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Änderungen von Dokument BPE 6.3 Graphisches Ableiten

Zuletzt geändert von Holger Engels am 2025/08/02 07:35
Von Version 1.1
bearbeitet von holger
am 2022/11/13 17:11
Änderungskommentar: Es gibt keinen Kommentar für diese Version
Auf Version 52.1
bearbeitet von Stephanie Wietzorek
am 2025/05/20 12:36
Änderungskommentar: Es gibt keinen Kommentar für diese Version

Zusammenfassung

Details

Seiteneigenschaften
Titel
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1 -Graphisches Ableiten
1 +BPE 6.3 Momentane Änderungsrate und graphisches Ableiten
Übergeordnete Seite
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1 -Main.WebHome
1 +Eingangsklasse.WebHome
Dokument-Autor
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1 -XWiki.holger
1 +XWiki.wies
Inhalt
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1 -{{box cssClass="floatinginfobox" title="**Contents**"}}
2 -{{toc start=2 depth=2 /}}
3 -{{/box}}
1 +{{seiteninhalt/}}
4 4  
5 -Die Schülerinnen und Schüler bestimmen grafisch Werte der Tangentensteigung, zeichnen davon ausgehend den Graphen und deuten diesen als Graphen der Ableitungsfunktion. Sie beschreiben Zusammenhänge zwischen den beiden Funktionsgraphen und entwickeln erste Hypothesen über einen möglichen algebraischen Zusammenhang.
3 +=== Kompetenzen ===
4 +[[Kompetenzen.K4]] [[Kompetenzen.K5]] Ich kann Werte der Tangentensteigung graphisch bestimmen
5 +[[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann aus Werten der Tangentensteigung einen Graphen zeichnen und diesen als Graphen der Ableitungsfunktion deuten
6 +[[Kompetenzen.K6]] Ich kann Zusammenhänge zwischen den beiden Funktionsgraphen beschreiben
7 +[[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann erste Hypothesen über einen möglichen algebraischen Zusammenhang zwischen Funktion und Ableitungsfunktion entwickeln
8 +
9 +{{lernende}}
10 +**Interaktiv Erkunden:** [[Graphisches Ableiten>>https://kmap.eu/app/browser/Mathematik/Differentialrechnung/Graphisches%20Ableiten#erkunden]]
11 +{{/lernende}}
12 +
13 +* Kann eine Tangente den Funktionsgraphen schneiden?
14 +
15 +* Bereiche mit positiver/ negativer Steigung schraffieren
16 +* Punktweise graphisch ableiten
17 +* Qualitativ graphisch ableiten
18 +* Zusammenhänge HP, TP, SP vorwärts und rückwärts
19 +
20 +* Funktionsterm der Ableitungsfunktion aus Tangentensteigungen aufstellen
21 +* Beobachtungen bei e^x
22 +
23 +{{aufgabe id="Tangenten einzeichnen" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
24 +Zeichne jeweils die Tangenten an den Stellen {{formula}}x\in\{-1, 0, 1\}{{/formula}} ein und bestimme deren Steigungen.
25 +[[image:Tangenten einzeichnen 1.svg|| width="350px"]] [[image:Tangenten einzeichnen 2.svg|| width="350px"]] [[image:Tangenten einzeichnen 3.svg|| width="350px"]] [[image:Tangenten einzeichnen 4.svg|| width="350px"]]
26 +{{/aufgabe}}
27 +
28 +{{aufgabe id="Rauf und runter" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
29 +Markiere jeweils auf der x-Achse Intervalle mit positiver Steigung blau und negativer Steigung rot. Markiere die Stellen mit Steigung Null.
30 +[[image:Tangenten einzeichnen 1.svg|| width="350px"]] [[image:Tangenten einzeichnen 2.svg|| width="350px"]] [[image:Tangenten einzeichnen 3.svg|| width="350px"]] [[image:Tangenten einzeichnen 4.svg|| width="350px"]]
31 +{{/aufgabe}}
32 +
33 +{{aufgabe id="Punkte mit gegebener Steigung finden" afb="?" kompetenzen="" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="3"}}
34 +Es ist das Schaubild {{formula}}K_f{{/formula}} einer Funktion {{formula}}f{{/formula}} gegeben. Kennzeichne Punkte auf {{formula}}K_f{{/formula}}, für die gilt:
35 + {{formula}}f'(x) =1{{/formula}}
36 + {{formula}}f'(x) =1,5{{/formula}}
37 + {{formula}}f'(x) =0{{/formula}}
38 + {{formula}}f'(x) = -\frac{17}{4}{{/formula}}
39 +[[image:Tangentensteigung.svg|| width="700px"]]
40 +{{/aufgabe}}
41 +
42 +{{aufgabe id="Zuordnung" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung"}}
43 +(% style="float:left; margin-right: 16px" %)
44 +| [[image:Polynome zuordnen f.svg||width=200]] | | | | | [[image:Polynome zuordnen A.svg||width=200]]
45 +| [[image:Polynome zuordnen g.svg||width=200]] | | | | | [[image:Polynome zuordnen B.svg||width=200]]
46 +| [[image:Polynome zuordnen h.svg||width=200]] | | | | | [[image:Polynome zuordnen C.svg||width=200]]
47 +| [[image:Polynome zuordnen i.svg||width=200]] | | | | | [[image:Polynome zuordnen D.svg||width=200]]
48 +{{/aufgabe}}
49 +
50 +{{aufgabe id="Skizzieren anhand Eigenschaften" afb="?" kompetenzen="" quelle=" Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung"}}
51 +(% style="float:left; margin-right: 16px" %)
52 +Es ist ein zur y-Achse symmetrisches Schaubild einer Funktion 4. Grades gesucht. Folgende Angaben sind bekannt:
53 +
54 +(% class="border" %)
55 +|= {{formula}}x{{/formula}} |-4|-1|0|1 |4
56 +|= {{formula}}f(x){{/formula}} |-2,5| |2 |0|
57 +|= {{formula}}f'(x){{/formula}} |-2| |0|-1 |
58 +{{/aufgabe}}
59 +
60 +{{aufgabe id="Aussagen Polynomfunktion I" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}}
61 +Prüfe die Aussagen! Welche sind wahr? Eine Polynomfunktion 3. Grades ..
62 +☐ hat immer zwei Extrempunkte!
63 +☐ kann auch mal nur einen Extrempunkt haben!
64 +☐ kann auch mal keinen Extrempunkt haben!
