Course Details

# Mathematics II

Academic Year 2023/24

GA04 course is part of 1 study plan

B-P-C-GK / GI Summer Semester 1st year

Primitive function, indefinite integrals, properties of indefinite integrals, overview of basic indefinite integrals, methods of integration. Integrating rational functions, trigonometric functions, selected types of irrational functions.

Newton integral, its properties and calculation. Defining the Riemann integral. Applications of the definite integral in geometry and physics.

Real two- and more-functions, composite functions. Limit of a function, continuous two- and more functions. Theorems on continuous functions. Partial derivatives of composite functions, higher-order partial derivatives. Transformations of differential expressions. Total differential of a function. Higher-order total differentials. Taylor polynomials of two-functions. Local maxima and minima of two-functions. One-functions defined implicitly. A two-function defined implicitly. Global maxima and minima. Finding global maxima and minima using realtive maxima and minima. Scalar field and its levels. Directional derivative of a scalar function, gradient. Tangent and normal plane to a 3D Curve. Tangent plane and normal to a surface defined implicitly.

Newton integral, its properties and calculation. Defining the Riemann integral. Applications of the definite integral in geometry and physics.

Real two- and more-functions, composite functions. Limit of a function, continuous two- and more functions. Theorems on continuous functions. Partial derivatives of composite functions, higher-order partial derivatives. Transformations of differential expressions. Total differential of a function. Higher-order total differentials. Taylor polynomials of two-functions. Local maxima and minima of two-functions. One-functions defined implicitly. A two-function defined implicitly. Global maxima and minima. Finding global maxima and minima using realtive maxima and minima. Scalar field and its levels. Directional derivative of a scalar function, gradient. Tangent and normal plane to a 3D Curve. Tangent plane and normal to a surface defined implicitly.

Course Guarantor

Institute

Objective

After the course, the students should understand the principles of integration of some more sophisticated elementary functions, some of the applications of teh definite integral.

They should acquaint themselves with the basics of calculus of two- and more-functions, including partial derivatives, implicit functions, understand the geometric interpretation of the total differential. Learn how to find local and glogal minima and maxima of two-functions, calculate directional derivatives.

They should acquaint themselves with the basics of calculus of two- and more-functions, including partial derivatives, implicit functions, understand the geometric interpretation of the total differential. Learn how to find local and glogal minima and maxima of two-functions, calculate directional derivatives.

Knowledge

Students will known methods of solving undefinite and definite integrals and will be able to use methods successfully to important applied problems. Except this students will understand basic calculus of functions of several variables and its application to analysis of behavior of functions in three-dimenesional space.

Syllabus

1. Notion of a primitive function. Properties of an indefinite integral. Integration methods for indefinite integral.

2. Integrating a rational function. Integrating a trigonometric function.

3. Integrating selected types of irrational functions. Newton integral, its properties and calculation. Riemann integral.

4. Applying calculus in geomery and physics.

5. Real functions two and more variables, composite functions. Limit and continuity of functions two and more variables. Theorems on continuous functions.

6. Partial derivatives, partial derivatives of a composite function, higher-order partial derivatives. Transformations of differential expressions.

7. The total differential of a function. Higher-order total differentials. Taylor polynomial of a two-function. Local maxima and minima of two-functions.

8. Functions defined implicitly. Two-functions defined implicitly.

9. Global maxima and minima. Simple problems in global maxima and minima using relative maxima and minima. Scalar field and its levels. Directional derivative of a scalar function, gradient.

10. Tangent and normal plane to a 3D curve. Tanget plane and normal to a surface defined explicitly.

2. Integrating a rational function. Integrating a trigonometric function.

3. Integrating selected types of irrational functions. Newton integral, its properties and calculation. Riemann integral.

4. Applying calculus in geomery and physics.

5. Real functions two and more variables, composite functions. Limit and continuity of functions two and more variables. Theorems on continuous functions.

6. Partial derivatives, partial derivatives of a composite function, higher-order partial derivatives. Transformations of differential expressions.

7. The total differential of a function. Higher-order total differentials. Taylor polynomial of a two-function. Local maxima and minima of two-functions.

8. Functions defined implicitly. Two-functions defined implicitly.

9. Global maxima and minima. Simple problems in global maxima and minima using relative maxima and minima. Scalar field and its levels. Directional derivative of a scalar function, gradient.

10. Tangent and normal plane to a 3D curve. Tanget plane and normal to a surface defined explicitly.

Prerequisites

Basics of the theory of one-functions(limit, continuous functions, graphs of functions, derivative, sketching the graph of a function).

Formulas used to calculate indefinite and definite integrals, and the basic integration methods.

Formulas used to calculate indefinite and definite integrals, and the basic integration methods.

Language of instruction

Czech

Credits

5 credits

Semester

summer

Forms and criteria of assessment

course-unit credit and examination

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

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

Offered to foreign students

Not to offer

Course on BUT site

Lecture

13 weeks, 2 hours/week, elective

Syllabus

1. Notion of a primitive function. Properties of an indefinite integral. Integration methods for indefinite integral.
2. Integrating a rational function. Integrating a trigonometric function.
3. Integrating selected types of irrational functions. Newton integral, its properties and calculation. Riemann integral.
4. Applying calculus in geomery and physics.
5. Real functions two and more variables, composite functions. Limit and continuity of functions two and more variables. Theorems on continuous functions.
6. Partial derivatives, partial derivatives of a composite function, higher-order partial derivatives. Transformations of differential expressions.
7. The total differential of a function. Higher-order total differentials. Taylor polynomial of a two-function. Local maxima and minima of two-functions.
8. Functions defined implicitly. Two-functions defined implicitly.
9. Global maxima and minima. Simple problems in global maxima and minima using relative maxima and minima. Scalar field and its levels. Directional derivative of a scalar function, gradient.
10. Tangent and normal plane to a 3D curve. Tanget plane and normal to a surface defined explicitly.

Exercise

13 weeks, 2 hours/week, compulsory

Syllabus

1. Integrating a rational function.
2. Integrating a trigonometric function.
3. Integrating selected types of irrational functions. Newton integral, its properties and calculation. Riemann integral.
4. Geometric and physical applications of calculus.
5. Real functions of two and more variables, composite function. Limit and continuity.
6. Seminar test I. Partial derivative, partial derivative of a composite function, higher-order partial derivatives. Transformations of differential expressions.
7. The total differential of a function. Higher-order total differentials. Taylor polynomial of functions of two variables. Local extreme of functions of two variables.
8. Functions defined implicitly.
9. Seminar test II. Global extreme. Scalar field and its levels. Directional derivative of a scalar function, gradient.
10. Tangent and normal plane to a 3D curve. Tangent plane and normal to a surface defined explicitly.