The teaching is organized as follows:

Learning outcomes

Module: 1 - lectures
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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.


Module: 2 - esercise lectures
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This module intends to complete the theoretic knowledge with the adequate calculus ability

Program

Module: 1 - lectures
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Part I (revisal)

Polinomials
Powers and logarithms
Equations and inequalities
Analytic geometry

Part II (Real analysis)

Theory of sets. Power set. Cartesian product. Numerical sets: natural, integer, rational and real numbers
Functions. Composition of functions. Inverse function
Real numbers. Sup and inf of a set of real numbers.
Real functions. Plot. Image and inverse image. Sup of a function. Monotone functions. Elementary functions and their graphics. Power, exponential and logarithmic function
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. Mention to Taylor's formula and convex functions
Integrals. Primitive of a function. Riemann integral. Some properties of the Riemann integral. 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. Convergence criteria for series with positive terms

Part III (Linear algebra)

Linear spaces Rn. Linear dependence and linear independence. Subspaces. Basis and dimension of a space. Inner product
Mention to linear transformations. Matrices. Kernel and image of a linear transformation. Rank
Determinant and its properties. Inverse matrix. Calculus of the rank
Systems of linear equations. Rouché-Capelli's theorem. Cramer's theorem

Part IV (Real analysis in more variables)

Functions of more than one variable. Sets in Rn. Restriction. Level curves
Quadratic forms. Sign of a quadratic form. Study of the sign with principal minors
Partial derivatives and gradient. Derivatives and continuity. Differentiability. Second derivatives and Schwarz's theorem
Maxima and minima. Non constrained and constrained search of minima


Module: 2 - esercise lectures
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The topics are the same of the lectures

Examination Methods

Module: 1 - lectures
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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.