## 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.

Furthermore, 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

First and second order equations and inequalities.

## Course unit content

Linear functions and models.

Linear systems and matrices. Economic applications.

Non-linear models.

Differential Calculus and economic applications.

Integrals. Economic applications.

Functions in several variables.

## 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.

## 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.

The Syllabus, the detailed program and the previous exam tests are

uploaded on Elly.

## Assessment methods and criteria

Written examination (60 minutes).

During the exam, the student can use a scientific calculator. Graphic

calculators, smartphone, tablet, laptops and smartwatch are not allowed.

The knowledge and the skill in comprehension will be tested through

three questions about elementary mathematics (1), three short problems

(2) and three theoretic/practical questions (3).

The quality of learning, the skill in the applications of the concepts and

the independence of judgement will be verified through the economic

problems (2). In order to solve such a problem, the student will

individuate an opportune mathematical model, by obtaining the solution

through the analytical tools presented in Course.

The knowledge of an appropriate technical language will be checked

through the three questions (3) about the theoretic subjects with

applications of the Course.

Indicative marks:

(1): 3/30

(2): 15-16/30

(3): 12-13/30

If a correct result is not justified in the paper given to the student for the

development of the exam, no mark will be assigned.

If the test, possibly integratd by an oral exam, is excellent, it will be

valuated with full marks cum laude.

The teacher could ask for an oral test and/or partially modify the written

examination, also according to the exigencies of integration with possible

online activities.

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

## Other information

Further teaching materials, the Syllabus, the detailed program of the

course and the exams already assigned will be published on Elly.

Furthermore, the teacher could modify some indications, in particular

about lessons

and examinations, also according to the evolution of pandemic situation.

Every change will be communicated well in advance to students through

the usual channels.