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a/Analisi matematica 1/Parte teorica/2023-03-31, 04, 18, Teoria sulle derivate/main.tex b/Analisi matematica 1/Parte teorica/2023-03-31, 04, 18, Teoria sulle derivate/main.tex index 9a55b3c..7d73f5a 100644 --- a/Analisi matematica 1/Parte teorica/2023-03-31, 04, 18, Teoria sulle derivate/main.tex +++ b/Analisi matematica 1/Parte teorica/2023-03-31, 04, 18, Teoria sulle derivate/main.tex @@ -4,7 +4,7 @@ \title{\textbf{Note del corso di Analisi Matematica 1}} \author{Gabriel Antonio Videtta} -\date{31 marzo, 4 e 18 aprile 2023} +\date{31 marzo, 4, 18 e 20 aprile 2023} \begin{document} @@ -483,4 +483,51 @@ Si consideri il rapporto incrementale $\frac{f(x) - f(x_0)}{x - x_0}$. Allora, per $x \to x_0$, per il teorema di de l'Hopital, $f'(x_0) = \lim_{x \to x_0} f'(x)$. \end{proof} + + \begin{theorem} (sullo sviluppo di Taylor) + Sia $I$ un intervallo e sia $\xbar \in I$. Sia $f : I \to \RR$ e + sia $d \in \NN$. Sia $f$ derivabile $d-1$ dappertutto e sia derivabile + $d$ volte in $\xbar$. Allora, detti + + \[ P_d(h) = f(\xbar) + f'(\xbar) h + \ldots + \frac{f^{(d)}(\xbar)}{d!} h^d, \] + + \[ R_d(h) = f(\xbar + h) - P_d(h), \] + + \begin{enumerate}[(a)] + \item $R_d(h) = o(h^d)$ per $h \to 0$, + + \item se $f$ è derivabile $d$ volte su $I$ e $d+1$ volte in $\xbar$, allora $R_d(h) = O(h^{d+1})$ per $h \to 0$ e $\frac{R_d(h)}{h^{d+1}} \to \frac{f^{(d+1)}(\xbar)}{(d+1)!}$, + + \item se $f$ è derivabile $d+1$ volte su $I$, allora + $\forall h \mid \xbar + h \in I$, $\exists \tilde x \in [\xbar, \xbar + h] \mid R_d(h) = \frac{f^{(d+1)}(\tilde x)}{(d+1)!}$ (\textit{formula del resto di Lagrange}), + + \item se $f \in C^{d+1}$, allora $R_d(h) = \frac{1}{d!} \int_0^h (h-t)^d f^{(d+1)}(\xbar + t) \, dt$ (\textit{formula integrale}). + \end{enumerate} + \end{theorem} + + \begin{proof} % TODO: dimostrazione farlocca, migliorarla + (d) Si assuma $\xbar = 0$ e $f \in C^{d+1}$. Innanzitutto + si osserva che la tesi è equivalente a mostrare che $f(h) = P_d(h) + \frac{1}{d!} \int_0^h (h-t)^d f^{(d+1)}(t) \, dt$. \\ + + Se $d=0$, $f(h) = f(0) + \int_0^h f'(t) \,dt$ (teorema fondamentale + del calcolo integrale). $f(h) = f(0) + \abs{-(h-t) f'(h)}_0^h + \int_0^h (h-t) f''(t) \, dt$. [...] % TODO: si continua per induzione + \end{proof} + + \begin{proposition} + Sia $f : I \to \RR$ derivabile. Sia $\xbar \in I$ tale che + $f'(\xbar) = 0$ ed esista $f''(\xbar)$. Allora: + + \begin{enumerate}[(i)] + \item $f''(\xbar) > 0 \implies \xbar$ è un punto di minimo locale stretto, + \item $f''(\xbar) < 0 \implies \xbar$ è un punto di massimo locale.stretto, + \item $\xbar$ è un punto di minimo locale $\implies f''(\xbar) \geq 0$, + \item $\xbar$ è un punto di massimo locale $\implies f''(\xbar) \leq 0$. + \end{enumerate} + \end{proposition} + + \begin{proof} + Per lo sviluppo di Taylor, $f(\xbar + h) = f(\xbar) + f'(\xbar) h + \frac{1}{2} f''(\xbar) h^2 + o(h^2) \implies \frac{f(\xbar + h) - f(\xbar)}{h^2} = \frac{1}{2} f''(\xbar) + o(1)$ (infinitesimo). + Allora $\frac{f(\xbar + h) - f(\xbar)}{h^2} \tendsto{h} L > 0$. + Quindi permanenza del segno. + \end{proof} \end{document} \ No newline at end of file diff --git a/Analisi matematica 1/Parte teorica/2023-04-21/main.pdf b/Analisi matematica 1/Parte teorica/2023-04-21/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7248bad0eec46a101e589c361f20c71e742f4654 GIT binary patch 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+\usepackage[italian]{babel} + +\title{\textbf{Note del corso di Analisi Matematica 1}} +\author{Gabriel Antonio Videtta} +\date{21 aprile 2023} + +\begin{document} + + \maketitle + + \begin{center} + \Large \textbf{Integrale secondo Riemann} + \end{center} + + \begin{definition} (partizione di un intervallo) + Preso $[a, b] \subset \RR$. Sia $\sigma = \{x_0, x_1, \ldots, x_n \}$ + con $n \in \NN$. Diciamo che $\sigma$ è una \textbf{partizione} + di $[a, b]$ se $a = x_0 < x_1 < \cdots < x_n = b$. + \end{definition} + + \begin{definition} (taglia di una partizione) + Si definisce $\delta(\sigma)$, con $\sigma$ partizione, + come la massima distanza tra due punti consecutivi della partizione $\sigma$, ed è detta \textbf{parametro di finezza} della partizione $\sigma$. + \end{definition} + + \begin{definition} (ordinamento sulle partizioni) + Siano $\sigma_1$, $\sigma_2$ due partizioni di $[a, b]$. Allora + $\sigma_2$ è più fine di $\sigma_1$ se $\sigma_1 \subset \sigma_2$. + \end{definition} + + \begin{remark} + Siano $\sigma_1$ e $\sigma_2$ sono due partizioni di $[a, b]$. \\ + + \li Chiaramente $\sigma_1 \cup \sigma_2$ è più fine sia di + $\sigma_1$ che di $\sigma_2$. + + \li Inoltre, se $\sigma_1$ è più fine di $\sigma_2$, + $\delta(\sigma_2) \geq \delta(\sigma_1)$. + \end{remark} + + \begin{definition} [somma di Riemann inferiore e superiore] + Sia $f : [a, b] \to \RR$ limitata e sia $\sigma = \{ x_0, \ldots, x_n \}$ una partizione + di $[a, b]$. Si definisce allora la \textbf{somma di Riemann inferiore} + $S'$ come: + + \[ S'(\sigma) = \sum_{i=1}^n \left( \inf_{x_{i-1} \leq x \leq x_i} f \right) (x_i - x_{i-1}), \] + + e si definisce la \textbf{somma di Riemann superiore} $S''$ come: + + \[ S''(\sigma) = \sum_{i=1}^n \left( \sup_{x_{i-1} \leq x \leq x_i} f \right) (x_i - x_{i-1}). \] + \end{definition} + + \begin{proposition} + Sia $f : [a, b] \to \RR$ limitata. Allora: + + \begin{enumerate}[(i)] + \item $\forall \sigma$ partizione di $[a, b]$, $S'(\sigma) \leq S''(\sigma)$, + + \item $\forall \sigma_1$, $\sigma_2$ partizioni di $[a, b]$ + con $\sigma_2$ più fine di $\sigma_1$, vale che + $S'(\sigma_1) \leq S'(\sigma_2) \leq S''(\sigma_1) \geq S'(\sigma_2)$. + + \item $\forall \sigma_1$, $\sigma_2$ partizioni di $[a, b]$, + $S'(\sigma_1) \leq S''(\sigma_2)$. + \end{enumerate} + \end{proposition} + + \begin{proof} + \begin{enumerate}[(i)] + \item ovvio. + + \item Sia $\sigma_1 = \{ x_0, \ldots, x_n \}$ e sia + $\sigma_2 = \sigma_1 \cup \{ \xi \}$. Aggiungi un elemento + e la disuguaglianza