## Learning objectives

The aim of the Mathematics class for SG is to enable students to autonomously elaborate and interprete, both qualitatively and quantitatively, experimental data and/or simple mathematical models of applied sciences.

The expected learning outcomes are:

(1) Knowledge and understanding: Knowledge of the basic techniques of differential calculus and linear programming

(2) Application capabilities: Apply the principles of differential calculus to the analysis of simple biological, physical and economic models.

Use and manipulate formulas and equations by easily managing their units of measurement

(3) Autonomy of judgment: Evaluation and interpretation of mathematical models. Evaluation and interpretation of experimental data.

(4) Communication skills: Written and oral communication through the scientifically correct vocabulary of the subject

(5) Ability to learn: Develop a scientific approach in the execution of experiments and in the mathematical formalization of their results.

Ability to successfully carry out the Master's Degree courses of the LM70 class and in particular the Master's Degree course in Food Science and Technology

## Prerequisites

High school mathematics.

## Course unit content

Basic concepts and methods of differential and integral calculus for functions of one real variable: number sets, sequences, limits, graphs of functions, derivatives and integrals. Introduction to the methods of Linear Programming in two and three variables. Although the presentation of the arguments privileges the understanding of the concepts and techniques rather than the formal rigor, some selected theorems with relative proof are presented.

## Full programme

Numbers and real functions. Basics of set theory. Natural, integer, rational, and real numbers. Functions and Cartesian representation.

Parabolas and circumferences. Injective and surjective functions, monotonic functions. Inverse function. Composition. Linear functions. Power functions, exponential and logarithmic functions. Trigonometric functions.

Limits and continuous functions. Limits of functions. Definition examples and properties of continuous functions. The existence of intermediate values theorem. Fundamental limits. The Euler number "e".

Derivatives. Definition of derivative. Geometric meaning of the derivative. Rules of derivation. Derivatives of some elementary functions. Higher derivatives.

Basic applications of differential calculus. Lagrange's theorem (mean value theorem). Consequences and applications. Points of increase, decrease, maximum and minimum of a function. Convex functions.

Elementary theory of integration. Summations. Integral as an area. Integral of a continuous function. Mean value of a function and mean value theorem. The fundamental theorem of integral calculus. Fundamental formula of integral calculus. Indefinite integrals. Integration by summation decomposition. Integration by parts. Integration by substitution.

Algebra of matrices; systems of linear equations and their solution; Rouché-Capelli theorem. Elements ol linear programming: linear programs in two and three variables: the diet problem; the product mix problem, the transportation problem.

## Bibliography

A. Guerraggio: Matematica per le scienze, seconda edizione, Pearson Editore.

Course notes.

## Teaching methods

The course will be in presence. During lectures, the material of the course is presented using formal definitions and proofs; abstract concepts are illustrated through significant examples, applications, and exercises. The discussion of examples and exercises is of fundamental importance for grasping the meaning of the abstract mathematical concepts; for this reason, besides lectures, guided recitation sessions to discuss and solve exercises and assignments will be provided within the “Progetto IDEA”.

## Assessment methods and criteria

Course grades will be based on a final exam which consists of a written text and an oral interview. There will be the possibility of two intermediate written exams to avoid the final exam. The written text is aimed at ascertaining the calculation and application skills of the methods. The oral exam is aimed at ascertaining the theoretical competences and the exposition skills of the student.

## Other information

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