chapter 0 preliminary knowledge

  0.1polar coordinate system

  0.1.1plotting points with polar coordinates

  0.1.2converting between polar and cartesian coordinates

  0.2complex numbers

  0.2.1the definition of the complex number

  0.2.2the complex plane

  0.2.3absolute value,conjugation and distance

  0.2.4polar form of complex numbers

chapter 1 theoretical basis of calculus

  1.1sets and functions

  1.1.1sets and their operations

  1.1.2mappings and functions

  1.1.3the primary properties of functions

  1.1.4composition of functions

  1.1.5elementary functions and hyperbolic functions

  1.1.6 modeling our real world

  exercises 1.1

  1.2limits of sequences of numbers

  1.2.1the sequence

  1.2.2convergence of a sequence

  1.2.3calculating limits of sequences

  exercises 1.2

  1.3limits of functions

  1.3.1speed and rates of change

  1.3.2the concept of limit of a function

  1.3.3properties and operation rules of functional limits

  1.3.4two important limits

  exercises 1.3

  1.4infinitesimal and infinite quantities

  1.4.1infinitesimal quantities and their order

  1.4.2infinite quantities

  exercises 1.4

  1.5continuous functions

  1.5.1continuous function and discontinuous points

  1.5.2operations on continuous functions and the continuity of elementary functions

  1.5.3properties of continuous functions on a closed interval

  exercises 1.5

chapter 2 derivative and differential

  2.1concept of derivatives

  2.1.1introductory examples

  2.1.2definition of derivatives

  2.1.3geometric interpretation of derivative

  2.1.4relationship between derivability and continuity

  exercises 2.1

  2.2rules of finding derivatives

  2.2.1derivation rules of rational operations

  2.2.2derivative of inverse functions

  2.2.3derivation rules of composite functions

  2.2.4derivation formulas of fundamental elementary functions

  exercises 2.2

  2.3higher-order derivatives

  exercises 2.3

  2.4derivation of implicit functions and parametric equations,related rates

  2.4.1derivation of implicit functions

  2.4.2derivation of parametric equations

  2.4.3related rates

  exercises 2.4

  2.5differential of the function

  2.5.1concept of the differential

  2.5.2geometric meaning of the differential

  2.5.3differential rules of elementary functions

  exercises 2.5

  2.6differential in linear approximate computation

  exercises 2.6

chapter 3 the mean value theorem and applications of derivatives

  3.1the mean value theorem

  3.1.1rolle's theorem

  3.1.2lagrange's theorem

  3.1.3cauchy s theorem

  exercises 3.1

  3.2l'hospital's rule

  exercises 3.2

  3.3taylor's theorem

  3.3.1 taylor's theorem

  3.3.2applications of taylor's theorem

  exercises 3.3

  3.4monotonicity and convexity of functions

  3.4.1monotonicity of functions

  3.4.2convexity of functions,inflections

  exercises 3.4

  3.5local extreme values,global maxima and minima

  3.5.1local extreme values

  3.5.2global maxima and minima

  exercises 3.5

  3.6graphing functions using calculus

  exercises 3.6

chapter 4 indefinite integrals

  4.1concepts and properties of indefinite integrals

  4.1.1antiderivatives and indefinite integrals

  4.1.2properties of indefinite integrals

  exercises 4.1

  4.2integration by substitution

  4.2.1integration by the first substitution

  4.2.2 integration by the second substitution

  exercises 4.2

  4.3integration by parts

  exercises 4.3

  4.4integration of rational fractions

  4.4.1integration of rational fractions

  4.4.2antiderivatives not expressed by elementary functions

  exercises 4.4

chapter 5 definite integrals

  5. 1concepts and properties of definite integrals

  5.1.1instances of definite integral problems

  5.1.2the definition of definite integral

  5.1.3properties of definite integrals

  exercises 5.1

  5.2the fundamental theorems of calculus

  exercises 5.2

  5.3integration by substitution and by parts in definite integrals

  5.3.1substituti