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Бакалавриат 2020/2021

# Математический анализ

Лучший по критерию «Полезность курса для Вашей будущей карьеры»
Статус:
Направление: 01.03.02. Прикладная математика и информатика
Когда читается: 1-й курс, 1-4 модуль
Формат изучения: без онлайн-курса
Язык: английский
Кредиты: 10
Контактные часы: 144

### Course Syllabus

#### Abstract

The discipline gives the fundamentals of mathematics, provides the foundation for mathematical modeling, and introduces the first concepts of data analysis. The prerequisites are high school algebra and trigonometry. Prior experience with calculus is helpful but not essential.

#### Learning Objectives

• Students will develop an understanding of fundamental concepts of the single and multi variable calculus and form a range of skills that help them work efficiently with these concepts.
• Students will gain knowledge of the derivatives of single-variable functions, their integral, and the derivatives of multi-variable functions.
• The course will give students an understanding of simple optimization problems.

#### Expected Learning Outcomes

• Students should be able to analyze functions represented in a variety of ways: graphical, numerical, analytical, or verbal, and understand the relationships between these various representations.
• Students should be able to understand and apply basic concepts of the theory of limits, continuous and differentiable single-variable functions, antiderivatives and integrals of single-variable functions, continuous and differentiable several-variable functions.
• Students should be able to represent a function as the Taylor polynomial and a remainder term.
• Students should be able to compute derivatives and antiderivatives.
• Students should be able to compute limits of sequences and functions.
• Students should be able to estimate the asymptotical behavior of functions.
• Students should be able to determine the convergence of improper integrals.
• Students should be able to apply the computation of the integrals to the determination of the length of parametric curve arcs, the area of domains, and the volume of solid revolutions.
• Students should be able to understand the relationship between the derivative and the definite integral, as expressed by the Fundamental Theorem of Calculus.
• Students should be able to describe the space of several variables, convergence in the space, and properties of the distance.
• Students should be able to find the extrema of single- and several-variable functions.
• Students should be able to formulate and solve simple optimization problems.
• Students should be able to understand basic principles of numerical algorithms that solve algebraic equations and compute derivatives and integrals.
• Students should be able to apply numerical algorithms that solve algebraic equations and compute derivatives and integrals, to model a written description of simple economic or physical phenomena with functions, differential equations, or an integral, use mathematical analysis to solve problems, interpret results, and verify conclusions, determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.

#### Course Contents

• Sequences. Limit of a sequence: numbers, bounded sets, limits, operations with limits, monotone sequences, number e, Bolzano-Weierstrass theorem, completeness of real numbers.
• Continuous functions: limit of a function, definition of a continuous function, operations with continuous functions, monotonicity, inverse function, properties of continuous functions (basic theorems), types of discontinuity, uniform continuity
• Differentiable functions: definition of the derivatives, properties of differentiable functions, inverse functions, big and little o-notation, the mean value theorem, the second mean value theorem, higher derivatives, l’Hospital’s rule, Taylor’s theorem, numerical solution of algebraic equations
• Integration: indefinite integral, antiderivative, properties of the integral, methods of integration, the Riemann integral, the fundamental theorem of calculus, mean value theorems, improper integrals, numerical computations of integrals.
• Space of several variables and continuous functions on it: n-dimensional space R^n, open and closed sets, limit points, convergence of point sequences, continuous functions in R^n and their properties.
• Differentiation of functions of several variables: partial derivatives, differentials, the chain rule, the mean value theorem and Taylor’s theorem, optimization, sufficient conditions of extrema, constrained optimization, implicit function theorem, inverse mapping and Jacobians.

#### Assessment Elements

• 2nd module Exam
The exam may be carried out online via distance learning platforms. At the end of the second and fourth modules the students pass a written exam.
• 4th module Exam
The exam may be carried out online via distance learning platforms. At the end of the second and fourth modules the students pass a written exam.
• 1st semester Regular activity
During the year students must also complete weekly home assignments. Professors can ask students to present their written solutions orally. Quizzes are held regularly in classes.
• 2nd semester Regular activity
During the year students must also complete weekly home assignments. Professors can ask students to present their written solutions orally. Quizzes are held regularly in classes.
• 1st semester Bonus points
• 2nd semester Bonus points

#### Interim Assessment

• Interim assessment (2 module)
G(rade)=roundoff (min (0.4*Regular activity + 0.6*Exam +Bonus points, 10))
• Interim assessment (4 module)
G(rade)=roundoff (min (0.4*Regular activity + 0.6*Exam +Bonus points, 10))

#### Recommended Core Bibliography

• Calculus early transcendentals, Stewart, J., 2012
• Friedman, A. (2007). Advanced Calculus (Vol. Dover edition). Mineola, N.Y.: Dover Publications. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1153250
• William H. Press, Saul A. Teukolsky, William T. Vetterling, & Brian P. Flannery. (1992). Numerical Recipes in C: The Art of Scientific Computing. Second Edition. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.9CFCD6AE

#### Recommended Additional Bibliography

• Курс дифференциального и интегрального исчисления. Т.1: ., Фихтенгольц, Г. М., 2001
• Никитин А. А., Фомичев В. В. - МАТЕМАТИЧЕСКИЙ АНАЛИЗ. УГЛУБЛЕННЫЙ КУРС 2-е изд., испр. и доп. Учебник и практикум для академического бакалавриата - М.:Издательство Юрайт - 2019 - 460с. - ISBN: 978-5-534-00464-9 - Текст электронный // ЭБС ЮРАЙТ - URL: https://urait.ru/book/matematicheskiy-analiz-uglublennyy-kurs-432899
• Сборник задач и упражнений по математическому анализу : учеб. пособие для вузов, Демидович, Б. П., 2004