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Except for the limited purpose of indicating that +material is shared under a Creative Commons public license or as +otherwise permitted by the Creative Commons policies published at +creativecommons.org/policies, Creative Commons does not authorize the +use of the trademark "Creative Commons" or any other trademark or logo +of Creative Commons without its prior written consent including, +without limitation, in connection with any unauthorized modifications +to any of its public licenses or any other arrangements, +understandings, or agreements concerning use of licensed material. For +the avoidance of doubt, this paragraph does not form part of the +public licenses. + +Creative Commons may be contacted at creativecommons.org. + diff --git a/an1e_zf.pdf b/an1e_zf.pdf Binary files differnew file mode 100644 index 0000000..e9f93f8 --- /dev/null +++ b/an1e_zf.pdf diff --git a/an1e_zf.tex b/an1e_zf.tex new file mode 100644 index 0000000..1587ce6 --- /dev/null +++ b/an1e_zf.tex @@ -0,0 +1,485 @@ +\documentclass[a4paper, twocolumn]{article} + +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{mathtools} + +\usepackage{float} +\usepackage{array} +\usepackage{booktabs} +\usepackage{multirow} +\usepackage{framed} + +\usepackage[german]{babel} + +\usepackage[margin=2cm]{geometry} +\usepackage{xcolor} +\usepackage{graphicx} + +\usepackage[colorlinks = true, + linkcolor = blue, + urlcolor = blue, + citecolor = blue, + anchorcolor = blue]{hyperref} + +\usepackage{tikz} +\usetikzlibrary{calc} + + +\title{An1E Zusammenfassung} +\author{Naoki Pross} +\date{Januar 2020} + + +\newcommand{\nset}[1]{\ensuremath{\mathbb{#1}}} +\newcommand{\heq}{\ensuremath{\stackrel{\hat{\texttt{H}}}{=}}} +\newcommand{\noticeq}{\ensuremath{\stackrel{!}{=}}} +\newcommand{\dd}[1]{\ensuremath{\mathrm{d}#1}} +\newcommand{\df}[2]{\ensuremath{\frac{\dd{#1}}{\dd{#2}}}} + +\newcommand{\brpage}[1]{\textcolor{red!70!black}{\small\texttt{S#1}}} + + +\begin{document} + +\section{Ungleichungen \brpage{31}} +\begin{tabular*}{\linewidth}{l >{\(}r<{\) } @{{\(\;\leq\;\)}} >{ \(}l<{\)}} + Bernoulli & 1 + na & (1+a)^n \\ + Binomische & |ab| & \frac{1}{2}(a^2 + b^2) \\ + Dreiecks & |a + b| & |a| + |b| \\ +\end{tabular*} +Mittel (\(\forall j: a_j \geq 0, n \in \nset{N}\)) +\begin{align*} +\begin{array}{*3{>{\displaystyle}l}} + \texttt{HM } &\leq \texttt{ GM } &\leq \texttt{ AM} \\ + \left[ \frac{1}{n}\sum_{j=1}^n\frac{1}{a_j}\right]^{-1} + &\leq + \sqrt[n]{\prod_{j=1}^n a_j} + &\leq + \frac{1}{n}\sum_{j=1}^{n} a_j +\end{array} +\end{align*} +Integral +\begin{align*} + \left| \int_a^b f(x) \;\dd{x} \right| \leq \int_a^b |f(x)| \;\dd{x} +\end{align*} + +\section{Zahlenfolgen und Reihen} +\subsection{Konvergenz \brpage{679}} +\textbf{Hinweise:} Induktion, Einschlie{\ss}ungsprinzip, Bolzano-Weierstrass. +\begin{align*} + \exists g \in \nset{R} : g = \lim_{n\to\infty} \langle f_n \rangle + \iff \langle f_n \rangle \text{ konvergiert} +\end{align*} + +\subsection{Divergenz \brpage{472}} +Divergent hei{\ss}t nicht konvergent: +\begin{align*} + \lim_{n\to\infty} \langle f_n \rangle = \pm\infty + &\implies \langle f_n \rangle \text{ divergiert \emph{bestimmt}} \\ + \nexists \lim_{n\to\infty} \langle f_n \rangle + &\implies \langle f_n \rangle \text{ divergiert} +\end{align*} + +\subsection{Binomischer Satz \brpage{12}} +\begin{align*} + (a+b)^n &= \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k + & + \binom{n}{k} &= \frac{n!}{k!(n-k)!} +\end{align*} + +\subsection{Folgen \brpage{470}} +\begin{center} + \begin{tabular}{l >{\(}l<{\)} >{\(}l<{\)}} + Arithmetisch & a_{n+1} = a_n + d & d = a_{n+1} - a_n \\ + Geometrisch & a_{n+1} = q a_n & q = a_{n+1} / a_n \\ + \end{tabular} +\end{center} +\begin{center} + Monotonie der Folge + \begin{tabular}{*3{>{\(}l<{\)}}} + \midrule + d > 0 & q > 1 &\implies \langle a_n\rangle\Uparrow \\ + d \geq 0 & q \geq 1 &\implies \langle a_n\rangle\uparrow \\ + d < 0 & q \in (0;1) &\implies \langle a_n\rangle\Downarrow \\ + d \leq 0 & q \in (0;1] &\implies \langle a_n\rangle\downarrow \\ + \end{tabular} +\end{center} + +\subsection{Reihen \brpage{20,477,1075}} +\begin{align*} + \sum_{k=1}^n k &= \frac{n(n+1)}{2} & + \sum_{k=1}^n k^2 &= \frac{n(n+1)(2n+1)}{6} \\ + \sum_{k=1}^n k^3 &= \frac{n^2(n+1)^2}{4} & + \sum_{k=0}^{n-1} ar^k &= a\left(\frac{1-r^n}{1-r}\right) (r \neq 1) +\end{align*} + +\section{Funktionen \brpage{49}} +\[ +f : \mathbb{D}_f \to \mathbb{W}_f \quad x \mapsto f(x) +\] + +\subsection{Lineare Transformationen} +Seien \(\mu,\lambda,\ell,o \geq 0\). +Mit \(< 0\) werte Streckungen sind Spiegelungen und Verschiebungen sind in Gegenrichtung. +\[ +\mathfrak{T}\{f\} = \mu f(\lambda x + \ell) + o +\] +Wobei +\(\mu = y\)-Streckung, +\(\lambda = x\)-Streckung, +\(\ell = \) Verschiebung nach links, +\(o = \) Verschiebung nach oben. + +\subsection{Monotonie \brpage{51,453}} +\begin{center} + \begin{tabular}{>{\(}c<{\)} l >{\(}l<{\)}} + \text{Zeichen} & \text{Bedeutung} & \text{Bedingung } (\forall\varepsilon > 0) \\ + \midrule + f \Uparrow & \text{streng wachsend} & f(x) < f(x + \varepsilon) \\ + f \uparrow & \text{wachsend} & f(x) \leq f(x + \varepsilon) \\ + f \Downarrow & \text{streng fallend} & f(x) > f(x + \varepsilon) \\ + f \downarrow & \text{fallend} & f(x) \geq f(x + \varepsilon) \\ + \end{tabular} \\ +\end{center} +\begin{center} + \begin{tabular}{*3{>{\(}c<{\)}}} + \text{Monotonie} & f'' \neq 0 & f^{(n)} \neq 0 \text{ und } n \text{ gerade} \\ + \midrule + f \Uparrow & f' > 0 & f^{(n-1)} > 0 \\ + f \uparrow & f' \geq 0 & f^{(n-1)} \geq 0 \\ + f \Downarrow & f' < 0 & f^{(n-1)} < 0 \\ + f \downarrow & f' \leq 0 & f^{(n-1)} \leq 0 \\ + \end{tabular} +\end{center} +\footnotesize{NB: Gilt auch f\"ur Zahlenfolgen (\(f(x) \leadsto f_n, f(x+\varepsilon) \leadsto f_{n+1}\)) + +\subsection{Symmetrien \brpage{52}} +\begin{center} + \resizebox{\linewidth}{!}{% + \begin{tabular}{l >{\(}r<{\)} @{\(\;=\;\)} >{\(}l<{\)} l} + \(f\)& \multicolumn{2}{l}{\text{Bedingung}} & Bedeutung \\ + \midrule + gerade & f(-x) & f(x) & \(y\)-Symmetrisch \\ + ungerade & f(-x) & -f(x) & Nullpunkt-Symmetrisch \\ + periodisch & f(x) & f(x\pm p) & \(p \in \nset{R}\) + \end{tabular} + } +\end{center} + +\subsection{Beschranktheit \brpage{52,676}} +Eine funktion hei{\ss}t nach unten oder oben beschr\"ankt, wenn ihre Werte nicht gr\"o{\ss}er oder kleiner als eine eine bestimmte Zahl \(K\) bzw. \(k\) sind. +\(f\) ist beschr\"ankt wenn \(\exists \sup f \wedge \exists \inf f \iff \forall x: k < f(x) < K\). +\begin{align*} + K = \sup f &\iff \exists K \in \nset{R} : \forall x : f(x) < K \\ + k = \inf f &\iff \exists k \in \nset{R} : \forall x : f(x) > k +\end{align*} + +\subsection{Stetigkeit \brpage{60}} +Eine funktion hei{\ss}t \emph{stetig} wenn: +\begin{align*} + \forall x \in \mathbb{D}_f : \lim_{u^{-} \to x} f(u) = \lim_{u^{+} \to x} f(u) = f(x) +\end{align*} + +\subsection{Nullstellen \brpage{40,47,48}} +\subsection{Extremstellen \brpage{455}} +\subsection{Wendepunkte \brpage{256}} +\subsection{Konvexit\"at \brpage{253}} +Auch als Kr\"ummungsverhalten bekannt. Sei \(P = (x,f(x))\) und kein Wendepunkt, +d.h. \(f''(x) \neq 0\). +\begin{align*} + f''(x) > 0 & \implies \text{ nach oben konkav, streng konvex} \\ + f''(x) < 0 & \implies \text{ nach unten konkav, streng konkav} +\end{align*} + +\subsection{Wendepunkte \brpage{256}} + +\subsection{Asymptoten \brpage{260}} +Sei \(a(x) = kx + b\) die allgemeine Asymptot von \(f(x)\), d.h. +\(\lim_{x\to\infty} f(x) - a(x) = 0\). Dann +\begin{align*} + k &= \lim_{x\to\infty}\frac{f(x)}{x} \heq \lim_{x\to\infty} f'(x) + & b &= \lim_{x\to\infty}\left( f(x) - kx \right) +\end{align*} + +\subsection{Umkehrfunktion \brpage{53}} +Umkehrbarkeit Bedingungen: +\begin{align*} + f^{-1} : \mathbb{W}_f \to \mathbb{D}_f \quad f(x) \mapsto x \\ + \exists f^{-1} \iff (f\Downarrow)\vee(f\Uparrow) +\end{align*} + +\subsection{Polynomen \brpage{65}} +\begin{align*} +P_n(x) = \sum_{i=0}^n a_i x^i = \prod_{i=1}^n (x-r_i) +\end{align*} +Nullstellen \brpage{40} (Wurzeln) \(r_i\) k\"onnen