Course contents: 

1. Real number systems: properties of real numbers, coordinate systems, relations and functions. 
2. Limit functions: motivation, definition of limits, properties of limits, left and right limits, limits of trigonometric functions, limits to infinity and limits to infinity, limits of sequences, natural numbers, limits related to natural numbers, continuity. 
3. Derivatives: motivation, definition of derivatives, derivative properties, derivatives of compositional functions (chain rule), derivatives of inverse functions, derivatives of trigonometric functions, derivatives of cyclometric functions, derivatives of logarithmic functions, derivatives of exponential functions, derivatives of hyperbolic functions, derivatives of implicit functions, partial derivatives, derivatives logarithmically. The geometric meaning of the derivative: the derivative as the gradient of the tangent line, the derivative as the rate of change. 
4. Differential: total differential and its uses. High derivation rate. Rolle's Theorem and Intermediate Value Theorem. 
5. The extreme value theorem. Functions: increasing/decreasing, maximum/minimum, convex/concave, inflection point/inflection, graph/curve painting. 
6. Application of extreme problems, related rate of change. Taylor series and MacLaurin series. Hospital Rule L. 
7. Indeterminate integrals: the meaning and nature of indeterminate integrals.