mirror of
https://github.com/janishutz/eth-summaries.git
synced 2026-06-12 11:31:20 +02:00
[PS] Add tables
This commit is contained in:
Binary file not shown.
|
After Width: | Height: | Size: 420 KiB |
@@ -80,4 +80,4 @@ Herleitung der Randdichte (``wegintegrieren''):
|
||||
\]
|
||||
Folglich sind die beiden Koordinaten unabhängig.
|
||||
|
||||
\bi{Einheitskreisscheibe} $f(x, y) \neq f_\cX(x) f_\cY(y)$, mit Dichten von oben. Also sind die beiden Koordinaten nicht unabhängig.
|
||||
\bi{Einheitskreisscheibe} $f(x, y) \neq f_\cX(x) f_\cY(y)$, mit Dichten von oben. Also sind die beiden Koordinaten nicht unabh.
|
||||
|
||||
@@ -0,0 +1,16 @@
|
||||
\subsection{Tips \& Tricks}
|
||||
Zur Bestimmung von W. wie (mit $\cX, \cY \sim \cU(0, 1)$)
|
||||
\[
|
||||
W(t) = \P[\cX + \cY \leq t] \ \forall t \in \R
|
||||
\]
|
||||
können wir dies via gemeinsamer Dichte ($f_{\cX, \cY}(x, y) = 1$ muss gegeben sein) und Menge
|
||||
\[
|
||||
A_t = \{ (x, y) \in [0, 1]^2 : x + y \leq t \}
|
||||
\]
|
||||
bestimmen. Dann ist für $t < 0$, $W(t) = 0$.
|
||||
|
||||
Für $0 \leq t \leq 1$ ist $A_t$ ein rechtwinkliges Dreieck mit Katheten der Länge $t$, also $W(t) = t^2 \div 2$.
|
||||
|
||||
Für $1 < t \leq 2$ ist $A_t$ Einheitsquadrat ohne rechtw. Dreieck mit Katheten der Länge $(2 - t)$, also $W(t) = 1 - \frac{(2 - t)^2}{2}$.
|
||||
|
||||
Für $t > 2$ ist $W(t) = 1$.
|
||||
@@ -0,0 +1,49 @@
|
||||
\rmvspace
|
||||
\subsection{Verschiedene Funktionen}
|
||||
\subsubsection{Logarithmen}
|
||||
\begin{itemize}
|
||||
\item \textit{(Basiswechsel)} $\log_a(x) = \frac{\ln(x)}{\ln(a)}$
|
||||
\item \textit{(Potenzen)} $\log_a(x^y) = y\log_a(x)$
|
||||
\item \textit{(Div, Mul)} $\log_a(x \cdot (\div) y) = \log_a(x) +(-) \log_a(y)$
|
||||
\item $\log_a(1) = 0 \smallhspace \forall a \in \N$
|
||||
\end{itemize}
|
||||
|
||||
|
||||
\subsubsection{Trigonometrie}
|
||||
$\cot(\xi) = \displaystyle\frac{\cos(\xi)}{\sin(\xi)}, \tan(\xi) = \frac{\sin(\xi)}{\cos(\xi)}$
|
||||
|
||||
$\sinh(x) := \frac{e^x - e^{-x}}{2} : \R \rightarrow \R$,
|
||||
$\cosh(x) := \frac{e^x + e^{-x}}{2} : \R \rightarrow [1, \infty]$,
|
||||
$\cosh(x) := \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} : \R \rightarrow [-1, 1]$
|
||||
|
||||
\begin{enumerate}
|
||||
\item $\cos(x) = \cos(-x)$ \trand $\sin(-x) = -\sin(x)$
|
||||
\item $\cos(\pi - x) = -\cos(x)$ \trand $\sin(\pi - x) \sin(x)$
|
||||
\item $\sin(x + w) = \sin(x) \cos(w) + \cos(x) \sin(w)$
|
||||
\item $\cos(x + w) = \cos(x) \cos(w) - \sin(x) \sin(w)$
|
||||
\item $\cos(x)^2 + \sin(x)^2 = 1$
|
||||
\item $\sin(2x) = 2 \sin(x) \cos(x)$
|
||||
\item $\cos(2x) = \cos(x)^2 - \sin(x)^2$
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