mithilfe von Faktorisierung, +der Quadratische Formel \(r = \frac{1}{2a}(-b \pm \sqrt{b^2 - 4ac}) \) +oder dem Hornerschema \brpage{966} gel\"ost werden. +\begin{align*} + P_n(x) = (x - u) P_{n-1}(x) + P_n(u) +\end{align*} +Seien \(a_i\) die Koeffizienten von \(P_n(x)\), \(b_i\) von \(P_{n-1}(x)\) und \(u \in \mathbb{D}_P\). +Wenn \(P_n(u) = 0\), dann ist \(u = r\) d.h. eine Nullstelle. +\begin{center} + \begin{tabular}{>{\(}c<{\)} | >{\(}c<{\)} >{\(}c<{\)} >{\(}c<{\)} >{\(}c<{\)} | >{\(}c<{\)} c} + & a_n & a_{n-1} & \cdots & a_1 & a_0 & \multirow{2}{*}{+} \\ + \times u & & u b_{n-1} & \cdots & u b_1 & u b_0 \\ + \midrule + & b_{n-1} & b_{n-2} & \cdots & b_0 & P_n(u) \\ + \end{tabular} +\end{center} + +\subsection{Gebrochene Funktionen \brpage{14}} +\begin{align*} +R(x) = \frac{P_m(x)}{Q_n(x)} = \frac{p_m x^m + \cdots + p_0}{q_n x^n + \cdots + q_0} +\end{align*} + +\subsubsection{Partialbruchzerlegung \brpage{15}} + +\subsection{Trigonometrische \brpage{77,80,147,165}} +\begin{center} + \begin{tikzpicture}[scale=4] + \draw[gray,dashed] (0,0) -- + node[pos=.7, sloped, above] {\(0\)} + node[pos=1, anchor=west, sloped] {\(\left(1,0,0\right)\)} + (1.1,0); + + \draw[gray,dashed] (0,0) -- + node[pos=.7, sloped, above] {\(\pi/2\)} + node[pos=1, anchor=west, sloped] {\(\left(0,1,\infty\right)\)} + (0,1.1); + + \draw[gray,dashed] (0,0) -- + node[pos=.7, sloped, above] {\(\pi/12\)} + node[pos=1, anchor=west, sloped] {\(\left(\frac{1+ \sqrt3}{2\sqrt 2},\frac{\sqrt3 -1}{2\sqrt 2}\right)\)} + ({1.1 *cos(15)}, {1.1 * sin(15)}); + + \draw[gray,dashed] (0,0) -- + node[pos=.7, sloped, above] {\(\pi/8\)} + node[pos=1, anchor=west, sloped] {\(\scriptscriptstyle\left(\frac{\sqrt{2 + \sqrt{2}}}{2},\frac{\sqrt{2-\sqrt{2}}}{2}\right)\)} + ({1.1 *cos(pi/8 r)}, {1.1 * sin(pi/8 r)}); + + \draw[dashed] (0,0) -- + node[pos=.7, sloped, above] {\(\pi/6\)} + node[pos=1, anchor=west, sloped] {\(\left(\frac{\sqrt 3}{2},\frac{1}{2},\frac{\sqrt3}{3}\right)\)} + ({1.1 *cos(30)}, {1.1 * sin(30)}); + + \draw[dashed] (0,0) -- + node[pos=.7, sloped, above] {\(\pi/4\)} + node[pos=1, anchor=west, sloped] {\(\left(\frac{\sqrt 2}{2},\frac{\sqrt 2}{2}, 1\right)\)} + ({1.1 *cos(45)}, {1.1 * sin(45)}); + + \draw[dashed] (0,0) -- + node[pos=.7, sloped, above] {\(\pi/3\)} + node[pos=1, anchor=west, sloped] {\(\left(\frac{1}{2},\frac{\sqrt 3}{2},\sqrt{3}\right)\)} + ({1.1 *cos(60)}, {1.1 * sin(60)}); + + \draw[black, thick] ({cos(-5)}, {sin(-5)}) arc (-5:100:1); + \end{tikzpicture} +\end{center} +Definitionen der grunds\"atzlichen Winkelfunktionen. +\begin{align*} + \sin(x) &= \frac{e^{ix} - e^{-ix}}{2i} = \sum_{n=0}^\infty (-1)^n \frac{x^{(2n+1)}}{(2n+1)!