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zq}>1RVf8Qy6*)TVoS<0xgE0C`M)-7sbOmOlll*}^`De5fflzYfDynb%q%D*!XhZb1`uUnVq#=tWBZAV6cu3; zk7H@cL>D#p4k z(P6h=ztzbI*GP02Za~DSnA-H<+i&CJ=)3FsndkbRrzTuHmv|4FG-y=6KvTpNim%yoSmxLRtuK+Q-Uw~g|8W1Fi<6->xn%M zPJYZ%L=Cn;pjpT_Amq8Pz&lO}&T$@^4c!k+$~UM}xAp;B{Rx=H4?(pPf*QN8TssaQ zQWX-}x9;e-`a4o(MGJy9NtSS&IEqL?gqDd95IqJ-KtGDSGB98vFvdnzAL_TH;V8o% zX&uR*!Ngsv%=X*WBeL2M9=lK2=8%9(YezAt6%C-s<3Jrcc_ze*YLFrqS51)L5~A_5 zR9-oTZr5_Pu_{kqeF5C|wMa=0&SCZZZJ9TR111DJkYe6D6lD}3$%}bsJf)^xMmSHm z*E_-(>>4n#hyEE5_HVeQKH&dI{M%jSzoD}Zv!+D?Low}n!L>($V(RiDFxDow+cAMP z-Q6cJ>`$r9eW2-jfYp5cE9iS!b)Q`TzJF00d4Mi?c1+PuM!{N+g4QeD!ji951HHZ> zkBB3TDKTGZrTfDlIDY(JzzAmndH?Sh!O7Xc(b?Vc=gI$JdRds+U`R 0$ bel., dann wähle $\delta := \epsilon$. Dann gilt + $\forall x \in \R^{\times }$ mit $|x| < \delta $: + \[ + \left| f_1(0) - f_1(x) \right| + = \left|x \cdot \sin\left( \frac{1}{x} \right) \right| + \le |x| < \delta = \epsilon + .\] $\implies f_1$ in $x_0 = 0$ stetig. + + Weiter gilt für $x_0 = 0$ + \[ + D_h f_1(0) = \frac{f_1(0+h) - f_1(0)}{h} = \frac{h \cdot \sin(1 / h)}{h} + = \sin\left( \frac{1}{h} \right) + .\] Nun sind offensichtlich $(h_n)_{n\in\N}$ mit $h_n := \frac{1}{2\pi n}$ + und $(l_n)_{n\in\N}$ mit $l_n := \frac{1}{2 \pi n + \frac{\pi}{2}}$ Nullfolgen, aber + \[ + \lim_{n \to \infty} D_{h_n} f_1(0) = \lim_{n \to \infty} \sin\left( 2 \pi n \right) + = 0 \neq 1 = \lim_{n \to \infty} \sin \left( 2\pi n + \frac{\pi}{2} \right) + = \lim_{n \to \infty} D_{l_n} f_1(0) + .\] $\implies f_1$ in $x_0$ nicht differenzierbar. + \end{proof} + \item Beh.: $f_2$ in $x_0 = 0$ differenzierbar, aber $f_2'$ in $x_0 = 0$ nicht stetig. + \begin{proof} + Es gilt + \[ + D_h f_2(x_0) = \frac{f_2(h)}{h} = \frac{h^2 \sin(1 / h)}{h} = h \cdot \sin(1 / h) + .\] $\implies$ Analog zum Stetigkeitsbeweis in (a), folgt $f_2$ in $x_0$ differenzierbar.\\ + $\implies \lim_{h \to 0} D_h f_2(x) = 0 = f_2'(0)$. + + Für $x \neq 0$ folgt mit Ableitungsregeln direkt: + \[ + f_2'(x) = 2x \cdot \sin(1 / x) - \cos ( 1 / x) + .\] Für $(x_n)_{n\in\N}$ mit $x_n := \frac{1}{2\pi n}$ folgt direkt + $x_n \xrightarrow{n \to \infty} 0 = x_0$, aber + \[ + f_2'(x_n) = 2 \cdot \frac{1}{2\pi n} \sin(2 \pi n) - \cos(2 \pi n) + = -1 \neq 0 = f_2'(0) + .\] $\implies$ $f_2'$ nicht stetig in $x_0 = 0$. + \end{proof} + \item Beh.: $f_3$ in $x_0 = 0$ nur einmal differenzierbar. + in $x_0 = 0$. + \begin{proof} + Für $x_0 = 0$ folgt direkt + \[ + D_h f_3(x_0) = \frac{f_3(h)}{h} = \frac{h^{3} \cdot \sin(1 / h)}{h} = h^2 \cdot \sin(1 / h) + \xrightarrow{h \to 0} 0 + .\] Für $x \neq 0$ gilt mit Ableitungsregeln direkt: + \[ + f_3'(x) = 3x^2 \cdot \sin(1 / x) - x \cdot \cos(1 / h) + .\] Damit folgt + \[ + \lim_{x \to 0} f_3'(x) = 0 = f'(0) + .\] $\implies f_3'$ in $x_0 = 0$ stetig. Aber wegen + \[ + D_h f_3'(0) = \frac{3 h^2 \sin(1 / h) - h \cos(1 / h)}{h} + = 3 h \sin(1 / h) - \cos (1 / h) + .\] $\implies f_3'$ in $x_0 = 0$ nicht differenzierbar, analog zu (a). + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Sei $f\colon R_{+} \to \R$, $x \mapsto \sqrt{x} $. + \begin{enumerate}[(a)] + \item Beh.: $f^{(k)}(x) = \left( -\frac{1}{2} \right)^{k} \prod_{n=0}^{k-1} (2n-1) \cdot x^{\frac{1}{2} - k} $ + \begin{proof} + durch vollständige Induktion nach $k$. + + I.A.: $k = 0$ + \[ + f^{(0)}(x) = \left( -\frac{1}{2} \right)^{0} \cdot x^{\frac{1}{2}} = \sqrt{x} = f(x) + .