Mathematical analysisTopics

Within the scope of the topic, the concept of numerical sequence, the methods of giving the sequence and its main properties are studied. the concept of the limit of a numerical sequence and the main features of the limit are considered. the basic principles of monotone and bounded sequences, their stacking and the existence of their limit are explained. also, the definition of the number e, which plays an important role in mathematical analysis, its main properties and determination through the limit of the sequence are investigated.

Within the scope of the topic, the concept of the limit of a function and the main properties of the limit are studied. the limit of a function at a point and at infinity, left and right one-sided limits and their relationship with the existence of a limit of a function are considered. prominent limits and their application in calculating various limits are explained. infinitely decreasing and infinitely increasing functions, their main properties, comparison and relation with the concept of limit are also investigated.

Within the scope of the topic, the concept of continuity of a function at a point and the main properties of continuous functions are studied. the relationship between the limit of the function and its value at that point and the main conditions of continuity are considered. breakpoints of the function, their formation characteristics and classification according to the main types are explained. continuity of elementary functions and the effect of basic operations on continuous functions on continuity are also investigated.

Within the scope of the topic, the concept of the derivative of a function, the calculation of the derivative and its main properties are studied. the geometric meaning of the derivative is considered in terms of the slope coefficient of the tangent, and the mechanical meaning is considered in terms of the instantaneous rate of change of the variable quantity. the relationship between differentiability and continuity of a function is explained. rules for calculating derivatives of complex and inverse functions, as well as methods of differentiation of parametrically given functions are investigated.

In the framework of the topic, high order derivatives of the function and methods of their calculation are studied. the leibniz formula for calculating the higher order derivative of the product of two functions is considered. the basic theorems of differential calculus, such as roll, lagrange and cauchy's theorems, are explained in terms of their mathematical essence and their relationship with each other. the application of these theorems in studying the properties of functions and solving various mathematical problems is also investigated.

In the framework of the topic, taylor's formula is studied, which ensures the approximation of functions around a certain point by means of a polynomial. the construction of the taylor polynomial, different forms of the residual limit, and the evaluation of the accuracy of the approximation are considered. maclauren's formula is explained as a special case of taylor's formula, and taylor and maclauren openings of basic elementary functions are investigated. the application of taylor's formula in calculating approximate values ​​of functions, determining limits and solving various mathematical problems is also studied.

In the framework of the topic, the methods of investigating the main properties of functions are studied using the derivative. the principles of determining the increasing and decreasing intervals, stationary and critical points of the function are considered. methods of using first and second order derivatives for finding local and absolute extrema of a function, determining intervals of convexity and depression, as well as inflection points are explained. also, the issues of determining the asymptotes of the function and conducting a complete analysis of the function and constructing a graph based on the obtained results are investigated.

Concepts of primitive function and indefinite integral are studied within the subject. the relationship between the set of elementary functions of a function and the indefinite integral and the essence of the integration operation as the inverse operation of differentiation are considered. the basic properties of the indefinite integral and the role of the constant of integration are explained. also, the table of basic integrals covering the integrals of the basic elementary functions and the rules for using these formulas in the calculation of simple integrals are examined.

In the framework of the subject, the concept of a certain integral, its definition and main properties are studied. a definite integral function with a variable upper bound and the derivative of this function with respect to the upper bound are considered. the basic principles of the calculation of the definite integral and the relationship between the primitive function and the definite integral are explained. the newton–leibnis formula and methods of calculating various definite integrals using this formula are also investigated.

In the framework of the topic, the concept of non-specific integral and its main types are studied. non-singular integrals with an infinite interval of integration and arising in cases where the subintegral function is unbounded are considered. the principles of defining non-particular integrals through the limit, differentiating integrals and dispersive integrals are explained. it also examines the application of prime comparison and other signs of convergence to determine the convergence of non-singular integrals.

Special derivatives of multivariable functions and the concept of gradient are studied within the subject. the calculation of special derivatives of the function with respect to individual variables and their role in the analysis of the change characteristics of multivariable functions are considered. the definition of a gradient vector, its relation to special derivatives, and determining the direction of fastest growth of a function are explained. the application of special derivatives and gradients in the analysis of multivariable functions and solving various mathematical problems is also investigated.

Extrema of multivariable functions and methods of their determination are studied within the subject. the concepts of local maximum and minimum, necessary and sufficient conditions for the existence of an extremum are considered. the determination of stationary and critical points by means of special derivatives and the analysis of the nature of these points using second order special derivatives are explained. also, the use of extrema of multivariate functions in solving various mathematical and applied optimization problems is investigated.

In the framework of the topic, the concept of numerical sequence, the main characteristics of sequences, and issues of collection are studied. partial sums of a series, convergent and divergent sequences, as well as a necessary condition for the convergence of a series are considered. the essence and conditions of application of dalamber's sign are explained for determining the summation of numerical series consisting of positive terms. methods of determining the accumulation or collapse of a sequence based on the limit of the ratio of successive terms are also investigated.