Course Details

Mathematics 2

BAA002 course is part of 5 study plans

Bc. full-t. program BPC-EVB compulsory Summer Semester 1st year 5 credits

Bc. full-t. program BPC-SI > spVS compulsory Summer Semester 1st year 5 credits

Bc. full-t. program BPC-MI compulsory Summer Semester 1st year 5 credits

Bc. full-t. program BPA-SI compulsory Summer Semester 1st year 5 credits

Bc. combi. program BKC-SI compulsory Summer Semester 1st year 5 credits

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.

Course Guarantor

doc. Ing. Vladislav Kozák, CSc.

Institute

Institute of Mathematics and Descriptive Geometry

Learning outcomes

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 (area of a plane figure, 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 3D. Basic notions of the theory of functions of one variable, derivatives of elementary functions.

Corequisites

Not required.

Planned educational 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.

Forms and criteria of assessment

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.

Objective

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 instruction, the form of instruction, and the form of compensation of the absences

Vymezení kontrolované výuky a způsob jejího provádění stanoví každoročně aktualizovaná vyhláška garanta předmětu.

Lecture

2 hours/week, 13 weeks, elective

Syllabus of lectures

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.

Practice

2 hours/week, 13 weeks, compulsory

Syllabus of practice

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.