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Some little improvements done
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‎.project

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<?xml version="1.0" encoding="UTF-8"?>
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<projectDescription>
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<name>An1E</name>
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<comment></comment>
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<projects>
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</projects>
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<buildSpec>
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<buildCommand>
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<name>net.sourceforge.texlipse.builder.TexlipseBuilder</name>
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<arguments>
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</arguments>
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</buildCommand>
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</buildSpec>
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<natures>
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<nature>net.sourceforge.texlipse.builder.TexlipseNature</nature>
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</natures>
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</projectDescription>
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<?xml version="1.0" encoding="UTF-8"?>
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<projectDescription>
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<name>An1E</name>
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<comment></comment>
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<projects>
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</projects>
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<buildSpec>
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<buildCommand>
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<name>net.sourceforge.texlipse.builder.TexlipseBuilder</name>
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<arguments>
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</arguments>
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</buildCommand>
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</buildSpec>
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<natures>
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<nature>net.sourceforge.texlipse.builder.TexlipseNature</nature>
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</natures>
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<linkedResources>
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<link>
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<name>Analysis1E.pdf</name>
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<type>1</type>
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<location>D:/Dropbox/HSR/Eclipse/An1E/Analysis1E.pdf</location>
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</link>
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</linkedResources>
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</projectDescription>

‎.texlipse

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#TeXlipse project settings
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#Fri Jan 11 13:42:21 CET 2013
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markTmpDer=true
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builderNum=2
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outputDir=
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makeIndSty=
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bibrefDir=
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outputFormat=pdf
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tempDir=tmp
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mainTexFile=document.tex
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outputFile=document.pdf
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langSpell=en
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markDer=true
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srcDir=
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#TeXlipse project settings
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#Fri Jan 11 14:16:12 CET 2013
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markTmpDer=true
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builderNum=6
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outputDir=
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makeIndSty=
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bibrefDir=
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outputFormat=pdf
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tempDir=tmp
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mainTexFile=document.tex
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outputFile=document.pdf
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langSpell=en
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markDer=true
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srcDir=

‎1_Einfuehrung.tex

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Einführung
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Einf"uhrung}
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\subsection{Zahlenmengen\formelbuchgreen{1,331}}
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\begin{minipage}[c]{6.5cm}
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$ \mathbb{N} = \left\{1,2,3,...\right\};\; $\\
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$ \mathbb{Q} = \left\{x|x \;=\; ^{p}/_{q} \text{ mit } p \in \mathbb{Z}
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\text{ und } (q \in \mathbb{Z} \smallsetminus \{0\})\right\};\;$
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\end{minipage}
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\begin{minipage}[c]{5cm}
