# MATHEMATICS

## Learning outcomes of the course unit

THE COURSE AIMS TO PROVIDE BASIC MATHEMATICAL KNOWLEDGE THAT WILL ENABLE THE STUDENT TO DESCRIBE AND ANALYSE ECONOMIC AND BUSINESS PROBLEMS IN A STRUCTURED WAY AND TO ADEQUATELY USE THE MOST SUITABLE TOOLS OF CALCULUS FOR SOLVING THEM.

COMPETENCES THAT CAN BE ACQUIRED

AT THE END OF THE COURSE THE STUDENT WILL HAVE ACQUIRED THE NECESSARY QUANTITATIVE TOOLS FOR DISCERNING THE STRUCTURE OF A PROBLEM FROM THE CONTEXT, FOR THE PURPOSE OF UNDERSTANDING AND SUCCESSFULLY COMMUNICATING WHAT IS NEEDED FOR TAKING THE MOST SUITABLE ECONOMIC DECISIONS IN A SUFFICIENTLY INFORMED WAY. IN PARTICULAR, THE STUDENT WILL BE ABLE TO READ A GRAPH, INTERPRET A TABLE, CORRECTLY DECIPHER QUANTITATIVE INFORMATION TAKEN FROM THE INTERNET, AND ALSO CONSTRUCT A USEFUL MATHEMATICAL MODEL (AT LEAST IN THE SIMPLEST CASES) FOR SOLVING MICRO AND MACRO-ECONOMIC PROBLEMS.

## Prerequisites

BASIC CALCULUS

## Course contents summary

- LINEAR FUNCTIONS AND MODELS.

- SYSTEMS OF LINEAR EQUATIONS AND MATRICES. MATRIX ALGEBRA AND APPLICATIONS.

- NON-LINEAR MODELS.

- THE DERIVATIVE. TECHNIQUES OF DIFFERENTIATION. APPLICATIONS OF THE DERIVATIVE.

- THE INTEGRAL. TECHNIQUES OF CALCULUS AND APPLICATIONS.

- FUNCTIONS IN SEVERAL VARIABLES.

- eCONOMIC APPLICATIONS.

## Course contents

Functions and Linear Models

The concepts of function and mathematical model.

Representation of a function.

Common types of function. Examples of mathematical economic models.

Linear functions.

Linear economic models.

Systems of linear equations and matrices

Systems of linear equations.

The reduction algorithm of Gauss-Jordan.

Economic applications of linear systems.

Linear algebra and applications

Concept of matrix and vector.

Matrix operations.

Matrix form of a linear system.

Inverse matrix and its use for the resolution of a linear system.

Determinant of a matrix calculation for arrays of size 2x2.

Non-linear models

General aspects: bounded functions, monotone functions, maxima and minima, infimum and supremum, even functions and odd functions, composite functions, inverse function, concave and convex functions (definition only).

Quadratic functions, exponential and logarithmic functions.

Economic models: quadratic, exponential and logarithmic.

The derivative

Average (or quotient) and instantaneous (or derivative) rate of change.

The derivative as the slope. Link between sign of the derivative and growth / decreasing function. Derivation rules.

Marginal analysis.

Limits: definition and examples of calculation. Continuity.

Techniques of differentiation

Rule of derivation of the product and ratio.

Rule of derivation of composite functions.

Derivatives of logarithmic and exponential functions.

Applications of the derivative

Maxima and minima. Applications.

Second derivative and study the graph.

Elasticity of demand.

The integral

The indefinite integral.

Integration by substitution.

Definite Integral.

The fundamental theorem of calculus.

Integrals: techniques and applications

Integration by parts.

Generalized integrals (notes).

Functions of several variables

Functions of several variables.

Notes on the graphs of functions of two variables.

Sections and contours.

Partial derivatives.

Maxima and minima.

Free and constrained optimization.

## Recommended readings

S. WANER, S.R. COSTENOBLE, STRUMENTI QUANTITATIVI PER LA GESTIONE AZIENDALE, APOGEO, MILANO, 2006.

FOR SOME IN-DEPTH STUDY, LECTURE NOTES WILL BE MADE AVAILABLE ON THE INTERNET.

## Teaching methods

Oral and practical lesson

## Assessment methods and criteria

Written exam.

The knowledge and comprehension will be tested with three questions related to the course prerequisites (1), a problem (2) and three theoretical questions (3).

The quality of learning, skills and ability to apply knowledge to practical problems will be checked through the problem (2) to solve which the student must identify an appropriate mathematical model, finally getting the solution using the analytical tools learned in the course.

The maximum score achievable through the problem is 15 points.

The ability to communicate with the appropriate technical language will be assessed through three open-ended questions (3) on the topics covered by the syllabus.

The maximum score achievable through the open questions is 12 points.