Essentials of High School Math Course 2

Course guarantor:

Lukáš Heriban

Lecturer:

Lukáš Heriban

Tutor:

Lukáš Heriban

Supervisor:

Department of Mathematics

Synopsis:

The course introduces the fundamental areas of mathematics essential for university studies and practical applications. It covers sets, logic, proofs, functions, derivatives, integrals, analytic geometry, combinatorics, and probability, with emphasis on understanding principles, rigor, and problem solving.

Requirements:

To receive credit, you need to earn 50% of the points from homework and tests.

Syllabus of lectures:

1. Sets: basic set operations, set identities

2. Mathematical logic: logical connectives, truth tables, tautologies

3. Proofs: direct proof, proof by contradiction, proof by contrapositive, mathematical induction

4. Inequalities: rules for working with inequalities, basic mathematical inequalities

5. Functions: domain, range, basic properties of functions

6. Derivatives: intuitive concept of a derivative, basic formulas for computation, derivative of a monomial

7. Integrals: introduction to the indefinite integral and Newtons integral, formulas for computation

8. Trigonometric functions: geometric definition, proofs of basic trigonometric identities

9. Logarithm and exponential function: definition of logarithm and exponential, properties of these functions

10. Complex numbers: introduction to complex numbers, De Moivres theorem

11. Vectors and matrices: standard scalar product, vector product, working with matrices

12. Analytic geometry: equations of geometric objects, distances, angles

13. Combinatorics and probability: permutations, variations and combinations, basic formulas, random variable

Syllabus of tutorials:

Practical demonstrations of problems from lectures.

Study Objective:

-Introduce students to the fundamental areas of mathematics forming the basis for further studies.

-Develop students logical and abstract thinking skills.

-Teach students to use various methods of mathematical proof.

-Strengthen skills in working with functions, inequalities, derivatives, and integrals.

-Master the basic concepts of analytic geometry, combinatorics, and probability.

-Promote the ability to apply acquired knowledge in solving both theoretical and practical problems.

Study materials:

Recommended references:

[1] J. Polák: Středoškolská matematika v úlohách I, Prometheus, 2007

[2] J. Polák: Středoškolská matematika v úlohách II, 2008

[3] I. Bušek: Řešené maturitní úlohy z matematiky, Prometheus 2004

[4] J. Polák: Přehled středoškolské matematiky, Prometheus 2010

[5] M. Gould, E. Hurst: Bridging the Gap to University Mathematics, Springer, 2009

Note:

Time-table for winter semester 2025/2026:

Time-table is not available yet

Time-table for summer semester 2025/2026:

Time-table is not available yet

The course is a part of the following study plans:

• Physical Engineering - Computational physics (elective course)
• Quantum Technologies (elective course)
• Nuclear and Particle Physics (elective course)
• Mathematical Engineering - Mathematical Physics (elective course)
• Physical Engineering - Plasma Physics and Thermonuclear Fusion (elective course)
• Mathematical Engineering - Mathematical Modelling (elective course)
• Mathematical Engineering - Mathematical Informatics (elective course)
• Physical Engineering - Solid State Engineering (elective course)
• Physical Engineering - Laser Technology and Photonics (elective course)