# 230-0201/01 – Mathematics (BcM1)

 Gurantor department Department of Mathematics Credits 5 Subject guarantor RNDr. Petr Volný, Ph.D. Subject version guarantor RNDr. Petr Volný, Ph.D. Study level undergraduate or graduate Requirement Compulsory Year 1 Semester winter Study language Czech Year of introduction 2018/2019 Year of cancellation Intended for the faculties FAST Intended for study types Bachelor
Instruction secured by
TUZ006 RNDr. Michaela Bobková, Ph.D. DUB02 RNDr. Viktor Dubovský, Ph.D. KRE40 doc. RNDr. Pavel Kreml, CSc.  VOL06 RNDr. Petr Volný, Ph.D.  Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 3+3

### Subject aims expressed by acquired skills and competences

Mathematics is an essential part of education on technical universities. It should be considered rather the method in the study of technical courses than a goal. Thus the goal of mathematics is train logical reasoning than mere list of mathematical notions, algorithms and methods. Students should learn how to analyse problems, distinguish between important and unimportant, suggest a method of solution, verify each step of a method, generalise achieved results, analyse correctness of achieved results with respect to given conditions, apply these methods while solving technical problems, understand that mathematical methods and theoretical advancements outreach the field mathematics.

### Teaching methods

Lectures
Individual consultations
Tutorials
Other activities

### Summary

Mathematics I is connected with secondary school education. It is divided in three parts, differential calculus of functions of one real variable, linear algebra and analytic geometry in the three dimensional Euclidean space E3. The aim of the first chapter is to handle the concept of a function and its properties, a limit of functions, a derivative of functions and its application. The second chapter emphasizes the systems of linear equations and the methods of their solution. The third chapter introduces the basics of vector calculus and basic linear objects in three dimensional space.

### Compulsory literature:

Doležalová, J.: Mathematics I. VŠB – TUO, Ostrava 2005, ISBN 80-248-0796-3, http://mdg.vsb.cz/portal/en/Mathematics1.pdf. Trench, W.F.: Introduction to real analysis, Free Edition 1.06, January 2011, ISBN 0-13-045786-8.

### Recommended literature:

Harshbarger, Ronald; Reynolds, James: Calculus with Applications, D.C. Heath and Company 1990, ISBN 0-669-21145-1.

### Way of continuous check of knowledge in the course of semester

Passing the course, requirements Course-credit -participation on tutorials is obligatory, 30% of absence can be apologized, -elaborate programs, -pass the written tests, Point classification: 5-20 points. Exam Practical part of an exam is classified by 0 - 60 points. Practical part is successful if student obtains at least 25 points. Theoretical part of the exam is classified by 0 - 20 points. Theoretical part is successful if student obtains at least 5 points. Point quantification ECTS grade 100 - 91 A 90 - 81 B 80 - 71 C 70 - 61 D 60 - 51 E 50 - 0 F Point quantification National grading scheme 100 - 86 excellent 85 - 66 very good 65 - 51 satisfactory 50 - 0 failed List of theoretical questions 1. Definition of real functions of one real variable 2. Monotonic functions 3. Bounded functions 4. Even, odd and periodic functions 5. Composite functions 6. One-to-one functions, inverse functions 7. Trigonometric functions, D(f), H(f), graph 8. Inverse trigonometric functions, D(f), H(f), graph 9. Limit of a function 10. One-side limit 11. Limit theorems 12. Continuity of functions 13. Definition of derivation of function at a point 14. Geometrical meaning of derivation of function at a point 15. Derivation rules 16. Derivation of composite functions 17. Derivation of function f(x)^g(x) 18. Derivation of parametric and implicit functions 19. Differential of functions 20. Taylor polynomial 21. l´Hospital rule 22. Extrema of functions 23. Concavity, convexity, inflection points 24. Asymptotes 25. Matrices 26. Matrices, algebraic operations 27. Rank of a matrix 28. Determinant of a matrix 29. Inverse 30. System of linear equations 31. Frobenius theorem 32. Cramer´s rule 33. Gaussian elimination algorithm 34. Scalar product of vectors 35. Triple product of vectors 36. Cross product of vectors 37. Equation of a line in a 3-dimensional space 38. Equation of a plane in a 3-dimensional space 39. Relative position of two lines 40. Relative position of a line and plane 41. Relative position of two planes 42. Distance of a point from a line 43. Distance of a point from a plane 44. Distance of a plane from a plane 45. Angle between lines 46. Angle between a line and a plane 47. Transversal of two skew lines 48. Common perpendicular of two skew lines

### E-learning

http://www.studopory.vsb.cz http://mdg.vsb.cz

### Další požadavky na studenta

There are no other requirements.

### Prerequisities

Subject has no prerequisities.

### Co-requisities

Subject has no co-requisities.

### Subject syllabus:

Syllabus of lecture 1. Real functions of one real variable. Definition, graph. Bounded function, monotonic functions, even, odd and periodic functions. One-to-one functions, inverse and composite functions. 2. Elementary functions (including inverse trigonometric functions). 3. Limit of a function, infinite limit of a function. Limit at an improper point. Continuous and discontinuous functions. 4. Differential calculus of functions of one real variable. Derivative of a function, its geometrical and physical meaning. Derivative rules. 5. Derivative of elementary functions. 6. Differential of a function. Derivative of higher orders. l’Hospital rule. 7. Relation between derivative and monotonicity, convexity and concavity of a function. 8. Extrema of a function. Asymptotes. Plot graph of a function. 9. Linear algebra. Matrices. Matrix operations. Rank of a matrix. Inverse. 10. Determinants, properties of a determinant. 11. Solution of systems of linear equations. Frobenius theorem. Cramer’s rule. Gaussian elimination algorithm. 12. Analytic geometry. Euclidean space. Scalar, cross and triple product of vectors, properties. 13. Equation of a plane, line in E3. Relative position problems. 14. Metric or distance problems. Syllabus of tutorial 1. Domain of a real function of one real variable. 2. Bounded function, monotonic functions, even, odd and periodic functions. 3. One-to-one functions, inverse and composite functions. Elementary functions. 4. Inverse trigonometric functions. Limit of functions. 5. Derivative and differential of functions. 6. l’Hospital rule. Monotonic functions, extrema of functions. 7. 1st test (properties of functions, limits). Concave up function, concave down function, inflection point. 8. Asymptotes. Course of a function. 9. 2nd test (derivative of a function). Matrix operations. 10. Elementary row operations, rank of a matrix, inverse. 11. Determinants. 12. Solution of systems of linear equations. Gaussian elimination algorithm. 13. 3rd test (linear algebra). Analytic geometry. 14. Reserve.

### Conditions for subject completion

Conditions for completion are defined only for particular subject version and form of study

### Occurrence in study plans

Academic yearProgrammeField of studySpec.FormStudy language Tut. centreYearWSType of duty 2019/2020 (B3607) Civil Engineering P Czech Ostrava 1 Compulsory study plan
2018/2019 (B3607) Civil Engineering P Czech Ostrava 1 Compulsory study plan

### Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner 