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 order to acquire the criteria, methods and limits of application of mathematical methods to real problems. In particular, the course aims to provide students with the knowledge of the basic elements of differential and integral calculus for functions of one variable and various applications.

In order to accustom the student to follow simple concatenations of various arguments and to develop the ability to apply mathematical methods to formalize, identify, and resolve problems of Information Engineering, classical results of Mathematical analysis for real functions of one 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 solution.

The resolution in class 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 and Fourier series

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

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

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.

To each test indicated it is assigned a score between zero and thirty. The student will be admitted to the second test only if he passed the first one with an evaluation equal to or greather than eighteen of thirty. 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 e P. Montecchiari, Note di Analisi Matematica uno, Esculapio (ristampa 2015)

- Ingegneria Informatica e dell'Automazione (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