• A
• A
• A
• АБВ
• АБВ
• АБВ
• А
• А
• А
• А
• А
Обычная версия сайта
02
Октябрь

# Algebra and Pre-Calculus: Introductory Course (Low Level)

2021/2022
Учебный год
ENG
Обучение ведется на английском языке
3
Кредиты
Статус:
Курс по выбору
Когда читается:
1-й курс, 1, 2 модуль

### Course Syllabus

#### Abstract

The discipline «Algebra and analysis: introductory course (base level)» has a purpose to study the following sections: «Sets and functions», «Vectors », «Straight line on the plane and in space. The plane in space», «Complex numbers» and «Polynomials and rational fractions». This subject allows one to further master the educational disciplines, such as «Linear algebra», «Mathematical analysis-I», «Mathematical analysis-II», «Microeconomics», «Macroeconomics», «Econometrics». The course of "Algebra and Analysis: Introductory course (basic level)" will be approached in the theory and application of the disciplines of the economics cycle. The course materials can be used to construct and study mathematical models in various subject areas, primarily in economics.

#### Learning Objectives

• The goal of the course is study of the sections “Sets and functions”, “Vectors”, “Straight line on a plane and in a space. A plane in a space”, “Complex numbers” and “Polynomials and rational fractions” that allow to further understanding of the following courses: “Linear algebra”, “Mathematical analysis-I”, “Mathematical analysis-II”, “Microeconomics”, “Macroeconomics”, “Econometrics”

#### Expected Learning Outcomes

• The knowledge and understanding of basic elementary functions, the ability to construct graphs of elementary functions by means of basic transformations on a plane.
• The knowledge and understanding of the module, solving of equations and inequalities using the properties of the module
• Acquirement of competence to find a domain, set of values of a function, to investigate a function on monotonicity, even/odd, periodicity.
• Acquirement of the ability to apply basic operations over vectors to solve practical and geometric problems
• The knowledge and understanding of the concept of the equation of a straight line in the plane, the ability to create the equation of a line, to construct a graph of a straight line, to apply conditions of mutual arrangement of lines in the plane in solving geometric and economic problems.
• Acquirement of the ability to work with complex numbers in an arbitrary form of writing, to solve algebraic equations with complex numbers, to perform arithmetic operations of elevation to degree and root extraction.
• Acquirement of the ability to apply Bézout's and Descartes' theorems to specific polynomials, and to factor them into factors. Isolate an entire part from a rational fraction. Represent the correct rational fraction in the sum of the simplest fractions.

#### Course Contents

• Sets and functions
Sets and subsets. An empty set. The set of all subsets of the set is set to a set. Sets are set to a set. Operations on sets. The Cartesian product of sets. It is a one-to-one correspondence. Equivalent sets, counting sets, and countable sets. Examples. Set of real numbers. Extended number axis. Axiomatic of real numbers. Subsets of set of real numbers. Display. Definition area and value range. Numerical function: graph of a numerical function, monotony, periodicity, even and odd functions. Arithmetic operations over numeric functions. The largest and smallest value of the monotone function. Elementary functions. Basic elementary functions, their properties and graphs. Transformations of graphs of elementary functions in the plane (axial shifts, scale transformations, reflections). Inverse mapping, its property. Composition of maps. Properties of composition. Injective, surjective and bijective mapping. Examples of numerical functions giving surjective, injective and bijective maps
• Vectors in R^n
Geometric vectors are repetition: definition, triangle rule and parallelogram, polygon rule. Arithmetic vectors. Linear operations. Vector coordinates. Linear dependence and vector independence in R n . Scalar product of vectors in R n and its properties. Angle between vectors. Vector projection to axis and direction. Projection properties of projections. Condition of collinearity and orthogonality of vectors. Division of a line in a given relation. Application: Calculation of parallelogram area and parallelepiped volume
• Straight line on the plane and in space. The plane in space
Equation of a line in the plane and in space: a line. Equation of a surface in space: a plane. The equation of a line on a point and a normal vector, the equation on a point and a guide vector, the canonical, general, parametric equation, the equation with an angular coefficient, the equation on two points. Reciprocal arrangement of two lines in the plane: angle between lines, orthogonal and parallel condition. The equation of the plane in space: point and normal vector, three points. Equation of a line in space: Canonical equation, parametric equation. Line as intersection of planes. The reciprocal arrangement of two lines in space: the angle between the lines, the condition of orthogonality and parallelism. The reciprocal position of the line and the plane: the angle between the line and the plane, the condition of orthogonal and parallelism. Distance from point to line. Distance from point to plane. Distance between lines. Distance between planes. Apply to geometric problems.
• Complex numbers
Definition, arithmetic operations and their properties. Module, complex conjugation. Algebraic form of writing. Geometric interpretation. The argument of a complex number. Trigonometric form of writing a complex number. An indicative form of writing a complex number. Euler formulas. Natural degree. Muavra’s formulas. Extraction of an n-degree root from a complex number. Equations and inequalities with complex numbers and their geometric interpretation.
• Polynomials and rational fractions
Definition, arithmetic operations and their properties. Module, complex conjugation. Algebraic form of writing. Geometric interpretation. The argument of a complex number. Trigonometric form of writing a complex number. An indicative form of writing a complex number. Euler formulas. Natural degree. Muavra’s formulas. Extraction of an n-degree root from a complex number. Equations and inequalities with complex numbers and their geometric interpretation

#### Assessment Elements

• Individual homework 1
The completed homework 1 should be scanned and attached to SmartLMS. When students specify a task, the answers must be entered into the system. Upon the decision of the seminar professor, some individual homework may consist of auditorian work and extracurricular activities.
• Individual homework 2
Assignment options are provided on the SmartLMS platform. The completed homework 2 should be scanned and attached to SmartLMS. When students specify a task, the answers must be entered into the system. Upon the decision of the seminar professor, some individual homework may consist of auditorian work and extracurricular activities.
• Control work
• Exam

#### Interim Assessment

• Interim assessment (2 module)
0.27 * Control work + 0.49 * Exam + 0.12 * Individual homework 1 + 0.12 * Individual homework 2

#### Recommended Core Bibliography

• Ильин В.А., Садовничий В.А., Сендов Б.Х. - МАТЕМАТИЧЕСКИЙ АНАЛИЗ Ч. 1 4-е изд., пер. и доп. Учебник для бакалавров - М.:Издательство Юрайт - 2016 - 660с. - ISBN: 978-5-9916-2733-7 - Текст электронный // ЭБС ЮРАЙТ - URL: https://urait.ru/book/matematicheskiy-analiz-ch-1-389342
• Кудрявцев Л. Д. - КУРС МАТЕМАТИЧЕСКОГО АНАЛИЗА В 3 Т. ТОМ 1 6-е изд., пер. и доп. Учебник для бакалавров - М.:Издательство Юрайт - 2019 - 703с. - ISBN: 978-5-9916-3701-5 - Текст электронный // ЭБС ЮРАЙТ - URL: https://urait.ru/book/kurs-matematicheskogo-analiza-v-3-t-tom-1-425369