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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}