regge. Fallo aggiungendo ogni elemento. + + \item Usa l'unione che è più fine. + \end{enumerate} + \end{proof} + + \begin{definition} [integrale di Riemann inferiore e superiore] + Si definisce l'\textbf{integrale di Riemann inferiore} di $f$ come: + + \[ I_- = \sup \{ S'(\sigma) \mid \sigma \text{ partizione di } [a, b] \}, \] + + e l'\textbf{integrale di Riemann superiore} di $f$ come: + + \[ I_+ = \inf \{ S''(\sigma) \mid \sigma \text{ partizione di } [a, b] \}. \] + \end{definition} + + \begin{remark} + Si osserva che $I_+ \geq I_-$. + \end{remark} + + \begin{definition} [integrale di Riemann] + Sia $f : [a, b] \to \RR$ limitata. Si dice che + $f$ è \textbf{integrabile secondo Riemann} in $[a, b]$ + se $I_+ = I_-$. + \end{definition} + + \begin{definition} [uniformemente continua] + Sia $X \subseteq \RR$ e sia $f : X \to \RR$. Si dice + che $f$ è \textbf{uniformemente continua} se $\forall \eps > 0$, + $\exists \delta(\eps) > 0$ tale che $\forall x, \xbar \in X, + \abs{x-\xbar} < \delta \implies \abs{f(x)-f(\xbar)} < \eps$. + \end{definition} + + \begin{remark} + Se $f$ è uniformemente continua, chiaramente $f$ è + continua, benché non sia vero il viceversa. + \end{remark} + + \begin{example} + Sia $f : [a, +\infty) \to \RR$ tale che $f(x) = \sqrt{x}$. Sia + $x > \xbar$, allora $\sqrt{x} > \sqrt{\xbar}$. Sia + $x = \xbar + h$. Si considera $\sqrt{\xbar + h} - \sqrt{\xbar} < \eps$, + allora $\sqrt{\xbar + h} < \eps + \sqrt{\xbar}$, da cui si + deduce che $\xbar + h < \eps^2 + \xbar + 2 \eps \sqrt{\xbar}$, + ossia $h < \eps^2 + \eps \sqrt{\xbar}$. Preso allora $h < \eps^2$, + si ha che $f$ è uniformemente continua. + \end{example} + + \begin{example} + Come prima, ma per $\sin(x)$. Per Lagrange $\exists \tilde x \in (x, \xbar) \mid \frac{\sin(x) - \sin(\xbar)}{x-\xbar}=\cos(\tilde x)$, + da cui $\sin(x) - \sin(\xbar) = \cos(\xbar) (x - \xbar)$, ossia + $\sin(x) - \sin(\xbar) \leq x - \xbar \leq \delta = \eps$. + (In realtà vale per ogni $f$ con $\abs{f'} \leq l$.) + \end{example} + + \begin{example} + Dimostra che non sono unif. continue: $e^x$ (con $\log(n+1)$ e $\log(n)$), $\log(x)$ (con $e^{-n+1}$ e $e^{-n}$), + $\sin(x^2)$ (con $\sqrt(2\pi n + \pi/2)$ e $\sqrt{2\pi n}$). + \end{example} + + \begin{theorem} + $f : [a, b] \to \RR$ continua. Allora $f$ è uniformemente continua. + \end{theorem} + + \begin{proof} + Per assurdo suppongo che $f$ non sia uniformemente continua. + Allora considero $x_n$ e $\xbar_n$ tale che + $\abs{x_n - \xbar_n} \leq \frac{1}{n}$ ma $\abs{f(x_n) - f(\xbar_n)} > \eps$ $\forall n$. Per Bolzano-Weierstrass, $\exists n_k$ sottosuccessione di tale che $x_{n_k} \to x_0 \in [a, b]$. + Anche $\xbar_{n_k} \to x_0 \in [a, b]$. Poiché $f$ è continua, + $f(x_{n_k}) \to f(x_0)$, $f(\xbar_{n_k}) \to 0$, e quindi che + $\abs{f(x_{n_k}) - f(\xbar_{n_k})} \to 0$, contraddizione perché + $> \eps$. + \end{proof} + + \begin{theorem} + $f : [a, b] \to \RR$ continua allora $f$ è integrabile + secondo Riemann. + \end{theorem} + +\end{document}