Band 2 Grafiken bis 2.4
This commit is contained in:
584
Band2/Band2.typ
584
Band2/Band2.typ
@@ -1,6 +1,3 @@
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#import "@preview/cetz:0.4.2": * //Grafiken
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#import "@preview/fletcher:0.5.8" as fletcher: diagram, node, edge
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#import "@preview/marge:0.1.0": sidenote
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#import "@preview/itemize:0.2.0" as el
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@@ -9,21 +6,41 @@
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#import "deckblatt.typ": deckblatt
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//#import "abhaengigkeit.typ": abhaengigkeit
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#import "abhaengigkeit.typ": abhaengigkeit
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#let einruecken(abstand, inhalt) = pad(left: abstand, inhalt)
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#let def-counter = counter("definition")
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// Zähler bei jedem neuen Kapi#import "@preview/fletcher:0.5.8" as fletcher: diagram, node, edge
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#import "@preview/marge:0.1.0": sidenote
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#import "@preview/itemize:0.2.0" as el
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#import "@preview/eqalc:0.1.3": *
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#import "grafiken.typ": *
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#import "deckblatt.typ": deckblatt
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#import "abhaengigkeit.typ": abhaengigkeit
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#let einruecken(abstand, inhalt) = pad(left: abstand, inhalt)
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#let def-counter = counter("definition")
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#let bsp-counter = counter("besipiel")
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// Zähler bei jedem neuen Kapitel (Ebene 1) zurücksetzen
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#let def-counter = counter("definition")
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#let hrule = line(length: 100%)
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// Zähler bei jedem neuen Kapitel zurücksetzen
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#show heading.where(level: 1): it => {
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it
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def-counter.update(0)
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def-counter.update(1)
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bsp-counter.update(1)
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}
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#let definition(title: none, body, label: none) = context {
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@@ -46,12 +63,42 @@
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]
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}
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#let beispiel(title: none, body, label: none) = context {
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bsp-counter.step()
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let h1-num = counter(heading).at(here()).first()
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let b-num = bsp-counter.at(here()).first()
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let full-num = [#h1-num.#b-num]
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// Die Figure wird erstellt und das Label direkt angehängt
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[
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#figure(
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block(width: 100%, align(left)[
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#hrule
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*Beispiel #full-num*: #if title != none [(#title)] #body
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#hrule]),
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kind: "beispiel",
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supplement: [B.],
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numbering: _ => full-num,
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) #label
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]
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}
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// Zähler bei jedem neuen Kapitel (Ebene 1) zurücksetzen
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#show heading.where(level: 1): it => {
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it
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def-counter.update(1)
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}
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#show figure.caption: it => {
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it.supplement // "Abbildung"
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[ ] // Ein Leerzeichen statt des Doppelpunkts
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context it.counter.display(it.numbering)
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[ ] // Noch ein Leerzeichen vor dem Text
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it.body // Der eigentliche Text der Caption
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}
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#let sn-base = sidenote
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@@ -67,9 +114,6 @@
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]
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//#import "definitionen.typ": *
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#set text(
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font: "Lato", //Serifenlose Schrift
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size: 8.5pt,
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@@ -91,9 +135,13 @@
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#let meinContent = [*Band 7* #sym.dot.c _Grundlagen_]
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// Einfach die Funktion aufrufen
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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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width:162mm,
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height: 230mm,
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@@ -105,6 +153,7 @@
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leading: 0.55em,
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)
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#set page(
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//background: grid(columns: (1mm,) * 162, rows: (1mm,) * 230, stroke: 0.1mm),
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margin: (
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@@ -123,361 +172,7 @@
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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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#let mynode(pos, zahl, inhalt, name) = {
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node(
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pos,
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name: name,
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shape: rect, // 1. Zwingt Fletcher zum Rechteck
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corner-radius: 0pt, // 2. Verhindert abgerundete Ecken
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fill: white, // Optional: Hintergrund füllen
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stroke: 1.25pt,
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// 3. WICHTIG: Festlegen einer Mindestgröße, damit alle gleich starten
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// Anstatt extrude nutzen wir feste Maße, die sich aber am Gitter ausrichten
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width: 12em,
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height: 5.8em,
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{
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// Dein Layout-Block
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set align(top + right)
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text(2.5em, fill: gray.darken(90%))[#zahl]
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place(bottom + left, inhalt)
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}
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)}
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#let myedge(start, end, type, ) = {
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let sep = 0.15em
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edge(start, end, type)
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edge(start, end)
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}
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#diagram(
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spacing: (11mm, 7mm),
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//debug: 3,
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edge-stroke: 1.25pt,
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mark-scale: 55%,
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mynode((0,0),"",[Vorbereitungsband], <A0>),
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mynode((1,0),"1", [Grundlagen], <A1>),
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mynode((2,0), "13",[Lineare Algebra], <A13>),
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mynode((0,1),"3",[Unendliche Reihen], <A3>),
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mynode((1,1), "2",[Differential- und Integralrechnung], <A2>),
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mynode((2,1), "14",[Lineare Optimierung], <A14>),
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mynode((0,2), [7$#sub(text(0.5em)[1])$],[Gewöhnliche Differentialgleichungen], <A7_1>),
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mynode((1,2), "4",[Differentialrechnung mit mehreren Variablen], <A4>),
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mynode((2,2), "15",[Nichtlineare Optimierung], <A15>),
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mynode((0,3), [7$#sub(text(0.5em)[2])$],[Gewöhnliche Differentialgleichungen], <A7_2>),
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mynode((1,3.), "5",[Integralrechnung mit mehreren Variablen], <A5>),
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mynode((2,3), "16",[Optimale Prozesse und Systeme], <A16>),
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mynode((0,4), "8",[Partielle Differentialgleichungen], <A8>),
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mynode((1,4), "6",[Differentialgeometrie], <A6>),
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mynode((2,4), "17",[Wahrscheinlichkeitsrechnung, math.Statistik], <A17>),
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mynode((0,5), "9",[Komplexe Funktionen], <A9>),
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mynode((1,5), "10",[Operatorenrechnung], <A10>),
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mynode((2,5), [21$#sub(text(0.5em)[1])$],[Spieltheorie], <A21_1>),
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mynode((0,6), "12",[Spezielle Funktionen], <A12>),
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mynode((1,6), "11",[Tensoralgebra und -analysis], <A11>),
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mynode((2,6), [21$#sub(text(0.5em)[2])$],[Graphentheorie], <A21_2>),
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mynode((0,7), [18],[Numerische Methoden], <A18>),
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mynode((1,7), [20],[Simulation], <A20>),
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mynode((2,7), [19$#sub(text(0.5em)[1])$],[Stochastische Prozesse und Modelle], <A19_1>),
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mynode((0,8), [22],[Funktionalanalysis], <A22>),
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mynode((1,8), [23],[Symmetriegruppen], <A23>),
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mynode((2,8), [19$#sub(text(0.5em)[2])$],[Statistische Versuchsplanung], <A19_2>),
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myedge(<A0.east>,<A1.west>,"-|>"),
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myedge(<A1.south>,<A2.north>,"-|>"),
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myedge(<A2.west>,<A3.east>,"-|>"),
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myedge(<A13.south>,<A14.north>,"-|>"),
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myedge(<A2.south>,<A4.north>,"-|>"),
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myedge(<A4.south>,<A5.north>,"-|>"),
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myedge(<A5.south>,<A6.north>,"-|>"),
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myedge(<A14.south>,<A15.north>,"-|>"),
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myedge(<A7_1.south>,<A7_2.north>,"-|>"),
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myedge(<A7_2.south>,<A8.north>,"-|>"),
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myedge(<A8.south>,<A9.north>,"-|>"),
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myedge(<A9.south>,<A12.north>,"-|>"),
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myedge(<A12.south>,<A18.north>,"-|>"),
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myedge(<A10.south>,<A11.north>,"-|>"),
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myedge(<A15.south>,<A16.north>,"-|>"),
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myedge(<A17.south>,<A21_1.north>,"-|>"),
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myedge(<A19_1.south>,<A19_2.north>,"-|>"),
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// NEUES Edge: 1,3 (n3_1) → 3,2 (n2_3)
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// Verlauf: rechts von n3_1 nach rechts, runter auf Höhe von n2_3, dann links zur Node
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//9 → 22
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edge(
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vertices: (
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<A9.east>,
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(0, 5.2), // deutlich nach rechts → sichtbarer Knick
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(0.5, 5.2), // dann nach links
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(0.5, 8), // runter
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<A22.east>,
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),
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corner: right,
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corner-radius: 0pt, // abgerundete Ecken!
