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\(\in \mathbb{R}\) \} \end{itemize} +\textbf{Für Intervalle}: +Runde Klammer schließt die Grenzen aus, Eckige Klammern ein. + +\subsection{Binomialkoeffizienten} +\begin{itemize} + \item ${n \choose j} = \frac{n!}{j!\cdot (n-j)!} \textbf{ ,für }j\leq n$ + \item ${n \choose j} = {n \choose n-j}$ + \item $(a+b)^n = \sum\limits_{k=0}^n {n\choose k} a^{n-k} b^k $ +\end{itemize} + + + \subsection{Binomische Formeln}\label{binomische-formeln} \begin{tablebox}{ll} @@ -110,6 +122,18 @@ Grundlage ist ein Polynom: $a{x}^{2} + bx + c = 0$ \end{align*} \end{minipage} +\subsection{Bruchrechnung}\label{bruchrechnung} +\begin{tablebox}{lll} +Division & $\frac{a}{b} : \frac{c}{d} = \frac{ad}{bc}$ & Multiplizieren mit dem Kehrwert \\ +Multiplikation & $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$ \\ +Kürzen & $\frac{2}{2 \cdot 3} = \frac{2}{2} \cdot \frac{1}{3} = \frac{1}{3}$ & Nur Faktoren, keine Summanden!! \\ +\end{tablebox} +\textbf{Trick 17:} $\frac{x-1}{x+4} = \frac{x+4-5}{x+4} = \frac{x+4}{x+4} - \frac{5}{x+4} = 1 - \frac{5}{x+4}$ + +\end{sectionbox} +% Manueller Spaltenumbruch +\begin{sectionbox} + \subsection{Wurzelrechnung} \begin{align*} @@ -119,18 +143,6 @@ Grundlage ist ein Polynom: $a{x}^{2} + bx + c = 0$ \frac{ \sqrt[n]{a} }{ \sqrt[n]{b} } &= \frac{ {a}^{ \frac{1}{n} } }{ {b}^{ \frac{1}{n} } } = { \left( \frac{a}{b} \right) }^{\frac{1}{n}} = \sqrt[n]{ \frac{a}{b} } \text{ wenn } b \neq 0 \end{align*} -\end{sectionbox} -% Manueller Spaltenumbruch -\begin{sectionbox} - -\subsection{Bruchrechnung}\label{bruchrechnung} -\begin{tablebox}{lll} -Division & $\frac{a}{b} : \frac{c}{d} = \frac{ad}{bc}$ & Multiplizieren mit dem Kehrwert \\ -Multiplikation & $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$ \\ -Kürzen & $\frac{2}{2 \cdot 3} = \frac{2}{2} \cdot \frac{1}{3} = \frac{1}{3}$ & Nur Faktoren, keine Summanden!! \\ -\end{tablebox} -\textbf{Trick 17:} $\frac{x-1}{x+4} = \frac{x+4-5}{x+4} = \frac{x+4}{x+4} - \frac{5}{x+4} = 1 - \frac{5}{x+4}$ - \subsection{Sinus \& Cosinus} \begin{tablebox}{c|c|c|c|c} Bogenmaß & Grad & $\sin{x}$ & $\cos{x}$ & $\tan{x}$ \\ \hline @@ -157,6 +169,9 @@ $\cfrac{1}{2}\sqrt{2} \cong 0.70710678$ und $\cfrac{1}{2}\sqrt{3} \cong 0.866025 \end{sectionbox} +% Manueller Spaltenumbruch +\columnbreak + % Mengenlehre % ---------------------------------------------------------------------- \section{Mengenlehre} @@ -167,7 +182,7 @@ $\cfrac{1}{2}\sqrt{2} \cong 0.70710678$ und $\cfrac{1}{2}\sqrt{3} \cong 0.866025 Ist $E$ eine Eigenschaft, die ein Element haben kann oder auch nicht, so beschreibt man die Menge der $E$ erfüllenden Elemente durch: A = $\lbrace x \vert x $ hat Eigenschaft $ E \rbrace$ - + \subsection{Teilmengen} Sind A und B Mengen, so heißt A Teilmenge oder auch Untermenge von B, wenn jedes Element von A auch Element von B ist. \begin{cookbox}{Merke zu Teilmengen} @@ -176,11 +191,7 @@ $\cfrac{1}{2}\sqrt{2} \cong 0.70710678$ und $\cfrac{1}{2}\sqrt{3} \cong 0.866025 \item Ist $A \subseteq B$ und $B \subseteq C$, so folgt $A \subseteq C$ \item Aus $A \subseteq B$ und $B \subseteq A$ folgt $A = B$ \end{cookbox} - -\end{sectionbox} -% Manueller Spaltenumbruch -\begin{sectionbox} \subsection{Operationen} \begin{tablebox}{lll} @@ -235,6 +246,11 @@ $\cfrac{1}{2}\sqrt{2} \cong 0.70710678$ und $\cfrac{1}{2}\sqrt{3} \cong 0.866025 $A \cup \emptyset = A $ & $\overline{A \cap B} = \overline{A} \cup \overline{B}$ & $ $ \\ \end{tablebox} + +\end{sectionbox} +% Manueller Spaltenumbruch +\begin{sectionbox} + \subsection{Regeln} \begin{tablebox}{ll} Kommutativ & $A \cup B = B \cup A$\\ @@ -255,10 +271,6 @@ $\cfrac{1}{2}\sqrt{2} \cong 0.70710678$ und $\cfrac{1}{2}\sqrt{3} \cong 0.866025 de Morganschen