\] + I.S.: $k \to k+1$. Es ex. ein festes, aber bel. $k \in \N$, für das die Beh. gilt. Damit folgt + \begin{align*} + f^{(k+1)}(x) = f^{(k)}'(x) + &\stackrel{\mathclap{\text{I.V.}}}{=} + \left( \left( -\frac{1}{2} \right)^{k} + \prod_{n=0}^{k-1} (2n-1) \cdot x^{\frac{1}{2} -k} \right)'(x) \\ + &\stackrel{\mathclap{\text{Potenzregel}}}{=} \qquad + \left( \frac{1}{2} -k \right) \left(-\frac{1}{2}\right)^{k} + \prod_{n=0}^{k-1} (2n-1) \cdot x^{\frac{1}{2} - k -1} \\ + &= -\frac{1}{2} (2k-1) + \left(-\frac{1}{2}\right)^{k} + \prod_{n=0}^{k-1} (2n-1) \cdot x^{\frac{1}{2} - (k +1)} \\ + &= \left( -\frac{1}{2} \right)^{k+1} \prod_{n=0}^{k} (2n-1) \cdot x^{\frac{1}{2} - (k+1)} + .\end{align*} + \end{proof} + \item Mit $(**)$ folgt dann direkt aus (a) + \[ + f^{(k)}(x) = \frac{\sqrt{\pi}}{2 \Gamma\left( \frac{3}{2} -k \right) } \cdot x^{\frac{1}{2}-k} + .\] + \item Aus (a) und (b) folgt damit direkt + \begin{align*} + T_{\infty}(x, x_0) &= \sum_{k=0}^{\infty} \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k \\ + &= \sum_{k=0}^{\infty} \frac{\sqrt{\pi}}{2 \Gamma\left( \frac{3}{2} -k \right) k! } + \cdot x_0^{\frac{1}{2}-k} (x - x_0)^{k}\\ + &= \frac{\sqrt{\pi} }{2}\sum_{k=0}^{\infty} + \frac{x_0^{\frac{1}{2}-k}}{\Gamma\left( \frac{3}{2} -k \right) k! } + (x - x_0)^{k} + .\end{align*} + \item Zunächst folgt aus (ii) direkt + \begin{align*} + \Gamma\left(\frac{3}{2} - k\right) + &\stackrel{\mathclap{\text{(ii)}}}{=} + (-1)^{k-1} \frac{\Gamma\left( -\frac{3}{2} \right) \Gamma\left( \frac{5}{2} \right) } + {\Gamma(k + 1 - \frac{3}{2})} \\ + &\stackrel{\mathclap{\text{(iii)}}}{=} + (-1)^{k-1} \frac{\pi}{\Gamma\left( k - \frac{1}{2} \right) } + \intertext{$\implies$} + \Gamma\left(k - \frac{1}{2}\right) &= (-1)^{k-1} \frac{\pi}{\Gamma\left( \frac{3}{2} -k \right) } + \intertext{Eingesetzt in (i) ergibt sich damit} + -\frac{1}{2^{k-1}} \prod_{n=0}^{k-1} (2n-1) &= + (-1)^{k-1} \frac{\pi}{\Gamma\left( \frac{3}{2} -k \right) \sqrt{\pi} } \\ + \implies \left( \frac{1}{2} \right)^{k-1} \; + \prod_{n=0}^{k-1} (2n-1) &= (-1)^{k} \frac{\sqrt{\pi} } + {2 \Gamma\left( \frac{3}{2} -k \right) } \\ + \implies \left( - \frac{1}{2} \right)^{k} \; + \prod_{n=0}^{k-1} (2n-1) &= (-1)^{k} \frac{\sqrt{\pi} }{\Gamma\left( \frac{3}{2} -k \right) } + .\end{align*} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Mit $A := \left( e^{3x} - 5x \right)^{\frac{1}{x}}$ folgt + \[ + \ln A = \frac{1}{x} \cdot \ln\left( e^{3x} - 5x \right) + .\] Damit folgt + \begin{align*} + \frac{\ln(e^{3x} - 5x)'}{x'} = \frac{1}{e^{3x} - 5x} \left( 3 e^{3x} - 5 \right) + \xrightarrow{x \to 0} -2 + .\end{align*} Mit de l'Hospital folgt damit + \[ + \lim_{x \to 0} \left( e^{3x} - 5x \right)^{\frac{1}{x}} = e^{-2} + .\] + \item Analog zu (a) mit + \[ + \frac{\ln(e^{3x} - 5x)'}{x'} = \frac{1}{e^{3x} - 5x} \left( 3 e^{3x} - 5 \right) + = \frac{3 - \frac{5}{e^{3x}}}{1-5 \frac{x}{e^{3x}}} + \xrightarrow{x \to \infty} 3 + .\] Damit folgt + \[ + \lim_{x \to \infty} \left( e^{3x} - 5x \right)^{\frac{1}{x}} = e^{3} + .