From c0b6c470f333f04ecc07acde9a1f927dcb13e8d9 Mon Sep 17 00:00:00 2001 From: christian Date: Thu, 16 Jan 2020 18:11:34 +0100 Subject: [PATCH] add ana21 --- lecture.cls | 1 + ws2019/ana/lectures/analysis.pdf | Bin 473389 -> 497992 bytes ws2019/ana/lectures/analysis.tex | 1 + ws2019/ana/lectures/analysis20.tex | 84 ++++++++++++ ws2019/ana/lectures/analysis21.pdf | Bin 0 -> 131370 bytes ws2019/ana/lectures/analysis21.tex | 204 +++++++++++++++++++++++++++++ 6 files changed, 290 insertions(+) create mode 100644 ws2019/ana/lectures/analysis21.pdf create mode 100644 ws2019/ana/lectures/analysis21.tex diff --git a/lecture.cls b/lecture.cls index 5f7f034..caff339 100644 --- a/lecture.cls +++ b/lecture.cls @@ -19,6 +19,7 @@ \RequirePackage{pgfplots} \RequirePackage[nobottomtitles]{titlesec} \RequirePackage{listings} +\RequirePackage{mathtools} \usetikzlibrary{quotes, angles} diff --git a/ws2019/ana/lectures/analysis.pdf b/ws2019/ana/lectures/analysis.pdf index 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b/ws2019/ana/lectures/analysis.tex index 23f8b38..23dab10 100644 --- a/ws2019/ana/lectures/analysis.tex +++ b/ws2019/ana/lectures/analysis.tex @@ -35,5 +35,6 @@ \input{analysis18.tex} \input{analysis19.tex} \input{analysis20.tex} +\input{analysis21.tex} \end{document} diff --git a/ws2019/ana/lectures/analysis20.tex b/ws2019/ana/lectures/analysis20.tex index b11eee5..fa3948c 100644 --- a/ws2019/ana/lectures/analysis20.tex +++ b/ws2019/ana/lectures/analysis20.tex @@ -260,6 +260,50 @@ \end{enumerate} \end{satz} +\begin{proof} + \begin{enumerate} + \item Z.z.: $(\alpha f + \beta g)'(x) = \alpha f'(x) + \beta g'(x)$ + \begin{align*} + \frac{(\alpha f + \beta g)(x_1) - (\alpha f + \beta g)(x_0)}{x_1 - x_0} + &= \alpha \left( \frac{f(x_1) - f(x_0)}{x_1 - x_0} \right) + + \beta \left( \frac{g(x_1) -g(x_0)}{x_1-x_0} \right) \\ + &\xrightarrow{x_1\to x_0} \alpha f'(x_0) + \beta g'(x_0) + .\end{align*} + \item Z.z.: $(f g)'(x) = f'(x) g(x) + f(x) g'(x)$ + \begin{align*} + \frac{f(x_1) g(x_1) - f(x_0) g(x_0)}{x_1 - x_0} + &= \frac{f(x_1) g(x_1) - f(x_0)g(x_1) + f(x_0)g(x_1) + - f(x_0) g(x_0)}{x_1 - x_0} \\ + &= g(x_1) \frac{f(x_1) - f(x_0)}{x_1 - x_0} + + f(x_0) \frac{g(x_1) - g(x_0)}{x_1 - x_0} \\ + &\xrightarrow[g \text{ stetig in } x_0]{x_1 \to x_0} g(x_0) f'(x_0) + f(x_0) g'(x_0) + .\end{align*} + \item Z.z.: $\left( \frac{f}{g} \right)'(x) = \frac{f'(x)g(x) + - f(x) g'(x)}{g^2(x)}$ + + Für $f \equiv 1$: + \begin{align*} + \left( \frac{1}{g} \right)'(x_0) + &= \lim_{x \to x_0} \frac{1}{x-x_0} + \left( \frac{1}{g(x)} - \frac{1}{g(x_0)}\right) \\ + &= \lim_{x \to x_0} \frac{1}{x - x_0} + \frac{g(x_0) - g(x)}{x - x_0}\\ + &\stackrel{\mathclap{g \text{ stetig}}}{=} \quad + \lim_{x \to x_0} \frac{1}{g(x) g(x_0)} + \cdot \lim_{x \to x_0} \frac{g(x_0) - g(x)}{x - x_0} \\ + &= \frac{1}{g(x_0)^2} \cdot (- g'(x_0)) + \intertext{Nun für $f$ beliebig mit Produktregel:} + \left( \frac{f}{g} \right)'(x_0) + &= (f\cdot \frac{1}{g})' (x_0) \\ + &= f'(x_0) \cdot \frac{1}{g(x_0)} + + f(x_0) \cdot \left(\frac{1}{g(x_0)}\right)' \\ + &= f'(x_0) \cdot \frac{1}{g(x_0)} + - f(x_0) \cdot \frac{g'(x_0)}{g(x_0)^2} \\ + &= \frac{f'(x_0) g(x_0) - f(x_0) g'(x_0)}{g(x_0)^2} + .