Paolo Marcellini e Carlo Sbordone
Elementi di analisi matematica uno

THE REVIEW

This book of Analisi Matematica Uno is the torment of a good part of students enrolled in the first year of Engineering. You immediately find yourself engaged, grappling with a challenging subject like infinitesimal calculus, in a new context like university. The subject covered is one of the most arduous to tackle in the engineering study path and is already quite a testing ground. Before the Moratti reform (the 3 + 2), the engineering degree courses had a duration of 5 years. The exam of Analisi Uno (together with Physics One and Physics Two) was the most difficult of the old two-year period, and there wasn't a single freshman or overdue student who didn't deal with this subject, which became the nightmare of entire generations of aspiring engineers. Today, with the new reform, it is simply a difficult exam. All newly enrolled students are required to take it, everyone, from civil to industrial to computer engineers. There is no crossing the ford, there is no other solution to taking it than to change faculty.

The book of Analisi Uno that I am most familiar with is the Marcellini-Sbordone, widely used in all universities, preferred by a significant number of professors. This "Elementi di Analisi matematica uno" is a comprehensive and exhaustive book, perhaps more focused on theoretical aspects and less on exercises (which are better done with all the books one can find). The Marcellini-Sbordone, when taken as a course book, is explained literally on the board and with its shrewd and refined quality, it chisels solutions and displays very high numerical virtuosity.

Of course, it starts with set theory, just a light brush-up to move on to functions, which are the first difficulties of the course, with all the Cartesian diagrams and all the study, including the domain and the absolute value. The most difficult are the trigonometric functions, although I also found logarithmic and hyperbolic functions challenging. The second chapter is devoted to the completion of real numbers, with the study of extremes, and right after, there's another hurdle: limits, in all their forms and with all the theorems, one of the most important is the uniqueness of limit theorem, another important theorem is the permanence of sign, attached with it all the monotonic sequences and that of Cauchy. After Cauchy, it's time for derivatives, in their most complete forms. The application of derivatives includes relative maxima and minima with the Fermat's theorem, along with those of Rolle and Lagrange, two other fundamental theorems. Then we move on to functions with multiple real variables, finally reaching the point of maximum difficulty: integrals, concluding with a few notes on sequences that will be the first topic of the Mathematics Two course.

In conclusion, it is recommended for those who are about to attend Engineering to study Analysis right away; if you have a feeling of having some gaps, it's better to catch up quickly to not fall behind as the course progresses. To pass this subject, you need commitment and study. The Marcellini-Sbordone became likable to me after making me spend days over it and sleepless nights. In the end, you realize that this is just a small step towards the final goal, and especially looking at the other exams, this one can even turn out to be easier than the others.