230-0202/09 – Mathematics II (BcM II)

Mathematics is essential part of education on technical universities. It should be considered rather the method in the study of technical subjects than a goal. The aim of the subject is therefore 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.

The course includes three chapters – integral calculus of functions of one real variable, introduction to differential calculus of functions of two real variables and ordinary differential equations. The aim of the first chapter is to master the basic techniques of integration and, above all, to become familiar with the geometric and physical applications of the definite integral. The second chapter deals with the basics of differential calculus of functions of two variables, creating a geometric idea of ​​a graph, determining local extrema and the tangent plane to a surface. The third chapter introduces the basic types of ordinary differential equations and their solutions.

Harshbarger, R.J.-Reynolds, J.J.: Calculus with Applications. D.C.Heath and Company, Lexington1990, ISBN 0-669-21145-1 James, G.: Modern Engineering Mathematics. Addison-Wesley, 1992.

Passing the course, requirements Course-credit -participation on tutorials is obligatory, -elaborate programs, 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 - 86 85 - 66 65 - 51 50 - 0 National grading scheme excellent very good satisfactory failed 1 2 3 4 List of theoretical questions: 1. Antiderivatives, primitive functions. 2. Integration by substitution. 3. Integration by parts. 4. Integration of rational functions, polynomials in denominator have different real roots. 5. Integration of rational functions, polynomials in denominator have k-fold roots. 6. Integration of rational functions, polynomials in denominator have complex conjugate roots. 7. Integration of functions of the type R(sin x)cos x. 8. Integration of functions of the type R(cos x)sin x. 9. Integration of functions of the type sin^m x cos^n x. 10. Integration of functions of the type R(sin x, cos x). Universal trigonometric substitution. 11. Newton-Leibnitz theorem for calculation of definite integrals. 12. Integration by substitution for definite integrals. 13. Integration by parts for definite integrals. 14. Application of definite integrals - area. Explicit and parametric representation. 15. Application of definite integrals - arc length. Explicit and parametric representation. 16. Application of definite integrals - volume of a solid of revolution. Explicit and parametric representation. 17. Application of definite integrals - lateral surface of a solid of revolution. Explicit and parametric representation. 18. Definition of functions of n real variables. 19. Partial derivatives. 20. Geometrical meaning of partial derivatives of functions of two variables. 21. Equation of a tangent plane to a graph of functions of two variables. 22. Equation of a normal to a graph of functions of two variables. 23. Second order partial derivative. 24. Total differential of functions of more variables. 25. Necessary condition for existence of extremum of functions of more variables, Fermat theorem. 26. Sufficient condition for existence of extremum of functions of more variables. 27. Implicit functions, derivation of implicit functions. 28. Ordinary differential equations. 29. General and particular solution of differential equations. 30. Separable differential equation, general form and method of solution. 31. Homogeneous differential equation, general form and method of solution. 32. Linear differential equation, general form and method of solution. 33. Linear differential equation, method of variation of arbitrary constant. 34. Linearly independent functions, Wronskian. 35. 2nd order linear differential equations with constant coefficients, general form, method of solution. 36. 2nd order linear differential equations with constant coefficients, characteristic equation. 37. LDE, independent solutions for different real roots of characteristic equation. 38. LDE, independent solutions for 2-fold real roots of characteristic equation. 39. LDE, independent solutions for complex conjugate roots of characteristic equation. 40. 2nd order linear differential equations with constant coefficients, method of variation of arbitrary constants. 41. 2nd order LDE, write a particular solution for a special right-hand side f(x)=Pm(x). 42. 2nd order LDE, write a particular solution for a special right-hand side f(x)=e^(ax) Pm(x). 43. 2nd order LDE, write a particular solution for a special right-hand side f(x)=x^2 e^x cos3x. 44. 2nd order LDE, write a particular solution for a special right-hand side f(x)=x e^x sin3x. 45. 2nd order LDE, write a particular solution for a special right-hand side f(x)=x sin3x. 46. 2nd order LDE, write a particular solution for a special right-hand side f(x)=x e^(5x). 47. 2nd order LDE, write a particular solution for a special right-hand side f(x)=e^2x sin2x. 48. 2nd order LDE, principle of superposition.