Single Variable Calculus (N1)

Cartesian and polar coordinates on the plane, introduction to the calculus of one variable and the method of the mathematical induction. Functions of one variable, the conception of representation, limit and continuity, Boltzano theorem. Derivative of first or higher order of functions and geometrical meaning, derivation rules and total differential. Inverse and composite functions, parametric equations, complex forms and L’ Hospital rule. Analysis, monotony and extremities of functions, asymptotes. Fermat theorem and theorems of mean value. Applications of derivatives with the use of method of extremities for functions of physical interest and finding the curvature of a plane curve. Sequences, number series and convergence criterions. Series of functions, uniform convergence criterions and power series. Binomial expansion, generalized mean value theorem or Taylor formula and local approximation of function. Taylor and Maclaurin series, binomial series and convergence. Fourier series and properties, even, odd and periodic functions, function extension, Fourier expansion of even and odd functions, convergence of Fourier series, derivation and integration of Fourier series, Bessel inequality and Parseval identity. Indefinite integral of functions and antiderivative, change of variable and analytic techniques of integration. Riemann integral and definite integral. Introduction to the numerical methods of integration and to the generalized integrals. Applications of integrals to the calculation of plane areas, curve’s length, surface areas and domain volumes by rotation.