\] + \item Hier folgt direkt + \[ + \frac{(5^{x} - 2^{x})'}{x'} = \frac{\ln(5) \cdot 5^{x} - \ln(2)2^{x}}{1} + \xrightarrow{x \to 0} \ln(5) - \ln(2) + \qquad \stackrel{\mathclap{\text{de l'Hospital}}}{=} \qquad + \lim_{x \to 0} \frac{5^{x} - 2^{x}}{x} + .\] + \item Abl. bedeutet hier, Zähler und Nenner seperat abgeleitet. + \[ + \frac{x - \sin(x)}{x^2} \xrightarrow{\text{Abl.}} \frac{1 - \cos(x)}{2x} + \xrightarrow{\text{Abl.}} \frac{\sin(x)}{2} + \xrightarrow{x \to 0} 0 + .\] de l'Hospital: $\implies \lim_{x \to 0} \left( \frac{1}{x} - \frac{\sin(x)}{x^2} \right) = 0 $. + \item + \[ + \frac{\ln(1+x) - \sin x}{x^2} \xrightarrow{\text{Abl.}} + \frac{\frac{1}{1+x} - \cos x}{2x} + \xrightarrow{\text{Abl.}} + \frac{-\frac{1}{(1+x)^2} + \sin x}{2} \xrightarrow{x \to 0} -\frac{1}{2} + .\] de l'Hospital: $\implies \lim_{x \to 0} \frac{\ln(1+x) - \sin(x)}{x^2} = -\frac{1}{2} $. + \item Mit $y := \frac{1}{x}$ und $A := (1+y)^{\frac{1}{y}}$, folgt + \[ + \ln A = \frac{\ln(1+y)}{y} \xrightarrow{\text{Abl.}} + \frac{1}{1+y} \xrightarrow{y \to \infty} 0 + .\] Damit folgt mit de l'Hospital $\lim_{y \to \infty} A = 1$. Insgesamt ergibt sich damit + \[ + \lim_{y \to \infty} \frac{1}{y} \cdot (1+y)^{\frac{1}{y}} - \frac{e}{y} = 0 + .\] + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Die Funktion ist konvex, d.h. die Bedingung $r'(m) = 0$ reicht bereits aus. Damit folgt + \begin{align*} + 0 &\stackrel{!}{=} r'(m) = - \frac{1}{2} \sum_{i=1}^{N} 2(y_i - m x_i - b^{*}) \\ + \implies m^{*} &= \frac{\sum_{i=1}^{N} x_i y_i - b^{*} \sum_{i=1}^{N} x_i}{\sum_{i=1}^{N} x_i^2} + .\end{align*} + Mit $\xi := \sum_{i=1}^{N} x_i y_i$ und $\zeta := \sum_{i=1}^{N} x_i^2$ folgt + \begin{align*} + m^{*} = \frac{\xi - b^{*} N \overline{x}}{\zeta} + .\end{align*} + Mit $b^{*} = \overline{y} - m \overline{x}$ folgt + \begin{align*} + m^{*} &= \frac{\xi - N(\overline{y} - m^{*} \overline{x})\overline{x}}{\zeta} \\ + \implies m^{*} &= \frac{\xi - N \overline{x} \overline{y}}{\zeta - N \overline{x}^2} + .\end{align*} + \item Mit $f(x) = m^{*} x + b^{*}$ und der in (a) gezeigten Formel folgt mit dem + Ansatz $f(x) = 0$, dass ab dem 32. Zettel niemand mehr abgibt. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $f_n(x) := \sin\left( \frac{1}{n} x \right)$, $x \in [-\pi, \pi]$ ist + gleichmäßig konvergent mit $f(x) = 0$. + + \begin{proof} + Der $\sin(\tau)$ hat auf $\tau \in [-\pi, \pi]$ genau zwei Extrema bei $\tau_1 = \frac{\pi}{2}$ + und $\tau_2 = -\frac{\pi}{2}$. + Für $n > 2$ gilt: $\left| \frac{1}{n} \cdot \pi \right| < \frac{\pi}{2}$. Damit + folgt $\forall x \in [-\pi, \pi]$: $\sin\left( \frac{1}{n} x \right) + \le \sin\left( \frac{1}{n} \pi \right) $. + + Sei nun $\epsilon > 0$ bel. Dann wähle + $n_{\epsilon} := \left\lceil \frac{\pi}{\arcsin(\epsilon)} \right\rceil > 2$. Dann folgt + $\forall x \in [-\pi, \pi]$ und alle $n \in \N$ mit $n \ge n_{\epsilon}$: + \[ + |f_n(x) - f(x)| = |f_n(x)| = \left|\sin\left( \frac{1}{n} x \right)\right| + \le \left| \sin\left( \frac{1}{n_{\epsilon}} \pi \right) \right| + \le \left| \sin\left( \frac{1}{\frac{\pi}{\arcsin(\epsilon)}} \pi \right) \right| + = \epsilon + .