Änderungen von Dokument BPE 6.3 Graphisches Ableiten

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

Von 47.1 nach 47.2 Von 171.1 nach 172.1

Von Version 47.2

bearbeitet von Holger Engels
am 2025/05/20 12:07

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Auf Version 171.1

bearbeitet von Holger Engels
am 2025/07/21 17:07

Änderungskommentar: Löschung des Anhangs Ableitungsfunktion.ggb

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@@ -1,6 +1,5 @@
  {{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
@@ -10,9 +10,6 @@
  **Interaktiv Erkunden:** [[Graphisches Ableiten>>https://kmap.eu/app/browser/Mathematik/Differentialrechnung/Graphisches%20Ableiten#erkunden]]
  {{/lernende}}
--* Kann eine Tangente den Funktionsgraphen schneiden?
--
--* Bereiche mit positiver/ negativer Steigung schraffieren
  * Punktweise graphisch ableiten
  * Qualitativ graphisch ableiten
  * Zusammenhänge HP, TP, SP vorwärts und rückwärts
@@ -20,50 +20,95 @@
  * 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.
++{{aufgabe id="Tangenten einzeichnen" afb="I" kompetenzen="K4, K5" 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.
++{{aufgabe id="Rauf und runter" afb="I" kompetenzen="K4, K5" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
++Markiere zuerst alle Stellen an denen die Kurve die Steigung null hat.
++Markiere dann auf der x-Achse Intervalle mit positiver Steigung blau und Intervalle mit negativer Steigung rot.
  [[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="3"}}
++{{aufgabe id="Punkte mit gegebener Steigung finden" afb="II" kompetenzen="K2, K4, K5" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="5"}}
  Es ist das Schaubild {{formula}}K_f{{/formula}} einer Funktion {{formula}}f{{/formula}} gegeben. Kennzeichne Punkte auf {{formula}}K_f{{/formula}}, für die gilt:
--  {{formula}}f'(x) =1{{/formula}}
--  {{formula}}f'(x) =1,5{{/formula}}
--  {{formula}}f'(x) =0{{/formula}}
--  {{formula}}f'(x) = -\frac{17}{4}{{/formula}}
++(%class=abc%)
++1. die Steigung der Tangente in diesem Punkt ist 1
++1. die Steigung der Tangente in diesem Punkt ist -1,5
++1. die Steigung der Tangente in diesem Punkt ist 0
  [[image:Tangentensteigung.svg|| width="700px"]]
  {{/aufgabe}}
--{{aufgabe id="Zuordnung" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung"}}
++{{aufgabe id="Steigungsfunktion zeichnen" afb="II" kompetenzen="K1, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5" interaktiv=}}
  (% 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]]
++Skizziere das Schaubild der Steigungsfunktion.
++[[image:Schaubild.svg||width=500]]
  {{/aufgabe}}
--{{aufgabe id="Skizzieren anhand Eigenschaften" afb="?" kompetenzen="" quelle=" Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung"}}
++
++{{aufgabe id="Beschleunigung" afb="II" kompetenzen="K1, K3, K4, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="6"}}
++Ein Auto soll auf freier Autobahn auf {{formula}}180\frac{km}{h}{{/formula}} beschleunigen. Die Geschwindigkeit wird annähernd durch {{formula}}v(t)=180\cdot(1-e^{-0,1t}){{/formula}} beschrieben.  {{formula}}v(t){{/formula}} beschreibt hierbei die momentante Geschwindigkeit zum Zeitpunkt {{formula}}t{{/formula}} in Sekunden. Der Verlauf der Geschwindigkeit ist dem Schaubild zu entnehmen.
++[[image:Beschleunigung.svg|| width="500px"]]
++
++(%class=abc%)
++1. Zu welchem Zeitpunkt wird die Höchstgeschwindigkeit von {{formula}}180\frac{km}{h}{{/formula}} erreicht?
++1. Wann ist die Beschleunigung am höchsten?
++1. Skizziere ein Schaubild, aus welchem die Beschleunigung zum Zeitpunkt t hervorgeht.
++{{/aufgabe}}
++
++{{aufgabe id="Zuordnung I" afb="I" kompetenzen="K4, K5" quelle="KMap" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung I"}}
++  Ordne jedem Funktionsgraph (grün) den Graphen ihrer Steigungsfunktion (blau) zu. Begründe deine Zuordnung.
++
  (% style="float:left; margin-right: 16px" %)
--Es ist ein zur y-Achse symmetrisches Schaubild einer Funktion 4. Grades gesucht. Folgende Angaben sind bekannt:
++| [[image:Polynome zuordnen f.svg||width=200]] | | | | | [[image:Polynome zuordnen C.svg||width=200]]
++| [[image:Polynome zuordnen g.svg||width=200]] | | | | | [[image:Polynome zuordnen D.svg||width=200]]
++| [[image:Polynome zuordnen h.svg||width=200]] | | | | | [[image:Polynome zuordnen B.svg||width=200]]
++| [[image:Polynome zuordnen i.svg||width=200]] | | | | | [[image:Polynome zuordnen A.svg||width=200]]
