Änderungen vorgenommen
This commit is contained in:
146
Band2/Band2.typ
146
Band2/Band2.typ
@@ -137,9 +137,6 @@
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#deckblatt(bandnr: "2", autor: tt, titel: "Differential- und Integralrechnung\nfür Funktionen\nmit einer Variablen")
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#deckblatt(bandnr: "2", autor: tt, titel: "Differential- und Integralrechnung\nfür Funktionen\nmit einer Variablen")
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#pagebreak()
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#pagebreak()
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#set page(
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#set page(
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@@ -164,6 +161,7 @@
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width:162mm,
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width:162mm,
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height: 230mm,
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height: 230mm,
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)
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)
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#v(9mm)
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#heading(outlined: false)[Abhängigkeitsgraph]
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#heading(outlined: false)[Abhängigkeitsgraph]
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/* #let mynode(pos, text1, text2, name) = {
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/* #let mynode(pos, text1, text2, name) = {
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node(pos, [#text( size:1.7em, text2) \ #text1], name:name, stroke: 1pt, corner-radius: 0mm,
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node(pos, [#text( size:1.7em, text2) \ #text1], name:name, stroke: 1pt, corner-radius: 0mm,
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@@ -171,8 +169,11 @@
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/* Dashed separator from east to west */
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/* Dashed separator from east to west */
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//edge(label(str(name)+".west"), label(str(name)+".east"),"-", stroke: 0.2mm, dash:(1mm, 0.8mm))
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//edge(label(str(name)+".west"), label(str(name)+".east"),"-", stroke: 0.2mm, dash:(1mm, 0.8mm))
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} */
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} */
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#v(-6mm)
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#scale(x:90%, y:90%)[#abhaengigkeit]
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#abhaengigkeit
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//#scale(factor: 90%, abhaengigkeit)
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//#abhaengigkeit
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#show heading: it => {
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#show heading: it => {
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set block(below: if it.level == 1 { 2em } else { 1em })
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set block(below: if it.level == 1 { 2em } else { 1em })
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@@ -430,11 +431,144 @@ Unter Verwendung von @eq22 und bekannten Grenzwertsätzen für Zahlenfolgen folg
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Die Gültigkeit von @eq24 wurde für eine beliebige und damit für jede Folge $\(x_n)$ mit den Eigenschaften @eq23 bewiesen. Daher gilt
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Die Gültigkeit von @eq24 wurde für eine beliebige und damit für jede Folge $\(x_n)$ mit den Eigenschaften @eq23 bewiesen. Daher gilt
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/* $$
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/*
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$$
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\lim _{x \rightarrow \frac{1}{2}} \frac{x^2-\frac{1}{4}}{x-\frac{1}{2}}=1,
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\lim _{x \rightarrow \frac{1}{2}} \frac{x^2-\frac{1}{4}}{x-\frac{1}{2}}=1,
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$$
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$$
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was in Einklang mit der Anschauung steht (Bild 2.2). */
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was in Einklang mit der Anschauung steht (Bild 2.2).
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Beispiel 2.3: Wir wollen den Grenzwert der Funktion
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$$
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f(x)=\left\{\begin{array}{lll}
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\frac{x^2-\frac{1}{4}}{x-\frac{1}{2}} & \text { für } & x \neq \frac{1}{2} \\
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2 & \text { für } & x=\frac{1}{2}
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\end{array}\right.
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$$
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für $x \rightarrow \frac{1}{2}$ ermitteln (s. Bild 2.5).
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Obwohl $f$ an der Stelle $x=\frac{1}{2}$ definiert ist, werden auch hier nur Folgen $\left(x_n\right)$ mit $\lim _{n \rightarrow \infty} x_n=\frac{1}{2}$ betrachtet, für die $x_n \neq \frac{1}{2}$ für alle $n$ gilt [vgl. (E 2)]. Für jede solche Folge erhält man wie in Beispiel 2.2
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$$
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\lim _{n \rightarrow \infty} f\left(x_n\right)=\lim _{n \rightarrow \infty} \frac{x_n^2-\frac{1}{4}}{x_n-\frac{1}{2}}=\lim _{n \rightarrow \infty}\left(x_n+\frac{1}{2}\right)=1,
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$$
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also ist
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$$
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\lim _{x \rightarrow \frac{1}{2}} f(x)=1
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$$
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* Aufgabe 2.1: Ermitteln Sie die folgenden Grenzwerte:
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a) $\lim _{x \rightarrow 0} \frac{3 x+2}{x-1}$,
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b) $\lim _{x \rightarrow-2} \frac{x^2-4}{x+2}$.
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Beispiel 2.4: Nun soll das Verhalten von $f(x)=\sin \frac{1}{x}(x \neq 0)$ für $x \rightarrow 0$ untersucht werden. Die Bildkurve von $f$ (Bild 2.6) schwankt für $x \rightarrow 0$ ständig zwischen -1 und 1 , wobei die Scheitel immer dichter aufeinander folgen. Wir wollen zeigen, daß $f$ für $x \rightarrow 0$ keinen Grenzwert hat. Dazu genügt es, eine Folge ( $x_n$ ) mit
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$$
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x_n \neq 0 \text { für alle } n \text { und } \lim _{n \rightarrow \infty} x_n=0
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$$
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anzugeben, für die die Folge $\left(f\left(x_n\right)\right)$ divergent ist. Setzen wir zum Beispiel $x_n=\frac{2}{(2 n-1) \pi}$, dann gilt (2.5), aber wegen
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$$
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f\left(x_n\right)=\sin (n \pi-\pi / 2)=(-1)^{n+1}
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$$
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ist die Folge $\left(f\left(x_n\right)\right)$ (unbestimmt) divergent.
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Man kann den Beweis auch dadurch führen, daß man zwei Folgen $\left(x_n\right)$ und $\left(\tilde{x}_n\right)$ mit den Eigenschaften (2.5) angibt, für die die Folgen $\left(f\left(x_n\right)\right)$ und $\left(f\left(\tilde{x}_n\right)\right)$ verschiedene Grenzwerte haben.
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Aufgabe 2.2: Führen Sie den Beweis in der soeben angedeuteten Weise durch!
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Beispiel 2.5: Abschließend betrachten wir noch die Funktion
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$$
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f(x)=(1+x)^{\frac{1}{x}} \quad(x>-1, x \neq 0) \text { für } \quad x \rightarrow 0 \text {. }
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$$
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Für die Folge $\left(x_n\right)$ mit $x_n=\frac{1}{n}$ gilt (vgl. Band 1, Abschnitt 10.7.)
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$$
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\lim _{n \rightarrow \infty} f\left(x_n\right)=\mathrm{e} .
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$$
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Ohne Beweis ${ }^1$ ) sei mitgeteilt, daß (2.6) sogar für jede Folge $\left(x_n\right)$ mit $x_n>-1, x_n \neq 0$ und $\lim _{n \rightarrow \infty} x_n=0$ gilt. Damit erhält man den für spätere Anwendungen wichtigen Grenzwert
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$$
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\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x}}=\mathrm{e}
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$$
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2.1.2. Die „ $\varepsilon-\delta$-Charakterisierung" des Grenzwertes
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Auf Grund der Anschauung wird man vermuten, daß man die Gleichung $\lim _{x \rightarrow x_0} f(x)=g$ auch folgendermaßen interpretieren kann:
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,,Der Abstand zwischen $f(x)$ und $g($ also $|f(x)-g|)$ ist beliebig klein, wenn nur der Abstand zwischen $x$ und $x_0$ hinreichend klein, aber von null verschieden ist." In geeigneter Präzisierung ist das der Inhalt des folgenden Satzes, den wir ohne Beweis angeben.
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Satz 2.1: Die Funktion $f$ sei (mindestens) in einer punktierten Umgebung der Stelle $x_0$
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S. 2.1 definiert. Genau dann gilt $\lim f(x)=g$, wenn $z u$ jeder (insbesondere jeder beliebig kleinen) Zahl $\varepsilon>0$ eine $\begin{gathered}\mathrm{Xahl} \quad \mathrm{Xax} \\ \text { Zahl }\end{gathered}=\delta(\varepsilon)>0$ existiert, so daß gilt
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$$
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|f(x)-g|<\varepsilon
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$$
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für alle $x$ mit
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$$
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0<\left|x-x_0\right|<\delta .
