From f1931528201c72703d82dfe85709d3a88bba84d7 Mon Sep 17 00:00:00 2001 From: flavis Date: Sat, 25 Apr 2020 14:19:06 +0200 Subject: [PATCH] update ana1 --- sose2020/ana/uebungen/ana1.pdf | Bin 124651 -> 135853 bytes sose2020/ana/uebungen/ana1.tex | 24 ++++++++++++++++++++++-- 2 files changed, 22 insertions(+), 2 deletions(-) diff --git a/sose2020/ana/uebungen/ana1.pdf b/sose2020/ana/uebungen/ana1.pdf index abd5ced01684bd7950706d839661ee6a7efdcf63..eb35868dc96f86dc153a8441e76464c576d87893 100644 GIT binary patch delta 65176 zcmYJZWl)|?(*=lIaBd{HdvJFP?(XjH4mYkrFWg;&LvSa!dvLd4!JU1+ceiT)b=6GI zoasK@HC=W7PQlldBGkX4!m#sjvjL(swBQSU9XA4Q%)phpj4qW!bV&SGRap(pk;)P- zz0QV5Do^5ysI<&ccg>>Zt+$tpVdscGa;Z|EgbcS6_AQqHxtYk#HSlsmA6Tgqwm+rw%+^ywubccJ{3t+0Y z-`rM*pN2A(Oqaj3T5(sZH*4P%bC|-RCvd@fz5Lj*-vX*}j5ap)YaE$`nO6Ltk1X9f z*MI&x1o!G~pD(X3kQn*<-7Lid;CzG)B4&veg3HW1<+r)13ZQES}e)%A;B-DSViTY3)v#W$VdyH^EPqUbiqa ztwZ-(r5(aw9x5YsDdg%AWV(eg-CQBFak<<%rO|#nF~z`i_+r`p?$FfJt_|KyG3wns 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z!ErV?;RA6FI=KjXQo$+SbHWKuSAjE)pw|uh65wnrILCMUxmEvfFn_QF2ETywT%ujpeFSwcyu5qi8HgKK&Zm{#sQE+R_E^HeqAcZQo-`QD_%4EGI zr^?JPOT{YRx-Gd>Q^JIeLK zI=>|U0jg+cy9#A)WF>QSa%Ev{3T19&Z(?c+b97;Hba--QW(qMfGdMUo3MC~)Peuwk CSP0$# diff --git a/sose2020/ana/uebungen/ana1.tex b/sose2020/ana/uebungen/ana1.tex index 6daf431..83e3c6a 100644 --- a/sose2020/ana/uebungen/ana1.tex +++ b/sose2020/ana/uebungen/ana1.tex @@ -79,7 +79,27 @@ = f(\psi(x)) \psi'(x) - f(\varphi(x)) \varphi'(x) .\end{align*} \end{proof} - \item + \item Sei $f\colon [a,b] \to \R$ stetig differenzierbar mit $f(a) = 0$. + + Beh.: + \[ + \int_{a}^{b} |f(x)f'(x)| \d x \le \frac{b-a}{2} \int_{a}^{b} f'(x)^2 \d x + .\] + \begin{proof} + Definiere $G(x) := \int_{a}^{x} |f'(t)| \d t $. Es folgt $G(a) = f(a) = 0$. Dann gilt + $\forall x \in [a,b]$: $G(x) \ge 0$ und $G(x) \ge f(x)$. + Außerdem ist $G'(x) = |f'(x)|$. + Damit folgt: + \begin{align*} + \int_{a}^{b} |f(x)f'(x)| \d x \qquad + &\le \qquad \int_{a}^{b} G(x)G'(x) \d x \\ + &= \qquad \frac{1}{2} \int_{a}^{b} (G^2(x))' \d x \\ + &\stackrel{G(a) = 0}{=} \qquad \frac{1}{2} G(b)^2 \\ + &\stackrel{G(a) = 0}{=} \qquad \frac{1}{2} \left| \int_{a}^{b} G'(x) \d x \right|^2 \\ + &\stackrel{\text{CSU}}{\le} \qquad \frac{1}{2} \int_{a}^{b} 1 \d x \cdot \int_{a}^{b} G'(x)^2 \d x \\ + &\stackrel{G'(x) = |f'(x)|}{=} \qquad \frac{b-a}{2} \cdot \int_{a}^{b} f'(x)^2 \d x + .\end{align*} + \end{proof} \end{enumerate} \end{aufgabe} @@ -114,7 +134,7 @@ Zu zeigen: $f_n$ nicht gleichmäßig konvergent. Sei $\epsilon > 0$ und $n \in \N$ mit $n > \frac{\epsilon \cdot 16}{\sqrt{27} }$. Dann wähle - $x_0 = \frac{1}{\sqrt{3} n}$. Damit folgt + $x_0 = \frac{1}{\sqrt{3} n}$. Mit $f_n(x_0) = \frac{\sqrt{27} n}{16}$ folgt \[ |f_n(x_0) - f(x_0)| = |f_n(x_0)| = \left| \frac{\sqrt{27} n}{16} \right|