Mathematics - 1 - lezione (2008/2009)

Learning outcomes

The aim of the course is to give the basic mathematical knowledge, necessary to the following courses in statistics and economics. The course provides the classical arguments from mathematical analysis and linear algebra.

Sets and subsets. Power set. Union and intersection of sets. Cartesian product. Numerical sets: natural, integer, rational and real numbers. Real intervals. Sup, inf, max, min of a set.

Real functions. Composition of functions. Monotone functions. Elementary functions and their graphics. Power, exponential and logarithmic function.

Analytical geometry. Curves and their equations.

Equations and inequalyties.

Limits and continuity. Calculus of limits. Landau symbols. Continuous functions. Weierstrass theorem.

Derivatives. Calculus of derivatives. Stationary points. Maxima and minima of functions. Lagrange theorem. Taylor formula.

Integrals. Primitive of a function. Riemann integral. Some properties of the Riemann integral. Sufficient conditions. Integral function and the fundamental theorem of calculus. Calculus of the Riemann integral. Elementary methods. Integration by parts. Change of variable in the integral. The Riemann generalized integral.

Series. Geometric series and armonic series.

Linear algebra topics. Linear spaces R^n. Linear dependence and linear independence. Subspaces. Basis and dimension of a space. Inner product.
Matrices. Multiplication of matrices. Determinant and its properties. Inverse matrix. Rank.
Systems of linear equations. Rouché-Capelli theorem. Cramer theorem.

Functions of more than one variable. Level curves. Continuity. Partial derivatives and gradient. Maxima and minima. Integral in two variables.

In order to pass the exam students are asked to pass first a multiple choice test. A written exam is then proposed. A final oral exam is required only in case of a non full sufficiency.