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$$
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Eine geometrische Deutung dieses Satzes gibt Bild 2.7. Mit den dort verwendeten Bezeichnungen bedeutet $\lim _{x \rightarrow x_0} f(x)=g$, daß zu jedem (noch so schmalen) „ $\varepsilon$-Streifen“
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${ }^1$ ) Beweise zu Teil 1 dieses Buches findet man, wenn nichts anderes gesagt ist, in [5] und [10].
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um $y=g$ ein „ $\delta$-Streifen" um $x=x_0$ existiert, so daß alle Punkte der Bildkurve von $f$, die in diesem " $\delta$-Streifen" - außer auf der Mittellinie $x=x_0{ }^1$ ) - liegen, auch dem vorgegebenen ,, $\varepsilon$-Streifen" angehören. Dabei ist offenbar $\delta$ im allgemeinen um so kleiner zu wählen, je kleiner $\varepsilon$ vorgegeben ist. Diesen Sachverhalt soll die Schreibweise $\delta=\delta(\varepsilon)$ zum Ausdruck bringen.
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Beispiel 2.6: Als Anwendung des Satzes wollen wir zeigen, daß
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$$
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\lim _{x \rightarrow x_0} \sqrt{x}=\sqrt{x_0} \quad\left(x_0>0\right)
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$$
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gilt (Bild 2.8). Es sei ein beliebiges $\varepsilon>0$ gegeben. Gemäß (2.8) ist $\left|\sqrt{x}-\sqrt{x_0}\right|$ abzuschätzen. Wir erweitern mit $\sqrt{x}+\sqrt{x_0}$ und erhalten
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$$
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\left|\sqrt{x}-\sqrt{x_0}\right|=\frac{\left|x-x_0\right|}{\sqrt{x}+\sqrt{x_0}} \leqq \frac{1}{\sqrt{x_0}}\left|x-x_0\right|<\varepsilon
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$$
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für alle $x \geqq 0$ mit $\left|x-x_0\right|<\sqrt{x_0} \varepsilon$. Daher setzen wir $\delta$ gleich der kleineren der beiden Zahlen $x_0$ und $\sqrt{x_0} \varepsilon$. Für alle $x$ mit $\left|x-x_0\right|<\delta$ gilt dann $x \geqq 0$ (warum?) und (2.11).
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2.2 Einseitige Grenzwerte
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Für die Existenz des Grenzwertes $\lim _{x \rightarrow x_0} \sqrt{x}$ ist die Voraussetzung $x_0>0$ wesentlich (s. (2.10)), denn für $x_0 \leqq 0$ gibt es keine punktierte Umgebung von $x_0$, in der die Funktion $f(x)=\sqrt{x}(x \geqq 0)$ definiert ist. Im Falle $x_0 \leqq 0$ existiert $\lim _{x \rightarrow x_0} \sqrt{x}$ also nicht. Der Stelle $x_0=0$ kann man sich aber immerhin noch „von rechts nähern“, ohne den Definitionsbereich von $f$ zu verlassen. Diese Überlegung führt zum Begriff der einseitigen Grenzwerte.
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D. 2.2 Definition 2.2: Die Funktion $f$ sei (mindestens) in einem Intervall ( $\left.x_0, x_0+c\right)^2$ ) $(c>0)$ definiert. Eine Zahl $g_{\mathrm{r}}$ heißt rechtsseitiger Grenzwert von $\boldsymbol{f}$ für $\boldsymbol{x}$ gegen $\boldsymbol{x}_0$, in Zeichen
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$$
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\left.\left.\lim _{x \rightarrow x_0+0} f(x)=g_r \quad \text { oder } \quad f(x) \rightarrow g_r \quad \text { für } \quad x \rightarrow x_0+0\right)^3\right),
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$$
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${ }^1$ ) Man beachte, daß $\left|x-x_0\right|>0$ äquivalent zu $x \neq x_0$ ist.
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${ }^2$ ) Ein solches Intervall nennt man auch punktierte rechtsseitige Umgebung von $x_0$.
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${ }^3$ ) Statt $\lim _{x \rightarrow 0+0} f(x)=g_r$ (bzw. $f(x) \rightarrow g_r$, für $x \rightarrow 0+0$ ) schreibt man kurz $\lim _{x \rightarrow+0} f(x)=g_r$ (bzw. $f(x) \rightarrow g_r$ für $x \rightarrow+0$ ).
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*/
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]
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]
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998
Band2/Grafiktest.typ
Normal file
998
Band2/Grafiktest.typ
Normal file
@@ -0,0 +1,998 @@
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// #import "@preview/cetz:0.4.0": canvas
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// #import "@preview/cetz-plot:0.1.2": plot
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// #canvas({
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// plot.plot(
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// size: (8, 8),
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// axis-style: "school-book", // nur Achsen, kein Rahmen[web:17]
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// x-min: -2, x-max: 2, // negativer und positiver x-Bereich[web:27]
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// y-min: -1, y-max: 3.5,
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// x-tick-step: none, // automatische Ticks aus[web:24]
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// y-tick-step: none,
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// // nur 1/2 und x auf der x-Achse:
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// x-ticks: (
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// (0.5, $1/2$),
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// (1.25, $x$),
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// ),
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// // z.B. ein paar y-Ticks, falls gewünscht:
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// y-ticks: (
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// (0.25, $1/4$),
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// (1.5625, $x^2$),
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// ),
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// shared-zero: false, // optional: 0 am Ursprung ausblenden[web:16]
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// // Parabel: y = x^2 auf [-2,2]
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// plot.add(
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// domain: (-3, 3),
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// (x) => x * x,
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// )
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// )
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// })
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// #import "@preview/cetz:0.4.2"
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// #import "@preview/cetz-plot:0.1.3": plot
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// #cetz.canvas({
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// //import cetz.plot
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// //import cetz.palette
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// import cetz.draw: *
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// plot.plot(
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// size: (6, 6),
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// x-tick-step: none,
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// y-tick-step: none,
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// axis-style: "school-book",
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// x-label: $x$,
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// y-label: $y$,
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// x-ticks: (
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// (0.5, $1/2$),
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// (1.25, $x$),
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// ),
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// y-ticks: (
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// (0.25, $1/4$),
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// (1.5625, $x^2$),
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// ),
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// {
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// let f = x => calc.pow(x, 2)
