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Dann $\exists \delta \in [0,1]$, s.d. + \[ + \sup_{x \in [0,1]} |f(x)| \cdot \delta < \frac{1}{4} \epsilon + .\] Auf $[\delta , 1] f(x)$ stetig und R.-integrierbar. Dann + ex. eine Zerlegung $Z_{\delta } \in \mathcal{Z}(\delta , 1)$, s.d. + \[ + |\overline{S}_{Z_\delta }(f) - \underline{S}_{Z_\delta }(f)| < \frac{1}{2} \epsilon + .\] Ergänze $Z_{\delta }$ um das Intervall $[0, \delta ]$ + $\implies Z \in \mathcal{Z}(0,1)$. Und es gilt + \[ + |\overline{S}_{Z}(f) - \underline{S}_{Z}(f)| + \le | \overline{S}_{Z_\delta }(f) - \underline{S}_{Z_\delta }(f)| + + 2 \sup_{x \in [0, \delta ]} |f(x)| \cdot \delta < \epsilon + .\] +\end{bsp} + +\begin{satz}[Linearität] + Seien $f, g\colon I = [a,b] \to \R$ (beschränkt) R.-integrierbar. + Dann ist $\alpha f + \beta g$, $\alpha, \beta \in \R$ über + $I$ R.-integrierbar und es gilt + \[ + \int_{a}^{b} (\alpha f + \beta g)(x) dx = + \alpha \int_{a}^{b} f(x) dx + \beta \int_{a}^{b} g(x) dx + .\] +\end{satz} + +\begin{proof} + Es ex. $RS_Z(f)$ und $RS_Z(g)$ s.d. + \begin{align*} + \lim_{h \to 0} RS_Z(f) &= + \int_{a}^{b} f(x) dx \\ + \lim_{h \to 0} RS_Z(g) &= + \int_{a}^{b} g(x) dx + .\end{align*} + o.B.d.A. $Z$ und $\xi_k$ sind gleich für $f$ und $g$. Damit folgt + \begin{align*} + RS_Z(\alpha f + \alpha g) &:= RS_Z(\alpha f) + RS_Z(\beta g) + = \alpha RS_Z(f) + \beta RS_Z(g) \\ + \implies \quad \alpha \int_{a}^{b} f(x) dx + \beta \int_{a}^{b} g(x) dx + &= \alpha \lim_{h \to 0} RS_Z(f) + \beta \lim_{h \to 0} RS_Z(g) \\ + &= \lim_{h \to 0} (\alpha \cdot RS_Z(f)) + + \lim_{h \to 0} (\beta \cdot RS_Z(g)) \\ + &= \lim_{h \to 0} RS_Z(\alpha f) + \lim_{h \to 0} RS_Z(\beta g) \\ + &= \lim_{h \to 0} RS_Z(\alpha f + \beta g) \\ + &= \int_{a}^{b} (\alpha f + \beta g)(x)dx \\ + .\end{align*} +\end{proof} + +\begin{satz}[Monotonie des Riemann-Integrals] + Seien $f, g\colon I = [a,b] \to \R$ (beschränkte) + R.-integrierbare Funktionen mit $g(x) \ge f(x)$ $\forall x \in [a,b]$. + Dann gilt + \[ + \int_{a}^{b} g(x) dx \ge \int_{a}^{b} f(x) dx + .\] + \label{satz:riemann-monoton} +\end{satz} + +\begin{proof} + Es gilt für Zerlegung $Z$ und $\xi_k \in I_k$ : + \begin{align*} + RS_Z(f) = \sum_{k=1}^{n} f(\xi_k)(x_k - x_{k-1}) + \le \sum_{k=1}^{n} g(\xi_k)(x_k - x_{k-1}) = RS_Z(g) + .\end{align*} Für $h \to 0$ folgt die Behauptung. +\end{proof} + +\begin{korrolar} + Sei $f\colon [a,b] \to \R$ (beschr.) R.-integrierbare Funktion, + $m \le f(x) \le M$. Dann gilt + \begin{align*} + m(b - a) \le \int_{a}^{b} f(x) dx \le M(b-a) + .\end{align*} +\end{korrolar} + +\begin{proof} + $g \equiv 1 \implies \int_{a}^{b} 1 dx = (b - a)$ Damit folgt + \begin{align*} + m (b-a) = \int_{a}^{b} m \cdot dx + \stackrel{\text{Satz \ref{satz:riemann-monoton}}}{\le} + \int_{a}^{b} f(x) dx + \le \int_{a}^{b} M dx + = M (b-a) + .