\] $\implies f_n$ gleichmäßig konvergent. + \end{proof} + \item Beh.: $f_n(x) := n x (1-x)^{n}$, $x \in [0,1]$ ist punktweise konvergent + mit $f(x) = 0$, aber nicht gleichmäßig konvergent. + \begin{proof} + Sei $x \in [0,1]$. Für $x = 0$ und $x = 1$ gilt + $f_n(1) = f_n(0) = f(0)$ $\forall n \in \N$. + + Für $x \in (0,1)$ gilt + \begin{align*} + \frac{f_{n+1}(x)}{f_n(x)} &= \frac{(n+1)(1-x)}{n} = \frac{n+1-x(n+1)}{n} + .\end{align*} + Wähle nun $n_{0} = \left\lceil \frac{1 - x}{x}\right\rceil $. Dann gilt $\forall n > n_0$: + \[ + x (n+1) > x \left( \frac{1-x}{x} + 1 \right) = 1 + \implies \frac{\overbrace{n+1 - x(n+1)}^{<\; n}}{n} < 1 + .\] + Damit ist $\forall n > n_0$ $f_n$ streng monoton fallend. + Da außerdem $f_n(x)$ nach unten beschränkt durch $0$, gilt damit + $f_n(x) \xrightarrow{n \to \infty} 0$. + $\implies f_n$ konvergiert punktweise gegen $f(x) = 0$.\\[2mm] + + Z.z.: $f_n$ nicht gleichmäßig konvergent. + $f_n(x)$ ist als Produkt von stetigen Funktionen stetig und nach Produktregel + differenzierbar. + Mit Produkt- und Potenzregel folgt demnach + \begin{align*} + f_n'(x) &= n \left[ (1-x)^{n} - nx (1-x)^{n-1} \right] \\ + &= n (1-x)^{n-1} \left[ 1-x - nx \right] \\ + f_n''(x) &= n^2 \left[ -(1-x)^{n-1} - (1-x)^{n-1} + x(n-1) (1-x)^{n-2} \right] \\ + &= n^2 \left( 1-x \right)^{n-2} + \left[ -2 +x + xn\right] + .\end{align*} + Für $\xi_n = \frac{1}{n+1}$ gilt + \begin{align*} + f_n'\left(\xi_n) + &= n \left( 1-\frac{1}{n+1} \right)^{n-1} \left[ 1 - \frac{1}{n+1} - \frac{n}{n+1} \right] + = n \left( 1- \frac{1}{n+1} \right)^{n-1} \left[ \frac{n+1 - 1 -n}{n+1}\right] = 0 \\ + f_n''(\xi_n) + &= n^2 \left( 1 - \frac{1}{n+1} \right)^{n-2} + \left[ -2 + \frac{1}{n+1} + \frac{n}{n+1} \right] + = \underbrace{n^2 \left( 1 - \frac{1}{n+1} \right)^{n-2}}_{> 0} + \underbrace{\left[ \frac{-n -1}{n+1} \right]}_{< 0} < 0 + .\end{align*} + Damit folgt, $f_n$ hat einen Hochpunkt bei $x = \xi_n$. Weiter gilt + \begin{align*} + f_n(\xi_n) &= \frac{1}{n+1} n \left( 1 - \frac{1}{n+1} \right)^{n} + = \frac{1}{\frac{1}{n} + 1} \left( 1 - \frac{1}{n+1} \right)^{n} + \xrightarrow{n \to \infty} \frac{1}{e} + .\end{align*} + + Sei nun $\frac{1}{2e} > \epsilon > 0$ beliebig. Dann ex. ein $N_0 \in \N$, s.d. gilt + $\forall n > N_0$: + \begin{align*} + \left| \frac{1}{e} - f_n(\xi_n) \right| < \epsilon + \implies \left| f(\xi_n) \right| + \epsilon > \left| \frac{1}{e} \right| + \implies |f_n(\xi_n)| > \frac{1}{e} - \epsilon + .\end{align*} + + Sei nun $N_0 < n_\epsilon \in \N$ beliebig. Dann wähle + $x = \xi_{n_\epsilon}$. Damit folgt + \[ + |f_{n_\epsilon}(\xi_{n_\epsilon}) - f(\xi_{n_\epsilon})| = |f_{n_{\epsilon}}(\xi_{n_\epsilon})| + > \frac{1}{e} - \epsilon + > 2 \epsilon - \epsilon = \epsilon + .\] $\implies f_n$ nicht gleichmäßig konvergent. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document} From e7b2d4b76852841114e8e76af4883b1df099ed3d Mon Sep 17 00:00:00 2001 From: flavis Date: Sat, 18 Jan 2020 14:18:45 +0100 Subject: [PATCH 3/5] update ana11 --- ws2019/ana/uebungen/ana11.pdf | Bin 142237 -> 142168 bytes ws2019/ana/uebungen/ana11.tex | 12 ++++++------ 2 files changed, 6 insertions(+), 6 deletions(-) diff --git a/ws2019/ana/uebungen/ana11.pdf b/ws2019/ana/uebungen/ana11.pdf index 9c6e92498b37b7c35f39210eadba7050f4e31351..30a6c2531968c9a06aaae3545a9f091fd709eae1 100644 GIT binary patch delta 13190 zcmai(LwleNu(V^_wr$(CoryEC{lvz^n%MTlwrx)|v90et-}wP&cQ3jZwdt#>I%gGT zZ3X5R1sDr^va=!