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"Mathematical analysis. Theory of functions of one variable (program of the Faculty of Computational Mathematics and Cybernetics) - course 9640 rub. from MSU, training 15 weeks. (4 months), Date: November 30, 2023.
Miscellaneous / / December 03, 2023
The course covers classical material on mathematical analysis, studied in the first year of university in the first semester. Sections “Elements of set theory and real numbers”, “Theory of numerical sequences", "Limit and continuity of a function", "Differentiability of a function", "Applications differentiability." We will get acquainted with the concept of a set, give a strict definition of a real number and study the properties of real numbers. Then we'll talk about number sequences and their properties. This will allow us to consider the concept of a numerical function, well known to schoolchildren, at a new, more rigorous level. We will introduce the concept of limit and continuity of a function, discuss the properties of continuous functions and their application to solve problems. In the second part of the course, we will define the derivative and differentiability of a function of one variable and study the properties of differentiable functions. This will allow you to learn how to solve such important applied problems as approximate calculation of values functions and solving equations, calculating limits, studying the properties of a function and constructing it graphic arts.
Form of study
Correspondence courses using distance learning technologies
Admission Requirements
Availability of VO or SPO
Lecture 1. Elements of set theory.
Lecture 2. The concept of a real number. Exact faces of numerical sets.
Lecture 3. Arithmetic operations on real numbers. Properties of real numbers.
Lecture 4. Number sequences and their properties.
Lecture 5. Monotonous sequences. Cauchy criterion for sequence convergence.
Lecture 6. The concept of a function of one variable. Function limit. Infinitely small and infinitely large functions.
Lecture 7. Continuity of function. Classification of break points. Local and global properties of continuous functions.
Lecture 8. Monotonous functions. Inverse function.
Lecture 9. The simplest elementary functions and their properties: exponential, logarithmic and power functions.
Lecture 10. Trigonometric and inverse trigonometric functions. Remarkable limits. Uniform continuity of function.
Lecture 11. The concept of derivative and differential. Geometric meaning of derivative. Rules of differentiation.
Lecture 12. Derivatives of basic elementary functions. Function differential.
Lecture 13. Derivatives and differentials of higher orders. Leibniz's formula. Derivatives of parametrically defined functions.
Lecture 14. Basic properties of differentiable functions. Rolle's and Lagrange's theorems.
Lecture 15. Cauchy's theorem. L'Hopital's first rule of disclosing uncertainties.
Lecture 16. L'Hopital's second rule for disclosing uncertainties. Taylor's formula with a remainder term in Peano form.
Lecture 17. Taylor's formula with a remainder term in general form, in Lagrange and Cauchy form. Expansion according to the Maclaurin formula of the main elementary functions. Applications of Taylor's formula.
Lecture 18. Sufficient conditions for an extremum. Asymptotes of the graph of a function. Convex.
Lecture 19. Inflection points. General scheme of function research. Examples of plotting graphs.
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