Applied Mathematics — Examination

Topics of the examination

1. Domain: find a domain of a given elementary real function of one real variable.
2. Derivative: compute derivatives of a given elementary function using formulas and rules of differentiation.
3. Tangent and normal lines: find equations of the tangent line and the normal line to the graf of a given function in a given point of the graph.
4. Monotonicity: find maximal intervals on which a given function is either decreasing or increasing (using the first derivative).
5. Extremal values: find all local extremal values of a given function.
6. Convexity and concavity: find maximal intervals on which a given function is either convex or concave and find points of inflection (using the second derivative).
7. Indefinite integral: find an antiderivative (or indefinite integral) of a given function by combination of the following methods — direct integration, integration by parts, integration by substitution.
8. Definite integral and its applications: evaluate a definite integral of a given function (by a combination of the direct method, inegration by parts and substitution and using The Fundamental Theorem of Calculus), compute the area of a plane region between graphs of given functions.
9. Differential equations: find a general solution of a given differential equation with separable variables.
10. Rank of a matrix: determine the rank of a given matrix (it may be also depending on a parameter).
11. Systems of linear algebraic equations: solve a given system by the Gaussian or Jordan method of elimination.
12. Determinants: evaluate a determinat of degree 4 and higher, solve a system of linear equation by the Cramer rule, solve equations or inequalities (of one real variable) with determinants.
13. Algebra of matrices: find the inverse to a given regular matrix, solve matrix equations (using matrix operations, identity matrix and the inverse to a matrix).

Composition of an exam

The exam consists of 6 problems which are equally distributed over the previous topics. The first one is to find a domain (topic 1), the following one or two problems are to compute a derivative or to use an application of derivatives (topics 2–6). The following one or two problems are to compute indefinite or definite integral or an area of a plane region or to solve a differential equation (topics 7–9). The last two problems are from linear algebra (topics 9–13).

Possible examinations

Several possible variants of examinations you find here.