Änderungen von Dokument BPE 6.3 Graphisches Ableiten

Zuletzt geändert von Holger Engels am 2025/08/02 07:35

Von 2.1 nach 1.1 Von 65.1 nach 64.1

Von Version 64.1

bearbeitet von Stephanie Wietzorek
am 2025/05/20 14:43

Änderungskommentar: Neues Bild Ableitungsfunktion 2.svg hochladen

Auf Version 2.1

bearbeitet von VBS
am 2022/11/25 08:48

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Zusammenfassung

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Details

Seiteneigenschaften

Titel

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--BPE 6.3 Momentane Änderungsrate und graphisches Ableiten
++Momentane Änderungsrate und graphisches Ableiten

Übergeordnete Seite

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--Eingangsklasse.WebHome
++Main.WebHome

Dokument-Autor

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--XWiki.wies
++XWiki.vbs

Inhalt

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--{{seiteninhalt/}}
++{{box cssClass="floatinginfobox" title="**Contents**"}}
++{{toc start=2 depth=2 /}}
++{{/box}}
--[[Kompetenzen.K4]] [[Kompetenzen.K5]] Ich kann Werte der Tangentensteigung graphisch bestimmen
--[[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann aus Werten der Tangentensteigung einen Graphen zeichnen und diesen als Graphen der Ableitungsfunktion deuten
--[[Kompetenzen.K6]] Ich kann Zusammenhänge zwischen den beiden Funktionsgraphen beschreiben
--[[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann erste Hypothesen über einen möglichen algebraischen Zusammenhang zwischen Funktion und Ableitungsfunktion entwickeln
--
--{{lernende}}
--**Interaktiv Erkunden:** [[Graphisches Ableiten>>https://kmap.eu/app/browser/Mathematik/Differentialrechnung/Graphisches%20Ableiten#erkunden]]
--{{/lernende}}
--
--* Punktweise graphisch ableiten
--* Qualitativ graphisch ableiten
--* Zusammenhänge HP, TP, SP vorwärts und rückwärts
--
--* Funktionsterm der Ableitungsfunktion aus Tangentensteigungen aufstellen
--* Beobachtungen bei e^x
--
--{{aufgabe id="Tangenten einzeichnen" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
--Zeichne jeweils die Tangenten an den Stellen {{formula}}x\in\{-1, 0, 1\}{{/formula}} ein und bestimme deren Steigungen.
--[[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"]]
--{{/aufgabe}}
--
--{{aufgabe id="Rauf und runter" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
--Markiere jeweils auf der x-Achse Intervalle mit positiver Steigung blau und negativer Steigung rot. Markiere die Stellen mit Steigung Null.
--[[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"]]
--{{/aufgabe}}
--
--{{aufgabe id="Punkte mit gegebener Steigung finden" afb="?" kompetenzen="" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="?"}}
--Es ist das Schaubild {{formula}}K_f{{/formula}} einer Funktion {{formula}}f{{/formula}} gegeben. Kennzeichne Punkte auf {{formula}}K_f{{/formula}}, für die gilt:
--  die Steigung der Tangente in diesem Punkt ist 1
--  die Steigung der Tangente in diesem Punkt ist 1,5
--  die Steigung der Tangente in diesem Punkt ist 0
--  die Steigung der Tangente in diesem Punkt ist {{formula}}-\frac{17}{4}{{/formula}}
--[[image:Tangentensteigung.svg|| width="700px"]]
--{{/aufgabe}}
--
--{{aufgabe id="Zuordnung" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung"}}
--(% style="float:left; margin-right: 16px" %)
--| [[image:Polynome zuordnen f.svg||width=200]] | | | | | [[image:Polynome zuordnen A.svg||width=200]]
--| [[image:Polynome zuordnen g.svg||width=200]] | | | | | [[image:Polynome zuordnen B.svg||width=200]]
--| [[image:Polynome zuordnen h.svg||width=200]] | | | | | [[image:Polynome zuordnen C.svg||width=200]]
--| [[image:Polynome zuordnen i.svg||width=200]] | | | | | [[image:Polynome zuordnen D.svg||width=200]]
--{{/aufgabe}}
--
--{{aufgabe id="Skizzieren anhand Eigenschaften" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="4"}}
