Selected Chapters from Mathematics for Optometrists

The course 17VRKM can be graded only after the course F7PBOVKM has been successfully completed.

Garant předmětu:

Jana Urzová

Lecturer:

Jana Urzová

Tutor:

Jana Urzová

Supervisor:

Department of Natural Sciences

Synopsis:

The course summarizes and systematizes the secondary school curriculum and builds on them. Students will get acquainted with the basics of linear algebra, differential and integral calculus of real functions of one real variable in applications. Emphasis is placed on the requirements of further study - solving equations of various types and their systems, modifications of trigonometric expressions and geometry of conic sections and the mutual position of the sphere and the plane.

Requirements:

Credit requirements:

1. Written demonstration of acquired knowledge in two tests during the semester (7th and 14th week), for successful passing of the credit it is necessary to obtain at least 50% of points from each of the two tests. The tests will each contain four numerical examples (scoring 0-5 points) of the material covered in lectures and seminars (maximum 20 + 20 points). The first test contains topics: solving equations, editing expressions, properties of functions. The second test contains topics: differential and integral calculus, analytical geometry. The number of points exceeding the minimum number to meet the credit (10 points) is included in the exam (ie 0-10 points from each test, a maximum of 20 points in total).

Test requirements:

1. The exam is only written, consisting of eight numerical examples supplemented by theoretical sub-questions and covers the subject matter in its entirety, the evaluation of individual examples is 0-10 points, the maximum is 80 points. Points from credit tests (points obtained above the minimum) are added to the point gain of the exam test and subsequent grading is standard.

Syllabus of lectures:

1. Numerical fields and their properties, basic concepts. Solution of linear and quadratic equations in the field of real and complex numbers.

2. Mathematical expressions with powers and roots and methods of their modifications. Expression unknown from the formula. Solution of linear and quadratic inequalities.

3. Systems of linear equations, Gaussian elimination method. System of quadratic and linear equations.

4. Matrices and determinants. Matrix operations, determinant calculation.

5. Functions and their properties. Types of functions: linear, quadratic, power, exponential.

6. Inverse function, logarithmic function, logarithm.

7. Goniometric functions, trigonometric expressions and their modifications.

8. Derivation of a function, properties, meaning, investigation of the course of a function using derivatives.

9. Limits and applications of derivatives - l´Hospital's rule. Problems about extremes,

10. Fundamentals of integral calculus - indefinite integral, integration methods.

11. A definite integral and its geometric applications.

12. Basics of solving differential equations.

13. Fundamentals of analytical geometry in E2, straight line and its input by equation and parametrically, conic sections, mutual position of conic section and straight line.

14. Fundamentals of analytical geometry in E3, plane and straight line and their input in E3, quadrics, mutual position of conic section and straight line, mutual position of sphere and plane, sphere and straight line.

Syllabus of tutorials:

1. Interval operations. Solution of linear and quadratic equations in the field of real and complex numbers.

2. Edit expressions with powers and roots. Expression unknown from the formula. Solution of inequalities.

3. Systems of linear equations, Gaussian elimination method. System of quadratic and linear equations.

4. Matrices and determinants. Matrix operations, determinant calculation.

5. Functions: linear, quadratic, power, exponential. Inverse function - logarithmic function. Logarithm operations.

6. Goniometric functions, trigonometric expressions and their modifications.

7. Derivation of a function, investigation of the course of a function using derivatives.

8. Application of derivatives - l´Hospital's rule. Problems about extremes,

9. Indefinite integral, integration methods.

10. A definite integral and its geometric applications.

11. Basics of solving differential equations.

12. Fundamentals of analytical geometry in E2, straight line and conic sections, their mutual position.

13. Fundamentals of analytical geometry in E3, plane, line and quadrics, their mutual position.

14. Test

Study Objective:

The course is to acquaint students of optometry with basic knowledge and numerical procedures in the fields of mathematical analysis and linear algebra to the extent necessary to master further studies, especially the physics of optics.

Study materials:

Required:

[1] Massachuttes Institute of Technology. http://ocw.mit.edu/courses/physics/

[2] The Princeton companion to mathematics. Editor Timothy GOWERS, editor June BARROW-GREEN, editor Imre LEADER. Princeton: Princeton University Press, c2008. ISBN 0691118809.

Recommended:

[1] RUDIN, Walter. Principles of mathematical analysis. 3d ed. New York: McGraw-Hill, 1976. ISBN 007054235x.

Note:

Time-table for winter semester 2023/2024:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon	roomKL:B-430 Urzová J. 14:00–15:50 (lecture parallel1 parallel nr.2) Kladno FBMI UčebnaroomKL:C-1 Urzová J. 16:00–17:50 (lecture parallel1) Kladno FBMI C1, Velký sál
Tue	roomKL:B-430 Urzová J. 08:00–09:50 (lecture parallel1 parallel nr.3) Kladno FBMI UčebnaroomKL:B-430 Urzová J. 10:00–11:50 (lecture parallel1 parallel nr.1) Kladno FBMI Učebna
Wed
Thu
Fri

Time-table for summer semester 2023/2024:

Time-table is not available yet

The course is a part of the following study plans:

• Optics and Optometry (compulsory course)