\end{align*} + \end{enumerate} +\end{proof} + \begin{satz}[Differenzierbarkeit der Umkehrfunktion] Sei $f\colon D \to B \subset \R$ stetige invertierbare Funktion mit Inverser @@ -272,6 +316,23 @@ .\] \end{satz} +\begin{proof} + Sei $y_n = f(x_n)$, $y_0 = f(x_0)$, $y_n \neq y_0$, + $y_n \to y_0$, $n \to \infty$. Wegen Stetigkeit + von $f^{-1}$ gilt $\underbrace{f^{-1}(y_n)}_{= x_n} + \xrightarrow{n \to \infty} \underbrace{f^{-1}(y_0)}_{= x_0}$, oder + $x_0 \xrightarrow{n \to \infty} x_n$. + + Berechne + \begin{align*} + \frac{f^{-1}(y_n) - f^{-1}(y_0)}{y_n - y_0} + = \frac{x_n - x_0}{f(x_n) - f(x_0)} + = \left( \frac{f(x_n) - f(x_0)}{x_n - x_0} \right)^{-1} + \xrightarrow{n \to \infty} + \left( f'(x_0) \right)^{-1} + .\end{align*} +\end{proof} + \begin{satz}[Kettenregel] Seien $f\colon D_f \to \R$, $g\colon D_g \to \R$ stetige Funktionen. $f \in x_0 \in D_f$ differenzierbar, $g \in y_0 = f(x_0) \in D_g$ @@ -282,6 +343,29 @@ .\] \end{satz} +\begin{proof} + Definiere die Funktion $\Delta g\colon D_g \to \R $, mit + $\Delta g(y) = \begin{cases} + \frac{g(y) - g(y_0)}{y - y_0} & y \neq y_0 = f(x_0) \\ + g'(y_0) & y = y_0 + \end{cases}$. + + $g$ in $y_0$ differenzierbar + $\implies \exists g'(y_0) \implies \lim_{y \to y_0} \Delta g(y) + = g'(y_0)$. + + Für $y \in D_g$ gilt $g(y) = g(y_0) + \Delta g(y)(y - y_0)$. Damit folgt + \begin{align*} + (g \circ f)'(x_0) + &\stackrel{\mathclap{\text{Def.}}}{=} + \lim_{x \to x_0} \frac{g(\overbrace{f(x)}^{y}) + - g(\overbrace{f(x_0)}^{y_0})}{x - x_0} \\ + &= \lim_{x \to x_0} \Delta g(f(x)) \cdot + \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x- x_0} \\ + &= g'(f(x_0)) \cdot f'(x_0) + .\end{align*} +\end{proof} + \begin{bsp} Für $x > 0$ \[ \frac{d}{dx}(\ln x) = \frac{1}{x} diff --git a/ws2019/ana/lectures/analysis21.pdf b/ws2019/ana/lectures/analysis21.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f65dea514ac73ae02b1760f88fbda701b667d648 GIT binary patch literal 131370 zcma%iQ*WeDh@lpR<5i%KhL;ck&eImxxRrlW0oSwbv`dmg=5IisELdu!|pfO#3K8- zUB>*;z0<W53b6Io|5!|M*}D@jJ=c(B$Llj&jaiuyJSD;r)0Ac)q~+y|-@hW}vwX zK<{Veuvsz#3v6HZct3ku{{4Abtj+z}W7xP#&P)`U)c0w1K@QAns_(4TyXxsl|Mm0< zsQs??ZvOaFo9u?^*ZqCFcYa)UBCe~`{c-z!KgclqLeUmQXrEUZrT%up0Qe}25}b9!DyzE6&? z5NC6G*LMBB-v&D?Ic8nEUhm+0v@-1=O5loDDw?nUBG=hrIYtWArswJEAIX2;^Wu!