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stroke: orange + 1.25pt,
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marks: "-|>",
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shift: (1pt, 1pt),
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),
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//10 → 22
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edge(
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vertices: (
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<A10.west>,
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(1, 5.2), // deutlich nach rechts → sichtbarer Knick
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(0.5, 5.2), // dann nach links
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(0.5, 8), // runter
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<A22.east>,
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),
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corner: left,
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corner-radius: 0pt, // abgerundete Ecken!
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stroke: orange + 1.25pt,
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marks: "-|>",
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shift: (1pt, 1pt),
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),
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//3 → 7.2
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edge(
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vertices: (
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<A3.west>,
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(-0.85, 1), // deutlich nach rechts → sichtbarer Knick
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(-0.85, 1), // dann nach links
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(-0.85, 3), // runter
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<A7_2.west>,
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),
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corner: left,
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corner-radius: 0pt, // abgerundete Ecken!
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stroke: red + 1.25pt,
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marks: "-|>",
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shift: (1pt, 1pt),
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),
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//13 → 4
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edge(
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vertices: (
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<A13.west>,
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(2, 0.18), // deutlich nach rechts → sichtbarer Knick
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(1.5, 0.2), // dann nach links
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(1.5, 2), // runter
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<A4.east>,
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),
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corner: left,
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corner-radius: 0pt, // abgerundete Ecken!
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stroke: green + 1.25pt,
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marks: "-|>",
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shift: (2pt, 2pt),
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),
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//1 → 13
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edge(
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(rel: (0pt, 5mm), to: <A1.east>),
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(rel: (0.1, 0.)),
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(rel: (0pt, 5mm), to: <A13.west>),
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"-|>",
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corner-radius: 0pt ,
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stroke: green.darken(30%) + 1.25pt,
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),
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//5 → 8
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edge(
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(rel: (0pt, 0mm), to: <A5.west>),
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(rel: (-0.15, 0.)),
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(rel: (0.08,0), to: <A8.east>),
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(rel: (0.,0), to: <A8.east>),
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"-|>",
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corner-radius: 0pt ,
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stroke: 1.25pt + olive,
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),
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//7_1 → 6
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edge(
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vertices: (
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(rel: (0pt, -2pt), to: <A7_1.east>),
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(0.25, 2.2),
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(0.52, 2.2),
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(0.52,4),
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(rel: (0pt, 0pt), to: <A6.west>)
|
||||
),
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corner-radius: 0pt,
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||||
stroke: gray + 1.25pt,
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||||
marks: "-|>",
|
||||
),
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//4 → 7_1
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edge(
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||||
(rel: (0pt, 5mm), to: <A4.west>),
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||||
|
||||
(rel: (-0.1, 0.)),
|
||||
|
||||
(rel: (0pt, 5mm), to: <A7_1.east>),
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||||
"-|>",
|
||||
corner-radius: 0pt ,
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||||
stroke: 1.25pt ,
|
||||
),
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//9 → 10
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edge(
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||||
(rel: (0pt, 5mm), to: <A9.east>),
|
||||
|
||||
(rel: (0.1, 0.)),
|
||||
|
||||
(rel: (0pt, 5mm), to: <A10.west>),
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt ,
|
||||
),
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//5 → 17
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||||
edge(
|
||||
<A5.east>,
|
||||
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||||
(rel: (0.07, 0.)),
|
||||
(rel: (0, 0.5)),
|
||||
|
||||
