INMBAKVM Quantitave Methods

Obchodně podnikatelská fakulta v Karviné
zima 2021
Rozsah
2/2/0. 7 kr. Ukončení: zk.
Vyučující
doc. RNDr. David Bartl, Ph.D. (přednášející)
Garance
doc. RNDr. David Bartl, Ph.D.
Katedra informatiky a matematiky – Obchodně podnikatelská fakulta v Karviné
Kontaktní osoba: Mgr. Radmila Krkošková, Ph.D.
Omezení zápisu do předmětu
Předmět je určen pouze studentům mateřských oborů.
Mateřské obory/plány
Osnova
  • 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.
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Předmět je zařazen také v obdobích zima 2015, zima 2016, zima 2017, zima 2018, zima 2022, zima 2023, zima 2024.