# MATHEMATICS

- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Bibliography
- Teaching methods
- Contacts/Info

None.

The course starts from the basics and can be profitably attended by possessing the minimum quantitative knowledge common to all high school graduated, regardless their major. It is however advisable to check one's previous skills and knowledge through the self-assessment test provided in the e-learning of the course and the assessment test. In case of difficulty with the concepts contained in these tests, it is recommended to promptly contact the instructors to identify possible ways to recover.

The exam is in written form. The use of calculator is allowed during the test.

The test is divided into three parts, designed to evaluate the skills of calculation, the knowledge of the terminology and the main statements presented during the course, the analytical skills developed by the student.

The exam can be taken in two ways:

General test.

At the end of the course, during the exam sessions, 90-minute exam sessions will be organized on the entire course program.

The first part of the test, evaluated up to 10 points, consists of short questions, with multiple choicess or with a short answer concerning the calculation abilities. Student must achieve at least 5 points for the exam to be evaluated.

The second part of the test assigns up to 11 points and focuses on the statements, definitions and proofs presented during the course. The student must rigorously answer to a score of questions. There is no minimum score for this part.

The third part of the test, evaluated 11 points, consists of more complex exercises, where students are required to use their calculus and theoretical skills to come up with the solution. There is no minimum score for this part.

The exam is passed if the sum of the scores obtained in the three parts is not less than 18 (eighteen), with a minimum of 5 (five) points in the first part. Scores above 30 provide "Honor" grade.

Partial exams.

At the end of the first cycle of lessons, during the teaching break, and at the end of the course in December, two partial tests will be organized mainly concerning the topics of the part of the course just ended. Each exam lasts 60 minutes and is divided into three parts.

The first part of the test, evaluated up to 5 points, consists of short questions, with multiple choicess or with a short answer concerning the calculation abilities. Student must achieve at least 2 points for the exam to be evaluated.

The second part of the test assigns up to 5 points and focuses on the statements, definitions and proofs presented during the course. The student must rigorously answer to a score of questions. There is no minimum score for this part.

The third part of the test, evaluated 6 points, consists of more complex exercises, where students are required to use their calculus and theoretical skills to come up with the solution. There is no minimum score for this part.

Each partial exam is passed with a minimum of 6 points (2 of which in the first part).

The exam is passed if, each partial exam is passed and the sum of the two grades is not less than 18 (eighteen). Honor grade is achieved with a sum greater than 30.

SLD students: Students affected by SLD are required to contact the SLD service (servizio.disabili@uninsubria.it) to define a Personalized Educational Project to be transmitted to the course'instructor within 10 days of each exam session.

The course aims to provide students with basic analytical tools to quantitative study of economic and management models.

At the end of the course, the student will be able to:

â€¢ Solve problems of a micro-economic nature with one or more decision variable;

â€¢ Resolve economic and management problems involving optimization with respect to one decision variable;

â€¢ Sketch the graph of functions of a real variable, studying main properties such as monotony, convexity and continuity;

â€¢ Understand discrete models, in economic, managerial and financial theory, involving sequences and series;

â€¢ Solve systems of linear equations, by means of linear algebra tools;

â€¢ Solve problems that require the use of integral calculus in one variable;

â€¢ Face the study of more advanced quantitative disciplines;

â€¢ Understand mathematical statement and basic mathematical proofs.

Numeric sets. (2h)

Linear Algebra. (11h)

Real valued functions of a real variable. (7h)

Sequences. (2h)

Limits of a real valued function. (9h)

Continuous functions. (5h)

Differential calculus for real functions of a real variable. (16h)

Function of multiple variables. (6h)

Integral calculus. (11h)

Series. (11h)

Linear Algebra. [Ch.16 Â§Â§ 1-6]

Algebra of vectors and matrices, determinant (Sarrus rule, Laplace theorem), inverse matrix, transposed, rank, linear systems (solution study and resolution).

Numeric set. [Ch.2 Â§Â§ 3; 5 Ch.4 Â§Â§ 2-5]

Set R: algebraic, metric structure. Distance, ordering, sup / inf, internal, external, isolated, accumulation, maximum and minimum points

Real functions of real variable. [Ch.2 Â§Â§ 3-4; Ch.3; Ch.4 Â§ 6]

Function definition. Elementary functions, graph, geometric transformations, graphically resolvable inequalities. Domain, bounded function, composition of functions, monotonicity, invertibility, concavity / convexity.

Limits of a function in a variable. [Ch.6 Â§Â§ 1-4.6; Ch.7 Â§Â§3-5]

Theorem of uniqueness of the limit, Theorem of existence of the limit for monotone functions, Theorem of permanence of the sign. Calculation of limits, notable limits, infinities and infinitesimal. The Landau' symbols.

Continuous functions. [Ch.7 Â§Â§1-2.6]

Weierstrass theorem, Zero values theorem, intermediate values â€‹â€‹theorem.

Sequences. [Ch.5 Â§Â§1; 3; Ch.6 Â§ 5]

Sequences defined by recurrence, limit of sequences.

Differential calculus for functions in a real variable. [Ch.8, 9]

Incremental ratio, derivative and its geometric meaning, points of non-derivability, calculation of derivatives, derivability and continuity, higher order derivatives, Taylor's theorem (order n), De Hospital's theorem. Rolle's theorem, Lagrange's theorem, Fermat's theorem. Monotonicity of differentiable functions, II test of recognition of stationary points, study of the graph of function.

Functions of multiple variables. [Ch.14 Â§Â§ 1-4]

Definition, domain, graph and level curves. Unconstrained extreme points. Partial derivatives and Fermat's theorem.

Integral Calculation. [Ch.10,11]

Undefined integral, immediate primitives, almost immediate, primitives of fractional rational functions, integrations by parts, by substitution. Definite integral, integral function. Mean value Theorem of integral calculus, fundamental theorem of integral calculus. Generalized integrals.

Series. [Ch.12]

Character of a series, geometric series. Necessary condition for convergence. Series with positive terms: generalized harmonic series, criterion of asymptotic comparison, of comparison. Series in terms of any sign: absolute convergence (sketch).

A. Guerraggio, Matematica 2/Ed.

Per gli argomenti di base, si consiglia di consultare

G. Anichini - A. Carbone - P. Chiarelli - G. Conti, Precorso di Matematica 2/Ed.

e-texts available!

The course is based on lectures. Tutoring sessions will also be offered.

Evaluation takes place through the results of the exams.