Bachelor
2021/2022

# Calculus

Type:
Compulsory course

Area of studies:
Economics

Delivered by:
International College of Economics and Finance

When:
1 year, 1-4 module

Mode of studies:
offline

Open to:
students of one campus

Language:
English

ECTS credits:
9

### Course Syllabus

#### Abstract

This course is designed to introduce students to the basic ideas and methods of mathematical analysis and their application to mathematical modeling. Four key concepts of the course, in order of appearance, are Limits, Derivatives, Series and Integrals. For each of the concepts the theoretical foundations are introduced and discussed, but the main focus is on the computational techniques and the applications. The course helps lay the foundation for the entire block of quantitative disciplines that are studied at ICEF, and it also provides some of the analytical tools that are required by advanced courses in economics. The course is taught in English.

#### Learning Objectives

- Levelling up the mathematical background of students coming from different high schools.
- Establishing a solid mathematical foundation for the entire block of quantitative disciplines that are studied at ICEF.
- Providing analytical tools required by ICEF courses in economics.
- Developing the analytical mindset and reasoning skillset required for the student’s course works and diploma.

#### Expected Learning Outcomes

- be able to analyze functions of one variable represented in a variety of ways: graphical, numerical, analytical, or verbal, and understand the relationships between these various representations
- Be able to apply integrals to solve a variety of problems: arc lengths, areas of plane regions, volumes and surface area of 3D solids, motion of a particle, work of a force, etc
- Be able to use derivatives to solve a variety of problems: curve sketching, optimization, related rates, implicit and inverse functions, parametric curves, motion of a particle, etc.
- Understand how the concept of definite integral extends to double and triple integrals. Be able to compute multiple integrals by reducing them to iterated integrals.
- Understand how the concept of the definite integral extends to the cases of unbounded intervals and unbounded functions. Be able to properly compute the improper integrals via limits.
- Understand the concept of an anti-derivative. Be able to compute indefinite integral using integration techniques: arithmetic rules, integration by parts, various substitutions. Be able to integrate rational functions via partial fractions decomposition.
- Understand the concept of an ODE and their classification by order. Understand the concept of the general solution, the initial value problem and the particular solution, the slope field for the 1-st order ODE. Understand the idea of modelling life processes by 1-st order ODE. Be able to make conclusions about the process based on a model. Be able to solve separable 1-st order ODE by separation of variables.
- Understand the concept of infinite series and the idea of approximating a function by its Taylor series. Be able to compute Taylor polynomials for a function near a given point. Be able to use Taylor polynomials to approximate function values. Understand the concept of the approximation error and be able to analyze it.
- Understand the concept of the continuity. Be able to analyze continuity of functions. Understand the properties of continuous functions and be able to apply them.
- understand the concepts of the limit of an infinite sequence, the limit of a function at a point and the limit of a function as its argument approaches infinity, the main properties of these limits. Be able to use the formal definition of the limit and to compute the limits of infinite sequences.
- Understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change. Be able to compute Riemann sums for functions on a given interval and use them to approximate the integral of the function. Understand the relationship between the derivative and the definite integral, as expressed by the Fundamental Theorem of Calculus.
- Understand the meaning of the derivative in terms of a rate of change and local linear approximation. Understand the geometrical meaning of the derivative. Be able to compute derivatives using the formal definition and via the arithmetic rules and the table of the derivatives of the elementary functions. Understand the concept of differentiable functions. Demonstrate the active knowledge of their properties.

#### Assessment Elements

- Written Mock examination - Fall
- Written examination - WinterOnline format.
- Written Mock examination - Spring
- Written examination - Summer (ICEF International Exam)

#### Interim Assessment

- 2021/2022 1st module
- 2021/2022 2nd module0.425 * Written Mock examination - Fall + 0.575 * Written examination - Winter
- 2021/2022 3rd module
- 2021/2022 4th module0.4 * 2021/2022 4th module + 0.19 * Written Mock examination - Spring + 0.41 * Written examination - Summer (ICEF International Exam)

#### Bibliography

#### Recommended Core Bibliography

- Calculus early transcendentals, Stewart, J., 2012
- Introduction to mathematical economics, Dowling, E. T., 2012

#### Recommended Additional Bibliography

- Fundamental methods of mathematical economics, Chiang, A. C., 2005
- Mathematics for economics and finance : methods and modelling, Anthony, M., 2012
- Mathematics for economists, Simon, C. P., 1994
- Высшая математика для экономистов : учебник для вузов, Кремер, Н. Ш., 2004
- Курс дифференциального и интегрального исчисления. Т.1: ., Фихтенгольц, Г. М., 2001
- Курс дифференциального и интегрального исчисления. Т.2: ., Фихтенгольц, Г. М., 2001
- Математический анализ. Т. 1: ., Зорич, В. А., 2015
- Математический анализ. Т. 2: ., Зорич, В. А., 2015
- Сборник задач и упражнений по математическому анализу : учеб. пособие для вузов, Демидович, Б. П., 2003