## Mathematics

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Course Unit Code Number of ECTS Credits Allocated Type of Course Unit * Level of Course Unit * Year of Study * 230-0201/01 5 ECTS credits Compulsory First Cycle First Year Winter Semester Face-to-face Czech, English There are no prerequisites or co-requisites for this course unit KRE40 doc. RNDr. Pavel Kreml, CSc. VOL06 RNDr. Petr Volný, Ph.D. 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. 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. 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. 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. Burda, Pavel; Havelek, Radim; Hradecká, Radoslava; Kreml, Pavel: Matematika I, VŠB – TUO, Ostrava 2006, 80-248-1199-5 (CD-R). Burda, Pavel; Kreml, Pavel: Diferenciální počet funkcí jedné proměnné. Matematika IIa, VŠB – TUO, Ostrava 2004, ISBN 80-248-0634-7. Burda, Pavel; Havelek, Radim; Hradecká, Radoslava: Algebra a analytická geometrie, 2. vyd., VŠB – TUO, Ostrava 2005, 80-248-0966-4. http://www.studopory.vsb.cz/studijnimaterialy/MatematikaI/MI.html http://www.studopory.vsb.cz/studijnimaterialy/MatematikaI/m1.pdf http://mdg.vsb.cz/portal Harshbarger, Ronald; Reynolds, James: Calculus with Applications, D.C. Heath and Company 1990, ISBN 0-669-21145-1. Vrbenská, Helena; Bělohlávková, Jana;: Základy matematiky pro bakaláře I, 2. vyd., VŠB – TUO, Ostrava 2003, 80-248-0519-7, 978-80-248-0519-1. Láníček, Josef; Mičulka, Břetislav; Píšová, Dagmar; Restl, Čestmír; Řehák, Miroslav: Cvičení z matematiky I. VŠB – TUO, Ostrava 1999, 80-7078-973-5. Dobrovská, Věra; Mičulka, Břetislav; Šarmanová, Jana; Žižka, Jan: Cvičení z matematiky II, 9. vyd., VŠB – TUO, Ostrava 1997, 80-7078-987-5. Lectures, Individual consultations, Tutorials, Other activities Credit and Examination Credit and Examination 100 (100) 51 Credit Credit 20 5 Examination Examination 80 (80) 30 Písemná zkouška Written examination 60 25 Ústní zkouška Oral examination 20 5
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