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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 2.1
bearbeitet von VBS
am 2022/11/25 08:48
Änderungskommentar: Es gibt keinen Kommentar für diese Version
Auf Version 92.1
bearbeitet von Holger Engels
am 2025/05/21 05:13
Änderungskommentar: Neues Bild Ableitungsfunktion 2.svg hochladen

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Titel
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1 -Momentane Änderungsrate und 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.vbs
1 +XWiki.holgerengels
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.K4]] [[Kompetenzen.K5]] Ich kann Werte der Tangentensteigung graphisch bestimmen
4 +[[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann aus Werten der Tangentensteigung einen Graphen zeichnen und diesen als Graphen der Ableitungsfunktion deuten
5 +[[Kompetenzen.K6]] Ich kann Zusammenhänge zwischen den beiden Funktionsgraphen beschreiben
6 +[[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann erste Hypothesen über einen möglichen algebraischen Zusammenhang zwischen Funktion und Ableitungsfunktion entwickeln
7 +
8 +{{lernende}}
9 +**Interaktiv Erkunden:** [[Graphisches Ableiten>>https://kmap.eu/app/browser/Mathematik/Differentialrechnung/Graphisches%20Ableiten#erkunden]]
10 +{{/lernende}}
11 +
12 +* Punktweise graphisch ableiten
13 +* Qualitativ graphisch ableiten
14 +* Zusammenhänge HP, TP, SP vorwärts und rückwärts
15 +
16 +* Funktionsterm der Ableitungsfunktion aus Tangentensteigungen aufstellen
17 +* Beobachtungen bei e^x
18 +
19 +{{aufgabe id="Tangenten einzeichnen" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
20 +Zeichne jeweils die Tangenten an den Stellen {{formula}}x\in\{-1, 0, 1\}{{/formula}} ein und bestimme deren Steigungen.
21 +[[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"]]
22 +{{/aufgabe}}
23 +
24 +{{aufgabe id="Rauf und runter" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
25 +Markiere jeweils auf der x-Achse Intervalle mit positiver Steigung blau und negativer Steigung rot. Markiere die Stellen mit Steigung Null.
26 +[[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"]]
27 +{{/aufgabe}}
28 +
29 +{{aufgabe id="Punkte mit gegebener Steigung finden" afb="?" kompetenzen="" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="?"}}
30 +Es ist das Schaubild {{formula}}K_f{{/formula}} einer Funktion {{formula}}f{{/formula}} gegeben. Kennzeichne Punkte auf {{formula}}K_f{{/formula}}, für die gilt:
31 + die Steigung der Tangente in diesem Punkt ist 1
32 + die Steigung der Tangente in diesem Punkt ist 1,5
33 + die Steigung der Tangente in diesem Punkt ist 0
34 + die Steigung der Tangente in diesem Punkt ist {{formula}}-\frac{17}{4}{{/formula}}
35 +[[image:Tangentensteigung.svg|| width="700px"]]
36 +{{/aufgabe}}
37 +
38 +{{aufgabe id="Steigungsfunktion zeichnen" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="3" interaktiv=}}
39 +(% style="float:left; margin-right: 16px" %)
40 +Skizziere das Schaubild der Steigungsfunktion.
41 +[[image:Schaubild.svg||width=500]]
42 +{{/aufgabe}}
43 +
44 +{{aufgabe id="Zuordnung I" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung"}}
45 +(% style="float:left; margin-right: 16px" %)
46 +| [[image:Polynome zuordnen f.svg||width=200]] | | | | | [[image:Polynome zuordnen A.svg||width=200]]
47 +| [[image:Polynome zuordnen g.svg||width=200]] | | | | | [[image:Polynome zuordnen B.svg||width=200]]
48 +| [[image:Polynome zuordnen h.svg||width=200]] | | | | | [[image:Polynome zuordnen C.svg||width=200]]
49 +| [[image:Polynome zuordnen i.svg||width=200]] | | | | | [[image:Polynome zuordnen D.svg||width=200]]
50 +{{/aufgabe}}
51 +
52 +{{aufgabe id="Skizzieren anhand Eigenschaften" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="4"}}
53 +a) Skizziere eine mögliche Parabel 2. Grades, welche eine waagrechte Tangente an der Stelle {{formula}}x = -2{{/formula}} hat. Welche Gemeinsamkeiten haben diese Parabeln?
54 +
55 +b) Skizziere das Schaubild einer möglichen Funktion, welches drei waagrechte Tangenten besitzt. Welchen minimalen Grad hat die Funktion?
56 +
57 +c) Eine Funktion f hat nur positive Steigungen. Skizziere das Schaubild der Ableitungsfunktion.
58 +
59 +d) Es ist ein zur y-Achse symmetrisches Schaubild einer Funktion 4. Grades gesucht. Folgende Angaben sind bekannt, fülle die Lücken und skizziere das Schaubild der Funktion.
