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Lecturer(s)
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Polášek Vladimír, Mgr. Ph.D.
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Cerman Zbyněk, Mgr. Ph.D.
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Volaříková Jana, Mgr. Ph.D.
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Běták Vojtěch, Ing. Ph.D.
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Course content
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- Introduction to the course - Propositions, operations with propositions, propositional functions, tautology, existential and universal quantifier. - Sets, operations with sets, Cartesian product, mappings. - Real function of one real variable, domain and codomain, properties of the functions, graphs of the functions. - Algebraic and transcendental functions. - Limit of a function, theorems about limits, infinite limit, limits in improper points, asymptote, continuity of a function. - Derivative of a function, enumeration of derivative, differential of a function, derivatives of higher orders, and L'Hospital's rule. - Monotonicity of a function, extrema of a function, convexity, concavity, inflex points. - Applications of the differential calculus in physics and economics. - Integral calculus of the functions of one real variable: Primitive function, indefinite integral, decomposition methods. - Integration by parts, substitution method. - Integration of rational and irrational functions, integration of goniometric functions. - Definite integral: Definition of definite integral, enumeration and properties of definite integral. - Applications of definite integral in geometry, physics, and economy.
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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
- Home preparation for classes
- 2 hours per semester
- Preparation for course credit
- 8 hours per semester
- Preparation for examination
- 34 hours per semester
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| prerequisite |
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| Knowledge |
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| Knowledge of secondary school mathematics |
| Knowledge of secondary school mathematics |
| learning outcomes |
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| Verbally define the term function (a real function of one real variable) and the related terms domain of definition and range of values. |
| Verbally define the term function (a real function of one real variable) and the related terms domain of definition and range of values. |
| Identify the elementary functions from the graphs. |
| Identify the elementary functions from the graphs. |
| Explain the geometric meaning of the derivative of a function at a point. |
| Explain the geometric meaning of the derivative of a function at a point. |
| Explain what a primitive function is to a given function. |
| Explain what a primitive function is to a given function. |
| Formulate the Newton-Leibniz formula. |
| Formulate the Newton-Leibniz formula. |
| Clarify the geometric meaning of a definite integral. |
| Clarify the geometric meaning of a definite integral. |
| Skills |
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| Determine and set the domain of definition of the function. |
| Determine and set the domain of definition of the function. |
| Sketch the graphs of elementary functions and describe their properties. |
| Sketch the graphs of elementary functions and describe their properties. |
| Calculate limits using algebraic adjustments and using L'Hospital's rule. |
| Calculate limits using algebraic adjustments and using L'Hospital's rule. |
| Differentiate elementary, composite, product and quotient functions. |
| Differentiate elementary, composite, product and quotient functions. |
| Calculate the stationary points of the function and decide on the types of possible extremes. |
| Calculate the stationary points of the function and decide on the types of possible extremes. |
| Find the inflection points of a function and the intervals on which the function is convex/concave. |
| Find the inflection points of a function and the intervals on which the function is convex/concave. |
| Find the equation of the tangent to the graph of the function and sketch it. |
| Find the equation of the tangent to the graph of the function and sketch it. |
| Compute simple indefinite integrals. |
| Compute simple indefinite integrals. |
| Using a definite integral, calculate the area content of bounded graphs of elementary functions. |
| Using a definite integral, calculate the area content of bounded graphs of elementary functions. |
| teaching methods |
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| Knowledge |
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| Practice exercises |
| Lecturing |
| Lecturing |
| Practice exercises |
| Methods for working with texts (Textbook, book) |
| Methods for working with texts (Textbook, book) |
| assessment methods |
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| Written examination |
| Grade (Using a grade system) |
| Written examination |
| Grade (Using a grade system) |
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Recommended literature
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Kluvánek, Mišík, Švec. Matematika 1. Bratislava, 1959.
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Kuben, J., Šarmanová, P. Diferenciální počet funkcí jedné proměnné. VŠB-TU Ostrava, 2006.
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Matejdes, M. Aplikovaná matematika. Matcentrum-Zvolen, 2005.
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Polák, J. Přehled středoškolské matematiky. Praha : Prometheus, 1995. ISBN 80-85849-78-X.
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Polák, J. Středoškolská matematika v úlohách II. Praha : Prometheus, 1999. ISBN 80-7196-166-3.
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Polák, Josef. Středoškolská matematika v úlohách I. 2., upr. vyd. Praha : Prometheus, 2006. ISBN 80-7196-337-2.
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Rektorys, K. Přehled užité matematiky. Praha : Prometheus, 1995.
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Škrášek, J., Tichý, Z. Základy aplikované matematiky I., II. Praha : SNTL, 1986.
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