--a) Skizziere eine mögliche Parabel 2. Grades, welche eine waagrechte Tangente an der Stelle {{formula}}x = -2{{/formula}} hat. Welche Gemeinsamkeiten haben diese Parabeln?
--
--b) Skizziere das Schaubild einer möglichen Funktion, welches drei waagrechte Tangenten besitzt. Welchen minimalen Grad hat die Funktion?
--
--c) Eine Funktion f hat nur positive Steigungen. Skizziere das Schaubild der Ableitungsfunktion.
--
--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.
--(% class="border" %)
--|x|-4|-1|0|1  |4
--|Funktionswert|-2,5| |2  |0|
--|Tangentensteigung|-2| |0|-1  |
--{{/aufgabe}}
--
--{{aufgabe id="Aussagen Polynomfunktion" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}}
--Prüfe die Aussagen! Welche sind wahr? Eine Polynomfunktion 3. Grades ..
--☐ hat immer zwei Extrempunkte!
--☐ kann auch mal nur einen Extrempunkt haben!
--☐ kann auch mal keinen Extrempunkt haben!
--☐ hat immer genau einen Wendepunkt!
--☐ hat entweder einen Sattelpunkt oder zwei Extrempunkte!
--{{/aufgabe}}
--
--{{aufgabe id="Aussagen Schaubild" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="?"}}
--Gegeben ist das Schaubild einer Funktion. Nimm Stellung zu folgenden Aussagen und begründe deine Antwort.
--[[image:Aussagen.svg|| width="500px"]]
--☐ {{formula}}f(-3)=3{{/formula}}
--☐ {{formula}}x = 3{{/formula}} ist dreifache Nullstell
--☐ die Tangentensteigungen sind negativ für {{formula}}x \in ]2;5[{{/formula}}
--☐ die Steigung der Tangente an der Stelle {{formula}}x = 1<-2{{/formula}}
--☐ an der Stelle {{formula}}x = 2{{/formula}} liegt eine waagrechte Tangente
--☐ die Tangentensteigungen sind negativ für {{formula}}-4 < x < 2{{/formula}}
--☐ die Tangentensteigungen haben einen Vorzeichenwechsel bei {{formula}}x=-4{{/formula}} von ⊝ ⇾ ⊕
--{{/aufgabe}}
--
--{{aufgabe id="Aussagen Sattelstelle" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}}
--Welche Aussagen treffen auf eine Sattelstelle zu?
--☐ Eine Sattelstelle hat eine waagrechte Asymptote
--☐ An einer Sattelstelle hat die Steigung ein Maximum oder ein Minimum
--☐ An einer Sattelstelle gibt es immer auch einen Krümmungswechsel
--☐ Eine Sattelstelle ist auch eine Wendestelle
--☐ Eine Sattelstelle kann auch eine Maximalstelle sein
--{{/aufgabe}}
--
--{{seitenreflexion bildungsplan="" kompetenzen="" anforderungsbereiche="" kriterien="" menge=""/}}
++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.

Ableitungsfunktion 1.svg

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1		-XWiki.wies

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Ableitungsfunktion 2.svg

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1		-XWiki.wies

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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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Polynome zuordnen C.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 D.svg

Author

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1		-XWiki.holgerengels

Größe

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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

Inhalt

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Polynome zuordnen h.svg

Author

...	...	@@ -1,1 +1,0 @@
1		-XWiki.holgerengels

Größe

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1		-30.6 KB

Inhalt

@@ -1,566 +1,0 @@
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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

Größe

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Inhalt

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Tangenten einzeichnen 2.svg

Author

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1		-XWiki.holgerengels

Größe

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Inhalt

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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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