Course detail
Mathematics 2
FAST-BAA002Acad. year: 2024/2025
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
Department
Institute of Mathematics and Descriptive Geometry (MAT)
Offered to foreign students
Of all faculties
Entry knowledge
Basics of linear algebra, vector calculus and analytical geometry in 3D. Basic notions of the theory of functions of one variable, derivatives of elementary functions.
Rules for evaluation and completion of the course
Extent and forms are specified by guarantor’s regulation updated for every academic year.
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.
Students will achieve the subject's objectives. They 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.
Students will achieve the subject's objectives. They 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.
Study aids
Not applicable.
Prerequisites and corequisites
Not applicable.
Basic literature
Eliáš, J., Horváth, J., Kajan, J., Śulka, R., Zbierka úloh z vzššej matamatiky 2, Alfa Bratislava 1979. (SK)
Jirásek, F., Čipera, S., Vacek, M., Sbírka řešených příkladů z matematiky I, SNTL Praha 1986. (CS)
Jirásek, F., Čipera, S., Vacek, M., Sbírka řešených příkladů z matematiky I, SNTL Praha 1986. (CS)
Recommended reading
Rektorys, K., Přehled užité matematiky, Prometheus, Praha 2000. (CS)
Classification of course in study plans
- Programme BPC-SI Bachelor's
specialization VS , 1 year of study, summer semester, compulsory
- Programme BPC-MI Bachelor's 1 year of study, summer semester, compulsory
- Programme BPC-EVB Bachelor's 1 year of study, summer semester, compulsory
- Programme BKC-SI Bachelor's 1 year of study, summer semester, compulsory
- Programme BPA-SI Bachelor's 1 year of study, summer semester, compulsory
- Programme CZV1-AKR Lifelong learning
specialization PBC , 1 year of study, summer semester, compulsory
Type of course unit
Lecture
26 hod., optionally
Teacher / Lecturer
Syllabus
1. Antiderivative, indefinite integral and their properties. Integration by parts and using substitutions.
2. Integrating rational functions.
3. Integrating trigonometric functions. Integrating irrational functions.
4. Newton and Riemann integral and their properties.
5. Integration methods for definite integrals. Applications of the definite integral.
6. Geometric and 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
RNDr. Karel Mikulášek, Ph.D.
RNDr. Jana Slaběňáková
prof. Ing. Jiří Vala, CSc.
Ing. Pavel Špaček, Dr.
RNDr. Veronika Chrastinová, Ph.D.
RNDr. Radko Odehnal
doc. Ing. Vladislav Kozák, CSc.
doc. Mgr. Irena Hinterleitner, Ph.D.
doc. Mgr. Ing. Miroslav Trcala, Ph.D.
Ing. Jan Holešovský, Ph.D.
Mgr. Darina Brothánková, Ph.D.
Ing. Matouš Cabalka
RNDr. Rudolf Schwarz, CSc.
Ing. Anna Derevianko, Ph.D.
RNDr. Jana Slaběňáková
prof. Ing. Jiří Vala, CSc.
Ing. Pavel Špaček, Dr.
RNDr. Veronika Chrastinová, Ph.D.
RNDr. Radko Odehnal
doc. Ing. Vladislav Kozák, CSc.
doc. Mgr. Irena Hinterleitner, Ph.D.
doc. Mgr. Ing. Miroslav Trcala, Ph.D.
Ing. Jan Holešovský, Ph.D.
Mgr. Darina Brothánková, Ph.D.
Ing. Matouš Cabalka
RNDr. Rudolf Schwarz, CSc.
Ing. Anna Derevianko, Ph.D.
Syllabus
1. Differentiating revision.
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. Geometric and 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.