Адаптационный курс по математическому анализу (преподается на английском языке)

2020/2021

Статус: Дисциплина общефакультетского пула

Кто читает: Департамент больших данных и информационного поиска

Где читается: Факультет компьютерных наук

Когда читается: 1, 2 модуль

Преподаватели: Орел Ольга Евгеньевна

Язык: русский

Кредиты: 2

Контактные часы: 24

Полная версия программы учебной дисциплины

Аннотация

The compulsory course in Calculus is one of the key mathematical courses for data scientists and analysts. To master the subject, students will need extensive knowledge of the school programme. The main purpose of the Adaptation Course in Calculus is to help students follow the material of the compulsory course in Calculus by discussing its most important and challenging topics. We also aim to teach our students the correct treatment of mathematical proof and mathematical definitions as well as the right logical reasoning behind proving statements and solving problems. The course topics follow those of the Data Science and Business Analytics’ basic course but will also prove useful for Applied Mathematics and Informatics and Software Engineering students as well. The structure of the course is as follows: method of mathematical induction, numerical sequences, limits of sequences, functions of a single variable, limits of functions, continuity of functions, derivatives of functions, analysis of functions and sketching their graphs.

Цель освоения дисциплины

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.

Планируемые результаты обучения

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.

Содержание учебной дисциплины

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.

Элементы контроля

Hometasks
seminar activities

Промежуточная аттестация

Промежуточная аттестация (2 модуль)
0.5 * Hometasks + 0.5 * seminar activities

Программа дисциплины