I. Differentiation and Continuity of function of a real variable
1. History of calculus and some elementary prerequisites in analytical geometry and algebra.
2. Concept of infinitesimal and the concept of differentiation.
3. Differentiability and Continuity. First order approximation of a function value near a known function value.
4. Differentiation rules, arithmetic rules, and chain rule of elementary functions. Differentiation of inverse function.
5. Roll’s theorem, mean value theorem, intermediate value theorem.
6 Graphing of rational functions, trigonometric and inverse functions.
7. Extrema problems of continuous and differentiable functions. Applications of this extremal calculus。
8 Implicit differentiation of functions. How to locate the tangent line to a conics.
II. Integration
9.Partition and integration of a continuous function, upper and lower sums,Riemann sums to prove arithmetic laws of integration. Fundamental theorem of calculus.
10. Elementary indefinite integrals of elementary functions.