Änderungen von Dokument BPE 6.3 Graphisches Ableiten

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

Von 13.2 nach 13.1 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 13.2

bearbeitet von Holger Engels
am 2025/05/20 10:46

Änderungskommentar: Es gibt keinen Kommentar für diese Version

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Zusammenfassung

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--XWiki.wies
++XWiki.holgerengels

Inhalt

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  {{seiteninhalt/}}
++=== Kompetenzen ===
  [[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
@@ -9,82 +9,27 @@
  **Interaktiv Erkunden:** [[Graphisches Ableiten>>https://kmap.eu/app/browser/Mathematik/Differentialrechnung/Graphisches%20Ableiten#erkunden]]
  {{/lernende}}
++* Markiere alle Stellen, wo die Tangente waagerecht ist
++* Kann eine Tangente den Funktionsgraphen schneiden?
++* Tangentensteigungen bestimmen
++
++* Bereiche mit positiver/ negativer Steigung schraffieren
  * Punktweise graphisch ableiten
  * Qualitativ graphisch ableiten
  * Zusammenhänge HP, TP, SP vorwärts und rückwärts
  * Funktionsterm der Ableitungsfunktion aus Tangentensteigungen aufstellen
++* Zuordnung Polynomfunktionen
  * 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.
++Zeichne jeweils die Tangente 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.
++{{aufgabe id="Tangenten einzeichnen" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
++Zeichne jeweils die Tangente 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="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=""/}}

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

Größe

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Inhalt

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Tangentensteigung.ggb

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Tangentensteigung.svg

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