Course detail
Mathematics I
FCH-BC_MAT1Acad. year: 2023/2024
Basics of calculus of functions of one real variable. Basics of linear algebra.
Language of instruction
Number of ECTS credits
Mode of study
Guarantor
Entry knowledge
Rules for evaluation and completion of the course
The exam is written. Students do not use any electronic devies during the exam, however they can use written preparation in the range of two A4 sheets.
The compulsory attendance at seminars. In the exercises are included 2 tests (each at most 12 points). In total the exercises can receive a maximum of 24 points. A student has to obtain at least 6 points from each test.
Aims
The knowledge and skills will appear on the following fields
1. Students will manage successfully a work with matrices.
2. Students will be endowed with the knowledge of elementary functions and their properties. Students are expected to manage the concept of a limit and derivative and comprehend their meaning.They master their computation applying basic rules including the L´Hospital rule. Students will also be able to investgate the course of a function of one variable.
3. Students will be endowed with the knowledge of the indefinite and definite integral including the improper integral. They learn the basic methods of integral computations and be aquaitanced with the basic applications.
4. Students obtain the ability of solving simple tasks of the physical character and tasks occuring in the advanced courses.
Study aids
Prerequisites and corequisites
Basic literature
Thomas G. B.: Calculus, Addison Wesley (EN)
Thomas G.B., Finney R.L.: Calculus and Analytic Geometry, Addison Wesley (EN)
Recommended reading
Rektorys K. a spol.: Přehled užité matematiky I,II ,SNTL (CS)
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Classification of course in study plans
- Programme BKCP_CHCHTE Bachelor's 1 year of study, winter semester, compulsory
- Programme BKCP_AAEFCH Bachelor's 1 year of study, winter semester, compulsory
- Programme BPCP_AAEFCH Bachelor's 1 year of study, winter semester, compulsory
- Programme BPCP_ECHBM Bachelor's 1 year of study, winter semester, compulsory
- Programme BKCP_ECHBM Bachelor's 1 year of study, winter semester, compulsory
- Programme BPCP_CHCHTE Bachelor's 1 year of study, winter semester, compulsory
- Programme BKCP_CHTM Bachelor's 1 year of study, winter semester, compulsory
- Programme BPCP_CHTM Bachelor's 1 year of study, winter semester, compulsory
- Programme BPCP_CHTOZP Bachelor's 1 year of study, winter semester, compulsory
- Programme BKCP_CHTOZP Bachelor's 1 year of study, winter semester, compulsory
- Programme BPCP_CHTPO Bachelor's
specialization CHPL , 1 year of study, winter semester, compulsory
specialization PCH , 1 year of study, winter semester, compulsory
specialization BT , 1 year of study, winter semester, compulsory - Programme BKCP_CHTPO Bachelor's
specialization PCH , 1 year of study, winter semester, compulsory
specialization BT , 1 year of study, winter semester, compulsory
specialization CHPL , 1 year of study, winter semester, compulsory - Programme BPCP_CHMA Bachelor's 1 year of study, winter semester, compulsory
- Programme BPCP_CHTN Bachelor's 1 year of study, winter semester, compulsory
Type of course unit
Lecture
Teacher / Lecturer
Syllabus
Sem. The recapitulation of selected themes of secondary schools. Introduction to matrices.
2. Linear independence, rank of matrices, determinants.
Sem. Matrix operations. Elementary row operations, rank.
3. Systems of linear equations. Frobenius theorem, Gaussian elimination, Cramer's rule.
Sem. Determinants to order 3. Systems of linear equations.
4. Geometry in E2 and E3: inner, outer and vector products. Lines and planes.
Sem. Systems of linear equations. Application of the products.
5. Geometry in E2 and E3: the role of angles and distances. Conics.
Sem. Parametric and general equations of lines and planes. Classification of conic sections and quadrics without mixed member (filling into a square).
6. Functions of one real variable. Basic features, graph. Inverse function.
Sem. TEST 1: 1) Matrix multiplication 2) Determinant 3) The system of linear equations 4) The geometry of lines and planes 5) Classification of conic sections and quadrics
7. Elementary functions: polynomials, rational functions, power functions, exponential and logarithmic functions, trigonometric functions.
Sem. Domains of elementary functions.
8. Derivative, geometric and physical meaning, calculation, chemical applications.
Sem.. Calculations of derivatives.
9. Calculations limits using of derivative (L'Hospital's rule). Taylor polynomial.
Sem. Taylor polynomial (briefly). Calculations of limits.
10. The determination of functions properties (with emphasis on the extremes).
Sem. Functions properties.
11. The method of least squares.
Cv. The method of least squares.
12. Interpolation polynomials and splines.
Cv. TEST 2: 1) Domain of functions 2) Derivative 3) [six-point example] Graphing functions
13. Summarizing lecture, discussion.
Cv. Interpolation polynomials and splines. Evaluation of seminars, granting credits.
Exercise
Teacher / Lecturer
Syllabus
Cv. 1. Stručné opakování vybraných témat středoškolské látky. Úvod do matic.
Cv. 2. Operace s maticemi. Elementární úpravy, hodnost.
Cv. 3. Determinant. Determinant stačí do řádu 3. Soustavy lineárních rovnic.
Cv. 4. Soustavy lineárních rovnic – dokončení. Aplikace součinů.
Cv. 5. Parametrické a obecné rovnice přímek a rovin. Klasifikace kuželoseček a kvadrik bez smíšeného členu (doplňování na čtverec).
Cv. 6. TEST 1: 1) Násobení matic 2) Determinant 3) Soustava lineárních rovnic 4) Geometrie přímek a rovin 5) Klasifikace kuželoseček a kvadrik
Cv. 7. Definiční obory elementárních funkcí.
Cv. 8. Výpočty derivací.
Cv. 9. Taylorův polynom (stručně). Výpočty limit.
Cv. 10. Průběh funkce.
Cv. 11. Metoda nejmenších čtverců.
Cv. 12. TEST 2: 1) Definiční obor 2) Derivace 3) [šestibodový příklad] Průběh funkce
Cv. 13. Interpolační polynomy a splajny. Vyhodnocení cvičení, udělení zápočtů.
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