Mathematics I (FSI-1M-A)

Academic year 2020/2021
Supervisor: prof. RNDr. Miroslav Doupovec, CSc., dr. h. c.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: English
Aims of the course unit:
The course aims to acquaint the students with the basics of linear algebra, vector calculus, analytic geometry and differential and integral calculus of functions of one variable. This will enable them attend engineering courses and deal with engineering problems. Another goal of the course is to develop the students' logical thinking.
Learning outcomes and competences:
Students will be made familiar with linear algebra, analytic geometry and differential and integral calculus of functions of one variable. They will be able to solve systems of linear equations and apply the methods of linear algebra and differential and integral calculus when dealing with engineering tasks. After completing the course students will be prepared for further study of technical disciplines.
Prerequisites:
Students are expected to have basic knowledge of secondary school mathematics.
Course contents:
Basic concepts of the set theory and mathematical logic.
Linear algebra: matrices, determinants, systems of linear equations.
Vector calculus and analytic geometry.
Differential calculus of functions of one variable: basic elementary functions, limits, derivative and its applications.
Integral calculus of functions of one variable: primitive function, proper integral and its applications.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:
COURSE-UNIT CREDIT REQUIREMENTS: The course includes seminars and exercises in the computer lab. There are two written tests within the seminars. Students may achieve max 12 points in each of these two tests, i.e. 24 points altogether. The course-unit credit is conditional on obtaining at least 6 points in each written test. If the minimum number of points is not achieved, students may repeat the test during the first two weeks of the examination period.

FORM OF EXAMINATIONS:
The exam has a written part (at most 75 points) and an oral part (at most 25 points)

WRITTEN PART OF EXAMINATION (at most 75 points)
In a 120-minute written test, students have to solve the following four problems:
Problem 1: Functions and their properties: domains, graphs (at most 10 points)
Problem 2: In linear algebra, analytic geometry (at most 20 points)
Problem 3: In differential calculus (at most 20 points)
Problem 4: In integral calculus (at most 25 points)
The above problems can also contain a theoretical question.

ORAL PART OF EXAMINATION (max 25 points)
• Discussion based on the written test: students have to explain how they solved each problem. Should the student fail to explain it sufficiently, the test results will not be accepted and will be classified by 0 points.
• Possible theoretic question.
• Possible simple problem to be solved straight away.
• The results achieved in the written tests in seminars may be taken into account within the oral examination.



FINAL CLASSIFICATION:
0-49 points: F
50-59 points: E
60-69 points: D
70-79 points: C
80-89 points: B
90-100 points: A
Controlled participation in lessons:
Attendance at lectures is recommended, attendance at seminars is required. The lessons are planned on the basis of a weekly schedule. The way of compensation for an absence is fully at the discretion of the teacher.
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.: Calculus (Addison Wesley, 2003)
1. Thomas G.B., Finney R.L.: Calculus and Analytic Geometry (7th edition)
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
Literature - recommended:
1. Rektorys K. a spol.: Přehled užité matematiky I,II (SNTL, 1988)
1. Nedoma J.: Matematika I., Část první. Algebra a geometrie (skriptum VUT)
2. Eliaš J., Horváth J., Kajan J.: Zbierka úloh z vyššej matematiky I, II, III, IV (Alfa Bratislava, 1985)
2. Nedoma J.: Matematika I. Část druhá. Diferenciální a integrální počet funkcí jedné proměnné (skriptum VUT)
2. Děmidovič B. P.: Sbírka úloh a cvičení z matematické analýzy
3. Nedoma J.: Matematika I. Část třetí, Integrální počet funkcí jedné proměnné (skriptum VUT)
4. Mezník I. - Karásek J. - Miklíček J.: Matematika I pro strojní fakulty (SNTL 1992)
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B-STI-A full-time study --- no specialisation -- Cr,Ex 9 Compulsory 1 1 W