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// // Definition der Punkte
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// let x0 = 0.5
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// let y0 = f(x0)
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// let x1 = 1.25
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// let y1 = f(x1)
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// // Berechnung der Steigung m = (y1 - y0) / (x1 - x0)
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// let m = (y1 - y0) / (x1 - x0)
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// // Sekantenfunktion: s(x) = m * (x - x0) + y0
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// let s = x => m * (x - x0) + y0
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// // 1. Die Parabel
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// plot.add(f, domain: (-1.3, 1.8), label: $f(x) = x^2$)
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// // 2. Die Sekante (etwas weiter gezeichnet für die Optik)
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// plot.add(s, domain: (0, 2), style: (stroke: blue))
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// // 3. Die Punkte markieren
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// plot.add(((x0, y0),), mark: "o", label: $P_0$)
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// plot.add(((x1, y1),), mark: "o", label: $P$)
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// // 4. Hilfslinien (gestrichelt)
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// plot.add(((x0, 0), (x0, y0)), style: (stroke: (dash: "dashed", paint: gray, thickness: 0.5pt)))
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// plot.add(((x1, 0), (x1, y1)), style: (stroke: (dash: "dashed", paint: gray, thickness: 0.5pt)))
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// plot.add(((x0, y0), (x1, y0)), style: (stroke: ( paint: gray.darken(80%), thickness: 0.5pt)))
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// plot.add-anchor("pt1", (2,5))
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// }
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// )
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// // content("plot.x", [asdf])
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// })
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// #import "@preview/cetz:0.4.2"
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// #import "@preview/cetz-plot:0.1.2": plot
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// #cetz.canvas({
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// plot.plot(
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// size: (6, 6),
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// x-tick-step: 1,
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// y-tick-step: 1,
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// axis-style: "school-book",
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// x-label: $x$,
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// y-label: $y$,
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// {
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// let f = x => calc.pow(x, 2)
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// let x0 = 0.5
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// let x1 = 1.5
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// // 1. Die Parabel
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// plot.add(f, domain: (-1.3, 1.8), label: $f(x)$)
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// // 2. Die Sekante
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||||||
|
// let m = (f(x1) - f(x0)) / (x1 - x0)
|
||||||
|
// plot.add(x => m * (x - x0) + f(x0), domain: (0, 2), style: (stroke: blue))
|
||||||
|
|
||||||
|
// // 3. Hilfslinien
|
||||||
|
// plot.add(((x0, 0), (x0, f(x0))), style: (stroke: (dash: "dashed", paint: gray)))
|
||||||
|
// plot.add(((x1, 0), (x1, f(x1))), style: (stroke: (dash: "dashed", paint: gray)))
|
||||||
|
|
||||||
|
// // 4. Beschriftungen OHNE den "bounds" Fehler
|
||||||
|
// // Wir nutzen plot.add mit leeren Markern oder speziellen Labels
|
||||||
|
|
||||||
|
// // Punkte markieren und direkt beschriften
|
||||||
|
// plot.add(((x0, f(x0)),), mark: "o", label: $P_0$)
|
||||||
|
// plot.add(((x1, f(x1)),), mark: "o", label: $P$)
|
||||||
|
|
||||||
|
// // Beschriftung an der x-Achse
|
||||||
|
// // Trick: Wir addieren einen "unsichtbaren" Punkt an der Achse mit Label
|
||||||
|
// plot.add(((x0, 0),), label: $x_0$, mark: none)
|
||||||
|
// plot.add(((x1, 0),), label: $x$, mark: none)
|
||||||
|
// }
|
||||||
|
// )
|
||||||
|
// })
|
||||||
|
|
||||||
|
// #import "@preview/cetz:0.4.2"
|
||||||
|
// #import "@preview/cetz-plot:0.1.2" as cetz_plot
|
||||||
|
|
||||||
|
// #cetz.canvas({
|
||||||
|
// let cp = cetz_plot.plot
|
||||||
|
|
||||||
|
// let x0 = 0.5
|
||||||
|
// let x1 = 1.5
|
||||||
|
// let f = x => calc.pow(x, 2)
|
||||||
|
// let y0 = f(x0)
|
||||||
|
// let y1 = f(x1)
|
||||||
|
|
||||||
|
// cp.plot(
|
||||||
|
// size: (6, 6),
|
||||||
|
// x-tick-step: 1,
|
||||||
|
// y-tick-step: 1,
|
||||||
|
// axis-style: "school-book",
|
||||||
|
// x-label: $x$,
|
||||||
|
// y-label: $y$,
|
||||||
|
// {
|
||||||
|
// // 1. Die Graphen
|
||||||
|
// cp.add(f, domain: (-1.3, 1.8))
|
||||||
|
|
||||||
|
// let m = (y1 - y0) / (x1 - x0)
|
||||||
|
// cp.add(x => m * (x - x0) + y0, domain: (-0.2, 2.2), style: (stroke: blue))
|
||||||
|
|
||||||
|
// // 2. Hilfslinien
|
||||||
|
// cp.add(((x0, 0), (x0, y0)), style: (stroke: (dash: "dashed", paint: gray)))
|
||||||
|
// cp.add(((x1, 0), (x1, y1)), style: (stroke: (dash: "dashed", paint: gray)))
|
||||||
|
|
||||||
|
// // 3. Punkte P0 und P
|
||||||
|
// cp.add(((x0, y0),), mark: "o")
|
||||||
|
// cp.add(((x1, y1),), mark: "o")
|
||||||
|
|
||||||
|
// // 4. BESCHRIFTUNG (Der Trick: add mit mark: none)
|
||||||
|
// // Wir setzen die Labels direkt an die Koordinaten
|
||||||
|
|
||||||
|
// // x-Achse Beschriftung (leicht unter y=0 verschoben mit 'label-offset')
|
||||||
|
// cp.add(((x0, 0),), label: $x_0$, mark: none)
|
||||||
|
// cp.add(((x1, 0),), label: $x$, mark: none)
|
||||||
|
|
||||||
|
// // Punkte beschriften
|
||||||
|
// cp.add(((x0 - 0.1, y0 + 0.3),), label: $P_0$, mark: none)
|
||||||
|
// cp.add(((x1 + 0.2, y1),), label: $P$, mark: none)
|
||||||
|
|
||||||
|
// // Funktionsnamen
|
||||||
|
// cp.add(((1.6, 3.2),), label: $f(x)$, mark: none)
|
||||||
|
// }
|
||||||
|
// )
|
||||||
|
// })
|
||||||
|
|
||||||
|
// #import "@preview/cetz:0.4.2"
|
||||||
|
// #import "@preview/cetz-plot:0.1.2" as cetz_plot
|
||||||
|
|
||||||
|
// #cetz.canvas({
|
||||||
|
// import cetz.draw: *
|
||||||
|
// let cp = cetz_plot.plot
|
||||||
|
|
||||||
|
// // 1. Parameter festlegen
|
||||||
|
// let (w, h) = (6, 6) // Größe des Plots
|
||||||
|
// let x-min = -1.5
|
||||||
|
// let x-max = 2.5
|
||||||
|
// let y-min = -1.0
|
||||||
|
// let y-max = 4.0
|
||||||
|
|
||||||
|
// let x0 = 0.5
|
||||||
|
// let x1 = 1.5
|
||||||
|
// let f = x => calc.pow(x, 2)
|
||||||
|
// let y0 = f(x0)
|
||||||
|
// let y1 = f(x1)
|
||||||
|
|
||||||
|
// // 2. Den Plot zeichnen (als Basis)
|
||||||
|
// cp.plot(
|
||||||
|
// size: (w, h),
|
||||||
|
// x-tick-step: 1,
|
||||||
|
// y-tick-step: 1,
|
||||||
|
// axis-style: "school-book",
|
||||||
|
// x-label: $x$,
|
||||||
|
// y-label: $y$,
|
||||||
|
// x-domain: (x-min, x-max),
|
||||||
|
// y-domain: (y-min, y-max),
|
||||||
|
// name: "p",
|
||||||
|
// {
|
||||||
|
// cp.add(f, domain: (x-min, x-max))
|
||||||
|
|
||||||
|
// let m = (y1 - y0) / (x1 - x0)
|
||||||
|
// cp.add(x => m * (x - x0) + y0, domain: (x-min, x-max), style: (stroke: blue))
|
||||||
|
|
||||||
|
// cp.add(((x0, 0), (x0, y0)), style: (stroke: (dash: "dashed", paint: gray)))
|
||||||
|
// cp.add(((x1, 0), (x1, y1)), style: (stroke: (dash: "dashed", paint: gray)))
|
||||||
|
|
||||||
|
// cp.add(((x0, y0),), mark: "o")
|
||||||
|
// cp.add(((x1, y1),), mark: "o")
|
||||||
|
// }
|
||||||
|
// )
|
||||||
|
|
||||||
|
// // 3. DER FIX: Manuelle Beschriftung über das "p.origin"
|
||||||
|
// // Wir nutzen die Größe des Plots, um die Koordinaten selbst zu setzen.
