# 37292 - MATHEMATICS

### Scheda insegnamento

• Docente Gian Luca Tassinari

• Crediti formativi 12

• SSD SECS-S/06

• Modalità didattica Convenzionale - Lezioni in presenza

• Lingua di insegnamento Inglese

• Materiale didattico

• Orario delle lezioni dal 04/11/2019 al 10/12/2019

### SDGs

L'insegnamento contribuisce al perseguimento degli Obiettivi di Sviluppo Sostenibile dell'Agenda 2030 dell'ONU.  ## Conoscenze e abilità da conseguire

At the end of the course the student will be capable of using the techniques of Linear Algebra; furthermore he will have acquired a working knowledge of First Year Calculus, together with the related applications in Finance and Economics.

## Programma/Contenuti

Course contents

A preliminary tutorial covers a number of introductory topics (so-called precalculus), including elementary set theory, sets of real numbers, complex numbers, polynomials, linear and quadratic equations and inequalities, systems of inequalities, absolute value and rational inequalities, Cartesian coordinate system, basic analytic geometry, basic concepts and definitions about functions, elementary functions (power, exponential and logarithmic), exponential and logarithmic equations and inequalities, trigonometric functions.

Course content - Calculus and Linear Algebra

Introduction to the course and crash review of preliminary mathematical notions

One-variable functions: basic definitions, graphs and elementary functions (linear, quadratic, polynomial, rational, irrational, power, exponential, logarithmic, absolute value). Odd and even functions. Composite functions. Inverse functions.

Limits and continuity.

Differentiation of one-variable functions: tangents and derivatives, rules of differentiation, chain rule, higher-order derivatives.

Derivatives in use: implicit differentiation and economic examples, differentiation of the inverse function, linear and quadratic approximations, Taylor's formula, elasticities; continuity and differentiability, intermediate-value theorem, L’Hôpital’s Rule.

Single-variable optimization: local and global extrema, stationary points and first-order condition, simple tests for extreme points, extreme points for concave and convex functions, second-order derivative and convexity, inflection points, study of the graph of a function, asymptotes.

Integration: the Riemann integral and its geometrical interpretation as area; primitives and indefinite integrals, fundamental theorems of integral calculus; rules and methods of integration: integration by parts, integration by substitution, improper integrals.

Integration in economics: continuous compounding and discounting, present values; 'stock' and 'flow' variables; probability in economics and finance; a glimpse at differential equations: separable and linear differential equations.

Infinite sequences and series; convergence criteria, geometric series; a glimpse at difference equations.

Linear algebra: vector spaces, bases and dimension; matrices and their properties, matrix operations, rank and determinant; linear maps and associated matrices, systems of equations, existence of solutions, cases of one solution and infinitely many solutions, Gaussian elimination, inverse of a matrix and Cramer's rule; eigenvalues and eigenvectors.

Multi-variable calculus: partial derivatives with two variables, geometric interpretation; partial elasticities; chain rules, implicit differentiation along a level curve; functions of more variables, gradient, differentials and linear approximations; economic applications.

Multi-variable optimization; maxima, minima and saddle points; tests based on second derivatives; constrained optimization and Lagrange multipliers.

## Testi/Bibliografia

R.A. ADAMS, C. ESSEX. Calculus, a complete course, 9th Edition, Pearson, 2018.

K. SYDSÆTER, P. HAMMOND (with A. STRØM). Essential Mathematics for Economic Analysis, 4th Edition. Pearson, 2012 (a student solutions manual is freely available from the publisher's website).

## Metodi didattici

Class lectures. During the class lectures (as well as in the additional exercise classes) each topic will be illustrated by examples and worked-out exercises.

## Modalità di verifica dell'apprendimento

Written exam.

The exam of the first (summer) session can be taken in three steps: a first midterm exam (after 1/3 of the course, during the mid-term session of January/February) with a duration 1 hour and 10 minutes, a second mid-term exam (after 2/3 of the course, during the mid-term session of April) on the second 1/3 of the course with a duration of 50 minutes and a final exam (also called third mid-term or third partial exam) with a duration of 50 minutes. In this case, the firts mid-term exam is weighted 40% in the final grade, the second partial exam is weighted 30% and the third partial 30%. In occasion of third mid-term exam, students who have not passed or who have not taken the first midterm exam can only take the total exam (duration 2 hours and 30 minutes). During the exam, students are allowed to use a pocket scientific calculator. Textbooks and other teaching materials are not allowed. Laptops, tablets and smartphones must be turned off.

The written exam aims at testing the student's ability to correctly and effectively apply the basic and advanced techniques learned in the course to specific problems in calculus and linear algebra. The written exam consists of a number of short exercises (routine exercises about basic concepts and calculations) and one or two more challenging 'review problems' (for instance, the complete study of a one-variable function, a problem of constrained optimization in two variables, the general and particular solutions of a differential equation, problems on matrices and linear systems depending on a parameter, etc.). Such review problems generally include questions of different levels of difficulty as well as connections to the several economic applications illustrated in the textbook.