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 Nonelementary 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 xx0（39）
2.2.2 Definition of infinite limits of functions as xx0（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 Lefthand derivative and righthand 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 Rolles theorem（116）
4.1.2 Lagranges theorem（118）
4.1.3 Cauchys theorem（121）
4.2 LHospitals 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 Antiderivative and indefinite integral（173）
5.1.1 Concept of antiderivatives（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 NewtonLeibniz 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）
內容試閱：
Calculus Ⅰ is generally aimed at first year undergraduate students who major in finance or economics. This book strives to provide students and teachers with perspectives and approaches in calculus.The book guides the reader to an appreciation of interrelation among different aspects of the subject. It features examples that illustrate key concept as well as exercises that strengthen understanding.Chapter 1 provides preliminary knowledge and describes the classical functions.Chapter 2 discusses limits and continuity of functions and presents basic facts about continuous functions. In Chapter 3, the derivative is defined and the basic rules of differentiation are presented.Chapter 4 describes the basic theory of differentiation and Taylor formula. It provides how the derivative used to discuss properties of functions and apply in economics. Chapter 5 defines anti-derivative and indefinite integral. Fundamental integral formulas, change of variablein integral and integration by parts are also discussed.Chapter 6 introduces, through examples of area and distance, the notion of the integral, and the approximate integrals leading to its definition.Fundamental theorem of calculus is proved and applications of integrals are provided.
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