Gesetz & $A \setminus B = A \cap \overline{B}$ \\ \end{tablebox} -\end{sectionbox} -% Manueller Spaltenumbruch -\begin{sectionbox} - \subsection{Kartesisches Produkt} Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten Paare $\left( a , b \right)$ mit $a \in A$ und $b \in B$ @@ -314,6 +326,8 @@ Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten P \end{sectionbox} +\newpage % Hilfestellung um die Aussagenlogik auf einer Seite zu haben. + % Aussagenlogik % ---------------------------------------------------------------------- \section{Aussagenlogik} @@ -327,9 +341,6 @@ Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten P $A \leftrightarrow B$ & A genau dann, wenn B & Äquivalenz oder Bijunktion \\ $A \rightarrow B$ & wenn A dann B & Implikation oder Subjunktion \\ \end{tablebox} - -\end{sectionbox} -\begin{sectionbox} \subsection{Regeln} \begin{tablebox}{ll} @@ -358,12 +369,6 @@ Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten P de Morganschen Regeln & $\neg \left( A \vee B \right) = \neg A \wedge \neg B$ \\ & $\neg \left( A \wedge B \right) = \neg A \vee \neg B$ \\ - \end{tablebox} -\end{sectionbox} -% Manueller Spaltenumbruch -\begin{sectionbox} - \begin{tablebox}{ll} - Umwandeln & $A \wedge B = \neg \left( A \rightarrow \neg B \right)$ \\ & $A \vee B = \neg A \rightarrow B $ \\ & $A \rightarrow B = \neg A \vee B$ \\ @@ -387,11 +392,6 @@ Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten P $\neg \left( P \vee \neg \left( P \vee G \right)\right)$ \\ \end{quote} - -\end{sectionbox} - -\begin{sectionbox} - \subsection{Wahrheitstafeln} \begin{minipage}{0.49\textwidth} \textbf{Konjunkiton} (UND) @@ -415,6 +415,12 @@ Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten P $1$ & $1$ & $1$ \\ \hline \end{tablebox} \end{minipage} + +\end{sectionbox} +% Manueller Spaltenumbruch +\begin{sectionbox} + + \begin{minipage}{0.49\textwidth} \textbf{Bijunktion} (ist richtig wenn beide gleich sind) \begin{tablebox}{|l|l|l|} @@ -453,8 +459,6 @@ Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten P \end{sectionbox} -% Manueller Spaltenumbruch -\columnbreak % Komplexe Zahlen % ---------------------------------------------------------------------- @@ -503,6 +507,10 @@ $a = 0, b = 0 $ & $\varphi = 0^\circ $ & $\varphi = 0 $ \\ ${ i }^{ 3 } = -i$ & ${ i }^{ 6 } = -1 $... \\ \end{tablebox} +\end{sectionbox} +% Manueller Spaltenumbruch +\begin{sectionbox} + \subsection{Rechenoperationen} \begin{minipage}{0.49\textwidth} \textbf{Addition} @@ -537,10 +545,6 @@ $a = 0, b = 0 $ & $\varphi = 0^\circ $ & $\varphi = 0 $ \\ &=\frac { ac+bd }{ { c }^{ 2 }+{ d }^{ 2 } } +\frac { \left( bc-ad \right) }{ { c }^{ 2 }+{ d }^{ 2 } } i \end{align*} -\end{sectionbox} -% Manueller Spaltenumbruch -\begin{sectionbox} - \textbf{Potenzierung} \begin{align*} { z }^{ n } &={ \left( a+bi \right) }^{ n } \\ @@ -581,6 +585,10 @@ c \cdot \left(A + B \right) &= cA + cB \item Nicht Kommutativ!! $A \cdot B \neq B \cdot A$ \end{itemize} +\end{sectionbox} +% Manueller Spaltenumbruch +\begin{sectionbox} + \subsubsection{Determinante} \begin{itemize} \item Ist $\det A \neq 0$ dann ist die Matrix invertierbar @@ -649,6 +657,9 @@ Ist das Skalarprodukt = 0 dann sind die Vektoren orthogonal (senkrecht) zueinand \end{sectionbox} + +\columnbreak % Manueller Zeilenumbruch + % Geraden und Ebenen % ---------------------------------------------------------------------- \section{Geraden und Ebenen} @@ -707,8 +718,6 @@ $\overrightarrow{p}$ = Stützvektor und $\overrightarrow{u}$,$\overrightarrow{v} \end{sectionbox} -% Manueller Spaltenumbruch -\columnbreak % Grenzwerte % ---------------------------------------------------------------------- @@ -720,7 +729,7 @@ Der Grenzwert oder Limes einer Folge ist eine