} \\ + \cos(x) &= \frac{e^{ix} + e^{ix}}{2} = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}\\ + \sinh(x) &= \frac{e^{x} - e^{-x}}{2} = \sum_{n=0}^\infty \frac{x^{(2n+1)}}{(2n+1)!} \\ + \cosh(x) &= \frac{e^{x} + e^{x}}{2} = \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}\\ +\end{align*} +Beziehungen und Identit\"aten. +\[ +\cos^2(x) + \sin^2(x) = 1 \quad \cosh^2(x) - \sinh^2(x) = 1 +\] + +\begin{center} + \begin{tabular}{>{\(}l<{\)} @{\(\;=\;\)} >{\(}r<{\)} >{\(}l<{\)} @{\(\;=\;\)} >{\(}r<{\)} } + \toprule + \cos(\alpha + 2\pi) & \cos(\alpha) & \sin(\alpha + 2\pi) & \sin(\alpha) \\ + \cos(-\alpha) & \cos(\alpha) & \sin(-\alpha) & -\sin(\alpha) \\ + \cos(\pi - \alpha) & -\cos(\alpha) & \sin(\pi - \alpha) & \sin(\alpha) \\ + \cos(\frac{\pi}{2} - \alpha) & \sin(\alpha) & \sin(\frac{\pi}{2} - \alpha) & \cos(\alpha) \\ + \midrule + \cos(\alpha + \beta) & \multicolumn{3}{l}{\(\cos\alpha\cos\beta - \sin\alpha\sin\beta\)} \\ + \sin(\alpha + \beta) & \multicolumn{3}{l}{\(\sin\alpha\cos\beta - \cos\alpha\sin\beta\)} \\ + \midrule + \cos(2\alpha) & \multicolumn{3}{l}{\(\cos^2{\alpha} - \sin^2{\alpha} \)} \\ + & \multicolumn{3}{l}{\(1 - 2\sin^2\alpha\)} \\ + & \multicolumn{3}{l}{\(2\cos^2\alpha - 1\)} \\ + \sin(2\alpha) & \multicolumn{3}{l}{\(2\sin\alpha\cos\alpha\)} \\ + \tan(2\alpha) & \multicolumn{3}{l}{\((2\tan\alpha)(1 + \tan^2\alpha)^{-1}\)} \\ + \bottomrule + \end{tabular} +\end{center} + +\section{Grenzwert \brpage{55}} +Bedingungen f\"ur die Existenz einer Grenzwert: +\begin{align*} + \exists \lim_{x\to a} f(x) = g \iff \lim_{x\to a^-} f(x) = \lim_{x\to a^+} f(x) +\end{align*} +Formell lautet der \(\delta - \varepsilon\) Kriterium: +\begin{align*} + \lim_{x\to a}f(x) \iff \forall \varepsilon > 0: \exists a: |f(a) - g| < \varepsilon +\end{align*} + + +\subsection{Unbestimmte Formen} +\begin{align*} +\frac{0}{0},\; \frac{\infty}{\infty},\; 0\cdot\infty,\; \infty - \infty,\; 0^0,\; \infty^0,\; 1^\infty +\end{align*} + +\subsection{Enschlie{\ss}ungsprinzip \brpage{56}} +Auch als ``Sandwitch'' bekannt. +\(\forall x : a(x) \leq f(x) \leq b(x)\) +\begin{align*} + \exists \left(\lim_{x\to\pm\infty} a(x) = \lim_{x\to\pm\infty} b(x) = g\right) + \implies + \lim_{x\to\pm\infty} f(x) = g +\end{align*} +\footnotesize{NB: gilt auch f\"ur folgen \(a_n, b_n, f_n\)} + +\subsection{Bolzano-Weierstrass \brpage{701}} +\begin{align*} + \begin{rcases} + \exists \sup f \wedge f\Uparrow \\ + \exists \inf f \wedge f\Downarrow + \end{rcases} + \implies f \text{ konvergiert} +\end{align*} + +\subsection{Bemerkenswerte Grenzwerte} +\begin{align*} + \setlength\extrarowheight{8pt} + \begin{array}{*2{>{\displaystyle}l}} + \lim_{x\to 0} \frac{\sin x}{x} = 1 & \lim_{x\to\infty} \left(1 + \frac{a}{x}\right)^x = e^a \\ + \lim_{x\to 0} \frac{a^x - 1}{x} = \ln a & \lim_{x\to\infty} \frac{(\ln x)^a}{x^b} = 0 \\ + \lim_{x\to 0} \frac{e^x - 1}{2} = 1 & \lim_{x\to\infty} \sqrt[x]{p} = 1\\ + \lim_{x\to 0} x\ln x = 0 & \lim_{x\to\infty} \sum_{k=0}^x q^k = \frac{1}{1-q} \quad (|q| < 1)\\ + \end{array} +\end{align*} + +\subsection{Bernoulli-l'H\^opitalsche Regel \brpage{57}} +Wenn \(f(x)/g(x) \to \pm\infty/\pm\infty\) oder \(f/g \to 0/0\) dann gilt: +\begin{align*} + \lim_{x\to a} \frac{f(x)}{g(x)} \heq \lim_{x\to a} \frac{f'(x)}{g'(x)} +\end{align*} +\textbf{Hinweise:} +\begin{align*} + \varphi\psi &= \frac{\varphi}{\psi^{-1}} = \frac{\psi}{\varphi^{-1}} + & 0\cdot\infty &\leadsto \frac{0}{0}, \frac{\infty}{\infty} \\ + \varphi - \psi &= \frac{\psi^{-1} - \varphi^{-1}}{(\varphi\psi)^{-1}} + & \infty - \infty &\leadsto \frac{0}{0} \\ + \varphi^\psi &= e^{\psi\ln\varphi} & (\varphi > 0) +\end{align*} + +\section{Differentialrechnung \brpage{444,446}} +\begin{align*} + f'(x) = \df{f}{x} = D_x f = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h} +\end{align*} + +\subsection{Differenzierbarkeit \brpage{444,445}} +Beide \(f'_+ \text{ und } f'_-\) mussen existieren und gleich sein. +\begin{align*} + \lim_{h\to 0^+} \frac{f(x+h) - f(x)}{h} = f'_+ \noticeq \lim_{h\to 0^-} \frac{f(x+h) - f(x)}{h} = f'_- +\end{align*} + +\subsection{Ableitungsregeln \brpage{445,450}} +\begin{alignat*}{3} + (af) &= af' &\quad&& (u(v(x)))' &= u'(v)v' \\ + (uv)' &= u'v + uv' &\quad&& \left(\frac{u}{v}\right)' &= \frac{u'v-uv'}{v^2} \\ + \left(\sum u_i\right)' &= \sum u'_i &\quad&& (\ln u)' &= \frac{u'}{u} \\ + (f^{-1})' &= \frac{1}{f'(f^{-1}(x))} +\end{alignat*} + +\subsection{Tangente und Normale Funktion} +Zur Funktion \(f(x)\) im Punkt \((p_x, p_y) = (z, f(z))\) +\begin{align*} + t(x) &= f'(p_x)(x - p_x) + p_y & + n(x) &= \frac{p_x - x}{f'(p_x)} + p_y +\end{align*} + +\subsection{Schnittwinkel} +Der Schnittpunkt \(S = (z,f(z)) = (z,g(z))\) findet man mit \(f(z) = g(z)\). Der Schnittwinkel ist dann +\begin{align*} + \tan\vartheta = \frac{g'(z) - f'(z)}{1 + f'(z)g'(z)} +\end{align*} + +\subsection{Mittlewertsatz (der DR) \brpage{454}} +\begin{align*} + f'(\xi) = \frac{f(b) - f(a)}{b-a} \qquad (\xi \in (a;b)) +\end{align*} + +\subsection{Taylor Polynom und Reihe \brpage{484}} +Der Taylor-Polynom approximiert eine Funktion um einen Entwicklungspunkt \(a\). +\begin{align*} + T_n(x, a) &= \sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k + R_n\\ + &= f(a) + \frac{f'(a)}{1!}(x-a)^1 + \frac{f''(a)}{2!}(x-a)^2 + \cdots +\end{align*} +Die Restgliede sind +\begin{align*} + R_n = \frac{f^{(n+1)}(\xi)}{(n+1)!