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@@ -1,6 +1,7 @@
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\documentclass[a4paper,10pt]{article}
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-\usepackage{amssymb}
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-\usepackage{amsmath}
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+\usepackage{amssymb, amsmath}
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+\DeclareMathOperator{\arcsinh}{arcsinh}
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+\DeclareMathOperator{\arccosh}{arccosh}
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\DeclareMathOperator{\arctanh}{arctanh}
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\usepackage[utf8]{inputenc} % this is needed for umlauts
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\usepackage[ngerman]{babel} % this is needed for umlauts
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@@ -30,12 +31,12 @@
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\begin{minipage}[b]{0.5\linewidth}\centering
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\begin{align*}
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-\lim_{x \to 0} \frac {\sin x}{x} &= 1 \\
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-\lim_{x \to 0} \frac {e^x - 1}{x} &= 1 \\
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-\lim_{h \to 0} \frac {e^{{x_0} + h} - e^{x_0}}{h} &= e^{x_0} \\
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-\sum_{n = 0}^{\infty} (-1)^n \frac {(-1)^{n + 1}}{n} &= \log 2 \\
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-\cos x &= \sum_{n = 0}^{\infty} (-1)^n \frac {x^{2n}}{(2n)!} \\
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-\sin x &= \sum_{n = 0}^{\infty} (-1)^n \frac {x^{2n + 1}}{(2n + 1)!}
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+ \lim_{x \to 0} \frac {\sin x}{x} &= 1 \\
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+ \lim_{x \to 0} \frac {e^x - 1}{x} &= 1 \\
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+ \lim_{h \to 0} \frac {e^{{x_0} + h} - e^{x_0}}{h} &= e^{x_0} \\
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+ \sum_{n = 0}^{\infty} (-1)^n \frac {(-1)^{n + 1}}{n} &= \log 2 \\
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+ \cos x &= \sum_{n = 0}^{\infty} (-1)^n \frac {x^{2n}}{(2n)!} \\
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+ \sin x &= \sum_{n = 0}^{\infty} (-1)^n \frac {x^{2n + 1}}{(2n + 1)!}
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\end{align*}
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\end{minipage}
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@@ -55,30 +56,47 @@ e^x &= \sum_{n = 0}^{\infty} \frac {x^n}{n!} \\
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\end{minipage}
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\end{table}
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-
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-
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\section{Zusammenhänge}
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\begin{align*}
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- (\cos x)^2 + (\sin x)^2 &= 1 \\
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- (\cosh x)^2 - (\sinh x)^2 &= 1 \\
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- \tan x &= \frac {\sin x}{\cos x} \\
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- \tanh x &= \frac {\sinh x}{\cosh x} \\
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+ (\cos x)^2 + (\sin x)^2 &= 1 \\
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+ (\cosh x)^2 - (\sinh x)^2 &= 1 \\
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+ \tan x &= \frac {\sin x}{\cos x} \\
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+ \tanh x &= \frac {\sinh x}{\cosh x} \\
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(x + y)^n &= \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
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\end{align*}
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\section{Ableitungen}
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+\begin{table}[ht]
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+\begin{minipage}[b]{0.5\linewidth}\centering
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+\begin{align*}
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+ (\sin x)' &= \cos x \\
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+ (\cos x)' &= -\sin x \\
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+ (\tan x)' &= \frac{1}{\cos^2 x} \\
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+ (\sinh x)' &= \cosh x \\
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+ (\cosh x)' &= \sinh x \\
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+\end{align*}
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+
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+\end{minipage}
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+\hspace{0.5cm}
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+\begin{minipage}[b]{0.5\linewidth}
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+\centering
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+
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\begin{align*}
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- (\arctan x)' &= \frac {1}{1 + x^2} \\
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- (\sin x)' &= \cos x \\
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- (\cos x)' &= -\sin x \\
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- (\arctanh x)' &= \frac {1}{1 - x^2}
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+ (\arcsin x)' &= \frac {1}{\sqrt{1-x^2}} \\
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+ (\arccos x)' &= - \frac {1}{\sqrt{1-x^2}} \\
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+ (\arctan x)' &= \frac {1}{1 + x^2} \\
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+ % (\arcsinh x)' &= \frac {1}{\sqrt{1+x^2}} \\
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+ % (\arccosh x)' &= \frac {1}{\sqrt{(1-x^2) \cdot (1+x^2)}} \\
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+ % (\arctanh x)' &= \frac {1}{1 - x^2}
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\end{align*}
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+\end{minipage}
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+\end{table}
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\section{Potenzreihen}
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Zuerst den Potenzradius r berechnen:
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\(
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- r = \frac {1}{\lim \text{sup} \sqrt[n]{|a_n|}}
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+ r = \frac {1}{\lim \text{sup} \sqrt[n]{|a_n|}}
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\)
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\end{document}
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