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Chapter 1 Functions£¨1£© 1.1 Preliminary knowledge£¨1£© 1.1.1 Inequalities and their properties£¨1£© 1.1.2 Absolute value and its properties£¨5£© 1.1.3 The range of variable£¨8£© 1.2 Functions£¨10£© 1.2.1 Concept of functions£¨10£© 1.2.2 Features of a function£¨12£© 1.2.3 Inverse functions£¨16£© 1.2.4 Composite functions£¨19£© 1.2.5 Elementary functions£¨20£© 1.2.6 Nonª²elementary functions£¨30£© 1.2.7 Implicit functions£¨33£© Exercise 1£¨33£© Chapter 2 Limit and Continuity£¨36£© 2.1 Limit£¨36£© 2.1.1 Definition of a sequence£¨36£© 2.1.2 Descriptive definition of limit of a sequence£¨36£© 2.1.3 Quantitative definition of limit of a sequence£¨38£© 2.2 Limits of functions£¨39£© 2.2.1 Definition of finite limits of functions as x→x0£¨39£© 2.2.2 Definition of infinite limits of functions as x→x0£¨42£© 2.2.3 Limits of functions as independent variable tending to infinity£¨44£© 2.2.4 Left limit and right limit£¨47£© 2.2.5 The properties of limits of functions£¨48£© 2.2.6 Operation rules of limits£¨50£© 2.2.7 Criteria of existence of limits and two important limits £¨54£© 2.2.8 Infinitesimal, infinity and their basic properties£¨58£© 2.2.9 Simple application of limit in economics£¨62£© 2.3 Continuity of functions£¨64£© 2.3.1 Continuity£¨64£© 2.3.2 Discontinuous points of a function£¨68£© 2.3.3 Operations and properties of continuous functions£¨69£© 2.3.4 Continuity of elementary functions£¨72£© 2.3.5 Continuity of the inverse functions£¨73£© 2.3.6 Properties of continuous functions on closed interval£¨73£© Exercise 2£¨76£© Chapter 3 Derivative and differential£¨80£© 3.1 Concept of derivative£¨80£© 3.1.1 Introduction of derivative£¨80£© 3.1.2 Definition of derivative£¨82£© 3.1.3 Leftª²hand derivative and rightª²hand derivative£¨84£© 3.1.4 The relationship between differentiability and continuity of functions£¨85£© 3.1.5 Applying the definition of derivative to find derivatives£¨87£© 3.1.6 Geometric interpretation of derivative£¨91£© 3.2 Rules of finding derivatives£¨91£© 3.2.1 Four arithmetic operation rules of derivatives£¨91£© 3.2.2 Derivative rules of composite functions£¨93£© 3.2.3 Derivative rules of inverse functions£¨95£© 3.2.4 Derivative rules of implicit functions£¨96£© 3.2.5 Derivative rules of function with parametric forms£¨97£© 3.2.6 Some special derivative rules£¨98£© 3.2.7 Basic differentiation formulas£¨100£© 3.2.8 Derivatives of higher order£¨102£© 3.3 Differentials of functions£¨104£© 3.3.1 Definition of differentials£¨104£© 3.3.2 The equations of a tangent and a normal£¨107£© 3.3.3 Formulas and operation rule of differentials£¨109£© 3.3.4 Application of differentials in approximating values£¨111£© Exercise 3£¨112£© Chapter 4 The mean value theorems and application of derivatives£¨116£© 4.1 The mean value theorems£¨116£© 4.1.1 Rolle’s theorem£¨116£© 4.1.2 Lagrange’s theorem£¨118£© 4.1.3 Cauchy’s theorem£¨121£© 4.2 L’Hospital’s rule£¨123£© 4.2.1 Evaluating limits of indeterminate forms of the type 00£¨124£© 4.2.2 Evaluating the limits of indeterminate forms of the type ∞∞£¨126£© 4.2.3 Evaluating the limits of other indeterminate forms£¨127£© 4.3 Taylor formula£¨129£© 4.4 Discuss properties of functions by derivatives£¨136£© 4.4.1 Monotonicity of functions£¨136£© 4.4.2 Concavity and Convexity£¨140£© 4.5 Extreme values£¨143£© 4.6 Absolute maxima (minima) and its application£¨148£© 4.6.1 Absolute maxima (minima)£¨148£© 4.6.2 Applied problems of absolute maxima (minima)£¨150£© 4.7 Graphing£¨152£© 4.7.1 Asymptotes lines of curves£¨152£© 4.7.2 Sophisticated graphing£¨154£© 4.8 Application of derivatives in economics£¨158£© 4.8.1 Marginal analysis£¨158£© 4.8.2 Elasticity of function£¨164£© Exercise 4£¨169£© Chapter 5 Indefinite integrals£¨173£© 5.1 Antiª²derivative and indefinite integral£¨173£© 5.1.1 Concept of antiª²derivatives£¨173£© 5.1.2 Concept of indefinite integrals£¨175£© 5.2 Fundamental integral formulas£¨177£© 5.3 Integral methods of substitution£¨180£© 5.3.1 The first kind of substitution£¨180£© 5.3.2 The second kind of substitution£¨185£© 5.4 Integration by parts£¨189£© 5.5 Evaluate indefinite integrals of some special type£¨194£© 5.5.1 Integrals of rational functions£¨194£© 5.5.2 Integrals of irrational functions£¨198£© 5.5.3 Integrals of trigonometric functions£¨199£© 5.5.4 Integral of piecewise defined function£¨201£© Exercise 5£¨202£© Chapter 6 Definite integrals£¨205£© 6.1 Definition of definite integrals£¨205£© 6.1.1 Two examples for definite integrals£¨205£© 6.1.2 Definition of definite integrals£¨207£© 6.1.3 Geometric meaning of definite integrals£¨211£© 6.2 Basic properties of definite integrals£¨212£© 6.3 Fundamental theorem of calculus£¨219£© 6.3.1 A function of upper limit of integral£¨219£© 6.3.2 Newtonª²Leibniz formula£¨222£© 6.4 Integration by substitution and by parts for definite integrals£¨224£© 6.4.1 Integration by substitution for definite integrals£¨225£© 6.4.2 Integration by parts for definite integrals£¨229£© 6.5 Improper integrals£¨231£© 6.5.1 Improper integrals on infinite intervals£¨231£© 6.5.2 Improper integrals of unbounded functions£¨239£© 6.6 Application of integrals£¨241£© 6.6.1 Computing areas of plan figures£¨241£© 6.6.2 Volume of a solid of revolution£¨245£© 6.6.3 Some economic applications of integrals£¨247£© Exercise 6£¨249£© Answers to exercises£¨256£© Answers to exercise1£¨256£© Answers to exercise2£¨257£© Answers to exercise3£¨257£© Answers to exercise4£¨259£© Answers to exercise5£¨261£© Answers to exercise6£¨262£©