Mathematics I 2

The objectives of the course are the following:

Knowledge and understanding: the student will learn some basic mathematical concepts and will develop both computing ability and the capacity of manipulating some common mathematical structures among which limits and derivatives for functions of real variable, integral calculus.

Applying knowledge and understanding: by examples related to applied sciences, the student will learn the central role of Mathematics within science and not only as an abstract topic. This will expand the cultural horizon. He will learn the fundamental techniques and will be able to apply them in some simple problems.

Making judgements: the student will reflect on the  meaningful methods of Mathematics to  sharpe  his logical ability. Many proofs will be exposed in an intuitive and schematic way, to make them more usable also to students that are not committed to Mathematics.

Communication skills: studying Mathematics and dedicating time to guided exercitations and seminars, the studente will learn to communicate with clarity and rigour both, verbally and in writing. The student will learn that using a properly structured language is the key point to clear and effective scientific, and non-scientific, communication.

Learning skills: the students, in particular the more willing, will be stimulated to examine in depth some arguments, alone or working in group.

The course is organized by lectures and practices. Home exercises will be assigned and then students will be asked to illustrate their solutions. Many hours will be dedicate to private colloquium with the students in order to clarify doubts both on the theoretical part and on the technical and to help them in their study method.

The prerequisites are those required for  a scientific degree course: algebraic calculus elements of 
trigonometry.

Numerical Sets: excursus from the set of natural numbers to the set of real numbers. Fundamental properties of the real set numbers. Some notions of topology in R and in R^2

Sequence of real numbers.

 Numerical series: character of a serie, series with positive terms. Leibnitz 's Theorem. Absolute convergence.

Real valued functions of real variable and their limits. .Continuity. Monotony. Inverse functions. Composition of functions

Differential calculus for real valued functions of one real variable and its applications Rules of derivations. Derivability of the geometri point of view. Local and global extremes. Fermat's Theorem, Rolle's Theorem.  Lagrange's Theorem and applications. Convex functions. Taylor's Formula.

Integral calculus: methods of integration

Rienman Integrability: Construction of Rienman integral and its properties. Functions that are not Rienman integrable and classes of functions that are Rienman integrable. Integral function. Fundamental Theorem of integral calculus. Torricell'?s Theorem.

First notions about the improper integrals

1) Giovanni Emmanuele Analisi Matematica I Pitagora editore

2) M. Bramanti, C.D. Pagani, S. Salsa: Matematica - calcolo infinitesimale e algebra lineare, ed. Zanichelli

3) S. Salsa, A. Squellati: Esercizi di Matematica 1, ed. Zanichelli

4) D. Benedetto, M. Degli Esposti, C. Maffei  Matematica per le scienze della vita. Casa editrice ambrosiana

5)  D. Greco, P. Valabrega: Cento pagine di algebra lineare

The exam consists of a written test and an oral test. The written test consists of technical exercises 
and theory questions.  The dates of the exams will be on the website of the degree course. 
You need to make a reservation on the student portal. 
Reservations are possible up to two days before the date of the exam.
 
Final grades will be assigned taking into account the following criteria:
Rejected: Basic knowledges have not been acquired. 
18-23: Basic knowledges have been acquired. The student has sufficient communications skills and making judgements. He is able to solve basic 
exercises without errors
24-27: All the  knowledges have been acquired. The student  has good communications skills and making 
judgements. He is able to solve most of  the assigned exercises with few errors. 
28-30
 cum laude: All the knowledges have been completely acquired. The 
student is able to apply  the knowledges to some  new problems. He has 
excellent communications skills, learning skills and making judgements. He is able to solve all the assigned exercises without errors

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs.

It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or DSA) of the Department.

On the page of Studium one can find the exercises assigned in the last examines. In the oral part of the
 exam  the student must  present the main results of each chapter showing 
that he has understood the concepts and that he is able to connect them. In addition to proving a
 theorem, the student is required to provide examples and counter-examples that make it clear the 
importance of the hypotheses of the theorem itself. Such examples will be provided in class but the 
student may be able to develop similar ones. For example I can require to demostrate Weierstrass Theorem 
for continuous functions and after some example of function that does not verify the 
hypotheses of the Theorem and that it does not admit minimum or maxim point. Other examples:
Theorem of the limit of an encreasing ( or decreasing ) sequence. The student must underline 
why it is important thae monotony of the sequence. Fermat's Theorem: after the proof the student 
can be required to provide an example of functions such that in a suitable point has derivative zero 
although  the point is not an extreme point.