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Regular version of the site
Bachelor 2020/2021

## Calculus

When: 1 year, 3, 4 module
Mode of studies: offline
Open to: students of one campus
Language: English
ECTS credits: 6

### Course Syllabus

#### Abstract

This course is designed to introduce students to the basic ideas and methods of mathematical analysis and their application in business management. This course serves as a basis for the entire block of quantitative disciplines studied at HSE, and it also provides some analytical tools required by advanced courses in information technologies. The course provides students with experience in the methods and applications of calculus to theoretical and practical problems. The course is taught in English. #### Learning Objectives

• providing students basic knowledge in calculus and ordinary differential equations;
• familiarizing with the applied problems of calculus;
• developing skills to solve typical problems of calculus. #### Expected Learning Outcomes

• analyze functions represented in a variety of ways: graphical, numerical, analytical, or verbal, and understand the relationships between these various representations.
• understand the meaning of the derivative in terms of a rate of change and marginal analysis, and use derivatives to solve various problems.
• understand the meaning of the definite integral as the net accumulation of change, and use integrals to solve various problems.
• communicate mathematics in well-written sentences and to explain the solutions to problems.
• model a written description of a simple situation with a function, differential equation, or an integral.
• use mathematical analysis to solve problems, interpret results, and verify conclusions.
• understand and explain the meaning of solutions, including sign, size, relative accuracy, and units of measurement. #### Course Contents

• Introduction
Functions used in economics: the demand, the supply, the revenue, the cost and the profit functions. Different forms of representation of functions. Elementary concepts: domain and range of a function, even and odd functions, periodic functions. Graphs of elementary functions. Continuous compounding. Composition of functions.
• Limits and continuity
Cost management: capacity of a factory. The limit of a function. Properties of limits. Computation of limits. Infinite limits. Long-term behaviour. Continuity. Continuity on an interval. The intermediate value property.
• The derivative
Marginal analysis (output of a business). The derivative. Techniques of differentiation. Approximations using increments. Product and quotient Rules. Higher-order derivatives. The chain rule. Implicit differentiation and related rates.
• Applications of the derivative
Elasticity of demand. Increasing and decreasing functions. Local extrema. Convexity and points of inflection. Curve sketching.
• Number series, power series, and Taylor expansions
Infinite series. The multiplier effect in economics. Marginal propensity to consume. Present value of a perpetual annuity. Tests for convergence. Functions as power series. Taylor series. Approximation by Taylor polynomials.
• The indefinite integral
The indefinite integral. Applied initial value problems. Integration by substitution. An application involving substitution. Integration by parts.
• The definite integral
Net value problem. Future value and present value of an income flow. Area as the limit of a sum. The definite integral. The fundamental theorem of calculus. Substitution and integration by parts.
• Applications of the definite integral
Applications of the definite integral in business and economics. Useful life of a machine. Future value and present value of an income flow. Consumer willingness to spend. Consumers’ and producers’ surplus.
• The double integral
Average output for a firm. Definition of double integrals. Reduction of double integrals to iterated integrals. Changing the order of integration in iterated integrals. The economic interpretation and main properties of double integrals.
• Improper Integrals
Present value of a perpetual income flow. Integrals with infinite bounds. Improper integrals of the first kind. Principle value. Convergence tests for improper integrals. Absolute and relative convergence of improper integrals.
• Functions of several variables
Fundamentals of decision analysis. Graphs of functions of two variables. Level curves. Level curves in economics: isoquants and indifference curves. Continuity. The main properties of continuous functions of several variables.
• Partial derivatives and related topics
Marginal productivity. Definition of the partial derivatives of the first order. Substitute and complementary commodities. Directional derivative and gradient. Partial derivatives of second order. Law of diminishing returns. Properties of mixed derivatives.
• Optimizing functions of two variables.
Maximization of utility. Allocation of resources. Critical points. Saddle points. Practical optimization problems. Constrained optimization: the method of Lagrange multipliers. The significance of the Lagrange multiplier.
• Differential equations and slope fields
Price adjustment model. First-order differential equations. Separable equations. Logistic growth. First-order linear differential equations.
• Cauchy initial value problem
A model of financial debt. Initial value problem for ordinary differential equations (ODE) of the first order. #### Assessment Elements

• Examination (Gfinal)
• Mid-term exam (Gmidterm)
• Tests (Gtest)
• Class activity (Gtest) #### Interim Assessment

• Interim assessment (4 module)
0.08 * Class activity (Gtest) + 0.6 * Examination (Gfinal) + 0.2 * Mid-term exam (Gmidterm) + 0.12 * Tests (Gtest) #### Recommended Core Bibliography

• Fundamental methods of mathematical economics, Chiang, A. C., 1984