(rel: (0pt, 10pt), to: <A17.north>),
|
||||
<A17.north>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: blue.lighten(20%) +1.25pt ,
|
||||
),
|
||||
//4 → 15
|
||||
edge(
|
||||
(rel: (0pt, -5mm), to: <A4.east>),
|
||||
|
||||
(rel: (0.1, 0.)),
|
||||
|
||||
(rel: (0pt, -5mm), to: <A15.west>),
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt ,
|
||||
),
|
||||
//8 → 16
|
||||
edge(
|
||||
(rel: (0.2,0), to: <A8.south>),
|
||||
(rel: (0., 0.1)),
|
||||
(rel: (1.3, 0.)),
|
||||
(rel: (-0.115, 0), to: <A16.west>),
|
||||
<A16.west>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + yellow.darken(20%),
|
||||
),
|
||||
//6 → 11
|
||||
edge(
|
||||
(rel: (0pt, -5mm), to: <A6.east>),
|
||||
|
||||
(rel: (0.065, 0.)),
|
||||
(rel: (0.065,0), to: <A11.east>),
|
||||
(rel: (0.,0), to: <A11.east>),
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + purple.lighten(20%),
|
||||
),
|
||||
//13 → 23
|
||||
edge(
|
||||
(<A13.east>),
|
||||
(rel:(0.75,0)),
|
||||
(rel:(0, 9.8)),// <----
|
||||
(rel:(-3.38, 0)),
|
||||
//(rel: (0.6, 0.), to: <A21_2.east>),
|
||||
<A23.south>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + orange.darken(10%),
|
||||
|
||||
),
|
||||
|
||||
//15 → 21_1 21_2
|
||||
edge(
|
||||
(<A15.east>),
|
||||
// Startpunkt
|
||||
(rel: (0.5, 0)),
|
||||
(rel: (0.5, 0.), to: <A21_1.east>),
|
||||
<A21_1.east>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + blue,
|
||||
),
|
||||
edge(
|
||||
(<A15.east>),
|
||||
// Startpunkt
|
||||
(rel: (0.5, 0)),
|
||||
(rel: (0.5, 0.), to: <A21_2.east>),
|
||||
<A21_2.east>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + blue,
|
||||
),
|
||||
|
||||
//16 → 19_1 19_2
|
||||
edge(
|
||||
(<A16.east>),
|
||||
(rel: (0.3, 0)),
|
||||
(rel: (0.3, 0.), to: <A19_1.east>),
|
||||
<A19_1.east>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + purple,
|
||||
),
|
||||
edge(
|
||||
(<A16.east>),
|
||||
(rel: (0.3, 0)),
|
||||
(rel: (0.3, 0.), to: <A19_2.east>),
|
||||
<A19_2.east>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + purple,
|
||||
),
|
||||
|
||||
//17 → 21_2
|
||||
edge(
|
||||
(rel: (-0.2,0), to: <A17.south>),
|
||||
(rel: (0, 0.2)),
|
||||
(rel: (-.325, 0.)),
|
||||
(rel: (-0.14, 0), to: <A21_2.west>),
|
||||
<A21_2.west>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + red,
|
||||
),
|
||||
//17 → 19_1, 19_2
|
||||
edge(
|
||||
(<A17.west>),
|
||||
(rel: (-0.075, 0)),
|
||||
(rel: (-0.075,-0.20), to: <A19_1.west>),
|
||||
(rel: (0.,-0.2), to: <A19_1.west>),
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + green,
|
||||
),
|
||||
edge(
|
||||
(<A17.west>),
|
||||
(rel: (-0.075, 0)),
|
||||
(rel: (-0.075, 0.), to: <A19_2.west>),
|
||||
<A19_2.west>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + green,
|
||||
),
|
||||
|
||||
//19_2 → 20
|
||||
edge(
|
||||
(<A19_2.north>),
|
||||
(rel: (-0.2, 0)),
|
||||
(rel: (0, -0.1)),
|
||||
(rel: (-0, 0.18), to: <A20.south>),
|
||||
<A20.south>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + blue,
|
||||
),
|
||||
)
|
||||
#abhaengigkeit
|
||||
|
||||
#show heading: it => {
|
||||
set block(below: if it.level == 1 { 2em } else { 1em })
|
||||
@@ -513,7 +208,7 @@ edge(
|
||||
|
||||
#set outline(indent: auto)
|
||||
|
||||
|
||||
#show figure.where(kind: image): set figure(supplement: [Bild]) //Abbildung zu Bild
|
||||
|
||||
#set page(
|
||||
header: context {
|
||||
@@ -564,6 +259,7 @@ edge(
|
||||
|
||||
)
|
||||
|
||||
|
||||
#show selector(<nonumber>): set heading(numbering: none)
|
||||
|
||||
= Vorwort <nonumber>
|
||||
@@ -611,22 +307,154 @@ Die mathematischen Möglichkeiten reichen jedoch über die unmittelbare Anwendba
|
||||
|
||||
#pagebreak()
|
||||
|
||||
= Grenzwerte
|
||||
#set math.equation(
|
||||
numbering: n => numbering("(2.1)", n),
|
||||
supplement: none,
|
||||
)
|
||||
#let numbered_eq(label_name, content) = [
|
||||
#math.equation(
|
||||
block: true,
|
||||
content
|
||||
)
|
||||
#label(label_name)
|
||||
]
|
||||
#show figure: set figure(numbering: n => {
|
||||
let chap = counter(heading).get().at(0)
|
||||
[#chap.#n]
|
||||
})
|
||||
|
||||
//== Grenzwert einer Funktion für $\boldsymbol{x} \rightarrow \boldsymbol{x}_0$
|
||||
// 2.1.1. Definition des Grenzwertes einer Funktion für $\boldsymbol{x} \rightarrow \boldsymbol{x}_0$
|
||||
|
||||
// Im folgenden bedeutet „Funktion“ stets „reellwertige Funktion einer reellen Variablen".
|
||||
|
||||
// Als Vorbereitung auf den Grenzwertbegriff für Funktionen behandeln wir das
|
||||
// Beispiel 2.1: An die Parabel $y=x^2$ werde die Sekante durch den festen Kurvenpunkt $P_0\left(\frac{1}{2}, \frac{1}{4}\right)$ und den variablen Kurvenpunkt $P\left(x, x^2\right)$ gelegt (s. Bild 2.1). Der Anstieg der Sekante ist eine Funktion $f$ von $x$ :
|
||||
= Grenzwerte *$x arrow.r x_0$*
|
||||
|
||||
// $$
|
||||
// f(x)=\frac{x^2-\frac{1}{4}}{x-\frac{1}{2}} \quad\left(x \neq \frac{1}{2}\right) .
|
||||
// $$
|
||||
== Grenzwert einer Funktion für *$x arrow.r x_0$*
|
||||
|
||||
$x arrow.r x_0$
|
||||
Im folgenden bedeutet "Funktion" stets "reellwertige Funktion einer reellen Variablen".
|
||||
|
||||
Als Vorbereitung auf den Grenzwertbegriff für Funktionen behandeln wir das
|
||||
|
||||
#beispiel[An die Parabel $y=x^2$ werde die Sekante durch den festen Kurvenpunkt $P_0(1/2, 1/4)$ und den variablen Kurvenpunkt $P(x, x^2)$ gelegt (s. @abb1). Der Anstieg der Sekante ist eine Funktion $f$ von $x$:
|
||||
|
||||
#table(
|
||||
columns: (1fr, 1fr),
|
||||
column-gutter: 1em,
|
||||
stroke: none,
|
||||
align: bottom + center,
|
||||
[
|
||||
#figure(
|
||||
scale(x: 50%, y: 50%, reflow: true, b2_1),
|
||||
caption: [],
|
||||
) <abb1>
|
||||
],
|
||||
[
|
||||
#figure(
|
||||
scale(x: 50%, y: 50%, reflow: true, b2_2),
|
||||
caption: [],
|
||||
) <abb2>
|
||||
]
|
||||
)
|
||||
|
||||
#numbered_eq("eq21", $ f(x)=(x^2 - 1/4)/(x - 1/2) $)
|
||||
|
||||
Auf Grund der Anschauung wird man vermuten, daß bei "Annäherung" von $x$ an die Stelle $ 1/2$ die Sekante in eine gewisse "Grenzlage" übergeht, also auch ihr Anstieg ( @eq21 ) einen gewissen "Grenzwert“ annimmt. Betrachten wir also die Funktion $f$. An der Stelle $x=1/2$ ist $f$ nicht definiert. Für $x eq.not 1/2$ gilt
|
||||
|
||||
#numbered_eq("eq22", $ f(x) = ((x - 1/2) (x + 1/2)) / (x - 1/2) = x + 1/2 quad quad (x eq.not 1/2)$)
|
||||
|
||||
Die Bildkurve von $f$ ist in @abb2 dargestellt #footnote[In @abb2 soll der kleine Kreis um den Punkt $(1)/2, 1)$ andeuten, daß dieser Punkt nicht zur Bildkurve von $f$ gehört. Analog wird in den folgenden Beispielen verfahren.]. Die Anschauung legt jetzt etwa die folgende Formulierung nahe: „Für $x$ gegen $1/2$ strebt $f(x)$ gegen $1$."
|
||||
|
||||
]
|
||||
|
||||
Unsere Aufgabe wird es nun sein, einer solchen Formulierung einen von der Anschauung unabhängigen, wohldefinierten Sinn zu geben.
