Master 2020/2021

Calculus

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

Area of studies: Psychology

Delivered by: School of Data Analysis and Artificial Intelligence

Where: Institute for Cognitive Neuroscience

When: 1 year, 1, 2 module

Mode of studies: offline

Instructors: Dmitry Frolenkov

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

Language: English

ECTS credits: 3

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

• The students should understand the definition of sequence and its limit. The students should be able to evaluate limit of a sequence.
• Students should be able to detect composition of functions and to find the inverse function.
• 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 evaluate derivatives of complicated functions
• 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
• The students should know basic properties of trigonometric and exponential functions. The students should know basic properties of logarithm
• The students should be able to prove basic properties of elementary functions using derivatives
• Stiudents should be able to evaluate differnt difficult limits using L'hopital rule
• 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 sketch graph of a function based on the analysis of this function via derivatives
• 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 evaluate different integrals using integration by parts or the change of variable.
• Students should be able to solve linear differential equations and separable differential equations.
• Students should be able to expand periodic functions in Fourier series.

Course Contents

• Sequences. Limit of sequence.
  The definition of sequence and its limit is discussed.
• Functions of one variable
  Definition of function, composition of functions, inverse function
• The Limit of a function. Continious functions
  Definition and basic properties of limit of a function. Definition and properties of continious functions.
• Elementary functions
  basic properties of powers, trigonometric functions, exponential and logarithm
• Introduction to Derivatives.
  The definition of derivative. Basic properties of derivatives.
• More Derivatives
  Derivative of composition of functions, derivative of inverse function, higher order derivatives
• Calculus of elementary functions
  Investigation elementary functions using derivatives
• Application of derivatives
  Analyzing the behaviour of a function using derivatives
• Graph sketching
  Sketching graphs of functions
• L'hopital rule
  Investigation of L'hopital rule for evaluating limits
• Taylor series
  Investigation of Taylor series of different functions . Evaluation limits using Taylor series
• Introduction to integrals
  Definition of indefinite and definite integrals. Area under the curve as a definite integral. Indefinite integrals of elementary functions.
• Evaluation of integrals
  Integration by parts and change of variable.
• Introduction to differentail equations
  Linear differential equations. Separable equations
• Introduction to Fourier analysis
  Fourier series expansion of periodic functions

Assessment Elements

• test
  The retake will consist of the similar tasks and will be evaluated in the same manner.
• test
  The retake will consist of the similar tasks and will be evaluated in the same manner.
• Written exam
  The first retake will consist of the similar tasks and will be evaluated in the same manner. The final mark of the course will also evaluate in the same manner. The second retake will consist of the similar tasks. In this case the final mark is equal to the mark obtained during the second retake.

Interim Assessment

• Interim assessment (2 module)
  0.1 * test + 0.3 * test + 0.6 * Written exam

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