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# Calculus

2019/2020
Учебный год
ENG
Обучение ведется на английском языке
9
Кредиты
Статус:
Курс обязательный
Когда читается:
1-й курс, 1-4 модуль

#### Преподаватели

Жуков Павел Владимирович

Сорокина Антонина Михайловна

### Course Syllabus

#### Abstract

Students are expected to have a firm grounding in elementary mathematics, algebra, trigonometry, and geometry on the coordinate plane, the properties and graphs of elementary functions at the level of Russian high school. This course is designed to introduce students to the basic ideas and methods of mathematical analysis and their application to mathematical modeling. Four key concepts of the course, in order of appearance, are Limits, Derivatives, Series and Integrals. For each of the concepts the theoretical foundations are introduced and discussed, but the main focus is on the computational techniques and the applications. The course helps lay the foundation for the entire block of quantitative disciplines that are studied at ICEF, and it also provides some of the analytical tools that are required by advanced courses in economics. The course is taught in English.

#### Learning Objectives

• to demonstrate how to analyze functions of one variable represented in a variety of ways: graphical, numerical, analytical, or verbal, and understand the relationships between these various representations
• to introduce the concepts of the limit of an infinite sequence, the limit of a function at a point and the limit of a function as its argument approaches infinity
• to explain the meaning of the derivative in terms of a rate of change and local linear approximation, and show how to use derivatives to solve a variety of problems
• to introduce the concept of infinite series and the idea of approximating a function by its Taylor series
• to ensure the students grasp meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change
• to outline the relationship between the derivative and the definite integral, as expressed by the Fundamental Theorem of Calculus;
• to explain how the concept of definite integral extends to double and triple integrals, and make sure students are able to compute multiple integrals by reducing them to iterated integrals
• to teach how to communicate mathematics in well-written sentences and to explain the solutions to problems
• to show how to model a written description of a simple economic or physical situation with a function, differential equation, or an integral
• to use mathematical analysis to solve problems, interpret results, and verify conclusions
• to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement

#### Expected Learning Outcomes

• be able to analyze functions of one variable represented in a variety of ways: graphical, numerical, analytical, or verbal, and understand the relationships between these various representations;
• be able to compute the limits of infinite sequences; be able to apply arithmetic limit laws, theorems on limit inequalities, The Squeeze Theorem.
• be able to compute the limits of functions at a point and at infinity, be able to apply arithmetic limit laws, theorems on limit inequalities, The Squeeze Theorem; be able to find linear asymptotes of functions;
• be able to demonstrate continuity or discontinuity of functions; be able to apply Intermediate Value Theorem
• apply the meaning of the derivative in terms of a rate of change and local linear approximation, be able to apply theorems about the properties of differentiable functions.
• be able to use derivatives to solve a variety of problems involving the rates of change; be able to solve optimization problems; be able to analyze the graph of a function with the help of its 1st and 2nd derivatives
• use Taylor polynomials to approximate function values
• be able to apply integration by parts and integration via substitution to compute anti-derivatives
• apply the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change, understand the relationship between the derivative and the definite integral, as expressed by the Fundamental Theorem of Calculus
• be able to use integrals to solve a variety of problems, including problems on finding areas of plane regions and volumes of solids, problems on computing accumulated quantities or finding quantities from their rates of change;
• be able to compute improper integrals
• be able to compute multiple integrals by reducing them to iterated integrals
• be able to apply integration techniques to solve separable 1st order ODE; be able to analyze slope fields for1st order ODE
• be able to communicate mathematics in well-written sentences and to explain the solutions to problems
• be able to model a written description of a simple economic or physical situation with a function, differential equation, or an integral
• be able to use mathematical analysis to solve problems, interpret results, and verify conclusions
• be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement

#### Course Contents

• Introduction. Functions of one variable
The application of mathematics to describing phenomena. The role of mathematics and mathematical modeling in economics. Different forms of representation of functions. Elementary concepts: domain and range of a function, even and odd functions, periodic functions. Graphs of elementary functions. Shifts and distortions of graphs. Implicit functions. Examples of functions in economics: utility function, production function, cost function, demand and supply functions.
• Sequences. Limit of a sequence.
Sequences: bounded and unbounded, infinitely small and infinitely large. Limit of a sequence. Limit theorems for sequences: arithmetic operations, sandwich theorem. Monotone sequences. Convergence of a monotone increasing sequence. The number e.
• Limit of a function
The limit of a function at infinity. Asymptotes of a function at infinity. The limit of a function at a point. Limit theorems for functions. Functions that tend to zero, functions that tend to infinity. First and Second Special Limits. Types of indeterminate forms. Finding limits. Left and right limits.
• Continuity.
Definition of continuity of a function at point and on an interval. Continuity of elementary functions. Properties of continuous functions. Points of discontinuity. Classification of points of discontinuity. Vertical asymptotes.
• The derivative
Definition of the derivative. Tangent lines and normal lines. Geometric, physical and economic interpretations of the derivative. Right and left derivatives. Differentiability at a point. Differentiability and continuity. Differentiation. Rules of differentiation. Derivatives of elementary functions. Differentiation of inverse functions. Logarithmic differentiation. Differentiation of implicit functions. Existence of a differentiable implicit function. Definition and geometric interpretation of differentials. Approximate calculations using differentials. The second derivative. The economic meaning of the second derivative. Higher-order derivatives and differentials. Properties of differentiable functions: Rolle's theorem, the Mean Value theorem, Cauchy’s theorem, and their geometric interpretation.
• Applications of the derivative
L’Hospital’s rule. Necessary and sufficient conditions for increasing/decreasing functions. Related rates. Concave and convex functions. Different ways of expressing concavity. Economic interpretation of concave and convex functions. Points of inflection. Local extrema. First-order necessary and sufficient conditions for a local extremum. Second-order necessary and sufficient conditions for a local extremum. Maximum and minimum values of a function on an interval. Geometric and economic applications of optimization. Curve sketching.
• Infinite series, power series, Taylor series
Necessary condition for convergence of a series. Harmonic series and power series. The ratio test. Comparing series to test for convergence. Alternating series. Sufficient condition for convergence of an alternating series. Absolute convergence. Radius and interval of convergence of a power series. Abel’s theorems. Taylor’s formula. Taylor and Maclaurin series. Taylor and Maclaurin expansions for elementary functions. Application of Taylor series for analyzing the behavior of a function at a point and for conducting approximate calculations.
• Anti-derivatives and the indefinite integral
Anti-derivatives. The indefinite integral and its properties. Table of indefinite integrals. Basic methods of integration: direct integration, substitution and integration by parts. Integration of rational functions.
• The definite integral
Problems that require the definite integral. Definition of the definite integral using Riemann sums. Sufficient condition for the existence of the definite integral. Approximate calculation of definite integrals using rectangles and trapezoids. Simpson’s rule. Properties of the definite integral. Differentiation of a definite integral with variable upper bound. The fundamental theorem of calculus. Substitution and integration by parts.
• Applications of the definite integral
Applications of the definite integral in geometry, economics and physics. Area of a flat region, volume of a solid of revolution, volume of a solid with known cross-sections. Use of definite integrals to solve separable differential equations.
• Improper integrals
Integrals with infinite bounds. Improper integrals of the first kind. Integration of unbounded functions. Improper integrals of the second kind. Principle value. Convergence tests for improper integrals. Absolute and relative convergence of improper integrals.
• Double integrals, triple integrals
Definition of double and triple integrals. Reduction of double integrals to iterated integrals. Changing the order of integration in iterated integrals. The geometric interpretation and main properties of double integrals.
• Introductions to differential equations
Definition of first order differential equations. General and particular solutions. Existence and uniqueness theorem. Isoclines and direction fields. Solution of separable differential equations. Solution of homogeneous differential equations and first-order linear equations. Application of differential equations to physics and economics.
• Course review

#### Assessment Elements

• Fall midterm exam
• Winter exam
• homework and in-class activities
• Final exam
Экзамен проводится в письменной форме с использованием асинхронного прокторинга. Экзамен проводится на платформе https://hse.student.examus.net). К экзамену необходимо подключиться за 10 минут до начала. Проверку настроек компьютера необходимо провести заранее, чтобы в случае возникших проблем у вас было время для обращения в службу техподдержки и устранения неполадок. Компьютер студента должен удовлетворять требованиям: 1. Стационарный компьютер или ноутбук (мобильные устройства не поддерживаются); 2. Операционная система Windows (версии 7, 8, 8.1, 10) или Mac OS X Yosemite 10.10 и выше; 3. Интернет-браузер Google Chrome последней на момент сдачи экзамена версии (для проверки и обновления версии браузера используйте ссылку chrome://help/); 4. Наличие исправной и включенной веб-камеры (включая встроенные в ноутбуки); 5. Наличие исправного и включенного микрофона (включая встроенные в ноутбуки); 6. Наличие постоянного интернет-соединения со скоростью передачи данных от пользователя не ниже 1 Мбит/сек; 7. Ваш компьютер должен успешно проходить проверку. Проверка доступна только после авторизации. Для доступа к экзамену требуется документ удостоверяющий личность. Его в развернутом виде необходимо будет сфотографировать на камеру после входа на платформу «Экзамус». Также вы должны медленно и плавно продемонстрировать на камеру рабочее место и помещение, в котором Вы пишете экзамен, а также чистые листы для написания экзамена (с двух сторон). Это необходимо для получения чёткого изображения. Во время экзамена запрещается пользоваться любыми материалами (в бумажном / электронном виде), использовать телефон или любые другие устройства (любые функции), открывать на экране посторонние вкладки. В случае выявления факта неприемлемого поведения на экзамене (например, списывание) результат экзамена будет аннулирован, а к студенту будут применены предусмотренные нормативными документами меры дисциплинарного характера вплоть до исключения из НИУ ВШЭ. Если возникают ситуации, когда студент внезапно отключается по любым причинам (камера отключилась, компьютер выключился и др.) или отходит от своего рабочего места на какое-то время, или студент показал неожиданно высокий результат, или будут обнаружены подозрительные действия во время экзамена, будет просмотрена видеозапись выполнения экзамена этим студентом и при необходимости студент будет приглашен на онлайн-собеседование с преподавателем. Об этом студент будет проинформирован заранее в индивидуальном порядке. Во время выполнения задания, не завершайте Интернет-соединения и не отключайте камеры и микрофона. Во время экзамена ведется аудио- и видео-запись. Процедура пересдачи проводится в соотвествии с нормативными документами НИУ ВШЭ.
• Spring midterm exam

#### Interim Assessment

• Interim assessment (2 module)
0.43 * Fall midterm exam + 0.57 * Winter exam
• Interim assessment (4 module)
0.55 * Final exam + 0.45 * Interim assessment (2 module)

#### Recommended Core Bibliography

• Mathematics for economists, Simon C. P., Blume L., 1994