CALCULUS - 2

This course belongs to the curricular activities of the Bachelor in Computer Science.
The course aims at providing students with the basic instruments of Calculus 2, concerning functions of two or more real variables, and the solution of ordinary differential equations of the first order.

The aim of this course is to develop skills one needs to solve differential and integral questions arising in technology, science, economics and business.

Regular and active participation in the teaching activities offered by the Course, together with independent learning activities, will enable students to:
1. (knowledge and understanding)
-- acquire knowledge and understanding regarding some basic concepts of Calculus 2 concerning two-variable functions.

2. (applying knowledge and understanding)
-- solve simple ODEs;
-- compute the points of minimum, maximum, and saddle points of a multi-variable function;
-- draw the graph of two-variable functions;
-- compute the integral of two-variable functions;

3. (making judgements)
-- correctly understand Math statements concerning multi-variable functions.

The students are supposed to know the main concepts of the courses Calculus Module 1 and Linear Algebra.

Students must pass the exam of Module 1 before attempting the exam of the Module 2.

Improper Integrals
Ordinary differential equations of the first order.
Vectorial calculus and parametric curves.
Real functions of two or more variables.
Limits and continuity in n-dimensions.
Directional derivatives.
Partial derivatives and gradient.
Higher order derivatives.
Maximum, minimum and saddle points.
Two-dimensional integrals and reduction methods; rule of change of variables.

It is recommended to study on the material provided in the moodle page of the course

The following books can be helpful:

M. Bertsch, R. Dal Passo, L. Giacomelli. Analisi Matematica. McGraw-Hill, Milano 2007
Bramanti Pagani Salsa. Analisi Matematica 2, Zanichelli
Salsa Squellati. Esercizi di Analisi Matematica 2. Zanichelli

Examination Module 2: it consists of a written test (duration: two hours) with open-ended problems aimed at verification of all the course contents in order to evaluate the ability of students in solving differential equations, analyzing parametric curves, optimizing multi-variable functions and integrating two-variable mappings.. The grade is given by the sum of the scores assigned to the single problems. The exam will be passed if and only if the minimum grade is 18. The maximum grade of the written test is 30. An oral exam is (absolutely) optional: in this case, the overall result may include failure and the final grade can at most be three points higher than the one of the written test. The maximum grade, 30 summa cum laude, is assigned exclusively to the students who have passed the written test with at least 28 points and have demonstrated mastery of the theoretical aspects in the oral exam..

Indications for the final grade of Calculus.
Only the students that already passed the exam of Module 1 can take the exam of Module 2. It is possible to take both the modules 1 and 2 in the same day, in this rigorous succession. The final grade of Calculus is the average of the grades of the two modules, with approximation by excess. The maximum grade of Calculus, 30 summa cum laude, is assigned with the full agreement of both the teachers only. In any case, students must pass the exams of both modules in the same academic year. Otherwise, any passing grade achieved in Module 1 will be cancelled.

Classroom lessons and exercises. The moodle platform is exploited, in order to deliver supplementary material.

written