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$ \mathbb{N}_0 = \left\{0,1,2,3,...\right\};\; $\\
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\end{minipage}
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\begin{minipage}[c]{5cm}
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$ \mathbb{Z} = \left\{...,-2,-1,0,1,2,..\right\}; $\\
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$ \mathbb{R} = zB \; \sqrt{2}, \pi,\phi$
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\end{minipage}
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\subsection{Mengenlehre\formelbuchgreen{334}}
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$A \;=\; \left\{-2,-1,0,1,2\right\} ,\; B\; =\; \left\{0,1,2,3,4\right\}$\\
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\begin{minipage}[c]{6.5cm}
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Schnittmenge:\\
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Vereinigungsmenge:\\
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Differenzmenge:\\
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Produktmenge:\\
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Kommutativgesetz:\\
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Assoziativgesetz:\\
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Distributivgesetz:
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\end{minipage}
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\begin{minipage}[c]{6.5cm}
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$A \; \cap B \;=\;\left\{x|x \in A \text{ und } x \in B \right\}$\\
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$A \; \cup B \;=\;\left\{x|x \in A \text{ oder } x \in B \right\}$\\
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$A \; \smallsetminus B \;=\;\left\{x|x \in A \text{ und } x \notin B \right\}$\\
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$A \; \times B\;=\;\left\{(a,b)|a \in A \text{ und } b \in B \right\}$\\
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$A \; \cap B \;=\;B \; \cap A$ \\
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$\left(A \cap B \right) \cap C\;=\;A \cap \left( B \cap C \right)$ \\
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$A\; \cap \left(B\cup C\right)\;=\;\left(A \cup B\right)\cap \left(A \cup C\right) $
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\end{minipage}
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\begin{minipage}[c]{7cm}
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$A \; \cap B \;=\; \left\{0,1,2\right\}$\\
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$A \; \cup B \;=\;\left\{-2,-1,0,1,2\right\}$\\
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$A \; \smallsetminus B \;=\;\left\{-2,-1\right\}$\\
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$ $\\
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$A \; \cup B \;=\;B \; \cup A$ \\
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$\left(A \cup B\right) \cup C\;=\;A \cup \left( B \cup C \right)$ \\
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$A\; \cup \left( B \cap C \right)\;=\;\left( A \cap B \right) \cup \left(A \cap C\right) $
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\end{minipage}
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\subsection{Beweismethoden\formelbuchgreen{5}}
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\subsection{Spezielle Ungleichungen\formelbuchgreen{30}}
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\begin{tabbing} xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx\=xxxxxxxxxxxxxxxxxxxxxx\=xxxxxxxxxxxxxxxxxxxxxx\=xxxxxxxxxxxxxxxxxxxxxx\=\kill
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Bernoulli-Ungleichung: \>
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$(1 + a)^n > 1 + n \cdot a$\>
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f"ur $n \in N, n \geq 2, a \in R, a > -1, a\neq0$\\
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Binomische Ungleichung: \>
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$|a\cdot b|\leq\frac{1}{2}(a^2 + b^2)$\\
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Dreiecksungleichung: \>
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$\left|a+b\right|\leq\left|a\right|+\left|b\right|$ \>
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$\left|a-b\right|\leq\left|a\right|+\left|b\right|$ \>
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$\left|a-b\right|\geq\left|\left|a\right|-\left|b\right|\right|$\\
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Geometrisches und arithmetisches Mittel:\\
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f"ur $a_i\geq0,\;n \in \mathbb{N},\;i \in \left\{1,2,...,n \right\}:$\>
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$\sqrt[n]{a_1 a_2 \ldots a_n}\leq \frac{1}{n} \cdot \sum\limits _{i=1}^n a_i = \frac{a_1+a_2+...+a_n}{n}$\>\>
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$\sqrt{ab}\leq \frac{a+b}{2}$, siehe Br. S.19/20 \\
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Minima/Maxima: \>
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$\min\{a_i\} \leq \sqrt[n]{a_1a_2 \ldots a_n} \leq \max\{a_i\}$\\
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Betragsungleichung:\>$-c<x<c\;\Leftrightarrow\;|x|<c$
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\end{tabbing}
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\subsection{Umgebung}
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\begin{minipage}[c]{14.5cm}
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Jedes offene Intervall, dass die Zahl a enth"alt, heisst eine Umgebung von a. \\
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Es sei $\epsilon >$ 0. Unter der $\epsilon$-Umgebung von a versteht man das offene Intervall $(a-\epsilon,a+\epsilon).$\\
76-
Eine $\epsilon$-Umgebung von a ohne die Zahl a selbst wird punktierte $\epsilon$-Umgebung von a genannt.