60 +(% class="border" %)
61 +|x|-4|-1|0|1 |4
62 +|Funktionswert|-2,5| |2 |0|
63 +|Tangentensteigung|-2| |0|-1 |
64 +{{/aufgabe}}
65 +
66 +{{aufgabe id="Aussagen Polynomfunktion" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}}
67 +Prüfe die Aussagen! Welche sind wahr? Eine Polynomfunktion 3. Grades ..
68 +☐ hat immer zwei Extrempunkte!
69 +☐ kann auch mal nur einen Extrempunkt haben!
70 +☐ kann auch mal keinen Extrempunkt haben!
71 +☐ hat immer genau einen Wendepunkt!
72 +☐ hat entweder einen Sattelpunkt oder zwei Extrempunkte!
73 +{{/aufgabe}}
74 +
75 +{{aufgabe id="Aussagen Schaubild" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="?"}}
76 +Gegeben ist das Schaubild einer Funktion. Nimm Stellung zu folgenden Aussagen und begründe deine Antwort.
77 +[[image:Aussagen.svg|| width="500px"]]
78 +☐ {{formula}}f(-3)=3{{/formula}}
79 +☐ {{formula}}x = 3{{/formula}} ist dreifache Nullstell
80 +☐ die Tangentensteigungen sind negativ für {{formula}}x \in ]2;5[{{/formula}}
81 +☐ die Steigung der Tangente an der Stelle {{formula}}x = 1<-2{{/formula}}
82 +☐ an der Stelle {{formula}}x = 2{{/formula}} liegt eine waagrechte Tangente
83 +☐ die Tangentensteigungen sind negativ für {{formula}}-4 < x < 2{{/formula}}
84 +☐ die Tangentensteigungen haben einen Vorzeichenwechsel bei {{formula}}x=-4{{/formula}} von ⊝ ⇾ ⊕
85 +{{/aufgabe}}
86 +
87 +{{aufgabe id="Aussagen Sattelstelle" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}}
88 +Welche Aussagen treffen auf eine Sattelstelle zu?
89 +☐ Eine Sattelstelle hat eine waagrechte Asymptote
90 +☐ An einer Sattelstelle hat die Steigung ein Maximum oder ein Minimum
91 +☐ An einer Sattelstelle gibt es immer auch einen Krümmungswechsel
92 +☐ Eine Sattelstelle ist auch eine Wendestelle
93 +☐ Eine Sattelstelle kann auch eine Maximalstelle sein
94 +{{/aufgabe}}
95 +
96 +{{aufgabe id="Zuordnung II" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="?"}}
97 +Es ist das Schaubild einer Steigungsfunktion gegeben. Zudem sind drei Schaubilder von drei Funktionen (A, B und C) gegeben. Welche Schaubilder (A, B oder C) können nicht zu der Steigungsfunktion gehören? Begründe deine Zuordnung.
98 +[[image:Ableitungsfunktion.svg|| width="500px"]]
99 +[[image:Ableitungsfunktion 1.svg||width="300px"]] [[image:Ableitungsfunktion 2.svg||width=300]] [[image:Ableitungsfunktion 3.svg||width=300]]
100 +{{/aufgabe}}
101 +
102 +{{seitenreflexion bildungsplan="" kompetenzen="" anforderungsbereiche="" kriterien="" menge=""/}}
Ableitungsfunktion 1.svg
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Ableitungsfunktion 2.svg
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Ableitungsfunktion 3.svg
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Ableitungsfunktion.ggb
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Ableitungsfunktion.svg
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Aussagen.ggb
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Aussagen.svg
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Polynome zuordnen A.svg
Author
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1 +XWiki.holgerengels
Größe
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1 +29.3 KB
Inhalt
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Polynome zuordnen B.svg
Author
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1 +XWiki.holgerengels
Größe
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1 +29.2 KB
Inhalt
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546 +<title>Function g&apos;</title>
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Polynome zuordnen C.svg
Author
... ... @@ -1,0 +1,1 @@
1 +XWiki.holgerengels
Größe
... ... @@ -1,0 +1,1 @@
1 +30.0 KB
Inhalt
... ... @@ -1,0 +1,566 @@
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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
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1 +29.6 KB
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546 +<title>Function g</title>
547 +<desc>g(x) = -x³</desc>
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Polynome zuordnen h.svg
Author
... ... @@ -1,0 +1,1 @@
1 +XWiki.holgerengels
Größe
... ... @@ -1,0 +1,1 @@
1 +30.6 KB
Inhalt
... ... @@ -1,0 +1,566 @@
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Polynome zuordnen i.svg
Author
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1 +XWiki.holgerengels
Größe
... ... @@ -1,0 +1,1 @@
1 +30.6 KB
Inhalt
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Tangenten einzeichnen 1.svg
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Inhalt
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Tangenten einzeichnen 2.svg
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Größe
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Inhalt
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Tangenten einzeichnen 3.svg
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Tangenten einzeichnen.ggb
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Tangentensteigung.ggb
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Tangentensteigung.svg
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