|
||||
|
||||
Soll allgemein das Verhalten einer Funktion $f$ bei "Annäherung" der unabhängigen Variablen $x$ an eine reelle Zahl $x_0$ untersucht werden, so ist es naheliegend, die Variable $x$ Zahlenfolgen $(x_n)$ mit folgenden Eigenschaften durchlaufen zu lassen:
|
||||
|
||||
|
||||
// #align ohne Nummerierung
|
||||
#align(left,[
|
||||
#set math.equation(numbering: none)
|
||||
$
|
||||
& ("E 1") &quad& x_n in D(f) ^#footnote[$D(f)$ bezeichnet den Definitionsbereich von $f$.] &quad& " für alle" n &quad& (n=1,2,3,dots) \
|
||||
& ("E 2") && x_n eq.not x_0 && " für alle" n \
|
||||
& ("E 3") && lim_(x->n)x_n = x_0
|
||||
$])
|
||||
|
||||
Die Eigenschaft (E 2) bedeutet, daß das Verhalten von $f$ an der Stelle $x_0$ selbst nicht in Betracht gezogen wird. Daher braucht $f$ auch nur in einer sog. _punktierten Umgebung_ von $x_0$ definiert zu sein. Das ist, mit einem $c>0$, die Menge aller $x$ mit
|
||||
#v(3mm)
|
||||
|
||||
#figure(
|
||||
{
|
||||
set math.equation(numbering: none)
|
||||
scale(x: 70%, y: 70%, reflow: true, b2_3)
|
||||
},
|
||||
caption: [],
|
||||
) <abb3>
|
||||
|
||||
Das Verhalten von $f$ in einer punktierten Umgebung von $x_0$ wird nun durch das Verhalten der Folge der Funktionswerte $f(x_n)$ charakterisiert.
|
||||
|
||||
|
||||
#definition(label: <D.21>)[Die #snl([@D.21]) Funktion $f$ sei (mindestens) in einer punktierten Umgebung von $x_0$ definiert. Eine Zahl $g$ heißt *Grenzwert von $f$ für $x$ gegen $x_0$*, in Zeichen
|
||||
#math.equation(
|
||||
block: true,
|
||||
numbering: none,
|
||||
$ lim_(x->x_0)f(x) = g " oder " f(x)->g " für " x->x_0 $
|
||||
)
|
||||
wenn für je de Folge ( $x_n$ ) mit den Eigenschaften $(E" "1)$, $(E" "2)$, $(E" "3)$ die Folge ( $f(x_n)$ ) gegen $g$ konvergiert.]
|
||||
|
||||
Damit ist der Begriff des Grenzwertes einer Funktion auf den Grenzwertbegriff für Zahlenfolgen zurückgeführt.
|
||||
|
||||
In @abb4 haben wir die ersten drei Glieder einer Folge ( $x_n$ ) und der zugehörigen Folge $f\(x_n))$ eingezeichnet.
|
||||
|
||||
#figure(
|
||||
{
|
||||
set math.equation(numbering: none)
|
||||
scale(x: 70%, y: 70%, reflow: true, b2_4)
|
||||
},
|
||||
caption: [],
|
||||
) <abb4>
|
||||
|
||||
#beispiel[Gesucht ist der Grenzwert der Funktion
|
||||
#math.equation(
|
||||
block: true,
|
||||
numbering: none,
|
||||
$ f(x)=(x^2-1/4)/(x-1/2) " für " x -> 1/2. $
|
||||
)
|
||||
|
||||
Es sei $(x_n)$ eine beliebige Folge mit
|
||||
|
||||
#numbered_eq("eq23", $ x_n eq.not 1/2 " für alle " n " und " lim_(n->infinity)x_n = 1/2 $ )
|
||||
|
||||
Unter Verwendung von @eq22 und bekannten Grenzwertsätzen für Zahlenfolgen folgt dann
|
||||
|
||||
#numbered_eq("eq24", $ lim_(n->infinity) f(x_n)=lim_(n->infinity}(x_n+1/2)=lim_(n->infinity) x_n + lim_(n->infinity) 1/2 = 1/2 + 1/2 =1 $)
|
||||
|
||||
|
||||
Die Gültigkeit von @eq24 wurde für eine beliebige und damit für jede Folge $\(x_n)$ mit den Eigenschaften @eq23 bewiesen. Daher gilt
|
||||
|
||||
/* $$
|
||||
\lim _{x \rightarrow \frac{1}{2}} \frac{x^2-\frac{1}{4}}{x-\frac{1}{2}}=1,
|
||||
$$
|
||||
|
||||
was in Einklang mit der Anschauung steht (Bild 2.2). */
|
||||
|
||||
]
|
||||
|
||||
|
||||
|
||||
|
||||
//{n \rightarrow \infty} x_n+\lim _{n \rightarrow \infty} \frac{1}{2}=\frac{1}{2}+\frac{1}{2}=1$
|
||||
|
||||
|
||||
/*
|
||||
$$
|
||||
\lim _{x \rightarrow x_0} f(x)=g \quad \text { oder } \quad f(x) \rightarrow g \quad f \ddot{u} r \quad x \rightarrow x_0,
|
||||
$$
|
||||
|
||||
|
||||
|
||||
*/
|
||||
|
||||
|
||||
/*
|
||||
|
||||
#import "@preview/cetz-plot:0.1.3": plot
|
||||
|
||||
@@ -759,9 +587,6 @@ $x arrow.r x_0$
|
||||
})
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
#import "@preview/simple-plot:0.3.0": plot
|
||||
|
||||
//#set page(width: auto, height: auto, margin: 0.5cm)
|
||||
@@ -794,4 +619,5 @@ $x arrow.r x_0$
|
||||
(fn: x => calc.pow(2, x), stroke: blue + 1.5pt, label: $2^x$),
|
||||
(fn: x => calc.pow(3, x), stroke: red + 1.5pt, label: $3^x$),
|
||||
)
|
||||
*/
|
||||
|
||||
|
||||
@@ -0,0 +1,72 @@
|
||||
|
||||
#import "@preview/cetz:0.4.2"
|
||||
#import "@preview/cetz-plot:0.1.3": plot
|
||||
|
||||
#cetz.canvas(length: 1.25cm,{
|
||||
import cetz.draw: *
|
||||
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")
|
||||
})
|
||||
|
||||
|
||||
@@ -0,0 +1,48 @@
|
||||
#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")
|
||||
})
|
||||
|
||||
|
||||
@@ -0,0 +1,135 @@
|
||||
#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),
|
||||
ymin: -2,
|
||||
ymax: 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, $0$),),
|
||||
{
|
||||
//berechnungen und plot
|
||||
|
||||
|
||||
plot.add(x => (calc.pow(x, 2) -2), domain: (-1.4, 1.8))
|
||||
plot.add(x => (calc.pow(x, 2) +2), domain: (-1.4, 1.8))
|
||||
}
|
||||
)
|
||||
})
|
||||
|
||||
#let linspace(start, end, n)=range(0,n).map(x => x / n * (end - start) + start)
|
||||
|
||||
#cetz.canvas({
|
||||
import cetz.draw: *
|
||||
line((-5,0), (5,0), mark: (end: "stealth", fill: black))
|
||||
content((), block(inset: 0.2em)[$x$], anchor: "west")
|
||||
line((0,-5), (0,5), mark: (end: "stealth", fill: black))
|
||||
content((), block(inset: 0.2em)[$p^0$], anchor: "east")
|
||||
let pe = linspace(-2, 2, 200).map(x => (calc.pow(x,2) +2, x))
|
||||
line(..pe, stroke: olive)
|
||||
// let draw-point(pos, text, color, anchor) = {
|
||||
// circle(pos, radius: 0.1, fill: color, stroke: none)
|
||||
// content((), block(inset: 0.3em)[#text], anchor: anchor)
|
||||
//}
|
||||
// draw-point((0, 3), "(a)", red, "east")
|
||||
// draw-point((0, -3), "(b)", blue, "east")
|
||||
// draw-point((1, 1), "(c)", red, "south-east")
|
||||
// draw-point((1, -1), "(d)", blue, "north-east")
|
||||
// draw-point((3, 0), "(e)", olive, "north-east")
|
||||
// draw-point((0, 0), "(f)", olive, "east")
|
||||
})
|
||||
|
||||
#let linspace(start, end, n)=range(0,n).map(x => x / n * (end - start) + start)