\end{align*} +\end{proof} + +\begin{korrolar} + Seien $f, g\colon I \to \R$ zwei beschr. R.-integrierbare + Funktionen. Dann gilt + \begin{enumerate}[(a)] + \item $f_+ := \text{max}\{f, 0\} $ und + $f_- := \text{min}\{f, 0\} $ sind R.-integrierbar + \item $|f|$ ist R.-integrierbar und es gilt + \[ + \left| \int_{a}^{b} f(x) dx \right| + \le \int_{a}^{b} |f(x)| dx + .\] + \item $\forall p \in [1, \infty)$ ist $|f|^{p}$ R.-integrierbar + \item $f \cdot g$ ist R.-integrierbar. + \end{enumerate} +\end{korrolar} + +\begin{proof} + \begin{enumerate}[(a)] + \item $Z \in \mathcal{Z}(a,b)$. + \begin{align*} + 0 &\le \overline{S}_Z(f_+) - \underline{S}_Z(f_+) + \le \overline{S}_Z(f) - \underline{S}_Z(f) \\ + 0 &\le \overline{S}_Z(f_-) - \underline{S}_Z(f_-) + \le \overline{S}_Z(f) - \underline{S}_Z(f) + .\end{align*} + Falls $|\overline{S}_Z(f) - \underline{S}_Z(f)| + \xrightarrow{h \to 0} 0$ + $\implies$ $|\overline{S}_Z(f_{\pm}) - \underline{S}_Z(f_\pm) + | \xrightarrow{h \to 0} 0 \implies$ Beh. + \item $|f| = f_+ - f_- \stackrel{\text{Linearität}}{\implies} + |f|$ R.-integrierbar. + $f \le |f|, - f \le |f| \stackrel{\text{Monotonie}}{\implies} + \int_{a}^{b} f(x) dx \le \int_{a}^{b} |f(x)| dx + \implies \left| \int_{a}^{b} f(x) dx \right| + \le \int_{a}^{b} |f(x)| dx $. + \item Sei $M = \sup_{x \in [a,b]} |f| \stackrel{\text{linear}}{\implies} \frac{|f|}{M}$ integrierbar. $0 \le \frac{|f|}{M} \le 1 + \implies$ z.Zg.: $|f|^{p}$ integr. für $0 \le f \le 1$. + + Sei $0 \le x \le y \le 1$. Aus dem Mittelwertsatz der + Differentialrechnung folgt + \begin{align*} + y^{p} - x^{p} &= p \cdot \xi^{p-1}(y - x) \\ + \implies |y|^{p} - |x|^{p} &= p \cdot |\xi|^{p-1}(|y| - |x|) + \le p \left( |y| - |x| \right) + .\end{align*} + Für $Z \in \mathcal{Z}(0,1)$ gilt + \begin{align*} + \underbrace{\overline{S}_Z(|f|^{p}) - \underline{S}_Z(|f|^{p})}_{\xrightarrow{h \to 0} 0} + \le p \underbrace{(\overline{S}_Z(|f|) - \underline{S}_Z(|f|))}_{ \impliedby \xrightarrow{h \to 0} 0} + .\end{align*} + \item $f\cdot g = \frac{1}{4}\left( (f+g)^2 - (f-g)^2 \right) $ und + c). + \end{enumerate} +\end{proof} + +\begin{bem} + Im Allgemeinen ist + \[ + \int_{a}^{b} f(x) g(x) dx \neq \left( \int_{a}^{b} f(x) dx \right) + \left( \int_{a}^{b} g(x) dx \right) + .\] +\end{bem} + +\begin{korrolar}[Definitheit des R.-Integrals] + Sei $f\colon I = [a,b] \to \R$ eine stetige Funktion mit + $f(x) \ge 0$, $x \in [a,b]$. Dann gilt + \begin{align*} + \int_{a}^{b} f(x) dx = 0 \implies f \equiv 0 + .\end{align*} +\end{korrolar} + +\begin{proof} + durch Kontraposition. + Sei $f \not\equiv 0$, d.h. $\exists x_0 \in [a,b]$ mit $f(x_0) > 0$. + $\stackrel{f\text{ stetig}}{\implies} \exists I_{\epsilon} + := [x_0, x_0 + \epsilon]$ oder $I_{\epsilon} := [x_0 - \epsilon, x_0]$, s.d. + $f(x) \ge \delta > 0$ $\forall x \in I_{\epsilon}$. + + Sei $Z \in \mathcal{Z}(a,b)$ mit $h$ klein genug, s.d. + für ein $k$ $I_k \subset I_{\epsilon}$. Dann gilt + \begin{align*} + 0 &< \delta (x_{k} - x_{k-1}) \le + \inf_{x \in I_k} f(x) (x_k - x_{k-1}) + \le \underline{S}_Z(f) \le \int_{a}^{b} f(x) dx + .