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Part 1
Differential calculus of one real variable
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1. Real functions of one real variable
(number of lectures: 2)
- domian and range of real functions, graphs of real functions, graphical interpretation of the graph of a function,
- basic properties of real functions (odd and even functions, monotonicity, boundedness, one-to-one maps),
- compositions of functions,
- inverse functions
2. Continuity and limits of real functions
(number of lectures: 2)
- delta-neighborhood of a real point, left and right delta-neighborhoods,
- continuity of real functions at real points and on closed intervals, properties of continuous functions on closed intervals (Weierstraß' theorem with consequences),
- improper points and their arithmetic, reduced delta-neighborhood of a point, limits of functions at proper and improper points, the algebra of limits,
3. The derivative of a function
(number of lectures: 1)
- the possibilities of measuring the slope of a curve at the point x=x_0, the transition from a secant of a curve to the tangent at x=x_0, the meaning of the indeterminate term [0/0] and of the theory of limits for computing the slope of the curve at x=x_0,
- the definition of the derivative of a function with the help of the derivative,
- general derivatives of elementary functions, basic rules for computing derivatives
4. The course of the graph of a function
(number of lectures: 2)
- investigation of monotonicity with the help of the derivative sign,
- local extremes of a function and their characteristics, investigation of extremes with the help of derivatives and in certain special cases also with the help of the definition,
- convexity and concavity of a function, points of inflection, mathematical and practical meaning,
- asymptotes, graphical and practical meaning
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Part 2
Integral calculus of real functions of one ral variable
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5. The indefinite integral of a real function
(number of lectures: 2)
- basic concepts
- basic integration rules and techniques,
- integration by substitution,
- integration by parts,
- decomposition of rational functions into partial fractions
6. Volume of a plane area, construction of the definite integral
(number of lectures: 1+)
- construction of the upper and lower estimations of an plane area,
- definition of an plane area by limiting procedure,
- sketch of the proof of the formula for computation of the volume of an plane area (optional), Newton-Leibniz formula,
- basic applications in the microeconomics
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Part 3
Linear algebra
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9. Introduction to matrix algebra
(number of lectures: 1)
- definition of real matrices and related concepts,
- basic classification of matrices by their type and by their values,
- basics of matrix algebra (addition, subtraction, scalar multiplication, multiplication of matrices, power of a matrix, transposition),
- stochastic matrices and their applications in the preference model
10. Number characteristics of real matrices, linear matrix equations, inverse matrices
(number of lectures: 2)
- rank of matrix, transformation to Gauß' form, related concepts,
- definition and computation of determinants of orders n=2,3,4, Sarrus' rule,
- properties of determinants,
- Laplace's expansion,
- basic geometric application of the determinant value,
- definition and computation of the inverse matrix, adjoint matrix and other related concepts,
- matrix equations of the form A+k*X=B, A*X=B, X*A=B,
11. Systems of linear equations a their applications in economics
(number of lectures: 1)
- definitions and basic concepts,
- matrix notation,
- Gauss' elimination and Frobenius' theorem,
- systems of linear equations involving real parameters,
- network analysis, polynomial curve fitting, Leontief input-output model (optional).