Mathematics I (FSI-1M)

Academic year 2024/2025
Supervisor: prof. RNDr. Miroslav Doupovec, CSc., dr. h. c.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Course type: departmental course
Aims of the course unit:
 
Learning outcomes and competences:
 
Prerequisites:
 
Course contents:
 
Teaching methods and criteria:
 
Assesment methods and criteria linked to learning outcomes:
 
Controlled participation in lessons:
 
Type of course unit:
    Lecture  13 × 4 hrs. optionally                  
    Exercise  11 × 4 hrs. compulsory                  
    Computer-assisted exercise  2 × 4 hrs. compulsory                  
Course curriculum:
    Lecture Week 1: Basics of mathematical logic and set operations, matrices and determinants (transposing, adding, and multiplying matrices, common matrix types).
Week 2: Matrices and determinants (determinants and their properties, regular and singular matrices, inverse to a matrix, calculating the inverse to a matrix using determinants), systems of linear algebraic equations (Cramer's rule, Gauss elimination method).
Week 3: More about systems of linear algebraic equations (Frobenius theorem, calculating the inverse to a matrix using the elimination method), vector calculus (operations with vectors, scalar (dot) product, vector (cross) product, scalar triple (box) product).
Week 4: Analytic geometry in 3D (problems involving straight lines and planes, classification of conics and quadratic surfaces), the notion of a function (domain and range, bounded functions, even and odd functions, periodic functions, monotonous functions, composite functions, one-to-one functions, inverse functions).
Week 5: Basic elementary functions (exponential, logarithm, general power, trigonometric functions and cyclometric (inverse to trigonometric functions), polynomials (root of a polynomial, the fundamental theorem of algebra, multiplicity of a root, product breakdown of a polynomial), introducing the notion of a rational function.
Week 6: Sequences and their limits, limit of a function, continuous functions.
Week 7: Derivative of a function (basic problem of differential calculus, notion of derivative, calculating derivatives, geometric applications of derivatives), calculating the limit of a function using L' Hospital rule.
Week 8: Monotonous functions, maxima and minima of functions, points of inflection, convex and concave functions, asymptotes, sketching the graph of a function.
Week 9: Differential of a function, Taylor polynomial, parametric and polar definitions of curves and functions (parametric definition of a derivative, transforming parametric definitions into polar ones and vice versa).
Week 10: Primitive function (antiderivative) (definition, properties and basic formulas), integrating by parts, method of substitution.
Week 11: Integrating rational functions (no complex roots in the denominator), calculating a primitive function by the method of substitution in some of the elementary functions.
Week 12: Riemann integral (basic problem of integral calculus, definition and properties of the Riemann integral), calculating the Riemann integral (Newton' s formula).
Week 13: Applications of the definite integral (surface area of a plane figure, length of a curve, volume and lateral surface area of a rotational body), improper integral.
    Exercise The first week will be devoted to revision of knowledge gained at secondary school. Following weeks: seminars related to the lectures given in the previous week.
    Computer-assisted exercise Seminars in a computer lab have the programme MAPLE as a computer support. Obligatory topics to go through: Elementary arithmetic, calculations and evaluation of expressions, solving equations, finding roots of polynomials, graph of a function of one real variable, symbolic computations.
Literature - fundamental:
1. Thomas G.B., Finney R.L.: Calculus and Analytic Geometry (7th edition)
1. Thomas G. B.: Calculus (Addison Wesley, 2003)
2. Sneall D.B., Hosack J.M.: Calculus, An Integrated Approach
3. Rektorys K. a spol.: Přehled užité matematiky I,II (SNTL, 1988)
4. Howard, A.A.: Elementary Linear Algebra, Wiley 2002
5. Satunino, L.S., Hille, E., Etgen, J.G.: Calculus: One and Several Variables, Wiley 2002
7. FRANCŮ, Jan. Matematika I. Učební texty vysokých škol (Vysoké učení technické v Brně). Brno: Akademické nakladatelství CERM, 2023. ISBN 978-80-214-6174-1
Literature - recommended:
1. Rektorys K. a spol.: Přehled užité matematiky I,II (SNTL, 1988)
2. Děmidovič B. P.: Sbírka úloh a cvičení z matematické analýzy
2. Nedoma J.: Matematika I. Část druhá. Diferenciální a integrální počet funkcí jedné proměnné (skriptum VUT)
2. Eliaš J., Horváth J., Kajan J.: Zbierka úloh z vyššej matematiky I, II, III, IV (Alfa Bratislava, 1985)
3. Nedoma J.: Matematika I. Část třetí, Integrální počet funkcí jedné proměnné (skriptum VUT)
5. Thomas G.B., Finney R.L.: Calculus and Analytic Geometry (7th edition)
6. Jan Franců: Matematika I (skripta VUT)
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B-ENE-P full-time study --- no specialisation -- Cr,Ex 9 Compulsory 1 1 W
B-MET-P full-time study --- no specialisation -- Cr,Ex 9 Compulsory 1 1 W
B-PDS-P full-time study --- no specialisation -- Cr,Ex 9 Compulsory 1 1 W
B-PRP-P full-time study --- no specialisation -- Cr,Ex 9 Compulsory 1 1 W
B-STR-P full-time study STR Engineering -- Cr,Ex 9 Compulsory 1 1 W
B-VTE-P full-time study --- no specialisation -- Cr,Ex 9 Compulsory 1 1 W
B-ZSI-P full-time study STI Fundamentals of Mechanical Engineering -- Cr,Ex 9 Compulsory 1 1 W
B-ZSI-P full-time study MTI Materials Engineering -- Cr,Ex 9 Compulsory 1 1 W
C-AKR-P full-time study CZS -- Cr,Ex 9 Elective 1 1 W
B-STG full-time study --- no specialisation -- Cr,Ex 9 Compulsory 1 1 W