Änderungen von Dokument BPE 6.3 Graphisches Ableiten
Zuletzt geändert von Holger Engels am 2025/08/02 07:35
Von Version 173.5
bearbeitet von Holger Engels
am 2025/08/02 07:35
am 2025/08/02 07:35
Änderungskommentar:
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Auf Version 71.1
bearbeitet von Stephanie Wietzorek
am 2025/05/20 14:54
am 2025/05/20 14:54
Änderungskommentar:
Löschung des Bildes Ableitungsfunktion 1.svg
Zusammenfassung
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Seiteneigenschaften (3 geändert, 0 hinzugefügt, 0 gelöscht)
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Anhänge (2 geändert, 0 hinzugefügt, 12 gelöscht)
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Objekte (0 geändert, 0 hinzugefügt, 2 gelöscht)
Details
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- Titel
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... ... @@ -1,1 +1,1 @@ 1 -BPE 6.3 Graphisches Ableiten1 +BPE 6.3 Momentane Änderungsrate und graphisches Ableiten - Dokument-Autor
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... ... @@ -1,1 +1,1 @@ 1 -XWiki. holgerengels1 +XWiki.wies - Inhalt
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... ... @@ -1,5 +1,6 @@ 1 1 {{seiteninhalt/}} 2 2 3 +[[Kompetenzen.K4]] [[Kompetenzen.K5]] Ich kann Werte der Tangentensteigung graphisch bestimmen 3 3 [[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann aus Werten der Tangentensteigung einen Graphen zeichnen und diesen als Graphen der Ableitungsfunktion deuten 4 4 [[Kompetenzen.K6]] Ich kann Zusammenhänge zwischen den beiden Funktionsgraphen beschreiben 5 5 [[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann erste Hypothesen über einen möglichen algebraischen Zusammenhang zwischen Funktion und Ableitungsfunktion entwickeln ... ... @@ -8,78 +8,48 @@ 8 8 **Interaktiv Erkunden:** [[Graphisches Ableiten>>https://kmap.eu/app/browser/Mathematik/Differentialrechnung/Graphisches%20Ableiten#erkunden]] 9 9 {{/lernende}} 10 10 11 -{{aufgabe id="Tangenten einzeichnen" afb="I" kompetenzen="K4, K5" quelle="Holger Engels" cc="BY-SA" zeit="3"}} 12 -Zeichne jeweils die Tangenten an den Stellen {{formula}}x\in\{-1; 0; 1\}{{/formula}} ein und bestimme deren Steigungen. 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. 13 13 [[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"]] 14 14 {{/aufgabe}} 15 15 16 -{{aufgabe id="Rauf und runter" afb="I" kompetenzen="K4, K5" quelle="Holger Engels" cc="BY-SA" zeit="3"}} 17 -Markiere zuerst alle Stellen an denen die Kurve die Steigung null hat. 18 -Markiere dann auf der x-Achse Intervalle mit positiver Steigung blau und Intervalle mit negativer Steigung rot. 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. 19 19 [[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"]] 20 20 {{/aufgabe}} 21 21 22 -{{aufgabe id="Punkte mit gegebener Steigung finden" afb=" I" kompetenzen="K2, K4, K5" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="5"}}29 +{{aufgabe id="Punkte mit gegebener Steigung finden" afb="?" kompetenzen="" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="?"}} 23 23 Es ist das Schaubild {{formula}}K_f{{/formula}} einer Funktion {{formula}}f{{/formula}} gegeben. Kennzeichne Punkte auf {{formula}}K_f{{/formula}}, für die gilt: 24 - (%class=abc%)25 - 1.die Steigung der Tangente in diesem Punkt ist 126 - 1.die Steigung der Tangente in diesem Punkt ist-1,527 - 1.die Steigung der Tangente in diesem Punkt ist031 + 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}} 28 28 [[image:Tangentensteigung.svg|| width="700px"]] 29 29 {{/aufgabe}} 30 30 31 -{{aufgabe id="Zuordnung" afb="I" kompetenzen="K4, K5" quelle="KMap" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung"}} 32 -Ordne jedem Funktionsgraph (grün) den Graphen ihrer Steigungsfunktion (blau) zu. Begründe deine Zuordnung. 