Chapter 1 Functions and limits 1.1 Mappings and functions 1.1.1 Sets 1.1.2 Mappings 1.1.3 Functions Exercises 1-1 1.2 Limits of sequences 1.2.1 Concept of limits of sequences 1.2.2 Properties of convergent sequences Exercises 1-2 1.3 Limits of functions 1.3.1 Definitions of limits of functions 1.3.2 The properties of functional limits Exercises 1-3 1.4 Infinitesimal and infinity quantity 1.4.1 Infinitesimal quantity 1.4.2 Infinity quantity Exercises 1-4 1.5 Rules of limit operations Exercises 1-5 1.6 Principle of limit existence—two important limits Exercises 1-6 1.7 Comparing with two infinitesimals Exercises 1-7 1.8 Continuity of functions and discontinuous points 1.8.1 Continuity of functions 1.8.2 Discontinuous points of functions Exercises 1-8 1.9 Operations on continuous functions and the continuity of elementary functions 1.9.1 Continuity of the sum，difference，product and quotient of continuous functions 1.9.2 Continuity of inverse functions and composite functions 1.9.3 Continuity of elementary functions Exercises 1-9 1.10 Properties of continuous functions on a closed interval 1.10.1 Boundedness and maximum-minimum theorem 1.10.2 Zero point theorem and intermediate value theorem *1.10.3 Uniform continuity Exercises 1-10 Exercises 1 Chapter 2 Derivatives and differential 2.1 Concept of derivatives 2.1.1 Examples 2.1.2 Definition of derivatives 2.1.3 Geometric interpretation of derivative 2.1.4 Relationship between derivability and continuity Exercises 2-1 2.2 Fundamental derivation rules 2.2.1 Derivation rules for sum，difference，product and quotient of functions 2.2.2 The rules of derivative of inverse functions 2.2.3 The rules of derivative of composite functions（The Chain Rule）
2.2.4 Basic derivation rules and derivative formulas Exercises 2-2 2.3 Higher-order derivatives Exercises 2-3 2.4 Derivation of implicit functions and functions defined by parametric equations 2.4.1 Derivation of implicit functions 2.4.2 Derivation of a function defined by parametric equations 2.4.3 Related rates of change Exercises 2-4 2.5 The Differentials of functions 2.5.1 Concept of the differential 2.5.2 Geometric meaning of the differential 2.5.3 Formulas and rules on differentials 2.5.4 Application of the differential in approximate computation Exercises 2-5 Exercises 2 Chapter 3 Mean value theorems in differential calculus and applications of derivatives 3.1 Mean value theorems in differential calculus Exercises 3-1 3.2 L』Hospital』s rules Exercises 3-2 3.3 Taylor formula Exercises 3-3 3.4 Monotonicity of functions and convexity of curves 3.4.1 Monotonicity of functions 3.4.2 Convexity of curves and inflection points Exercises 3-4 3.5 Extreme values of functions, maximum and minimum 3.5.1 Extreme values of functions 3.5.2 Maximum and minimum of function Exercises 3-5 3.6 Differentiation of arc and curvature 3.6.1 Differentiation of an arc 3.6.2 curvature Exercises 3-6 Exercises 3 Chapter 4 Indefinite integral 4.1 Concept and property of indefinite integral 4.1.1 Concept of antiderivative and indefinite integral 4.1.2 Table of fundamental indefinite integrals 4.1.3 Properties of the indefinite integral Exercises 4-1 4.2 Integration by substitutions 4.2.1 Integration by substitution of the first kind 4.2.2 Integration by substitution of the second kind Exercises 4-2 4.3 Integration by parts Exercises 4-3 4.4 Integration of rational function 4.4.1 Integration of rational function
4.4.2 Integration which can be transformed into the integration of rational function Exercises 4-4 Exercises 4 Chapter 5 Definite integrals 5.1 Concept and properties of definite integrals 5.1.1 Examples of definite integral problems 5.1.2 The definition of define integral 5.1.3 Properties of definite integrals Exercises 5-1 5.2 Fundamental formula of calculus 5.2.1 The relationship between the displacement and the velocity 5.2.2 A function of upper limit of integral 5.2.3 Newton-Leibniz formula Exercises 5-2 5.3 Integration by substitution and parts for definite integrals 5.3.1 Integration by substitution for definite integrals 5.3.2 Integration by parts for definite integral Exercises 5-3 5.4 Improper integrals 5.4.1 Improper integrals on an infinite interval 5.4.2 Improper integrals of unbounded functions Exercises 5-4 5.5 Tests for convergence of improper integrals Γ-function 5.5.1 Test for convergence of infinite integral 5.5.2 Test for convergence of improper integrals of unbounded functions 5.5.3 Γ-function Exercises 5-5 Exercises05 Chapter 6 Applications of definite integrals 6.1 Method of elements for definite integrals 6.2 The applications of the definite integral in geometry 6.2.1 Areas of plane figures 6.2.2 The volumes of solid 6.2.3 Length of plane curves Exercises 6-1 6.3 The applications of the definite Integral in physics 6.3.1 Work done by variable force 6.3.2 Force by a liquid 6.3.3 Gravity Exercises 6-2 Exercises 6 Chapter 7 Differential equations 7.1 Differential equations and their solutions Exercises 7-1 7.2 Separable equations Exercises 7-2 7.3 Homogeneous equations 7.3.1 Homogeneous equations 7.3.2 Reduction to homogeneous equation Exercises 7-3
7.4 A first-order linear differential equations 7.4.1 Linear equations 7.4.2 Bernoulli』s equation Exercises 7-4 7.5 Reducible second-order equations Exercises 7-5 7.6 Second-order linear equations 7.6.1 Construction of solutions of second-order linear equation 7.6.2 The method of variation of parameters Exercises 7-6 7.7 Homogeneous linear differential equation with constant coefficients Exercises 7-7 7.8 Nonhomogeneous linear differential equation with constant coefficients Exercises 7-8 7.9 Euler』s differential equation Exercises 7-9 Exercises 7 Appendix References
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