|
||||
|
||||
#cetz.canvas({
|
||||
import cetz.draw: *
|
||||
line((0,0), (5,0), mark: (end: "stealth", fill: black))
|
||||
content((), block(inset: 0.2em)[$abs(bold(p))$], anchor: "west")
|
||||
line((0,-5), (0,5), mark: (end: "stealth", fill: black))
|
||||
content((), block(inset: 0.2em)[$p^0$], anchor: "east")
|
||||
let pa = linspace(0, 4, 100).map(x => (x, calc.sqrt(9+x*x)))
|
||||
line(..pa, stroke: red)
|
||||
let pb = linspace(0, 4, 100).map(x => (x, -calc.sqrt(9+x*x)))
|
||||
line(..pb, stroke: blue)
|
||||
line((0,0), (5,5), stroke: red)
|
||||
line((0,0), (5,-5), stroke: blue)
|
||||
let pe = linspace(-4, 4, 100).map(x => (calc.sqrt(9+x*x), x))
|
||||
line(..pe, stroke: olive)
|
||||
let draw-point(pos, text, color, anchor) = {
|
||||
circle(pos, radius: 0.1, fill: color, stroke: none)
|
||||
content((), block(inset: 0.3em)[#text], anchor: anchor)
|
||||
}
|
||||
draw-point((0, 3), "(a)", red, "east")
|
||||
draw-point((0, -3), "(b)", blue, "east")
|
||||
draw-point((1, 1), "(c)", red, "south-east")
|
||||
draw-point((1, -1), "(d)", blue, "north-east")
|
||||
draw-point((3, 0), "(e)", olive, "north-east")
|
||||
draw-point((0, 0), "(f)", olive, "east")
|
||||
})
|
||||
|
||||
#import "@preview/cetz:0.3.2"
|
||||
#import "@preview/cetz-plot:0.1.1"
|
||||
#cetz.canvas(
|
||||
{
|
||||
import cetz.draw: *
|
||||
import cetz-plot: *
|
||||
|
||||
|
||||
plot.plot(
|
||||
//size: (5,5),
|
||||
xmin: -2cm,
|
||||
xmax: 2cm,
|
||||
ymin: -2cm,
|
||||
ymax: 4cm,
|
||||
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: ((2, $2$),(1, $1$)),
|
||||
plot.add(
|
||||
domain: (-5,5),
|
||||
x => calc.pow(x, 2)+2
|
||||
)
|
||||
)
|
||||
}
|
||||
)
|
||||
|
||||
#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,2)+2)
|
||||
plot.plot(size: (6,9), axis-style: "school-book", 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: ((2, $2$),), {
|
||||
plot.add(domain: (-1.5, 1.5), f)
|
||||
|
||||
// Hilfslinie damit Koordinatensystem nicht verrutscht
|
||||
plot.add(((0, 0), (0, 0.1)), style: (stroke: none))
|
||||
})
|
||||
})
|
||||
|
||||
@@ -506,12 +506,14 @@
|
||||
)
|
||||
})
|
||||
|
||||
|
||||
|
||||
#import "@preview/cetz:0.4.2"
|
||||
#import "@preview/cetz-plot:0.1.3": plot
|
||||
|
||||
#cetz.canvas(length: 1.25cm,{
|
||||
import cetz.draw: *
|
||||
set-style(
|
||||
set-style(
|
||||
axes: (
|
||||
// Basisstil beibehalten
|
||||
stroke: (thickness: 0.5pt),
|
||||
@@ -635,3 +637,280 @@
|
||||
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: *
|
||||
set-style(
|
||||
axes: (
|
||||
stroke: (thickness: 0.5pt),
|
||||
x: (mark: (end: "stealth", fill: black)),
|
||||
y: (mark: (end: "stealth", fill: black)),
|
||||
),
|
||||
)
|
||||
|
||||
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))
|
||||
|
||||
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)))
|
||||
|
||||
// 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, 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 - 0.05,f(x2)+0.1))
|
||||
})
|
||||
|
||||
content("plot.F", text(0.85em)[$y=f(x)$], anchor: "east", 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: ">"))
|
||||
|
||||
})
|
||||
)
|
||||
|
||||
|
||||
|
||||
#import "@preview/cetz-plot:0.1.3": plot
|
||||
|
||||
#let intersection(plotA, plotB, style: (stroke: red + 2pt)) = {
|
||||
let dataA = plotA.first().data
|
||||
let dataB = plotB.first().data
|
||||
|
||||
let points = dataA.filter(x => x in dataB)
|
||||
|
||||
for ((i, point)) in points.enumerate() {
|
||||
plot.add-anchor("i_" + str(i),point )
|
||||
}
|
||||
|
||||
plotA
|
||||
plotB
|
||||
}
|
||||
|
||||
#canvas(
|
||||
length: 1cm,
|
||||
{
|
||||
import draw: *
|
||||
plot.plot(
|
||||
name: "my_plot",
|
||||
size: (5, 5),
|
||||
axis-style: "school-book",
|
||||
x-label: [$x$],
|
||||
y-label: [$y$],
|
||||
{
|
||||
intersection(
|
||||
plot.add(
|
||||
style: (stroke: black + 1.5pt),
|
||||
domain: (-1, 1),
|
||||
x => calc.pow(x, 4) - 2 * calc.pow(x, 2) + 1,
|
||||
),
|
||||
plot.add(
|
||||
style: (stroke: black + 1.5pt),
|
||||
domain: (-1, 1),
|
||||
x => -(calc.pow(x, 4) - 2 * calc.pow(x, 2) + 1),
|
||||
)
|
||||
)
|
||||
}
|
||||
)
|
||||
|
||||
// reference the point with {plot name}.i_{number}
|
||||
line("my_plot.i_1", ((), "|-", (0,3.5)), mark: (start: ">"), name: "line")
|
||||
content("line.end", [Here], anchor: "south", padding: .1)
|
||||
},
|
||||
)
|
||||
|
||||
|
||||
#let intersection(plotA, plotB, style: (stroke: red + 2pt)) = {
|
||||
import draw: *
|
||||
let dataA = plotA.first().data
|
||||
let dataB = plotB.first().data
|
||||
|
||||
plot.annotate({
|
||||
hide({
|
||||
intersections("i", line(..plotA.first().data), line(..plotB.first().data))
|
||||
})
|
||||
for-each-anchor("i", (name) => {
|
||||
circle("i." + name, radius: 0.05, fill: red)
|
||||
})
|
||||
})
|
||||
|
||||
plotA
|
||||
plotB
|
||||
}
|
||||
|
||||
#canvas(
|
||||
length: 1cm,
|
||||
{
|
||||
import draw: *
|
||||
plot.plot(
|
||||
name: "plot",
|
||||
size: (10, 10),
|
||||
axis-style: "school-book",
|
||||
x-label: [$x$],
|
||||
y-label: [$y$],
|
||||
{
|
||||
intersection(
|
||||
plot.add(
|
||||
style: (stroke: black + 1.5pt),
|
||||
domain: (-1, 1),
|
||||
x => calc.pow(x, 4) - 2 * calc.pow(x, 2) + 1,
|
||||
),
|
||||
plot.add(
|
||||
style: (stroke: black + 1.5pt),
|
||||
domain: (-1, 1),
|
||||
x => -(calc.pow(x, 4) - 2 * calc.pow(x, 2) + 0.5),
|
||||
),
|
||||
)
|
||||
},
|
||||
)
|
||||
},
|
||||
)
|
||||
|
||||
|
||||
#import "@preview/cetz:0.4.0"
|
||||
#import "@preview/cetz-plot:0.1.2"
|
||||
|
||||
#let my-plot = cetz-plot.plot.plot.with(axis-style: "school-book")
|
||||
|
||||
#cetz.canvas({
|
||||
import cetz.draw: *
|
||||
import cetz-plot: *
|
||||
set-style(
|
||||
axes: (
|
||||
stroke: (dash: "dotted", paint: gray),
|
||||
x: (mark: (start: ">", end: ">"), padding: 1),
|
||||
