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#LyX 2.3 created this file. For more info see http://www.lyx.org/
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\begin_body
\begin_layout Title
Analysis 1 Zusammenfassung
\end_layout
\begin_layout Part
Folgen und Reihen
\end_layout
\begin_layout Section
Formeln
\end_layout
\begin_layout Description
Arithmetisches
\begin_inset space ~
\end_inset
Bildungsgesetz
\begin_inset Formula $a_{n}=a_{1}+(n-1)*d$
\end_inset
\end_layout
\begin_layout Description
Summe
\begin_inset space ~
\end_inset
Arithmetischer
\begin_inset space ~
\end_inset
Folgen
\begin_inset Formula $S_{n}=n\cdot\frac{a_{1}+a_{n}}{2}$
\end_inset
\end_layout
\begin_layout Description
Geometrisches
\begin_inset space ~
\end_inset
Bildungsgesetz
\begin_inset Formula $a_{n}=a_{1}\cdot q^{n-1}$
\end_inset
\end_layout
\begin_layout Description
Summe
\begin_inset space ~
\end_inset
Geometrischer
\begin_inset space ~
\end_inset
Folgen
\begin_inset Formula $S_{n}=a_{1}\cdot\frac{q^{n}-1}{q-1}$
\end_inset
\end_layout
\begin_layout Description
Gauss
\begin_inset space ~
\end_inset
Formel
\begin_inset Formula $1+2+...+n=\frac{(n+1)\cdot n}{2}$
\end_inset
\end_layout
\begin_layout Description
Unentliche
\begin_inset space ~
\end_inset
geometrische
\begin_inset space ~
\end_inset
Reihe
\begin_inset Formula $\overset{\infty}{\underset{k=0}{\sum}=\frac{a_{0}}{1-q}}$
\end_inset
\end_layout
\begin_layout Subsection
Summen
\end_layout
\begin_layout Standard
\begin_inset Formula $\underset{i=1}{\overset{n}{\sum}}i=\frac{1}{2}\cdot n\cdot(n+1)$
\end_inset
\begin_inset Newline newline
\end_inset
\begin_inset Formula $\text{\ensuremath{\overset{n}{\underset{i=1}{\sum}}i^{2}=\frac{1}{6}\cdot n\cdot(n+1)\cdot(2\cdot n+1)}}$
\end_inset
\begin_inset Newline newline
\end_inset
\begin_inset Formula $\overset{n}{\underset{n=1}{\sum}}i^{3}=\frac{1}{4}\cdot n^{2}\cdot(n+1)^{2}$
\end_inset
\end_layout
\begin_layout Section
Grenzwerte
\end_layout
\begin_layout Subsection
Grenzwertsatz
\end_layout
\begin_layout Standard
Es sei eine Folge
\begin_inset Formula $a_{n}$
\end_inset
mit Grenzwert
\begin_inset Formula $a$
\end_inset
und eine Folge
\begin_inset Formula $b_{n}$
\end_inset
mit Grenzwert
\begin_inset Formula $b$
\end_inset
.
Dann gilt:
\end_layout
\begin_layout Standard
\begin_inset Formula $\lim_{n\to\infty}(a_{n}\pm b_{n})=a\pm b$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\lim_{n\to\infty}(a_{n}\cdot b_{n})=a\cdot b$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\lim_{n\to\infty}(\frac{a_{n}}{b_{n}})=\frac{a}{b}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\lim_{n\to\infty}\frac{a}{n}=0$
\end_inset
für
\begin_inset Formula $a\in\mathbb{R}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\lim_{n\to\infty}\sqrt[n]{a}=1$
\end_inset
für
\begin_inset Formula $a\in\mathbb{R^{+}}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\lim_{n\to\infty}a^{n}=0$
\end_inset
für
\begin_inset Formula $|a|<1$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\lim_{n\to\infty}(1+\frac{1}{n})^{n}=e$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\lim_{n\to\infty}(1-\frac{1}{n})^{n}=\frac{1}{e}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\lim_{n\to\infty}(1+\frac{k}{n})^{n}=e^{k}$
\end_inset
\end_layout
\begin_layout Section
Induktion
\end_layout
\begin_layout Enumerate
Induktionsanfang: Zeigen, dass die Annahme für ein beliebiges n, zB.
