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+\begin{tabular}{|c|m{1cm}|m{1cm}|m{1cm}|m{1cm}|m{1cm}|m{1cm}|@{}m{0cm}@{}} + \hline + Aufgabe & \centering A1 & \centering A2 & \centering A3 & \centering A4 & \centering A5 & \centering $\sum$ & \\[5mm] \hline + Punkte & & & & & & & \\[5mm] \hline +\end{tabular} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Sei $f\colon [a,b] \to \R$ stetige Funktion mit $f(a) = 0$ und $f(x) \neq 42$ $\forall x \in [a,b]$. + + Beh.: $f(b) < 42$. + \begin{proof} + Angenommen: $f(b) \ge 42$. Dann folgt mit ZWS: $\exists x \in [a,b]$, s.d. + $f(x) = 42$. Widerspruch zu $f(x) \neq 42 \; \forall x \in [a,b]$. + \end{proof} + \item Seien $f, g\colon \R \to \R$ stetige Funktionen mit + $f(42) > g(42)$ und $f(x) \neq g(x)$ $\forall x \in \R$. + + Beh.: $f(x) > g(x)$ $\forall x \in \R$ + \begin{proof} + Angenommen: $\exists a \in \R$, s.d. $f(a) < g(a)$. Dann definiere + $d(x) := f(x) - g(x)$. Wegen $f, g$ stetig, ist auch $d$ stetig. + + O.B.d.A. $a < 42$. Dann wähle $b := 42$. Wegen + $f(b) > g(b) \implies d(b) > 0$ und da + $f(a) < g(a) \implies d(a) < 0$.\\ + $\stackrel{\text{ZWS}}{\implies} \exists x' \in [a,b]$ mit + $d(x') = 0 \implies f(x') = g(x')$. Widerspruch. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: Jede Lipschitz-stetige Funktion ist gleichmäßig-stetig. + \begin{proof} + Sei $f\colon D \to \R$ Lipschitz-stetig auf $D \subset \R$. Dann + $\exists L > 0$, s.d. $|f(x) - f(y)| \le L |x - y|$ $\forall x,y \in D$. + + Sei nun $\epsilon > 0$ beliebig. Wähle $\delta := \frac{\epsilon}{L}$. Dann gilt + $\forall x, y \in D$ mit $|x - y| < \delta = \frac{\epsilon}{L}$ + \[ + |f(x) - f(y)| \le L |x-y| < L \cdot \frac{\epsilon}{L} = \epsilon + .\] + \end{proof} + \item Beh.: $f\colon \R^{+} \to \R$ mit $f(x) = \sqrt{x}$ + ist gleichmäßig stetig, aber nicht Lipschitz-stetig. + \begin{proof} + Sei $\epsilon > 0$ bel. dann wähle $\delta := \left( \frac{\epsilon}{2} \right)^2$. + Dann gilt $\forall x, y \in \R^{+}$ mit $|x-y| < \delta$: + \[ + |\sqrt{x} - \sqrt{y}| \stackrel{\sqrt{x} \text{ monoton steigend}}{<} + |\sqrt{x + \delta } - \sqrt{x}| + = \left| \frac{\delta}{\sqrt{x + \delta } + \sqrt{x} } \right| + < \left| \frac{\delta}{\sqrt{\delta }} \right| + = \frac{\epsilon^2}{4} \cdot \frac{2}{\epsilon} + = \frac{\epsilon}{2} + < \epsilon + .\] $\implies$ $f$ gleichmäßig-stetig. + + Sei nun $L \in \R$ beliebig und wähle $x := 0$. Dann gilt + \[ + \frac{|\sqrt{x} - \sqrt{y}| }{|x-y|} = \frac{\sqrt{y} }{|y|} + = \frac{1}{\sqrt{y}} \xrightarrow{y \searrow 0} \infty > L + .\] $\implies$ $f$ in $x_0 = 0$ nicht Lipschitz-stetig. + \end{proof} + \item Beh.: Jede gleichmäßig stetige Funktion ist stetig. + \begin{proof} + Sei $f\colon D \to \R$ gleichmäßig stetig auf $D \subset \R$. + + Sei $\epsilon > 0$ bel. dann ex. ein $\delta > 0$ mit + $|f(x) - f(y)| < \epsilon$ $\forall x, y \in D$ mit $|x - y| < \delta $. + + Insbesondere für ein beliebiges $a \in D$ gilt also + $\forall x \in D$ mit $|x - a| < \delta $: + \[ + |f(x) - f(a)| < \epsilon + .