\shade{teal}{\tr{Values of trigonometric functions}{Werte der trigonometrischen Funktionen}}
|
||||
\begin{tables}{ccccc}{° & rad & $\sin(\xi)$ & $\cos(\xi)$ & $\tan(\xi)$}
|
||||
0° & $0$ & $0$ & $1$ & $1$ \\
|
||||
\hline
|
||||
30° & $\frac{\pi}{6}$ & $\frac{1}{2}$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{3}}{2}$ \\
|
||||
\hline
|
||||
45° & $\frac{\pi}{4}$ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{2}}{2}$ & $1$ \\
|
||||
\hline
|
||||
60° & $\frac{\pi}{3}$ & $\frac{\sqrt{3}}{3}$ & $\frac{1}{2}$ & $\sqrt{3}$ \\
|
||||
\hline
|
||||
90° & $\frac{\pi}{2}$ & $1$ & $0$ & $\varnothing$ \\
|
||||
\hline
|
||||
120° & $\frac{2\pi}{3}$ & $\frac{\sqrt{3}}{2}$ & $-\frac{1}{2}$ & $-\sqrt{3}$ \\
|
||||
\hline
|
||||
135° & $\frac{3\pi}{4}$ & $\frac{\sqrt{2}}{2}$ & $-\frac{\sqrt{2}}{2}$ & $-1$ \\
|
||||
\hline
|
||||
150° & $\frac{5\pi}{6}$ & $\frac{1}{2}$ & $-\frac{\sqrt{3}}{2}$ & $-\frac{\sqrt{3}}{2}$ \\
|
||||
\hline
|
||||
180° & $\pi$ & $0$ & $-1$ & $0$ \\
|
||||
\end{tables}
|
||||
@@ -0,0 +1,53 @@
|
||||
\subsection{Tabelle von Auf- und Ableitungen}
|
||||
\begin{scriptsize}
|
||||
\begin{tables}{lll}{\tr{Antiderivative}{Stammfunktion} & \tr{Function}{Funktion} & \tr{Derivative}{Ableitung}}
|
||||
$\displaystyle \frac{x^{n + 1}}{n + 1}$ & $x^n$ & $n \cdot x^{n - 1}$ \\
|
||||
$\ln|x|$ & $\displaystyle \frac{1}{x} = x^{-1}$ & $\displaystyle -x^{-2} = -\frac{1}{x^2}$ \\[0.2cm]
|
||||
$\frac{2}{3} x^{\frac{3}{2}}$ & $\displaystyle \sqrt{x} = x^{\frac{1}{2}}$ & $\displaystyle \frac{1}{2 \cdot \sqrt{x}}$ \\[0.3cm]
|
||||
$\frac{n}{n + 1} x^{\frac{1}{n} + 1}$ & $\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}$ & $\frac{1}{n} x^{\frac{1}{n} - 1}$ \\[0.3cm]
|
||||
\hline \\[-0.2cm]
|
||||
$e^x$ & $e^x$ & $e^x$ \\
|
||||
$\exp(x)$ & $\exp(x)$ & $\exp(x)$ \\
|
||||
$\frac{1}{a \cdot (n + 1)}(ax + b)^{n + 1}$ & $(ax + b)^n$ & $n\cdot (ax + b)^{n - 1} \cdot a$ \\
|
||||
$x \cdot (\ln|x| - 1)$ & $\ln(x)$ & $\frac{1}{x} = x^{-1}$ \\
|
||||
$\displaystyle \frac{1}{\ln(a)}\cdot a^x$ & $a^x$ & $a^x \cdot \ln(a)$ \\
|
||||
$\frac{x}{\ln(a)} \cdot (\ln|x| - 1)$ & $\log_a|x|$ & $\displaystyle \frac{1}{x \cdot \ln(a)}$ \\[0.3cm]
|
||||
\hline \\[-0.2cm]
|
||||
$-\cos(x)$ & $\sin(x)$ & $\cos(x)$ \\
|
||||
$\sin(x)$ & $\cos(x)$ & $-\sin(x)$ \\
|
||||
$-\ln|\cos(x)|$ & $\tan(x)$ & $\displaystyle \frac{1}{\cos^2(x)}$ \\[0.3cm]
|
||||
$x \cdot \arcsin(x) + \sqrt{1 - x^2}$ & $\arcsin(x)$ & $\displaystyle\frac{1}{\sqrt{1 - x^2}}$ \\
|
||||
$x \cdot \arccos(x) - \sqrt{1 - x^2}$ & $\arccos(x)$ & $\displaystyle -\frac{1}{\sqrt{1 - x^2}}$ \\
|
||||
$\displaystyle x \cdot \arctan(x) - \frac{\ln(x^2 + 1)}{2}$ & $\arctan(x)$ & $\displaystyle \frac{1}{x^2 + 1}$ \\[0.2cm]
