## Learning objectives

In an ever increasing number of contexts it is advisable that a graduate in

economic disciplines is able to use quantitative measurements and tools.

The main objective of the course is to allow the student to "take possession" of these tools in order to separate the structure of a problem

from the context so that he/she can understand and communicate as

effectively as possible what is needed to take informed and justified

economic decisions.

Furthermore, at the end of the course:

- as far as concerns knowledge and understanding: the student should

understand and properly treat the main models presented in the course;

- as far as concerns applying knowledge and understanding: the student

should be able to solve practical problems;

- as far as concerns making judgements: the student should show good

skills in making judgement, developing reasoning and critical capacities;

- as far as concerns communication skills: the student should clearly

communicate what he/she has learned;

- as far as concerns learning skills: the student should update and

consolidate his/her quantitative knowledge and relate this knowledge and

competence to other disciplines in the degree course.

Finally, the student should be able to formalize in quantitative

terms some economic problems, by individuating initial data and the more adequate mathematical instruments in order to obtain an efficient

and rigorous solution and to provide an economic interpretation of the

obtained results.

## Prerequisites

BASIC CALCULUS

## Course unit content

- 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.

## Full programme

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.

## Bibliography

S. WANER, S.R. COSTENOBLE, STRUMENTI QUANTITATIVI PER LA GESTIONE AZIENDALE, APOGEO, MILANO, 2019.

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

## Teaching methods

1) Knowledge and understanding: teacher-fronted sessions.

2) Applying knowledge and understanding: practice sessions.

3) Making judgements: teacher-fronted sessions and practice sessions.

4) Communication skills: practice sessions.

5) Learning skills: teacher-fronted sessions and practice sessions.

## Assessment methods and criteria

Written exam.

During the exam, the student can use a scientific

calculator. Graphic calculators, smartphone, tablet,

laptops and smartwatch are not allowed.

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.

A further oral examination may be requested by the teacher, if necessary.

The honors will be awarded to those particularly deserving students who, in addition to having complied with the requisites necessary to obtain the full evaluation, have overall demonstrated an appreciable systematic knowledge of the topic, an excellent ability to apply the knowledge acquired to the specific problem in question, a considerable autonomy of judgment, as well as a particular care in the formal drafting of the exam.

The University will send to the students an email message to their

University email address with the result of the exam (through Essetre

system). The students can reject the result in a week, through an online

procedure, clearly described in the message.

## Other information

The Syllabus, the detailed program and the previous exam tests are uploaded on Elly.