} (x-a)^{(n+1)} \qquad (\xi \in (x;a)) +\end{align*} +Wenn \(\lim_{n\to\infty}R_n = 0\) dann \(f(x) \noticeq T(x,a)\), d.h. die Taylor Rehie zu \(f\) identisch ist. Sonst berechnet man der \emph{worst case} Fehler \(\epsilon \geq |R_n|\) und der dazugeh\"orig \(\hat{\xi} = \underset{\xi}{\arg}\max|R_n|\): +\begin{align*} + \epsilon + = \max |R_n| + = \max \left[\frac{|f^{(n+1)}(\xi)|}{(n+1)!} |x-a|^{(n+1)}\right] +\end{align*} + +\subsection{Fehlerrechnung \brpage{862,866} und Fortpflanzung \brpage{869}} +Sei \(\mathbf{y}\) eine direkte Messerung von eine Funktion \(y\) von \(x\). Ist dann \(\Delta y\) der \emph{absolute} Fehler und \(\delta y\) der \emph{relative} Fehler. +\begin{align*} + \mathbf{y} = y \pm \Delta y = y(1 \pm \delta y) +\end{align*} +Der Messerungsfehler kann mithilfe von einer lineare Approximation fortgepflanzt werden. +\begin{align*} + \lim_{\Delta x\to 0} \frac{\Delta y}{\Delta x} \noticeq \df{y}{x} + \implies &\Delta y \approx y'\Delta x \\ + & \delta y = \frac{\Delta y}{y} \approx \frac{y'\Delta x}{y} + = k\delta x +\end{align*} + +\section{Integralrechnung \brpage{493}} +\subsection{Riemann Itegrierbarkeit \brpage{507}} +Sei \(f \text{ in } [a;b]\) stetig, \(x_0 = a, \dots, x_n = b\) und \(\xi_i \in [x_{i-1};x_{i}]\). +\begin{align*} + \int_b^a f(x) \;\dd{x} = \mathfrak{Ri}\{f\} + = \lim_{\substack{n\to\infty\\ \Delta x_i\to0 }} + \sum_{i=1}^n f(\xi_i) \underbrace{(x_i - x_{i-1})}_{\Delta x_i} +\end{align*} +Bedingungen f\"ur \(f\): stetig oder monoton oder beschr\"ankt und an h\"ochstens endlich vielen Stellen unstetig. + +\subsection{Aufwendungen} +\begin{align*} + \text{Fl\"acheninhalt} && A &= \int_a^b |f(x)| \;\dd{x} \\ + \text{Bogenl\"ange} && \ell &= \int_a^b \sqrt{1 + (f'(x))^2} \;\dd{x} +\end{align*} + +\subsection{Bestimmte Integral \brpage{509}} +\begin{align*} + \int_a^b f(t)\;\dd{t} &= F(b) - F(a) \\ + \int_a^b f(t)\;\dd{t} &= \int_a^0 f(t)\;\dd{t} + \int_0^b f(t)\;\dd{t} +\end{align*} + +\subsection{Mittlewertsatzt \brpage{510}} +Sei \(f(x)\) in \([a;b]\) stetig, dann \(\exists \xi \in (a;b) : f(\xi) = \mu\) (Mittelwert). +\begin{align*} + \frac{1}{b-a}\int_a^b f(t) \;\dd{t} = f(\xi) = \mu \qquad (\xi\in (a,b)) +\end{align*} + +\subsection{Differenzierbarkeit \brpage{509}} +\begin{align*} + \df{}{x} \int f(t) \;\dd{t} &= f(x) \\ + \df{}{x} \int_{a(x)}^{b(x)} f(t) \;\dd{t} &= f(b(x)) b'(x) - f(a(x))a'(x) +\end{align*} + +\section*{License} +{ \tt +An1E-ZF (c) by Naoki Pross +\\\\ +An1E-ZF is licensed under a Creative Commons Attribution-ShareAlike 4.0 Unported License. +\\\\ +You should have received a copy of the license along with this work. If not, see +\\\\ +\url{http://creativecommons.org/licenses/by-sa/4.0/} +} + +\end{document} |