|
||||
$\ln|\sin(x)|$ & $\cot(x)$ & $\displaystyle -\frac{1}{\sin^2(x)}$ \\
|
||||
$\cosh(x)$ & $\sinh(x)$ & $\cosh(x)$ \\
|
||||
$\sinh(x)$ & $\cosh(x)$ & $\sinh(x)$ \\
|
||||
$\ln|\cosh(x)|$ & $\tanh(x)$ & $\displaystyle \frac{1}{\cosh^2(x)}$ \\
|
||||
& $\arcsinh(x)$ & $\frac{1}{\sqrt{1 + x^2}}$ \\
|
||||
& $\arccosh(x)$ & $\frac{1}{\sqrt{x^2 - 1}}$ \\
|
||||
& $\arctanh(x)$ & $\frac{1}{1 - x^2}$ \\
|
||||
\end{tables}
|
||||
\end{scriptsize}
|
||||
|
||||
\shade{teal}{Weitere Ableitungen}
|
||||
\begin{tables}{cc}{$F(x)$ & $f(x)$}
|
||||
$\frac{1}{a} \ln|ax + b|$ & $\frac{1}{ax + b}$ \\
|
||||
$\frac{ax}{c} - \frac{ad - bc}{c^2} \ln|cx + d|$ & $\frac{a (cx + d) - c(ax + b)}{(cx + d)^2}$ \\
|
||||
$\frac{x}{2} f(x) + \frac{a^2}{2} \ln|x + f(x)|$ & $\sqrt{a^2 + x^2}$ \\
|
||||
$\frac{x}{2} f(x) - \frac{a^2}{2} \arcsin\left( \frac{x}{|a|} \right)$ & $\sqrt{a^2 - x^2}$ \\
|
||||
$\frac{x}{2} f(x) - \frac{a^2}{2} \ln|x + f(x)|$ & $\sqrt{x^2 - a^2}$ \\
|
||||
$\ln(x + \sqrt{x^2 \pm a^2})$ & $\frac{1}{\sqrt{x^2 \pm a^2}}$ \\
|
||||
$\arcsin \left( \frac{x}{|a|} \right)$ & $\frac{1}{\sqrt{x^2 - a^2}}$ \\
|
||||
$\frac{1}{a}\arctan \left( \frac{x}{|a|} \right)$ & $\frac{1}{a^2 - x^2}$ \\
|
||||
\end{tables}
|
||||
\begin{tables}{cc}{$F(x)$ & $f(x)$}
|
||||
$-\frac{1}{a} \cos(ax + b)$ & $\sin(ax + b)$ \\
|
||||
$\frac{1}{a} \sin(ax + b)$ & $\cos(ax + b)$ \\[1mm]
|
||||
\hline
|
||||
$x^x$ & $x^x \cdot (1 + \ln|x|)$ \\
|
||||
$(x^x)^x$ & $(x^x)^x \cdot (x + 2x\ln|x|)$ \\
|
||||
$x^{(x^x)}$ & $x^{(x^x)} \cdot (x^{x - 1} + \ln|x| \cdot x^x (1 + \ln|x|))$ \\
|
||||
\hline \\[-3mm]
|
||||
$\frac{1}{2}(x - \frac{1}{2} \sin(2x))$ & $\sin(x)^2$ \\[1mm]
|
||||
$\frac{1}{2}(x + \frac{1}{2} \sin(2x))$ & $\cos(x)^2$ \\
|
||||
\end{tables}
|
||||
@@ -0,0 +1,2 @@
|
||||
\subsection{Limits}
|
||||
\includegraphics[width=1\columnwidth]{./assets/limits.png}
|
||||
Binary file not shown.
@@ -78,6 +78,7 @@
|
||||
\section{Gemeinsame Verteilungen}
|
||||
\input{parts/04_joint-distribution/00_discrete.tex}
|
||||
\input{parts/04_joint-distribution/01_continuous.tex}
|
||||
\input{parts/04_joint-distribution/02_remarks.tex}
|
||||
% \input{parts/04_joint-distribution/}
|
||||
|
||||
|
||||
@@ -122,8 +123,11 @@
|
||||
|
||||
|
||||
\newsectionNoPB
|
||||
\section{Kombinatorik}
|
||||
\section{Tabellen}
|
||||
\input{parts/10_tips-and-tricks/00_phi-func-table.tex}
|
||||
\input{parts/10_tips-and-tricks/01_various.tex}
|
||||
\input{parts/10_tips-and-tricks/02_diff-table.tex}
|
||||
\input{parts/10_tips-and-tricks/03_limits.tex}
|
||||
% \input{parts/10_tips-and-tricks/}
|
||||
|
||||
\end{document}
|
||||
|
||||
Reference in New Issue
Block a user