guida - Facoltà di Ingegneria - Università Politecnica delle Marche
guida - Facoltà di Ingegneria - Università Politecnica delle Marche
guida - Facoltà di Ingegneria - Università Politecnica delle Marche
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GUIDA DELLO STUDENTE<br />
Aims<br />
(english version)<br />
Knowledge of the language of Mathematical Analysis. Knowledge of basic elements of <strong>di</strong>fferential calculus for functions of one variable and<br />
applications.<br />
Topics<br />
Sets, Relations and Functions. Natural, Integer, Rational and Real numbers. Complex numbers, trigonometric and exponential representation.<br />
De Moivre Formula. The Induction principle. Modulus and powers. Exponential, logaritmic and angular functions. Limit of real sequences and<br />
its properties. Indeterminate forms. Monotone sequences. The Neper's number and related limits. Asymptotic comparison. Limits of real<br />
function of real variale. Properties. Indeterminate forms. Asymptotic comparison. Monotone functions. Continuity; The Weierstrass's and the<br />
Interme<strong>di</strong>ate Values Theorems. Derivative and Derivative Formulas. Successive Derivative. The Fermat's, Rolle's, Lagrange's and Cauchy's<br />
Theorems. Derivative and monotonicity. Convexity. Primitives. The De L'Hospital's Theorems. Taylor Formulas. Asymptots and the study of<br />
the graphs of functions. Definite Integral and its properties. Fundamental Theorem and Formula of the Integral Calculus. Indefinite Integral and<br />
integration methods: sum decomposition, by parts and sostitution. Improper integral and convergence tests.<br />
Series. The Geometric and Harmonic Series. Convergence tests. Absolute convergence. Leibnitz Theorem.<br />
Taylor series and sufficient con<strong>di</strong>tions for the convergence. Power series. Ra<strong>di</strong>us of Convergence and Abel Theorem. Term by term<br />
<strong>di</strong>fferentiantion and integration of a power series. Taylor Series of elementary functions.<br />
Fourier Series. Bessel Inequality and Riemman Lenesgue Lemma. Punctual Convergence and Integration term by term of Fourier series. An<br />
introduction to Lapace and Fourier Transform.<br />
Exam<br />
written and oral<br />
Textbooks<br />
P. Marcellini, C. Sbordone, Elementi <strong>di</strong> Analisi Matematica I, Liguori E<strong>di</strong>tore<br />
M. Bertsch, R. Dal Passo, L. Giacomelli. Analisi Matematica, (seconda E<strong>di</strong>zione) McGraw-Hill<br />
Lecture notes of the course<br />
Tutorial session<br />
two hours per week to be established with the students<br />
35
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