33 - 38 +{{aufgabe id="Zuordnung I" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung"}} 34 34 (% style="float:left; margin-right: 16px" %) 35 -| [[image:Polynome zuordnen f.svg||width=200]] | | | | | [[image:Polynome zuordnen C.svg||width=200]]36 -| [[image:Polynome zuordnen g.svg||width=200]] | | | | | [[image:Polynome zuordnen D.svg||width=200]]37 -| [[image:Polynome zuordnen h.svg||width=200]] | | | | | [[image:Polynome zuordnen B.svg||width=200]]38 -| [[image:Polynome zuordnen i.svg||width=200]] | | | | | [[image:Polynome zuordnen A.svg||width=200]]40 +| [[image:Polynome zuordnen f.svg||width=200]] | | | | | [[image:Polynome zuordnen A.svg||width=200]] 41 +| [[image:Polynome zuordnen g.svg||width=200]] | | | | | [[image:Polynome zuordnen B.svg||width=200]] 42 +| [[image:Polynome zuordnen h.svg||width=200]] | | | | | [[image:Polynome zuordnen C.svg||width=200]] 43 +| [[image:Polynome zuordnen i.svg||width=200]] | | | | | [[image:Polynome zuordnen D.svg||width=200]] 39 39 {{/aufgabe}} 40 40 41 -{{aufgabe id="Steigungsfunktion zeichnen" afb="II" kompetenzen="K1, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5" interaktiv=}} 42 -(% style="float:left; margin-right: 16px" %) 43 -Skizziere das Schaubild der Steigungsfunktion. 44 -[[image:Schaubild.svg||width=500]] 45 -{{/aufgabe}} 46 - 47 -{{aufgabe id="Beschleunigung" afb="II" kompetenzen="K1, K3, K4, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="6"}} 48 -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. 49 -[[image:Beschleunigung.svg|| width="500px"]] 50 - 51 -(%class=abc%) 52 -1. Zu welchem Zeitpunkt wird die Höchstgeschwindigkeit von {{formula}}180\frac{km}{h}{{/formula}} erreicht? 53 -1. Wann ist die Beschleunigung am höchsten? 54 -1. Skizziere ein Schaubild, aus welchem die Beschleunigung zum Zeitpunkt t hervorgeht. 55 -{{/aufgabe}} 56 - 57 -{{aufgabe id="Algebraischer Zusammenhang I" afb="III" kompetenzen="K1, K2, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="8" interaktiv=}} 58 -Das blaue Schaubild zeigt eine Funktion, das rote Schaubild zeigt ihre Steigungsfunktion. 59 -(%class=abc%) 60 -1. Bestimme die Gleichungen der beiden Schaubilder. 61 -1. Welchen Grad besitzen die beiden Funktionen? 62 -1. Stelle eine Hypothese auf, welchen Grad die Steigungsfunktion einer Funktion 4. Grades hat und überlege dir, wie du die Hypothese überprüfen kannst. 63 - 64 -[[image:algebra.png||width=300]] 65 -{{/aufgabe}} 66 - 67 -{{aufgabe id="Algebraischer Zusammenhang II" afb="II" kompetenzen="K1, K2, K4, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5" interaktiv=}} 68 -Das blaue Schaubild zeigt eine Funktion, die roten Schaubilder zeigen ihre möglichen Steigungsfunktionen. 69 -[[image:algebra2.png||width=200]] 46 +{{aufgabe id="Skizzieren anhand Eigenschaften" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="4"}} 47 +a) Skizziere eine mögliche Parabel 2. Grades, welche eine waagrechte Tangente an der Stelle {{formula}}x = -2{{/formula}} hat. Welche Gemeinsamkeiten haben diese Parabeln? 70 70 71 -[[image:algebra3.png||width=200]] [[image:algebra4.png||width=250]] 72 -(%class=abc%) 73 -1. Ordne dem blauen Schaubild seine Steigungsfunktion begründet zu. 74 -1. Welchen (möglichen) Grad besitzen die drei Funktionen? 75 -{{/aufgabe}} 76 - 77 -{{aufgabe id="Skizzieren anhand Eigenschaften" afb="III" kompetenzen="K2, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="10"}} 78 -(%class=abc%) 79 -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? 80 -1. Skizziere das Schaubild einer möglichen Funktion, welches drei waagrechte Tangenten besitzt. Welchen Grad hat diese Funktion mindestens? 81 -1. Eine Funktion f hat nur positive Steigungen. Skizziere das Schaubild einer möglichen Funktion. 