Syllabus
*1. Introduction Logic, mappings, countable sets, real numbers, supremum property, complex numbers. *2. Limit of sequences Convergence of sequences, the limit of a monotone sequence, values of accumulation, limsup, liminf, Bolzano-Weierstrass theorem, Cantor principle, Bolzano-Cauchy condition. *3. Limit of functions and continuity Basic notions, limit, the neighborhood of a point, limit and continuity at a point (one-sided versions included), theorems on limits and arithmetics, ordering, the limit of a composed function, Heine theorem, the limit of a monotone function, continuous functions on an interval (intermediate value theorem, continuous image, boundedness, attaining of extrema, continuity of an inverse function. *4. Elementary functions Exponential, logarithmic, goniometric and cyclometric, and power function (without proofs). *5. Derivative Definition and basic properties, arithmetics of derivatives, the derivative of a composed function, derivative of an inverse function, derivative of elementary functions, Rolle theorem, Lagrange theorem, Cauchy theorem, l'Hospital rule, the limit of a derivative at a point, monotonicity and sign of a derivative, convex and concave functions, inflection point, derivative and convexity, asymptote, investigation of a function. *6. Taylor polynomial Taylor polynomial, Peano theorem, Lagrange theorem, Cauchy theorem, the symbol "little o" and its properties, Taylor polynomial of elementary functions.
Annotation
The first part of a four-semester course in mathematical analysis for bachelor's programs General Mathematics and Information Security.