Delivered at:: Big Data and Information Retrieval School

Course type:: Compulsory course

When:: 2 year, 1, 2 module

Instructors

Abstract

This course covers specific topics of advanced calculus, such as numeric and functional series, infinite products, Eulerian integrals, multiple integrals. The convergence and functional properties of power series are considered along with their applications to some problems of discrete mathematics involving the generating functions. Prerequisites: High school algebra and trigonometry, basic concepts of calculus (e. g., sequences, limits and continuity, derivatives, integrals).

Learning Objectives

Students will understand the concept of сonvergence and divergence of infinite series and infinite products; the rate of convergence.
Students will understand the concept of pointwise and uniform convergence of the functional series; the functional properties of their sums.
Students will understand the concept of representing functions by power series; Taylor series of the most common elementary functions.
Students will understand the concept of generating functions and their applications for solving linear recurrence relations.
Students will understand the concept of representing functions by trigonometric Fourier series.
Students will understand the concept of integrals depending on a parameter; beta and gamma functions.
Students will understand the concept of double and triple integrals; reduction to iterated integrals (Fubini’s theorem).
Students will understand the concept of the change of variables in multiple integrals; polar, cylindrical and spherical coordinate systems.
Studentswill understand the concept of improper integrals.

Expected Learning Outcomes

Students should be able to calculate sums of series using the methods of partial sums, power series, Fourier series.
Students should be able to compute double and triple integrals by means of Fubini’s theorem, using a change of variables if necessary.
Students should be able to determine the radius and the domain of convergence of power series.
Students should be able to apply the properties of the Eulerian integrals for the calculation of specific integrals.
Students should be able to calculate areas of regions, volumes of solids and surface areas.
Students should be able to calculate sums of series using the methods of partial sums, power series, Fourier series.
Students should be able to compute double and triple integrals by means of Fubini’s theorem, using a change of variables if necessary.
Students should be able to determine the radius and the domain of convergence of power series.

Course Contents

Infinite series.
Series of Nonnegative Terms. Convergence Tests.
Alternating series. Absolute and conditional convergence.
Uniform convergence.
Power series.
Generating functions.
Fourier series.
Integrals depending on a parameter.
Rearrangement theorems.
Multiple integral.
Improper integrals.
Beta and gamma functions.

Assessment Elements

Quizzes
A quiz consists of a selection of problems similar to those from the seminar/homework exercise lists. Students solve the problems in written form during ~30 min and pass the solutions to the teacher.
Written test (=midterm)
At the beginning of the second module the students pass a written test. The test consists of a selection of problems similar to those from the seminar/homework exercise lists. Students solve the problems in written form during 2 academic hours and pass the solutions to the teacher.
Written exam
At the end of the second module the students pass a written exam. The exam consists of a selection of problems similar to those from the seminar/homework exercise lists. Students solve the problems in written form during 2 or 3 academic hours and pass the solutions to the teacher.
Homeworks
Oral Test
At the end of the second module the students pass an oral test. The oral test is a list of theoretical questions (such as: formulate a statement of a theorem or provide a proof of a theorem). Students have ~30 minutes to prepare, after which they discuss their answer with a teacher.

Interim Assessment

2023/2024 2nd module
F = Round(7/10*C + 3/10*E) Where: C = 4/17*H + 4/17*Q + 4/17*W + 5/17*O is the cumulative course grade H is the average mark of all the homework assignments Q is is the average mark of all the quizzes W is the in-class written test (=midterm) mark O is the in-class oral test mark E is the exam mark All grades (namely, F, C, H, Q, W, O, and E) are real numbers from 0 to 10. The function Round(x) is defined as follows: if the decimal part of x is less than 0.2, the grade is rounded downwards; if the decimal part of x is greater than 0.7, the grade is rounded upwards; if the decimal part of x is from the interval [0.2;0.7] and the student's seminar attendance during the first semester is not below 66%, the grade is rounded upwards; otherwise the grade is rounded downwards. Only value 7/10*C + 3/10*E is rounded off.

Bibliography

Recommended Core Bibliography

Calculus early transcendentals, Stewart, J., 2012
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

Recommended Additional Bibliography

Ronald L. Graham, Donald E. Knuth, & Oren Patashnik. (1994). Concrete Mathematics : A Foundation for Computer Science. [N.p.]: Addison-Wesley Professional. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1601594

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