|
||||||
|
// // Da der Ursprung im school-book Stil bei (0,0) liegt:
|
||||||
|
// let plot-coords(x, y) = {
|
||||||
|
// let px = (x / (x-max - x-min)) * w
|
||||||
|
// let py = (y / (y-max - y-min)) * h
|
||||||
|
// return (rel: (px, py), to: "p.origin")
|
||||||
|
// }
|
||||||
|
|
||||||
|
// // Beschriftungen (Diese liegen nun außerhalb des Plot-Berechnungs-Logik)
|
||||||
|
// content(plot-coords(x0, -0.4), $x_0$)
|
||||||
|
// content(plot-coords(x1, -0.4), $x$)
|
||||||
|
// content(plot-coords(x0 - 0.3, y0 + 0.3), $P_0$)
|
||||||
|
// content(plot-coords(x1 + 0.3, y1), $P$)
|
||||||
|
// content(plot-coords(1.6, 3.2), $f(x)$)
|
||||||
|
// })
|
||||||
|
|
||||||
|
|
||||||
|
// #import "@preview/cetz:0.4.2"
|
||||||
|
|
||||||
|
// #cetz.canvas({
|
||||||
|
// import cetz.draw: *
|
||||||
|
|
||||||
|
// // 1. Koordinatensystem manuell skalieren (1 Einheit = 1.5cm)
|
||||||
|
// scale(1.5)
|
||||||
|
|
||||||
|
// // Parameter
|
||||||
|
// let x0 = 0.5
|
||||||
|
// let x1 = 1.5
|
||||||
|
// let f(x) = calc.pow(x, 2)
|
||||||
|
// let y0 = f(x0)
|
||||||
|
// let y1 = f(x1)
|
||||||
|
|
||||||
|
// // 2. Achsen zeichnen (School-Book)
|
||||||
|
// line((-1.5, 0), (2.5, 0), mark: (end: ">"), name: "xaxis")
|
||||||
|
// line((0, -1), (0, 4), mark: (end: ">"), name: "yaxis")
|
||||||
|
|
||||||
|
// content((2.5, -0.3), $x$)
|
||||||
|
// content((-0.3, 4), $y$)
|
||||||
|
|
||||||
|
// // 3. Parabel zeichnen (Sampling-Methode: Sicherster Weg ohne Extra-Module)
|
||||||
|
// let points = ()
|
||||||
|
// for i in range(-13, 19) {
|
||||||
|
// let x = i / 10
|
||||||
|
// points.push((x, f(x)))
|
||||||
|
// }
|
||||||
|
// line(..points, stroke: black)
|
||||||
|
|
||||||
|
// // 4. Sekante zeichnen
|
||||||
|
// let m = (y1 - y0) / (x1 - x0)
|
||||||
|
// // Gerade: y = m*(x - x0) + y0
|
||||||
|
// line((-0.5, m * (-0.5 - x0) + y0), (2.2, m * (2.2 - x0) + y0), stroke: blue)
|
||||||
|
|
||||||
|
// // 5. Hilfslinien & Punkte
|
||||||
|
// line((x0, 0), (x0, y0), stroke: (dash: "dashed", paint: gray))
|
||||||
|
// line((x1, 0), (x1, y1), stroke: (dash: "dashed", paint: gray))
|
||||||
|
|
||||||
|
// circle((x0, y0), radius: 0.05, fill: black)
|
||||||
|
// circle((x1, y1), radius: 0.05, fill: black)
|
||||||
|
|
||||||
|
// // 6. BESCHRIFTUNGEN (Direkt an den Objekten)
|
||||||
|
// content((x0, -0.4), $x_0$)
|
||||||
|
// content((x1, -0.4), $x$)
|
||||||
|
|
||||||
|
// content((x0 - 0.3, y0 + 0.2), $P_0$)
|
||||||
|
// content((x1 + 0.3, y1), $P$)
|
||||||
|
|
||||||
|
// content((1.8, f(1.8)), $f(x)$, anchor: "west", padding: .1)
|
||||||
|
// content((2.1, 3.5), [Sekante], fill: white, padding: .1)
|
||||||
|
// })
|
||||||
|
|
||||||
|
|
||||||
|
// #import "@preview/cetz:0.4.2"
|
||||||
|
|
||||||
|
// #cetz.canvas({
|
||||||
|
// import cetz.draw: *
|
||||||
|
|
||||||
|
// // 1. Koordinatensystem manuell skalieren (1 Einheit = 1.5cm)
|
||||||
|
// scale(1.5)
|
||||||
|
|
||||||
|
// // Parameter
|
||||||
|
// let x0 = 0.5
|
||||||
|
// let x1 = 1.5
|
||||||
|
// let f(x) = calc.pow(x, 2)
|
||||||
|
// let y0 = f(x0)
|
||||||
|
// let y1 = f(x1)
|
||||||
|
|
||||||
|
// // 2. Achsen zeichnen (School-Book)
|
||||||
|
// line((-1.5, 0), (2.5, 0), mark: (end: ">"), name: "xaxis")
|
||||||
|
// line((0, -1), (0, 4), mark: (end: ">"), name: "yaxis")
|
||||||
|
|
||||||
|
// content((2.5, -0.3), $x$)
|
||||||
|
// content((-0.3, 4), $y$)
|
||||||
|
|
||||||
|
// // 3. Parabel zeichnen (Sampling-Methode: Sicherster Weg ohne Extra-Module)
|
||||||
|
// let points = ()
|
||||||
|
// for i in range(-13, 19) {
|
||||||
|
// let x = i / 10
|
||||||
|
// points.push((x, f(x)))
|
||||||
|
// }
|
||||||
|
// line(..points, stroke: black)
|
||||||
|
|
||||||
|
// // 4. Sekante zeichnen
|
||||||
|
// let m = (y1 - y0) / (x1 - x0)
|
||||||
|
// // Gerade: y = m*(x - x0) + y0
|
||||||
|
// line((-0.5, m * (-0.5 - x0) + y0), (2.2, m * (2.2 - x0) + y0), stroke: blue)
|
||||||
|
|
||||||
|
// // 5. Hilfslinien & Punkte
|
||||||
|
// line((x0, 0), (x0, y0), stroke: (dash: "dashed", paint: gray))
|
||||||
|
// line((x1, 0), (x1, y1), stroke: (dash: "dashed", paint: gray))
|
||||||
|
|
||||||
|
// circle((x0, y0), radius: 0.05, fill: black)
|
||||||
|
// circle((x1, y1), radius: 0.05, fill: black)
|
||||||
|
|
||||||
|
// // 6. BESCHRIFTUNGEN (Direkt an den Objekten)
|
||||||
|
// content((x0, -0.4), $x_0$)
|
||||||
|
// content((x1, -0.4), $x$)
|
||||||
|
|
||||||
|
// content((x0 - 0.3, y0 + 0.2), $P_0$)
|
||||||
|
// content((x1 + 0.3, y1), $P$)
|
||||||
|
|
||||||
|
// content((1.8, f(1.8)), $f(x)$, anchor: "west", padding: .1)
|
||||||
|
// content((2.1, 3.5), [Sekante], fill: white, padding: .1)
|
||||||
|
// })
|
||||||
|
|
||||||
|
|
||||||
|
// #import "@preview/cetz:0.4.2"
|
||||||
|
|
||||||
|
// #align(center,
|
||||||
|
// cetz.canvas({
|
||||||
|
// // import cetz.plot
|
||||||
|
// import cetz.palette
|
||||||
|
// import cetz.draw: *
|
||||||
|
|
||||||
|
// plot.plot(size: (5,5),
|
||||||
|
// name: "plot",
|
||||||
|
// x-tick-step: 2,
|
||||||
|
// x-minor-tick-step: 1,
|
||||||
|
// y-tick-step: 6,
|
||||||
|
// y-minor-tick-step: 1,
|
||||||
|
// axis-style: "school-book", {
|
||||||
|
// plot.add(domain: (-2.5, 2.5),
|
||||||
|
// x => (4-calc.pow(x, 2)),
|
||||||
|
// style: palette.tango-light)
|
||||||
|
// plot.add(domain: (-2.5, 2.5),
|
||||||
|
// x => (4-2*x),
|
||||||
|
// style: palette.tango-light)
