Mathematics A

obligatory courses

Basic properties of elementary functions like the sine, cosine, logarithm and exponential function. Derivatives of elementary functions and techniques of differentiation. Basic properties of vectors and matrices, introduction to linear algebra.

Mathematics 0 differs from Mathematics A only in the number of problem sessions. Students who have chosen Mathematics 0 take part

in problem sessions three times a week, while those who have chosen Mathematics A --- twice a week.

Definitions of elementary functions: sine, cosine, tangent, exponential function, logarithms, root functions. Inverse trigonometric functions. Limits of functions. Continuity: intermediate value theorem, Weierstrass Maximum Principle. Definition of the derivative. Tangent to the graph of a function. Differentiation rules. Sketching graphs of functions: monotonicity, convexity, inflection points, locally extreme values. Taylor formula with Lagrange remainder and Peano remainder, approximation, error estimates. Taylor series. Primitive function. Simple integrals. Basic techniques of integration: substitution and integration by parts. Vectors in the plane and in the 3-dimensional space. Scalar product and vector product. Matrices: addition, multiplication by a number and by a matrix. Determinants. Linearity with regard to columns or rows, skew symmetry, row reduction, Cramer's formulae, eigenvalues and eigenvectors. Orthogonal matrices, symmetric matrices, transposed matrix.

Prerequisities: basic high school algebra, solving of linear systems with 2 or 3 unknowns, canonical form of a quadratic trinomial, second degree equations and inequalities, polynomial factorization by finding rational roots or by grouping terms in simple cases.