Real numbers. Sequences. Limit of a sequence. Limits and continuity of f?unctions. Intermediate value theorem. Weierstrass's theorem. The derivative. Techniques of differentiation. Fermat, Rolle and Lagrange theorems. L'H?pital's rule. Taylor polynomial approximations. The derivative in graphing and applications. The definite and indefinite integral. Principles on integral evaluation: integration by parts and by substitution. The fundamental theorem of calculus (Newton-Leibniz). Improper integral.
Infinite series. Convergence tests. Series of f?unctions; convergence and uniform convergence. Power series; representation of f?unctions by power series. Taylor series. F?unctionsof several variables- limits and continuity, partial and directional derivatives, Linear approximation, Gradient. The chain rule. Higher order partial derivatives and second degree Taylor polynomial. Relative/absolute maximum and minimum values. Lagrange multipliers. Multiple integrals. Fubini's theorem. Change of variables. Polar, cylindrical and spherical coordinates. Line integrals of scalar f?unctions. Line integrals of vector fields. Independence of path and Green theorem. Surface integrals of scalar f?unctions. Oriented surfaces and surface integrals of vector fields. The divergence theorem (Gauss-Ostrogradsky). Stokes' theorem. Applications.