From ef257c975d72d1d0b99dec8e6d9028f370f3b7e5 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Sat, 11 Jul 2026 12:58:09 +0200 Subject: [PATCH] AB zum Beweis --- content/Arbeitsmaterialien.tex | 19 +- content/Mathematischer_Hintergrund.tex | 2 +- galton.pdf | Bin 453107 -> 457287 bytes images/AB3_Galton.png | Bin 0 -> 41471 bytes images/AB3_GaltonVersuche.png | Bin 0 -> 46785 bytes images/AB3_X.png | Bin 0 -> 8084 bytes material/ab2.tex | 13 + material/ab3.pdf | Bin 0 -> 275904 bytes material/ab3.tex | 402 +++++++++++++++++++++++++ material/ab3_Zusatzblatt.pdf | Bin 0 -> 106047 bytes material/ab3_Zusatzblatt.tex | 252 ++++++++++++++++ 11 files changed, 686 insertions(+), 2 deletions(-) create mode 100644 images/AB3_Galton.png create mode 100644 images/AB3_GaltonVersuche.png create mode 100644 images/AB3_X.png create mode 100644 material/ab3.pdf create mode 100644 material/ab3.tex create mode 100644 material/ab3_Zusatzblatt.pdf create mode 100644 material/ab3_Zusatzblatt.tex diff --git a/content/Arbeitsmaterialien.tex b/content/Arbeitsmaterialien.tex index 92276c7..00a73db 100644 --- a/content/Arbeitsmaterialien.tex +++ b/content/Arbeitsmaterialien.tex @@ -14,4 +14,21 @@ Ein paar Anregungen für die Arbeitsmaterialien (kommt natürlich auf das Thema \end{itemize} -Bitte bedenken Sie, zumindest Lösungsskizzen zu den Aufgaben zu geben! Besser noch vollständige Musterlösungen. Als Lehrkraft ist es nicht immer einfach, die Lösungen schnell zu finden und das Dokument soll ja den Lehrkräften die Arbeit erleichtern. \ No newline at end of file +Bitte bedenken Sie, zumindest Lösungsskizzen zu den Aufgaben zu geben! Besser noch vollständige Musterlösungen. Als Lehrkraft ist es nicht immer einfach, die Lösungen schnell zu finden und das Dokument soll ja den Lehrkräften die Arbeit erleichtern. + +% 1. Einführung/ Motivation (lustig/ spielerisch) +% Spiel von Tobi (gibts nicht auch so nen Glücksspiel?) +% https://plinko-web-game.netlify.app/ : wieso macht man im Schnitt minus? +% Pachinko-Automat: Wo einwefen für max Gewinn + +% 2. Überleitung (interaktiv) +% Wabenaufgabe und Einführung über Pascallsches Dreieck, überführung Approx durch Normalverteilung: Tobi-AB + +% 3. Arbeitsmaterial ZSG (mathematisch Verständnisorientiert) +% wollen über eine Diskrete Verteilung eine stetige Kurve legen! +% Die Verteilung ist durch die Fakultät diskret, da diese nur für n\in N definiert ist +% 1 -> n! -> gamma Funktion -> Sterling Approx geben (Visuell!) +% 2 -> Vorfaktor "abschätzen" +% 3 -> potenzen Killen um E funktion zu gewinnen! + % ln und umformen (Klassenstufe! -> Anschätzung und Ergebnis vorgeben, nicht rechnen) + % Rücksubstitution machen \ No newline at end of file diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index d479566..310105b 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -168,7 +168,7 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung aufgrund der \item[(\ref{eq5:3})] Gleichheit für $k$ aus der Standardisierung einsetzen: $k = np + z\sqrt{npq}$. \item[(\ref{eq5:4})] Distributivgesetz für $-1$ anwenden und Brüche durch Aufteilen der Addition und Einfügen der multiplikativen Identität $\frac{\sqrt{np}}{\sqrt{np}}$ (selbes für $nq$) vereinfachen. \end{itemize} - Die \cref{def:taylorpolynom} beweist die Taylor-Approximation für den Logarithmus: $ln(1-\alpha) = -\alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ und $ln(1+\alpha) \sim \alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ für $\alpha \in (-\infty, 1)$. Hierbei steht $O(\alpha^3)$ für die Landau-Notation. + Die \cref{def:taylorpolynom} beweist die Taylor-Approximation für den Logarithmus: $ln(1-\alpha) = -\alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ und $ln(1+\alpha) = \alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ für $\alpha \in (-\infty, 1)$. Hierbei steht $O(\alpha^3)$ für die Landau-Notation. 