Zahl, der die Folge beliebig nah k \subsection{Berechnung} Bei $n \rightarrow \infty$ teilt man durch die variable mit der höchsten Potenz, das Ergebnis ist dann der Grenzwert. \begin{align*} -&\lim\limits_{n \rightarrow \infty}{\frac{2{n}^{2} -1}{{n}^{2} + 1}} = \lim\limits_{n \rightarrow \infty}{ \frac{ 2 - \frac{1}{ {n}^{2} } }{1 + \frac{ 1 }{ {n}^{2} }} } =\frac{ \lim\limits_{n \rightarrow \infty}{ 2{n}^{2} -1 } }{ \lim\limits_{n \rightarrow \infty}{ {n}^{2} + 1} } = \frac{2-0}{1+0} = 2 +&\lim\limits_{n \rightarrow \infty}{\frac{2{n}^{2} -1}{{n}^{2} + 1}} = \lim\limits_{n \rightarrow \infty}{ \frac{ 2 - \frac{1}{ {n}^{2} } }{1 + \frac{ 1 }{ {n}^{2} }} } =\frac{ \lim\limits_{n \rightarrow \infty}{ 2{n}^{2} -1 } }{ \lim\limits_{n \rightarrow \infty}{ {n}^{2} + 1} } = \frac{2}{1} = 2 \end{align*} \textbf{Ergebnisse} @@ -733,6 +742,7 @@ $\frac{1}{1} = 1 $ & $\frac{1}{0} = \infty $ & $\frac{0}{1} = 0 $ & $\frac{1}{17 &\lim\limits_{x \rightarrow 1}{ \frac{ {x}^{3} - 6{x}^{2} + 5x }{ 2{x}^{2} + 32x - 34 } } = \lim\limits_{x \rightarrow 1}{ \frac{ {x} \left( x - 1 \right) \left( x - 5 \right) }{ 2 \left( x -1 \right) \left( x + 17 \right) } } \\ = &\lim\limits_{x \rightarrow 1}{ \frac{ x \left(x-5 \right) }{ 2 \left(x+17 \right) } } = \frac{-4}{36} = -\frac{1}{9} \end{align*} + \begin{cookbox}{Ablauf bei $\lim\limits_{n \rightarrow a}$} \item Schauen ob man etwas ausklammern kann oder muss \item Anwendung der p-q Formel um die Nullstellen zu berechnen @@ -743,117 +753,551 @@ $\frac{1}{1} = 1 $ & $\frac{1}{0} = \infty $ & $\frac{0}{1} = 0 $ & $\frac{1}{17 \end{cookbox} +\end{sectionbox} +% Manueller Spaltenumbruch +\begin{sectionbox} + + +\textbf{Der Satz von l'Hospital} Ist Anwendbar wenn im Zähler und Nenner 0 oder beide $\infty$ sind. \\ +Hierbei wird der Zähler und Nenner separat abgeleitet und der Limes vom somit entstandenen Bruch berechnet. Hierbei gilt wieder die Ergebnistabelle oben. + + +\begin{align*} + & \lim_{x \rightarrow 0}{\cfrac{x \cdot \sin{2x}}{e^x - e^(-x)}} &= \cfrac{0}{0} & \\ + & \lim_{x \rightarrow 0}{\cfrac{\sin{2x} + 2x \cdot \cos{2x}}{e^x + e^(-x)}} &= \cfrac{0}{1} &= 0 \\ +\end{align*} + + + \end{sectionbox} % ====================================================================== % Ableitungen % ====================================================================== -\section{Ableitungshilfe} +\section{Konvergenz von Reihen} \begin{sectionbox} + +Es sei eine Reihe $\sum_{i=0}^{\infty} a_i$ gegeben. So ist diese (absolut) Konvergent nach folgenden Kriterien (siehe auch Formelsammlung S. 76): + +\begin{tablebox}{ll} + Quotientenkriterium & $q = \lim{i \rightarrow \infty}{\left| \cfrac{a_{i+1}}{a_i} \right|}$ \\ + Wurzelkriterium & $q = \lim{i \rightarrow \infty}{\sqrt[i]{|a_i|}}$ \\ + Leibnitzkriterium & siehe \ref{leibnitzkriterium} \\ + Majorantenkriterium & siehe \ref{minorantenkriterium} \\ + Minorantenkriterium & siehe \ref{majorantenkriterium} \\ +\end{tablebox} +\begin{tablebox}{lll} + $q < 1 \Rightarrow$ Reihe konvergiert & $q > 1 \Rightarrow$ Reihe divergiert & $q = 1 $ keine Aussage möglich! +\end{tablebox} +\end{sectionbox} + + +\begin{sectionbox} +Gegeben sind die Reihen: + \begin{align*} + \sum_{i=0}^{\infty} a_i &; \sum_{i=0}^{\infty} b_i + \end{align*} + \subsection{Majorantenkriterium}\label{majorantenkriterium} + ist $b_i \geq 0$ konvergent für alle $i$. Gilt dann $|a_i| \leq b_i$ für alle $i$, so konvergiert die Reihe mit $a_i$ absolut. - \subsection{Potenzfunktionen} - \begin{minipage}{0.49\textwidth} - - \begin{tablebox}{|l|l|} - \hline - Funktion & Ableitung \\ \hline - $ {x}^{n} $ & $ n \cdot {x}^{n-1} $ \\ \hline - $ 1 $ & $ 0 $ \\ \hline - $ x $ & $ 1 $ \\ \hline - $ x^2 $ & $ 2x $ \\ \hline - $ x^3 $ & $ 3x^2 $ \\ \hline - $ x^4 $ & $ 4x^3 $ \\ \hline - $ \frac{1}{x} $ & $ -\frac{1}{x^2} $ \\ \hline - $ \frac{1}{x^2} $ & $ -\frac{ 2 }{ x^3 }$ \\ \hline - $ \frac{1}{x^3} $ & $ - \frac{3}{x^4} $ \\ \hline - \end{tablebox} - \end{minipage} - \begin{minipage}{0.49\textwidth} - - \begin{tablebox}{|l|l|} - \hline - Funktion & Ableitung \\ \hline - $ {x}^{\frac{1}{2}} $ & $ \frac{ 1 }{ 2 } x^{-\frac{1}{2}}$ \\ \hline - $ \sqrt{x} $ & $ \frac{ 1 }{2 \sqrt{x} }$ \\ \hline - $ \frac{1}{ \sqrt{x} } $ & $ -\frac{1}{2x\sqrt{x}} $ \\ \hline - $ x^{\frac{3}{2}} $ & $ -\frac{ 3 }{ 2 } x^{ \frac{1}{2} }$ \\ \hline - $ \frac{1}{x^3} $ & $ - \frac{3}{x^4} $ \\ \hline - $ x \sqrt{x} $ & $ \frac{3}{2} \sqrt{x} $ \\ \hline - %$ \arccot{x} $ & $ \frac{ 1 }{ 1+x^2 }$ & & \\ \hline - $ x^{-\frac{3}{2}} $ & $ - \frac{3}{2} x^{ -\frac{5}{2} } $ \\ \hline - \end{tablebox} - \end{minipage} - - \subsection{Exponention und Logarithmus} - \begin{minipage}{0.49\textwidth} - \begin{tablebox}{|l|l|} - \hline - Funktion & Ableitung \\ \hline - $ e^x $ & $ e^x $ \\ \hline - $ e^{kx} $ & $ k e^{kx} $ \\ \hline - $ a^x $ & $ \ln{a} a^x $ \\ \hline - \end{tablebox} - \end{minipage} - \begin{minipage}{0.49\textwidth} - - \begin{tablebox}{|l|l|} - \hline - Funktion & Ableitung \\ \hline - $ \ln{x} $ & $ \frac{1}{x} $ \\ \hline - $ a_{\log{x}} $ & $ \frac{ 1 }{ x \ln{a} }$ \\ \hline - \end{tablebox} - \end{minipage} + + +\subsection{Minorantenkriterium}\label{minorantenkriterium} + ist $b_i$ eine divergente Reihe mit $b_i \geq 0$ für alle $i$. Gilt dann $|a_i| \geq b_i$ für alle $i$, so konvergiert die Reihe mit $a_i$ nicht. + +\subsection{Leibnitzkriterium}\label{leibnitzkriterium} + +\begin{cookbox}{Bedingungen} + \item Eine Alternierende Reihe + \item $\lim\limits_{i \rightarrow \infty}{u_i} = 0$ + \item strenge Monotonie $\forall i \Rightarrow u_i > u_{i+1}$ wenn $u_i$ positiv oder $u_i < u_{i+1}$ falls sie negativ sind. +\end{cookbox} +Es sei $u_i$ eine Folge von Zahlen, die entweder alle positiv oder alle negativ sind. Somit entspricht die folgende Reihe einer \textbf{alternierenden} Reihe: +\begin{align*} + \sum_{i = 1}^{\infty} (-1)^i u_i +\end{align*} + +\begin{cookbox}{Vorsicht!} + Dies bedeutet nicht \dots + Ausführliche Konvergenzbedingngen Seite 73 / 74 in Formelsammlung +\end{cookbox} + \end{sectionbox} -\begin{sectionbox} - - \subsection{Winkelfunktionen} - \begin{minipage}{0.48\textwidth} - \begin{tablebox}{|l|l|} - \hline - Funktion & Ableitung \\ \hline - $ \sin{x} $ & $ \cos{x} $ \\ \hline - $ \cos{x} $ & $ - \sin{x} $ \\ \hline - $ \tan{x} $ & $ \frac{ 1 }{ \cos^{2}{x} }$ \\ \hline - $ \cot{x} $ & $ - \frac{ 1 }{ \sin^{2}{x} }$ \\ \hline - \end{tablebox} - \end{minipage} - \begin{minipage}{0.49\textwidth} - \begin{tablebox}{|l|l|} - \hline - Funktion & Ableitung \\ \hline - $ \arcsin{x} $ & $ \frac{ 1 }{ \sqrt{ 1-x^2 } }$ \\ \hline - $ \arccos{x} $ & $ \frac{ 1 }{ \sqrt{ 1 - x^2 } }$ \\ \hline - $ \arctan{x} $ & $ \frac{ 1 }{ 1 + x^2 }$ \\ \hline - $ \arccot{x} $ & $ - \frac{ 1 } { 1 + x^2 } $ \\ \hline - \end{tablebox} - \end{minipage} - - \begin{minipage}{0.48\textwidth} - \begin{tablebox}{|l|l|} - \hline - Funktion & Ableitung \\ \hline - $ \sinh{x} $ & $ \cosh{x} $ \\ \hline - $ \cosh{x} $ & $ \sinh{x} $ \\ \hline - $ \tanh{x} $ & $ \frac{ 1 }{ \cosh^{2}{x} }$ \\ \hline - $ \coth{x} $ & $ - \frac{ 1 }{ \sinh^{2}{x} }$ \\ \hline - \end{tablebox} - \end{minipage} - \begin{minipage}{0.49\textwidth} - \begin{tablebox}{|l|l|} - \hline - Funktion & Ableitung \\ \hline - $ \arcsinh{x} $ & $ \frac{ 1 }{ \sqrt{ 1 + x^2 } }$ \\ \hline - $ \arccosh{x} $ & $ \frac{ 1 }{ \sqrt{ x^2 - 1 } }$ \\ \hline - $ \arctanh{x} $ & $ \frac{ 1 }{ 1 - x^2 }$ \\ \hline - $ \arccoth{x} $ & $ - \frac{ 1 } { 1 - x^2 } $ \\ \hline - \end{tablebox} - \end{minipage} +\section{Potenzreihen} +\begin{sectionbox} + +Es sei: + \begin{align*} + P(x) = \sum_{i=0}^{\infty} c_i \left( x-x_0\right)^i + \end{align*} + + \subsection{Konvergenzradius} + Der Konvergenzradius $r$ beschreibt den maximalen Abstand von $x_0$ zu einem Konvergenzpunkt. Konvergent sind alle $r$ für die gilt: + + \begin{align*} + + \end{align*} + + Das verhalten im Rendpunkten muss seperat bestimmt werden. Die spitzen Klammern sund dann gegen $]$ (wenn die Grenze enthalten ist) oder $)$ (wenn die Grenze nicht enthalten ist) auszutauschen. + + Zur Berechnung gibt es die folgenden beiden Methoden: + \begin{align*} + r = \lim\limits_{i \rightarrow \infty} \left|\cfrac{c_i}{c_{i + 1}}\right| + \end{align*} + \begin{align*} + r = \cfrac{1}{\lim\limits_{i \rightarrow \infty} \sqrt[i]{|c_i|} } + \end{align*} + + \subsection{Operationen} + \begin{align*} + P_1(x) = \sum_{i=0}^{\infty} c_i \left( x-x_0\right)^i + \end{align*} + \begin{align*} + P_2(x) = \sum_{i=0}^{\infty} b_i \left( x-x_0\right)^i + \end{align*} + + + %Zusatz von Juliane + + \subsection{Taylorreihen} + Die Taylorreihe von f ist definiert durch: + $$ T_f(x) = \sum_{i = 0}^{\infty} \frac{f^{(i)}(x_0) }{i!} \cdot (x-x_0)^i$$ + Voraussetzung ist das f unendlich eof differenzierbar ist. + Bricht man die Summation nach n Summanden ab erhält man das Taylorpolynom vom Grad n. + \textbf{Restglieddarstellungen} + $$ R_n(x) = \frac{1}{n!} \int_{x_0}^{x}(x-t)^n f^{n+1} (t) dt $$ + und + $$ R_n(x) = \frac{f^{n+1}(\xi)}{(n+1)!}\cdot (x - x_0)^{n+1}$$ + + % Ende Zusatz + +\end{sectionbox} + + + +% ==================================================================== +% Kurvendiskussion +% ==================================================================== + \section{Funktionen} +\begin{sectionbox} + +\subsection{Grundlagen} +\begin{tablebox}{ll} + Tangente & $y(x) = mx + b$ \\ + Tangente & $y(x) = f'(x_0) \cdot x + f(x_0) - f'(x_0) - x_0 $ \\ + Lineare Funktion & $y(x) = ax^2 + 2bx + c$ \\ +\end{tablebox} + +\end{sectionbox} +% Manueller Spaltenumbruch +\begin{sectionbox} + + + \subsection{Polynomdivision} + \begin{align*} + \cfrac{x^2+3x+16}{x-2} + \end{align*} + \begin{cookbox}{Ablauf Polynomdivision} + \item Größte Exponent aus beiden Polynomen ermitteln + \item Dividieren und zurückmultiplizieren + \item Substrahieren und von vorne beginnen + \item Rest aufschreiben + \end{cookbox} + + \subsection{Partialbruchzerlegung} + \begin{align*} + \cfrac{x^2+16}{x\cdot (x-2)} + \end{align*} + \begin{cookbox}{Partialbruchzerlegung bei reelen Nullstellen} + \item Nennerpolynom in Linearfaktoren aufteilen + \item Aufteilen der Linearfaktoren auf die Partialbrüche + \item Im Zähler steht jeweils eine Konstante + \item Nun mit dem Nennerpolynom multiplizieren + \item Die Nenner aus den Brüchen kürzen + \item Die Nullstellen der Linearfaktoren einsetzen + \item Die Variablen ausklammern und nach Exponent sortieren. + \item Restliche Konstanten mit einem LGS bestimmen + \end{cookbox} + + \subsection{Flächenberechnung} + Die Fläche unter einer Kurve entspricht ihrem Integral. + \begin{cookbox}{zwichen zwei Funktionen} + \item die Funktionen gleich setzen: $f(x) - g(x) = h(x)$ + \item Schnittpunkte ermitteln $h(x) = 0$ + \item $h(x)$ Integrieren und die Nullstellen als Grenzen einsetzen. + \item Mehrere Schnittpunkte müssen einzeln berechnet werden. + \item Ergebnisse zusammenrechnen. + \end{cookbox} + + +\subsection{Kurvendiskussion} + +\begin{tablebox}{ll} + In $x_0$ gilt: & \\ + lokales Max-/Minimum in & $f'(x_0)=0$\\ + lokales Minimum in & $f'(x_0)=0 \Rightarrow f''(x_0) > 0$ \\ + lokales Maximum in & $f'(x_0)=0 \Rightarrow f''(x_0) < 0$ \\ + lokales Minimum in & n ist gerade $\Rightarrow f^n(x_0) > 0$ \\ %wenn