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Zuletzt geändert von Stephanie Wietzorek am 2025/05/21 10:29
Von Version 91.1
bearbeitet von Holger Engels
am 2025/05/21 07:13
Änderungskommentar: Neues Bild Ableitungsfunktion 1.svg hochladen
Auf Version 47.1
bearbeitet von Holger Engels
am 2025/05/20 14:07
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1 1  {{seiteninhalt/}}
2 2  
3 +=== Kompetenzen ===
3 3  [[Kompetenzen.K4]] [[Kompetenzen.K5]] Ich kann Werte der Tangentensteigung graphisch bestimmen
4 4  [[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann aus Werten der Tangentensteigung einen Graphen zeichnen und diesen als Graphen der Ableitungsfunktion deuten
5 5  [[Kompetenzen.K6]] Ich kann Zusammenhänge zwischen den beiden Funktionsgraphen beschreiben
... ... @@ -9,6 +9,9 @@
9 9  **Interaktiv Erkunden:** [[Graphisches Ableiten>>https://kmap.eu/app/browser/Mathematik/Differentialrechnung/Graphisches%20Ableiten#erkunden]]
10 10  {{/lernende}}
11 11  
13 +* Kann eine Tangente den Funktionsgraphen schneiden?
14 +
15 +* Bereiche mit positiver/ negativer Steigung schraffieren
12 12  * Punktweise graphisch ableiten
13 13  * Qualitativ graphisch ableiten
14 14  * Zusammenhänge HP, TP, SP vorwärts und rückwärts
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26 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 27  {{/aufgabe}}
28 28  
29 -{{aufgabe id="Punkte mit gegebener Steigung finden" afb="?" kompetenzen="" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="?"}}
33 +{{aufgabe id="Punkte mit gegebener Steigung finden" afb="?" kompetenzen="" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="3"}}
30 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 + {{formula}}f'(x) =1{{/formula}}
36 + {{formula}}f'(x) =1,5{{/formula}}
37 + {{formula}}f'(x) =0{{/formula}}
38 + {{formula}}f'(x) = -\frac{17}{4}{{/formula}}
35 35  [[image:Tangentensteigung.svg|| width="700px"]]
36 36  {{/aufgabe}}
37 37  
38 -{{aufgabe id="Steigungsfunktion zeichnen" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="3" interaktiv=}}
42 +{{aufgabe id="Zuordnung" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung"}}
39 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 46  | [[image:Polynome zuordnen f.svg||width=200]] | | | | | [[image:Polynome zuordnen A.svg||width=200]]
47 47  | [[image:Polynome zuordnen g.svg||width=200]] | | | | | [[image:Polynome zuordnen B.svg||width=200]]
48 48  | [[image:Polynome zuordnen h.svg||width=200]] | | | | | [[image:Polynome zuordnen C.svg||width=200]]
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49 49  | [[image:Polynome zuordnen i.svg||width=200]] | | | | | [[image:Polynome zuordnen D.svg||width=200]]
50 50  {{/aufgabe}}
51 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?
50 +{{aufgabe id="Skizzieren anhand Eigenschaften" afb="?" kompetenzen="" quelle=" Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung"}}
51 +(% style="float:left; margin-right: 16px" %)
52 +Es ist ein zur y-Achse symmetrisches Schaubild einer Funktion 4. Grades gesucht. Folgende Angaben sind bekannt:
54 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 60  (% class="border" %)
61 -|x|-4|-1|0|1 |4
62 -|Funktionswert|-2,5| |2 |0|
63 -|Tangentensteigung|-2| |0|-1 |
55 +|= {{formula}}x{{/formula}} |-4|-1|0|1 |4
56 +|= {{formula}}f(x){{/formula}} |-2,5| |2 |0|
57 +|= {{formula}}f'(x){{/formula}} |-2| |0|-1 |
64 64  {{/aufgabe}}
65 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!
60 +{{aufgabe id="Aussagen" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}}
61 +Prüfe die Aussagen! Welche sind wahr?
62 +Eine Polynomfunktion 3. Grades ..
63 +☐ hat immer zwei Extrempunkte!<br/>
64 +☐ kann auch mal nur einen Extrempunkt haben!<br/>
65 +☐ kann auch mal keinen Extrempunkt haben!<br/>
66 +☐ hat immer genau einen Wendepunkt!<br/>
67 +☐ hat entweder einen Sattelpunkt oder zwei Extrempunkte!<br/>
73 73  {{/aufgabe}}
74 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 -
102 102  {{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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