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+\usetikzlibrary{circuits.logic.IEC} +\usepackage{tabularray} +\usepackage{tikz} +\usepackage{tkz-graph} +\usepackage{tabularx} +\usepackage{listings} +\usepackage{xcolor} +\usepackage{setspace} +\usepackage{letltxmacro} +\usepackage{wrapfig} + +\usetikzlibrary{calc} + +\setlist{itemsep=0.3em, topsep=0.5em, parsep=0pt} + +\newcommand{\bpf}[1]{% + \par\vspace{0.8\baselineskip}% Abstand vor der Überschrift + \noindent\textbf{#1}% Fettgedruckte Überschrift + \par\vspace{0.3\baselineskip}% Abstand nach der Überschrift +} + + + +\renewcommand{\contentsname}{Inhaltsverzeichnis} +\renewcommand{\figurename}{Grafik} +\renewcommand{\partname}{Teil} +\renewcommand{\epsilon}{\varepsilon} + +\definecolor{darkgrey}{HTML}{232327} + +% Zählerdefinition mit AUTOMATISCHEM SUBSECTION-RESET +\newcounter{commoncounter}[subsection] +\renewcommand{\thecommoncounter}{\thesection.\arabic{commoncounter}} + +\usepackage{xparse} + +\newcommand{\answer}[1]{% + \ifhmode\newline\newline\fi + \foreach \n in {1,...,#1}{\noindent\rule{\linewidth}{0.4pt}\par\vspace{1em}} +} + + +\newcommand{\currentsection}{} +\newcommand{\currentsubsection}{} + +\LetLtxMacro{\oldsection}{\section} +\LetLtxMacro{\oldsubsection}{\subsection} + +\RenewDocumentCommand{\section}{s o m}{% + \IfBooleanTF{#1}% + { \oldsection*{#3} }% Stern-Variante → unverändert + {% + \IfValueTF{#2}% + { \oldsection[#2]{#3} }% Optionales TOC-Argument + { \oldsection{#3} }% + \renewcommand{\currentsection}{#3}% + }% +} + +\RenewDocumentCommand{\subsection}{s o m}{% + \IfBooleanTF{#1}% + { \oldsubsection*{#3} }% + {% + \IfValueTF{#2}% + { \oldsubsection[#2]{#3} }% + { \oldsubsection{#3} }% + \renewcommand{\currentsubsection}{#3}% + }% +} + + + +% Zähler direkt bei Dokumentstart initialisieren +\AtBeginDocument{\setcounter{commoncounter}{0}} + +% BOX-DEFINITIONEN mit korrigierter Zählerlogik +\newtcolorbox{definitionbox}[1][]{ + before title={\refstepcounter{commoncounter}}, % KRITISCH: vor dem Titel! + title={Definition \thecommoncounter: #1}, + colback=white, + colframe=white!75!darkgrey, + fonttitle=\bfseries, + boxrule=0.6mm, + coltitle=black, + rounded corners, + before skip=10pt, + after skip=10pt +} + +\newtcolorbox{extrataskbox}[1][]{ + before title={\refstepcounter{commoncounter}}, % KRITISCH: vor dem Titel! + title={Zusatzaufgabe: #1}, + colback=white, + colframe=white!75!red, + fonttitle=\bfseries, + boxrule=0.6mm, + coltitle=black, + rounded corners, + before skip=10pt, + after skip=10pt +} + +\newtcolorbox{merkbox}[1][]{ + title={Merken: #1}, + colback=white, + colframe=white!75!red, + fonttitle=\bfseries, + boxrule=0.6mm, + coltitle=black, + rounded corners, + before skip=10pt, + after skip=10pt +} + +\definecolor{SatzColor}{RGB}{66,133,244} +\newtcolorbox{satzbox}[1][]{ + before title={\refstepcounter{commoncounter}}, % KRITISCH: vor dem Titel! + title={Satz \thecommoncounter: #1}, + colback=SatzColor!5, + colframe=white!75!SatzColor, + fonttitle=\bfseries, + boxrule=0.6mm, + coltitle=black, + rounded corners, + before skip=10pt, + after skip=10pt +} + +\newcommand{\schnipsel}[2][4cm]{% + \fbox{% + \parbox[c][3em][c]{#1}{\centering #2}% + }% +} + +\definecolor{codegreen}{rgb}{0,0.6,0} +\definecolor{codeblue}{rgb}{0,0,0.8} +\definecolor{codered}{rgb}{0.8,0,0} +\definecolor{lightgray}{rgb}{0.95,0.95,0.95} + +\lstdefinestyle{CStyle}{ + language=C, + basicstyle=\ttfamily\small, % Monospace-Schrift + keywordstyle=\color{blue}\bfseries, % Schlüsselwörter fett und blau + stringstyle=\color{red}, % Strings rot + commentstyle=\color{codegreen}, % Kommentare grün + backgroundcolor=\color{lightgray}, % Hintergrundfarbe + numbers=left, % Zeilennummern links + numbersep=10px, % Abstand zwischen Zeilennummern und Code + numberstyle=\color{gray}\texttt, + stepnumber=1, % Zeilennummerierung Schrittweite 1 + frame=single, % Rahmen um den Code + tabsize=4, % Tabulatorgröße + breaklines=true, % Zeilenumbruch aktivieren + captionpos=below, + showstringspaces=false, + xleftmargin=15pt, % Linker Rand für den Code (verschiebt alles nach rechts) +} + + +\lstdefinestyle{ShellStyle}{ + language=Bash, + basicstyle=\ttfamily\small, % Monospace-Schrift + keywordstyle=\color{black}, % Schlüsselwörter fett und blau + stringstyle=\color{black}, % Strings rot + commentstyle=\color{codegreen}, % Kommentare grün + backgroundcolor=\color{lightgray}, % Hintergrundfarbe + numbers=left, % Zeilennummern links + numbersep=10px, % Abstand zwischen Zeilennummern und Code + numberstyle=\color{gray}\texttt, + stepnumber=1, % Zeilennummerierung Schrittweite 1 + frame=single, % Rahmen um den Code + tabsize=4, % Tabulatorgröße + breaklines=true, % Zeilenumbruch aktivieren + captionpos=below, + showstringspaces=false, + xleftmargin=15pt, % Linker Rand für den Code (verschiebt alles nach rechts) +} + + +% Umgebung für Listings mit Titel und Zähler +\newenvironment{codeexample}[1][]{ + \refstepcounter{commoncounter} % Zähler erhöhen + \lstset{ + style=CSharpStyle, + caption={Listing \thecommoncounter: #1}, % Titel mit Zähler + label={listing:\thecommoncounter} + } +}{} + +\makeatletter +% Internen Zähler von subsubsection auf commoncounter umbiegen +\let\c@subsubsection\c@commoncounter + +% Anzeige der Nummer angleichen +\renewcommand{\thesubsubsection}{\thecommoncounter} +\makeatother + +\setlength{\parindent}{0pt} + +\sethlcolor{black!10} % Farbe des "Textmarkers" + +\newcommand{\curdate}{09.07.2026} +\newcommand{\curtitle}{ZSG Movire Laplace} + +\title{\curtitle} +\author{Fierke, Janik, Weidlich} +\pagestyle{scrheadings} +\date{\curdate} +\begin{document} + \ihead{\normalfont\normalsize Mathematik} + \chead{\normalfont\normalsize \curtitle} + \ohead{\normalfont\normalsize \curdate} + \ofoot{\normalfont\normalsize Seite {\pagemark} von \pageref{LastPage}} + \ifoot{\normalfont\normalsize Fierke, Janik, Weidlich} + + \section{\curtitle} + Dieses Dokument liefert die Textbausteine für Aufgabe 3 von dem Arbeitsblatt: ``ZSG Movire-Laplace'' + \vspace{3em} \\ + \schnipsel[6cm]{$\frac{n!}{k!(n-k)!} p^k q^{n-k}$} \vspace{1em} \\ + \schnipsel[6cm]{$\frac{1}{\sqrt{2\pi npq}}e^{\left(-\frac{(k-np)^2}{2npq}\right)}$} \vspace{1em} \\ + \schnipsel[6cm]{$\frac{k-np}{\sqrt{npq}}$} \vspace{1em} \\ + \schnipsel[6cm]{$np + z\sqrt{npq}$} \vspace{1em} \\ + \schnipsel[6cm]{$nq - z\sqrt{npq}$} \vspace{1em} \\ + \schnipsel[8cm]{$\frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{n^2pq}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k}$} \vspace{1em} \\ + \schnipsel[8cm]{$\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{npq}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k}$} \vspace{1em} \\ + \schnipsel[8cm]{$\frac{1}{\sqrt{2\pi npq}}e^{-\frac{\left(\frac{k-np}{\sqrt{npq}}\right)^2}{2}}$} \vspace{1em} \\ + \schnipsel[8cm]{$\frac{1}{\sqrt{2\pi npq}}e^{\left(-\frac{(k-np)^2}{2npq}\right)}$} + \label{LastPage} +\end{document} \ No newline at end of file