alle vorherigen Ableitungen gleich null sind! + locales Maximum in & n ist gerade $\Rightarrow f^n(x_0) < 0$ \\ + Wendepunkt & Bedingung: $f''(x_0) = 0$ \\ + & Vorzeichenwechsel: $f' \rightarrow f'''(0) \neq 0$ \\ + Sattelpunkt & zusätzlich zu Wendepunkt-bed.: $f'(x) = 0$ \\ + Pol & $\lim\limits{x \rightarrow x_0}{f(x)} = \pm \infty$ \\ + & \\ + Auf dem Intervall I gilt: & \\ + konvex & $ f''(x) \geq 0 \forall x \in I $\\ + konkav & $ f''(x) \leq 0 \forall x \in I $\\ + monoton steigend & $ f'(x) \geq 0 \forall x \in I $ \\ + monoton fallend & $ f'(x) \leq 0 \forall x \in I$ \\ + streng monoton steigend & $ f'(x) > 0 \forall x \in I $ \\ + streng monoton fallend & $ f'(x) < 0 \forall x \in I$ \\ +\end{tablebox} + +\textcolor{red}{Wendepunkt} bedeutet der Drehsinn der Kurventangente wird geändert. + \end{sectionbox} +\section{Integrale} +\begin{sectionbox} + +Lineare Gleichung $\rightarrow $ einfache Ableitung + +\subsection{Analyse des Integrals} +\begin{tablebox}{ll} + Produkt & Partielle Integration\\ + & Substitution\\ + Brüche & Umformen zu einem \textbf{Produkt}\\ + & Partialbruchzerlegung \\ + & Polynomdifision \\ + +\end{tablebox} + + +\subsection{Substitutionsregel} + +\textbf{Voraussetzung} ist das es eine innere Funktion gibt! + +Gegeben sind zwei Funktionien bei der die eine aus einer inneren und äußeren Funktion besteht und die andere als Ableitung der inneren Funktion geschrieben werden kann. + +\begin{align*} + \int f(x) dx = \int f(g(t)) \cdot g'(t) dt +\end{align*} + +So lässt sich das Integral durch Substitution vereinfachen und das Ergebnis zurück substituieren. + +\textbf{Beispiel:} +\begin{align*} + \int \cfrac{x^2}{x^3-17} dx &= \int x^2 \cdot \cfrac{1}{x^3-17} \\ + g(x) = x^3 - 17 & \Rightarrow g'(x) = 3x^2 \\ + \cfrac{1}{3} \int 3x^2 \cdot \cfrac{1}{x^3-17} dx &\Rightarrow \cfrac{1}{3} \int g'(x) \cdot f(g(x)) dx \\ + g'(x) = \cfrac{dg}{dx} &\Rightarrow dg = g'(x)dx \\ + \Rightarrow \cfrac{1}{3} \int \cfrac{1}{g} dg &= \cfrac{1}{3} ln(g) + c \\ + &\Rightarrow \cfrac{1}{3} ln(x^3-17) + c +\end{align*} + + +\subsection{Partielle Integration} + + \begin{align*} + \int f(x) * g(x) dx + \end{align*} + +Gegeben sei ein Integral mit den Funktionen f(x) und g(x). Bei der Substitution setzt man nun eine der beiden Funktionen als Abgeleitet vorraus und Integriert diese. Im folgenden wird f(x) als abgeleitet gesetzt. Somit ist $h'(x) = f(x)$ und $h(x) = \int f(x)$. + + \begin{align*} + \int h'(x) * g(x) dx = h(x) \cdot g(x) - \int g'(x) \cdot h(x) dx + \end{align*} + +Nicht immer ist der erste Ansatz zielführend. + +%Zusatz von Juliane +\subsection{Anwendungen} +\begin{cookbox} {Uneigentliches Integral} +Ist eine der Grenzen des Integrals mit $\infty$ gegeben, so gilt für das Integral: \\ +$$\int_{-\infty}^{b} f(x)dx = \lim_{a\to-\infty}\int_{a}^{b}f(x)dx $$ +und analog auch im positiven Bereich + +\end{cookbox} + +\end{sectionbox} +% Manueller Spaltenumbruch +\begin{sectionbox} + +\begin{cookbox}{Volumen eines Rotationskörper} +$$V = \pi\int_{a}^{b}f^2(x)dx$$ +\end{cookbox} + +\begin{cookbox}{Bogenlänge} +$$L = \int_{a}^{b}\sqrt{1+(f'(x))^2 } dx$$ +\end{cookbox} + +\subsection{Numerische Integration} +Es gibt verschiedene Verfahren zur Berechnung von Näherungen von Integralen. Eines davon ist die \textbf{Trapezregel}: +$$T_n = \frac{h}{2}\cdot (y_0+2y_1+2y_2+ \dots + 2y_{n-1}+y_n)$$ +Die Schrittweite zwischen den betrachteten Punkten: $$ h = \frac{b-a}{n}$$ +Für den Abstand zwischen Näherung und Integralwert gilt: +$$ | T_n - \int_{a}^{b}f(x) dx | \leq h^2 \cdot \frac{b-a}{12} \cdot \max_{x \in [a,b]}| f''(x) | +$$ + +Eine genauere Methode bietet die \textbf{Simpsonregel}: +$$ h = \frac{b-a}{2n}$$ +$$S_n = \frac{h}{3}(\Sigma_1 + 4 \cdot\Sigma_2 + 2 \cdot\Sigma_3 )$$ +Wobei gilt: \\ +$\Sigma_1 = $ Die Summe von $f_0$ und $f_{2n}$\\ +$\Sigma_2 = $ Die Summe von $f_i$ mit ungeradem i\\ +$\Sigma_3 = $ Die Summe von $f_i$ mit geradem i\\ + +Für die Genauigkeit gilt hier: +$$ | S_n = \int_a^b f(x) dx | \leq h^4 \cdot \frac{b-a}{180} \cdot \max_{x \in [a,b] } | f^{(4)}(x)| $$ + +%Ende Zusatz + +\end{sectionbox} + + + +\section{Differenzialgleichung} +\begin{sectionbox} + +\subsection{Differenzierbarkeit einer Funktion} +Gegeben ist eine Funktion f(x) deren Differenzierbarkeit in einem Punkt $x_0 $ +bestimmt werden soll. Dann gilt, wenn der $\lim\limits_{x \rightarrow x_0} $ +existiert, ist die Funktion für in $x_0 $ Differenzierbar. +\begin{align*} + \lim\limits_{x \rightarrow x_0}{ \cfrac{f(x)-f(x_0)}{x-x_0}} +\end{align*} + +\subsection{Differenzialgleichung 1. Ordnung} + +\subsubsection{Trennung der Variablen}\label{trennung-variablen} +Gegeben ist eine Differenzialgleichung der Form: +\begin{align*} + y' = g(x) * h(y) +\end{align*} +dann schreibt man diese um zu: +\begin{align*} + \int \cfrac{dy}{h(y)} = \int g(x) dx +\end{align*} +und löst zu y auf. + + +\subsubsection{Variation der Konstanten} +mit reelen Funktionen $g_1(x)$ und $g_2(x)$ wird eine Differenzialgleichung erster Ordung folgender Form \textbf{variierbare} Differenzialgleichung genannt: +\begin{align}\label{variation-gleichung} + y'=y \cdot g_1(x) + g_2(x) +\end{align} + +\end{sectionbox} +% Manueller Spaltenumbruch +\begin{sectionbox} + +\begin{cookbox}{Lösungsweg mit Beispiel} + \item\label{anfangsdifferenzial} Untersuchen der Differenzialgleichung \\ + $y' = -2xy + x \cdot e^{-x^2}$ + \item Man erstellt die Lösung des homogenen Problems gemäß \ref{trennung-variablen} :\\ + $g_1(x) = -2x \Rightarrow y_h(x) = C \cdot e^{-x^2} $ + \item\label{yx} Ersetzen (variieren) der Konstante $C$ durch $C(x)$ \\ + $y(x) = C(x) * e^{-x^2}$ + \item\label{y1x} Bestimmung der Ableitung $y'(x)$ \\ + $y'(x) = C(x)(-2x)e^{-x^2} + C'(x) e^{-x^2}$ + \item Einsetzen der Gleichungen von Punkt \ref{yx} \ref{y1x} in die Differenzialgleichug \ref{anfangsdifferenzial} \\ + $C(x)(-2x)e^{-x^2} + C'(x)e^{-x^2} = (-2x)C(x)\cdot e^{-x^2} + x \cdot e^{-x^2}$\\ + $\Rightarrow C'(x)e^{-x^2} = x \cdot e^{-x^2}$ + \item Bestimmung von $C'(x)$ \\ $C'(x) = x$ + \item Integrieren von $C'(x)$ \\ + $C(x) = \cfrac{x^2}{2} + K$ + \item Einsetzen in die Gleichung von Punkt \ref{yx} \\ + $y(x) = \left( \cfrac{x^2}{2} + K \right) \cdot e^{-x^2}$ +\end{cookbox} + +\subsection{Lineare Differenzialgleichung - Homogener Fall} + +\begin{cookbox}{Lösungsweg} + \item Differenzialgleichung analysieren + \item Charakteristische Polynom aufschreiben + \item Die Nullstellen finden + \item Fundamentalmenge aufschreiben + \item Allgemeine Lösung aufschreiben / ableiten + \item Anfangswerte einsetzen + \item Lösung des Anfangswerteproblems aufschreiben +\end{cookbox} + + +Sei die Differenzialgleichung der Form: +\begin{align*} + y^{n} + a_{n-1} y^{n-1} + \dots + a_1 y' + a_0 y = 0 +\end{align*} + +Dann bestimmt man das \textbf{charakteristische Polynom} +\begin{align*} + p(\lambda) = \lambda^n + a_{n-1}\lambda^{n-1} + \dots + a_1 \lambda + a_0 +\end{align*} + +Das charakteristische Polynom n-ter Ordnung hat nun k verschiedene \textbf{reelle Nullstellen} $\lambda_1, \dots \lambda_k$ mit der jeweiligen Vielfachheit $\mu_1 \dots \mu_k$, wobei gilt $ \mu_1+ \dots \mu_k = n$ +\newline +Dann bildet die Funktionenmenge +\begin{align*} + e^{\lambda_1 x} , xe^{\lambda_1x}, ..., x^{\mu_1-1}e^{\lambda_1x}, \dots , e^{\lambda_k x},\dots x^{\mu_k-1}e^{\lambda_kx} +\end{align*} + +ein \textbf{Fundamentalsystem} dieser Differenzielgleichung. + + +\textbf{Achtung bei komplexen Nullstellen des charakteristischen Polynoms!!