In order to pass the course (grade 3) the student should be able to
give an account of important concepts and definitions in the area of the course;
exemplify and interpret important concepts in specific cases;
formulate important results and theorems covered by the course;
describe the main features of the proofs of important theorems;
express problems from relevant areas of applications in a mathematical form suitable for further analysis;
use the theory, methods and techniques of the course to solve mathematical problems;
present mathematical arguments to others.
Higher grades, 4 or 5, require a higher level of proficiency. The student should be able to solve problems of greater complexity, i.e. problems requiring a combination of ideas and methods for their solution, and be able to give a more detailed account of the proofs of important theorems and by examples and counter-examples be able to motivate the scope of various results. Requirements concerning the students ability to present mathematical arguments and reasoning are greater.
Arithmetic for rational and real numbers, inequalities, absolute value. Permutations and combinations. Induction. Polynomials: factorisation and division, completing squares, simple algebraic equations. The binomial theorem. Complex numbers: real and imaginary part, polar form, the complex plane, second degree equations and binomial equations. Elementary functions: the exponential function, logarithms (in different bases), logarithmic rules, and trigonometric functions. Trigonometric formulas. Simple exponential, logarithmic and trigonometric equations. Coordinate systems in the plane. The distance formula. Equations for the line and the circle. Equations for the ellipse, hyperbola and parabola in standard form.
Lectures and problem solving sessions.
Written examination at the end of the course. Moreover, compulsory assignments may be given during the course.