$aAY9wirxhZB zXNQ6tCX;ZfrvF|~B4^uf$CO=uK0ZAqEPDdpYqcH;Rp)Q%-#QM&>DiOUbpc&3UuS=J zu@Tw}A%C{zrv)AjZ$4w!0VTtXUuUDie-jjL=b8PCVfrZXgll88vA^wRm;ElU%Hv3| zzI`pF9d7>t9=#ULVU~a97EnrD_2+`=2iht;Wnk7(%rpy9DGfvzkIFjV&$B0!vWFX) zQ6^zn{NXlY<;WnOhYBcX-Y9UL#FQn}_pW_8yn{^XjXf9)e$H@KfW z>sogK2xjpb(~X{waH!QHK;eeJS4-fg2`HM_c^&qL#x=?)PD+EFmPw!T4JmC_Gp$@a znR*N3_VT%PLFBSYmwBWpi5eSOb*vXx!OI-+Et$@JxLg(qTwA8~%iGefG@8Zc?g3Gx zRU21~)$Doq4_-Z{`5qX^R7zWzVPs0U;n!2(mz&#MG#AQn_-sX_Mi)(pWalzfKYlK{ zGdkJ|qE+*;0yG9@BU20-qWaP~8cX2MnDCg6sJ5V!?zTRC?wUsJD|P|Xrqo6zjRqZz zc}8m}NHd9rCq7#FQ#Q53a)S4e0J*a!mFjklWo$b8fak6bd*+Hl4UCehn;i%?{aq*E z%!iT>Epxsi(od$)jDu2}#uQ#=o=4$)b(d*Z1tlBv@8R0ZE%6pC;hfQ9ZTEmwj8S^h z!f?zHxOdu~0Z5`AclGuoCPlpuL&Dns<5WL(zYE9cj?j=Du45z5K2%8|ROHT~sska9KV4WX633q;2uJ$v^S z&7ro{kQf1Fr$BPsU3+X^LbIleIgL2I%5cRKe5cqh_1}^Ecb|gKn544pqvhNgF35cO zBiwe=a29AsEvG#@n$ zuf_9OqXa`SM(cvsf#o4%rR!%)hj9%t`hyDXgHTsS3NtBMN00ehdfdpd@fjgW8&X`P zM7Sx`?DTu9D%}#*)rE>#hjn(z6gB(IYrc?e@>fG{a|d=-5V-*PfVly07A$9fSOW#d z+U{t5P9QvyMsu}G?%mc%eu*^z#(_?Z$yNfgk1d@%t0x!2BqkqaVAhlxWal?jJ-G(P zxQMfIcg?Pkff9IQKI>*d_BR9%-3Ygllur0VpH2^aU+S=#KiIf}ktJQVh8jpiC>$tE zWF;nODGwQ}I-(m+xPzOsMPJ0c29&&vQ|KMjOM?-@?IiL!Tx5Q#y@vps=Zc0mZ~qKZ zTvd?IErKZ-BTE@fXipQ19cWv|+;rx?OlW&0B|F14wz}&s z+Q`GLUq9*G>5kcmxOA`M7q(=WfJzK|pqF)bp>rTUEn-QobPif&zh%qtkz)iJC0CZT zYpnzhlT;vr%O-@Ya{#~)qAxXsL2Iz+&f^F3@>p9itW`?7e?-Q-yw$f&*z-^yFO-i= zO^!pcjfeiCGo!8hQGpv%QW;$`Se}e8j>D838G)Ih+t9ooU1VkX!-#?Nq`k4d0b}N0 zqQV6+p)28|(mziW=iJ9u+IGplX@hBLRK!~cL*2Dd$;LHVIA0)qi!-(Hhf?87`izdO zW-3d0!icxsY5EJ~cLztkPAyS3#|pIL_o+iUT=_4n+Oiy3%h}?AM*R~7Ld^Z&(w^UJ zs*tYJu{A1=D#HB^OM`WMciU3S>;wD9Dv*7_hEAFXGWGgG=3-LlGJ{ z3&C-rY?vD^R06=gP5J)ZVhd+w+g4yMC^vJcm}5<298u_)!N1=p&<}Ctn_?UOHe93( zphI#;U?7pPR%Q~NNfy5A$fqiF!=?pGupx+B&U-)6{fbC!b~=v&NH}SpVec?zs5fH; zKw &L<3dPecb7dKVVc%1f4zfvf+`co1q2)YSSy(j_nnR7b|&kf76Ase$T6IP#6 zC-J&B$8K^&q$~vkP#V)0@+ksU|DW|Yi|RQE1qhB+xG~7JMBd|AlhY2G+J9I|{0z8D zCPRCi9<)(trfB2>Wd2aQ5g>EXev|XKDdbWB#oP((<1;vy!m!HcH~Q0XfYFKv zJiQX19|M5vY`oKA<1ek`tcdvJ_8_eEPQ953)%rdJ>{K0?C77}JUY-|4i|QHDZV!5q zy7d%I?>nh#BtJNl7x3jBIf2Q2{gg-z{Btj{<%^#=KNA|nQyg+kV&2PdfP8mp;+=}^ zj1}jh=%sjTLK;yDvNKq{2oxny{FM%mDqTJ7D{H`Cr2>0pZcK|6vl0XWrvDtFk%+o9AY$GKC~2SIEswP9Q!u<1dOa~m@G|W9bQ6@(XPMFE;DyG zmIj@v%QT+xk*d3bVciZjgl@->s-*XExdV`_P_+e;%n_rjPtEA zr3S)!I-sI!#_@(Vz|@IYa+n?uYWJK)JY zmZ9^>rbnu#mAnP+JX4tB7NR~Vpr+wqrk%n4tRD*P)47sY;J}2wIh%Acl-of?nT^KJ z!FKag;^s72I0-UoGLq(-nd8V%zpu8^Bm!Ke9H@iW)V%qzF<7cSk#K|56~mJ2&<)60 z)8L3UQq9B?U$eQvxsB7kdwyr@K5xeo|K>|gpTAGqxT2kJF3=xKAP~Sgqn?txYTTGr zL1kB``*9pg*a)i7JkDs-x^YLPomWTPD z88KV}#4&bocSuVmN1)be%CYR|+1b#$cAGM({??gO`l15`N7w2(&iQA8YDh;8T+oAH zwrJw@X_NNtv&$|=77tVZNc;BHsG!?jPP@&MCnl?njaIhL)c2UYY?*Kf6#y%Smnlrk zuz%XhXcZ#$)TEnL!@oicJQRvWqVD3-yJU)t4z=}amCM_vz$BxImgDR@ycI@}q@x?z zUzXHd>)wU7w1w`T%bu4bw08u1U6Fj-eYTe8ugv~a;j&-V!Lc1rwAwJQ$-VH%l)fDq z0z&sa0b=We*QVcsI)dLa2~-)7Ozfnd?xvmwzj;iHXoYWp^R2VTHL+za~;2 zk8;4%GjSw`D3=d1F32!H$I-W{%0XZnX zl4a*bXiICZo6Cr85^1lG?jb1NBgXX}1fL6^n z%c_glu?g5x<6_nRgkSZoION-*?)PfWE*8Q>Dk~Cuk0~-%!dTHGOdK^rPDsSdR zpFfXl69^P`P8QT^5AB9l@nN!6Zu=uEqnwXGYXu_l68KnXfwS^t2|`v?bfnAr2%M<0 zY-JKs;sBI#7LB(2h@=V53SUvEW$j%5rnri@7R$Df1-r0Bvu`O&hwL$HnB)b%=(~9u zzKx5;vZ&H7bj66i(A4{lUs0nV|6~tgBF3>jepvTYR74ipc=F>gtWKNw38;JJoiRAq}0^jOx2}G#?rK&rQ<_xj|6~J zaun`OY8|d3)o2(IzV{S&CX$d$={STDD~=cnI#=cn(Jfbwi?d2H?3Z-50=^ZJ^5z6l z8vW|NAJCmU_hmQXRB=MO^sO2(>s?y@DX_4PdX)eGQ@9zwcOCRIf148;>e?j46#IK9kR1D)1AFVa&N%!Iy4Mjj*h z*u6FZSg2eBq%u7TQYD3t!