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Advanced mathematics (2nd edition) I

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Table of Contents
Chapter 1 Fundamental Knowledge of Calculus
1.1 Mappings and Functions
1.1.1 Sets and Their Operations
1.1.2 Mappings and Functions
1.1.3 Elementary Properties of Functions
1.1.4 Composite Functions and Inverse Functions
1.1.5 Basic Elementary Functions and Elementary Functions
Exercises 1.1 A
Exercises 1.1 B
1.2 Limits of Sequences
1.2.1 The Definition of Limit of a Sequence
1.2.2 Properties of Limits of Sequences
1.2.3 Operations of Limits of Sequences
1.2.4 Some Criteria for Existence of the Limit of a Sequence
Exercises 1.2 A
Exercises 1.2 B
1.3 The Limit of a Function
1.3.1 Concept of the Limit of a Function
1.3.2 Properties and Operations of Limits for Functions
1.3.3 Two Important Limits of Functions
Exercises 1.3 A
Exercises 1.3 B
1.4 Infinitesimal and Infinite Quantities
1.4.1 Infinitesimal Quantities
1.4.2 Infinite Quantities
1.4.3 The Order of Infinitesimals and Infinite Quantities
Exercises 1.4 A
Exercises 1.4 B
1.5 Continuous Functions
1.5.1 Continuity of Functions
1.5.2 Properties and Operations of Continuous Functions
1.5.3 Continuity of Elementary Functions
1.5.4 Discontinuous Points and Their Classification
1.5.5 Properties of Continuous Functions on a Closed Interva
Exercises 1.5 A
Exercises 1.5 B

Chapter 2 Derivative and Differentia
2.1 Concept of Derivatives
2.1.1 Introductory Examples
2.1.2 Definition of Derivatives
2.1.3 Geometric Meaning of the Derivative
2.1.4 Relationship between Derivability and Continuity
Exercises 2.1 A
Exercises 2.1 B
2.2 Rules of Finding Derivatives
2.2.1 Derivation Rules of Rational Operations
2.2.2 Derivation Rules of Composite Functions
2.2.3 Derivative of Inverse Functions
2.2.4 Derivation Formulas of Fundamental Elementary Functions
Exercises 2.2 A
Exercises 2.2 B
2.3 Higher Order Derivatives
Exercises 2.3 A
Exercises 2.3 B
2.4 Derivation of Implicit Functions and Parametric Equations,
Related Rates
2.4.1 Derivation of Implicit Functions
2.4.2 Derivation of Parametric Equations
2.4.3 Related Rates
Exercises 2.4 A
Exercises 2.4 B
2.5 Differential of the Function
2.5.1 Concept of the Differential
2.5.2 Geometric Meaning of the Differential
2.5.3 Differential Rules of Elementary Functions
2.5.4 Differential in Linear Approximate Computation
Exercises 2.5

Chapter 3 The Mean Value Theorem and Applications of Derivatives
3.1 The Mean Value Theorem
3.1.1 Rolle's Theorem
3.1.2 Lagrange's Theorem
3.1.3 Cauchy's Theorem
Exercises 3.1 A
Exercises 3.1 B
3.2 L'Hospital's Rule
Exercises 3.2 A
Exercises 3.2 B
3.3 Taylor's Theorem
3.3.1 Taylor's Theorem
3.3.2 Applications of Taylor's Theorem
Exercises 3.3 A
Exercises 3.3 B
3.4 Monotonicity, Extreme Values, Global Maxima and Minima of Functions
3.4.1 Monotonicity of Functions
3.4.2 Extreme Values
3.4.3 Global Maxima and Minima and Its Application
Exercises 3.4 A
Exercises 3.4 B
3.5 Convexity of Functions, Inflections
Exercises 3.5 A
Exercises 3.5 B
3.6 Asymptotes and Graphing Functions
Exercises 3.6

Chapter 4 Indefinite Integrals
4.1 Concepts and Properties of Indefinite Integrals
4.1.1 Antiderivatives and Indefinite Integrals
4.1.2 Formulas for Indefinite Integrals
4.1.3 Operation Rules of Indefinite Integrals
Exercises 4.1 A
Exercises 4.1 B
4.2 Integration by Substitution
4.2.1 Integration by the First Substitution
4.2.2 Integration by the Second Substitution
Exercises 4.2 A
Exercises 4.2 B
4.3 Integration by Parts
Exercises 4.3 A
Exercises 4.3 B
4.4 Integration of Rational Functions
4.4.1 Rational Functions and Partial Fractions
4.4.2 Integration of Rational Fractions
4.4.3 Antiderivatives Not Expressed by Elementary Functions
Exercises 4.4

Chapter 5 Definite Integrals
5.1 Concepts and Properties of Definite Integrals
5.1.1 Instances of Definite Integral Problems
5.1.2 The Definition of the Definite Integral
5.1.3 Properties of Definite Integrals
Exercises 5.1 A
Exercises 5.1 B
5.2 The Fundamental Theorems of Calculus
5.2.1 Fundamental Theorems of Calculus
5.2.2 NewtonLeibniz Formula for Evaluation of Definite Integrals
Exercises 5.2 A
Exercises 5.2 B
5.3 Integration by Substitution and by Parts in Definite Integrals
5.3.1 Substitution in Definite Integrals
5.3.2 Integration by Parts in Definite Integrals
Exercises 5.3 A
Exercises 5.3 B
5.4 Improper Integral
5.4.1 Integration on an Infinite Interval
5.4.2 Improper Integrals with Infinite Discontinuities
Exercises 5.4 A
Exercises 5.4 B
5.5 Applications of Definite Integrals
5.5.1 Method of Setting up Elements of Integration
5.5.2 The Area of a Plane Region
5.5.3 The Arc Length of Plane Curves
5.5.4 The Volume of a Solid by Slicing and Rotation about an Axis
5.5.5 Applications of Definite Integral in Physics
Exercises 5.5 A
Exercises 5.5 B

Chapter 6 Differential Equations
6.1 Basic Concepts of Differential Equations
6.1.1 Examples of Differential Equations
6.1.2 Basic Concepts
Exercises 6.1
6.2 FirstOrder Differential Equations
6.2.1 FirstOrder Separable Differential Equation
6.2.2 Equations can be Reduced to Equations with Variables Separable
6.2.3 FirstOrder Linear Equations
6.2.4 Bernoulli's Equation
6.2.5 Some Examples that can be Reduced to FirstOrder Linear Equations
Exercises 6.2
6.3 Reducible SecondOrder Differential Equations
Exercises 6.3
6.4 HigherOrder Linear Differential Equations
6.4.1 Some Examples of Linear Differential Equation of HigherOrder
6.4.2 Structure of Solutions of Linear Differential Equations
Exercises 6.4
6.5 Linear Equations with Constant Coefficients
6.5.1 HigherOrder Linear Homogeneous Equations with Constant Coefficients
6.5.2 HigherOrder Linear Nonhomogeneous Equations with Constant Coefficients
Exercises 6.5
6.6 *Euler's Differential Equation
Exercises 6.6
6.7 Applications of Differential Equations
Exercises 6.7
Bibliography
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Advanced mathematics (2nd edition) I
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