The course aims at providing the fundamental basic notions and technical skills of Mathematical Analysis.
Prerequisites
Basic algebraic calculus. Trigonometry. Logarithms. Polynomial, rational, irrational, logarithmic, exponential, trigonometric equations and inequalities. Elementary analytic geometry in the plane.
Teaching Methods
Lectures and training sessions.
Type of Assessment
Intermediate written tests or final written exam. Final oral exam.
Course program
Real number. Axioms of the system of real numbers. Supremum of a set. Real powers, logarithms.
Sequences of real numbers. Limits of sequences. Fundamental theorems on limits. Monotone sequences and the Nepero number. Cauchy sequences. Limsup and liminf.
Real-valued functions of one real variable. Linking theorem between limits of sequences and limits of functions. Fundamental theorems on limits. Limits from the right and from the left. Monotone functions. Limsup and liminf.
Continuous functions. Continuity of elementary functions. Discontinuity points. Fundamental theorem on continuous functions. Theorems on continuous functions on intervals. Uniformly continuous functions, Lipschitz and Hoelder continuous functions.
Differential calculus. Derivative of a function. Derivatives of elementary functions. Differentiation rules. Fundamental theorems of differential calculus for functions on intervals. De l’Hopital rule. Taylor formula. Convex functions.
Integral calculus. Definition of Riemann integral. Elementary properties of integral. Criteria for the existence of integrals. Fundamental theorem of integral calculus. Indefinite integrals. Integrations by parts and change of variable formula. Integrals of rational functions.
Series. Convergence criteria for series with positive terms. Convergence criteria for series with terms of arbitrary sign.