Course detail
Mathematics I/1
FAST-BA06Acad. year: 2012/2013
Real function of one real variable. Sequences, limit of a function, continuous functions. Derivative of a function, its geometric and physical applications, basic theorems on derivatives, higher-order derivatives, differential of a function, Taylor expansion of a function, sketching the graph of a function.
Linear algebra (basics of the matrix calculus, rank of a matrix, Gauss elimination method, inverse to a matrix, determinants and their applications). Eigenvalues and eigenvectors of a matrix. Basics of vectors, vector spaces. Linear spaces. Analytic geometry (dot, cross and mixed product of vectors, affine and metric problems for linear bodies in E3).
Language of instruction
Number of ECTS credits
Mode of study
Guarantor
Department
Learning outcomes of the course unit
They should know how to perform operations with matrices, elementary transactions, calculate determinants, solve systems of algebraic equations using Gauss elimination method.
Prerequisites
Co-requisites
Planned learning activities and teaching methods
Assesment methods and criteria linked to learning outcomes
Course curriculum
Sem. Absolute value of a function. Quadratic equations in complex field. Conics. (50 min.) - Graphs of selected elementary functions. Basic properties of functions. (50 min.)
2. Some elementary functions, inverse trigonometric functions. Hyperbolic functions. (50 min.) – Polynomial and the basic properties of its roots, decomposition of a polynomial in the field of real numbers. (50 min.)
Sem. Composite function and inverse to a function (inverse trigonometric functions, logarithmic functions). (50 min.) Polynomial. (50 min.)
3. Rational functions. (50 min.) – Sequence and its limit. (50 min.)
Sem. Initial test. (50 min.) – Polynomial, sign of a polynomial. (50 min.)
4. Limit of a function, continuous functions, basic theorems. (50 min.) – Derivative of a function, its geometric and physical applications, differentiating rules. (50 min.)
Sem. Rational function, sign of a rational function, decomposition into partial fractions. (2 times 50 min.)
5. Derivatives of composite and inverse functions. (50 min.) – Differential of a function. Rolle and Lagrange theorem. (50 min.)
Sem. Limit of a function. Derivative of a function (basic calculation) and its geometric applications, basic formulas and rules for differentiating. (2 times 50 min.)
6. Higher-order derivatives, higher-order differentials. (50 min.) – Taylor theorem. (50 min.)
Sem. Derivative of an inverse function. Basic differentiation formulas and rules, simplification of the results of differentiation. (2 times 50 min.)
7. L`Hospital's rule. Asymptotes of the graph of a function. Sketching the graph of a function. (2 times 50 min.)
Sem. Higher-order derivatives. Taylor theorem. (50 min.) L` Hospital's rule. (50 min.)
8. Basics of matrix calculus, elementary transformations of a matrix, rank of a matrix. (50 min.) – Solutions to systems of linear algebraic equations by Gauss elimination method. (50 min.)
Sem. Test I. (20 min.) Asymptotes of the graph of a function. Sketching the graph of a function. (80 min.)
9. Second-order determinants. Higher-order determinants calculated by Laplace expansion. (50 min.) – Rules for calculating with determinants. Cramer's rule of solving a system of linear algebraic equations. (50 min.)
Sem. Basic operations with matrices. Elementary transformations of a matrix, rank of a matrix, solutions to systems of linear algebraic equations by Gauss elimination method. (2 times 50 min.)
10. Inverse to a matrix. Jordan's method of calculation. Matrix equations. (50 min.) - Real linear space, base and dimension of a linear space. Linear spaces of arithmetic and geometric vectors. (50 min.)
Sem. Calculating determinants using Laplace expansion and rules for calculating with determinants. (50 min.) Calculating the inverse to a matrix for A(2,2), A(3,3) matric
Work placements
Aims
They should know how to perform operations with matrices, elementary transactions, calculate determinants, solve systems of algebraic equations using Gauss elimination method.
Specification of controlled education, way of implementation and compensation for absences
Recommended optional programme components
Prerequisites and corequisites
Basic literature
LARSON, R.- HOSTETLER, R.P.- EDWARDS, B.H.: Calculus (with Analytic Geometry). Brooks Cole, 2005. (EN)
STEIN, S. K: Calculus and analytic geometry. New York, 1989. (EN)
Recommended reading
DLOUHÝ, O., TRYHUK, V.: Diferenciální počet I. CERM, 2009. (CS)
kolektiv: Elektronické studijní opory. FAST VUT, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)
NOVOTNÝ, J.: Základy lineární algebry. CERM, 2004. (CS)
TRYHUK, V. - DLOUHÝ, O.: Modul GA01_M01 studijních opor předmětu GA01. FAST VUT, Brno, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)
Classification of course in study plans
- Programme B-P-E-SI Bachelor's
branch VS , 1 year of study, winter semester, compulsory
- Programme B-P-C-SI Bachelor's
branch VS , 1 year of study, winter semester, compulsory
- Programme B-P-C-ST Bachelor's
branch VS , 1 year of study, winter semester, compulsory
- Programme B-K-C-SI Bachelor's
branch VS , 1 year of study, winter semester, compulsory
Type of course unit
Lecture
Teacher / Lecturer
Syllabus
2. Some elementary functions, inverse trigonometric functions. Hyperbolic functions. Polynomial and the basic properties of its roots, decomposition of a polynomial in the field of real numbers.
3. Rational functions. Sequence and its limit.
4. Limit of a function, continuous functions, basic theorems. Derivative of a function, its geometric and physical applications, differentiating rules.
5. Derivatives of composite and inverse functions. Differential of a function. Rolle and Lagrange theorem.
6. Higher-order derivatives, higher-order differentials. Taylor theorem.
7. L`Hospital's rule. Asymptotes of the graph of a function. Sketching the graph of a function.
8. Basics of matrix calculus, elementary transformations of a matrix, rank of a matrix. Solutions to systems of linear algebraic equations by Gauss elimination method.
9. Second-order determinants. Higher-order determinants calculated by Laplace expansion. Rules for calculating with determinants. Cramer's rule of solving a system of linear algebraic equations.
10. Inverse to a matrix. Jordan's method of calculation. Matrix equations. Real linear space, base and dimension of a linear space. Linear spaces of arithmetic and geometric vectors.
11. Eigenvalues and eigenvectors of a matrix. Coordinates of a vector. Dot and cross product of vectors, calculating with coordinates.
12. Mixed product of vectors. Plane in E3. Straight line in E3, positional problems.
13. Metric problems in E3. Surfaces in E3.
Exercise
Teacher / Lecturer
Syllabus
2. Composite function and inverse to a function (inverse trigonometric functions, logarithmic functions). Polynomial.
3. Initial test. Polynomial, sign of a polynomial.
4. Rational function, sign of a rational function, decomposition into partial fractions.
5. Limit of a function. Derivative of a function (basic calculation) and its geometric applications, basic formulas and rules for differentiating.
6. Derivative of an inverse function. Basic differentiation formulas and rules, simplification of the results of differentiation.
7. Higher-order derivatives. Taylor theorem. L` Hospital's rule.
8. Test I. Asymptotes of the graph of a function. Sketching the graph of a function.
9. Basic operations with matrices. Elementary transformations of a matrix, rank of a matrix, solutions to systems of linear algebraic equations by Gauss elimination method.
10. Calculating determinants using Laplace expansion and rules for calculating with determinants.Calculating the inverse to a matrix for A(2,2), A(3,3) matrices using Jordan's method.
11. Test II. Matrix equations. Eigenvalues and eigenvectors of a matrix.
12. Using dot and cross products in solving problems in 3D analytic geometry.
13. Mixed product. Seminar evaluation.