Chapter 1: The definition and representation of real valued functions. Inverse functions and the composition of functions.
Chapter 2: Real numbers. The limits of sequences and functions.
Chapter 3: Tangent line and derivative of functions.
Chapter 4: Implicit differentiation. The differentiation of parametric equations. The geometric meaning of derivative. First order approximation.
Chapter 5: Mean value theorem. Second order and higher order derivatives. Cauchy Theorem. Indeterminate form. L’Hopital rule.
Chapter 6: The graphs of functions and approximate lines. Taylor polynomials.
Chapter 7: Definite integral. The fundamental theorem of calculus.
Chapter 8: The method to calculate indefinite integral.
Chapter 9: Application of definite integral. Approximate integral and general integral.
Chapter 10: Functions of several variables.
Chapter 11: Partial differentiation, tangent planes, gradients and directional derivatives.
Chapter 12: Limit of several variables functions. Lagrange multipliers.
Chapter 13: Multiple integration.
Appendix: Infinite serials, change variable of polar coordinate, some simple differential equation.