Facoltà di Ingegneria - Guida degli insegnamenti (Syllabus)

Algebric calculus and analytic geometry.

The aim of the course is that of providing the theoretical, methodological and practical elements of mathematical analysis with the objective of understanding physical and chemical phenomena and of providing (together with the course of Calculus 2) the mathematical tools commonly employed in the engineering sciences. In particular, the course aims at providing the student with the basic knowledge of the differential and integral calculus for real functions of one variable with a number of applications.

The main classical results of analysis will be introduced, in order to develop the students ability to follow rigorous mathematical thought and to use mathematical methods towards the formulation of models, the analysis and the solution of problems. The theoretical notions will be accompanied by numerous applications. This path will lead the student to achieving the capability of: 1. analyzing problems; 2. detecting the methods of solution; 3. choosing the best solving technique.

The expertise acquired in this course will be needed in order to study the material of later courses. Individual and collective problem-solving sessions will improve the ability to develop independent thought and learning capabilities. Oral presentations of the topics taught in the course, with the language proper of the basic disciplines of the degree course will help developing communication skills.

Elements of set theory . The set of the real numbers and its properties. Complex numbers. Numerical sequences and definition of limit. Numerical series and their behavior. Functions of one variable: elementary functions. Limit of a function. Continuous functions and their properties. Differential calculus for functions of one variable. Graph of a function. Some optimization problems. Taylor polynomial . Taylor series . Complex exponential. Integral calculus for functions of one variable: primitive of a function. Improper integral and convergence criteria. Sequences and series of functions: pointwise and uniform convergence. Power series and Fourier series.

The student will be assessed through on a written test and an oral test. The written test will assess the ability to solve problems by using the learned techniques. The oral test will assess the learning of the theory and the exposition skills.

In the exams the student have to prove that he understands the concepts presented in the course, that he knows the results and methods presented in the lectures, and finally that he is able to set a problem and solve it properly through the learned methods.

For each of the tests indicated above it is assigned a score between zero and thirty. The student will be admitted to the oral exam only if he passed the written test. The highest rating, equal to thirty of thirty, is achieved by demonstrating in-depth knowledge of the course contents and full autonomy in the performing the test. The minimum assessment, equal to eighteen of thirty, is assigned to students who manage to solve the proposed problems and who demonstrate sufficient knowledge of the topics of matter.

The overall grade, out of thirty, is derived from the comparative evaluation of both tests. The praise is reserved to the students who, having carried out the tests correctly and completely, has shown a special independence and excellence.

Marco Bramanti, Carlo D. Pagani, Sandro Salsa, "Analisi matematica 1", Zanchelli - F.G. Alessio-P. Montecchiari, Note di Analisi Matematica Uno, Esculapio

- Ingegneria Meccanica (Corso di Laurea Triennale (DM 270/04))

**Università Politecnica delle Marche**

P.zza Roma 22, 60121 Ancona

Tel (+39) 071.220.1, Fax (+39) 071.220.2324

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