y: (mark: none),
|
||||
tick: (stroke: gray + .5pt),
|
||||
),
|
||||
)
|
||||
my-plot(
|
||||
size: (5, 4),
|
||||
y-tick-step: none,
|
||||
{
|
||||
plot.add(calc.sin, domain: (0, calc.pi * 2))
|
||||
},
|
||||
)
|
||||
})
|
||||
|
||||
#let plot-cfg = (size: (5, 4), axis-style: "school-book")
|
||||
#cetz.canvas({
|
||||
import cetz.draw: *
|
||||
import cetz-plot: *
|
||||
plot.plot(..plot-cfg, {
|
||||
plot.add(calc.sin, domain: (0, calc.pi * 2))
|
||||
})
|
||||
})
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
#import "@preview/simple-plot:0.3.0": plot
|
||||
|
||||
//#set page(width: auto, height: auto, margin: 0.5cm)
|
||||
|
||||
// Showcases: major grid only, grid-label-break, axis extension, integer ticks
|
||||
#plot(
|
||||
xmin: -3.1, xmax: 3.1,
|
||||
ymin: -1, ymax: 9,
|
||||
xlabel: $x$,
|
||||
ylabel: $y$,
|
||||
//show-grid: "major",
|
||||
(fn: x => calc.pow(x, 2), stroke: blue + 1.5pt, label: $x^2$, label-pos: 0.7, label-side: "below-right"),
|
||||
)
|
||||
|
||||
#plot(
|
||||
xmin: -2 * calc.pi, xmax: 2 * calc.pi,
|
||||
ymin: -2, ymax: 2,
|
||||
width: 10, height: 5,
|
||||
show-grid: "major",
|
||||
xtick: (-2*calc.pi, -calc.pi, 0, calc.pi, 2*calc.pi),
|
||||
xtick-labels: ($-2pi$, $-pi$, $0$, $pi$, $2pi$),
|
||||
(fn: x => calc.sin(x), stroke: blue + 1.2pt),
|
||||
(fn: x => calc.cos(x), stroke: red + 1.2pt),
|
||||
)
|
||||
|
||||
#plot(
|
||||
xmin: -2, xmax: 5,
|
||||
ymin: 0, ymax: 32,
|
||||
show-grid: true,
|
||||
(fn: x => calc.pow(2, x), stroke: blue + 1.5pt, label: $2^x$),
|
||||
(fn: x => calc.pow(3, x), stroke: red + 1.5pt, label: $3^x$),
|
||||
)
|
||||
360
Band2/abhaengigkeit.typ
Normal file
360
Band2/abhaengigkeit.typ
Normal file
@@ -0,0 +1,360 @@
|
||||
#import "@preview/fletcher:0.5.8" as fletcher: diagram, node, edge
|
||||
|
||||
|
||||
|
||||
#let mynode(pos, zahl, inhalt, name) = {
|
||||
node(
|
||||
pos,
|
||||
name: name,
|
||||
shape: rect, // 1. Zwingt Fletcher zum Rechteck
|
||||
corner-radius: 0pt, // 2. Verhindert abgerundete Ecken
|
||||
fill: white, // Optional: Hintergrund füllen
|
||||
stroke: 1.25pt,
|
||||
|
||||
// 3. WICHTIG: Festlegen einer Mindestgröße, damit alle gleich starten
|
||||
// Anstatt extrude nutzen wir feste Maße, die sich aber am Gitter ausrichten
|
||||
width: 12em,
|
||||
height: 5.8em,
|
||||
|
||||
{
|
||||
// Dein Layout-Block
|
||||
set align(top + right)
|
||||
text(2.5em, fill: gray.darken(90%))[#zahl]
|
||||
place(bottom + left, inhalt)
|
||||
}
|
||||
)}
|
||||
|
||||
#let myedge(start, end, type, ) = {
|
||||
let sep = 0.15em
|
||||
edge(start, end, type)
|
||||
edge(start, end)
|
||||
}
|
||||
|
||||
|
||||
#let abhaengigkeit = diagram(
|
||||
spacing: (11mm, 7mm),
|
||||
//debug: 3,
|
||||
edge-stroke: 1.25pt,
|
||||
|
||||
mark-scale: 55%,
|
||||
mynode((0,0),"",[Vorbereitungsband], <A0>),
|
||||
mynode((1,0),"1", [Grundlagen], <A1>),
|
||||
mynode((2,0), "13",[Lineare Algebra], <A13>),
|
||||
|
||||
mynode((0,1),"3",[Unendliche Reihen], <A3>),
|
||||
mynode((1,1), "2",[Differential- und Integralrechnung], <A2>),
|
||||
mynode((2,1), "14",[Lineare Optimierung], <A14>),
|
||||
|
||||
mynode((0,2), [7$#sub(text(0.5em)[1])$],[Gewöhnliche Differentialgleichungen], <A7_1>),
|
||||
mynode((1,2), "4",[Differentialrechnung mit mehreren Variablen], <A4>),
|
||||
mynode((2,2), "15",[Nichtlineare Optimierung], <A15>),
|
||||
|
||||
mynode((0,3), [7$#sub(text(0.5em)[2])$],[Gewöhnliche Differentialgleichungen], <A7_2>),
|
||||
mynode((1,3.), "5",[Integralrechnung mit mehreren Variablen], <A5>),
|
||||
mynode((2,3), "16",[Optimale Prozesse und Systeme], <A16>),
|
||||
|
||||
mynode((0,4), "8",[Partielle Differentialgleichungen], <A8>),
|
||||
mynode((1,4), "6",[Differentialgeometrie], <A6>),
|
||||
mynode((2,4), "17",[Wahrscheinlichkeitsrechnung, math.Statistik], <A17>),
|
||||
|
||||
mynode((0,5), "9",[Komplexe Funktionen], <A9>),
|
||||
mynode((1,5), "10",[Operatorenrechnung], <A10>),
|
||||
mynode((2,5), [21$#sub(text(0.5em)[1])$],[Spieltheorie], <A21_1>),
|
||||
|
||||
mynode((0,6), "12",[Spezielle Funktionen], <A12>),
|
||||
mynode((1,6), "11",[Tensoralgebra und -analysis], <A11>),
|
||||
mynode((2,6), [21$#sub(text(0.5em)[2])$],[Graphentheorie], <A21_2>),
|
||||
|
||||
mynode((0,7), [18],[Numerische Methoden], <A18>),
|
||||
mynode((1,7), [20],[Simulation], <A20>),
|
||||
mynode((2,7), [19$#sub(text(0.5em)[1])$],[Stochastische Prozesse und Modelle], <A19_1>),
|
||||
|
||||
mynode((0,8), [22],[Funktionalanalysis], <A22>),
|
||||
mynode((1,8), [23],[Symmetriegruppen], <A23>),
|
||||
mynode((2,8), [19$#sub(text(0.5em)[2])$],[Statistische Versuchsplanung], <A19_2>),
|
||||
|
||||
|
||||
myedge(<A0.east>,<A1.west>,"-|>"),
|
||||
myedge(<A1.south>,<A2.north>,"-|>"),
|
||||
myedge(<A2.west>,<A3.east>,"-|>"),
|
||||
myedge(<A13.south>,<A14.north>,"-|>"),
|
||||
myedge(<A2.south>,<A4.north>,"-|>"),
|
||||
myedge(<A4.south>,<A5.north>,"-|>"),
|
||||
myedge(<A5.south>,<A6.north>,"-|>"),
|
||||
myedge(<A14.south>,<A15.north>,"-|>"),
|
||||
myedge(<A7_1.south>,<A7_2.north>,"-|>"),
|
||||
myedge(<A7_2.south>,<A8.north>,"-|>"),
|
||||
myedge(<A8.south>,<A9.north>,"-|>"),
|
||||
myedge(<A9.south>,<A12.north>,"-|>"),
|
||||
myedge(<A12.south>,<A18.north>,"-|>"),
|
||||
myedge(<A10.south>,<A11.north>,"-|>"),
|
||||
myedge(<A15.south>,<A16.north>,"-|>"),
|
||||
myedge(<A17.south>,<A21_1.north>,"-|>"),
|
||||
myedge(<A19_1.south>,<A19_2.north>,"-|>"),
|
||||
// NEUES Edge: 1,3 (n3_1) → 3,2 (n2_3)
|
||||
// Verlauf: rechts von n3_1 nach rechts, runter auf Höhe von n2_3, dann links zur Node
|
||||
|
||||
//9 → 22
|
||||
edge(
|
||||
vertices: (
|
||||
<A9.east>,
|
||||
(0, 5.2), // deutlich nach rechts → sichtbarer Knick
|
||||
(0.5, 5.2), // dann nach links
|
||||
(0.5, 8), // runter
|
||||
<A22.east>,
|
||||
),
|
||||
corner: right,
|
||||
corner-radius: 0pt, // abgerundete Ecken!