(TT-zt@N$+f|SH9X+qUXnFL``f89&s`;g z@x7|{jn?QAMs1QM12jt>KGFM%=ub0+3oj=*i=Y0_vXNGqZbjq|P_;~h9U zDsU$%T#){hlRkiSJ^I2Q-=6^A^qT3O)Quh0=8vPv%z1f!U!3c7(f?Aaa(k;|(X=#! 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b/ws2019/ana/uebungen/ana11.tex @@ -16,7 +16,7 @@ Für $k \in \N$ ist \[ f_k(x) := \begin{cases} - x^{k} \sin\left(\frac{1}{x)}\right) & x \neq 0 \\ + x^{k} \sin\left(\frac{1}{x}\right) & x \neq 0 \\ 0 & x = 0 \end{cases} \] definiert. @@ -73,7 +73,7 @@ \xrightarrow{h \to 0} 0 .\] Für $x \neq 0$ gilt mit Ableitungsregeln direkt: \[ - f_3'(x) = 3x^2 \cdot \sin(1 / x) - x \cdot \cos(1 / h) + f_3'(x) = 3x^2 \cdot \sin(1 / x) - x \cdot \cos(1 / x) .\] Damit folgt \[ \lim_{x \to 0} f_3'(x) = 0 = f'(0) @@ -140,9 +140,9 @@ (-1)^{k-1} \frac{\pi}{\Gamma\left( \frac{3}{2} -k \right) \sqrt{\pi} } \\ \implies \left( \frac{1}{2} \right)^{k-1} \; \prod_{n=0}^{k-1} (2n-1) &= (-1)^{k} \frac{\sqrt{\pi} } - {2 \Gamma\left( \frac{3}{2} -k \right) } \\ + {\Gamma\left( \frac{3}{2} -k \right) } \\ \implies \left( - \frac{1}{2} \right)^{k} \; - \prod_{n=0}^{k-1} (2n-1) &= (-1)^{k} \frac{\sqrt{\pi} }{\Gamma\left( \frac{3}{2} -k \right) } + \prod_{n=0}^{k-1} (2n-1) &= \frac{\sqrt{\pi} }{2 \Gamma\left( \frac{3}{2} -k \right) } .\end{align*} \end{enumerate} \end{aufgabe} @@ -229,7 +229,7 @@ \begin{proof} Der $\sin(\tau)$ hat auf $\tau \in [-\pi, \pi]$ genau zwei Extrema bei $\tau_1 = \frac{\pi}{2}$ und $\tau_2 = -\frac{\pi}{2}$. - Für $n > 2$ gilt: $\left| \frac{1}{n} \cdot \pi \right| < \frac{\pi}{2}$. Damit + Für $n > 2$ gilt: $\frac{\pi}{n} < \frac{\pi}{2}$. Damit folgt $\forall x \in [-\pi, \pi]$: $\sin\left( \frac{1}{n} x \right) \le \sin\left( \frac{1}{n} \pi \right) $. @@ -253,7 +253,7 @@ \begin{align*} \frac{f_{n+1}(x)}{f_n(x)} &= \frac{(n+1)(1-x)}{n} = \frac{n+1-x(n+1)}{n} .\end{align*} - Wähle nun $n_{0} = \left\lceil \frac{1 - x}{x}\right\rceil $. Dann gilt $\forall n > n_0$: + Wähle nun $n_{0} := \left\lceil \frac{1 - x}{x}\right\rceil $. Dann gilt $\forall n > n_0$: \[ x (n+1) > x \left( \frac{1-x}{x} + 1 \right) = 1 \implies \frac{\overbrace{n+1 - x(n+1)}^{<\; n}}{n} < 1 From 5b5d5911cf71dc156740c780ca942332eae540c2 Mon Sep 17 00:00:00 2001 From: flavis Date: Sat, 18 Jan 2020 15:28:03 +0100 Subject: [PATCH 4/5] add la11 --- ws2019/la/uebungen/la11.pdf | Bin 0 -> 114092 bytes ws2019/la/uebungen/la11.tex | 162 ++++++++++++++++++++++++++++++++++++ 2 files changed, 162 insertions(+) create mode 100644 ws2019/la/uebungen/la11.pdf create mode 100644 ws2019/la/uebungen/la11.tex diff --git a/ws2019/la/uebungen/la11.pdf b/ws2019/la/uebungen/la11.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f19943e34601f796c255bdae832c93d862b3c01a GIT binary patch literal 114092 zcmbrmbzBr$w>~~}cjwR@GYl==-6aCj0@B^xQqn0M64DANFmyosL(*(ZbV4&c@2c+6GfX0@K6O-NwQR)Az%s@ki&6 z3iv@k4e@tmZIGAt_k(E2s}A_(XGxWa$hh&hrIYM)UevavCJ>)rOt>o_3F@$^wi5>t zu)B+7HA5cRW(5Y`9HTI$4IpT;U}y#l3yVwoUSWHF&JpTm+I|;DsP?2k%-Y<^JvieH 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z98)F3K6E)8SuV_|%&E-Di5o>VXBS86GBXm<;K`^y3zyImWX19Sl7AE(_2?