Calculus

Bachelor 2023/2024

Type: Compulsory course (International Programme in Economics and Finance)

Area of studies: Economics

Delivered by: International College of Economics and Finance

Where: International College of Economics and Finance

When: 1 year, 1-4 module

Mode of studies: offline

Open to: students of one campus

Instructors: Кирьянов Владислав Владимирович, Ковальцун Ксения Валерьевна, Anatoly Evgenievich Patrik, Andrey Vorchik, Pavel Zhukov, Мамаева Полина Игоревна

Language: English

ECTS credits: 8

Contact hours: 136

Full Syllabus Ask Question

Abstract

Calculus allows one to answers two basic questions: how fast something is changing (via calculating derivatives) and how much of something is accumulated (via calculating integrals). Calculus permits the computation of instantaneous rates of change (for example, velocities) by a limiting procedure called differentiation. The same procedure is also used to solve optimization problems: how to maximize profits or how to minimize losses. A second limiting procedure, called integration, permits the computation of areas, volumes, probabilities, or income accumulation. The Fundamental Theorem of Calculus is a kind of miracle that connects the two key procedures – differentiation and integration. Since both of them are limiting procedures, Calculus begins with a study of limits. This course will also solve equations involving derivatives – differential equations – for population growth or epidemic spread. Exponential, logarithmic, trigonometric, and inverse trigonometric functions will play an important role in all parts of the course. Calculus lays down the foundation for the block of quantitative disciplines that are studied at ICEF. It also develops analytical tools that are required in intermediate and advanced microeconomics courses. Calculus is studied and taught in English.

Learning Objectives

Levelling up the mathematical and analytical capabilities of students coming from different secondary schools and having different educational backgrounds.
Building a foundation for the block of quantitative disciplines studied at ICEF.
Developing analytical tools required for intermediate and advanced microeconomics courses.
Developing student's analytical skills, logical reasoning, critical thinking and attention to detail.

Expected Learning Outcomes

Students learn how to find the domain and range of a function, how to decide if the function is odd, even, or neither, how to determine if the function periodic or not.
Students learn how to calculate limits of simple sequences using the definition; how to calculate limits of rational sequences, sequences containing roots, and sequences containing logarithms using properties of limits and algebraic transformations. Also, students learn how to apply the Squeeze Theorem for calculating limits of more complicated sequences.
Students learn how to compute limits of functions at a point and at infinity; find slant asymptotes of functions (or conclude that a function does not have slant asymptotes).
Students will learn how to classify points of discontinuity: removable, Type I (also known as Jump discontinuity), Type II. Students will also learn how to apply main theorem about continuous functions: The Extreme Value Theorem and The Intermediate Value Theorem.
Students will learn how to calculate derivatives of elementary functions using the sum, product, quotient, and chain rules; how to determine if a function is differentiable or not; how to calculate derivatives of inverse and implicitly defined functions.
Students will learn to solve various optimization problems, where it is necessary to find maximum or minimum values of functions modelling real-life situations. For example, finding the shortest path, or the shape of a field having maximum area. Students will also learn to calculate rates of change of dependent (related) variables. For example, to calculate the rate of change of the volume of a ball when its radius is increasing according to a certain rule or formula.
Students will learn how to justify L'Hospital's rule using the Mean Value Theorem, and how to calculate tough limits using the L'Hospital's rule. Students will also learn how to investigate shapes of graphs and how to classify critical points (maximum or minimum) using the second derivative test.
Students will learn how to calculate higher derivatives of simple and convenient functions, and how to use the Leibniz formula for that purpose. Students will also learn how to apply Taylor's formula for important functions, and how to use Taylor's formula for calculating tough limits.
Students will learn about two main methods of finding anti-derivatives: integration by substitution and integration by parts. Students will also learn how to integrate any rational function, and any trigonometric-rational function.
Students will learn about the definition, the main properties, and the geometric meaning of definite integrals. Students will also learn about properties of Reimann sums and about the main theorem of this course - the Fundamental Theorem of Calculus.
Students will learn how to calculate improper integrals of the first kind (infinite integration interval) and of the second kind (unbounded integrand). Students will also learn how to calculate the principal value of improper integrals, and how to determine whether an improper integral converges or not.
Students will learn how to use the necessary condition, the ratio test, the limit comparison test, and the integral test for convergence of infinite series. They will also learn how to use Abel's and Dirichlet's tests for convergence of alternating series.
Student will learn about the definition, geometric interpretations, and properties of double integrals. They will learn how to reduce the double integrals to iterated integrals. Students will also learn how to write down and calculate double integrals in Polar coordinates.
Students will learn how to model real-life situations by differential equations, and how to solve separable, homogeneous, and first-order linear differential equations. Students will also learn how to use slope fields for analysing complicated differential equations.

Course Contents

Introduction. Elementary functions of a single variable.
Infinite sequences. Limit of a sequence. Special limits.
Limit of a function. Slant asymptotes. One-sided limits.
Types of discontinuities. Main theorems about continuous functions.
Her Majesty the Derivative: differentiable functions, derivatives of inverse and implicit functions.
Applications of derivative: maximum, minimum, rates of change, related rates.
The Mean Value Theorem. The second derivative test. Functions concave up and concave down. Inflection points.
Higher derivatives. Taylor’s formula. Limits of complicated functions.
Anti-derivatives and the indefinite integral.
His Majesty the Definite Integral.
Applications of the definite integral: areas of plane regions, volumes of solids.
Improper integral of the first and the second kind.
Convergence of infinite series.
Double and iterated integrals. Polar coordinates.
Modelling with differential equations. Solving differential equations.

Assessment Elements

Spring Interim Assessment
Spring interim assessment: 30 multiple choice questions, 80 minutes; 6 free response questions 100 minutes.
Summer Final Examination
Final (summer) examination: 30 multiple choice questions, 80 minutes; 6 free response questions 120 minutes.
Winter Examination
Winter Examination: 30 multiple choice questions, 80 minutes; 5 free response questions 90 minutes.
Autumn Interim Assessment
Autumn Interim Assessment: 30 multiple choice questions, 80 minutes; 4 free response questions 80 minutes.

Interim Assessment

2023/2024 2nd module
0.4 * Autumn Interim Assessment + 0.6 * Winter Examination
2023/2024 4th module
0.3 * Spring Interim Assessment + 0.7 * Summer Final Examination

Bibliography

Recommended Core Bibliography

Calculus early transcendentals, Stewart, J., 2012
Calculus, Stewart, J., 2012
Дифференциальное и интегральное исчисление в примерах и задачах. Функции одной переменной : учеб. пособие для вузов, Марон, И. А., 2008
Математический анализ. Ч. 1: ., Ильин, В. А., 2015
Основы математического анализа. Ч.1: ., Ильин, В. А., 2002

Recommended Additional Bibliography

Calculus, Anton, H., 2002

Authors

Patrik Anatolii Evgenevich

Course Syllabus