Facoltà di Ingegneria - Guida degli insegnamenti (Syllabus)

Basic elements of Calculus and Analytic Geometry

The aim of the course is that of providing the theoretical, methodological and practical elements of mathematical analysis needed for the understanding of physical and chemical phenomena and of providing 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. analizing 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.

Natural, Integer, Rational and Real numbers. The Induction principle. Limit of real sequences and its properties. Indeterminate forms. Monotone sequences. The Neper's number and related limits. Limits of real function of real variable and its properties. Indeterminate forms. Asymptotic comparison. Monotone functions. Continuity. The Weierstrass's and the Intermediate Values Theorems. Derivative and Derivative Formulas. Successive Derivative. The Fermat's, Rolle's, Lagrange's and Cauchy's Theorems. Derivative and monotonicity. Convexity. The De L'Hospital's Theorems and Taylor's Formula. Asymptots and study of the graphs of functions. Definite integral and its properties. Fundamental Theorem and Formula of the integral calculus. Indefinite Integral and integration methods: by sum decomposition, by parts and substitution. Improper integral and convergence tests. Numerical series and convergence criteria. Power and Taylor series.

The student will be assessed through two written tests and an oral test. The first written test will assess the learning of the theory, the second one the ability to solve problems by using the learned techniques. The oral test will focus on a discussion of the two written tests.

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

The first written test will assess the knowledge of the concepts and results presented in the lectures, the presentation skills and the ability to make connections between the various concepts introduced. The second test will assess the ability to set up and properly solve the posed problems, by using the learned techniques. The oral test will focus on a discussion of the two written tests. In the oral exam

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

F.G. Alessio-P. Montecchiari, Note di Analisi Matematica Uno, Esculapio

- Ingegneria Civile e Ambientale (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

P.I. 00382520427