\begin_inset Formula $n_{1}$
\end_inset
, wahr ist.
\end_layout
\begin_layout Enumerate
Induktionsschritt: Man nimmt an, die Aussage sei für ein gewisses nicht
präzisiertes n∈N wahr und zeigt davon ausgehend die Aussage für n + 1.
\end_layout
\begin_layout Section
Definitionen
\end_layout
\begin_layout Description
Konvergenz Wenn sich eine Folge einem Grenzwert annähert ist sie
\series bold
konvergent
\series default
, ansonsten ist sie
\series bold
divergent
\series default
.
Konvergiert eine Folge
\begin_inset Formula $x_{n}$
\end_inset
zu einem Grenzwert
\begin_inset Formula $g$
\end_inset
, so liegen alle Glieder
\begin_inset Formula $n$
\end_inset
ab einer bestimmten Distanz
\begin_inset Formula $\epsilon$
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innerhalb der Toleranzgrenze.
Alle Glieder
\begin_inset Formula $n$
\end_inset
grösser als
\begin_inset Formula $|x_{n}-a|<\epsilon$
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liegen innerhalb des Bereichs von
\begin_inset Formula $\epsilon$
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\begin_inset Formula $.$
\end_inset
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\begin_layout Description
Monotonie Wenn der Funktionswert immer steigt, so heißt die Funktion
\series bold
streng monoton steigend
\series default
, steigt der Funktionswert immer
\series bold
oder bleibt er gleich
\series default
, heißt sie
\series bold
monoton steigend
\series default
.
Analog heißt eine Funktion
\series bold
streng monoton
\series default
fallend, wenn ihr Funktionswert
\series bold
immer fällt
\series default
und monoton fallend, wenn er immer fällt oder
\series bold
gleich bleibt.
\end_layout
\begin_layout Description
Beschränktheit Eine Folge heisst nach oben beschränkt, wenn es eine reelle
Zahl M gibt, so dass
\begin_inset Formula $a_{n}\leq M$
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für alle natürlichen
\begin_inset Formula $n$
\end_inset
.
\end_layout
\begin_layout Part
Kurvendiskussion
\end_layout
\begin_layout Description
Liniarfaktoren
\begin_inset space ~
\end_inset
von
\begin_inset space ~
\end_inset
Polynomen
\begin_inset space ~
\end_inset
(Produktedarstellung)
\begin_inset Formula $a_{n}(x-x_{1})(x-x_{2})(x-x_{3})$
\end_inset
\end_layout
\begin_layout Description
Qualitativ Was passiert grob, kommt von qualis: wie beschaffen (von welcher
Art)
\end_layout
\begin_layout Description
Quantitaiv Mit Zahlen, kommt von quantus: wie groß (von welchem Ausmaß)
\end_layout
\begin_layout Description
Newton
\begin_inset space ~
\end_inset
Verfahren
\begin_inset Formula $x_{n+1}=x_{n}-\frac{f(x_{n})}{f'(x_{n})}$
\end_inset
\end_layout
\begin_layout Description
Optimieren
\begin_inset space ~
\end_inset
einer
\begin_inset space ~
\end_inset
Fläche 1.
Seitenlängen in
\begin_inset Formula $x$
\end_inset
und
\begin_inset Formula $f(x)$
\end_inset
ausdrücken 2.
Formel für Fläche aufstellen 3.
Maximum der Funktion im gegebenen Bereich berechnen
\end_layout
\begin_layout Enumerate
Extremstellen:
\begin_inset Formula $f(x)'=0\wedge f(x)''\neq0$
\end_inset
\end_layout
\begin_deeper
\begin_layout Enumerate
Wenn
\begin_inset Formula $f(x)'=0\wedge f(x)''=0\Rightarrow$
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Sattelpunkt
\end_layout
\end_deeper
\begin_layout Enumerate
Maximum & Minimum:
\begin_inset Formula $f(x)''<0\Rightarrow$
\end_inset
Min.