\] $\implies f$ stetig. + \end{proof} + \item Beh.: Die Funktion $f\colon \R \to \R$ mit $f(x) = x^2$ ist stetig, + aber nicht gleichmäßig stetig. + \begin{proof} + $f$ ist als Polynom stetig. Wähle nun $\epsilon := 1$ und $\delta > 0$ beliebig. + Dann wähle $x > \frac{1}{\delta} - \frac{\delta }{4}$ und + $y := x + \frac{\delta}{2}$. Dann gilt $|x - y| = \frac{\delta}{2} < \delta $, aber + \[ + \left| \left( x + \frac{\delta}{2} \right)^2 - x^2 \right| + = \left| x^2 + \delta x + \frac{\delta^2}{4} - x^2 \right| + = \left|\delta x + \frac{\delta ^2}{4} \right| + > \left| 1 - \frac{\delta ^2}{4} + \frac{\delta ^2}{4} \right| = 1 = \epsilon + .\] $\implies f$ nicht gleichmäßig stetig. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item + \begin{proof} + \begin{itemize} + \item $\sin(0) = \sum_{k=0}^{\infty} (-1)^{k} \cdot 0 = 0$ + \item $\cos(0) = \sum_{k=0}^{\infty} (-1)^{k} \cdot \frac{0^{2k}}{(2k)!} = 1$ + \item $\sin(\pi) = \sin \frac{\pi}{2} \cdot \underbrace{\cos \frac{\pi}{2}}_{= 0} + + \underbrace{\cos \frac{\pi}{2}}_{= 0} \cdot \sin \frac{\pi}{2} = 0$ + \item $\sin(2\pi) = \underbrace{\sin \pi}_{=0} \cdot \cos \pi + + \cos \pi \cdot \underbrace{\sin \pi}_{=0} = 0$ + \item $\cos \pi = \underbrace{\cos \frac{\pi}{2} \cdot \cos \frac{\pi}{2}}_{= 0} + - \sin \frac{\pi}{2} \cdot \sin \frac{\pi}{2} (*)$ + \begin{align*} + &\cos \pi - \cos 0 = - 2\cdot \sin \frac{\pi}{2} \cdot \sin \frac{\pi}{2} \\ + \implies& \cos \pi - 1 = - 2 \left( \sin \frac{\pi}{2} \right)^2 \\ + \stackrel{(*)}{\implies}& - \left( \sin \frac{\pi}{2} \right)^2 + - 1 = - 2 \left( \sin \frac{\pi}{2} \right)^2 \\ + \implies & \left( \sin \frac{\pi}{2} \right)^2 = 1 \\ + \stackrel{(*)}{\implies} & \cos \pi = -1 + .\end{align*} + \item $\cos 2\pi = \underbrace{\cos \pi \cdot \cos \pi}_{= 1} + - \underbrace{\sin \pi \cdot \sin \pi}_{= 0} = 1$ + \item $\sin \frac{3}{2}\pi - \sin \frac{\pi}{2} = 2 \cos(\pi) \cdot \sin(\frac{\pi}{2}) + \stackrel{\cos \pi = -1}{\implies} \sin \frac{3}{2} \pi = - \sin \frac{\pi}{2}$ + \item $\sin 2\pi - \sin \pi = 0 = 2 \cos \frac{3}{2} \pi \cdot \sin \frac{\pi}{2} + \stackrel{\sin \frac{\pi}{2} \neq 0}{\implies} \cos \frac{3}{2} \pi = 0$ + \item Aus VL folgt $\sin x > 0$ $\forall x \in ]0,2[$ und $\frac{\pi}{2} \in ]0,2[$ + $\implies \sin \frac{\pi}{2} = 1$ und damit $\sin \frac{3}{2} \pi = -1$. + \end{itemize} + \end{proof} + \item + \begin{proof} Mit $e^{ix} = \cos x + i \sin x$ und (a) folgen direkt + \begin{itemize} + \item $e^{i\frac{\pi}{2}} = \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} = i$ + \item $e^{i\pi} = \cos \pi + i \sin \pi = -1$ + \item $e^{i\frac{3\pi}{2}} = \cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2} = -i$ + \item $e^{i 2\pi} = \cos 2\pi + i \sin 2\pi = 1$ + \end{itemize} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item $f(x) = (x^{x})^{x} = e^{\ln(x) \cdot x^2}$, $x > 0$. Mit Ketten- und Produktregel folgt + \[ + f'(x) = e^{\ln(x)\cdot x^2} (x + 2x \cdot \ln(x)) = (x^{x})^{x} (x + 2 x \ln x) + .