# MATHEMATICS

No theoretical prerequisite.

Final exam consists of a two hours written test. To pass, students must solve correctly both theoretical and practical exercises, covering all the course topics. The exam can be divided into two partial exams organized accordingly the academic calendar. Topics for each partial test will be declared during the course.

The course aims to provide students with basic analytical tools to address quantitative study of economics and management.

At the end of the course, students will be able to:

• Solve simple micro-economic problems with one decision variable;

• Solve simple management problems through optimization with respect to one decision variable;

• Graph functions of one variable, studying their main properties of monotonicity, convexity and continuity; • Understand discrete models of economics, finance and management, involving sequences and series;

• Solve problems formalized through systems of linear equations, using tools of linear algebra;

• Solve problems that require the use of integral calculus in one variable;

• Addressing the study of more advanced quantitative disciplines;

• Understand the mathematical formalization of statements and their proof.

Lectures cover both theoretical arguments and solution techniques for the main problems that students are required to be able to handle. Practice hours are devoted to additional exercises to sharpen students calculus skills.

Active participation to lectures and practice hours is highly recommended.

Lectures will mostly cover the following topics:

Number sets.

The set of real numbers: algebraic and metric structure. Distance, ordering, supremum and infimum, interior, exterior and boundary points, cluster points, maximum and minimum.

Real functions of one real variable.

Definition of function. Elementary functions, graphic, geometric transformations, graphical methods to solve inequalities. Domain, bounded functions, composition of function, monotonicity, invertibility, concavity and convexity.

Limit.

Theorems (Uniqueness of the limit, sign permanence - proofs) and calculus techniques, notable limits, infinite and infinitesimal.

Continuous functions.

Theorems (Weierstrass, Intermediate Zero Theorem, Intermediate Value Theorem - proofs).

Sequences.

Recursively defined sequences, limits of sequences.

Linear Algebra.

Vectors and matrices algebra, determinant (Sarrus’s Rule, Laplace expansion), inverse matrix (proof of the uniqueness of the inverse matrix), transposed, rank, linear systems (study of solutions and resolution).

One variable differential calculus.

Difference Quotient, derivative and its geometrical interpretation, points of non differentiability, calculus of derivatives, differentiability and continuity, higher order derivatives, Taylor's expansion (order n), De Hospital’s Theorem. Rolle's Theorem (proof), Mean Value Theorem (proof), Fermat's Theorem for stationary points (proof). Testing monotonicity for differentiable functions (proof), Testing convexity for twice

differentiable functions, Second order sufficient condition for stationary points (proof), deducing the graph of a function.

Integral Calculus.

Indefinite integral, primitives of elementary functions, primitives of rational functions, integration by parts and by substitution. Definite integral, integral function. Mean Value Theorem of integral calculus (proof), Fundamental Theorem of integral calculus (proof). Generalized integrals.

Series.

Convergence of a series, convergence of the geometric series (proof). Necessary condition for convergence (proof). Series with positive terms: harmonic series, convergence criteria: comparison, asymptotic comparison, ratio, root, integral. A dash on absolute convergence of a general series.

Angelo Guerraggio, "Matematica – Seconda edizione", Pearson, Milano, 2009.

Lectures.