} +Für Gleichung 2. Ordnung sei die Form: +\begin{align*} + y'' + \textcolor{green}{a_1}y' + \textcolor{orange}{a_0} y = 0 +\end{align*} + + +Dann gilt: $D = \cfrac{\textcolor{green}{a_1}^2}{4} - \textcolor{orange}{a_0} < 0$ +(=komplexe Lösung) + +Dann lässt sich der Lösungsweg wie folgt abkürzen: + +$$ \textcolor{red}{\alpha} = - \cfrac{\textcolor{green}{a_1}}{2} \text{und} \textcolor{blue}{\beta} = \sqrt{-\left(\cfrac{\textcolor{green}{a_1}^2}{4} - \textcolor{orange}{a_0} \right) } $$ \newline +$$(x) = c_1 e^{\textcolor{red}{\alpha} x} \cos{\textcolor{blue}{\beta} x} + c_2 e^{\textcolor{red}{\alpha} x} \sin{\textcolor{blue}{\beta} x}$$ + +\end{sectionbox} + + +\begin{sectionbox} + + \subsection{Lineare Differenzialgleichung - Inhomogener Fall} +Allgemein hat die Differenzialgleichung dann die Form: +$$ y^{(n)} + a_{n-1}y^{(n-1)}+ \dots + a_1y'+a_0 y = b(x) $$ + +Allgemeine Lösung: +\begin{itemize} + \item Bestimmen der allgemeinen Lösung der zugehörigen homogenen Gleichung + \item addieren einer speziellen Lösung $y_p(x)$ (\textbf{partikuläre Lösung}) + Hierfür benötigt man meist eine \textbf{Ansatzfunktion}: +\end{itemize} + +\begin{enumerate} + \item $b(x)$ ist in der Form $f(x) \cdot e^{ax}$ + \begin{itemize} + \item Dabei ist f(x) ein Polynom m-ten Grades und a eine reelle Zahl + \item Dann gilt: Ist a eine k-fache Nullstelle des charakteristischen Polynoms so gibt es eine partikuläre Lösung der Form $$ y_p(x) = x^k \cdot q(x) \cdot e^{ax}$$ Mit einem Polynom q(x) vom Grad m. Ist a keine Nullstelle so ist k = 0 zu setzen. + \item Koeffizientenvergleich um q(x) zu bestimmen. + \item Die allgemeine Lösung setzt sich zusammen aus der Lösung der homogenen Gleichung $y_h(x)$ und der partikulären Lösung $y_p(x)$ + + \end{itemize} + \item Die Differenzialgleichung hat die Form $$ y' + ay = d_1sin(\omega x) + d_2 cos(\omega x)$$ + \begin{itemize} + \item Feste reelle Zahlen $a, d_1, d_2, \omega$ + \item Dann gibt es eine partikuläre Lösung der Form + $$ y_p(x)= b_1 sin(\omega x) + b_2 cos(\omega x) $$ mit demselben $\omega$ + \end{itemize} + + \item Die Differenzialgleichung hat die Form + $$ y'' + a_1 y' + a_0y = d_1 sin(\omega x) + d_2cos(\omega x) $$ + \begin{itemize} + \item mit festen reellen Zahlen $a_0, a_1, d_1, d_2, \omega$ + \item Dann gilt: + \begin{itemize} + \item Ist $i\omega$ keine Nullstelle des charakt. Polynoms dann gibt es eine Lösung der Form: + $$ y_p(x) = b_1 sin(\omega x) + b_2 cos(\omega x)$$ + \item sonst + $$ y_p(x) = x\cdot (b_1 sin(\omega x) + b_2 cos(\omega x))$$ + + \end{itemize} + \end{itemize} +\end{enumerate} + +\end{sectionbox} + +\columnbreak % Manueller Spaltenumbruch + +\section{Warscheinlichkeiten} +\begin{sectionbox} + +\subsection{Wahrscheinlichkeitsraum} +Sei F ein Ereignisfeld, p eine Wahrscheinlichkeit auf F und $\Omega$ ein Ergebnisraum von F, so nenn man das Tripel $(\Omega, F, p )$ einen Wahrscheinlichkeitsraum. $(F,p)$ ist ein Wahrscheinlichkeitsfeld. + +\subsection{Zufallsgröße} +Sei $(\Omega, F, p)$ ein \textbf{Wahrscheinlichkeitsraum}. Dann ist: \\ +Eine auf $\Omega $ definierte reelle Funktion X heißt \textbf{Zufallsgröße}, wenn für jede reele Zahl x gilt: +$\{\omega \in \Omega; X(\omega) \leq x\} \in F $ (Abkürzung: $(X\leq x) \in F)$ \\ + +Außerdem heißt für alle $x\in R$ durch +$F_{X}(x) = p(X \leq x) $ definierte Funktion \textbf{Verteilungsfunktion} (kurz: Verteilung) von X +\textbf{Eigenschaften einer Verteilungsfunktion:} +\begin{itemize} + \item $0 \leq F_{X}(x) \leq 1 $ für alle $x \in R$ + \item $ F_{X}(x)$ ist auf ganz R monoton steigend. + \item $$\lim_{x \to -\infty}F_{X}(x) = 0 $$ und $$\lim_{x \to +\infty}F_{X}(x) = 1 $$ + \item $F_X(x)$ ist auf ganz R rechtsseitig stetig + \item Für alle $a < b \in R$ ist: + $$ p(a