2020/2021

Спецкурс по математическому анализу

Статус: Дисциплина общефакультетского пула

Кто читает: Международный институт экономики и финансов

Где читается: Международный институт экономики и финансов

Когда читается: 1-4 модуль

Преподаватели: Патрик Анатолий Евгеньевич

Язык: английский

Кредиты: 6

Контактные часы: 64

Course Syllabus

Abstract

It is expected that students attend classes and lectures of the main ICEF 1st year Calculus subject, and that they successfully completed “Algebra and Introduction to Calculus” subject at the level of secondary school programme. · Module I: rational and irrational numbers, the method of mathematical induction, limits of sequences, properties of continuous functions, differentiable functions, derivatives. · Module II: higher derivatives, the Taylor formula, convergence of infinite series, anti-derivatives and indefinite integrals. · Module III: functions of two real variables, extreme values, constraint maximization, absolute maxima and minima, classical inequalities, double and triple integrals, calculation of volumes and surface areas. · Module IV: topics from AP Calculus BC: differential and functional equations, Taylor series.

Learning Objectives

• enable students to to solve advanced problems from selected chapters of the main Calculus course
• show how to deal with double and triple integrals and use them for calculating volumes, surface areas and for solving various problems from probability theory

Expected Learning Outcomes

• Calculate limits of sequences and solve advanced related problems
• Apply the main theorems (IVT, EVT)
• Apply mean value theorem
• Apply Taylor formula, estimate approximation error
• Investigate convergence for infinite series
• Calculate anti-derivatives, integrate rational functions
• Use differentiation, extreme values, Lagrange method, Polar coordinates
• Write down iterated integrals, change variables
• Calculate volumes, masses, and areas of surfaces
• Solve Bernoulli equations, second-order equations with constant coefficients
• Solve functional equations

Course Contents

• Prologue
  Rational and irrational numbers. Properties of the Dirichlet function. The method of mathematical induction.
• Sequences. Limit of a sequence.
  The definition of a divergent sequence. The sandwich theorem. Sequences defined recursively. The Stolz lemma. Fundamental sequences and the Cauchy criterion. Advanced questions on convergence of sequences. Sequences of functions. Uniform convergence.
• Properties of continuous functions
  Limit of a function. The extreme value theorem. The definition of a discontinuous function.
• Differentiable functions.
  The chain rule. Derivatives of inverse functions.
• Higher derivatives. Taylor formula.
  Calculating derivatives of n-th order. The Taylor formulae for elementary functions. Irrationality of the number e. Application of the Taylor formula for calculating limits.
• Infinite series.
  Telescoping series. The necessary condition for convergence. The Cauchy criterion. Convergence of the generalized harmonic series. Dirichlet’s test.
• Anti-derivatives.
  Integration by parts. The method of undetermined coefficients. Decimation formulae. Integration of rational and trigonometric functions. Euler’s substitutions.
• Functions of two and several variables.
  Polar, Cylinder and Spherical coordinates. The chain rule. Stationary points. The second derivative test. Lagrange method.
• Definite integrals
  Riemann sums. Non-integrable functions. Properties of definite integrals.
• Double integrals.
  Reduction of double integrals to iterated integrals. Changing the integration order in iterated integrals. Changing variables in double integrals.
• Applications of double integrals
  Volumes and Surface Areas. Double integrals and probability.
• Triple integrals
  Reduction of triple integrals to iterated integrals. Using cylinder and spherical coordinate for calculating triple integrals
• Differentials Equations
  Separable differential equations. Bernoulli differential equations. Solution of differential equations in terms of power series.
• Calculus BC APT questions
  Infinite series. Functional equations. Taylor series.

Assessment Elements

• midterm exam 1
• weakly quizzes
• April exam
• midterm exam 2
• midterm exam 3
• midterm exam 4

Interim Assessment

• Interim assessment (4 module)
  0.5 * April exam + 0.1 * midterm exam 1 + 0.1 * midterm exam 2 + 0.1 * midterm exam 3 + 0.1 * midterm exam 4 + 0.1 * weakly quizzes

Bibliography

Recommended Core Bibliography

• Spivak, M. (1998). Calculus On Manifolds : A Modern Approach To Classical Theorems Of Advanced Calculus. New York: CRC Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=421137

Recommended Additional Bibliography

• Binmore, K. G. (1982). Mathematical Analysis : A Straightforward Approach (Vol. Second edition). Cambridge [Cambridgeshire]: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=510997
• Дифференциальное и интегральное исчисление в примерах и задачах. Функции одной переменной : учеб. пособие для вузов, Марон, И. А., 2008