*This course is not currently conducted!*

*Erasmus code:*14.3

*ISCED code:*0311

*ECTS credits:*unknown

*Language:*English

*Organized by:*Faculty of Economic Sciences

*Related to study programmes:*

# Mathematical Analysis II 2400-FIM1AMII

Winter semester:

1. Elements of mathematical logic, sets, mathematical induction (1 lecture)

2. Repetition of secondary school mathematics (absolute value, powers, roots, system of coordinates, polynomials, trigonometric functions) (1 lecture)

3. Averages (different kinds), interest rates of loans and deposits, compound interest, continuous compounding, e number. Exponential function, logarithmic function, power function. (2 lectures)

4. Sequences of real numbers, limits of sequences, basic properties of limits, examples, improper limits. Monotone and bounded sequences, Bolzano-Weierstrass theorem, Stolz theorem. Recursive sequences, difference equations, rate of converegence. (2 lectures)

5. Real series, (notion of convergence, sum of a series, examples), convergence criteria for series of nonnegative terms (d'Alembert, Cauchy). Series of arbitrary numbers (criteria of Abel, Dirichlet, Leibniz), product of series, power series, formula for radius of convergence. (2 lectures)

6. Notion of function, limit and continuity of a function of a real single variable, basic properties of limits, examples, improper limits, one-sided limits, Weierstrass theorem. Properties of continuous functions, Darboux property, Lipschitz condition, fixed point theorem (2 lectures)

7. Definition of derivative of the function of a single real variable, physical and geometric interpretation. Basic differentiation rules, derivatives of elementary functions. Intermediate value (Lagrange) value theorem. Derivative and monotonicity of the function. The de l'Hospital rule. Elasticity and asymptotes. (2 lectures)

8. Finding Extremes of Functions with Derivatives. Examples.

9. Primary function (indefinite integral), integration methods. Integration of some elementary functions, examples. Definite (Newton) Integral. (2 lectures)

Summer semester:

1. Higher order derivatives, Taylor formula. [1 lecture]

2. Expanding functions into power series, examples. Convex functions. Jensen inequality. [1 lecture]

3. Structure of the multidimensional Euclidean space R^k (norm, scalar product). [1 lecture]

4. Open, closed, convex, connected sets in space R^k. The limit of sequence in R^k. [1 lecture]

5. Vector and scalar functions in R^k. Limit and continuity of functions of several variables. [1 lecture]

6. Directional and partial derivatives. Gradient and Jacobi matrix. The concept of the differential of functions of several variables. The differential of composition of function and differential of the inverse function. Conditions of differentiability. [2 lectures]

7. Extremes of functions of several variables. Partial derivatives of higher orders. Sylvester criterion - conditions for the existence of local extremes.

Finding extremes of functions of several variables, examples. [3 lectures]

8. Invertibility of functions in R^k, diffeomorphisms. Implicit function theorem. manifolds, tangent space and normal space. [1 lecture]

9. Conditional extremes, the Lagrange multipliers method, Kuhn-Tucker theorem. [2 lectures]

10. Calculation of the length of curves, surface areas, volumes, with the help of the definite integral. [1 lecture]

11. Double and triple integral. Change of variables for integrals of multivariate functions. Calculation of the Poisson's integral. [1 lecture]

## Type of course

## Learning outcomes

Knowledge of basic concepts, theorems and methods of Mathematical Analysis. The ability to apply these methods to solve problems occurring in economic issues, in particular in optimization tasks. Conceptual preparation

to learn the Probability and Statistics.

## Assessment criteria

Assessment based on a written exam, tests carried out during the semester and activity during tutorials.

Detailed rules for passing the winter semester:

1. The final grade of the course is calculated on the basis of the sum of points obtained during the semester (max. 100p). Points are granted for tutorials (max 20p), for two joint mid-term exams during the semester (max 15 + 15 = 30p) and for the exam after the end of the semester (max 50p). Only those who have received at least 15points from the exam can receive a positive final grade.

2. Points for tutorials are obtained during short tests carried out in particular groups and for the activity during tutorials. During the semester there will be 8 short tests (every two weeks, during tutorials) checking knowledge of the current material. For each

test max. 2p can be obtained. Points for activity (max 4p in each semester) are awarded by the tutor according to his

own assessment.

3. Joint mid-term examstests (carried out apart from tutorials) consist in solving a number of problems from the materials taught during previous lectures and tutorials.

4. After completing the course (before the examination session), the student receives the grade from the exercises, calculated on the basis of the sum of points obtained during tutorials and joint mid-term exams (max 20 + 30 = 50p), according to the following scale: sufficient = 20p, sufficient + = 25p, good = 30p, good

+ = 35p, very good = 40p, very good + = 45p.

5. In order to take the exam, the student must complete the tutorials, i.e. Obtain at least sufficient grade for the tutorials.

6. The final exam is written and consists in solving a certain number of problems covering the whole material taught during the semester both during lectures and

tutorials. The rules for calculating the final grade are the same in each of the two terms of the exam.

7. In case of failing to pass the tutorials, the student has the opportunity to pass the correction test (the form is decided by the tutor. The student who passes the correction test, may take the final exam during the second term, however with the same number of points credited for tutorials (excluding the result of the correction test).

## Bibliography

Claudio Canuto, Anita Tabacco Mathematical Analysis I, Springer

Claudio Canuto, Anita Tabacco Mathematical Analysis II, Springer

## Additional information

Information on *level* of this course, *year of study* and semester when the course
unit is delivered, types and amount of *class hours* - can be found in course structure
diagrams of apropriate study programmes. This course is related to
the following study programmes:

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*localization and schedules* of classes), might be available in the USOSweb system: