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z0;4|fYk{~Ytt~D$C#5_LhCB8GvGn?90;!hwm$mp}&LdHEA^oY?TLQ2#p#w=^7#G%W zOCtc?s3k}SHh(kOr~`1u<`_Y35x(^#$h!BKSZx>DL3}5@ekjfg2lz!g{N1tt=Eu92 zWbf9ZKEaP>&@lq$4k04J8Z^SPKZ(lDiJUwGGVVRl$b4`3^$w~PRVJ8hMsFIad?Iv& zQ=dk2bjcU?vAnxgXWR<^yv~`SXx#J(w zwa<9o+?{kgcg_io%^Or&=rXldc6|25ayOUd^%n!)aeyn8gjmdcMM-{tAT13ComO|Z z0dt=MEY(@j2lPo*FE-4n;SO0ff5Ml1FM)LK@8CG{@7vBO!>6OUPm6u#eFEGQ42)Y# zlMTCA1sAq)IA6ouSI-Y<7@Jf|;)us{&Y2}i{ioY82nHpK}!n6m56 zcNhcDRkFZ_srk;>6r!Sevi^p+vU5~n!VwPC0^tg_)nA}YXSD?X=YVu_HgI%ycl;TF S05&!ND+3G}nTVVy%>M`F*sy~D literal 0 HcmV?d00001 diff --git a/ws2019/ana/lectures/analysis22.tex b/ws2019/ana/lectures/analysis22.tex new file mode 100644 index 0000000..e016c56 --- /dev/null +++ b/ws2019/ana/lectures/analysis22.tex @@ -0,0 +1,297 @@ +\documentclass{../../../lecture} + +\usepackage[]{subcaption} + +\begin{document} + +\begin{satz}[Satz von Taylor] + Jede Funktion $f \in C^{n+1}(D,\R)$ lässt sich für + $x, x_0 \in D$ nach Potenzen von $(x-x_0)$ + entwickeln: + \begin{align*} + f(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2}(x-x_0)^2 + \ldots + f^{(n)}(x_0)\frac{(x - x_0)^{n}}{n!} + R_{n+1}(x) + .\end{align*} Dabei ist das Restglied $R_{n+1}(x)$: + \begin{align*} + R_{n+1}(x) = f^{(n+1)}(\xi) \frac{(x-x_0)^{n+1}}{(n+1)!} + .\end{align*} mit $\xi$ ein Punkt zwischen $x_0$ und $x$ + ($\xi = x_0 + \tau(x-x_0)$, $\tau \in (0,1)$). + + Der Ausdruck + \[ + T_n(x) = T_{n}(f, x, x_0) := \sum_{k=0}^{n} \frac{f^{(k)}(x_0)}{k!} (x-x_0)^{k} + .\] heißt Taylorpolynom $n$-ter Ordnung von $f$ bei $x_0$. +\end{satz} + +\begin{proof} + Für $x = x_0$: klar. + + Sei $x \neq x_0$. Betrachte $R = R(x, x_0)$ definiert durch + $f(x) = T_N(f, x, x_0) + \frac{(x-x_0)^{n+1}}{(n+1)!}\cdot R$. + + Für $y \in D$ definiere + \[ + \varphi(y) := f(x) - \sum_{k=0}^{n} \frac{f^{(k)}(y)}{k!}(x-y)^{k} - \frac{(x-y)^{n+1}}{(n+1)!}R + .\] + Dann folgt $\varphi(x_0) = 0 = \varphi(x)$, $\varphi \in C^{1}$.\\ + $\stackrel{\text{Satz von Rolle}}{\implies}$ $\exists \xi$ zwischen $x$ und $x_0$ mit $\varphi'(\xi) = 0$. + \begin{align*} + 0 = \varphi'(\xi) &= - \sum_{k=0}^{n} \frac{f^{(k+1)}(\xi)}{k!} (x-\xi)^{k} + - \sum_{k=0}^{n} \frac{f^{(k)}(\xi)}{k!}k(x-\xi)^{k-1}(-1) + - \frac{(n+1)(x-\xi)^{n}}{(n+1)!} R \\ + &\stackrel{\mathclap{\text{Teleskop}}}{=} \quad - \frac{f^{n+1}(\xi)}{n!}(x-\xi)^{n} + \frac{(x-\xi)^{n}}{n!} R \\ + &= \frac{(x-\xi)^{n}}{n!} \left( - f^{(n+1)}(\xi) + R \right) + .\end{align*} $\implies R = f^{(n+1)}(\xi)$, $\xi$ zwischen $x$ und $x_0$. +\end{proof} + +\begin{satz} + Sei $f$ auf einem beschränkten Intervall $(a,b)$ eine + $C^{\infty}$ Funktion mit gleichmäßig beschränkten Ableitungen. + \[ + \sup_{x \in (a,b)} \left| f^{(n)}(x) \right| \le M < \infty + .\] Dann ist $f$ auf $(a,b)$ analytisch, d.h. + $\forall x, x_0 \in (a,b)$ konvergiert die Taylor-Reihe von $f$ und + es gilt: + \[ + f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(x_0)}{k!}(x-x_0)^{k} + .\] +\end{satz} + +\begin{proof} + \[ + |f(x) - T_n(f, x, x_0)| \le \frac{\left| f^{(n+1)}(\xi)\right|}{(n+1)!}|x-x_0|^{n+1} + \le \frac{M}{(n+1)!(b-a)^{n+1}} + .\] $\implies \forall \epsilon > 0$ $\exists n_{\epsilon}$, s.d. $\frac{M}{(n+1)!}(b-a)^{n+1} < \epsilon$ +\end{proof} + +\begin{bem} + \begin{enumerate} + \item Eine $C^{\infty}$ Funktion muss nicht analytisch sein. + \item Ist $f \in C^{n}$ mit $f(n) \equiv 0$ auf $D$, dann ist + $f$ ein Polynom vom Grad kleiner oder gleich $n-1$, da + \[ + f(x) = T_{n-1}(f, x, x_0) + \underbrace{R_n(x, x_0)}_{= \;0} + .\] + \end{enumerate} +\end{bem} + +\begin{korrolar}[Lokale Extrema] + Sei $f \in C^{n}(D, \R)$, $D = (a,b)$ und + für $x_0 \in (a,b)$ gelte + \[ + f'(x_0) = f''(x_0) = \ldots = f^{(n-1)}(x_0) \quad \text{und} \quad f^{(n)}(x_0) \neq 0 + .\] Dann gilt + \begin{enumerate} + \item Ist $n$ gerade und $f^{(n)}(x_0) < 0$ bzw. $f^{(n)}(x_0) > 0$, dann ist + $x_0$ ein lokales Maximum bzw. lokales Minimum von $f$. + \item Ist $n$ ungerade, dann ist $x_0$ kein + lokales Extremum von $f$ (Sattelpunkt, Wendepunkt). + \end{enumerate} +\end{korrolar} + +\begin{figure} + \centering + \begin{subfigure}{.4\textwidth} + \caption{$f(x) = x^3$} + \begin{tikzpicture} + \begin{axis}% + [grid=both, + minor tick num=4, + grid style={line width=.1pt, draw=gray!10}, + major grid style={line width=.2pt,draw=gray!50}, + axis lines=middle, + enlargelimits={abs=0.2}, + ymax=3, + ymin=-3, + width=.9\textwidth + ] + \addplot[domain=-2:2,samples=100,smooth,red] {x^3}; + \end{axis} + \end{tikzpicture} + \end{subfigure} + \begin{subfigure}{.4\textwidth} + \caption{$f(x) = x^2$} + \begin{tikzpicture} + \begin{axis}% + [grid=both, + minor tick num=4, + grid style={line width=.1pt, draw=gray!10}, + major grid style={line width=.2pt,draw=gray!50}, + axis lines=middle, + enlargelimits={abs=0.2}, + ymax=3, + ymin=-3, + width=.9\textwidth + ] + \addplot[domain=-2:2,samples=100,smooth,red] {x^2}; + \end{axis} + \end{tikzpicture} + \end{subfigure} +\end{figure} + +\begin{proof} + Taylor Satz $\implies$ $f(x) = f(x_0) + \frac{f^{(n)}(\xi)}{n!