|
||||
stroke: orange + 1.25pt,
|
||||
marks: "-|>",
|
||||
shift: (1pt, 1pt),
|
||||
),
|
||||
//10 → 22
|
||||
edge(
|
||||
vertices: (
|
||||
<A10.west>,
|
||||
(1, 5.2), // deutlich nach rechts → sichtbarer Knick
|
||||
(0.5, 5.2), // dann nach links
|
||||
(0.5, 8), // runter
|
||||
<A22.east>,
|
||||
),
|
||||
corner: left,
|
||||
corner-radius: 0pt, // abgerundete Ecken!
|
||||
stroke: orange + 1.25pt,
|
||||
marks: "-|>",
|
||||
shift: (1pt, 1pt),
|
||||
),
|
||||
//3 → 7.2
|
||||
edge(
|
||||
vertices: (
|
||||
<A3.west>,
|
||||
(-0.85, 1), // deutlich nach rechts → sichtbarer Knick
|
||||
(-0.85, 1), // dann nach links
|
||||
(-0.85, 3), // runter
|
||||
<A7_2.west>,
|
||||
),
|
||||
corner: left,
|
||||
corner-radius: 0pt, // abgerundete Ecken!
|
||||
stroke: red + 1.25pt,
|
||||
marks: "-|>",
|
||||
shift: (1pt, 1pt),
|
||||
),
|
||||
|
||||
//13 → 4
|
||||
edge(
|
||||
vertices: (
|
||||
<A13.west>,
|
||||
(2, 0.18), // deutlich nach rechts → sichtbarer Knick
|
||||
(1.5, 0.2), // dann nach links
|
||||
(1.5, 2), // runter
|
||||
<A4.east>,
|
||||
),
|
||||
corner: left,
|
||||
corner-radius: 0pt, // abgerundete Ecken!
|
||||
stroke: green + 1.25pt,
|
||||
marks: "-|>",
|
||||
shift: (2pt, 2pt),
|
||||
),
|
||||
|
||||
//1 → 13
|
||||
edge(
|
||||
(rel: (0pt, 5mm), to: <A1.east>),
|
||||
(rel: (0.1, 0.)),
|
||||
(rel: (0pt, 5mm), to: <A13.west>),
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: green.darken(30%) + 1.25pt,
|
||||
),
|
||||
|
||||
//5 → 8
|
||||
edge(
|
||||
(rel: (0pt, 0mm), to: <A5.west>),
|
||||
|
||||
(rel: (-0.15, 0.)),
|
||||
(rel: (0.08,0), to: <A8.east>),
|
||||
(rel: (0.,0), to: <A8.east>),
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + olive,
|
||||
),
|
||||
|
||||
//7_1 → 6
|
||||
edge(
|
||||
vertices: (
|
||||
(rel: (0pt, -2pt), to: <A7_1.east>),
|
||||
(0.25, 2.2),
|
||||
(0.52, 2.2),
|
||||
(0.52,4),
|
||||
(rel: (0pt, 0pt), to: <A6.west>)
|
||||
),
|
||||
corner-radius: 0pt,
|
||||
stroke: gray + 1.25pt,
|
||||
marks: "-|>",
|
||||
),
|
||||
//4 → 7_1
|
||||
edge(
|
||||
(rel: (0pt, 5mm), to: <A4.west>),
|
||||
|
||||
(rel: (-0.1, 0.)),
|
||||
|
||||
(rel: (0pt, 5mm), to: <A7_1.east>),
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt ,
|
||||
),
|
||||
//9 → 10
|
||||
edge(
|
||||
(rel: (0pt, 5mm), to: <A9.east>),
|
||||
|
||||
(rel: (0.1, 0.)),
|
||||
|
||||
(rel: (0pt, 5mm), to: <A10.west>),
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt ,
|
||||
),
|
||||
//5 → 17
|
||||
edge(
|
||||
<A5.east>,
|
||||
|
||||
(rel: (0.07, 0.)),
|
||||
(rel: (0, 0.5)),
|
||||
|
||||
(rel: (0pt, 10pt), to: <A17.north>),
|
||||
<A17.north>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: blue.lighten(20%) +1.25pt ,
|
||||
),
|
||||
//4 → 15
|
||||
edge(
|
||||
(rel: (0pt, -5mm), to: <A4.east>),
|
||||
|
||||
(rel: (0.1, 0.)),
|
||||
|
||||
(rel: (0pt, -5mm), to: <A15.west>),
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt ,
|
||||
),
|
||||
//8 → 16
|
||||
edge(
|
||||
(rel: (0.2,0), to: <A8.south>),
|
||||
(rel: (0., 0.1)),
|
||||
(rel: (1.3, 0.)),
|
||||
(rel: (-0.115, 0), to: <A16.west>),
|
||||
<A16.west>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + yellow.darken(20%),
|
||||
),
|
||||
//6 → 11
|
||||
edge(
|
||||
(rel: (0pt, -5mm), to: <A6.east>),
|
||||
|
||||
(rel: (0.065, 0.)),
|
||||
(rel: (0.065,0), to: <A11.east>),
|
||||
(rel: (0.,0), to: <A11.east>),
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + purple.lighten(20%),
|
||||
),
|
||||
//13 → 23
|
||||
edge(
|
||||
(<A13.east>),
|
||||
(rel:(0.75,0)),
|
||||
(rel:(0, 9.8)),// <----
|
||||
(rel:(-3.38, 0)),
|
||||
//(rel: (0.6, 0.), to: <A21_2.east>),
|
||||
<A23.south>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + orange.darken(10%),
|
||||
|
||||
),
|
||||
|
||||
//15 → 21_1 21_2
|
||||
edge(
|
||||
(<A15.east>),
|
||||
// Startpunkt
|
||||
(rel: (0.5, 0)),
|
||||
(rel: (0.5, 0.), to: <A21_1.east>),
|
||||
<A21_1.east>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + blue,
|
||||
),
|
||||
edge(
|
||||
(<A15.east>),
|
||||
// Startpunkt
|
||||
(rel: (0.5, 0)),
|
||||
(rel: (0.5, 0.), to: <A21_2.east>),
|
||||
<A21_2.east>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + blue,
|
||||
),
|
||||
|
||||
//16 → 19_1 19_2
|
||||
edge(
|
||||
