Questions in Mathematics I

 1. Map, function, domain, range, graph, function properties (bounded, even,
    odd, periodic, increasing, decreasing, invertible), bounded sets in R,
    supremum, infimum.
 2. Inverse function, composition of functions, domain, range.
 3. Elementary functions.
 4. Continuous function in a point and in an interval. Theorems on 
    continuous functions. Function continuous in an closed interval.
 5. Proper and improper limit of a function. Limit in infinity.
    Single side limit. Theorems on limits.
 6. Sequence and limit. Monotonic and bounded sequence. Number e.
 7. Series, convergence, power series.
 8. Derivative, definition, geometrical and physical application,
    differential.
 9. Derivative of sum, product and fraction, derivative of composed function.
    Derivative of elementary functions. Higher order derivative.
10. Mean value theorems, application. L'Hospital rule.
11. Taylor formula and application, error estimate, Taylor series.
12. Function investigation.
13. Newton method for solving the equation f(x)=0.
14. Parametric curves in a plane. Tangent vector.
15. Polar coordinates. Goniometric form of complex numbers.
    Curves in polar coordinates.
16. Antiderivative and its properties.
17. Newton definition of definite integral and its properties.
    Geometric meaning. Mean value theorem.
18. Evaluation of definite and indefinite integral by substitution and 
    by parts.
19. Polynomials, algebraic equations, polynomial decomposition. Rational
    function, its domain, range, graph. Integration of rational functions.
20. Riemann definition of definite integral, its properties.
21. Improper integral.
22. Geometrical and physical applications of definite integral.
23. Numerical integration, Richardson extrapolation.
24. Differential equation, terms, the case y'=f(x,y).
25. Separation of variables for the differential equation y'=f(x,y).
26. Linear space, linear independence, base, dimension. Examples of linear
    spaces, R^n, C(I), subspace.
27. Matrix, matrix algebra, rank, linear map.
28. Determinant of a matrix and its properties.
29. Scalar product of vectors, vector product of vectors.
30. Inverse matrix, its evaluation, matrix equations.
31. Systems of linear algebraic equations.
                                                  11. May 2005 
                                                 Pavel Pokorny