-d5G))1 z7{(GfB$N<52&<)HU1`nrZ#|#qY4LfmO}U?QL6ijd8m7vvNH_=6u_#E(*tP$bWbA}x zGQr@hSRv6&!o~IV4t@hsnAxzdtA#gi+m1+uDW>`^&bj9RvQ*U6(rPqhp;?29yvdrtIU3@ptz*G5Eh*e^;YHgRwpl+1CQ zVl0;7A+DwZx{nQtD7|r)mEV{WGaY=6i}k^rr-&l-A0+D?{>=aE&+dE2eG#F1l2w+r z0J>gf|IqZ-oXCs~%G#7CO}>BnDbeSor}vnc-o|J`JdE}w6?(F1GbbE;m&oYFjLwSi zW(=+L&$jQ3?FjTO&jnF!LmX_2m@A03PKqCnS$Yc4w(O6yRkrS=&yNb`*c0kBzu9y# zrT2SHGqvaLu!XC**^Oiqj$g~fN*0_%DsHR$rh(ZN%L@(+ ogSWdsOyi6Im2dv%y>xLkad!1^ez=HWFh2w&fXB=%qb!T}Z_$w;egFUf literal 0 HcmV?d00001 diff --git a/ws2019/la/uebungen/la11.tex b/ws2019/la/uebungen/la11.tex new file mode 100644 index 0000000..2912547 --- /dev/null +++ b/ws2019/la/uebungen/la11.tex @@ -0,0 +1,162 @@ +\documentclass[uebung]{../../../lecture} + +\title{Lineare Algebra I: Übungsblatt 11} +\author{Christian Merten} + +\begin{document} + +\punkte + +\begin{aufgabe} + Die Adjunkte zu $A$ erhalten wir nach Berechnung zahlreicher Unterdeterminanten und + lästigem Ausrechnen. Damit ergibt sich + \begin{align*} + \widetilde{A} = \begin{pmatrix} + 8 & 8 & -32 & -8 \\ + 3 & 3 & -12 & -3 \\ + 1 & 1 & -4 & -1 \\ + -5 & -5 & 20 & 5 + \end{pmatrix} + .\end{align*} + Weiter folgt + \begin{align*} + A \cdot \widetilde{A} = \begin{pmatrix} + 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 \\ + \end{pmatrix} + .\end{align*} + Damit folgt mit der ersten Cramerschen Regel $\text{det}(A) = 0$. +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item + \begin{align*} + \intertext{(1)} + \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ + 4 & 3 & 2 & 1 & 5 \end{pmatrix} + &= \begin{pmatrix} 1 & 4 \end{pmatrix} + \circ \begin{pmatrix} 3 & 2 \end{pmatrix} \\ + \intertext{Permutationsmatrix} + \begin{pmatrix} + 0 & 0 & 0 & 1 & 0\\ + 0 & 0 & 1 & 0 & 0 \\ + 0 & 1 & 0 & 0 & 0 \\ + 1 & 0 & 0 & 0 & 0\\ + 0 & 0 & 0 & 0 & 1 + \end{pmatrix} + \intertext{(2)} + \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ + 2 & 4 & 5 & 1 & 3\end{pmatrix} + &= \begin{pmatrix} 3 & 5 \end{pmatrix} \circ \begin{pmatrix} 1 & 4 \end{pmatrix} + \circ \begin{pmatrix} 1 & 2 \end{pmatrix} + \intertext{Permutationsmatrix} + \begin{pmatrix} + 0 & 0 & 0 & 1 & 0\\ + 1 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 1 \\ + 0 & 1 & 0 & 0 & 0\\ + 0 & 0 & 1 & 0 & 0 + \end{pmatrix} + .\end{align*} + \item Beh.: siehe Aufgabe + \begin{proof} + Sei $F := \{ i \in \{1, \ldots, n\} \mid \sigma(i) = i \} $ + \begin{align*} + \text{Sp}(\varphi(\sigma)) &= \sum_{i=1}^{n} \varphi(\sigma)_{ii} \\ + &= \sum_{i=1}^{n} \left( e_{\sigma(i)} \right)_i \\ + &= \sum_{i = 1, i \in F}^{n} (e_i)_i + + \sum \{ (e_j)_i \mid i \in \{1, \ldots, n\} \setminus F, \; j = \sigma(i) + \implies j \neq i\} \\ + &= \sum_{i = 1, i \in F}^{n} 1 + \sum_{i = 1, i \not\in F}^{n} 0 = (\# F)_K + .