Facoltà di Ingegneria - Guida degli insegnamenti (Syllabus)

Basic elements of Calculus and Analityc Geometry

The course aims to provide theoretical, methodological and applicative knowledge of mathematical analysis in the differential and integral calculus in order to gain expertise in the application of mathematical methods to real problems. In particular, the course aims to provide students with the knowledge of the basic elements of differential calculus and of the theory of integration for functions of one variable with various applications.

In order to accustom the student to follow simple concatenation of various arguments and to develop the ability to apply the methods of differential and integral calculus for solving engineering problems, the classical results of mathematical analysis for real functions of real variable, accompanied by many applications, will be introduced. This path will lead the student to the attainment of the following skills: 1. Ability to analyze problems; 2. ability to identify various solution methods; 3. ability to choose the best solving route.

The skills acquired during the course will be necessary to address the study of future courses. The resolution in the classroom and individually of many problems and exercises will improve the learning ability and independent judgment. The exposure of the learned topics and the specificity of the language of the basic sciences will develop the ability to communicate.

Sets, Relations and Functions. Natural, Integer, Rational and Real numbers. Complex numbers, trigonometric and exponential representation. De Moivre Formula. The Induction principle. Modulus and powers. Exponential, logaritmic and angular functions. Limit of real sequences and its properties. Indeterminate forms. Monotone sequences. The Neper's number and related limits. Asymptotic comparison. Limits of real function of real variale. 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. Primitives. The De L'Hospital's Theorems. Taylor Formulas. Asymptots and the study of the graphs of functions. Riemann integral and integrability. Definite Integral and its properties. Fundamental Theorem and Formula of the Integral Calculus. Indefinite Integral and integration methods: sum decomposition, by parts and sostitution. Improper integral and convergence tests. Series. The Geometric and Harmonic Series. Convergence tests. Absolute convergence. Leibnitz Theorem. Introduction to Taylor 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 has 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.

In the written test is evaluated the ability to set up and properly solve the posed problems, using their own methods of the course. In the oral exam it is assessed 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.

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.

F.G. Alessio e P. Montecchiari, Note di Analisi Matematica uno, Esculapio (ristampa 2015)

- Ingegneria Gestionale (Corso di Laurea Triennale Fuori Sede (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