} (x-x_0)^{n}$ + $f^{(n)}$ stetig in $x_0$, $f^{(n)} \neq 0$ in $x_0$ \\ + $\implies \exists \delta > 0$, s.d. $f^{(n)}(x) \neq 0$ für + $x \in \; ]x_0 - \delta , x_0 + \delta [$ und hat das gleiche + Vorzeichen wie $f^{(n)}(x_0)$. + \begin{enumerate} + \item $n$ gerade $\implies (x-x_0)^{n} > 0$, falls + $x \neq x_0$. + \[ + f(x) = f(x_0) + \frac{f^{(n)}(\xi)}{n!}\underbrace{(x-x_0)^n}_{> 0} + .\] $\implies$ $f(x) > f(x_0)$, falls $f^{(n)}(\xi) > 0$, dann + $f^{(n)}(x_0) > 0$. + $f(x) < f(x_0)$, falls $f^{(n)}(x_0) < 0$. + \item $n$ ungerade, wechselt $(x-x_0)^{n}$ das Vorzeichen. + \end{enumerate} +\end{proof} + +\begin{korrolar} + Sei $f \in C^{2}\left( (a,b), \R \right) $ und + $x_0 \in (a,b)$. Dann folgt + \begin{enumerate}[(i)] + \item $x_0$ ist ein lokales Minimum von $f$ + $\implies f'(x_0) = 0$, $f''(x_0) \ge 0$ + \item $x_0$ ist ein lokales Maximum von $f$ + $\implies f'(x_0) = 0, f''(x_0) \le 0$. + \end{enumerate} +\end{korrolar} + +\begin{proof} + klar. +\end{proof} + +\begin{korrolar}[Hinreichende Optimalitätsbedingung] + Sei $x_0$ mit $f'(x_0) = 0$, $f''(x_0) > 0$. Dann ist + $x_0$ ein lokales Minimum. +\end{korrolar} + +\begin{proof} + trivial. +\end{proof} + +\subsection{Die Regeln von de l'Hospital} + +Ziel: Grenzwerte zu berechnen für $x \to \pm \infty$ oder +$f(x) \to \pm \infty$. Grenzwerte vom Typ: +\[ + \left( \frac{0}{0}, \frac{\infty}{\infty}, 0 \cdot \infty, \infty - \infty, \infty^{\infty}, \ldots \right) +.\] + +\begin{lemma}[Verallgemeinerter Mittelwertsatz] + Seien $f, g$ im Intervall $[a,b]$ stetig und in $(a,b)$ + differenzierbar, $g'(x) \neq 0$ $\forall x \in (a,b)$. + Dann $\exists $ ein $c \in (a,b)$ s.d. gilt + \[ + \frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)} + .\] +\end{lemma} + +\begin{proof} + ohne Beweis. +\end{proof} + +\begin{satz}[1. Regel von de l'Hospital] + Seien $-\infty \le a < b \le + \infty$, $I := (a,b)$ und + $f, g\colon I \to \R$ differenzierbare Funktionen + mit + \[ + \lim_{x \nearrow b} f(x) = 0 = \lim_{x \nearrow b} g(x) + .\] Es gelte + $g'(x) \neq 0$ $\forall x \in I$ und + $\lim_{x \nearrow b} \frac{f'(x)}{g'(x)} = c$. $c \in \R \cup \{\pm \infty\} $. + Dann gilt: + \[ + g(x) \neq 0 \quad \forall x \in I \; \text{und} \; + \lim_{x \nearrow b} \frac{f(x)}{g(x)} = c + .