(<A16.east>),
|
||||
(rel: (0.3, 0)),
|
||||
(rel: (0.3, 0.), to: <A19_1.east>),
|
||||
<A19_1.east>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + purple,
|
||||
),
|
||||
edge(
|
||||
(<A16.east>),
|
||||
(rel: (0.3, 0)),
|
||||
(rel: (0.3, 0.), to: <A19_2.east>),
|
||||
<A19_2.east>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + purple,
|
||||
),
|
||||
|
||||
//17 → 21_2
|
||||
edge(
|
||||
(rel: (-0.2,0), to: <A17.south>),
|
||||
(rel: (0, 0.2)),
|
||||
(rel: (-.325, 0.)),
|
||||
(rel: (-0.14, 0), to: <A21_2.west>),
|
||||
<A21_2.west>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + red,
|
||||
),
|
||||
//17 → 19_1, 19_2
|
||||
edge(
|
||||
(<A17.west>),
|
||||
(rel: (-0.075, 0)),
|
||||
(rel: (-0.075,-0.20), to: <A19_1.west>),
|
||||
(rel: (0.,-0.2), to: <A19_1.west>),
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + green,
|
||||
),
|
||||
edge(
|
||||
(<A17.west>),
|
||||
(rel: (-0.075, 0)),
|
||||
(rel: (-0.075, 0.), to: <A19_2.west>),
|
||||
<A19_2.west>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + green,
|
||||
),
|
||||
|
||||
//19_2 → 20
|
||||
edge(
|
||||
(<A19_2.north>),
|
||||
(rel: (-0.2, 0)),
|
||||
(rel: (0, -0.1)),
|
||||
(rel: (-0, 0.18), to: <A20.south>),
|
||||
<A20.south>,
|
||||
"-|>",
|
||||
corner-radius: 0pt ,
|
||||
stroke: 1.25pt + blue,
|
||||
),
|
||||
)
|
||||
215
Band2/grafiken.typ
Normal file
215
Band2/grafiken.typ
Normal file
@@ -0,0 +1,215 @@
|
||||
#import "@preview/cetz:0.4.2"
|
||||
#import "@preview/cetz-plot:0.1.3": plot
|
||||
|
||||
#let b2_1 = 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")
|
||||
})
|
||||
|
||||
#let b2_2 = 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")
|
||||
})
|
||||
|
||||
#let b2_3 = cetz.canvas(length: 3cm, {
|
||||
import cetz.draw: *
|
||||
|
||||
set-style(
|
||||
mark: (fill: black, scale: 2),
|
||||
stroke: (thickness: 0.4pt, cap: "round"),
|
||||
angle: (
|
||||
radius: 0.3,
|
||||
label-radius: .22,
|
||||
fill: green.lighten(80%),
|
||||
stroke: (paint: green.darken(50%))
|
||||
),
|
||||
content: (padding: 1pt)
|
||||
)
|
||||
|
||||
//grid((-.25, -.25), (2, 2 ), step: 0.5, stroke: gray + 0.2pt)
|
||||
|
||||
line((0, 0.25), (3, 0.25), mark: (end: "stealth", scale: 0.8))
|
||||
content((), $ x $, anchor: "west")
|
||||
|
||||
line((0.5,0.25),(2.5,0.25), stroke: 2pt)
|
||||
|
||||
circle((0.5,0.25), radius: 0.025, fill: white, stroke: (paint: black, thickness: 1.25pt), name: "01")
|
||||
|
||||
//content("01.end", anchor: "north", padding: (left: 3mm, bottom: 2mm), text(green.darken(30%))[Punkt $(a_1, a_2)$])
|
||||
|
||||
content("01.south", padding: (left: 0mm, bottom: -5mm), text([$x_0 -c$]))
|
||||
|
||||
circle((1.5,0.25), radius: 0.025, fill: white, stroke: (paint: black, thickness: 1.25pt), name: "02")
|
||||
|
||||
content("02.south", padding: (left: 0mm, bottom: -5mm), text([$x_0$]))
|
||||
|
||||
circle((2.5,0.25), radius: 0.025, fill: white, stroke: (paint: black, thickness: 1.25pt), name: "03")
|
||||
|
||||
content("03.south", padding: (left: 0mm, bottom: -5mm), text([$x_0 +c$]))
|
||||
|
||||
circle((1.5,0), radius: 0.0005, fill: white, stroke: (paint: white, thickness: 1.25pt), name: "02")
|
||||
|
||||
})
|
||||
|
||||
#let b2_4 = 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")
|
||||
|
||||
|
||||
})
|
||||
29
Band2/snippets.typ
Normal file
29
Band2/snippets.typ
Normal file
@@ -0,0 +1,29 @@
|
||||
#align(left, $
|
||||
& 1 &&=& a_1 &+& a_2 &+& a_3 &+& a_4 \
|
||||
& 1 + 2^3 &&=& 2a_1 &+& 2^2a_2 &+& 2^3a_3 &+& 2^4a_4 \
|
||||
& 1 + 2^3 + 3^3 &&=& 3a_1 &+& 3^2a_2 &+& 3^3a_3 &+& 3^4a_4 \
|
||||
& 1 + 2^3 + 3^3 + 4^4 &&=& 4a_1 &+& 4^2a_2 &+& 4^3a_3 &+& 4^4a_4
|
||||
$)
|
||||
|
||||
|
||||
$
|
||||
cases(
|
||||
1 & = thin a_1 &+ a_2 &+ a_3 &+ a_4 ,
|
||||
1 + 2^3 & = thin 2a_1 &+ 2^2a_2 &+ 2^3a_3 &+ 2^4a_4,
|
||||
1 + 2^3 + 3^3 & = thin 3a_1 &+ 3^2a_2 &+ 3^3a_3 &+ 3^4a_4,
|
||||
1 + 2^3 + 3^3 + 4^4 & = thin 4a_1 &+ 4^2a_2 &+ 4^3a_3 &+ 4^4a_4,
|
||||
)
|
||||
$
|
||||
|
||||
|
||||
$ cases(
|
||||
2y_1 +&& y_2 =&& 4x_1\
|
||||
y_1 +&& 2y_2 =&& -12x_2\
|
||||
-2y_1 +&& 3y_2 =&& x_3
|
||||
) $
|
||||
|
||||
$
|
||||
2y_1 +&& y_2 =&& 4x_1\
|
||||
y_1 +&& 2y_2 =&& -12x_2\
|
||||
-2y_1 +&& 3y_2 =&& x_3
|
||||
$
|
||||
Reference in New Issue
Block a user