82 -1. Es ist ein achsensymmetrisches Schaubild einer Funktion 4. Grades gesucht. Folgende Angaben sind bekannt, fülle die Lücken und skizziere das Schaubild der Funktion. 49 +b) Skizziere das Schaubild einer möglichen Funktion, welches drei waagrechte Tangenten besitzt. Welchen minimalen Grad hat die Funktion? 50 + 51 +c) Eine Funktion f hat nur positive Steigungen. Skizziere das Schaubild der Ableitungsfunktion. 52 + 53 +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. 83 83 (% class="border" %) 84 84 |x|-4|-1|0|1 |4 85 85 |Funktionswert|-2,5| |2 |0| ... ... @@ -86,14 +86,38 @@ 86 86 |Tangentensteigung|-2| |0|-1 | 87 87 {{/aufgabe}} 88 88 89 -{{aufgabe id="Aussagen Schaubild" afb="I" kompetenzen="K1, K4, K5, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5"}} 60 +{{aufgabe id="Aussagen Polynomfunktion" 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 Schaubild" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="?"}} 90 90 Gegeben ist das Schaubild einer Funktion. Nimm Stellung zu folgenden Aussagen und begründe deine Antwort. 91 91 [[image:Aussagen.svg|| width="500px"]] 72 +☐ {{formula}}f(-3)=3{{/formula}} 73 +☐ {{formula}}x = 3{{/formula}} ist dreifache Nullstell 92 92 ☐ die Tangentensteigungen sind negativ für {{formula}}x \in ]2;5[{{/formula}} 93 -☐ die Steigung der Tangente an der Stelle {{formula}}x = 1 {{/formula}} ist kleiner als {{formula}}-2{{/formula}}75 +☐ die Steigung der Tangente an der Stelle {{formula}}x = 1<-2{{/formula}} 94 94 ☐ an der Stelle {{formula}}x = 2{{/formula}} liegt eine waagrechte Tangente 95 -☐ die Funktionswerte sindpositiv für {{formula}}-4 < x < 2{{/formula}}77 +☐ die Tangentensteigungen sind negativ für {{formula}}-4 < x < 2{{/formula}} 96 96 ☐ die Tangentensteigungen haben einen Vorzeichenwechsel bei {{formula}}x=-4{{/formula}} von ⊝ ⇾ ⊕ 97 97 {{/aufgabe}} 98 98 99 -{{seitenreflexion bildungsplan="5" kompetenzen="4" anforderungsbereiche="4" kriterien="5" menge="3"/}} 81 +{{aufgabe id="Aussagen Sattelstelle" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}} 82 +Welche Aussagen treffen auf eine Sattelstelle zu? 83 +☐ Eine Sattelstelle hat eine waagrechte Asymptote 84 +☐ An einer Sattelstelle hat die Steigung ein Maximum oder ein Minimum 85 +☐ An einer Sattelstelle gibt es immer auch einen Krümmungswechsel 86 +☐ Eine Sattelstelle ist auch eine Wendestelle 87 +☐ Eine Sattelstelle kann auch eine Maximalstelle sein 88 +{{/aufgabe}} 89 + 90 +{{aufgabe id="Zuordnung II" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="?"}} 91 +Es ist das Schaubild einer Steigungsfunktion gegeben. Zudem sind drei Schaubilder von drei Funktionen (A, B und C) gegeben. Zu welcher Funktion (A, B oer C) gehört diese Steigungsfunktion? Begründe deine Zuordnung. 92 + 93 +{{/aufgabe}} 94 +{{seitenreflexion bildungsplan="" kompetenzen="" anforderungsbereiche="" kriterien="" menge=""/}}
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... ... @@ -1,1 +1,0 @@ 1 -XWiki.holgerengels - Größe
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- Ableitungsfunktion 3.svg
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- Author
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... ... @@ -1,1 +1,1 @@ 1 -XWiki. holgerengels1 +XWiki.wies - Größe
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... ... @@ -1,1 +1,1 @@ 1 - 16.8KB1 +23.1 KB - Inhalt
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... ... @@ -1,1 +1,1 @@ 1 -XWiki. holgerengels1 +XWiki.wies - Größe
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... ... @@ -1,1 +1,0 @@ 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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... ... @@ -1,1 +1,0 @@ 1 -Die Aufgaben "Aussagen Polynomfunktion" und "Aussagen Sattelstelle" wurden nach 12.6 verschoben.