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Lecturer(s)
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Sedláček Lubomír, Mgr. Ph.D.
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Krňávek Jan, Mgr. Ph.D.
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Včelař František, RNDr. CSc.
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Course content
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1. Function of one real variable and its properties. 2. Limits and continuity of functions. One-sided limit, improper limit, limit at improper point. Asymptotes of the graph of a function. 3. Derivative of a function and its meaning. Derivatives of elementary functions. Derivative of a compound function. 4. Differential and its use. Derivatives of higher orders. Taylor polynomial. 5. Extremes of a function, intervals of monotonicity, convexity, concavity, inflex points. 6. Behaviour of a function. Using of derivatives in applications. 7. Primitive function, indefinite integral. 8. Basic methods of integration. Direct integration, method per partes, substitution. 9. Integration of rational functions, partial fractions decomposition, integration of partial fractions. 10. Proper integral. Method per partes and substitution for calculation of a proper integral. 11. Applications of proper integral. 12. Improper integral.
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Learning activities and teaching methods
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Lecturing, Methods for working with texts (Textbook, book), Demonstration, Projection (static, dynamic), Practice exercises, Individual work of students
- Preparation for course credit
- 20 hours per semester
- Preparation for examination
- 40 hours per semester
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| prerequisite |
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| Knowledge |
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| Standard knowledge of mathematical apparatus gained in the course Seminar of Mathematics is supposed. |
| Standard knowledge of mathematical apparatus gained in the course Seminar of Mathematics is supposed. |
| learning outcomes |
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| Students will gain basic knowledge and skills needed to analyze the behavior of a real function of one real variable. They will be able to independently analyze the occurrence of extrema of these functions, find intervals of monotonicity of the function, intervals of convexity, concavity, and find possible asymptotes of the function. Furthermore, students should be able to define basic concepts of integral calculus, use the method per partes, substitution method and decomposition into partial fractions, define and calculate a definite integral, improper integral, apply a definite integral in geometry and physics. |
| Students will gain basic knowledge and skills needed to analyze the behavior of a real function of one real variable. They will be able to independently analyze the occurrence of extrema of these functions, find intervals of monotonicity of the function, intervals of convexity, concavity, and find possible asymptotes of the function. Furthermore, students should be able to define basic concepts of integral calculus, use the method per partes, substitution method and decomposition into partial fractions, define and calculate a definite integral, improper integral, apply a definite integral in geometry and physics. |
| define the concept of a function (a real function of one real variable) and the related concepts of definition domain and value domain |
| define the concept of a function (a real function of one real variable) and the related concepts of definition domain and value domain |
| identify the basic elementary functions based on the graph |
| identify the basic elementary functions based on the graph |
| 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 function primitive to a given function is |
| explain what a function primitive to a given function is |
| formulate the Newton-Leibniz theorem |
| formulate the Newton-Leibniz theorem |
| explain the geometric meaning of a definite integral |
| explain the geometric meaning of a definite integral |
| Skills |
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| determine and write the defining domain of a function |
| determine and write the defining domain of a function |
| sketch graphs of basic elementary functions and describe their properties |
| sketch graphs of basic elementary functions and describe their properties |
| calculate limits using algebraic adjustments and L'Hospital's rule |
| calculate limits using algebraic adjustments and L'Hospital's rule |
| derive elementary, composite, product and quotient functions |
| derive elementary, composite, product and quotient functions |
| find the stationary points of the function and decide on the type of possible extreme |
| find the stationary points of the function and decide on the type of possible extreme |
| nalézt inflexní body funkce a intervaly, na kterých je funkce konvexní/konkávní |
| nalézt inflexní body funkce a intervaly, na kterých je funkce konvexní/konkávní |
| 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 |
| calculate simple indefinite integrals |
| calculate simple indefinite integrals |
| using a certain integral, calculate the content of the area bounded by the graphs of elementary functions |
| using a certain integral, calculate the content of the area bounded by the graphs of elementary functions |
| teaching methods |
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| Knowledge |
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| Methods for working with texts (Textbook, book) |
| Projection (static, dynamic) |
| Projection (static, dynamic) |
| Methods for working with texts (Textbook, book) |
| Demonstration |
| Demonstration |
| Lecturing |
| Lecturing |
| Individual work of students |
| Practice exercises |
| Practice exercises |
| Individual work of students |
| assessment methods |
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| Grade (Using a grade system) |
| Grade (Using a grade system) |
| Written examination |
| Written examination |
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Recommended literature
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Bear, H. S. Understanding calculus. 2nd ed. Piscataway : IEEE Press ; Hoboken : Wiley-Interscience, 2003. ISBN 0-471-43307-1.
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BOELKINS, Matt, David AUSTIN and Steve SCHLICKER. Active Calculus 2.0. Grand Valley State University, 2017. ISBN 978-1974206841.
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Černý, Ilja. Úvod do inteligentního kalkulu : 1000 příkladů z elementární analýzy. Vyd. 1. Praha : Academia, 2002. ISBN 80-200-1017-3.
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Demidovič, Boris Pavlovič. Sbírka úloh a cvičení z matematické analýzy. 1. vyd. Havlíčkův Brod : Fragment, 2003. ISBN 80-7200-587-1.
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KREML, Pavel. Mathematics II. Ostrava: VŠB - Technical University of Ostrava, 2005. ISBN 802480798x.
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RILEY, K. F., M. P. HOBSON a S. J. BENCE. Mathematical methods for physics and engineering. 3rd ed.. New York: Cambridge University Press, 2006. ISBN 9780521679718.
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WEIR, Maurice D., Joel. HASS, George B. THOMAS a Ross L. FINNEY. Thomas' calculus. 11th ed., media upgrade.. Boston: Pearson Addison Wesley, 2008. ISBN 9780321489876.
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