## Learning objectives

Intended learning outcomes:

1. Knowing the main concepts of multivariable calculus and optimization. Understanding the methods adopted to study functions of several variables and optimization problems. Understanding mathematical models arising in economics and based on the topics taught in the course. (Knowledge and understanding.)

2. Formulating a mathematical model involving functions of several variables starting from a problem in economics. Analyzing a mathematical model with multivariable calculus and optimization techniques. (Applying knowledge and understanding.)

3. Describing the steps leading to the formulation of a mathematical model involving functions of several variables, starting from a problem in economcs. Explaining the properties of a mathematical model and the methods to study it. Describing the results obtained. (Communication skills.)

4. Assessing the quality and the adequacy of a mathematical model involving functions of several variables, according to the underlying economic problem. Verifying the soundness of the results obtained and interpreting them. (Making judgements.)

5. Finding connections between the concepts taught in the course and other topics studied during the Bachelor's degree program, aiming at analyzing problems arising in economics or management with a quantitative approach, whenever possible. (Learning skills.)

## Prerequisites

Solving first-degree, second-degree, irrational, and absolute value equations and inequalities.

Solving systems of equations and inequalities.

Matrix algebra, determinant, and inverse matrix.

Exponential and logarithmic functions.

Differential calculus for real functions of one real variable.

Optimization of real functions of one real variable.

## Course unit content

The course aims at providing students with adequate mathematical foundations of multivariable calculus and optimization, and at introducing them to quantitative methods used in economics. Particular attention is devoted to discussing mathematical models arising in agricultural economics and food supply chain management.

## Full programme

1) Functions of several real variables: domain, image, graph, level sets, level curves, sections, restrictions. Applications to economics.

2) An introduction to geometry and topology of Euclidean spaces. An introduction to limits and continuity of functions of several variables.

3) Directional and partial derivatives, gradient vector. Tangent plane and approximation of functions. Implicit functions, Dini's theorem (a.k.a. implicit function theorem), and applications to economics.

4) Second-order directional and partial derivatives, Hessian matrix and its properties. Convex and concave functions, and their properties.

5) Unconstrained optimization and applications to economics. An introduction to the least-squares method.

6) Constrained optimization and applications to economics. Linear and quadratic programming.

## Bibliography

- E. Castagnoli, M. Cigola, L. Peccati, "La matematica in azienda 2: complementi di analisi", Egea, Milano, 2010.

If necessary, additional teaching material will be made available on the course page (Elly) or mentioned during the lessons.

## Teaching methods

Lectures and exercise sessions.

The teacher will use slides during lectures, which will be uploaded in advance on the course webpage (Elly).

These slides are part of the course bibliography.

## Assessment methods and criteria

Written exam for the whole course (module 1 and module 2), with a total duration of 60 minutes.

The part of the written exam which pertains to module 1 of the course consists of:

1) An exercise on differential calculus to verify, on one hand, the understanding of the concepts and methods specific to this subject and, on the other hand, the ability to explain the methods used to solve the exercise and to communicate and interpret the results obtained (6 points);

2) An exercise on optimization to verify, on one hand, the understanding of the concepts and methods specific to this subject and, on the other hand, the ability to apply the knowledge through the formulation of a mathematical model based on an economic problem. Additionally, the ability to explain the methods used to solve the exercise and to communicate and interpret the results obtained will be evaluated (6 points);

3) An open-ended question on the theoretical content of the course to verify, on one hand, the understanding of the concepts and methods illustrated during the lectures and, on the other hand, to verify the ability to describe the steps leading to the formulation of a mathematical model involving functions of several variables and based on an economic problem, and the ability to illustrate the properties of a mathematical model and the methods used for its study. The open-ended question may require stating definitions and theorems and illustrating appropriate examples (4 points).

During the exam, students may use a scientific calculator, as long as it is not programmable or graphical.

Smartphones, tablets, laptops, smartwatches, and any other electronic device that allow internet connection or communication with others are not allowed.

The part of the written exam which pertains to module 1 of the course is graded on a scale of 0-16. This score is added to the score obtained in the part of the written exam which pertains to module 2 of the course. The passing grade is 18 or higher.

The teachers may award "30 e lode" at their sole discretion and only if the total score of the written exam is 31 or higher.

In any case, the instructor reserves the right to request an oral exam if deemed necessary.

The written exams will take place during the exam sessions scheduled in the academic calendar. No mid-term exams will be held.

Students will learn the outcome of the exam through an email message sent by the University to their institutional email address via the Essetre system. This message will also provide the procedures and deadlines for any refusal of the grade received.

## Other information

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## 2030 agenda goals for sustainable development

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