++{{/aufgabe}}
++
++{{aufgabe id="algebraischer Zusammenhang I" afb="III" kompetenzen="K1, K2, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="8" interaktiv=}}
++Das blaue Schaubild zeigt eine Funktion, das rote Schaubild zeigt ihre Steigungsfunktion.
++(%class=abc%)
++1. Bestimme die Gleichungen der beiden Schaubilder.
++1. Welchen Grad besitzen die beiden Funktionen?
++1. Stelle eine Hypothese auf, welchen Grad die Steigungsfunktion einer Funktion 4. Grades hat und überlege dir, wie du die Hypothese überprüfen kannst.
++[[image:algebra.png||width=300]]
++{{/aufgabe}}
++
++{{aufgabe id="algebraischer Zusammenhang II" afb="II" kompetenzen="K1, K2, K4, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5" interaktiv=}}
++Das blaue Schaubild zeigt eine Funktion, die roten Schaubilder zeigen ihre möglichen Steigungsfunktionen.
++[[image:algebra2.png||width=200]]
++
++[[image:algebra3.png||width=200]]  [[image:algebra4.png||width=250]]
++(%class=abc%)
++1. Ordne dem blauen Schaubild seine Steigungsfunktion begründet zu.
++1. Welchen (möglichen) Grad besitzen die drei Funktionen?
++{{/aufgabe}}
++
++
++
++{{aufgabe id="Skizzieren anhand Eigenschaften" afb="III" kompetenzen="K2, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="10"}}
++(%class=abc%)
++1. Skizziere eine mögliche Parabel 2. Grades, welche eine waagrechte Tangente an der Stelle {{formula}}x = -2{{/formula}} hat. Welche Gemeinsamkeiten haben alle Parabeln mit dieser Eigenschaft?
++1. Skizziere das Schaubild einer möglichen Funktion, welches drei waagrechte Tangenten besitzt. Welchen Grad hat diese Funktion mindestens?
++1. Eine Funktion f hat nur positive Steigungen. Skizziere das Schaubild einer möglichen Funktion.
++1. Es ist ein achsensymmetrisches Schaubild einer Funktion 4. Grades gesucht. Folgende Angaben sind bekannt, fülle die Lücken und skizziere das Schaubild der Funktion.
  (% class="border" %)
--|= {{formula}}x{{/formula}} |-4|-1|0|1  |4
--|= {{formula}}f(x){{/formula}} |-2,5| |2  |0|
--|= {{formula}}f'(x){{/formula}} |-2| |0|-1  |
++|x|-4|-1|0|1  |4
++|Funktionswert|-2,5| |2  |0|
++|Tangentensteigung|-2| |0|-1  |
  {{/aufgabe}}
--{{aufgabe id="Aussagen" 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 id="Aussagen Schaubild" afb="I" kompetenzen="K1, K4, K5, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5"}}
++Gegeben ist das Schaubild einer Funktion. Nimm Stellung zu folgenden Aussagen und begründe deine Antwort.
++[[image:Aussagen.svg|| width="500px"]]
++☐ die Tangentensteigungen sind negativ für {{formula}}x \in ]2;5[{{/formula}}
++☐ die Steigung der Tangente an der Stelle {{formula}}x = 1{{/formula}} ist kleiner als {{formula}}-2{{/formula}}
++☐ an der Stelle {{formula}}x = 2{{/formula}} liegt eine waagrechte Tangente
++☐ die Funktionswerte sind positiv für {{formula}}-4 < x < 2{{/formula}}
++☐ die Tangentensteigungen haben einen Vorzeichenwechsel bei {{formula}}x=-4{{/formula}} von ⊝ ⇾ ⊕
  {{/aufgabe}}
  {{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.svg

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Aussagen.ggb

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Aussagen.svg

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Beschleunigung.ggb

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Beschleunigung.svg

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algebra II.ggb

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algebra.ggb

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algebra2.png

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algebra3.png

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algebra4.png

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	1	+XWiki.dirktebbe

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	1	+Bei Aufgabe "Aussagen Sattelstelle" haben wir die Frage, ob diese Aufgabe hier an der richtigen Stelle ist. Sattelpunkt, Wendepunkt, Minimum und Maximum sind Begriffe, die erst in TGJ1 eingeführt werden.

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	1	+2025-06-27 12:08:40.853

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	1	+Die Aufgaben "Aussagen Polynomfunktion" und "Aussagen Sattelstelle" wurden nach 12.6 verschoben.

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	1	+2025-06-27 12:51:04.585

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