¸ßµÈÊýѧ(ÉÏ) ADVANCED MATHEMATICS 1

Chapter 1 Function and limit 1.1 Function 1.1.1 Set 1.1.2 Function 1.1.3 Elementary functions Exercises 1-1 1.2 Limit 1.2.1 Limit of sequence 1.2.2 Limit of function Exercises 1-2 1.3 Rules of limit operations 1.3.1 Rational operation rules 1.3.2 Operation rule for composite function Exercises 1-3 1.4 Principle of limit existence and two important limits 1.4.1 Principle of limit existence 1.4.2 Two important limits Exercises 1-4 1.5 Infinitesimal and infinity 1.5.1 Infinitesimal 1.5.2 Infinity 1.5.3 Comparison of infinitesimal order Exercises 1-5 1.6 Continuity of function and the properties 1.6.1 Continuity of function and discontinuity 1.6.2 Operations on continuous functions and the continuity of elementary functions 1.6.3 Properties of continuous functions on a closed interval Exercises 1-6 Summary Quiz ExercisesChapter 2 Derivative and differential 2.1 Definition of derivatives 2.1.1 Cited examples 2.1.2 Definition of derivatives 2.1.3 Geometric interpretation of derivative 2.1.4 Use the unit to explain derivative 2.1.5 Relationship between derivability and continuity 2.1.6 The application of derivative in natural science Exercises 2-1 2.2 Rules of finding derivatives 2.2.1 Derivative formulas for several fundamental elementary functions 2.2.2 Derivation rules of rational operations 2.2.3 Derivative of inverse functions 2.2.4 Derivation rules of composite functions Exercises 2-2 2.3 Higher-order derivatives 2.3.1 The concept of higher order derivative 2.3.2 The linear property and Leibniz formula Exercises 2-3 2.4 Derivation of implicit functions and parametric equations 2.4.1 Derivation of implicit functions 2.4.2 Logarithmic derivation method 2.4.3 Derivation of parametric equations 2.4.4 Derivative of the function expressed by the polar equation Exercises 2-4 2.5 Differential of the function 2.5.1 Concept of the differential 2.5.2 The necessary and sufficient condition for differential 2.5.3 Geometric meaning of differential 2.5.4 Differential rules of elementary functions 2.5.5 Application of the differential Exercises 2-5 2.6 The application of derivative in economic analysis 2.6.1 Marginal analysis 2.6.2 Elasticity analysis Exercises 2-6 Summary Quiz ExercisesChapter 3 The mean value theorem 3.1 Mean value theorems the differential calculus 3.1.1 Format Lemma 3.1.2 Rolle's Theorem 3.1.3 Lagrange's Theorem 3.1.4 Cauehy's Theorem Exercises 3-1 3.2 Taylor's Theorem Exercises 3-2 Summary Quiz ExercisesChapter 4 Applications of derivatives 4.1 Indeterminate form limit 4.1.1 The indeterminate form of type 00 4.1.2 The indeterminate form of type ¡Þ¡Þ 4.1.3 Other indeterminate forms Exercises 4-1 4.2 Monotonicity and local extreme value 4.2.1 Monotonieity of functions 4.2.2 The extreme value of function 4.2.3 Maximum value and minimum value of function Exercises 4-2 4.3 The convexity, concavity and inflection point of function Exercises4-3 4.4 Description of the function graph 4.4.1 Asymptote of curve 4.4.2 Description of the function graph Exercises 4-4 4.5 Curvature 4.5.1 Differentiation of an arc 4.5.2 Calculation formula of curvature 4.5.3 Curvature circle Exercises 4-5 4.6 Approximation of equation 4.6.1 Bisection method 4.6.2 Tangent method Exercises 4-6 4.7 The application of extreme value of function in economic management 4.7.1 Demand analysis 4.7.2 Problems of minimum average cost 4.7.3 Problems of minimizing inventory cost 4.7.4 Problems of maximal profit 4.7.5 Problems of maximal revenue 4.7.6 Problems of maximal tax 4.7.7 Problems of compound interest Exercises4-7 Summary Quiz ExercisesChapter 5 Indefinite integrals 5.1 Indefinite integrals 5.1.1 Antiderivative 5.1.2 The conception of indefinite integral 5.1.3 Table of basic indefinite integrals 5.1.4 Properties of the indefinite integral Exercises 5-1 5.2 Integration by substitution 5.2.1 Integration by the first substitution (Gather differential together) 5.2.2 Integration by the second substitution Exercises 5-2 5.3 Integration by parts Exercises 5-3 5.4 Integration of rational functions 5.4.1 Integration of rational functions 5.4.2 Integration of rational trigonometric functions 5.4.3 Integration of simply irrationa4-functions Exercises 5-4 5.5 Application of indefinite integral in economy Exercises 5-5 Summary Quiz ExercisesChapter 6 Definite integrals 6.1 The concept of definite integral 6.1.1 Examples of definite integral problems 6.1.2 The definition of definite integral Exercises 6-1 6.2 Properties of definite integrals Exercises 6-2 6.3 Fundamental formula of calculus 6.3.1 The relationship between the displacement and the velocity 6.3.2 A function of upper limit of integral 6.3.3 Newton-Leibnitz formula Exercises 6-3 6.4 Integration by substitution and parts for definite integrals 6.4.1 Integration by substitution for definite integrals 6.4.2 Integration by parts for definite integrals Exercises 6-4 6.5 Improper integrals 6.5.1 Improper integrals on an infinite interval 6.5.2 Improper integral of unbounded functions Exercises 6-5 Summary Quiz ExercisesChapter 7 Applications of definite integral 7.1 The element method of definite integral 7.2 Application of definite integral in geometry 7.2.1 The area of a region 7.2.2 Volume 7.2.3 Arc length of plane curve Exercises 7-2 7.3 Application of definite integral in economy 7.3.1 Calculate total output if its rate of change is given 7.3.2 Calculate amount function when marginal function is given 7.3.3 Discount problems Exercises 7-3 Summary Quiz ExercisesAnswers