37292 - Mathematics

Learning outcomes

This course aims to introduce students to differential calculus: limits, derivatives, extremes of a function and convexity, the qualitative study of a function, integral calculus: definitions and properties, integration techniques and differential equations and elements of linear algebra: linear systems, linear applications and matrices.

Course contents

Linear Algebra

Linear algebra: 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.

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.

Limits and continuity.

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

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; primitives and indefinite integrals, fundamental theorems of integral calculus. Rules and methods of integration: immediate integrals, integration by parts.

Applications to Calculus of Probabilities: Normal Distribution and Cumulative Distribution Function

Two-variables functions: Introduction to functions of multiple variables. Graphs and elementary functions: linear and quadratic functions.

Limits and continuity.

Partial Derivatives. Tangent planes, Linear Approximations and the Total Differential

Unconstrained optimization: Directional Derivatives and the Gradient. Higher-order Derivatives. Hessian Matrix. Maxima, minima and saddle points, tests based on second derivatives.

Readings/Bibliography

K. Sydsaeter, P. Hammond, A. Strom and A. Carvajal

Essential Mathematics for Economic Analysis

6th Edition, Pearson

Teaching methods

Lectures and exercises at the blackboard. Online Presentations.

 

Assessment methods

Mark range:

• 18-19: knowledge of a very limited number of topics covered in the course and analytical skills that emerge only with the help of the teacher, expressed in an overall correct language;
• 20-24: knowledge of a limited number of topics covered in the course and ability to autonomous analysis only on purely executive matters, expression in correct language;
• 25-29: good knowledge of a large number of topics covered in the course, ability to make independent choices of critical analysis, mastery of specific terminology;
• 30-30 cum laude: excellent knowledge of the topics covered in the course, ability to make autonomous choices of critical analysis and connection, full mastery of specific terminology and ability to argue and self-reflection.

Teaching tools

Online presentations. Links to website with resolution of exercises. Use of Software Geogebra for function analysis and graphs

Office hours

See the website of Donatella Giuliani