% 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z=;6J?VPF4MYuc56-CFMqZE{q(6}m>H 0$, s.d. $f(x_0) \ge f(x)$ bzw. + $f(x_0) \le f(x)$ + $\forall x \in B_\delta(x_0) \cap D = \;]x_0-\delta, x_0 + \delta [$. +\end{definition} + +\begin{figure}[htpb] + \centering + \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=.7\textwidth + ] + \addplot[domain=0:10,samples=100,smooth,red] {2*sin(deg(x)) -cos(deg(x/2))}; + \addplot[mark=*] coordinates {(0,-1.2)} node[pin=30:{lokales Minimum}]{} ; + \addplot[mark=*] coordinates {(1.76,1.35)} node[pin=85:{lokales Maximum}]{} ; + \addplot[mark=*] coordinates {(4.47,-1.35)} node[pin=270:{lokales Minimum}]{} ; + \addplot[mark=*] coordinates {(7.7,2.7)} node[pin=270:{globales Maximum}]{} ; + \addplot[mark=*] coordinates {(10,-1.45)} node[pin=160:{globales Minimum}]{} ; + \end{axis} + \end{tikzpicture} + \caption{Beispiel für Extrema} +\end{figure} + +\begin{satz} + Sei $f\colon (a,b) \to \R$, $a < x_0 < b$. + Ist $f$ in $x_0$ differenzierbar, und ist $x_0$ ein lokales + Extremum, dann gilt $f'(x_0) = 0$. + + \label{satz:lokalableitung0} +\end{satz} + +\begin{proof} + Sei $x_0$ ein lokales Maximum, dann $\exists \delta > 0$, s.d. + $f(x) - f(x_0) \le 0$ + $\forall x \in \;]x_0 - \delta , x_0 +\delta [ \cap (a, b)$: + \begin{align*} + f'(x_0) = \lim_{x \searrow x_0} + \underbrace{\left( \frac{f(x) - f(x_0)}{x - x_0} \right)}_{\le 0} + = \lim_{x \nearrow x_0} \underbrace{\left( \frac{f(x) - f(x_0)}{x- x_0} \right) }_{\ge 0} + .\end{align*} + $\implies f'(x_0) = 0$. Analog für Minimum +\end{proof} + +\begin{bem} + \begin{enumerate} + \item $a < x_0 < b$ ist wichtig! z.B.: $f\colon [0,1] \to \R$ + $f(x) := x$. Maximum bei $x = 1$, Minimum bei $x = 0$ mit + Ableitung $f'(x) = 1 \neq 0$. + \item $f'(x_0)$ nur eine notwendige Bedingung für ein + lokales Extremum, z.B.: $f(x) = x^{3}$, + $f'(x) = 3x^2 \implies f'(0) = 0$, aber $x = 0$ ist + kein lokales Extremum. + \begin{figure}[h] + \centering + \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=5, + ymin=-5 + ] + \addplot[domain=-3:3,samples=50,smooth,red] {x^3}; + \end{axis} + \end{tikzpicture} + \caption{$x^3$ hat bei $x = 0$ kein lokales Extremum} + \end{figure} + \end{enumerate} +\end{bem} + +\begin{satz}[Satz von Rolle] + Es sei $a < b$, $f\colon [a,b] \to \R$ stetig mit + $f(a) = f(b)$, $f$ auf $(a,b)$ differenzierbar. + + Dann ex. ein $\xi \in (a,b)$ mit $f'(\xi) = 0$. + \label{satz:rolle} +\end{satz} + +\begin{proof} + Ist $f(x) = f(a)$ $\forall x \in [a,b] \implies$ Behauptung + $\forall \xi \in \;]a,b\,[$. + + Nun $f$ nicht konstant. $[a,b]$ ist kompakt und + $f$ ist stetig, d.h. $f$ nimmt Minimum und Maximum an. + $\implies \exists x_{\text{min}}, x_{\text{max}} \in [a,b]$ mit + $f(x_{\text{min}}) = \text{min}\{f(x) \mid x \in [a,b]\} $ und + $f(x_{\text{max}}) = \text{max}\{f(x) \mid x \in [a,b]\} $. Da + $f$ nicht konstant $\implies$ $f(x_{\text{min}}) < f(a) = f(b)$ oder + $f(x_{\text{max}}) > f(a) = f(b)$.