77-
\end{minipage}
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\begin{minipage}[c]{5cm}
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Schreibweise: U(a)\\
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Schreibweise: $U_\epsilon(a)$\\
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Schreibweise: $\dot{U}_\epsilon(a)=U_\epsilon(a)\smallsetminus{a}$
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\end{minipage}
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\subsection{Summenzeichen\formelbuchgreen{7}}
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\begin{minipage}[c]{4.75cm}
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$\text{mit 1}\leq m\leq n $
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\end{minipage}
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\begin{minipage}[c]{16cm}
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Die Laufvariable $i$ wird immer um 1 aufaddiert. $i$ immer kleiner-gleich $n$ (z.B. wenn $i \in \mathbb{R}$)
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\end{minipage}
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\begin{minipage}[c]{4.75cm}
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$\sum\limits _{i=1}^n a_i = \sum\limits _{i=1}^m a_i + \sum\limits _{i=m+1}^n a_i;$
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\end{minipage}
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\begin{minipage}[c]{4.25cm}
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$\sum\limits _{i=1}^n a_i = \sum\limits _{i=1-j}^{n-j} a_{i+j};$
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\end{minipage}
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\begin{minipage}[c]{4.25cm}
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$\sum\limits _{i=1}^n a = n\cdot a;$
99-
\end{minipage}
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\begin{minipage}[c]{8cm}
101-
$\sum\limits _{i=1}^n \left(\lambda a_i + \beta b_i \right) = $
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$\lambda \sum\limits _{i=1}^n a_i + \beta \sum\limits _{i=1}^n b_i$
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\end{minipage}
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105-
\subsection{Spezielle endliche Reihen\formelbuchgreen{19}}
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\begin{minipage}[c]{4.25cm}
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$\sum\limits _{i=1}^n i = \frac{n(n+1)}{2}$
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\end{minipage}
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\begin{minipage}[c]{4.25cm}
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$\sum\limits _{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$
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\end{minipage}
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\begin{minipage}[c]{4.25cm}
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$\sum\limits _{i=1}^n i^3 = \frac{n^2(n+1)^2}{4}$
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\end{minipage}
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116-
\subsection{Produktzeichen\formelbuchgreen{7}}
117-
$a_n\prod\limits _{i=1}^n \left(x-x_i\right)=
118-
a_n\cdot\left(x-x_1\right)\cdot\left(x-x_2\right)\cdot...\cdot\left(x-x_n\right)$
119-
120-
\subsection{Fakult"at\formelbuchgreen{13}}
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\begin{minipage}[c]{6cm}
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$n! = 1\cdot2\cdot3\cdot...\cdot n $
123-
\end{minipage}
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\begin{minipage}[c]{6cm}
125-
$\text{f"ur n} \in \mathbb{N}, n \geq 3$
126-
\end{minipage}
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\begin{minipage}[c]{6cm}
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$n!>2^{n-1}$
129-
\end{minipage}
130-
131-
\subsection{Binomischer Satz\formelbuchgreen{12}}
132-
\begin{minipage}[c]{6cm}
133-
$\left(a+b\right)^n = \sum\limits _{i=0}^n \left(\stackrel{n}{i}\right)a^{n-i}\cdot b^i$;\\
134-
$\left(\stackrel{n}{i-1}\right)+\left(\stackrel{n}{i}\right)=\left(\stackrel{n+1}{i}\right)$;
135-
\end{minipage}
136-
\begin{minipage}[c]{6cm}
