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Lecturer(s)
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Fajkus Martin, RNDr. Ph.D.
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Course content
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1. Complex numbers. Operations with complex numbers. Trigonometric and exponential form of complex number. DeMoivre's Theorem. Roots of Complex Numbers. 2. Vector spaces, linearly dependent and independent vectors, dimension and bases, subspace. 3. The notion of matrix and the special types of matrices, operations with matrices. Row elementary operations. 4. Determinants and operations with determinants, determinant of regular/singular matrix, calculating of inverse matrix by determinants. 5. Systems of linear equations, the methods of solving of linear equations and their applications. Eigenvalues and Eigenvectors. 6. Real function of one real variable. Domain of definition and image of functions. Graphs of functions. Properties of functions. 7. Classification of elementary functions. Inverse and composed functions. Solving equations and inequalities. 8. Limit of a function. Differentiation of a function, formulas for differentiations of elementary functions. 9. Behavior of a funcion, approximate solution of equations. 10. Primitive function, indefinite integral, integration by parts, integration by substitution. 11. Integration of rational functions, integration of goniometric functions. 12. Definite integral, integration by parts, integration by substitution for definite integral. Applications of definite integral. 13. Arithmetic and geometric sequences. Limit of a sequence. Arithmetic and geometric series. 14. Infinite series. Power series. Taylor and Maclaurin series.
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Learning activities and teaching methods
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Lecturing, Practice exercises, Individual work of students
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| prerequisite |
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| Knowledge |
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| Basic initial knowledge and skills of secondary school mathematics are assumed. |
| Basic initial knowledge and skills of secondary school mathematics are assumed. |
| learning outcomes |
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| - defines a complex number |
| - defines a complex number |
| - defines operations with complex numbers |
| - defines operations with complex numbers |
| - clarifies the terms of vector, linear combination, linear dependence |
| - clarifies the terms of vector, linear combination, linear dependence |
| - explains the concepts of matrix, matrix rank, inverse matrix |
| - explains the concepts of matrix, matrix rank, inverse matrix |
| - defines the terms determinant, system of linear equations, matrix equation |
| - defines the terms determinant, system of linear equations, matrix equation |
| - defines the function of one real variable |
| - defines the function of one real variable |
| - explains and clarifies the possible properties of a function |
| - explains and clarifies the possible properties of a function |
| - explains the concepts of limit of a function and derivative of a function |
| - explains the concepts of limit of a function and derivative of a function |
| - clarifies the geometric meaning of the first and second derivatives |
| - clarifies the geometric meaning of the first and second derivatives |
| - defines the basic terms of integral calculus |
| - defines the basic terms of integral calculus |
| - clarifies basic integration methods: simplification of integrand, substitution, per partes |
| - clarifies basic integration methods: simplification of integrand, substitution, per partes |
| - defines a definite integral |
| - defines a definite integral |
| - clarifies the geometric meaning of a definite integral |
| - clarifies the geometric meaning of a definite integral |
| - defines an infinite sequence and an infinite series |
| - defines an infinite sequence and an infinite series |
| - explains the concept of convergence of an infinite series |
| - explains the concept of convergence of an infinite series |
| - defines the sum of an infinite series |
| - defines the sum of an infinite series |
| Skills |
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| - converts an algebraic form of a complex number to its goniometric form and vice versa |
| - converts an algebraic form of a complex number to its goniometric form and vice versa |
| - calculates all basic operations with complex numbers |
| - calculates all basic operations with complex numbers |
| - solves linear and quadratic equations with complex coefficients |
| - solves linear and quadratic equations with complex coefficients |
| - creates a vector that is a linear combination of given vectors |
| - creates a vector that is a linear combination of given vectors |
| - decides whether a given vector is a linear combination of given vectors |
| - decides whether a given vector is a linear combination of given vectors |
| - finds out whether the given vectors are linearly dependent |
| - finds out whether the given vectors are linearly dependent |
| - determines the rank of a matrix |
| - determines the rank of a matrix |
| - calculates the inverse matrix |
| - calculates the inverse matrix |
| - calculates the determinant of a matrix |
| - calculates the determinant of a matrix |
| - solves a system of linear equations |
| - solves a system of linear equations |
| - solves a matrix equation |
| - solves a matrix equation |
| - determines the domain of a function and draws it |
| - determines the domain of a function and draws it |
| - determines the properties of a function |
| - determines the properties of a function |
| - calculates the limit of a function at the specified point |
| - calculates the limit of a function at the specified point |
| - calculates the first and second derivatives of a function |
| - calculates the first and second derivatives of a function |
| - sketches a graph of a specified function |
| - sketches a graph of a specified function |
| - computes simple integrals by simplifying the integrand |
| - computes simple integrals by simplifying the integrand |
| - calculates integrals by substitution methods and per partes |
| - calculates integrals by substitution methods and per partes |
| - calculates definite and improper integral |
| - calculates definite and improper integral |
| - determines the area and volume of a rotational body using a definite integral |
| - determines the area and volume of a rotational body using a definite integral |
| - decides about the convergence of an infinite series |
| - decides about the convergence of an infinite series |
| - calculates the sum of an infinite series |
| - calculates the sum of an infinite series |
| - calculates the sum of a finite arithmetic and geometric sequencies |
| - calculates the sum of a finite arithmetic and geometric sequencies |
| teaching methods |
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| Knowledge |
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| Lecturing |
| Individual work of students |
| Individual work of students |
| Practice exercises |
| Practice exercises |
| Lecturing |
| assessment methods |
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| Written examination |
| Written examination |
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Recommended literature
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Barnett, R. A., Kearns, T. J. Intermediate Algebra: Structure and Use. McGraw-Hill, 1999.
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Lial, M. L. et al. Finite. Mathematics with Applications: in the Management, Natural, and Social Sciences. Pearson, 2006.
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Matejdes, M. Aplikovaná matematika. Matcentrum-Zvolen, 2005. ISBN 80-89077-01-3.
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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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Polášek, V., Sedláček, l. & Kozáková, L. Matematický seminář. Zlín: Nakladatelství UTB., 2018.
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Riley, K. F. et al. Mathematical Methods for Physics and Engineering. Cambridge University Press, 2015.
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