|
||||||
|
// plot.add-fill-between(domain: (0, 2),
|
||||||
|
// x => (4-calc.pow(x, 2)),
|
||||||
|
// x => (4-2*x),
|
||||||
|
// style: palette.tango-light)
|
||||||
|
// plot.add(((0,4), (2,0)),
|
||||||
|
// mark: "o",
|
||||||
|
// mark-style: (stroke: none, fill: black),
|
||||||
|
// style: (stroke: none))
|
||||||
|
// plot.add-anchor("pt1", (2,5))
|
||||||
|
// plot.add-anchor("pt2", (-2,5.5))
|
||||||
|
// plot.add-anchor("parab", (1, 4-calc.pow(1, 2)+0.2))
|
||||||
|
// plot.add-anchor("rline", (-1.5, 4-2*(-1.5)-0.2))
|
||||||
|
// })
|
||||||
|
|
||||||
|
// content("plot.pt1", [$y=4-2x^2$], anchor: "south", name: "prb")
|
||||||
|
// line("plot.parab", "prb", mark: (start: ">"))
|
||||||
|
|
||||||
|
// content("plot.pt2", [$y=4-2x$], anchor: "north", name: "curve2")
|
||||||
|
// line("plot.rline", "curve2", mark: (start: ">"))
|
||||||
|
|
||||||
|
// })
|
||||||
|
// )
|
||||||
|
|
||||||
|
// #align(center,
|
||||||
|
// cetz.canvas({
|
||||||
|
// // import cetz.plot
|
||||||
|
// // import cetz.palette
|
||||||
|
// import cetz.draw: *
|
||||||
|
|
||||||
|
// plot.plot(
|
||||||
|
// size: (6,6),
|
||||||
|
// name: "plot",
|
||||||
|
// x-tick-step: none,
|
||||||
|
// y-tick-step: none,
|
||||||
|
// x-label: $x$,
|
||||||
|
// y-label: $y$,
|
||||||
|
// x-ticks: ((0.5, $1/2$),(1.25, $x$),),
|
||||||
|
// y-ticks: ((0.25, $1/4$),(1.5625, $x^2$),),
|
||||||
|
// axis-style: "school-book",
|
||||||
|
|
||||||
|
// plot.add(
|
||||||
|
// domain: (-1.3, 1.8),
|
||||||
|
|
||||||
|
// x => (calc.pow(x, 2)),
|
||||||
|
|
||||||
|
// )
|
||||||
|
// )}))
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
// #import "@preview/cetz:0.4.2"
|
||||||
|
// #import "@preview/cetz-plot:0.1.3": plot
|
||||||
|
|
||||||
|
// #cetz.canvas({
|
||||||
|
// //import cetz.plot
|
||||||
|
// //import cetz.palette
|
||||||
|
// import cetz.draw: *
|
||||||
|
// plot.plot(
|
||||||
|
// size: (6, 6),
|
||||||
|
// x-tick-step: none,
|
||||||
|
// y-tick-step: none,
|
||||||
|
// axis-style: "school-book",
|
||||||
|
// x-label: $x$,
|
||||||
|
// y-label: $y$,
|
||||||
|
// x-ticks: (
|
||||||
|
// (0.5, $1/2$),
|
||||||
|
// (1.25, $x$),
|
||||||
|
// ),
|
||||||
|
// y-ticks: (
|
||||||
|
// (0.25, $1/4$),
|
||||||
|
// (1.5625, $x^2$),
|
||||||
|
// ),
|
||||||
|
// {
|
||||||
|
// let f = x => calc.pow(x, 2)
|
||||||
|
|
||||||
|
// // Definition der Punkte
|
||||||
|
// let x0 = 0.5
|
||||||
|
// let y0 = f(x0)
|
||||||
|
// let x1 = 1.25
|
||||||
|
// let y1 = f(x1)
|
||||||
|
|
||||||
|
// // Berechnung der Steigung m = (y1 - y0) / (x1 - x0)
|
||||||
|
// let m = (y1 - y0) / (x1 - x0)
|
||||||
|
// // Sekantenfunktion: s(x) = m * (x - x0) + y0
|
||||||
|
// let s = x => m * (x - x0) + y0
|
||||||
|
|
||||||
|
// // 1. Die Parabel
|
||||||
|
// plot.add(f, domain: (-1.3, 1.8), label: $f(x) = x^2$)
|
||||||
|
|
||||||
|
// // 2. Die Sekante (etwas weiter gezeichnet für die Optik)
|
||||||
|
// plot.add(s, domain: (0, 2), style: (stroke: blue))
|
||||||
|
|
||||||
|
// // 3. Die Punkte markieren
|
||||||
|
// plot.add(((x0, y0),), mark: "o", label: $P_0$)
|
||||||
|
// plot.add(((x1, y1),), mark: "o", label: $P$)
|
||||||
|
|
||||||
|
// // 4. Hilfslinien (gestrichelt)
|
||||||
|
// plot.add(((x0, 0), (x0, y0)), style: (stroke: (dash: "dashed", paint: gray, thickness: 0.5pt)))
|
||||||
|
// plot.add(((x1, 0), (x1, y1)), style: (stroke: (dash: "dashed", paint: gray, thickness: 0.5pt)))
|
||||||
|
|
||||||
|
// plot.add(((x0, y0), (x1, y0)), style: (stroke: ( paint: gray.darken(80%), thickness: 0.5pt)))
|
||||||
|
|
||||||
|
|
||||||
|
// plot.add-anchor("pt1", (2,5))
|
||||||
|
// }
|
||||||
|
|
||||||
|
// )
|
||||||
|
// // content("plot.x", [asdf])
|
||||||
|
// })
|
||||||
|
|
||||||
|
|
||||||
|
#import "@preview/cetz:0.4.2"
|
||||||
|
#import "@preview/cetz-plot:0.1.3": plot
|
||||||
|
|
||||||
|
#cetz.canvas({
|
||||||
|
import cetz.draw: *
|
||||||
|
plot.plot(
|
||||||
|
//definitionen
|
||||||
|
{
|
||||||
|
//berechnungen und plot
|
||||||
|
}
|
||||||
|
)
|
||||||
|
})
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
#import "@preview/cetz:0.4.2"
|
||||||
|
#import "@preview/cetz-plot:0.1.3": plot
|
||||||
|
|
||||||
|
#cetz.canvas(length: 1.25cm,{
|
||||||
|
import cetz.draw: *
|
||||||
|
set-style(
|
||||||
|
axes: (
|
||||||
|
// Basisstil beibehalten
|
||||||
|
stroke: (thickness: 0.5pt),
|
||||||
|
// x-Achse: stealth-Pfeil am Ende
|
||||||
|
x: (mark: (end: "stealth", fill: black)),
|
||||||
|
// y-Achse: stealth-Pfeil am Ende
|
||||||
|
y: (mark: (end: "stealth", fill: black)),
|
||||||
|
),
|
||||||
|
)
|
||||||
|
plot.plot(
|
||||||
|
//definitionen
|
||||||
|
name: "plot",
|
||||||
|
size: (6, 6),
|
||||||
|
x-tick-step: none,
|
||||||
|
y-tick-step: none,
|
||||||
|
axis-style: "school-book",
|
||||||
|
x-label: $x$,
|
||||||
|
y-label: $y$,
|
||||||
|
x-ticks: ((0.5, $1/2$),(1.25, $x$),),
|
||||||
|
y-ticks: ((0.25, $1/4$),(1.5625, $x^2$),),
|
||||||
|
{
|
||||||
|
//berechnungen und plot
|
||||||
|
let f = x => calc.pow(x, 2)
|
||||||
|
let x0 = 0.5
|
||||||
|
let y0 = f(x0)
|
||||||
|
let x1 = 1.25
|
||||||
|
let y1 = f(x1)
|
||||||
|
// Berechnung der Steigung m = (y1 - y0) / (x1 - x0)
|
||||||
|
let m = (y1 - y0) / (x1 - x0)
|
||||||
|
|
||||||
|
|
||||||
|
// Sekantenfunktion: s(x) = m * (x - x0) + y0
|
||||||
|
let s = x => m * (x - x0) + y0
|
||||||
|
|
||||||
|
let x1_1 = x1 + 0.1
|
||||||