137-
$\left(\stackrel{n}{i}\right)=\left(\stackrel{n}{n-i}\right)$\\
138-
$\left(\stackrel{n}{i}\right)=\frac{n!}{i!\left(n-i\right)!}$;
139-
\end{minipage}
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\begin{minipage}[c]{6cm}
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$\left(\stackrel{n}{0}\right)=1$
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\end{minipage}
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144-
\subsection{Einige Wurzeln}
1+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3+
% Einführung
4+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
5+
\section{Einf"uhrung}
6+
\subsection{Zahlenmengen\formelbuchgreen{1,331}}
7+
\begin{minipage}[c]{6.5cm}
8+
$ \mathbb{N} = \left\{1,2,3,...\right\};\; $\\
9+
$ \mathbb{Q} = \left\{x|x \;=\; ^{p}/_{q} \text{ mit } p \in \mathbb{Z}
10+
\text{ und } (q \in \mathbb{Z} \smallsetminus \{0\})\right\};\;$
11+
\end{minipage}
12+
\begin{minipage}[c]{5cm}
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$ \mathbb{N}_0 = \left\{0,1,2,3,...\right\};\; $\\
14+
\end{minipage}
15+
\begin{minipage}[c]{5cm}
16+
$ \mathbb{Z} = \left\{...,-2,-1,0,1,2,..\right\}; $\\
17+
$ \mathbb{R} = zB \; \sqrt{2}, \pi,\phi$
18+
\end{minipage}
19+
20+
\subsection{Mengenlehre\formelbuchgreen{334}}
21+
$A \;=\; \left\{-2,-1,0,1,2\right\} ,\; B\; =\; \left\{0,1,2,3,4\right\}$\\
22+
\begin{minipage}[c]{6.5cm}
23+
Schnittmenge:\\
24+
Vereinigungsmenge:\\
25+
Differenzmenge:\\
26+
Produktmenge:\\
27+
Kommutativgesetz:\\
28+
Assoziativgesetz:\\
29+
Distributivgesetz:
30+
\end{minipage}
31+
\begin{minipage}[c]{6.5cm}
32+
$A \; \cap B \;=\;\left\{x|x \in A \text{ und } x \in B \right\}$\\
33+
$A \; \cup B \;=\;\left\{x|x \in A \text{ oder } x \in B \right\}$\\
34+
$A \; \smallsetminus B \;=\;\left\{x|x \in A \text{ und } x \notin B \right\}$\\
35+
$A \; \times B\;=\;\left\{(a,b)|a \in A \text{ und } b \in B \right\}$\\
36+
$A \; \cap B \;=\;B \; \cap A$ \\
37+
$\left(A \cap B \right) \cap C\;=\;A \cap \left( B \cap C \right)$ \\
38+
$A\; \cap \left(B\cup C\right)\;=\;\left(A \cup B\right)\cap \left(A \cup C\right) $
39+
\end{minipage}
40+
\begin{minipage}[c]{7cm}
41+
$A \; \cap B \;=\; \left\{0,1,2\right\}$\\
42+
$A \; \cup B \;=\;\left\{-2,-1,0,1,2\right\}$\\
43+
$A \; \smallsetminus B \;=\;\left\{-2,-1\right\}$\\
44+
$ $\\
45+
$A \; \cup B \;=\;B \; \cup A$ \\
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$\left(A \cup B\right) \cup C\;=\;A \cup \left( B \cup C \right)$ \\
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$A\; \cup \left( B \cap C \right)\;=\;\left( A \cap B \right) \cup \left(A \cap C\right) $
48+
\end{minipage}
49+
50+
\subsection{Beweismethoden\formelbuchgreen{5}}
51+
52+
\subsection{Spezielle Ungleichungen\formelbuchgreen{30}}
53+
\begin{tabbing} xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx\=xxxxxxxxxxxxxxxxxxxxxx\=xxxxxxxxxxxxxxxxxxxxxx\=xxxxxxxxxxxxxxxxxxxxxx\=\kill
54+
Bernoulli-Ungleichung: \>
55+
$(1 + a)^n > 1 + n \cdot a$\>
56+
f"ur $n \in N, n \geq 2, a \in R, a > -1, a\neq0$\\
57+
Binomische Ungleichung: \>
58+
$|a\cdot b|\leq\frac{1}{2}(a^2 + b^2)$\\
59+
Dreiecksungleichung: \>
60+
$\left|a+b\right|\leq\left|a\right|+\left|b\right|$ \>
61+
$\left|a-b\right|\leq\left|a\right|+\left|b\right|$ \>
62+
$\left|a-b\right|\geq\left|\left|a\right|-\left|b\right|\right|$\\
63+
Geometrisches und arithmetisches Mittel:\\
64+
f"ur $a_i\geq0,\;n \in \mathbb{N},\;i \in \left\{1,2,...,n \right\}:$\>
65+
$\sqrt[n]{a_1 a_2 \ldots a_n}\leq \frac{1}{n} \cdot \sum\limits _{i=1}^n a_i = \frac{a_1+a_2+...+a_n}{n}$\>\>
66+
$\sqrt{ab}\leq \frac{a+b}{2}$, siehe Br. S.19/20 \\
67+
Minima/Maxima: \>
68+
$\min\{a_i\} \leq \sqrt[n]{a_1a_2 \ldots a_n} \leq \max\{a_i\}$\\
69+
Betragsungleichung:\>$-c<x<c\;\Leftrightarrow\;|x|<c$
70+
\end{tabbing}
71+
72+
\subsection{Umgebung}
73+
\begin{minipage}[c]{14.5cm}
74+
Jedes offene Intervall, dass die Zahl a enth"alt, heisst eine Umgebung von a. \\
75+
Es sei $\epsilon >$ 0. Unter der $\epsilon$-Umgebung von a versteht man das offene Intervall $(a-\epsilon,a+\epsilon).$\\
76+
Eine $\epsilon$-Umgebung von a ohne die Zahl a selbst wird punktierte $\epsilon$-Umgebung von a genannt.
