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Магистратура 2017/2018

## Математический анализ

Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Статус: Курс адаптационный
Направление: 37.04.01. Психология
Когда читается: 1-й курс, 1, 2 модуль
Формат изучения: Full time
Прогр. обучения: Когнитивные науки и технологии: от нейрона к познанию
Язык: английский
Кредиты: 3

### Программа дисциплины

#### Аннотация

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 &quot;Cognitive Sciences and Technologies: From Neuron to Cognition&quot;. 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.

#### Цель освоения дисциплины

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

#### Результаты освоения дисциплины

• 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 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
• 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 expand periodic functions in Fourier series.
• Students should be able to solve linear differential equations and separable differential equations.
• Students should be able to evaluate different integrals using integration by parts or the change of variable.
• 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.

#### Содержание учебной дисциплины

• 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

#### Элементы контроля

• test (контрольная)
• test (контрольная)
• Written exam (письменный экзамен)

#### Промежуточная аттестация

• Промежуточная аттестация (2 модуль)
0.25 * test (контрольная) + 0.25 * test (контрольная) + 0.5 * Written exam (письменный экзамен)

#### Рекомендуемая основная литература

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