\begin_inset Formula $f(x)''>0\Rightarrow$
\end_inset
Max.
\end_layout
\begin_layout Enumerate
Wendepunkte:
\begin_inset Formula $f(x)''=0\wedge f(x)'''\neq0$
\end_inset
\end_layout
\begin_layout Enumerate
Krümmung von Wendepunkten:
\begin_inset Formula $f(x)'''<0\Rightarrow$
\end_inset
Links-Rechts Krümmung,
\begin_inset Formula $f(x)'''>0\Rightarrow$
\end_inset
Rechts-Links Krümmung
\end_layout
\begin_layout Section
Tangenten und Normalen
\end_layout
\begin_layout Standard
\begin_inset Formula $m_{tangente}\cdot m_{normale}=-1$
\end_inset
\begin_inset Newline newline
\end_inset
Tangente für einen Punkt im einer Funktion bestimmen:
\end_layout
\begin_layout Enumerate
\begin_inset Formula $x$
\end_inset
in Funktion einsetzen um
\begin_inset Formula $y$
\end_inset
zu berechnen.
\end_layout
\begin_layout Enumerate
Steigung der Gesuchten Stelle
\begin_inset Formula $f(x)'=m_{t}$
\end_inset
berechnen.
\end_layout
\begin_layout Enumerate
Werte
\begin_inset Formula $y$
\end_inset
,
\begin_inset Formula $x$
\end_inset
und
\begin_inset Formula $f(x)'$
\end_inset
in Funktion
\begin_inset Formula $y=m_{t}\cdot x+b$
\end_inset
einsetzen, um
\begin_inset Formula $b$
\end_inset
zu berechnen.
\end_layout
\begin_layout Standard
Normale bestimmen:
\end_layout
\begin_layout Enumerate
Steigung der Tangente
\begin_inset Formula $f(x)'=m_{t}$
\end_inset
in die Gleichung
\begin_inset Formula $m_{t}\cdot m_{n}=-1$
\end_inset
einsetzen und nach
\begin_inset Formula $m_{n}$
\end_inset
auflösen.
\end_layout
\begin_layout Enumerate
Gleichung
\begin_inset Formula $y=m_{n}\cdot x+b$
\end_inset
nach
\begin_inset Formula $b$
\end_inset
auflösen
\end_layout
\begin_layout Enumerate
Werte in
\begin_inset Formula $y=m_{n}\cdot x+b$
\end_inset
einsetzen
\end_layout
\begin_layout Section
Ableitungsfunktionen & Regeln
\end_layout
\begin_layout Standard
\begin_inset Tabular
<lyxtabular version="3" rows="11" columns="2">
<features tabularvalignment="middle">
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<column alignment="center" valignment="top" width="0pt">
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $f(x)$
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Formula $f(x)'$
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\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $x^{r}$
\end_inset
mit
\begin_inset Formula $r\in\mathbb{R}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
\begin_inset Formula $r\cdot x^{r-1}$
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_inset Formula $\sin(x)$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\cos(x)$
\end_inset
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
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\begin_inset Formula $-\sin(x)$
\end_inset
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\end_inset
</cell>
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_inset Formula $\tan(x)$
\end_inset
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\begin_inset Text
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\end_inset
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $e^{x}$
\end_inset
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
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\begin_inset Formula $e^{x}$
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</cell>
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\ln(x)$
\end_inset
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\frac{1}{x}$
\end_inset
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</cell>
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
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\begin_inset Formula $a^{x}$
\end_inset
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\ln(a)\cdot a^{x}$
\end_inset
\end_layout
\end_inset
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
\begin_inset Formula $\arcsin(x)$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\pm\frac{1}{\sqrt{1-x^{2}}}$
\end_inset
\end_layout
\end_inset
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\begin_inset Formula $\arccos(x)$
