Course detail

# Mathematics I

FSI-1M-AAcad. year: 2017/2018

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.

Department

Learning outcomes of the course unit

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.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Thomas G. B.: Calculus (Addison Wesley, 2003)

Thomas G.B., Finney R.L.: Calculus and Analytic Geometry (7th edition)

Rektorys K. a spol.: Přehled užité matematiky I,II (SNTL, 1988)

Nedoma J.: Matematika I., Část první. Algebra a geometrie (skriptum VUT)

Nedoma J.: Matematika I. Část druhá. Diferenciální a integrální počet funkcí jedné proměnné (skriptum VUT)

Sneall D.B., Hosack J.M.: Calculus, An Integrated Approach

Děmidovič B. P.: Sbírka úloh a cvičení z matematické analýzy

Eliaš J., Horváth J., Kajan J.: Zbierka úloh z vyššej matematiky I, II, III, IV (Alfa Bratislava, 1985)

Nedoma J.: Matematika I. Část třetí, Integrální počet funkcí jedné proměnné (skriptum VUT)

Rektorys K. a spol.: Přehled užité matematiky I,II (SNTL, 1988)

Mezník I. - Karásek J. - Miklíček J.: Matematika I pro strojní fakulty (SNTL 1992)

Howard, A.A.: Elementary Linear Algebra, Wiley 2002

Satunino, L.S., Hille, E., Etgen, J.G.: Calculus: One and Several Variables, Wiley 2002

Planned learning activities and teaching methods

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: There are two written tests (each at most 12 points) within the seminars and a semestral work from the computer support (at most 1 point).

The student can obtain at most 25 points alltogether within the seminars. Condition for the course-unit credit: to obtain at least 6 points from each written test. Students, who do not fulfil conditions for the course-unit credit, can repeat the written test during first two weeks of examination time.

FORM OF EXAMINATIONS:

The exam has an obligatory written (and possible oral) part.

In a 120-minute written test, students have to solve the following four problems:

Problem 1: Functions and their properties, equations, inequalities (at most 10 points)

Problem 2: In linear algebra, analytic geometry (at most 20 points)

Problem 3: In differential calculus (at most 25 points)

Problem 4: In integral calculus (at most 20 points)

Above problems can also contain a theoretical question.

RULES FOR CLASSIFICATION

1. Results from seminars (at most 25 points)

2. Results from the written examination (at most 75 points)

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

Language of instruction

English

Work placements

Not applicable.

Aims

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.

Specification of controlled education, way of implementation and compensation for absences

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

52 hours, optionally

Teacher / Lecturer

Syllabus

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

44 hours, compulsory

Teacher / Lecturer

Syllabus

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

8 hours, compulsory

Teacher / Lecturer

Syllabus

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.

eLearning

**eLearning:** currently opened course