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Lecturer(s)
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Cerman Zbyněk, Mgr. Ph.D.
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Polášek Vladimír, Mgr. Ph.D.
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Course content
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DIFFERENTIAL CALCULUS OF A FUNCTION OF ONE REAL VARIABLE: - Definition domain and value domain (graph of a function, operations on functions - compound functions) - Properties of functions (injectivity, monotonicity, boundedness, parity, periodicity) - Inversion of a function - Elementary functions (constant, power, exponential, logarithmic, goniometric, cyclometric, special) - Continuity of a function (neighborhood of a point, points of discontinuity, one-sided continuity, continuity on an interval) - Limit of a function (one-sided limits, types of limits - (in)proper limit at (in)proper point, relation to continuity, operations with limits - indefinite expressions, special formulas, asymptotic equality, rate of functions, graphical recognition of limits) - Derivative of a function (physical and mathematical meaning, important functions and their derivatives, proper and improper derivatives, derivatives on an interval, basic formulas and rules, higher order derivatives, L'Hospital's rule, local approximation) - Function progression (monotonicity, convexity and concavity intervals, local function extremes and inflection points, asymptotes) INTEGRAL CALCULUS OF A FUNCTION OF ONE REAL VARIABLE: - Indefinite integral (primitive function and its existence, methods of calculation - substitution method and per-partes method, basic formulas and rules) - Definite integral (geometric meaning, Riemann and Newton integrals, properties and calculation of definite integral, area content between functions, volume of a rotating body, numerical integration) LINEAR ALGEBRA: - Matrices (sum of matrices, matrix group, matrix multiplication by a scalar, matrix product, square matrix ring, transposed matrix) - Systems of linear equations (elementary row operations, reduced triangular matrix, rank of a matrix, solvability of a system - Frobenius theorem, Gauss elimination method, homogeneous system - fundamental solution system) - Vector spaces (left outer product, linear combinations of vectors, linear dependence and independence, subspace, linear span, intersection and sum of subspaces, set of generators and basis, dimension)
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Learning activities and teaching methods
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Lecturing, Methods for working with texts (Textbook, book), Practice exercises
- Participation in classes
- 56 hours per semester
- Preparation for examination
- 45 hours per semester
- Home preparation for classes
- 4 hours per semester
- Preparation for course credit
- 20 hours per semester
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| prerequisite |
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| Knowledge |
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| Have a basic understanding of high school mathematics |
| Have a basic understanding of high school mathematics |
| Have basic logical thinking |
| Have basic logical thinking |
| Read the materials provided and consult if there is any confusion |
| Read the materials provided and consult if there is any confusion |
| Skills |
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| Show interest and effort in the subject |
| Show interest and effort in the subject |
| Regularly attend lectures and exercises |
| Regularly attend lectures and exercises |
| Be active in the exercises and answer questions during the lecture (every answer is appreciated) |
| Be active in the exercises and answer questions during the lecture (every answer is appreciated) |
| learning outcomes |
|---|
| Knowledge |
|---|
| Prohloubit logické myšlení |
| Prohloubit logické myšlení |
| Define and explain the basic concepts of mathematical analysis and linear algebra |
| Define and explain the basic concepts of mathematical analysis and linear algebra |
| Explain the concept of a real function of one real variable and the related concepts of definitional domain and value domain |
| Explain the concept of a real function of one real variable and the related concepts of definitional domain and value domain |
| List the basic elementary functions of one real variable, including their properties and graphs |
| List the basic elementary functions of one real variable, including their properties and graphs |
| Characterize the four basic types of limit functions of one real variable |
| Characterize the four basic types of limit functions of one real variable |
| Describe the geometric meaning of the derivative of a function at a point |
| Describe the geometric meaning of the derivative of a function at a point |
| Describe the procedure for investigating the progress of a function of one real variable |
| Describe the procedure for investigating the progress of a function of one real variable |
| Explain the concept of integral and distinguish between definite and indefinite integrals |
| Explain the concept of integral and distinguish between definite and indefinite integrals |
| Define a matrix over real numbers and describe matrix operations (sum, product, scalar multiplication, transpose). |
| Define a matrix over real numbers and describe matrix operations (sum, product, scalar multiplication, transpose). |
| Charakterizovat třídimenzionální vektorový prostor a popsat pojem báze prostoru. |
| Charakterizovat třídimenzionální vektorový prostor a popsat pojem báze prostoru. |
| Skills |
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| Use active knowledge of the concept of functional value in calculations and sketches of graphs of functions |
| Use active knowledge of the concept of functional value in calculations and sketches of graphs of functions |
| Rozpoznat z grafu funkce intervaly, na kterých je funkce rostoucí, klesající, prostá, konvexní, konkávní |
| Rozpoznat z grafu funkce intervaly, na kterých je funkce rostoucí, klesající, prostá, konvexní, konkávní |
| Calculate limits using algebraic adjustments and L'Hospital's rule |
| Calculate limits using algebraic adjustments and L'Hospital's rule |
| Derive functions of one real variable using basic formulas and five rules for derivation. |
| Derive functions of one real variable using basic formulas and five rules for derivation. |
| Investigate the course of a function of one real variable (definition domain, limits, local extrema, inflections, asymptotes) |
| Investigate the course of a function of one real variable (definition domain, limits, local extrema, inflections, asymptotes) |
| Recognize the difference between the substitution method and the per-partes method for integrating a function of one real variable and apply it appropriately to a given type of example |
| Recognize the difference between the substitution method and the per-partes method for integrating a function of one real variable and apply it appropriately to a given type of example |
| Calculate the content of any figure bounded by curves |
| Calculate the content of any figure bounded by curves |
| Solve a system of linear equations with exactly one solution, using elementary row transformations. |
| Solve a system of linear equations with exactly one solution, using elementary row transformations. |
| Determine the linear dependence and independence of the vectors and, if necessary, the base of the space or subspace |
| Determine the linear dependence and independence of the vectors and, if necessary, the base of the space or subspace |
| teaching methods |
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| Knowledge |
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| Lecturing |
| Lecturing |
| Students working in pairs |
| Students working in pairs |
| Projection (static, dynamic) |
| Projection (static, dynamic) |
| Monologic (Exposition, lecture, briefing) |
| Monologic (Exposition, lecture, briefing) |
| Dialogic (Discussion, conversation, brainstorming) |
| Dialogic (Discussion, conversation, brainstorming) |
| Skills |
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| Dialogic (Discussion, conversation, brainstorming) |
| Dialogic (Discussion, conversation, brainstorming) |
| Individual work of students |
| Individual work of students |
| Teamwork |
| Teamwork |
| Practice exercises |
| Practice exercises |
| assessment methods |
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| Knowledge |
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| Written examination |
| Written examination |
| Grade (Using a grade system) |
| Grade (Using a grade system) |
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Recommended literature
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HOŠKOVÁ, Š., KUBEN, J., RAČKOVÁ, P. Integrální počet funkcí jedné proměnné. 2006.
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KREML, P., VLČEK, J., VOLNÝ, P., KRČEK, J., POLÁČEK, J. Matematika II. ISBN 978-80-248-1316-5.
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Matejdes M. Aplikovaná matematika. MAT-Centrum Zvolen, 2005.
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Ostravský J., Polášek V. Diferenciální a integrální počet funkce jedné proměnné: vybrané statě. Zlín, 2011. ISBN 978-80-7454-124-7.
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Pavlíková P., Schmidt O. Základy matematiky. Praha, 2006. ISBN 80-7080-615-X.
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