Course detail
Mathematics I/1
FAST-MA06Acad. year: 2014/2015
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
Czech
Number of ECTS credits
5
Mode of study
Not applicable.
Guarantor
Department
Institute of Mathematics and Descriptive Geometry (MAT)
Learning outcomes of the course unit
After the course, students should understand the basics of calculus of functions of one variable and the basic interpretations of some of the concepts. They should master differentiating and sketching the graph of a function.
They should know how to perform operations with matrices, elementary transactions, calculate determinants, solve systems of algebraic equations using Gauss elimination method.
They should know how to perform operations with matrices, elementary transactions, calculate determinants, solve systems of algebraic equations using Gauss elimination method.
Prerequisites
Basic secondary-school mathematics. Graphs of essential elementary functions (powers and roots, quadratic function, direct and indirect proportion, absolute value, trigonometric functions) and the basic properties of such functions. Simplification of algebraic expressions. Definition of a geometric vector and basics of 3D analytic geometry (parametric equations of a straight line, dot product of vectors and its applications to metric and positional problems). Identifying the the types and basic properties of conics, sketching graphs of conics).
Co-requisites
Not applicable.
Planned learning activities and teaching methods
Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations - lectures, seminars.
Assesment methods and criteria linked to learning outcomes
Successful completion of the scheduled tests and submission of solutions to problems assigned by the teacher for home work. Unless properly excused, students must attend all the workshops.
Course curriculum
1. Real function of one real variable, explicit and parametric definition of a function. Composite function and inverse to a function.
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.
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.
Work placements
Not applicable.
Aims
After the course, students should understand the basics of calculus of functions of one variable and the basic interpretations of some of the concepts. They should master differentiating and sketching the graph of a function.
They should know how to perform operations with matrices, elementary transactions, calculate determinants, solve systems of algebraic equations using Gauss elimination method.
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
Extent and forms are specified by guarantor’s regulation updated for every academic year.
Recommended optional programme components
Not applicable.
Prerequisites and corequisites
Not applicable.
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)
STEIN, S. K.: Calculus and analytic geometry. New York, 1989. (EN)
Recommended reading
DANĚČEK, J. a kolektiv: Sbírka příkladů z matematiky I. Akademické nakladatelství CERM, s.r.o. Brno, 2003. (CS)
DLOUHÝ, O. - TRYHUK, V.: Matematika I, Diferenciální počet funkce jedné reálné proměnné. CERM, s.r.o. Brno, 2009. (CS)
Kolektiv: Elektronické studijní opory předmětu BA06. FAST VUT, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)
Kolektiv: Sbírka příkladů z matematiky I. Akademické nakladatelství CERM Brno, 2003. (CS)
Kolektiv: Studijní opory předmětu BA01, moduly M07,08,09,10. FAST VUT, Brno, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)
NOVOTNÝ, J.: Základy lineární algebry. CERM, 2004. (CS)
DLOUHÝ, O. - TRYHUK, V.: Matematika I, Diferenciální počet funkce jedné reálné proměnné. CERM, s.r.o. Brno, 2009. (CS)
Kolektiv: Elektronické studijní opory předmětu BA06. FAST VUT, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)
Kolektiv: Sbírka příkladů z matematiky I. Akademické nakladatelství CERM Brno, 2003. (CS)
Kolektiv: Studijní opory předmětu BA01, moduly M07,08,09,10. FAST VUT, Brno, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)
NOVOTNÝ, J.: Základy lineární algebry. CERM, 2004. (CS)
Classification of course in study plans
Type of course unit
Lecture
26 hod., optionally
Teacher / Lecturer
Syllabus
1. Real function of one real variable, explicit and parametric definition of a function. Composite function and inverse to a function.
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.
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
26 hod., compulsory
Teacher / Lecturer
Syllabus
1. Absolute value of a function. Quadratic equations in complex field. Conics. Graphs of selected elementary functions. Basic properties of functions.
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.
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.