\end{align*} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Gesucht sind zunächst die Nullstellen des charakteristischen Polynoms. Mit + \begin{align*} + \lambda E_3 - A &= + \begin{pmatrix} + \lambda - 4 & -5 & -6 \\ + 0 & \lambda - 3 & 0 \\ + 3 & 5 & \lambda + 5 + \end{pmatrix} + \intertext{gilt für die Determinante} + \text{det}(\lambda E_3 - A) &= (\lambda - 4)(\lambda - 3)(\lambda + 5) +18(\lambda -3) \\ + &= (\lambda -3) \left[ \lambda^2 + \lambda -2 \right]. + \intertext{Diese Polynom hat die Nullstellen} + \lambda_1 &= 1, \lambda_2 = -2, \lambda_3 = 3. + \intertext{Für $\lambda_1$ ergibt sich das homogene GLS} + \begin{pmatrix} + -3 & -5 & - 6\\ + 0 & -2 & 0 \\ + 3 & 5 & 6 + \end{pmatrix} + &\to + \begin{pmatrix} + 1 & 0 & 2 \\ + 0 & 1 & 0 \\ + 0 & 0 & 0 + \end{pmatrix} + \intertext{Damit folgt direkt} + \text{ker}(\lambda_1 E_3 - A) &= \text{Lin}\left[ \begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix} \right] + \intertext{Für $\lambda_2$ und $\lambda_3$ analog} + \text{ker}(\lambda_2 E_3 - A) &= \text{Lin}\left[ \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix} \right] \\ + \text{ker}(\lambda_3 E_3 - A) &= \text{Lin}\left[ \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} \right] + .\end{align*} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $(-)^{t}$ ist linear. + \begin{proof} + Seien $A, B \in M_{n,n}(\Q)$ und $\lambda \in Q$. Dann folgt direkt + \[ + (A + \lambda B)^{t} = A^{t} + (\lambda B)^{t} = A^{t} + \lambda B^{t} + .\] + \end{proof} + \item Beh.: $\text{dim}_\Q\left( \text{ker}(\lambda \cdot \text{id}_{M_{n,n}(\Q)}) - (-)^{t}) \right) + = \begin{cases} + \frac{n^2 + n}{2} & \lambda = 1 \\ + \frac{n^2 - n}{2} & \lambda = -1 \\ + 0 & \text{sonst} + \end{cases}$ + \begin{proof} + Sei $A \in \text{ker}(\lambda \cdot \text{id}_{M_{n,n}(\Q)}) - (-)^{t})$. + für ein $\lambda \in \Q$. + + Falls $\lambda = 0 \implies A^{t} = 0 \implies A = 0 + \implies \text{dim}_\Q\left( \text{ker}(\lambda \cdot \text{id}_{M_{n,n}(\Q)}) - (-)^{t}) \right) = 0$. + + Sei $\lambda \neq 0$. + \begin{align*} + \lambda A - A^{t} &= 0 \implies \lambda a_{ij} = a_{ji} \land \lambda a_{ji} = a_{ij} + \implies \lambda^2 a_{ij} = a_{ij} + \quad \forall i, j \in \{1, \ldots, n\}. + \intertext{Für $A \neq 0$ gilt, es ex. mind. ein $a_{ij} \neq 0$. Für diese $i$ und $j$ + gilt weiter} + \lambda^2 &= 1 \implies \lambda = \pm 1 + .\end{align*} + Für $\lambda \neq \pm 1 \implies A = 0 \implies \text{dim}_\Q\left( \text{ker}(\lambda \cdot \text{id}_{M_{n,n}(\Q)}) - (-)^{t}) \right) = 0$ + + Sei nun $\lambda = 1$. Dann ist $A = A^{t}$, also ist $A$ symmetrisch. + Damit bleiben $n$ Diagonaleinträge frei und die Hälfte + der restlichen Einträge, damit folgt: + \[ + \text{dim}_\Q\left( \text{ker}(\text{id}_{M_{n,n}(\Q)}) - (-)^{t}) \right) + = n + \frac{n^2 - n}{2} = \frac{n^2 + n}{2} + .\] Für $\lambda = -1$ ist $A = -A^{t}$, also $A$ antisymmetrisch. Damit + sind alle Diagonaleinträge $0$. Es folgt analog + \[ + \text{dim}_\Q\left( \text{ker}(-\cdot \text{id}_{M_{n,n}(\Q)}) - (-)^{t}) \right) = + \frac{n^2 - n}{2} + .\] + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document} From 316e3d19eabde2d7153a4db7b932b32f208b5fc1 Mon Sep 17 00:00:00 2001 From: flavis Date: Sat, 18 Jan 2020 16:17:53 +0100 Subject: [PATCH 5/5] update lecture cls --- lecture.cls | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/lecture.cls b/lecture.cls index caff339..cf7c3de 100644 --- a/lecture.cls +++ b/lecture.cls @@ -105,3 +105,9 @@ % activate my colour style \lstset{style=mystyle} + +% better stackrel +\let\oldstackrel\stackrel +\renewcommand{\stackrel}[2]{% + \oldstackrel{\mathclap{#1}}{#2} +}%