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.



11. Region viewed as consists of line segments and hence the computation of the



area based on the length of these line segments. Volume decided by cross sections. Volume



of revolution of a region of function. Cavalieri principle.



12 Generalized mean value theorem and L’Hopital’s rule.



13. Center of mass of volume of revolution and the theorem of Pappus.



14. Exponential functions, logarithmic functions, their derivatives



and integrations



15. Techniques of integration including substitution and integration by



parts, trigonometric substitutions, partial fractions.



16. Numerical integrations. Trapezoidal rule, Simpson’s rule, error of these numerical estimations.



17. Differential equations and its application in physics, biology and ecology.



18. Taylor polynomial, Taylor remainders, Taylor series of a function relative to a point.



19. Taylor expansion and numerical approximation of higher order.



20. Partial derivative of function more than 1 variable. Gradient vector and the criteria of extrema of function of two variables. Lagrange method.



21. Double and triple integration. Repeated integration.



22. (option) Theory of series, criteria of convergence and divergence