Mathematical AnalysisSillabus

This topic introduces the concept of a numerical sequence, methods of defining sequences, and their fundamental properties. The concept of the limit of a numerical sequence and the main properties of limits are examined. Monotone and bounded sequences, their convergence, and the fundamental principles related to the existence of their limits are explained. The definition and fundamental properties of the number e, which plays an important role in mathematical analysis, are also examined, including its definition through the limit of a sequence.

This topic examines the concept of the limit of a function and the fundamental properties of limits. The limit of a function at a point and at infinity, left-hand and right-hand limits, and their relationship with the existence of a function’s limit are considered. Remarkable limits and their applications in evaluating various limits are explained. Infinitesimal and infinite functions, their fundamental properties, comparison, and relationship with the concept of limits are also studied.

This topic examines the concept of continuity of a function at a point and the fundamental properties of continuous functions. The relationship between the limit of a function and its value at a given point, as well as the main conditions for continuity, are considered. Points of discontinuity, their causes, and their classification into major types are explained. The continuity of elementary functions and the effect of basic operations on continuous functions are also examined.

This topic introduces the concept of the derivative of a function, methods of calculating derivatives, and their fundamental properties. The geometric meaning of the derivative is considered as the slope of the tangent line, while its mechanical meaning is explained as the instantaneous rate of change of a variable quantity. The relationship between differentiability and continuity is discussed. The rules for finding derivatives of composite and inverse functions, as well as methods for differentiating parametrically defined functions, are examined.

This topic examines higher-order derivatives and methods for calculating them. The Leibniz formula for calculating higher-order derivatives of the product of two functions is studied. The fundamental theorems of differential calculus, including Rolle’s theorem, Lagrange’s Mean Value Theorem, and Cauchy’s Mean Value Theorem, are examined in terms of their conditions, mathematical significance, and relationships. Applications of these theorems to the analysis of function properties and the solution of various mathematical problems are also considered.

This topic examines the Taylor formula, which provides an approximation of functions around a given point by means of a polynomial. The construction of Taylor polynomials, different forms of the remainder term, and the evaluation of approximation accuracy are considered. The Maclaurin formula is introduced as a special case of the Taylor formula, and Taylor and Maclaurin expansions of fundamental elementary functions are examined. Applications of the Taylor formula to approximate calculations, evaluation of limits, and solution of various mathematical problems are also studied.

This topic examines methods for analyzing the fundamental properties of functions using derivatives. The principles for determining intervals of increase and decrease, stationary points, and critical points are considered. Methods for determining local and absolute extrema, intervals of concavity and convexity, and inflection points using first- and second-order derivatives are explained. The determination of asymptotes and the complete analysis and graphical representation of a function based on the obtained results are also studied.

This topic introduces the concepts of an antiderivative and an indefinite integral. The relationship between the set of antiderivatives of a function and the indefinite integral, as well as the nature of integration as the inverse operation of differentiation, are examined. The fundamental properties of the indefinite integral and the role of the constant of integration are explained. A table of basic integrals of elementary functions and the rules for applying these formulas to evaluate simple integrals are also studied.

This topic examines the concept of improper integrals and their main types. Improper integrals arising from infinite intervals of integration and unbounded integrands are considered. The definition of improper integrals through limits and the distinction between convergent and divergent integrals are explained. The application of comparison tests and other convergence criteria to determine the convergence of improper integrals is also studied.

This topic examines partial derivatives of functions of several variables and the concept of the gradient. The calculation of partial derivatives with respect to individual variables and their role in analyzing the behavior of multivariable functions are considered. The definition of the gradient vector, its relationship with partial derivatives, and its role in determining the direction of the greatest rate of increase of a function are explained. Applications of partial derivatives and gradients in the analysis of multivariable functions and the solution of various mathematical problems are also examined.

This topic examines extrema of functions of several variables and methods for determining them. The concepts of local maxima and minima, as well as the necessary and sufficient conditions for the existence of extrema, are considered. Methods for identifying stationary and critical points using partial derivatives and analyzing the nature of these points using second-order partial derivatives are explained. Applications of extrema of multivariable functions to various mathematical and applied optimization problems are also studied.

This topic examines the concept of a numerical series, its fundamental properties, and convergence. Partial sums, convergent and divergent series, and the necessary condition for convergence are considered. D’Alembert’s Ratio Test and its conditions of application for determining the convergence of series with positive terms are explained. Methods for determining whether a series converges or diverges based on the limit of the ratio of consecutive terms are also examined.