\end{align*} +\end{proof} + +\begin{definition} + Sei $a \le b$ Dann ist + \begin{align*} + \int_{b}^{a} f(x) dx &:= - \int_{a}^{b} f(x) dx \\ + \int_{a}^{a} f(x) dx &:= 0 + .\end{align*} +\end{definition} + +\begin{satz}[1. Mittelwertsatz] + Sei $f\colon I = [a,b] \to \R$ stetig, $g \colon I \to \R$ + R.-integrierbar. $g$ habe in $I$ keinen Vorzeichenwechsel. + Dann $\exists \xi \in [a,b]$ s.d. gilt + \[ + \int_{a}^{b} f(x) g(x) dx = f(\xi) \int_{a}^{b} g(x) dx + .\] +\end{satz} + +\begin{proof} + Sei $g \ge 0$ (o.B.d.A.). $f$ stetig + $\implies$ $\exists m = \min_{x \in I} f(x)$, $M = \max_{x \in I} f(x)$. + Dann folgt + \begin{align*} + m \int_{a}^{b} g(x) dx \le \int_{a}^{b} f(x) g(x) dx + \le M \int_{a}^{b} g(x) dx + .\end{align*} + Betrachte $\varphi(t) := (m (1-t) + M \cdot t) \int_{a}^{b} g(x) dx $, + $t \in [0,1]$. Nach ZWS $\exists \theta \in [0,1]$, s.d. + \begin{align*} + \varphi(\theta) &= y = (m(1-\theta) + M\cdot \theta) \int_{a}^{b} g(x) dx \\ + \varphi(0) &\le y \le \varphi(1) \\ + m \int_{a}^{b} g(x) dx &\le y \le M \int_{a}^{b} g(x) dx \\ + \implies \int_{a}^{b} f(x) g(x) dx &= \mu \int_{a}^{b} g(x) dx + .\end{align*} Nach dem ZWS für $f$ $\exists \xi \in [a,b]$, s.d. + $f(\xi) = \mu \in [m, M]$. +\end{proof} + +\begin{korrolar} + Sei $f\colon I \to \R$ stetig. + \begin{enumerate} + \item $\exists \xi \in I$, s.d. $\int_{a}^{b} f(x) dx = + f(\xi)(b-a)$ + \item Sei $f\colon [a,b] \to \R$ stetig mit $m \le f(x) \le M$. + $x \in I$. Sei $g\colon I \to \R$ R.-integrierbar mit $g \ge 0$. + Dann gilt + \[ + m \int_{a}^{b} g(x) dx \le \int_{a}^{b} f(x) g(x) dx + \le M \int_{a}^{b} g(x) dx + .\] + \end{enumerate} +\end{korrolar} + +\begin{bem} + Voraussetzungen sind unverzichtbar! + + Stetigkeit: $f(x) = \begin{cases} + 0 & 0 \le x < 1 \\ + 1 & 1 \le x \le 2 + \end{cases}$ unstetig. + \begin{align*} + f(\xi)(b-a) = f(\xi) \cdot 2 = \begin{cases} + 0 & 0 \le \xi < 1 \\ + 2 & 1 \le \xi \le 2 + \end{cases} \neq 1 = \int_{0}^{2} f(x) dx + .\end{align*} + + Positivität: $f(x) = x$, $g(x) = \begin{cases} + -1 & 0 \le x < 1 \\ + 1 & 1 \le x \le 2 + \end{cases}$. + \begin{align*} + \int_{0}^{1} f(x) g(x) dx = \int_{0}^{1} (-x) dx + + \int_{1}^{2} x dx + = - \left( \frac{1}{2} - 0 \right) + \left( \frac{4}{2} - \frac{1}{2} \right) = 1 + .\end{align*} aber + \begin{align*} + \xi \cdot \int_{0}^{2} g(x) dx = \xi \cdot 0 = 0 \quad \forall \xi \in [0,2] + .