77+
\end{minipage}
78+
\begin{minipage}[c]{5cm}
79+
Schreibweise: U(a)\\
80+
Schreibweise: $U_\epsilon(a)$\\
81+
Schreibweise: $\dot{U}_\epsilon(a)=U_\epsilon(a)\smallsetminus{a}$
82+
\end{minipage}
83+
84+
\subsection{Summenzeichen\formelbuchgreen{7}}
85+
\begin{minipage}[c]{4.75cm}
86+
$\text{mit 1}\leq m\leq n $
87+
\end{minipage}
88+
\begin{minipage}[c]{16cm}
89+
Die Laufvariable $i$ wird immer um 1 aufaddiert. $i$ immer kleiner-gleich $n$ (z.B. wenn $i \in \mathbb{R}$)
90+
\end{minipage}
91+
\begin{minipage}[c]{4.75cm}
92+
$\sum\limits _{i=1}^n a_i = \sum\limits _{i=1}^m a_i + \sum\limits _{i=m+1}^n a_i;$
93+
\end{minipage}
94+
\begin{minipage}[c]{4.25cm}
95+
$\sum\limits _{i=1}^n a_i = \sum\limits _{i=1-j}^{n-j} a_{i+j};$
96+
\end{minipage}
97+
\begin{minipage}[c]{4.25cm}
98+
$\sum\limits _{i=1}^n a = n\cdot a;$
99+
\end{minipage}
100+
\begin{minipage}[c]{8cm}
101+
$\sum\limits _{i=1}^n \left(\lambda a_i + \beta b_i \right) = $
102+
$\lambda \sum\limits _{i=1}^n a_i + \beta \sum\limits _{i=1}^n b_i$
103+
\end{minipage}
104+
105+
\subsection{Spezielle endliche Reihen\formelbuchgreen{19}}
106+
\begin{minipage}[c]{4.25cm}
107+
$\sum\limits _{i=1}^n i = \frac{n(n+1)}{2}$
108+
\end{minipage}
109+
\begin{minipage}[c]{4.25cm}
110+
$\sum\limits _{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$
111+
\end{minipage}
112+
\begin{minipage}[c]{4.25cm}
113+
$\sum\limits _{i=1}^n i^3 = \frac{n^2(n+1)^2}{4}$
114+
\end{minipage}
115+
116+
\subsection{Produktzeichen\formelbuchgreen{7}}
117+
$a_n\prod\limits _{i=1}^n \left(x-x_i\right)=
118+
a_n\cdot\left(x-x_1\right)\cdot\left(x-x_2\right)\cdot...\cdot\left(x-x_n\right)$
119+
120+
\subsection{Fakult"at\formelbuchgreen{13}}
121+
\begin{minipage}[c]{6cm}
122+
$n! = 1\cdot2\cdot3\cdot...\cdot n $
123+
\end{minipage}
124+
\begin{minipage}[c]{6cm}
125+
$\text{f"ur n} \in \mathbb{N}, n \geq 3$
126+
\end{minipage}
127+
\begin{minipage}[c]{6cm}
128+
$n!>2^{n-1}$
129+
\end{minipage}
130+
131+
\subsection{Binomischer Satz\formelbuchgreen{12}}
132+
\begin{minipage}[c]{6cm}
133+
$\left(a+b\right)^n = \sum\limits _{i=0}^n \left(\stackrel{n}{i}\right)a^{n-i}\cdot b^i$\\
134+
$\left(\stackrel{n}{i-1}\right)+\left(\stackrel{n}{i}\right)=\left(\stackrel{n+1}{i}\right)$
135+
\end{minipage}
136+
\begin{minipage}[c]{6cm}
137+
$\left(\stackrel{n}{i}\right)=\left(\stackrel{n}{n-i}\right)$\\
138+
$\left(\stackrel{n}{i}\right)=\frac{n!}{i!\left(n-i\right)!}$
139+
\end{minipage}
140+
\begin{minipage}[c]{6cm}
141+
$\left(\stackrel{n}{0}\right)=1$\\
142+
$2^n = \sum\limits _{i=0}^n \left(\stackrel{n}{i}\right)$
143+
\end{minipage}
144+
145+
\subsection{Einige Wurzeln}
145146
$\sqrt{2} = 1.414; \qquad \sqrt{3} = 1.732; \qquad \sqrt{5} = 2.236; \qquad \sqrt{6} = 2.449; \qquad \sqrt{7} = 2.645; \qquad \sqrt{8} = 2.828;$

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