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# Mathematics.Mathematical Analysis I

2021/2022
Учебный год
ENG
Обучение ведется на английском языке
7
Кредиты
Статус:
Курс по выбору
Когда читается:
1-й курс, 1, 2 модуль

### Course Syllabus

#### Abstract

“Mathematical Analysis 1” is a basic discipline included in the mathematical cycle of fundamental training of students of the “Economics” study program. In this discipline the students get acquainted with the topics “Limits of functions” and “Differential calculus of functions of one and several variables” that will be used in the theory and applications of disciplines of the mathematical cycle. The material of the course can be used in development and applications of numerical methods of solving the problems in many areas of knowledge, and for creating and analyzing the mathematical models in various subjects, first and foremost in economics. The discipline is a model for studying of the mathematical component of “Economics” students’ education.

#### Learning Objectives

• The goal of studying the discipline “Mathematical Analysis 1” is learning the basic course of Mathematical Analysis, getting familiar with the terminology, theoretical principles and problem solving skills based on the fundamental notions of the course such as mapping/function, limit, derivative and several others, and forming of the theoretical foundations and a set of mathematical tools for the disciplines of the Economics cycle.As the result of having finished the course the student must: 1. Know the elementary functions, basic notions, theorems and methods of differential calculus 2. Have the skills of interpreting the results obtained in mathematical research.

#### Expected Learning Outcomes

• Demonstrates the knowledge of basic notions of set theory: sets, functions, basic elementary functions, ability to sketch their graphs using basic substitutions.
• Demonstrates the knowledge of the notions of limit of a function, continuity, ability to compute the limits and tell whether a function is continuous.
• Demonstrates the ability to differentiate functions, find the limits using derivatives, study the functions and sketch their graphs using derivatives
• Demonstrates the ability to work with functions of several variables: find their domains of definition, level lines and surfaces
• Demonstrates the ability to work with functions of several variables: find first and second order partial derivatives, extrema, find the directional derivatives and the gradient of a function of several variables.