|
let x0_1 = x0 - 0.2
|
||||||
|
let y0_1 = y0 + 0.2
|
||||||
|
let x2 = ((x1 - x0)/2) + x0
|
||||||
|
let y3 = ((y1 - y0)/2) +y0
|
||||||
|
|
||||||
|
plot.add(x => calc.pow(x, 2), domain: (-1.4, 1.8))
|
||||||
|
//Sekante
|
||||||
|
plot.add(s, domain: (0.3, 1.7), style: (stroke: red))
|
||||||
|
plot.add-anchor("pt1", (-1.4,y1))
|
||||||
|
plot.add-anchor("P", (x1_1,y1))
|
||||||
|
plot.add-anchor("P0", (x0_1,y0_1))
|
||||||
|
|
||||||
|
plot.add-anchor("F1", (x2,0.14))
|
||||||
|
plot.add-anchor("F2", (x1+0.1,y3))
|
||||||
|
|
||||||
|
|
||||||
|
plot.add(((x0, 0), (x0, y0)), style: (stroke: (dash: "dashed", paint: gray, thickness: 0.75pt)))
|
||||||
|
plot.add(((x1, 0), (x1, y1)), style: (stroke: (dash: "dashed", paint: gray, thickness: 0.75pt)))
|
||||||
|
plot.add(((0, y1), (x1, y1)), style: (stroke: (dash: "dashed", paint: gray, thickness: 0.75pt)))
|
||||||
|
plot.add(((0, y0), (x0, y0)), style: (stroke: (dash: "dashed", paint: gray, thickness: 0.75pt)))
|
||||||
|
|
||||||
|
plot.add(((x0, y0), (x1, y0)), style: (stroke: ( paint: gray.darken(80%), thickness: 0.5pt)))
|
||||||
|
plot.add(((x1, y0), (x1, y1)), style: (stroke: ( paint: gray.darken(80%), thickness: 0.5pt)))
|
||||||
|
|
||||||
|
plot.add(((x0, y0),), mark: "o")
|
||||||
|
plot.add(((x1, y1),), mark: "o")
|
||||||
|
}
|
||||||
|
)
|
||||||
|
//Der Plot muss einen Namen haben
|
||||||
|
content("plot.pt1", text(0.85em)[$y=x^2$], anchor: "east", name: "pt")
|
||||||
|
content("plot.P", text(0.75em)[$P$], anchor: "west", name: "p")
|
||||||
|
content("plot.P0", text(0.75em)[$P_0$], anchor: "west", name: "p0")
|
||||||
|
content("plot.F1", text(0.75em)[$x- 1/2$], anchor: "center", name: "f1")
|
||||||
|
content("plot.F2", text(0.75em)[$x^2- 1/4$], anchor: "west", name: "f2")
|
||||||
|
})
|
||||||
|
|
||||||
|
#import "@preview/cetz:0.4.2"
|
||||||
|
#import "@preview/cetz-plot:0.1.3": plot
|
||||||
|
|
||||||
|
#cetz.canvas({
|
||||||
|
|
||||||
|
import cetz.draw: *
|
||||||
|
set-style(
|
||||||
|
axes: (
|
||||||
|
// Basisstil beibehalten
|
||||||
|
stroke: (thickness: 0.5pt),
|
||||||
|
// x-Achse: stealth-Pfeil am Ende
|
||||||
|
x: (mark: (end: "stealth", fill: black)),
|
||||||
|
// y-Achse: stealth-Pfeil am Ende
|
||||||
|
y: (mark: (end: "stealth", fill: black)),
|
||||||
|
),
|
||||||
|
)
|
||||||
|
plot.plot(
|
||||||
|
name: "plot",
|
||||||
|
size: (5, 5),
|
||||||
|
x-tick-step: none,
|
||||||
|
y-tick-step: none,
|
||||||
|
axis-style: "school-book",
|
||||||
|
x-label: $x$,
|
||||||
|
y-label: $y$,
|
||||||
|
x-ticks: ((0.5, $1/2$),),
|
||||||
|
y-ticks: ((1, $1$),),
|
||||||
|
|
||||||
|
//definitionen
|
||||||
|
{
|
||||||
|
let f = x => if x != 0.5 { (calc.pow(x, 2) - 0.25) / (x - 0.5) } else { none }
|
||||||
|
let x0 = 0.5
|
||||||
|
let y0 = 1
|
||||||
|
plot.add(f, domain: (-0.6, 0.75), style: (stroke: blue))
|
||||||
|
|
||||||
|
plot.add(((x0, 0), (x0, y0)), style: (stroke: (dash: "dashed", paint: gray, thickness: 0.75pt)))
|
||||||
|
|
||||||
|
plot.add(((0, y0), (x0, y0)), style: (stroke: (dash: "dashed", paint: gray, thickness: 0.75pt)))
|
||||||
|
|
||||||
|
plot.add(((x0, y0),), mark: "o")
|
||||||
|
|
||||||
|
plot.add-anchor("F1", (-.7,0.5))
|
||||||
|
plot.add-anchor("F2", (-.7,0.35))
|
||||||
|
}
|
||||||
|
)
|
||||||
|
content("plot.F1", text(0.85em)[$y=(x^2-1/4)/(x-1/2)$], anchor: "west", name: "pt")
|
||||||
|
content("plot.F2", text(0.85em)[$(x eq.not 1/2)$], anchor: "west", name: "pt")
|
||||||
|
})
|
||||||
|
|
||||||
|
#import "@preview/cetz:0.4.2"
|
||||||
|
#import "@preview/cetz-plot:0.1.3": plot
|
||||||
|
|
||||||
|
#cetz.canvas({
|
||||||
|
import cetz.draw: *
|
||||||
|
plot.plot(
|
||||||
|
//definitionen
|
||||||
|
{
|
||||||
|
//berechnungen und plot
|
||||||
|
}
|
||||||
|
)
|
||||||
|
})
|
||||||
|
|
||||||
|
#import "@preview/cetz:0.4.1"
|
||||||
|
#import "@preview/cetz-plot:0.1.2"
|
||||||
|
|
||||||
|
#cetz.canvas({
|
||||||
|
import cetz.draw: *
|
||||||
|
import cetz-plot: *
|
||||||
|
|
||||||
|
let f = x => (calc.pow(x,3)+0.5)
|
||||||
|
let x1 = 0.35
|
||||||
|
let x3 = 0.5
|
||||||
|
let x0 = 0.7
|
||||||
|
let x2 = 0.8
|
||||||
|
let y1 = f(x1)
|
||||||
|
|
||||||
|
|
||||||
|
plot.plot(size: (6,7), name: "plot", axis-style: "school-book", x-tick-step:none, y-tick-step:none, x-label: $x$,
|
||||||
|
y-label: $y$,
|
||||||
|
x-ticks: ((x1, $x_1$),(x3, $x_3$),(x0, $x_0$),(x2, $x_2$),),
|
||||||
|
y-ticks: ((f(x1), $f(x_1)$),(f(x2), $f(x_2)$),(f(x0), $g$),(f(x3), $f(x_3)$)), {
|
||||||
|
plot.add(domain: (0.25, 1), f, style: (stroke: red))
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
// Hilfslinie damit Koordinatensystem nicht verrutscht
|
||||||
|
plot.add(((0, 0), (0, 0.1)), style: (stroke: none))
|
||||||
|
//circle((x1, y1), radius: 0.05, fill: black)
|
||||||
|
|
||||||
|
|
||||||
|
plot.add(((x1, 0), (x1, f(x1))), style: (stroke: (dash: "dashed", paint: gray, thickness: 1pt)))
|
||||||
|
plot.add(((0, f(x1)), (x1, f(x1))), style: (stroke: (dash: "dashed", paint: gray, thickness: 1pt)))
|
||||||
|
|
||||||
|
plot.add(((x3, 0), (x3, f(x3))), style: (stroke: (dash: "dashed", paint: gray, thickness: 1pt)))
|
||||||
|
plot.add(((0, f(x3)), (x3, f(x3))), style: (stroke: (dash: "dashed", paint: gray, thickness: 1pt)))
|
||||||
|
|
||||||
|
plot.add(((x0, 0), (x0, f(x0))), style: (stroke: (dash: "dashed", paint: gray, thickness: 1pt)))
|
||||||
|
