Title:

# Mathematical Analysis

Code:IMA
Ac.Year:2015/2016
Sem:Summer
Curriculums:
ProgrammeField/
Specialization
YearDuty
IT-BC-3BIT1stCompulsory
Language of Instruction:Czech
Credits:5
Completion:examination (written)
Type of
instruction:
Hour/semLecturesSeminar
Exercises
Laboratory
Exercises
Computer
Exercises
Other
Hours:26100106
ExamsTestsExercisesLaboratoriesOther
Points:60025015
Guarantor:Krupková Vlasta, RNDr., CSc. (DMAT)
Lecturer:Hliněná Dana, doc. RNDr., Ph.D. (DMAT)
Krupková Vlasta, RNDr., CSc. (DMAT)
Instructor:Fuchs Petr, RNDr., Ph.D. (DMAT)
Faculty:Faculty of Electrical Engineering and Communication BUT
Department:Department of Mathematics FEEC BUT
Prerequisites:
 Discrete Mathematics (IDA), DMAT
Follow-ups:
 Numerical Methods and Probability (INM), 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:
Limit and continuity, derivative of a function. Partial derivatives. Basic differentiation rules. Elementary functions. 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.
Knowledge and skills required for the course:
Secondary shool mathematics and the kowledge from Discrete Mathematics course.
Learning outcomes and competencies:
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. Function of one variable, limit, continuity.
2. Differential calculus of functions of one variable I: derivative, differential, Taylor theorem.
3. Differential calculus of functions of one variable II: maximum, minimum, behaviour of the function.
4. Integral calculus of functions of one variable I: indefinite integral, basic methods of integration.
5. Integral calculus of functions of one variable II: definite Riemann integral and its application.
6. Infinite number and power series.
7. Taylor series.
8. Functions of two and three variables, geometry and mappings in three-dimensional space.
9. Differential calculus of functions of more variables I: directional and partial derivatives, Taylor theorem.
10. Differential calculus of functions of more variables II: funcional extrema, absolute and bound extrema.
11. Integral calculus of functions of more variables I: two and three-dimensional integrals.
12. Integral calculus of functions of more variables II: method of substitution in two and three-dimensional integrals.
Syllabus of numerical exercises:
The class work is prepared in accordance with the lecture.
Syllabus of computer exercises:
Trained tasks are prepared to follow and complete study matter from lectures and seminar practice.
Syllabus - others, projects and individual work of students:

• Limit, continuity and derivative of a function. Partial derivative. Derivative of a composite function.
• 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.
• Primitive function and undefinite integral. Basic methods of integration. Definite one-dimensional and multidimensional integral.
• Methods for solution of definite integrals (Newton-Leibnitz formula, Fubini theorem).
• Indefinite number series. Convergence of series. Sequences and series of functions. Taylor theorem. Power series.
Fundamental literature:

• Edwards, C.H., Penney, D.E., Calculus with Analytic Geometry, Prentice Hall, 1993.
• Fong, Y., Wang, Y., Calculus, Springer, 2000.
• Ross, K.A., Elementary analysis: The Theory of Calculus, Springer, 2000.
• Small, D.B., Hosack, J.M., Calculus (An Integrated Approach), Mc Graw-Hill Publ. Comp., 1990.
• Thomas, G.B., Finney, R.L., Calculus and Analytic Geometry, Addison-Wesley Publ. Comp., 1994.
• Zill, D.G., A First Course in Differential Equations, PWS-Kent Publ. Comp., 1992.
Study literature:

• Brabec B., Hrůza,B., Matematická analýza II, SNTL, Praha, 1986.
• Švarc, S. a kol., Matematická analýza I, PC DIR, Brno, 1997.
• Krupková, V. Matematická analýza pro FIT, electronical textbook, 2007.
Progress assessment:
Practice tasks: 25 points.
Homeworks: 15 points.
Semestral examination: 60 points.