# MATHEMATICS, SECOND PART

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

None.

However it is advisable to have passed the Mathematics I exam, or to possess the calculation skills presented in it.

Exam is a written test designed to assess the achievement of each goal of the course. Both theoretical and practical skills are tested. Exercises require both theoretical issues and practical (e.g. calculus) skills.

Students can take the exam with two alternatives:

General Exam.

Partial exams.

At the end of the first part of lectures and at the end of the lectures, students can take part to partial exams, covering mainly the topics of the lectures delivered during the midterm ended.

Please, be advised: students must enroll to attend the exam (partial and/or general)

Students with learning disabilities are required to contact the “servizio disabili” (servizio.disabili@uninsubria.it) to define the individualized training project to be sent within 10 days of each exam session they intend to take.

More details on test structure will be provided at the beginning of the course through the University e-learning system (and on the basis of the state of emergency).

The purpose of this course is to teach the student the basic notions of calculus together with the basic techniques and applications that accompany them.

LEARNING OUTCOMES

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

• solve one-decision variable microeconomic problems;

• solve economic and business problems that involve optimization with respect to one variable;

• graphically represent real variable functions (studying monotonicity, convexity and continuity);

• understand discrete models, in economic, managerial and financial theory, involving succession and series;

• solve formalized problems through systems of linear equations, using the tools of linear algebra;

• solve problems that require the use of integral calculus in one variable;

• tackle the study of more advanced quantitative disciplines;

• understand the mathematical formalization of sentences and their proof.

The main topics covered in the course will be:

• limits of functions and continuity;

• differential calculation of one-variable functions;

• numerical series;

• integrals.

1) Limits and continuity for functions:

* limits of functions of a real variable (definition of limit, of right limit, left limit);

* limits of elementary functions with proofs; vertical, horizontal and oblique asymptotes;

* computing limits: the symbols ~ and o;

* continuity for functions of a real variable; points of discontinuity and their characterization; continuity of elementary functions;

* properties of continuous functions: Weierstrass theorem, theorem of intermediate values (a.k.a.

Darboux Theorem), Zeros Theorem.

2) Differential calculation in one variable:

* difference quotient, derivative at a point; geometric interpretation; the equation of the tangent line;

* left and right derivatives, critical points;

* derivative function, derivatives of elementary functions with proofs;

* review of calculus;

* first order differential, relationships between derivability, differentiability and continuity;

* stationary points, a necessary condition for local maximum / minimum points, Fermat's theorem;

* Lagrange mean value theorem;

* Monotonicity test, First order sufficient condition for optimality;

* Taylor's Theorem with Peano's Form of Remainder; Maclaurin's formula;

* Convexity test, Second order sufficient condition for optimality;

* Study of the graph of a function; function study.

3) Series:

* definition; properties and sum;

* necessary condition for convergence;

* series with positive terms, convergence criteria, series with alternate sign terms, absolute convergence.

4) Integrals:

* definite and indefinite integrals, integrals calculation.

Angelo Guerraggio, Matematica – Terza edizione, Pearson, Milano, 2020.

However, it is possible to refer to previous editions (2004, 2009).

Elisa Mastrogiacomo, Esercizi di Matematica per l'Economia. Serie, Integrali, Algebra Lineare, Programmazione Lineare Ledizioni 2018 ISBN: 9788867058471

Lectures will discuss both theoretical arguments and solution techniques for most common exercises related to the subject. Additional exercise will be assigned during classes both for individual study and to be solved in the next lecture.

Additional materials will be provided through the e-learning platform