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# Calculus

2020/2021
Учебный год
ENG
Обучение ведется на английском языке
5
Кредиты
Статус:
Курс обязательный
Когда читается:
1-й курс, 1 модуль

### Course Syllabus

#### Abstract

This course will cover calculus fundamentals which are essential for more advanced courses in data science. We will begin with a basic introduction to concepts related to functional mappings. Then we will study limits (in relation to sequences, single- and multivariate functions), differentiability (again, starting from a single variable and building up to multiple variables), and integration. This will become our foundation before we proceed to the introduction to basic optimization. At the end of the course, you will be expected to complete a final programming project showcasing the use of optimization routine in machine learning. This will allow you to test your practical skills and give you relevant hands-on experience in programming. You will also have access to additional materials, including interactive plots in GeoGebra environment used during lectures, bonus PDF files with more information on the general methods and further insights into the discussed topics, as well as optional programming tasks, which can be used to deepen your knowledge and get a taste of real-life cases. Course topics: ● Introduction: numerical sets, functions, limits ● Limits and multivariate functions ● Derivatives and linear approximations: single variate functions ● Derivatives and linear approximations: multivariate functions ● Integrals: anti-derivative, area under curve, multivariate functions ● Optimization

#### Learning Objectives

• 1. Students will develop an understanding of fundamental concepts of the single and multi variable calculus and form a range of skills that help them work efficiently with these concepts. 2. Students will gain knowledge of the derivatives of single-variable functions, their integral, and the derivatives of multi-variable functions. 3. The course will give students an understanding of simple optimization problems.

#### Expected Learning Outcomes

• Students should be able to understand and apply basic concepts of the theory of limits, continuous and differentiable single-variable functions, antiderivatives and integrals of single-variable functions, continuous and differentiable several-variable functions.
• Students should be able to analyze functions represented in a variety of ways: graphical, numerical, analytical, or verbal, and understand the relationships between these various representations.
• Students should be able to determine the convergence of improper integrals.
• Students should be able to compute limits of sequences and functions.
• Students should be able to compute derivatives and antiderivatives.
• Students should be able to estimate the asymptotical behavior of functions.
• Students should be able to apply the computation of the integrals to the determination of the length of parametric curve arcs, the area of domains, and the volume of solid revolutions.
• Students should be able to describe the space of several variables, convergence in the space, and properties of the distance.
• Students should be able to find the extrema of single- and several-variable functions.
• Students should be able to formulate and solve simple optimization problems.
• Students should be able to understand basic principles of numerical algorithms that solve algebraic equations and compute derivatives and integrals.

#### Course Contents

• Sequences. Limit of a sequence: numbers, bounded sets, limits, operations with limits, monotone sequences, number e, Bolzano-Weierstrass theorem, completeness of real numbers.
• Continuous functions: limit of a function, the definition of a continuous function, operations with continuous functions, monotonicity, inverse function, properties of continuous functions (basic theorems), types of discontinuity, uniform continuity
• Differentiable functions: definition of the derivatives, properties of differentiable functions, inverse functions, big and little o-notation, the mean value theorem, the second mean value theorem, higher derivatives, l’Hospital’s rule, Taylor’s theorem, numerical solution of algebraic equations
• Integration: indefinite integral, antiderivative, properties of the integral, methods of integration, the Riemann integral, the fundamental theorem of calculus, mean value theorems, improper integrals, numerical computations of integrals.
• Space of several variables and continuous functions on it: n-dimensional space R^n, open and closed sets, limit points, a convergence of point sequences, continuous functions in R^n, and their properties.
• Differentiation of functions of several variables: partial derivatives, differentials, the chain rule, the mean value theorem and Taylor’s theorem, optimization, sufficient conditions of extrema, constrained optimization, implicit function theorem, inverse mapping, and Jacobians.

#### Assessment Elements

• Weekly Test
• Open Questions
• Projects

#### Interim Assessment

• Interim assessment (1 module)
0.06 * Open Questions + 0.38 * Projects + 0.56 * Weekly Test

#### Recommended Core Bibliography

• Calculus early transcendentals, Stewart, J., 2012