#### Course Contents

• Introduction. Elements of sets and functions theory
Introduction to the discipline The subject of mathematical analysis. Elements of mathematical logic. Elements of set theory The notion of a set and a subset. Empty set. The set of all subsets of a set. Operations on sets. Cartesian product of sets. Correspondence, relation, binary relation. One-to-one and onto correspondence. Bijections between sets. Countable and uncountable sets. Examples. Mathematical induction. The set of real numbers. Properties of real numbers. Subsets of the set of real numbers. Bounded (unbounded) (above, below) sets. The greatest (the least) elements. The least upper bound, the greatest lower bound. Infimum and supremum. Existence of infimum and supremum. The notion of a neighborhood and a punctured neighborhood of a point on the real line. The notion of a limit point of a set of real numbers. Interior and boundary points. Open and closed sets. Maps The notion of a map (function), its domain and range. Elementary functions. Composition of maps. Inverse map. Injective, surjective and bijective maps.
• Limits and continuity of functions of one real variable
Examples of number sequences. Limit of a sequence. Existence of the limit of a monotonous bounded above sequence. Lemma on nested intervals. Subsequences. Bolzano-Weierstrass theorem on existence of a convergent subsequence of a bounded sequence. Lemma on existence of a limit point of an infinite bounded subset of the real line. Limit of a function. One sided limits. Infinitesimals. Properties of limits. Limits of composite functions. The 1st and the 2nd useful limits. O- and o-notation. Resolving indeterminacies. Continuity of a function at appoint and on a set. One sided continuity. Discontinuities and their classification. Arithmetic operations on continuous functions. Continuity of basic elementary functions. Continuity of composition. Infimu/supremum and minimum/maximum of a function on its domain of definition.
• Differentiation of functions of one variable
The notion of derivative of a function of one variable. Geometric and economical interpretation of the derivative. Equation of the tangent line. The notion of a differentiable function. The necessary and sufficient conditions of differentiability. The relation of continuity and differentiability. Derivative of the sum, the product . the quotient, composite function, inverse function. Differentiation of functions given parametrically. Derivatives of the basic elementary functions. Higher order derivatives. The notion of an extremum of a function of one variable. Local extremum (interior and boundary). The fundamental theorems of differential calculus. The necessary condition of interior local extremum (Fermat’s theorem). Mean value theorems (Rolle’s, Lagrange’s and Cauchy’s theorems) and their geometric interpretation. L’Hospital rule. Taylor and Maclaurin formulas. Monotonicity and extrema. Sufficient conditions of strict monotonicity of a function on an interval. Sufficient conditions of local extremum of a function of one variable. Convex and concave functions of one variable. Necessary and sufficient conditions of convexity/concavity. Inflexion points. Necessary and sufficient conditions of an inflexion point. Vertical and skew asymptotes of the graph of a function of one variable. The study of a function using the first and the second derivatives and sketching its graph.
• Point sets and sequences in n-dimensional space
The set of all two-dimensional vectors. Geometric interpretation of two-dimensional vectors. n-dimensional vectors. Addition and multiplication by scalars. Properties of these operations. Scalar product, n-dimensional Euclidean space. The norm of an n-dimensional vector and its properties. The notion of a neighborhood of a point, punctured neighborhood. The notions of a limit point, interior and boundary points in n-dimensional space. Open and closed sets on a plane and in n-dimensional space. The notions of linear combination, nonnegative combination, convex combination of points on a plane and in n-dimensional space. Convex sets in on a plane and in n-dimensional space. The notion of distance. Cauchy-Buniakowski inequality, triangle inequality. Connected sets, bounded sets. Compact sets. The notion of a domain. A sequence of points on a plane and in n-dimensional space. The notions of a bounded and an unbounded sequence of points. The connection to coordinate-wise convergence. Bolzano-Weierstrass theorem. Lemma on the limit points.
• Differentiable functions of several variables
Partial derivatives. Gradient. Differentiability of functions of several variables. Differential. Geometric interpretation of partial derivatives. Tangent plane to the graph of a function of several variables. Differentiability of functions of several variables. Directional derivative. Orthogonality of the gradient and the level sets of a differentiable function of several variables at a point. Higher order partial derivatives. The theorem about the equality of mixed partial derivatives. Extrema of functions of several variables. Sign definiteness of a quadratic form. A sufficient condition of local extremum. Convex and strictly convex functions. Extremum of a convex function.
• Functions of several variables
Functions of two variables. The notion of the level set of a function of two variables. Generalization to the case of more than two variables.

#### Assessment Elements

• Test 1
The tests are being held in the auditorium or distance remotely (in case of distance learning format) in a written form, the duration is determined by the lecturer and the students are notified of it in advance. The possibility of the tests being held for all groups at the same time reserved. The requirements for the test procedure in distance format are being made available to the students via an instruction in LMS (and or via the University email) in advance
• Test 2
The tests are being held in the auditorium or distance remotely (in case of distance learning format) in a written form, the duration is determined by the lecturer and the students are notified of it in advance. The possibility of the tests being held for all groups at the same time reserved. The requirements for the test procedure in distance format are being made available to the students via an instruction in LMS (and or via the University email) in advance .
• Test 3
The tests are being held in the auditorium or distance remotely (in case of distance learning format) in a written form, the duration is determined by the lecturer and the students are notified of it in advance. The possibility of the tests being held for all groups at the same time reserved. The requirements for the test procedure in distance format are being made available to the students via an instruction in LMS (and or via the University email) in advance.
• Self Work
• Exam
The exam is being held in the auditorium or distance remotely (in case of distance learning format) in a written form, the duration is determined by the lecturer and the students are notified of it in advance. The possibility of the exam being held for all groups at the same time reserved. The requirements for the exam procedure in distance format are being made available to the students via an instruction in LMS (and or via the University email) in advance.

#### Interim Assessment

• Interim assessment (2 module)
0.39 * Exam + 0.1 * Self Work + 0.17 * Test 1 + 0.17 * Test 2 + 0.17 * Test 3

#### Recommended Core Bibliography

• Mangatiana A. Robdera, A Concise Approach to Mathematical Analysis, 2003, [electronic resource] link: https://link.springer.com/book/10.1007/978-0-85729-347-3
• V. A. Zorich. (2016). Mathematical Analysis I (Vol. 2nd ed. 2015). Springer.