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\begin_inset Formula $\mp\frac{1}{\sqrt{1-x^{2}}}$
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\begin_inset Formula $\arctan(x)$
\end_inset
\end_layout
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</cell>
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\begin_inset Formula $\frac{1}{1+X^{2}}$
\end_inset
\end_layout
\end_inset
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\begin_inset space ~
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\begin_inset space ~
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\begin_inset Tabular
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<column alignment="center" valignment="top" width="0pt">
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\begin_inset Text
\begin_layout Plain Layout
Faktorregel
\end_layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
\begin_inset Formula $((C\cdot f(x))'=C\cdot f(x)'$
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\end_layout
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<row>
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\begin_layout Plain Layout
Summenregel
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $(f(x)+g(x))'=f(x)'+g(x)'$
\end_inset
\end_layout
\end_inset
</cell>
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Produktregel
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $(f(x)\cdot g(x))'=f(x)'\cdot g(x)+f(x)\cdot g(x)'$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Quotientenregel
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $(\frac{f(x)}{g(x)})'=\frac{1}{g(x)^{2}}\cdot(f(x)'\cdot g(x)-f(x)\cdot g(x)'$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Kettenregel
\end_layout
\end_inset
</cell>
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $f(u(x))'=F(u)'\cdot u(x)'$
\end_inset
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Section
Trigonometrie
\end_layout
\begin_layout Description
Trigonometrischer
\begin_inset space ~
\end_inset
Pythagoras
\begin_inset Formula $\sin^{2}x+\cos^{2}x=1$
\end_inset
\end_layout
\begin_layout Description
Weitere
\begin_inset space ~
\end_inset
Beziehungen
\begin_inset Formula $\tan x=\frac{\sin x}{\cos x}=\frac{1}{\cot x}$
\end_inset
|
\begin_inset Formula $\cot x=\frac{\cos x}{\sin x}=\frac{1}{\tan x}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Graphics
filename Images/Trig_funct.png
lyxscale 50
scale 60
\end_inset
\begin_inset Newline newline
\end_inset
Weiteres: Papula S.
94 - 97
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
newpage
\end_layout
\end_inset
\end_layout
\begin_layout Part
Referenzen
\end_layout
\begin_layout Subsection
Geometrie
\end_layout
\begin_layout Description
Geometrische
\begin_inset space ~
\end_inset
Körper 28 - 41
\end_layout
\begin_layout Description
Lehrsätze
\begin_inset space ~
\end_inset
Geometrie 26 - 28
\end_layout
\begin_layout Subsection
Funktionen
\end_layout
\begin_layout Description
Umkehrfunktion 70
\end_layout
\begin_layout Description
Grenzwerte 71
\end_layout
\begin_layout Description
Funtkionen Liniar: 76 Quadratisch: 78 Polynome: 79
\end_layout
\begin_layout Description
Hoerner-Schema 80
\end_layout
\begin_layout Description
Nullstellen,
\begin_inset space ~
\end_inset
Definitionslücken
\begin_inset space ~
\end_inset
Pole 87
\end_layout
\begin_layout Description
Potenzfunktionen 89
\end_layout
\begin_layout Description
Wurzelfunktion 90
\end_layout
\begin_layout Description
Sin,
\begin_inset space ~
\end_inset
Cos,
\begin_inset space ~
\end_inset
Tan 92 - 95
\end_layout
\begin_layout Description
E-Funktion 104
\end_layout
\begin_layout Description
Logarithmusfunktionen 107
\end_layout
\begin_layout Description
Ableitungsregeln 132 - 134
\end_layout
\begin_layout Description
Tangente
\begin_inset space ~
\end_inset
und
\begin_inset space ~
\end_inset
Normale 139
\end_layout
\begin_layout Description
Maximum
\begin_inset space ~
\end_inset
Minimum
\begin_inset space ~
\end_inset
Wendepunkte 142
\end_layout
\begin_layout Description
Integralrechnung 148
\end_layout
\end_body
\end_document
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