37292 - Mathematics

Learning outcomes

The course aims at giving the student a basic knowledge of differential and integral calculus, and linear algebra for the study of economics, financel and statistical analysis. By the end of the course students have the ability to perform basic operations with vectors and matrices, to compute determinants, and to solve linear systems. As far as calculus is concerned, they can apply the methods of differential and integral calculus to plot the graph of functions, to compute the area of plane domains, and to find and classify critical points of functions of two variables.

Course contents

Calculus

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. Translation of functions.

Limits and continuity.

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

Implicit differentiation, differentiation of the inverse function, Taylor's formula, elasticities; continuity and differentiability, intermediate-value theorem, De L’Hôpital’s Rule.

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.

Sequences and series; convergence criteria; geometric series; Taylor's series. Sequences and series in financial mathematics.

Integration: the Riemann integral and its geometrical interpretation; primitives and indefinite integrals, fundamental theorems of integral calculus. Rules and methods of integration: immediate integrals, integration of rational functions, integration by parts, integration by substitution. Improper integrals.

Linear algebra

Vector spaces, bases and dimension; matrices and their properties, matrix operations, rank and determinant; linear 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.

Readings/Bibliography

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

Chapters: preliminaries, 1, 2, 3, 4, 5, 6, 7.9, 9, 10, 12, 13

K. SYDSÆTER, P. HAMMOND, A. STRØM, A. CARVAJAL. Essential Mathematics for Economic Analysis, 6th Edition. Pearson, 2021.

Chapters: 1, 2,3, 4, 5, 6, 7, 8, 9, 10, 12, 13

Teaching methods

Class lectures

Assessment methods

Written exam: students have to solve different exercises on the course topics. To each exercise a given maximum number of points is associated, and to get it the student has to solve correctly the exercise and all the steps must be justified.

The theoretical maximum number of points atteinable in case of a perfect exam is 32.

Office hours

See the website of Fabio Gobbi