Contents Chapter 1 Fundamental Knowledge of Calculus1 1.1 Mappings and Functions1 1.1.1 Sets and Their Operations1 1.1.2 Mappings and Functions6 1.1.3 Elementary Properties of Functions11 1.1.4 Composite Functions and Inverse Functions14 1.1.5 Basic Elementary Functions and Elementary Functions16 Exercises 1.1 A23 Exercises 1.1 B25 1.2 Limits of Sequences26 1.2.1 The Definition of Limit of a Sequence26 1.2.2 Properties of Limits of Sequences31 1.2.3 Operations of Limits of Sequences35 1.2.4 Some Criteria for Existence of the Limit of a Sequence38 Exercises 1.2 A44 Exercises 1.2 B46 1.3 The Limit of a Function46 1.3.1 Concept of the Limit of a Function47 1.3.2 Properties and Operations of Limits for Functions53 1.3.3 Two Important Limits of Functions58 Exercises 1.3 A61 Exercises 1.3 B63 1.4 Infinitesimal and Infinite Quantities63 1.4.1 Infinitesimal Quantities63 1.4.2 Infinite Quantities65 1.4.3 The Order of Infinitesimals and Infinite Quantities67 Exercises 1.4 A71 Exercises 1.4 B73 1.5 Continuous Functions73 1.5.1 Continuity of Functions74 1.5.2 Properties and Operations of Continuous Functions76 1.5.3 Continuity of Elementary Functions78 1.5.4 Discontinuous Points and Their Classification80 1.5.5 Properties of Continuous Functions on a Closed Interval83 Exercises 1.5 A87 Exercises 1.5 B89 Chapter 2 Derivative and Differential91 2.1 Concept of Derivatives91 2.1.1 Introductory Examples 91 2.1.2 Definition of Derivatives92 2.1.3 Geometric Meaning of the Derivative96 2.1.4 Relationship between Derivability and Continuity96 Exercises 2.1 A98 Exercises 2.1 B99 2.2 Rules of Finding Derivatives99 2.2.1 Derivation Rules of Rational Operations100 2.2.2 Derivation Rules of Composite Functions101 2.2.3 Derivative of Inverse Functions103 2.2.4 Derivation Formulas of Fundamental Elementary Functions104 Exercises 2.2 A105 Exercises 2.2 B107 2.3 Higher Order Derivatives107 Exercises 2.3 A110 Exercises 2.3 B111 2.4 Derivation of Implicit Functions and Parametric Equations， Related Rates111 2.4.1 Derivation of Implicit Functions111 2.4.2 Derivation of Parametric Equations114 2.4.3 Related Rates118 Exercises 2.4 A120 Exercises 2.4 B122 2.5 Differential of the Function123 2.5.1 Concept of the Differential123 2.5.2 Geometric Meaning of the Differential125 2.5.3 Differential Rules of Elementary Functions126 2.5.4 Differential in Linear Approximate Computation127 Exercises 2.5128 Chapter 3 The Mean Value Theorem and Applications of Derivatives130 3.1 The Mean Value Theorem130 3.1.1 Rolle's Theorem 130 3.1.2 Lagrange's Theorem132 3.1.3 Cauchy's Theorem135 Exercises 3.1 A137 Exercises 3.1 B138 3.2 L'Hospital's Rule138 Exercises 3.2 A144 Exercises 3.2 B145 3.3 Taylor's Theorem145 3.3.1 Taylor's Theorem145 3.3.2 Applications of Taylor's Theorem149 Exercises 3.3 A150 Exercises 3.3 B151 3.4 Monotonicity, Extreme Values, Global Maxima and Minima of Functions151 3.4.1 Monotonicity of Functions151 3.4.2 Extreme Values153 3.4.3 Global Maxima and Minima and Its Application156 Exercises 3.4 A158 Exercises 3.4 B160 3.5 Convexity of Functions， Inflections160 Exercises 3.5 A165 Exercises 3.5 B166 3.6 Asymptotes and Graphing Functions166 Exercises 3.6170 Chapter 4 Indefinite Integrals172 4.1 Concepts and Properties of Indefinite Integrals172 4.1.1 Antiderivatives and Indefinite Integrals172 4.1.2 Formulas for Indefinite Integrals174 4.1.3 Operation Rules of Indefinite Integrals175 Exercises 4.1 A176 Exercises 4.1 B177 4.2 Integration by Substitution177 4.2.1 Integration by the First Substitution177 4.2.2 Integration by the Second Substitution181 Exercises 4.2 A184 Exercises 4.2 B186 4.3 Integration by Parts186 Exercises 4.3 A193 Exercises 4.3 B194 4.4 Integration of Rational Functions194 4.4.1 Rational Functions and Partial Fractions194 4.4.2 Integration of Rational Fractions197 4.4.3 Antiderivatives Not Expressed by Elementary Functions201 Exercises 4.4201 Chapter 5 Definite Integrals202 5.1 Concepts and Properties of Definite Integrals202 5.1.1 Instances of Definite Integral Problems202 5.1.2 The Definition of the Definite Integral205 5.1.3 Properties of Definite Integrals208 Exercises 5.1 A213 Exercises 5.1 B214 5.2 The Fundamental Theorems of Calculus215 5.2.1 Fundamental Theorems of Calculus215 5.2.2 NewtonLeibniz Formula for Evaluation of Definite Integrals217 Exercises 5.2 A219 Exercises 5.2 B221 5.3 Integration by Substitution and by Parts in Definite Integrals222 5.3.1 Substitution in Definite Integrals222 5.3.2 Integration by Parts in Definite Integrals225 Exercises 5.3 A226 Exercises 5.3 B228 5.4 Improper Integral229 5.4.1 Integration on an Infinite Interval229 5.4.2 Improper Integrals with Infinite Discontinuities232 Exercises 5.4 A235 Exercises 5.4 B236 5.5 Applications of Definite Integrals237 5.5.1 Method of Setting up Elements of Integration237 5.5.2 The Area of a Plane Region239 5.5.3 The Arc Length of Plane Curves243 5.5.4 The Volume of a Solid by Slicing and Rotation about an Axis 247 5.5.5 Applications of Definite Integral in Physics249 Exercises 5.5 A252 Exercises 5.5 B254 Chapter 6 Differential Equations256 6.1 Basic Concepts of Differential Equations256 6.1.1 Examples of Differential Equations256 6.1.2 Basic Concepts258 Exercises 6.1259 6.2 FirstOrder Differential Equations260 6.2.1 FirstOrder Separable Differential Equation260 6.2.2 Equations can be Reduced to Equations with Variables Separable262 6.2.3 FirstOrder Linear Equations266 6.2.4 Bernoulli's Equation269 6.2.5 Some Examples that can be Reduced to FirstOrder Linear Equations270 Exercises 6.2272 6.3 Reducible SecondOrder Differential Equations273 Exercises 6.3276 6.4 HigherOrder Linear Differential Equations277 6.4.1 Some Examples of Linear Differential Equation of HigherOrder277 6.4.2 Structure of Solutions of Linear Differential Equations279 Exercises 6.4282 6.5 Linear Equations with Constant Coefficients283 6.5.1 HigherOrder Linear Homogeneous Equations with Constant Coefficients283 6.5.2 HigherOrder Linear Nonhomogeneous Equations with Constant Coefficients287 Exercises 6.5294 6.6 *Euler's Differential Equation295 Exercises 6.6296 6.7 Applications of Differential Equations296 Exercises 6.7301 Bibliography 303