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1 Functions1

1．1 Sets1

1．1．1 Definition of Set1

1．1．2 Operations upon Sets1

1．1．3 The Set of Real Numbers2

1．2 Functions3

1．2．1 Definition3

1．2．2 Some Properties of Functions4

1．3 Composite Functions and Inverse Functions6

1．3．1 Composite Functions6

1．3．2 Inverse Functions8

1．4 Elementary Functions9

1．4．1 Constant functions9

1．4．2 Power functions9

1．4．3 Exponential functions10

1．4．4 Logarithmic functions10

1．4．5 Trigonometric functions10

1．4．6 Anti-trigonometric functions12

1．5 Exercises12

2 Limits and Continuity15

2．1 Limits of Sequences15

2．1．1 Definition15

2．2 Limits of Functions18

2．2．1 A Limit of a Function f(x)as x Tends to a Real Number x018

2．2．2 Limits Involving Infinity23

2．3 Techniques for Finding Limits27

2．4 Continuous Functions33

2．5 Exercises39

3 The Derivative43

3．1 Tangent lines and Rates of Change43

3．2 Definition of Derivative46

3．3 Differentiation Formulas49

3．4 Derivatives of Logarithmic Functions52

3．5 Derivatives of Trigonometric Functions53

3．6 The Chain Rule56

3．7 Derivatives of Inverse Functions and Implicit Differentiation59

3．8 Higher Derivatives64

3．9 Differentials and Linear Approximations65

3．9．1 Diffcrentials65

3．9．2 Linear Approximations67

3．10 Exercises68

4 Applications of Derivative71

4．1 The Mean Value Theorem71

4．2 Indeterminate Forms and L HOSPITAL S Rule74

4．2．1 The Forms 0/0 and ∞/∞74

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

4．3 Monotonic Functions78

4．4 Concavity and Points of Inflection80

4．5 Extrema of Functions83

4．6 Applications to Economics89

4．7 Exercises92

5 Indefinite Integrals95

5．1 Antiderivatives and the Indefinite Integral95

5．2 Substitution Rules100

5．3 Integration by Parts106

5．4 Exercises110

6 Definite Integrals113

6．1 Area and the Definite Integral113

6．2 Properties of the Definite Integral117

6．3 The Fundamental Theorem of Calculus121

6．4 Techniques of Integration125

6．4．1 Formula for integration by substitution125

6．4．2 Formula for integration by parts126

6．5 Improper Integrals127

6．5．1 Type 1:Infinite Intervals127

6．5．2 Type 2:Discontinuous Integrand131

6．6 Exercises133

7 Applications of Definite Integrals137

7．1 Area between Curves137

7．2 Volume140

7．3 Are Length144

7．4 Area of a Surface of Revolution148

7．5 Work150

7．6 Applications in Business and Economics151

7．6．1 Continuous Income Stream152

7．6．2 Consumers and Producers Surplus152

7．7 Exercises156

8 Series159

8．1 Numerical Series159

8．1．1 Fundamental Concepts159

8．1．2 Elementary Properties160

8．1．3 Infinite Series of Nonnegative Terms162

8．1．4 Alternating Series164

8．1．5 Absolute and Conditional Convergence165

8．2 Functional Series167

8．2．1 Power Series168

8．2．2 Properties of Power Series170

8．3 Taylor Series171

8．4 Exercises174

9 Vector Algebra and Space Ana？ytic Geometry177

9．1 Rectangular Coordinates in Space177

9．2 Vector Algebra179

9．2．1 Operations of Vectors179

9．2．2 The Coordinates of a Vector181

9．2．3 The Scalar Product185

9．2．4 The Vector Product187

9．3 The Planes and Lines in Space189

9．3．1 The Point-Normal Form Equations of a Plane189

9．3．2 Distance from a Point to a Plane190

9．3．3 The Angle between Two Planes191

9．3．4 The General Equation of a Line in Space192

9．3．5 Equations of a Line192

9．3．6 The Angle between Two Lines193

9．4 Equations for a Surface or a Curve194

9．4．1 The Equation for a Sphere194

9．4．2 The Equation of a Cylindrical Surface with Generators Paralleling to a Coordinate Axis194

9．4．3 Equation for the Intersection of Two Curved Surfaces195

9．4．4 The Parametric Equation of a Space Curve196

9．4．5 Equation for the Projecting Curve on a Coordinate Plane of a Space Curve196

9．5 Surfaces of Revolution197

9．6 Quadratic Surfaces198

9．6．1 Ellipsoids198

9．6．2 Hyperboloids of One Sheet199

9．6．3 Hyperboloids of Two Sheets199

9．6．4 Quadratic Cones200

9．6．5 Paraboloids201

9．6．6 Quadratic Cylinders201

9．7 Exercises202

10 Functions of Several Variables203

10．1 Fundamental Concepts203

10．2 Limits and Continuity205

10．3 Partial Derivatives207

10．4 The Chain Rule211

10．5 Approximation and Total Differential215

10．6 Applications of Partial Derivatives217

10．6．1 Geometric App？ication217

10．6．2 Extreme Values of Functions of Two Variables221

10．7 Exercises228

11 Multiple Integrals231

11．1 Double Integrals231

11．2 Properties of Double Integral232

11．3 Evaluation of Double Integrals235

11．4 Triple Integrals240

11．4．1 The Mass of an Object of Nonhomogeneous Density240

11．4．2 The Definition of Triple Integral240

11．4．3 Evaluation of Triple Integrals in Rectangular Coordinates241

11．5 Exercises244