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anno/Analisi matematica 1/2023-03-31, 04, 18, Teoria sulle derivate/main.tex +++ b/Primo anno/Analisi matematica 1/2023-03-31, 04, 18, Teoria sulle derivate/main.tex @@ -447,24 +447,15 @@ Allora $\frac{f(x)}{g(x)} \tendsto{x_0} L$. \end{theorem} - \begin{proof} - Si consideri il caso (a) per $x_0$ finito. Si ponga $f(x_0) = g(x_0) := 0$. Senza perdità di generalità si + \begin{proof}[Dimostrazione del caso $(a)$ con $x_0$] + Senza perdità di generalità si ponga $f(x_0) = g(x_0) := 0$ e si assuma che $I$ sia un intorno destro di $x_0$. Sia $x \in I \setminus \{x_0\}$, da cui si ricava che $x > x_0$. \\ Si osserva che $\frac{f(x)}{g(x)} = \frac{f(x) - f(x_0)}{g(x)} - g(x_0)$. Per il teorema di Cauchy, esiste allora $\tilde{x} \in (x_0, x)$, in funzione di $x$, tale che $\frac{f(x)}{g(x)} = \frac{f'(\tilde x)}{g'(\tilde x)}$. Allora $\lim_{x \to x_0} \frac{f(x)}{g(x)} = \lim_{x \to x_0} \frac{f'(\tilde x)}{g'(\tilde x)} = \lim_{\tilde x \to x_0} \frac{f'(\tilde x)}{g'(\tilde x)} = L$, dove si è utilizzato che $\tilde x \tendsto{x_0} x_0$ per il teorema del confronto applicato sulla - relazione $x_0 < \tilde x < x$. \\ - - Si consideri ora il caso (b) per $x_0$ finito. Siano $x_1 > x_0$ - tali che $x_1 > x > x_0$. Allora vale la seguente identità: - - \[ \frac{f(x)}{g(x)} = \left( \frac{f(x) - f(x_1)}{g(x) - g(x_1)} + \frac{f(x_1)}{g(x) - g(x_1)} \right) \frac{g(x) - g(x_1)}{g(x)}. \] - - Si osserva allora che: - - \[ \lim_{x \to x_0} \frac{f(x)}{g(x)} = \lim_{x \to x_0} \frac{f(x) - f(x_1)}{g(x) - g(x_1)}. \] + relazione $x_0 < \tilde x < x$. \end{proof} \begin{remark} @@ -483,51 +474,4 @@ 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/Primo anno/Aritmetica/README.md b/Primo anno/Aritmetica/README.md index bc70e5c..af22840 100644 --- a/Primo anno/Aritmetica/README.md +++ b/Primo anno/Aritmetica/README.md @@ -7,4 +7,6 @@ Le varie cartelle contengono alcuni estratti di quelle che sarebbero dovute essere delle dispense completamente sostitutive del corso. Accorgendomi tuttavia della precarietà di alcuni capitoli, e influenzato anche dall'esistenza di dispense che seguono la stessa filosofia (come quelle di [Diego Monaco](https://github.com/diego-unipi/Appunti-Aritmetica)), ho deciso di procedere all'abbandono del progetto. Gli -appunti già creati -- spesso e volentieri tratti interamente da alcuni [miei scritti](https://scritti.hearot.it) -- sono stati tuttavia revisionati e integrati in una cartella unica assieme ai PDF degli appunti che ho preso durante le lezioni con [Microsoft Journal](https://apps.microsoft.com/store/detail/microsoft-journal/9N318R854RHH?hl=it-it&gl=it). \ No newline at end of file +appunti già creati -- spesso e volentieri tratti interamente da alcuni [miei scritti](https://scritti.hearot.it) -- sono stati tuttavia revisionati e integrati in una cartella unica assieme ai PDF degli appunti che ho preso durante le lezioni con [Microsoft Journal](https://apps.microsoft.com/store/detail/microsoft-journal/9N318R854RHH?hl=it-it&gl=it). + +In ogni caso, il corso di Aritmetica è terminato, e quindi questa cartella non vedrà più aggiornamenti. \ No newline at end of file diff --git a/Primo anno/Fond. di programmazione/README.md b/Primo anno/Fond. di programmazione/README.md index b02bd21..cdddde3 100644 --- a/Primo anno/Fond. di programmazione/README.md +++ b/Primo anno/Fond. di programmazione/README.md @@ -5,4 +5,5 @@ - [Sito web 🔗](http://pages.di.unipi.it/bodei/CORSO_FP_22/FP/index.html) Gli appunti relativi al corso di Fondamenti di programmazione sono stati presi con [Microsoft Journal](https://apps.microsoft.com/store/detail/microsoft-journal/9N318R854RHH?hl=it-it&gl=it). Seppur carenti per quanto riguarda la parte relativa al linguaggio C, tali appunti approfondiscono in modo particolare -il programma riguardante la semantica, gli automi a stati finiti (DFA, NFA, ε-NFA) e le grammatiche. \ No newline at end of file +il programma riguardante la semantica, gli automi a stati finiti (DFA, NFA, ε-NFA) e le grammatiche. +Il corso è terminato, e quindi questa cartella non vedrà più aggiornamenti. \ No newline at end of file