微积分学 英文PDF电子书下载

1 Functions 1

1．1 Sets 1

1．1．1 Definition of Set 1

1．1．2 Operations upon Sets 1

1．1．3 The Set of Real Numbers 2

1．2 Functions 3

1．2．1 Definition 3

1．2．2 Some Properties of Functions 4

1．3 Composite Functions and Inverse Functions 6

1．3．1 Composite Functions 6

1．3．2 Inverse Functions 8

1．4 Elementary Functions 9

1．4．1 Constant functions 9

1．4．2 Power functions 9

1．4．3 Exponential functions 10

1．4．4 Logarithmic functions 10

1．4．5 Trigonometric functions 10

1．4．6 Anti-trigonometric functions 12

1．5 Exercises 12

2 Limits and Continuity 15

2．1 Limits of Sequences 15

2．1．1 Definition 15

2．2 Limits of Functions 18

2．2．1 A Limit of a Function f(x)as x Tends to a Real Number x0 18

2．2．2 Limits Involving Infinity 23

2．3 Techniques for Finding Limits 27

2．4 Continuous Functions 33

2．5 Exercises 39

3 The Derivative 43

3．1 Tangent lines and Rates of Change 43

3．2 Definition of Derivative 46

3．3 Differentiation Formulas 49

3．4 Derivatives of Logarithmic Functions 52

3．5 Derivatives of Trigonometric Functions 53

3．6 The Chain Rule 56

3．7 Derivatives of Inverse Functions and Implicit Differentiation 59

3．8 Higher Derivatives 64

3．9 Differentials and Linear Approximations 65

3．9．1 Diffcrentials 65

3．9．2 Linear Approximations 67

3．10 Exercises 68

4 Applications of Derivative 71

4．1 The Mean Value Theorem 71

4．2 Indeterminate Forms and L HOSPITAL S Rule 74

4．2．1 The Forms 0/0 and ∞/∞ 74

4．2．2 The Forms 0·∞,00, ∞0,1∞ and ∞-∞ 76

4．3 Monotonic Functions 78

4．4 Concavity and Points of Inflection 80

4．5 Extrema of Functions 83

4．6 Applications to Economics 89

4．7 Exercises 92

5 Indefinite Integrals 95

5．1 Antiderivatives and the Indefinite Integral 95

5．2 Substitution Rules 100

5．3 Integration by Parts 106

5．4 Exercises 110

6 Definite Integrals 113

6．1 Area and the Definite Integral 113

6．2 Properties of the Definite Integral 117

6．3 The Fundamental Theorem of Calculus 121

6．4 Techniques of Integration 125

6．4．1 Formula for integration by substitution 125

6．4．2 Formula for integration by parts 126

6．5 Improper Integrals 127

6．5．1 Type 1:Infinite Intervals 127

6．5．2 Type 2:Discontinuous Integrand 131

6．6 Exercises 133

7 Applications of Definite Integrals 137

7．1 Area between Curves 137

7．2 Volume 140

7．3 Are Length 144

7．4 Area of a Surface of Revolution 148

7．5 Work 150

7．6 Applications in Business and Economics 151

7．6．1 Continuous Income Stream 152

7．6．2 Consumers and Producers Surplus 152

7．7 Exercises 156

8 Series 159

8．1 Numerical Series 159

8．1．1 Fundamental Concepts 159

8．1．2 Elementary Properties 160

8．1．3 Infinite Series of Nonnegative Terms 162

8．1．4 Alternating Series 164

8．1．5 Absolute and Conditional Convergence 165

8．2 Functional Series 167

8．2．1 Power Series 168

8．2．2 Properties of Power Series 170

8．3 Taylor Series 171

8．4 Exercises 174

9 Vector Algebra and Space Ana？ytic Geometry 177

9．1 Rectangular Coordinates in Space 177

9．2 Vector Algebra 179

9．2．1 Operations of Vectors 179

9．2．2 The Coordinates of a Vector 181

9．2．3 The Scalar Product 185

9．2．4 The Vector Product 187

9．3 The Planes and Lines in Space 189

9．3．1 The Point-Normal Form Equations of a Plane 189

9．3．2 Distance from a Point to a Plane 190

9．3．3 The Angle between Two Planes 191

9．3．4 The General Equation of a Line in Space 192

9．3．5 Equations of a Line 192

9．3．6 The Angle between Two Lines 193

9．4 Equations for a Surface or a Curve 194

9．4．1 The Equation for a Sphere 194

9．4．2 The Equation of a Cylindrical Surface with Generators Paralleling to a Coordinate Axis 194

9．4．3 Equation for the Intersection of Two Curved Surfaces 195

9．4．4 The Parametric Equation of a Space Curve 196

9．4．5 Equation for the Projecting Curve on a Coordinate Plane of a Space Curve 196

9．5 Surfaces of Revolution 197

9．6 Quadratic Surfaces 198

9．6．1 Ellipsoids 198

9．6．2 Hyperboloids of One Sheet 199

9．6．3 Hyperboloids of Two Sheets 199

9．6．4 Quadratic Cones 200

9．6．5 Paraboloids 201

9．6．6 Quadratic Cylinders 201

9．7 Exercises 202

10 Functions of Several Variables 203

10．1 Fundamental Concepts 203

10．2 Limits and Continuity 205

10．3 Partial Derivatives 207

10．4 The Chain Rule 211

10．5 Approximation and Total Differential 215

10．6 Applications of Partial Derivatives 217

10．6．1 Geometric App？ication 217

10．6．2 Extreme Values of Functions of Two Variables 221

10．7 Exercises 228

11 Multiple Integrals 231

11．1 Double Integrals 231

11．2 Properties of Double Integral 232

11．3 Evaluation of Double Integrals 235

11．4 Triple Integrals 240

11．4．1 The Mass of an Object of Nonhomogeneous Density 240

11．4．2 The Definition of Triple Integral 240

11．4．3 Evaluation of Triple Integrals in Rectangular Coordinates 241

11．5 Exercises 244