\] + \item $f(x) = \ln(x)^{x} = e^{\ln(\ln x) \cdot x}$, $x > 0$. Mit Ketten- und Produktregel folgt + \[ + f'(x) = \ln(x)^{x} \left(\frac{1}{\ln(x)} \cdot \frac{1}{x} \cdot x + \ln(\ln(x))\right) + = \ln(x)^{x}\left(\frac{1}{\ln(x)} + \ln(\ln(x))\right) + .\] + \item $f(x) = \frac{x^{4} + 2x^{3} -x}{x^{3} + 1}$, $x \neq -1$. Mit Quotientenregel folgt + \[ + f'(x) = \frac{(4 x^{3} + 6x^2 - 1)(x^{3}+1)-(x^{4}+2x^{3}-x)(3x^2)}{(x^{3}+1)^2} + = \frac{x^{6}+6x^{3}+6x^2-1}{x^{6}+2x^{3}+1} + .\] + \item $f(x) = \left( \sqrt{x} +1 \right) \left( \frac{1}{\sqrt{x} } - 1 \right) $, $x > 0$. + Mit Produktregel folgt + \[ + f'(x) = \frac{1}{2 \sqrt{x} } \left( \frac{1}{\sqrt{x} } -1 \right) + + \left( \sqrt{x} +1 \right) \left( -\frac{1}{2x^{\frac{3}{2}}} \right) + = \frac{1}{2x} - \frac{1}{2\sqrt{x} } - \frac{\sqrt{x} }{2x^{\frac{3}{2}}} - \frac{1}{2x^{\frac{3}{2}}} = \frac{-x-1}{2x^{\frac{3}{2}}} + .\] + \item $f(x) = \frac{\ln(x)}{1+x^2}$, $x > 0$. Mit Quotientenregel folgt + \[ + f'(x) = \frac{\frac{1}{x}(1+x^2) - \ln(x)\cdot 2x}{(1+x^2)^2} + = \frac{\frac{1}{x} + x - \ln(x)\cdot 2x}{(1+x^2)^2} + = \frac{1 + x^2(1 - \ln(x))}{x(1+x^2)^2} + .\] + \item $f(x) = \sin(x)^{\cos(x)} = e^{\ln(\sin(x)) \cdot \cos(x)}$, + $x \in \{ 0 < x - 2\pi k < \pi \mid k \in \Z\} $. + Mit Ketten- und Produktregel folgt + \begin{align*} + f'(x) &= \sin(x)^{\cos(x)} \left(\frac{1}{\sin(x)} \cdot \cos(x) \cdot \cos(x) - \ln(\sin(x)) \cdot \sin(x)\right) \\ + &= \sin(x)^{\cos(x)} \left( \cot(x) \cdot \cos(x) - \ln(\sin(x)) \cdot \sin(x) \right) \\ + &= \sin(x)^{\cos(x)+1} (\cot(x)^2 - \ln(\sin(x))) + .\end{align*} + Für $x = 2\pi k$, $k \in \Z$: + \begin{align*} + D_h f(2\pi k) = D_h f(0) &= \frac{\sin(h)^{\cos(h)} - \sin(0)^{\cos(0)}}{h} \\ + &= \frac{e^{\ln(\sin(h)) \cdot \cos(h)}}{e^{\ln(h)}} \\ + &= e^{\underbrace{\ln(\sin(h))}_{\to \ln(h)} \cdot \underbrace{\cos(h)}_{\to 1} - \ln(h)} + \xrightarrow{h \to 0} e^{0} = 1 + .\end{align*} $\implies f'(2\pi k) = 1$. + + Für $\pi \le x - 2\pi k < 2\pi$: $\sin(x) < 0 \implies \sin(x)^{\cos(x))} \not\in \R$. + \item $f(x) = \ln(\tan(x)) - \frac{\cos(2x)}{\sin^2(2x)}$, $x \in \left( 0, \frac{\pi}{2} \right) $. + Mit Quotientenregel folgt für + $(\tan(x))' = \left(\frac{\sin(x)}{\cos(x)}\right)' = \frac{\cos^2(x)+\sin^2(x)}{\cos^2(x)} = \frac{1}{\cos^2(x)}$. Damit folgt mit Ketten- und Quotientenregel: + \begin{align*} + f'(x) &= \frac{1}{\tan(x)} \cdot \frac{1}{\cos^2(x)} - + \frac{2 \sin(2x) \cdot \sin^2(2x) - \cos(2x) \cdot 2 \sin(2x) \cdot 2 \cos(2x)}{\sin^{4}(2x)} \\ + &= 2 \cdot \frac{\cos^2(2x) + 1}{\sin^{3}(2x) \sin(x)\cos(x)} + .