Mathematical Analysis 1
- The course cannot be taken simultaneously with:
- Elements of Calculus (BIK-ZMA)
- Garant předmětu:
- Department of Applied Mathematics
We begin the course by introducing students to the set of real numbers and its properties, and we note its differences with the set of machine numbers. Then we study real sequences and real functions of a real variable. We gradually introduce the notions of limits of sequences and functions, continuous functions, and derivatives of functions. This theoretical foundation is then applied to root-finding problems (iterative method of bisection and Newton’s method), construction of cubic interpolation (spline), and formulation and solution of simple optimization problems (i.e., the issue of finding extrema of functions). The course is closed with the Landau’s asymptotic notation and methods of mathematical description of complexity of algorithms.
Knowledge of high school mathematics, basics of mathematical logic (BIE-DML.21), and BIE-LA1.21.
- Syllabus of lectures:
1. Extended real number line: rational and irrational numbers, completeness axiom, neighborhood, infinity. Relation to machine numbers.
2. Basic properties of functions and sequences. Elementary functions (polynomials, trigonometric functions, exponential, and logarithm).
3. Limit of a sequence and limit of a function: definition, meaning, and illustrations.
4. Computation of limits: algebraic properties of limits, squeeze theorem, examples.
5. The continuity of a function, continuity of elementary functions, implications for root finding (the bisection method as an example of iterative numerical method).
6. The derivative of a function, geometric meaning, linearity of differentiation, product and quotient rule. Derivative of inverse function. Differentiation of elementary functions.
7. Newton’s method for root finding.
8. Cubic interpolation (splines). L’Hospital’s rule.
9. Lagrange’s mean value theorem, implications for monotony and convexity/concavity of functions.
10. Local extrema of functions. Sufficient conditions for their existence.
11. Analytical graph plotting: examples. The notion of an optimization problem.
12. Landau’s asymptotic notation.
13. Mathematical description of the complexity of algorithms.
- Syllabus of tutorials:
This is an outline of proseminars and subsequent exercises.
1. Functions and sequences, basic properties.
2. Elementary functions (polynomials, trigonometric functions, exponential and logarithm).
3. Limits of sequences and functions.
4. Continuity of functions.
5. Derivative of a function.
6. Analytical graph sketching (monotonicity, local exrtrema, asymptotes, etc.).
- Study Objective:
- Study materials:
1. Oberguggenberger M., Ostermann A. : Analysis for Computer Scientists. Springer, 2018. ISBN 978-0-85729-445-6.
2. Stewart J. : Calculus (8th Edition). Cengage Learning, 2015. ISBN 978-1285740621.
3. Bittinger M.L., Ellenbogen D.J., Surgent S.A. : Calculus and Its Applications (11th Edition). Pearson, 2015. ISBN 978-0321979391.
4. Kopáček J. : Matematická analýza nejen pro fyziky I. Matfyzpress, 2016. ISBN 978-80-7378-353-4.
- Further information:
- No time-table has been prepared for this course
- The course is a part of the following study plans:
- Bachelor specialization Information Security, part-time, in Czech, 2021 (compulsory course in the program)
- Bachelor specialization Software Engineering, part-time, in Czech, 2021 (compulsory course in the program)
- Bachelor specialization Computer Networks and Internet, part-time, in Czech, 2021 (compulsory course in the program)
- Bachelor specialization Computer Systems and Virtualization, part-time, in Czech, 2021 (compulsory course in the program)
- Bachelor program, unspecified specialization, part-time, in Czech, 2021 (compulsory course in the program)
- Bachelor branch Security and Information Technology, in Czech, part-time, 2015 commuters (compulsory course in the program)