.2 Rules of Finding Derivatives99

2.2.1 Derivation Rules of Rational Operations100

2.2.2 Derivation Rules of Composite Functions101

2.2.3 Derivative of Inverse Functions103

2.2.4 Derivation Formulas of Fundamental Elementary Functions104

Exercises 2.2 A105

Exercises 2.2 B107

2.3 Higher Order Derivatives107

Exercises 2.3 A110

Exercises 2.3 B111

2.4 Derivation of Implicit Functions and Parametric Equations��
Related Rates111

2.4.1 Derivation of Implicit Functions111

2.4.2 Derivation of Parametric Equations114

2.4.3 Related Rates118

Exercises 2.4 A120

Exercises 2.4 B122

2.5 Differential of the Function123

2.5.1 Concept of the Differential123

2.5.2 Geometric Meaning of the Differential125

2.5.3 Differential Rules of Elementary Functions126

2.5.4 Differential in Linear Approximate Computation127

Exercises 2.5128

Chapter 3 The Mean Value Theorem and Applications of Derivatives130

3.1 The Mean Value Theorem130

3.1.1 Rolle's Theorem 130

3.1.2 Lagrange's Theorem132

3.1.3 Cauchy's Theorem135

Exercises 3.1 A137

Exercises 3.1 B138

3.2 L'Hospital's Rule138

Exercises 3.2 A144

Exercises 3.2 B145

3.3 Taylor's Theorem145

3.3.1 Taylor's Theorem145

3.3.2 Applications of Taylor's Theorem149

Exercises 3.3 A150

Exercises 3.3 B151

3.4 Monotonicity, Extreme Values, Global Maxima and Minima of Functions151

3.4.1 Monotonicity of Functions151

3.4.2 Extreme Values153

3.4.3 Global Maxima and Minima and Its Application156

Exercises 3.4 A158

Exercises 3.4 B160

3.5 Convexity of Functions�� Inflections160

Exercises 3.5 A165

Exercises 3.5 B166

3.6 Asymptotes and Graphing Functions166

Exercises 3.6170

Chapter 4 Indefinite Integrals172

4.1 Concepts and Properties of Indefinite Integrals172

4.1.1 Antiderivatives and Indefinite Integrals172

4.1.2 Formulas for Indefinite Integrals174

4.1.3 Operation Rules of Indefinite Integrals175

Exercises 4.1 A176

Exercises 4.1 B177

4.2 Integration by Substitution177

4.2.1 Integration by the First Substitution177

4.2.2 Integration by the Second Substitution181

Exercises 4.2 A184

Exercises 4.2 B186

4.3 Integration by Parts186

Exercises 4.3 A193

Exercises 4.3 B194

4.4 Integration of Rational Functions194

4.4.1 Rational Functions and Partial Fractions194

4.4.2 Integration of Rational Fractions197

4.4.3 Antiderivatives Not Expressed by Elementary Functions201

Exercises 4.4201

Chapter 5 Definite Integrals202

5.1 Concepts and Properties of Definite Integrals202

5.1.1 Instances of Definite Integral Problems202

5.1.2 The Definition of the Definite Integral205

5.1.3 Properties of Definite Integrals208

Exercises 5.1 A213

Exercises 5.1 B214

5.2 The Fundamental Theorems of Calculus215

5.2.1 Fundamental Theorems of Calculus215

5.2.2 Newton��Leibniz Formula for Evaluation of Definite Integrals217

Exercises 5.2 A219

Exercises 5.2 B221

5.3 Integration by Substitution and by Parts in Definite Integrals222

5.3.1 Substitution in Definite Integrals222

5.3.2 Integration by Parts in Definite Integrals225

Exercises 5.3 A226

Exercises 5.3 B228

5.4 Improper Integral229

5.4.1 Integration on an Infinite Interval229

5.4.2 Improper Integrals with Infinite Discontinuities232

Exercises 5.4 A235

Exercises 5.4 B236

5.5 Applications of Definite Integrals237

5.5.1 Method of Setting up Elements of Integration237

5.5.2 The Area of a Plane Region239

5.5.3 The Arc Length of Plane Curves243

5.5.4 The Volume of a Solid by Slicing and Rotation about an Axis 247

5.5.5 Applications of Definite Integral in Physics249

Exercises 5.5 A252

Exercises 5.5 B254

Chapter 6 Differential Equations256

6.1 Basic Concepts of Differential Equations256

6.1.1 Examples of Differential Equations256

6.1.2 Basic Concepts258

Exercises 6.1259

6.2 First��Order Differential Equations260

6.2.1 First��Order Separable Differential Equation260

6.2.2 Equations can be Reduced to Equations with Variables Separable262

6.2.3 First��Order Linear Equations266

6.2.4 Bernoulli's Equation269

6.2.5 Some Examples that can be Reduced to First��Order Linear Equations270

Exercises 6.2272

6.3 Reducible Second��Order Differential Equations273

Exercises 6.3276

6.4 Higher��Order Linear Differential Equations277

6.4.1 Some Examples of Linear Differential Equation of Higher��Order277

6.4.2 Structure of Solutions of Linear Differential Equations279

Exercises 6.4282

6.5 Linear Equations with Constant Coefficients283

6.5.1 Higher��Order Linear Homogeneous Equations with Constant Coefficients283

6.5.2 Higher��Order Linear Nonhomogeneous Equations with Constant Coefficients287

Exercises 6.5294

6.6 *Euler's Differential Equation295

Exercises 6.6296

6.7 Applications of Differential Equations296

Exercises 6.7301

Bibliography
303