\end{align*} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $f_n(x) := \left| \cos^{n}(x) \right| $, $n \in \N$ konvergiert auf dem + Intervall $D := [0, \pi]$ punktweise gegen $f(x) := \begin{cases} + 1 & x \in \{0, \pi\} \\ + 0 & \text{sonst} + \end{cases}$, aber nicht gleichmäßig. + \begin{proof} + Sei $0 < \epsilon < 1$ und $x \in [0, \pi]$ beliebig. Für $x \in \{0, \pi\} $ + gilt $\forall n \in \N$ + \[ + |f_n(x) - f(x)| = |\cos^{n}(x) - f(x)| = |1^{n} - 1| = 0 < \epsilon + .\] Für $0 < x < \pi$: Wähle $n_{\epsilon} := \left\lceil \frac{\ln(\epsilon)}{\ln|\cos(x)|}\right\rceil$. + Dann gilt $\forall n \in \N$, $n > n_{\epsilon}$: + \[ + |f_n(x) - f(x)| = |\cos^{n}(x)| + \stackrel{|\cos(x)| < 1} < |\cos^{n_{\epsilon}}(x)| + \le \left|\cos^{\frac{\ln(x)}{\ln|\cos(x)|}}\right| + = e^{\frac{\ln(\epsilon)}{\ln|\cos(x)|} \ln|\cos(x)|} + = \epsilon + .\] $\implies f_n(x)$ punktweise konvergent + + Sei nun $n_{\epsilon} \in \N$ beliebig. Definiere + $\zeta = e^{\frac{\ln(\epsilon)}{n_{\epsilon}}} > 0$. Wegen + $\epsilon < 1 \implies \ln(\epsilon) < 0 \implies \zeta < 1$. + Damit definiere $\xi := \arccos\left( \zeta \right)$. + Dann gilt + $\forall x \in (0, \frac{\pi}{2}]$ mit $x < \xi$: + \begin{align*} + |f_{n_{\epsilon}}(x) - f(x)| + = \left| \cos^{n_{\epsilon}}(x) \right| + > \left| \cos^{n_{\epsilon}}\left( \arccos\left( e^{\frac{\ln(\epsilon)}{n_{\epsilon}}} \right) \right) \right| + = e^{\frac{\ln(\epsilon)}{n_{\epsilon}} \cdot n_{\epsilon}} + = \epsilon + .\end{align*} + + $\implies f_n$ nicht gleichmäßig konvergent. + \end{proof} + \item Beh.: $f_n(x)$ konvergiert auf dem Intervall + $\widetilde{D} := \left[ \frac{\pi}{4}, \frac{3\pi}{4} \right] $ gleichmäßig + mit $f(x) = 0$. + \begin{proof} + Sei $\epsilon > 0$ beliebig. Dann wähle + $n_{\epsilon} := \left\lceil \frac{\ln(\epsilon)}{\ln\left(\cos\left(\frac{\pi}{4}\right)\right)} \right\rceil $. Sei nun $x \in \widetilde{D}$ bel. Dann + gilt $\forall n \in \N$, $n > n_{\epsilon}$ + \begin{align*} + |f_n(x) - f(x)| \qquad \qquad + &= \qquad \qquad |\cos^{n}(x)| \\[2mm] + &\stackrel{\mathclap{|\cos(x)| < 1}}{<} \qquad \qquad |\cos^{n_{\epsilon}(x)}| \\[2mm] + &\le \qquad \qquad e^{\ln|\cos(x)| \frac{\ln(\epsilon)}{\ln\left( \cos\left( \frac{\pi}{4} \right) \right) }} \\[2mm] + &\stackrel{\mathclap{\cos\left( \frac{\pi}{4} \right) \;>\; |\cos(x)|}}{\le} \qquad \qquad + e^{\ln\left( \cos\left( \frac{\pi}{4} \right) \right) \cdot \frac{\ln(\epsilon)}{\ln\left( \cos\left( \frac{\pi}{4} \right) \right) } } \\[2mm] + &= \qquad \qquad \epsilon + .\end{align*} + $\implies f_n$ gleichmäßig konvergent. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}