\\ + $\implies x_{\text{min}} \in (a,b)$ oder $x_{\text{max}} \in (a,b)$ \\ + $\stackrel{\mathclap{\text{Satz \ref{satz:lokalableitung0}}}}{\implies} f'(x_{\text{min}}) = 0$ + oder $f'(x_{\text{max}}) = 0$ \\ + $\implies$ Behauptung gilt mit $\xi = x_{\text{min}}$ oder + $\xi = x_{\text{max}}$. +\end{proof} + +\begin{satz}[Mittelwertsatz] + Sei $f\colon [a,b] \to \R$ stetig auf $[a,b]$ und differenzierbar + auf $]a,b[$. Dann ex. ein $\xi \in \;]a, b[$ mit + $f'(\xi) = \frac{f(b) -f(a)}{b - a}$. + \label{satz:mittelwert} +\end{satz} + +\begin{proof} + Hilfsfunktion $g(x) := f(x) - f(a) - m(x-a)$ mit + $m := \frac{f(b) - f(a)}{b - a}$. Damit folgt + $g(a) = g(b) = 0 \stackrel{\mathclap{\text{Satz \ref{satz:rolle}}}}{\implies}$ + $\exists \xi \in \; ]a,b[$ mit + $g'(\xi) = 0 \implies f'(\xi) - \frac{f(b) - f(a)}{b - a} = 0$. +\end{proof} + +\begin{satz}[Monotoniekriterium] + \label{satz:monotonie} + Sei $f\colon D \to \R$ auf $D \subset \R$ differenzierbar. Dann gilt + \begin{enumerate}[(i)] + \item $f$ monoton wachsend $\iff$ $f'(x) \ge 0$ $\forall x \in D$ + \item $f$ streng monoton wachsend $\iff$ $f'(x) > 0$ $\forall x \in D$ + \end{enumerate} + Für fallende Funktionen ersetze $f$ durch $-f$. +\end{satz} + +\begin{proof} + Es gilt $\forall a < b \in D$ nach Satz \ref{satz:mittelwert} + \[ + f(b) - f(a) = \underbrace{(b-a)}_{> 0} f'(x) \qquad (*) + .\] für ein $x \in \; ]a, b[$. Daraus folgt direkt (ii) und + (i, ,,$\impliedby$'') + + Für (i, ,,$\implies$''): Betrachte in $(*)$: + \[ + \lim_{b \to a} \underbrace{\frac{f(b) - f(a)}{b - a}}_{\ge 0} = f'(a) \ge 0 + .\] +\end{proof} + +\begin{satz} + Sei $f\colon D \to \R$ differenzierbar. Dann gilt + $f$ ist konstant auf $D$ $\iff$ $f' \equiv 0$ auf $D$ +\end{satz} + +\begin{proof} + ,,$\implies$'' klar. + + ,,$\impliedby$ '': Für $a < b \in D$ gilt + $f(b) - f(a) = (b-a)\underbrace{f'(x)}_{= 0}$, $x \in \;]a,b[ \implies + f(b) = f(a)$. +\end{proof} + +\subsection{Höhere Ableitungen und Satz von Taylor} + +\begin{definition} + Ist $f\colon D \to \R$ differenzierbar und $f'\colon D \to \R$ + differenzierbar, dann sagt man: $f$ ist zweimal differenzierbar + und $f''\colon D \to \R$ heißt die zweite Ableitung von $f$. + + Analog wird die $n$-te Ableitung $f^{(n)}$ von $f$ definiert + mit $f^{(0)} := f$, die ,,nullte'' Ableitung. Schreibe + \[ + \frac{d^{n}f}{dx^{n}} \quad \text{oder} \quad \left( \frac{d}{dx} \right)^{n} f \quad \text{oder} \quad \frac{d^{n}}{dx^{n}}f + .\] +\end{definition} + +\begin{definition}[$C^{n}(D, \R)$] + Eine Funktion $f\colon D \to \R$ heißt stetig differenzierbar + auf $D$, falls $f$ auf $D$ differenzierbar und $f'\colon D \to \R$ stetig + ist. Schreibe $f \in C^{1}(D, \R)$. + + $f$ heißt $n$-mal stetig differenzierbar auf $D$ ($f \in C^{n}(D, \R)$), + falls $f$ $n$-mal differenzierbar und $f^{(n)}\colon D \to \R$ + stetig ist. +\end{definition} + +\begin{bem} + \begin{enumerate} + \item Ist $f$ beliebig oft differenzierbar, dann + gilt $f \in C^{\infty}(D, \R)$ bzw. $f$ ist glatt auf $D$. + \item $f$ stetig auf $D$, dann gilt $f \in C^{0}(D, \R)$. + \item $f \in C^{n}(D, \R) \implies f^{(k)}$ stetig auf $D$ + $\forall 0 \le k \le n$. + \end{enumerate} +\end{bem} +\end{document}