Title:

# Mathematical Analysis

Code:IMA
Ac.Year:2002/2003
Term:Summer
Study plans:
ProgramBranchYearDuty
IT-BC-3BIT1stCompulsory
Language:Czech
Credits:5
Completion:accreditation+exam (written)
Type of
instruction:
Hour/semLecturesSem. ExercisesLab. exercisesComp. exercisesOther
Hours:26130130
ExaminationTestsExercisesLaboratoriesOther
Points:00000
Guarantee:Krupková Vlasta, RNDr., CSc., DMATH
Lecturer:Krupková Vlasta, RNDr., CSc., DMATH
Faculty:Faculty of Electrical Engineering and Communication BUT
Department:Department of Mathematics FEEC BUT
Prerequisites:
 Discrete Mathematics (IDM-IT), DMAT
Follow-ups:
 Numerical Methods and Probability (INM-IT), DMAT

Learning objectives:
The main goal of the calculus course is to explain the basic principles and methods of higher mathematics that are necessary for the study of computer science. The practical aspects of applications of these methods and their use in solving concrete problems (including the application of contemporary mathematical software in the laboratories) are emphasized.
Description:
The limit and the continuity of a function. The derivative. Partial derivatives. Basic differentiation rules. The chain rule. The elementary functions. Applications of derivatives. Extrema for functions (of one and of several variables). Indefinite integral. Techniques of integration. The Riemann (definite)integral. Multiple integrals. Applications of integrals. Infinite sequences and infinite series. Taylor polynomials. Fourier series.
Learning outcomes and competences:
The ability of orientation in the basic problems of higher mathematics and the ability to apply the basic methods. Solving problems in the areas cited in the annotation above by using basic rules. Solving these problems by using modern mathematical software.
Syllabus of lectures:
1. Limit and continuity of a function of one and of several variables.
2. Derivative of a function. Partial derivative.
3. Derivative rules. Derivative of composite function.
4. Differential of functions of one and several variables.
5. Mean value theorem. L'Hospital's rule.
6. Behaviour of continuous and differentiable function.
7. Extrema of function of one and several variables, implicit functions (informatively).
8. Primitive function and indefinite integral. Basic integration methods (informatively).
9. Definite integrals one-dimensional and multidimensional.
10. Mathod for computation of definite integrals (Newton-Leibnitz formula, Fubini theorem).
11. Infinite number series. Convergence of series (informatively).
12. Sequences and series of functions. Taylor theorem. Power series.
13. Fourier series.
Syllabus of numerical exercises:
1. Limit and continuity of a function of one and of several variables.
2. Derivative of a function. Partial derivative.
3. Derivative rules. Derivative of composite function.
4. Differential of functions of one and several variables.
5. Mean value theorem. L'Hospital's rule.
6. Behaviour of continuous and differentiable function.
7. Extrema of function of one and several variables, implicit functions (informatively).
8. Primitive function and indefinite integral. Basic integration methods (informatively).
9. Definite integrals one-dimensional and multidimensional.
10. Mathod for computation of definite integrals (Newton-Leibnitz formula, Fubini theorem).
11. Infinite number series. Convergence of series (informatively).
12. Sequences and series of functions. Taylor theorem. Power series.
13. Fourier series.
Syllabus of computer exercises:
1. Graphs of elementary functions, exponential functions, logarithmic functin. Inverse function. Parametric functions.
2. Three-dimensional graphs, dependence of a graph of a function on parameters, illustration of types of second-order surfaces.
3. Definion of derivative, behaviour of a function. Methods of integration, notion of definite integral.
4. Extrema of functions. Characteristic points of surfaces.
5. Definite integral one-dimensional and multidimensional. Computational methods for definite integrals.
6. Taylor theorem. Power series. Decomposition of functions into Taylor and Maclaurin series. Fourier series.
Syllabus - others, projects and individual work of students:
1. Limit, continuity and derivative of a function. Partial derivative. Derivative of a composite function.
2. Differential of function of one and several variables. L'Hospital's rule. Behaviour of continuous and differentiable function. Extrema of functions of one and several variables.
3. Primitive function and undefinite integral. Basic methods of integration. Definite one-dimensional and multidimensional integral.
4. Methods for solution of definite integrals (Newton-Leibnitz formula, Fubini theorem).
5. Indefinite number series. Convergence of series. Sequences and series of functions. Taylor theorem. Power series.
6. Fourier series.
Fundamental literature:
1. Brabec B., Hrůza,B., Matematická analýza II, SNTL, Praha, 1986.
2. Diblík, J., Baštinec, J., Matematika III, ES VUT, Brno, 1991.
3. Edwards, C.H., Penney, D.E., Calculus with Analytic Geometry, Prentice Hall, 1993.
4. Fong, Y., Wang, Y., Calculus, Springer, 2000.
5. Ross, K.A., Elementary analysis: The Theory of Calculus, Springer, 2000.
6. Small, D.B., Hosack, J.M., Calculus (An Integrated Approach), Mc Graw-Hill Publ. Comp., 1990.
7. Švarc, S. a kol., Matematická analýza I, PC DIR, Brno, 1997.
8. Thomas, G.B., Finney, R.L., Calculus and Analytic Geometry, Addison-Wesley Publ. Comp., 1994.
9. Zill, D.G., A First Course in Differential Equations, PWS-Kent Publ. Comp., 1992.
Study literature:
1. Brabec B., Hrůza,B., Matematická analýza II, SNTL, Praha, 1986.
2. Diblík, J., Baštinec, J., Matematika III, ES VUT, Brno, 1991.
3. Edwards, C.H., Penney, D.E., Calculus with Analytic Geometry, Prentice Hall, 1993.
4. Fong, Y., Wang, Y., Calculus, Springer, 2000.
5. Ross, K.A., Elementary analysis: The Theory of Calculus, Springer, 2000.
6. Small, D.B., Hosack, J.M., Calculus (An Integrated Approach), Mc Graw-Hill Publ. Comp., 1990.
7. Švarc, S. a kol., Matematická analýza I, PC DIR, Brno, 1997.
8. Thomas, G.B., Finney, R.L., Calculus and Analytic Geometry, Addison-Wesley Publ. Comp., 1994.
9. Zill, D.G., A First Course in Differential Equations, PWS-Kent Publ. Comp., 1992.
Progress assessment:
Submission of projects (homework) in ruled terms.