230-0201/05 – Mathematics (BcM1)
| Gurantor department | Department of Mathematics | Credits | 4 |
| Subject guarantor | RNDr. Petr Volný, Ph.D. | Subject version guarantor | RNDr. Petr Volný, Ph.D. |
| Study level | undergraduate or graduate | | |
| | Study language | Czech |
| Year of introduction | 2018/2019 | Year of cancellation | |
| Intended for the faculties | FAST | Intended for study types | Bachelor |
Subject aims expressed by acquired skills and competences
The aim of the subject is to teach students not only basic mathematical knowledge, procedures and methods, but also to deepen their logical thinking. Students should learn to analyze a problem, distinguish the essential from the unessential, propose a solution procedure, check individual steps of the solution, generalize the conclusions, evaluate the correctness of the results with respect to the given conditions, apply tasks to solving technical problems, and understand that mathematical methods and thought processes are applicable in areas other than mathematics.
Teaching methods
Lectures
Individual consultations
Tutorials
Other activities
Summary
The subject is divided into three chapters - differential calculus of functions of one variable, linear algebra and analytical geometry in three-dimensional Euclidean space E3. The student will become familiar with the fundamental mathematical concept of differential calculus with a focus on civil engineering. In linear algebra, the student will learn to solve systems of linear equations by means of methods used in modeling building structures. In analytical geometry of the three-dimensional space, the student will learn mathematical description of elementary objects, point, line, plane and the students will acquire the skills to solve positional and metric problems.
Compulsory literature:
Recommended literature:
Additional study materials
Way of continuous check of knowledge in the course of semester
Passing the course, requirements
Course-credit
-participation on tutorials is obligatory, 20% 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 in the interval 100 - 91 90 - 81 80 - 71 70 - 61 60 - 51 50 - 0
ECTS grade A B C D E F
Point quantification in the interval 100 - 86 85 - 66 65 - 51 50 - 0
National grading scheme excellent very good satisfactory 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 and triple product of vectors
35. Cross product of vectors
36. Equation of a line in a 3-dimensional space
37. Equation of a plane in a 3-dimensional space
38. Relative position of two lines
39. Relative position of a line and plane
40. Relative position of two planes
41. Distance of a point from a line
42. Distance of a point from a plane
43. Angle between lines
44. Angle between a line and a plane
45. Transversal and common perpendicular of two skew lines
E-learning
http://www.studopory.vsb.cz
http://mdg.vsb.cz
(in Czech language)
Other requirements
At least 70% attendance at the exercises. Absence, up to a maximum of 30%, must be excused and the apology must be accepted by the teacher (the teacher decides to recognize the reason for the excuse).
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
Occurrence in study plans
Occurrence in special blocks
Assessment of instruction