Course detail

Mathematics

ÚSI-ESMATAcad. year: 2024/2025

Basic mathematical concepts. Differential calculus of one variable, limit, continuity, derivative of a function. Derivatives of higher orders, l´Hospital rule, behavior of a function. Integral calculus of fuctions of one variable, indefinite integral. Integration by parts, substitution methods. Definite integral and its applications. Introduction to descriptive statistics. Introduction to probability, conditional probability, dependence and independence of random events. Total probability rule and Bayes theorem. Discrete random variables (probability mass function, cumulative distribution function, mean and variance). Discrete probability distributions (binomial, hypergeometric, Poisson, uniform). Continuous random variables (probability density function, distrubution function, mean, variance, quantiles). Exponencial distribution. Normal distribution. Central limit theorem. Testing of statistical hypotheses (t-test).

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

doc. RNDr. Edita Kolářová, Ph.D.

Department

Department of Mathematics (UMAT)

Entry knowledge

Knowledge within the scope of standard secondary school requirements.

Rules for evaluation and completion of the course

  • Tests during the semester: 30 points.
  • Final exam: 70 points. 
  • Exam prerequisites: get at least 10 points during the semester.

Class attendance. If students are absent due to medical reasons, they should contact their lecturer. 

Aims

The main goal of the course is to explain the basic principles and methods of higher mathematics and probability and statistics that are necessary for the study at Brno University of Technology. The practical aspects of application of these methods and their use in solving concrete problems are emphasized.

After completing the course, students should be able to:
- estimate the domains and sketch the grafs of elementary functions;
- compute limits and asymptots for the functions of one variable, use the L’Hospital rule to evaluate limits;
- differentiate and find the tangent to the graph of a function, find the Taylor ploynomial of a function near a given point;
- sketch the graph of a function including extrema, points of inflection and asymptotes;
- integrate using technics of integration, such as substitution and integration by parts;
- evaluate a definite integral including integration by parts and by a substitution for the definite integral;
- compute the area of a region using the definite integral;
- compute the basic characteristics of statistical data (mean, median, modus, variance, standard deviation)
- choose the correct probability model (classical, discrete, geometrical probability) for a given problem and compute the probability of a given event
- compute the conditional probability of a random event A given an event B
- recognize and use the independence of random events when computing probabilities
- apply the total probability rule and the Bayes' theorem
- work with the cumulative distribution function, the probability mass function of a discrete random variable and the probability density function of a continuous random variable
- construct the probability mass functions (in simple cases)
- choose the appropriate type of probability distribution in model cases (binomial, hypergeometric, exponential, etc.) and work with this distribution
- compute mean, variance and standard deviation of a random variable and explain the meaning of these characteristics
- perform computations with a normally distributed random variable X: find probability that X is in a given range or find the quantile/s for a given probability
- perform a simple hypothesis testing (t-test)

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Fajmon, B., Hlavičková, I., Novák, M. Matematika 3. Elektronický text FEKT VUT, Brno, 2014
Krupková, V., Fuchs, P.,: Matematika 1. Elektronický text FEKT VUT, Brno, 2014

Recommended reading

Brabec B., Hrůza,B., Matematická analýza II, SNTL, Praha, 1986.
Casella, G., Berger, R. L.: Statistical Inference. Pacific Grove, CA: Duxbury Press, 2001.
Kolářová, E: Matematika 1 - Sbírka úloh, 2010
Likeš, J., Machek, J.: Počet pravděpodobnosti. Praha: SNTL - Nakladatelství technické literatury, 1981.
Neubauer, J., Sedlačík, M., Kříž, O.: Základy statistiky. Praha: Grada Publishing, 2012.
Švarc, S. a kol., Matematická analýza I, PC DIR, Brno, 1997.

Elearning

eLearning: currently opened course

Classification of course in study plans

  • Programme EID_P Master's 1 year of study, winter semester, compulsory
  • Programme REI_P Master's 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

Ing. Michal Fusek, Ph.D.

Syllabus

1. Basic mathematical concepts. High school math summary.
2. Concept of a function (basic properties and graphs). Operations with functions.
3. Differential calculus of one variable, limit, continuity.
4. Derivative of a function. Derivatives of higher orders.
5. l´Hospital rule. Behavior of a function, extremes.
6. Integral calculus of fuctions of one variable, indefinite integral. Integration by parts, substitution methods.
7. Definite integral and its applications.
8. Introduction to descriptive statistics.
9. Introduction to probability. Some probability models (classical, discrete, geometrical), conditional probability, dependence and independence of random events. Total probability rule and Bayes theorem.
10. Discrete random variables (probability mass function, cumulative distribution function, mean and variance). Discrete probability distributions (binomial, hypergeometric, Poisson, uniform).
11. Continuous random variables (probability density function, distrubution function, mean, variance). Exponencial distribution.
12. Normal distribution. Central limit theorem.
13. Testing of statistical hypotheses (t-test). 

Computer-assisted exercise

26 hod., compulsory

Teacher / Lecturer

Ing. Jan Novotný

Elearning

eLearning: currently opened course