65 +☐ hat immer genau einen Wendepunkt!
66 +☐ hat entweder einen Sattelpunkt oder zwei Extrempunkte!
67 +{{/aufgabe}}
68 +
69 +{{aufgabe id="Aussagen Polynomfunktion II" afb="I" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="3"}}
70 +Eine weitere Funktion hat folgendes Schaubild. Nimm Stellung zu folgenden Aussagen und begründe deine Antwort.
71 +- {{formula}}f(-3)=3{{/formula}}
72 +- {{formula}}x = 3{{/formula}} ist dreifache Nullstell
73 +- {{formula}}f'(x)<0{{/formula}} für {{formula}}x \in ]2;5[{{/formula}}
74 +- {{formula}}f'(1)<-2{{/formula}}
75 +- {{formula}}f'(2)=0{{/formula}}
76 +- {{formula}}f'(x)\ge 0{{/formula}} für {{formula}}-4 \le x \le 2{{/formula}}
77 +- {{formula}}f'(x){{/formula}} hat einen Vorzeichenwechsel bei {{formula}}x=-4{{/formula}} von - \to +
78 +
79 +{{/aufgabe}}
80 +
81 +
82 +
83 +{{aufgabe id="Aussagen Sattelstelle" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}}
84 +Welche Aussagen treffen auf eine Sattelstelle zu?
85 +☐ Eine Sattelstelle hat eine waagrechte Asymptote
86 +☐ An einer Sattelstelle hat die Steigung ein Maximum oder ein Minimum
87 +☐ An einer Sattelstelle gibt es immer auch einen Krümmungswechsel
88 +☐ Eine Sattelstelle ist auch eine Wendestelle
89 +☐ Eine Sattelstelle kann auch eine Maximalstelle sein
90 +{{/aufgabe}}
91 +
92 +{{seitenreflexion bildungsplan="" kompetenzen="" anforderungsbereiche="" kriterien="" menge=""/}}
Aussagen.ggb
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Polynome zuordnen A.svg
Author
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1 +XWiki.holgerengels
Größe
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Inhalt
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Author
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1 +XWiki.holgerengels
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Inhalt
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Polynome zuordnen C.svg
Author
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1 +XWiki.holgerengels
Größe
... ... @@ -1,0 +1,1 @@
1 +30.0 KB
Inhalt
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Polynome zuordnen D.svg
Author
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1 +XWiki.holgerengels
Größe
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1 +30.0 KB
Inhalt
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Polynome zuordnen f.svg
Author
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1 +XWiki.holgerengels
Größe
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1 +29.6 KB
Inhalt
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Polynome zuordnen g.svg
Author
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1 +XWiki.holgerengels
Größe
... ... @@ -1,0 +1,1 @@
1 +29.6 KB
Inhalt
... ... @@ -1,0 +1,566 @@
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393 +<clipPath id="clip55dea87b-6844-4194-8fd3-f87d57f4ffce">
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396 +<g clip-path="url(#clip48)">
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398 + <path d="M 0.0000 305.61 L 395.00 305.61"/>
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400 +</g> <!-- clip48 -->
401 +<clipPath id="clip9c3c3adc-5e1a-4384-ba6d-333db24a9c00">
402 + <path d="M 0.0000 0.0000 L 0.0000 361.00 L 395.00 361.00 L 395.00 0.0000 z"/>
403 +</clipPath>
404 +<g clip-path="url(#clip49)">
405 +<g stroke-linejoin="round" stroke-linecap="round" fill="none" stroke-opacity=".23529" stroke="#c0c0c0">
406 + <path d="M 0.0000 320.64 L 395.00 320.64"/>
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408 +</g> <!-- clip49 -->
409 +<clipPath id="clipf9f96851-1b90-43d5-8597-4cfd565932c3">
410 + <path d="M 0.0000 0.0000 L 0.0000 361.00 L 395.00 361.00 L 395.00 0.0000 z"/>
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420 +<g clip-path="url(#clip51)">
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495 +</g> <!-- transform -->
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535 +</g> <!-- transform -->
536 +</g> <!-- clip65 -->
537 +<g id="misc">
538 +</g><!-- misc -->
539 +<g id="layer0">
540 +<clipPath id="clipaca1c80e-8e88-4f30-986b-bb0797fd488b">
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546 +<title>Function g</title>
547 +<desc>g(x) = -x³</desc>
548 +
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Polynome zuordnen h.svg
Author
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1 +XWiki.holgerengels
Größe
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1 +30.6 KB
Inhalt
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Polynome zuordnen i.svg
Author
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1 +XWiki.holgerengels
Größe
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1 +30.6 KB
Inhalt
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Tangenten einzeichnen 1.svg
Author
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1 +XWiki.holgerengels
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Inhalt
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Tangenten einzeichnen 2.svg
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Tangenten einzeichnen 4.svg
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Tangenten einzeichnen.ggb
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Tangentensteigung.ggb
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