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
5.3.1substitution in definite integrals
5.3.2integration by parts in definite integrals
exercises 5.3
5.4improper integral
5.4.1integration on an infinite interval
5.4.2improper integrals with infinite discontinuities
exercises 5.4
5.5applications of definite integrals
5.5.1method of setting up elements of integration
5.5.2the area of a plane region
5.5.3the arc length of a curve
5.5.4the volume of a solid
5.5.5applications of definite integral in physics
exercises 5.5
chapter 6 infinite series
6. 1concepts and properties of series with constant terms
6.1.1 examples of the sum of an infinite sequence）
6.1.2concepts of series with constant terms
6.1.3properties of series with constant terms
exercises 6.1
6.2convergence tests for series with constant terms
6.2.1convergence tests of series with positive terms
6.2.2convergence tests for alternating series
6.2.3absolute and conditional convergence
exercises 6.2
6.3power series
6.3.1functional series
6.3.2power series and their convergence
6.3.3operations of power series
exercises 6.3
6.4expansion of functions in power series
6.4.1taylor and maclaurin series
6.4.2expansion of functions in power series
6.4.3applications of power series expansion of functions
exercises 6.4
6.5fourier series
6.5.1orthogonality of the system of trigonometric functions
6.5.2fourier series
6.5.3convergence of fourier series
6.5.4sine and cosine series
exercises 6.5
6.6fourier series of other forms
6.6.1fourier expansions of periodic functions with period 2l
6.6.2 * complex form of fourier series
exercises 6.6
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