## INMBAKVM Quantitave Methods

Winter 2023
Extent and Intensity
2/2/0. 7 credit(s). Type of Completion: zk (examination).
Teacher(s)
Guaranteed by
doc. RNDr. David Bartl, Ph.D.
Department of Informatics and Mathematics - School of Business Administration in Karvina
Contact Person: Mgr. Radmila Krkošková, Ph.D.
Timetable
Mon 14:45–16:20 A422
• Timetable of Seminar Groups:
INMBAKVM/01: Mon 16:25–18:00 A422, R. Krkošková
Prerequisites (in Czech)
FAKULTA ( OPF ) && TYP_STUDIA ( B ) && FORMA ( P )
Course Enrolment Limitations
The course is only offered to the students of the study fields the course is directly associated with.

The capacity limit for the course is 30 student(s).
Current registration and enrolment status: enrolled: 2/30, only registered: 1/30
fields of study / plans the course is directly associated with
Course objectives (in Czech)
To acquaint the students with basic knowledge and terminology in algebra and mathematical analysis to be able to use established concepts, theoretical results, and calculation procedures in other, especially economically oriented courses.
Learning outcomes (in Czech)
After completing the course, the student will be able to:
- treat simple logical formulas (conjunction, disjunction, implication, equivalence, negation);
- treat simple set operations (intersection, union, set difference);
- understand logical formulas with quantifiers;
- grasp the notion of number domains (natural numbers, integer numbers, rational numbers, real numbers);
- grasp the notion of arithmetic vector space;
- multiply matrices;
- calculate the determinant of a matrix;
- solve a system of linear equations;
- compose linear mappings (matrix multiplication);
- calculate the limit of some sequences;
- sum up the first elements of a geometric progression;
- sum up geometric series;
- perform operations with function (addition, subtraction, multiplication, division);
- compose functions;
- calculate the limit of some functions at some points;
- calculate the derivative of any differentiable function;
- calculate the limit of some functions at some points by using L'Hospital's Rule;
- find the local extrema and inflection points of some differentiable functions;
- find the intervals of monotonicity and convexity/concavity of some differentiable functions;
- sketch the graph of some differentiable functions;
- calculate the Riemann integral of a function by using the Newton-Leibniz formula;
- calculate the Newton integral of a function;
- find the indefinite integral (antiderivative) of some functions;
- integrate by substitution;
- integrate by parts.
Syllabus (in Czech)
• 1. Introduction and a Brief History of Mathematics
Mathematics in ancient Greece, mathematics in Europe. Emergence of centres of knowledge in the 17th century, mathematical analysis in the 18th century, development of mathematics in the 19th and the 20th century. Calculator, computer, and mathematics. Sets, set notation, propositions and logical operations, set relations and operations. Mapping. Number sets.
• 2. Vector Spaces, Matrices and Matrix Algebra
Arithmetic vector space. Linear combination of vectors, linear dependence and independence of vectors. Basis of a vector space, elementary properties and characterizations of a basis, the dimension of a vector space. Matrices. Addition of matrices and multiplication of a matrix by a constant, vector space of matrices. Square matrices. Transformation of a matrix into (upper) triangular form. The rank of a matrix. Identity matrix, singular and non-singular and matrices. Multiplication of matrices and its properties. Inverse matrix. Solution of matrix equations.
• 3. Systems of Linear Algebraic Equations, Determinants
The matrix of the coefficients of the system of linear equations, the augmented matrix. Frobenius theorem and its corollary. Gauss elimination method. Gauss-Jordan elimination method. Homogeneous systems of linear equations and its solution set (linear subspace). Determinant of a matrix, definition, elementary properties, geometrical meaning, Laplace expansion of a determinant and Leibniz formula, computation of the determinant (elimination method, other methods). Determinant of a singular and non-singular matrix. Cramer's rule. Adjugate (classical adjoint) matrix and finding the inverse matrix.
• 4. Sequences and the Limits of Sequences, Series and Infinite Sums
Arithmetic and geometric sequence (progression). Finite and infinite sequence. Bounded and unbounded sequence. Monotone sequence. Convergent and divergent sequence. The limit of a sequence and its properties. Bolzano-Weierstrass Theorem. Infinite series and its sum. Convergent and divergent series, geometric series. Necessary condition for the convergence, the tail (remainder) of a series. Series with positive terms, alternating (oscillating) series, convergence criterions. Alternating series, Leibniz criterion.
