目錄
Chapter 1 Functions and Limits
1.1 Mapping and Function
1.1.1 Mapping
1.1.2 Function
1.1.3 Elementary Functions
1.2 Properties of Function
1.2.1 Boundness
1.2.2 Monotonicity
1.2.3 Symmetry
1.2.4 Periodicity
1.3 Form a New Function from Known Functions
1.3.1 Arithmetic Operations of Functions
1.3.2 Composite Function
1.3.3 Inverse Function
1.3.4 Primary Function
1.4 The Limit of a Function
1.4.1 The Definition of a Limit
1.4.2 The Precise Definition of a Limit
1.4.3 One-Sided Limit
1.4.4 Infinite Limits
1.5 Properties of Function Limits
1.5.1 Properties of Function Limits
1.5.2 Relationship Between Function Limit and Sequence Limit
1.6 Calculating Limits Using the Limit Laws
1.6.1 Limit Laws
1.6.2 Two Important Limits
1.7 Infinitesimals and Infinite Limits
1.7.1 Infinitesimals
1.7.2 Infinite Limits
1.7.3 Comparison of Infinitesimals
1.8 Continuity of Functions
1.8.1 Concept of Continuity
1.8.2 Properties of Continuous Functions
1.8.3 Discontinuities of Functions and Their Classification
1.9 Properties of Continuous Functions on Closed Intervals
1.9.1 Extreme Value Theorem
1.9.2 Intermediate Value Theorem
Comprehensive Review 1
Chapter 2 Derivatives and Differentiation
2.1 Concept of Derivative
2.1.1 Examples and Definitions
2.1.2 Elementary Functions' Derivative
2.1.3 Geometrical Meaning of the Derivative
2.1.4 Relationship Between Differentiability and Continuity
2.2 Rules of Differentiation
2.2.1 Rules for Differentiating Arithmetic Operations
2.2.2 Derivative Rules for Inverse Functions
2.2.3 Chain Rules for Composite Functions
2.2.4 Basic Derivative Formulas
2.3 Higher-Order Derivatives
2.4 Implicit Differentiation
2.4.1 Derivative of Implicit Functions
2.4.2 Related Rates of Change
2.5 Differential of a Function
2.5.1 Concept of Differentiation
2.5.2 Differential Formulas and Rules
2.5.3 Application of Differentials in Approximation
Comprehensive Review 2
Chapter 3 Applications of Derivatives
3.1 Mean Value Theorems
3.1.1 Rolle's Mean Value Theorem
3.1.2 Lagrange's Mean Value Theorem
3.2 L'Hospital's Rule
3.3 Monotonicity and Extremum of Functions
3.3.1 Monotonicity of Functions
3.3.2 Extreme Values of Functions
3.4 Maximum and Minimum Values of Functions and Their Applications
3.5 Concavity and Inflection Points of Curves
3.6 Sketch Functions
3.6.1 Asymptotes of Curves
3.6.2 Graphical Representation of Functions
Comprehensive Review 3
Chapter 4 Indefinite Integrals
4.1 Concept and Properties of Indefinite Integrals
4.1.1 Primitive Function and Indefinite Integral
4.1.2 Properties of Indefinite Integrals
4.1.3 Basic Integration Formulas
4.1.4 Direct Integration Method
4.2 Integration by Substitution
4.2.1 The First Type of Substitution Method
4.2.2 The Second Type of Substitution Method
4.3 Integration by Parts
4.4 Integration of Rational Functions and Functions Reducible to Rational Functions
4.4.1 Integration of Rational Functions
4.4.2 Integration of Rational Trigonometric Functions
4.4.3 Integration of Simple Irrational Functions
4.4.4 Use of Integration Tables
Comprehensive Review 4
Chapter 5 Definite Integrals and Their Applications
5.1 Concept and Properties of Definite Integrals
5.1.1 Examples of Definite Integral Problems
5.1.2 Definition of the Definite Integral
5.1.3 Properties of Definite Integrals
5.2 Fundamental Theorem of Calculus
5.2.1 The Relationship Between Position Function and Velocity Function
5.2.2 Function of the Upper Limit of Integration and Its Derivative
5.2.3 Newton-Leibniz Formula
5.3 Integration by Substitution and by Parts for Definite Integrals
5.3.1 Integration by Substitution for Definite Integrals
5.3.2 Integration by Parts in Definite Integrals
5.4 Improper Integrals
5.4.1 Improper Integrals over Infinite Intervals
5.4.2 Improper Integrals of Unbounded Functions
5.5 Area of Plane
5.5.1 The Element Method of Definite Integrals
5.5.2 Area of Plane
5.6 Volume of Solids
5.6.1 Volume of Solids of Revolution
5.6.2 Volume of a Solid with Known Cross - Sectional Area
5.7 Applications of Definite Integrals in Physics
5.7.1 Work
5.7.2 Pressure
5.7.3 Force
Comprehensive Review 5
Chapter 6 Differential Equations
6.1 Basic Concepts of Differential Equations
6.2 Separable Differential Equations and Homogeneous Equations
6.2.1 Separable Differential Equations
6.2.2 Homogeneous Equations
6.3 First - Order Linear Differential Equations
6.3.1 Solution Method for First - Order Linear Equations
6.3.2 Bernoulli Equation
6.4 Total Differential Equations
Comprehensive Review 6
Chapter 7 Infinite Series
7.1 Concepts and Properties of Constant Term Series
7.1.1 Constant Term Series
7.1.2 Basic Properties of Convergent Series
7.2 Positive Series
&nb
[