\end{align*} +\end{bem} + +\subsection{Hauptsatz der Differential- und Integralrechnung} + +\begin{satz}[Hauptsatz der Differential- und Integralrechnung] + Es sei $a < b$, $f \colon [a,b] \to \R $ stetig. + \begin{enumerate}[(a)] + \item Die Funktion $F_0 \colon [a,b] \to \R$ definiert durch + $F_0(x) := \int_{a}^{x} f(t) dt $ ist + stetig differenzierbar auf $[a,b]$ und + \[ + F_0'(x) = f(x) \quad x \in [a,b] \quad (*) + .\] + Jede Funktion $F \in C^{1}([a,b], \R)$ welche $(*)$ erfüllt, + heißt Stammfunktion von $f$. + \item Jede Stammfunktion $F$ von $f$ hat die Form + \[ + F(x) = C + \int_{a}^{x} f(t) dt = C + F_0(x) + .\] + \item Ist $F$ eine Stammfunktion von $f$, dann gilt + \[ + \int_{a}^{b} f(t) dt = F(b) - F(a) \; \text{d.h. insb.} + \; \int_{a}^{b} F'(t) dt = F(b) - F(a) + \quad \forall F \in C^{1}([a,b], \R) + .\] + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate}[(a)] + \item + Für $h \neq 0$, $x \in [a,b]$ + \begin{align*} + \frac{F_0(x+h) - F_0(x)}{h} + = \frac{1}{h} \left( \int_{a}^{x+h} f(t) dt + - \int_{a}^{x} f(t) dt \right) + = \frac{1}{h} \int_{x}^{x+h} f(t) dt + \quad \stackrel{\text{1. MWS}}{=} + \quad \frac{1}{h} f(\xi_h) \cdot h + \xrightarrow[h \to 0]{f\text{ stetig}} f(x) \\ + \implies F_0'(x) = \lim_{h \to 0} \frac{F_0(x+h) - F_0(x)}{h} = f(x) + .\end{align*} + \item Sei $F$ eine Stammfunktion ovn $f$, dann gilt + $(F - F_0)' = f - f = 0 + \stackrel{\text{MWS Diff.}}{\implies} F - F_0 \equiv + \text{konstant} = C \implies F = F_0 + C$ für ein $C \in \R$. + \item $F(b) - F(a) \stackrel{\text{(b)}}{=} F_0(b) - F_0(a) + = \int_{a}^{b} f(t) dt $ + \end{enumerate} +\end{proof} + +\begin{bem} + \begin{enumerate} + \item $F(b) - F(a) =: F(x) \Big|_a^b$ + $\implies \int_{a}^{b} F'(t) dt = F(x) \Big|_a^b $. + + Man bezeichnet eine Stammfunktion auch als unbestimmtes Integral + \[ + F(x) = \int f(x) dx + .\] (math. nicht korrekte Bezeichnung) + \item Integration und Differentiation sind inverse zu einander + \begin{align*} + \frac{d}{dx} \int_{a}^{x} f(t) dt = f(x) \\ + F(x) = F(a) + \int_{a}^{x} F'(t) dt + .\end{align*} + \end{enumerate} +\end{bem} + +\end{document} diff 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z&$qk_3}4{wImUnUQ`ja?jIkbXosZ!o#xgRpTTrIPw2{29wuaqj#jYIf;?-eulA literal 0 HcmV?d00001 diff --git a/ws2019/la/uebungen/la12.tex b/ws2019/la/uebungen/la12.tex new file mode 100644 index 0000000..beb6634 --- /dev/null +++ b/ws2019/la/uebungen/la12.tex @@ -0,0 +1,12 @@ +\documentclass{../../../lecture} + +\begin{document} + +\begin{align*} + \left[\begin{matrix}1 & 1 & -1 & -0.5\\0 & -2 & 6 & 6\\0 & 0 & -6 & -15\\0 & 0 & 0 & 10\end{matrix}\right] +\left[\begin{matrix}1 & 0 & 0 & 0\\0 & 0.333333333333333 & 2.22044604925031 \cdot 10^{-16} & 4.44089209850063 \cdot 10^{-16}\\0 & 0 & 0.200000000000001 & 3.33066907387547 \cdot 10^{-15}\\0 & 0 & 1.33226762955019 \cdot 10^{-15} & 0.0357142857142889\end{matrix}\right] +.\end{align*} + +sympy integrate((-1+6*x+6*x^2)*(-0.5 +6*x + -15*x^2 + 10*x^3), (x, 0, 1)) sympy + +\end{document}