# Course syllabus MAT125C - Mathematics (VŠPP - SS 2019/2020)

Czech          English

Course code:
MAT125C
Course title in Czech:
Mathematics
Course title in English:
Mathematics
Mode of completion and number of credits: Exam (5 credits)
Mode of delivery:
full-time, 2/2 (hours of lectures per week / hours of seminars per week)
part-time, 8/0 (lectures per period / seminars per period)
Language of instruction:
Czech
Course supervisor:
RNDr. Michal Bejček, Ph.D.
Name of lecturer: RNDr. Michal Bejček, Ph.D. (supervisor)
Mgr. Dana Sobotová (examiner, instructor, lecturer)
Prerequisites:
none
Annotation:
The course covers key concepts of logic and differential and integral calculus as prerequisites to understanding mathematical methods in the subsequent courses. Its aim is to further develop the student's ability to reason logically and precisely, and to make decisions based on evidence, while increasing the knowledge base to be built on in other education and in real-life situations.

Course contents:
 1 Sets and numbers. Statement and its negation, composite statements. Quantified statements and their negation. (allowance 0/0) 2 Quadratic inequalities. Absolute value inequalities. Function. Linear and quadratic function. (allowance 0/0) 3 Exponential function. Monotonic function. One-to-one function. Inverse function. Logarithmic function. (allowance 0/0) 4 Composite function. Continuity and limit of a function. Limit computation. Indeterminate forms. (allowance 0/0) 5 Derivative of a function, its geometric interpretation. Tangent equation at a given point. Derivatives of elementary functions. (allowance 0/0) 6 Derivative of addition, subtraction, and division. Derivative of a composite function. (allowance 0/0) 7 Second derivative and its interpretation. Asymptotes. L'Hospital's rule. (allowance 0/0) 8 Course of a function, application of differential calculus (allowance 0/0) 9 Primitive functions. (allowance 0/0) 10 Methods of integration. Integration by substitution. Integration by parts. (allowance 0/0) 11 Definite integral, its geometric interpretation. (allowance 0/0) 12 Arithmetic sequence. Geometric sequence. Their application in financial mathematics. (allowance 0/0) 13 Limit of a sequence. Infinite geometric series. Series with non-negative terms. (allowance 0/0)

Learning outcomes:
Expert knowledge; you should be able to:

- solve a system of linear equations using matrices and determinants.

- solve a system of linear inequalities graphically and explain its use in linear programming.

- describe the way of finding local extrema (also constraint extrema) of a function of two variables.

Expert skills; you should be able to:

- use the rank of a matrix to determine vector dependence or independence.

- solve a system of multiple equations with multiple unknowns using matrices

- find local extrema (also constraint extrema) of a function of two variables.

General competences; you should be able to:

- understand and appreciate the importance of Mathematics in everyday life.

- show how to implement the acquired aspects of Mathematics in your own field of study.

Input knowledge:
No preliminary courses precede this course in the curriculum.
This course assumes basic skills in high school Mathematics.

Learning activities and teaching methods:
Attend tutorials. Study relevant lecture notes (or even basic or advanced literature) before attending a tutorial. Complete assignments on your own. If needed, request an appointment with the lecturer. Mastering basic definitions, mathematical terms, and algorithms is essential.

Rámcové podmínky zápočtu:
To earn course credit, students are required to complete five assignments (pass mark: 60%) and to pass the final test (pass mark: 60%).

Rámcové podmínky zkoušky:
The examination is oral and covers the following: 1. Basic mathematical statements, composite and quantified statements, their negation.
2. Elementary functions: linear, absolute value linear, quadratic, logaritmic, exponential, goniometric. Graphs, the domain and the range of a function.
3. Function characteristics: continuity at a given point, one-sided and two-sided. Function: increasing, decreasing, limited, even, odd, simple, inverse.
4. The limit of a function one-sided, two sided, proper and improper, limit at infinity. The limit of a sum, the product and quotient of two functions.
5. The derivative of a function, definition, geometric interpretation. The derivative of a sum, product, and quotient. The derivative of a composite function.
6. Using derivatives in examining the course of a function: tangent, monotony intervals, stationary points, local extrema. L'Hospital's rule.
7. Integrals, primitive function, basic integrals, integrating by parts, by substitution. Definite integrals and their application.
8. Sequences, their definition. Arithmetic and geometric sequences, basic formulas and their deduction. The limit of a sequence, the limit of addition, multiplication and division. Converging geometric series.
9. Sequences in financial Mathematics, simple and compound interest.
10. Infinite series. The sum of an infinite geometric series.