plot.add(((0, f(x0)), (x0, f(x0))), style: (stroke: (dash: "dashed", paint: gray, thickness: 1pt)))
|
||||||
|
|
||||||
|
plot.add(((x2, 0), (x2, f(x2))), style: (stroke: (dash: "dashed", paint: gray, thickness: 1pt)))
|
||||||
|
plot.add(((0, f(x2)), (x2, f(x2))), style: (stroke: (dash: "dashed", paint: gray, thickness: 1pt)))
|
||||||
|
|
||||||
|
plot.add(((x1, f(x1)),), mark: "o", mark-style:(stroke: (paint: black, thickness: 0.75pt), fill: black))
|
||||||
|
plot.add(((x3, f(x3)),), mark: "o", mark-style:(stroke: (paint: black, thickness: 0.75pt), fill: black))
|
||||||
|
plot.add(((x0, f(x0)),), mark: "o", mark-style:(stroke: (paint: black, thickness: 0.75pt), fill: white))
|
||||||
|
plot.add(((x2, f(x2)),), mark: "o", mark-style:(stroke: (paint: black, thickness: 0.75pt), fill: black))
|
||||||
|
|
||||||
|
plot.add-anchor("F", (x2 + 0.1,f(x2)))
|
||||||
|
})
|
||||||
|
|
||||||
|
content("plot.F", text(0.85em)[$y=x^2$], anchor: "west", name: "pt")
|
||||||
|
|
||||||
|
|
||||||
|
})
|
||||||
|
|
||||||
|
#import "@preview/cetz:0.4.2"
|
||||||
|
|
||||||
|
#align(center,
|
||||||
|
cetz.canvas({
|
||||||
|
import cetz-plot: *
|
||||||
|
import cetz.palette
|
||||||
|
import cetz.draw: *
|
||||||
|
|
||||||
|
plot.plot(size: (5,5),
|
||||||
|
name: "plot",
|
||||||
|
x-tick-step: 2,
|
||||||
|
x-minor-tick-step: 1,
|
||||||
|
y-tick-step: 2,
|
||||||
|
y-minor-tick-step: 1,
|
||||||
|
axis-style: "school-book", {
|
||||||
|
plot.add(domain: (-1, 2.5),
|
||||||
|
x => x - 1,
|
||||||
|
style: (color: red))
|
||||||
|
plot.add(domain: (-1, 2.5),
|
||||||
|
x => x - 1 + calc.exp(-x),
|
||||||
|
style: (color: yellow))
|
||||||
|
plot.add(domain: (-1, 2.5),
|
||||||
|
x => x - 1 - calc.exp(-x),
|
||||||
|
style: (color: red))
|
||||||
|
plot.add-anchor("pt1", (1,1.3))
|
||||||
|
plot.add-anchor("pt2", (1.8,-2))
|
||||||
|
plot.add-anchor("exp1", (1, calc.exp(-1)+0.2))
|
||||||
|
plot.add-anchor("exp2", (0.5, 0.5-1-calc.exp(-0.58)-0.2))
|
||||||
|
|
||||||
|
})
|
||||||
|
|
||||||
|
content("plot.pt1", [$y=t-1+e^(-t)$], anchor: "south", name: "prb")
|
||||||
|
line("plot.exp1", "prb", mark: (start: ">"))
|
||||||
|
|
||||||
|
content("plot.pt2", [$y=t-1-e^(-t)$], anchor: "north", name: "curve2")
|
||||||
|
line("plot.exp2", "curve2", mark: (start: ">"))
|
||||||
|
|
||||||
|
})
|
||||||
|
)
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
/////////////////////////////
|
||||||
|
|
||||||
|
#cetz.canvas(length: 1.25cm,{
|
||||||
|
import cetz.draw: *
|
||||||
|
set-style(
|
||||||
|
axes: (
|
||||||
|
// Basisstil beibehalten
|
||||||
|
stroke: (thickness: 0.5pt),
|
||||||
|
// x-Achse: stealth-Pfeil am Ende
|
||||||
|
x: (mark: (end: "stealth", fill: black)),
|
||||||
|
// y-Achse: stealth-Pfeil am Ende
|
||||||
|
y: (mark: (end: "stealth", fill: black)),
|
||||||
|
),
|
||||||
|
)
|
||||||
|
plot.plot(
|
||||||
|
//definitionen
|
||||||
|
name: "plot",
|
||||||
|
size: (6, 10),
|
||||||
|
x-tick-step: none,
|
||||||
|
y-tick-step: none,
|
||||||
|
axis-style: "school-book",
|
||||||
|
x-label: $x$,
|
||||||
|
y-label: $y$,
|
||||||
|
x-ticks: ((0.5, $1/2$),(1.25, $x$),),
|
||||||
|
y-ticks: ((0.25, $1/4$),(1.5625, $x^2$),),
|
||||||
|
{
|
||||||
|
//berechnungen und plot
|
||||||
|
//let f = x => calc.pow(x, 2)
|
||||||
|
//let g = x => calc.pow(x - 2,3)
|
||||||
|
//let h = x => -1 *calc.pow(0.1 * g(x) -2,2)
|
||||||
|
// -(0.1(x-2)^(3)-2)^(2)+6
|
||||||
|
let f = x => -1 * (calc.pow(0.1 * calc.pow((x - 2),3) -2,2)) - 0.2
|
||||||
|
let x0 = 0.5
|
||||||
|
let y0 = f(x0)
|
||||||
|
let x1 = 1.25
|
||||||
|
let y1 = f(x1)
|
||||||
|
// Berechnung der Steigung m = (y1 - y0) / (x1 - x0)
|
||||||
|
let m = (y1 - y0) / (x1 - x0)
|
||||||
|
|
||||||
|
|
||||||
|
// Sekantenfunktion: s(x) = m * (x - x0) + y0
|
||||||
|
let s = x => m * (x - x0) + y0
|
||||||
|
|
||||||
|
let x1_1 = x1 + 0.1
|
||||||
|
let x0_1 = x0 - 0.2
|
||||||
|
let y0_1 = y0 + 0.2
|
||||||
|
let x2 = ((x1 - x0)/2) + x0
|
||||||
|
let y3 = ((y1 - y0)/2) +y0
|
||||||
|
plot.add(f, samples: 300, domain: (1,3))
|
||||||
|
//plot.add(x => calc.pow(x, 2), domain: (0.5, 4))
|
||||||
|
//Sekante
|
||||||
|
|
||||||
|
plot.add-anchor("pt1", (-1.4,y1))
|
||||||
|
plot.add-anchor("P", (x1_1,y1))
|
||||||
|
plot.add-anchor("P0", (x0_1,y0_1))
|
||||||
|
|
||||||
|
plot.add-anchor("F1", (x2,0.14))
|
||||||
|
plot.add-anchor("F2", (x1+0.1,y3))
|
||||||
|
|
||||||
|
|
||||||
|
plot.add(((x0, 0), (x0, y0)), style: (stroke: (dash: "dashed", paint: gray, thickness: 0.75pt)))
|
||||||
|
plot.add(((x1, 0), (x1, y1)), style: (stroke: (dash: "dashed", paint: gray, thickness: 0.75pt)))
|
||||||
|
plot.add(((0, y1), (x1, y1)), style: (stroke: (dash: "dashed", paint: gray, thickness: 0.75pt)))
|
||||||
|
plot.add(((0, y0), (x0, y0)), style: (stroke: (dash: "dashed", paint: gray, thickness: 0.75pt)))
|
||||||
|
|
||||||
|
plot.add(((x0, y0), (x1, y0)), style: (stroke: ( paint: gray.darken(80%), thickness: 0.5pt)))
|
||||||
|
plot.add(((x1, y0), (x1, y1)), style: (stroke: ( paint: gray.darken(80%), thickness: 0.5pt)))
|
||||||
|
|
||||||
|
plot.add(((x0, y0),), mark: "o")
|
||||||
|
plot.add(((x1, y1),), mark: "o")
|
||||||
|
}
|
||||||
|
)
|
||||||
|
//Der Plot muss einen Namen haben
|
||||||
|
content("plot.pt1", text(0.85em)[$y=x^2$], anchor: "east", name: "pt")
|
||||||
|
content("plot.P", text(0.75em)[$P$], anchor: "west", name: "p")
|
||||||
|
content("plot.P0", text(0.75em)[$P_0$], anchor: "west", name: "p0")
|
||||||
|
