Master
2023/2024

# Calculus

Category 'Best Course for New Knowledge and Skills'

Type:
Elective course (Cognitive Sciences and Technologies: From Neuron to Cognition)

Area of studies:
Psychology

Delivered by:
School of Data Analysis and Artificial Intelligence

When:
1 year, 1, 2 module

Mode of studies:
offline

Open to:
students of one campus

Instructors:
Dmitry Frolenkov

Master’s programme:
Cognitive Sciences and Technologies: From Neuron to Cognition

Language:
English

ECTS credits:
6

Contact hours:
40

### Course Syllabus

#### Abstract

The course covers basic definitions and methods of calculus. This course, together with other mathematical courses, provides sufficient condition for students to be ready to participate in quantitative and computational modeling at the Master's Program "Cognitive Sciences and Technologies: From Neuron to Cognition". Students study the theory and applications of continuous functions their derivatives and integrals; solve optimization and approximation problems; study complex numbers and Fourier series as well as some basic material of the theory of differential equations.

#### Learning Objectives

- know basic skills to start quantitative and computational modeling
- know basic principles of calculus

#### Expected Learning Outcomes

- Stiudents should be able to evaluate differnt difficult limits using L'hopital rule
- Students should be able to detect composition of functions and to find the inverse function.
- Students should be able to evaluate derivatives of complicated functions
- Students should be able to evaluate different integrals using integration by parts or the change of variable.
- Students should be able to evaluate limit of a function and to check if a function is continious or not
- Students should be able to expand a function in Taylor series. Students should be able to evaluate limits using Taylor series
- Students should be able to expand periodic functions in Fourier series.
- Students should be able to find critical points of a function, to find segment where it is increasing (decreasing), to find maximum (minimum) of a function
- Students should be able to rewrite the area under the curve as a definite integral. Students should be able to evaluate indefinite integrals of elementary functions.
- Students should be able to sketch graph of a function based on the analysis of this function via derivatives
- Students should be able to solve linear differential equations and separable differential equations.
- The students should be able to prove basic properties of elementary functions using derivatives
- The students should know basic properties of trigonometric and exponential functions. The students should know basic properties of logarithm
- The students should understand the definition of sequence and its limit. The students should be able to evaluate limit of a sequence.

#### Course Contents

- The Limit of a function. Continious functions
- Elementary functions
- Introduction to Derivatives.
- More Derivatives
- Calculus of elementary functions
- Application of derivatives
- Graph sketching
- L'hopital rule
- Taylor series
- Introduction to integrals
- Evaluation of integrals
- Introduction to differentail equations
- Introduction to Fourier analysis

#### Bibliography

#### Recommended Core Bibliography

- Calculus : concepts and methods, Binmore, K., 2001
- Calculus early transcendentals, Stewart, J., 2012

#### Recommended Additional Bibliography

- Anton, H., Bivens, I. C., & Davis, S. (2016). Calculus (Vol. 11th ed). New York: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1639210
- Friedman, A. (2007). Advanced Calculus (Vol. Dover edition). Mineola, N.Y.: Dover Publications. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1153250