Course detail

Mathematics I/2

FAST-BA07Acad. year: 2012/2013

Antiderivative, indefinite integral, its properties and methods of calculation. Newton integral, its properties and calculation. Definition of Riemann integral. Applications of integral calculus in geometry and physics - area of a plane figure, length of a curve, volume and surface of a rotational body, static moments and the centre of gravity.
Functions in two and more variables. Limit and continuity, partial derivatives, implicit function, total differential, Taylor expansion, local minima and maxima, relative maxima and minima, maximum and minimum values of a function; directional derivative, gradient. Tangent to a 3-D curve, Tangent plane and normal to a surface.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. RNDr. Václav Tryhuk, CSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Students will get an understanding of the basics of integral calculus of functions of one variable, know how to integrate elementary functions and apply these (length of a curve, volume and surface area of a rotational body, static momentums and centre of gravity).
Students will acquaint themselves with the basic concepts of the calculus of functions of two and more variables. They will be able to calculate partial derivatives of functions of several variables. They will get an understanding of the total differential of a function and its geometrical meaning. They will also learn how to find local and global minima and maxima of two-functions. They will get acquainted with directional derivatives of functions of several variables and their calculation.

Prerequisites

Basics of linear algebra, vector calculus and analytical geometry in E3. Basic notions of the theory of functions of one variable, derivatives of elementary functions.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations - lectures, seminars.

Assesment methods and criteria linked to learning outcomes

Successful completion of the scheduled tests and submission of solutions to problems assigned by the teacher for home work. Unless properly excused, students must attend all the workshops.

Course curriculum

1. Antiderivative, indefinite integral and their properties. Integration by parts and using substitutions.
2. Integrating rational functions, formulas needed to integrate trigonometric functions. Integrating trigonometric functions.
3. Integrating trigonometric functions. Integrating irrational functions. Newton and Riemann integral and their properties.
4. Integrating irrational functions. Newton and Riemann integral and their properties.
5. Integration methods for definite integrals. Geometric applications of the definite integral.
6. Engineering applications of the definite integral.
7. Real function of several variables. Basic notions, composite function. Limits of sequences, limit and continuity of two-functions.
8. Partial derivative, partial derivative of a composite function, higher-order partial derivatives. Total differential of a function, higher-order total differentials.
9. Taylor polynomial of a function of two variables. Local maxima and minima of functions of two variables.
10. Function in one variable defined implicitly. Function of two variables defined implicitly.
11. Some theorems of continuous functions, relative and global maxima and minima.
12. Tangent to a 3-D curve, Tangent plane and normal to a surface.
13. Scalar field, directional derivative, gradient.

Work placements

Not applicable.

Aims

Understand and know how to integrate elementary functions, understand some applications of the definite integral (length of a curve, volume and surface area of a rotational body, static momentums and centre of gravity). Students should acquaint themselves with the basic concepts of calculus of two and more-functions. They should be able to calculate partial derivatives, acquaint themselves with the concept of an implicit function. To understand the geometric interpretation of the total differential of a function. Learn how to find local and global minima and maxima of two-functions. To learn about the directional derivative and its calculation.

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

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Lang, S.: Calculus of several variables. Springer Verlag, New York, 1988. (EN)
Stein, S. K.: Calculus and analytic geometry. New York, 1989. (EN)

Recommended reading

BUDÍNSKÝ, B. - CHARVÁT, J,: Matematika I. SNTL Praha, 1987. (CS)
Čermáková, Hana a kol.: Sbírka příkladů z matematiky II. CERM Brno, 1994. (CS)
J. Daněček a kol.: Sbírka příkladů z matematiky I. CERM Brno, 2006. (CS)
J. Daněček, O. Dlouhý, O. Přibyl: Neurčitý integrál. CERM Brno, 2007. (CS)
J. Daněček, O. Dlouhý, O. Přibyl: Určitý integrál. CERM Brno, 2007. (CS)
J. Slaběňáková a kolektiv: Sbírka příkladů z matematiky II. 2008. (CS)
Kolektiv: Elektronické studijní opory. FAST VUT, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)
TRYHUK,V.- DLOUHÝ, O.: Diferenciální počet II. CERM, Brno, 2004. (CS)

Classification of course in study plans

  • Programme B-P-E-SI Bachelor's

    branch VS , 1 year of study, summer semester, compulsory

  • Programme B-P-C-SI Bachelor's

    branch VS , 1 year of study, summer semester, compulsory

  • Programme B-P-C-ST Bachelor's

    branch VS , 1 year of study, summer semester, compulsory

  • Programme B-K-C-SI Bachelor's

    branch VS , 1 year of study, summer semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Václav Tryhuk, CSc.

Syllabus

1. Antiderivative, indefinite integral and their properties. Integration by parts and using substitutions.
2. Integrating rational functions, formulas needed to integrate trigonometric functions. Integrating trigonometric functions.
3. Integrating trigonometric functions. Integrating irrational functions. Newton and Riemann integral and their properties.
4. Integrating irrational functions. Newton and Riemann integral and their properties.
5. Integration methods for definite integrals. Geometric applications of the definite integral.
6. Engineering applications of the definite integral.
7. Real function of several variables. Basic notions, composite function. Limits of sequences, limit and continuity of two-functions.
8. Partial derivative, partial derivative of a composite function, higher-order partial derivatives. Total differential of a function, higher-order total differentials.
9. Taylor polynomial of a function of two variables. Local maxima and minima of functions of two variables.
10. Function in one variable defined implicitly. Function of two variables defined implicitly.
11. Some theorems of continuous functions, relative and global maxima and minima.
12. Tangent to a 3-D curve, Tangent plane and normal to a surface.
13. Scalar field, directional derivative, gradient.

Exercise

26 hod., compulsory

Teacher / Lecturer

doc. RNDr. Václav Tryhuk, CSc.

Syllabus

1. Antiderivative, indefinite integral and their properties.
2. Integration by parts and using substitutions.
3. Integrating rational functions.
4. Integrating trigonometric functions.
5. Integrating irrational functions.
6. Integration methods for definite integrals.
7. Geometric applications of the definite integral. Test 1.
8. Engineering applications of the definite integral.
9. Partial derivative, partial derivative of a composite function, higher-order partial derivatives.
10. Total differential of a function, higher-order total differentials. Taylor polynomial of a function of two variables.
11. Local maxima and minima of functions of two variables. Test 2.
12. Functions defined implicitly. Global maxima and minima.
13. Tangent plane and normal to a surface. Seminar evaluation.