content("plot.F1", text(0.75em)[$x- 1/2$], anchor: "center", name: "f1")
|
||||||
|
content("plot.F2", text(0.75em)[$x^2- 1/4$], anchor: "west", name: "f2")
|
||||||
|
})
|
||||||
|
|
||||||
|
#import "@preview/cetz:0.4.2": canvas
|
||||||
|
#import "@preview/cetz-plot:0.1.3": plot
|
||||||
|
|
||||||
|
#let theta = 45 * calc.pi / 180
|
||||||
|
#let c = calc.cos(theta)
|
||||||
|
#let s = calc.sin(theta)
|
||||||
|
|
||||||
|
#canvas({
|
||||||
|
import cetz.draw: *
|
||||||
|
|
||||||
|
plot.plot(
|
||||||
|
size: (12, 8),
|
||||||
|
axis-style: "school-book",
|
||||||
|
x-min: -8, x-max: 12,
|
||||||
|
y-min: -3, y-max: 9,
|
||||||
|
x-tick-step: 2,
|
||||||
|
y-tick-step: 1,
|
||||||
|
// Original
|
||||||
|
plot.add(x => {
|
||||||
|
let u = x - 2
|
||||||
|
calc.pow( -(0.1 * calc.pow(u, 3) - 2), 2) + 6
|
||||||
|
}, domain: (0, 4),
|
||||||
|
style: (stroke: (paint: blue, thickness: 1pt)), samples: 300),
|
||||||
|
// Rotierter Graph: Sampling über x_old und Rotation
|
||||||
|
plot.add(x_old => {
|
||||||
|
let u = x_old - 2
|
||||||
|
let y_old = calc.pow( -(0.1 * calc.pow(u, 3) - 2), 2) + 6
|
||||||
|
let x_new = x_old * c - y_old * s
|
||||||
|
let y_new = x_old * s + y_old * c
|
||||||
|
(x_new, y_new)
|
||||||
|
}, domain: (-1, 9),
|
||||||
|
style: (stroke: (paint: red, thickness: 3pt))),
|
||||||
|
)
|
||||||
|
})
|
||||||
|
|
||||||
|
|
||||||
|
#import "@preview/cetz:0.4.2": canvas
|
||||||
|
#import "@preview/cetz-plot:0.1.3": plot
|
||||||
|
|
||||||
|
#let theta = -45 * calc.pi / 180 // 45 Grad für deutlichen Effekt
|
||||||
|
#let c = calc.cos(theta)
|
||||||
|
#let s = calc.sin(theta)
|
||||||
|
|
||||||
|
#canvas({
|
||||||
|
import cetz.draw: *
|
||||||
|
|
||||||
|
plot.plot(
|
||||||
|
size: (14, 10),
|
||||||
|
axis-style: "school-book",
|
||||||
|
x-min: -10, x-max: 15,
|
||||||
|
y-min: -5, y-max: 12,
|
||||||
|
x-tick-step: 2,
|
||||||
|
y-tick-step: 2,
|
||||||
|
// Original
|
||||||
|
plot.add(x => {
|
||||||
|
let u = x - 2
|
||||||
|
calc.pow( -(0.1 * calc.pow(u, 3) - 2), 2) + 6
|
||||||
|
}, domain: (-3, 9),
|
||||||
|
style: (stroke: (paint: blue, thickness: 3pt))),
|
||||||
|
// Rotierter Graph: Erweiterte Domain für x_old
|
||||||
|
plot.add(x_old => {
|
||||||
|
let u = x_old - 2
|
||||||
|
let y_old = calc.pow( -(0.1 * calc.pow(u, 3) - 2), 2) + 6
|
||||||
|
x_old * c - y_old * s // Plotte x_new vs x_old (parametrisch projiziert)
|
||||||
|
}, domain: (-5, 11),
|
||||||
|
style: (stroke: (paint: red, thickness: 3pt))),
|
||||||
|
// Y_new vs x_old für vollständige Rotation (grün)
|
||||||
|
plot.add(x_old => {
|
||||||
|
let u = x_old - 2
|
||||||
|
let y_old = calc.pow( -(0.1 * calc.pow(u, 3) - 2), 2) + 6
|
||||||
|
x_old * s + y_old * c // y_new
|
||||||
|
}, domain: (-5, 11),
|
||||||
|
style: (stroke: (paint: green, thickness: 2pt))),
|
||||||
|
)
|
||||||
|
})
|
||||||
|
|
||||||
|
#import "@preview/cetz:0.4.2": canvas
|
||||||
|
#import "@preview/cetz-plot:0.1.3": plot
|
||||||
|
|
||||||
|
#let theta = 15deg
|
||||||
|
|
||||||
|
#let f = x => {
|
||||||
|
let u = x - 7
|
||||||
|
calc.pow(-(0.005 * calc.pow(u, 3) + 2), 2)
|
||||||
|
}
|
||||||
|
|
||||||
|
#canvas({
|
||||||
|
import cetz.draw: *
|
||||||
|
|
||||||
|
// Original
|
||||||
|
plot.plot(
|
||||||
|
size: (14, 10),
|
||||||
|
axis-style: "school-book",
|
||||||
|
x-min: -12, x-max: 16,
|
||||||
|
y-min: -8, y-max: 14,
|
||||||
|
x-tick-step: 4,
|
||||||
|
y-tick-step: 2,
|
||||||
|
{
|
||||||
|
plot.add(
|
||||||
|
domain: (-4, 12),
|
||||||
|
f,samples: 400,
|
||||||
|
style: (stroke: (paint: blue, thickness: 3pt)),
|
||||||
|
)
|
||||||
|
}
|
||||||
|
)
|
||||||
|
|
||||||
|
// Rotierter zweiter Graph, ohne zweite Achsen
|
||||||
|
group({
|
||||||
|
rotate(theta, origin: (0, 0))
|
||||||
|
plot.plot(
|
||||||
|
size: (14, 10),
|
||||||
|
axis-style: none,
|
||||||
|
x-min: -12, x-max: 16,
|
||||||
|
y-min: -8, y-max: 14,
|
||||||
|
{
|
||||||
|
plot.add(
|
||||||
|
domain: (-4, 12),
|
||||||
|
f, samples: 400,
|
||||||
|
style: (stroke: (paint: red, thickness: 3pt)),
|
||||||
|
)
|
||||||
|
}
|
||||||
|
)
|
||||||
|
})
|
||||||
|
})
|
||||||
|
|
||||||
|
//orig
|
||||||
|
|
||||||
|
#import "@preview/cetz:0.4.2": canvas
|
||||||
|
#import "@preview/cetz-plot:0.1.3": plot
|
||||||
|
|
||||||
|
#let theta = 45deg
|
||||||
|
|
||||||
|
#let f = x => {
|
||||||
|
let u = x - 2
|
||||||
|
calc.pow(-(0.1 * calc.pow(u, 3) - 2), 2) + 6
|
||||||
|
}
|
||||||
|
|
||||||
|
#canvas({
|
||||||
|
import cetz.draw: *
|
||||||
|
|
||||||
|
// Original
|
||||||
|
plot.plot(
|
||||||
|
size: (14, 10),
|
||||||
|
axis-style: "school-book",
|
||||||
|
x-min: -12, x-max: 16,
|
||||||
|
y-min: -8, y-max: 14,
|
||||||
|
x-tick-step: 4,
|
||||||
|
y-tick-step: 2,
|
||||||
|
{
|
||||||
|
plot.add(
|
||||||
|
domain: (-4, 12),
|
||||||
|
f,
|
||||||
|
style: (stroke: (paint: blue, thickness: 3pt)),
|
||||||
|
)
|
||||||
|
}
|
||||||
|
)
|
||||||
|
|
||||||
|
// Rotierter zweiter Graph, ohne zweite Achsen
|
||||||
|
group({
|
||||||
|
rotate(theta, origin: (0, 0))
|
||||||
|
plot.plot(
|
||||||
|
size: (14, 10),
|
||||||
|
axis-style: none,
|
||||||
|
x-min: -12, x-max: 16,
|
||||||
|
y-min: -8, y-max: 14,
|
||||||
|
{
|
||||||
|
plot.add(
|
||||||
|
domain: (-4, 12),
|
||||||
|
f,
|
||||||
|
style: (stroke: (paint: red, thickness: 3pt)),
|
||||||
|
)
|
||||||
|
}
|
||||||
|
)
|
||||||
|
})
|
||||||
|
})
|
||||||
Reference in New Issue
Block a user