# Mathematical analysis

Objectives and outcomes

Students acquire basic knowledge of mathematical analysis. Upon completion of the course, students will adopt the basic concepts of mathematical analysis, as well as the most important theorems covered during the course and they will be able to apply the above to other courses.

Lectures

Countable and uncountable sets. Open, closed and compact sets. Real functions – overview of basic concepts. Polynomial and rational function – decomposition of a rational function into partial fractions. Real series – limit of series, properties of convergent series, infinite limits, Squeeze theorem, Stolz’s theorem, monotone series, convergent series, subseries, limit values ​​of subseries, Cauchy’s criterion of convergence of series, Banach’s fixed point theorem and iteration method. Limit values ​​and continuity of real functions of a real variable-connection between limit values ​​of series and functions, intermediate value theorem, the bisection method, continuous functions on compact sets, uniform continuity, speed of convergence and infinitesimal sizes. Differential calculus – tangent problem and derivative definition, formalism of differentiation, differential, mean value theorems and their application, derivatives and differentials of higher order, Taylor’s formula, flow testing and sketching graphs of functions. Integral calculus—squaring problem and definition of definite integral, notion of integrable function and properties of integral, notion of primitive function and definition of indefinite integral, basic methods of integration (shift method, method of partial integration) for definite and indefinite integrals, improper integral, integration of some classes of functions ( rational, irrational, trigonometric), application of certain integrals. Series—introduction and basic concepts, positive series, series with terms of arbitrary sign, power series. Differential equations – introductory concepts, differential equations of the first order.

Practical classes

Computational exercises, solving tasks that follow the units covered in lectures. Test preparation.