• 5. The Limit and Continuity of a Function
A real function of a real variable. Least upper bound (supremum), greatest lower bound (infimum), bounded function, monotone function, convex and concave function. Injective and inverse function. Elementary functions, their domains, properties and graphs. Continuous functions and their properties. The limit of a function at a point and the properties of the limit.
• 6. Differential Calculus, Sketching the Graph of a Function, L'Hospital's Rule
The derivative of a function, geometrical meaning, continuity of a function having a bounded derivative. Derivative of arithmetic operations, chain rule, derivative of the inverse function. Differential, derivatives of higher orders. Sketching the graph of a function. Taylor's series and Taylor's polynomial. L'Hospital's rule.
• 7. Indefinite Integral (Antiderivative) and Definite Integral
Primitive function (antiderivative), integration by parts, integration by substitution. Riemann definite integral. Newton integral. Newton-Leibniz formula. Geometrical meaning (area of a planar region). Improper integral, convergence and divergence of the improper integral.
Literature
required literature
• HUGHES-HALLETT, Deborah, Andrew M. GLEASON, Patti Frazer LOCK and Daniel E. FLATH. Applied Calculus. 7th Edition. Wiley, 2021. ISBN 978-1-119-79906-1. info
• HUGHES-HALLETT, Deborah, Andrew M. GLEASON and William G. MCCALLUM. Calculus: Single and Multivariable. 8th Edition. Wiley, 2021. ISBN 978-1-119-69655-1. info
• STEWART, James, Daniel K. CLEGG and Saleem WATSON. Calculus. 9th Edition. Cengage, 2020. ISBN 978-0-357-11346-2. info
recommended literature
• STEWART, James, Lothar REDLIN and Saleem WATSON. Precalculus: Mathematics for Calculus. 8th Edition. Cengage, 2023. ISBN 979-8-214-03181-1. info
• STEWART, James, Daniel K. CLEGG and Saleem WATSON. Calculus: Early Transcendentals. 9th Edition. Cengage, 2020. ISBN 978-0-357-11351-6. info
• STEWART, James. Calculus: Concepts and Contexts. 4th Edition. Cengage, 2018. ISBN 978-1-337-68766-9. info
• MECKES, Elizabeth S. and Mark W. MECKES. Linear Algebra. Cambridge University Press, 2018. ISBN 978-1-107-17790-1. info
• TRENCH, William F. Introduction to Real Analysis. 2013. URL info
• STEWART, James and Daniel CLEGG. Brief Applied Calculus. 1st Edition. Cengage, 2011. ISBN 978-0-534-42382-7. info
• LANG, Serge. Short Calculus: The Original Edition of “A First Course in Calculus”. Springer, 2002. ISBN 978-0-387-95327-4. info
• COURANT, Richard. Differential and Integral Calculus, Volume 1. 2nd Edition. Wiley, 1988. ISBN 978-0-471-60842-4. info
• COURANT, Richard. Differential and Integral Calculus, Volume 2. 2nd Edition. Wiley, 1988. ISBN 978-0-471-60840-0. info
• LANG, Serge. Introduction to Linear Algebra. Second Edition. Springer, 1986. ISBN 978-1-4612-7002-7. info
not specified
• GARCIA, Stephan Ramon and Roger A. HORN. Matrix Mathematics: A Second Course in Linear Algebra. 2nd Edition. Cambridge University Press, 2023. ISBN 978-1-108-83710-1. info
• LANG, Serge. Basic Mathematics. Springer, 1988. ISBN 978-0-387-96787-5. info
• LANG, Serge. Calculus of Several Variables. Third Edition. Springer, 1987. ISBN 978-1-4612-7001-0. info
Teaching methods (in Czech)
lectures and seminars (exercises, problems and examples)
Assessment methods
Requirements for the student: regular study, attendance at seminars min. 70 %, final test.
Assessment: attendance at seminars, test (30 % of assessment), written test (70 % of assessment).
Assessment methods: final written test (several problems from the areas taught).
Language of instruction
English