\] Analoge Aussagen gelten für $x \searrow a$. +\end{satz} + +\begin{proof} + $f, g$ in $b$ stetig $\implies f(b) = g(b) = 0$. + $g'(x) \neq 0 \implies$ keine weiteren Nullstellen von + $g$ in $(a,b)$, d.h. $g(x) \neq 0$ $\forall x \in (a,b)$. + + Wir nutzen den verallgemeinerten Mittelwertsatz. + $\implies \exists \xi \in (a,b)$ mit + \[ + \frac{f'(\xi)}{g'(\xi)} = \frac{f(x)}{g(x)} + .\] Aus $x \nearrow b$ folgt $\xi \nearrow b$ $\implies$ Behauptung. +\end{proof} + +\begin{bsp} + \begin{align*} + \intertext{1.} + \lim_{x \to 0} \frac{\sin x}{x} &= + \lim_{x \to 0} \frac{\cos x}{1} = 1 + \intertext{2.} + \lim_{x \searrow 0} \frac{\sin x}{x^2} + &= \lim_{x \searrow 0} \frac{\cos x}{2x} = + \infty + \intertext{3.} + \lim_{x \searrow 0} \frac{(\sin x)^2}{x^2} + &= \lim_{x \searrow 0} \frac{2 \sin(x) \cdot \cos(x)}{2x} + = \lim_{x \searrow 0} \frac{2(\cos^2 x - \sin^2(x)}{2} + = 1 + .\end{align*} +\end{bsp} + +\begin{satz}[2. Regel von de l'Hospital] + Seien $- \infty \le a < b \le + \infty$, $I := (a,b)$ und + $f,g \colon I \to \R$ differenzierbar mit + $\lim_{x \nearrow b} g(x) = \pm \infty$, + $g'(x) \neq 0$ $\forall x \in I$. + \[ + \lim_{x \nearrow b} f(x) = \pm \infty \quad \text{und} + \quad \lim_{x \nearrow b} \frac{f'(x)}{g'(x)} = c \in \R \cup \{ \pm \infty\} + .\] Dann gilt: $\exists x_0 \in I$ mit $g(x) \neq 0$ für + $a < x_0 \le x < b$ und + \[ + \lim_{x \nearrow b} \frac{f(x)}{g(x)} = c + .\] Analoge Aussage für $x \searrow a$. +\end{satz} + +\begin{proof} + Rannacher. +\end{proof} + +\begin{bem} + \begin{enumerate} + \item + Grenzprozesse für $x \to \pm \infty$. + \[ + \lim_{x \to \pm \infty} \frac{f(x)}{g(x)} + .\] Substitution $y = \frac{1}{x}$, $(y \to 0$ für $x \to \pm \infty$). + \[ + \lim_{x \to \pm \infty} \frac{f(x)}{g(x)} = \lim_{y \to 0} + \frac{f(\frac{1}{y}}{g(\frac{1}{y})} + .\] +\item $0 - \infty$ + \[ + \lim_{x \nearrow b} f(x) = 0, \lim_{x \nearrow b} g(x) = \infty + .\] + \[ + \lim_{x \nearrow b} f(x) \cdot g(x) = \lim_{x \nearrow b} + \frac{f(x)}{g(x)^{-1}} \sim \frac{0}{0} + .\] +\item $\infty - \infty$ + \[ + \lim_{x \to b} f(x) = \infty, \lim_{x \to b} g(x) = \infty + .\] + \[ + \lim_{x \to b} (f(x) - g(x)) = \lim_{x \to b} \left( \frac{1}{\frac{1}{f(x)}}- \frac{1}{\frac{1}{g(x)}} \right) + = \lim_{x \to b} \frac{\frac{1}{g(x)} - \frac{1}{f(x)}}{\frac{1}{f(x)} \cdot \frac{1}{g(x)}} \sim \frac{0}{0} + .\] + +\item $f(x) \to 1$, $g(x) \to \infty$\\ + $f(x) \to \infty$, $g(x) \to 0$ \\ + $f(x) \to 0$, $g(x) \to 0$ \\ + $\lim f(x)^{g(x)}$? + + Logarithmiere $f(x)^{g(x)} = A$. + \[ + \ln A = g(x) \cdot \ln f(x) + .\] $\lim A = \exp(\lim g(x) \cdot \ln(f(x)))$ + \end{enumerate} +\end{bem} + +\end{document}