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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

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

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

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

Assessment Elements

  • non-blocking 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.
  • non-blocking 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.
  • non-blocking Control work
    Upon the decision of the seminar professor, control work may consist of auditorian work and extracurricular activities, according to the agreement of seminar professors and lecturer control work is carried out for the whole student flow on the platform of Smart LMS. The control work is conducted remotely in writing. The control work must be completed in 80 minutes and students have 10 minutes for loading the tasks. Work is conducted on the SmartLMS platform with additional use of Zoom or MS Teams. The control work session must be started 15 minutes before the start, upon the teacher’s signal to start solving tasks in SmartLMS. The student’s computer must meet the following requirements: presence of a working camera and microphone, high-speed Internet, support Zoom/MS Teams. The answers to the task must be written on white paper sheets A4, with a black pen, the sheets must be numbered, and when it is specified in the task, the answers must be additionally entered in the answer window. After finishing the control work,students must take a photo/scan his decision and upload it to SmartLMS. The photos should be vertical so that the text is not blurred and read unambiguously. The answers and the task numbers should be selected. The camera and microphone must be switched on during the whole work. The camera must be positioned side-by-side or frontally from itself in such a way that it is aimed at the working field - the sheet on which the work is performed - on the student and the space around the student (the room must be well lit). It is permitted to use the login to Zoom / MS Teams from a mobile phone with its camera. At the professor’s request, the student is obliged to switch to the transmission of his screen: turn on the back camera of the mobile phone or turn the phone to the computer screen within 5 seconds, or start a display of the screen. You cannot leave the room during inspection work. On the table you can have only writing accessories, no pencils, blank sheets of paper and water. The presence of any media around the workplace of the student, as well as other people, is considered as a violation and ends with the removal of the student from work and with a score of «0». Students are not allowed to turn off the camera and microphone during the test: until the end of the control work, the video and sound must remain active, including time to scan the completed work and send it for inspection. A short-term interruption of communication during control work is considered to be less than 5 minutes and not more than once. Long-term disruption of communication during control work is considered to be a 5-minute or more disruption. In case of a long-term disruption of communication, the student may continue to participate in the writing of the work at the professor’s discretion.
  • non-blocking Exam
    The exam is conducted remotely on the platform of Smart LMS in writing. The examk must be completed in 90 minutes and students have 15 minutes for loading the tasks. The exam is conducted on the SmartLMS platform with additional use of Zoom or MS Teams. The exam session must be started 15 minutes before the start, upon the teacher’s signal to start solving tasks in SmartLMS. The student’s computer must meet the following requirements: presence of a working camera and microphone, high-speed Internet, support Zoom/MS Teams. The answers to the task are written on white paper sheets A4, with a black pen, the sheets must be numbered, and when it is specified in the task, the answers must be additionally entered in the answer window. After finishing the exam,students must take a photo/scan his decision and upload it to SmartLMS. The photos should be vertical so that the text is not blurred and read unambiguously. The answers and the task numbers should be selected. The camera and microphone shall be switched on during the whole exam. The camera must be positioned side-by-side or frontally from itself in such a way that it is aimed at the working field - the sheet on which the work is performed - on the student and the space around the student (the room must be well lit). It is permitted to use the login to Zoom / MS Teams from a mobile phone with its camera. At the professor’s request, the student is obliged to switch to the transmission of his screen: turn on the back camera of the mobile phone or turn the phone to the computer screen within 5 seconds, or start a display of the screen. Students cannot leave the room during the inspection exam. On the table students can have only writing accessories, no pencils, blank sheets of paper and water. The presence of any media around the workplace of the student, as well as other people, is considered as a violation and ends with the removal of the student from the exam and with a score of «0». Students are not allowed to turn off the camera and microphone during the exam: until the end of the exam, the video and sound must remain active, including time to scan the completed work and send it for inspection. A short-term interruption of communication during control work is considered to be less than 5 minutes and not more than once. Long-term disruption of communication during exam is considered to be a 5-minute or more disruption. In case of a long-term disruption of communication, the student may continue to participate in the writing of the exam at the professor’s discretion.
Interim Assessment

Interim Assessment

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

Bibliography

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

Recommended Additional Bibliography

  • Путко Б.А., Тришин И.М., Кремер Н.Ш. - под ред. - МАТЕМАТИЧЕСКИЙ АНАЛИЗ В 2 Т. Учебник и практикум для академического бакалавриата - М.:Издательство Юрайт - 2016 - 634с. - ISBN: 978-5-9916-6238-3 - Текст электронный // ЭБС ЮРАЙТ - URL: https://urait.ru/book/matematicheskiy-analiz-v-2-t-388079