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Änderungen von Dokument BPE 6.3 Graphisches Ableiten

Zuletzt geändert von Holger Engels am 2025/08/02 07:35
Von Version 169.3
bearbeitet von Holger Engels
am 2025/06/27 10:51
Änderungskommentar: Kommentar hinzugefügt
Auf Version 173.3
bearbeitet von Holger Engels
am 2025/07/29 10:07
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Zusammenfassung

Details

Seiteneigenschaften
Inhalt
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1 1  {{seiteninhalt/}}
2 2  
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
6 6  [[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann erste Hypothesen über einen möglichen algebraischen Zusammenhang zwischen Funktion und Ableitungsfunktion entwickeln
... ... @@ -9,13 +9,6 @@
9 9  **Interaktiv Erkunden:** [[Graphisches Ableiten>>https://kmap.eu/app/browser/Mathematik/Differentialrechnung/Graphisches%20Ableiten#erkunden]]
10 10  {{/lernende}}
11 11  
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 19  {{aufgabe id="Tangenten einzeichnen" afb="I" kompetenzen="K4, K5" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
20 20  Zeichne jeweils die Tangenten an den Stellen {{formula}}x\in\{-1; 0; 1\}{{/formula}} ein und bestimme deren Steigungen.
21 21  [[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"]]
... ... @@ -27,7 +27,7 @@
27 27  [[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"]]
28 28  {{/aufgabe}}
29 29  
30 -{{aufgabe id="Punkte mit gegebener Steigung finden" afb="II" kompetenzen="K2, K4, K5" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="5"}}
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"}}
31 31  Es ist das Schaubild {{formula}}K_f{{/formula}} einer Funktion {{formula}}f{{/formula}} gegeben. Kennzeichne Punkte auf {{formula}}K_f{{/formula}}, für die gilt:
32 32  (%class=abc%)
33 33  1. die Steigung der Tangente in diesem Punkt ist 1
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36 36  [[image:Tangentensteigung.svg|| width="700px"]]
37 37  {{/aufgabe}}
38 38  
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 +
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]]
39 +{{/aufgabe}}
40 +
39 39  {{aufgabe id="Steigungsfunktion zeichnen" afb="II" kompetenzen="K1, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5" interaktiv=}}
40 40  (% style="float:left; margin-right: 16px" %)
41 41  Skizziere das Schaubild der Steigungsfunktion.
... ... @@ -42,7 +42,6 @@
42 42  [[image:Schaubild.svg||width=500]]
43 43  {{/aufgabe}}
44 44  
45 -
46 46  {{aufgabe id="Beschleunigung" afb="II" kompetenzen="K1, K3, K4, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="6"}}
47 47  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.
48 48  [[image:Beschleunigung.svg|| width="500px"]]
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53 53  1. Skizziere ein Schaubild, aus welchem die Beschleunigung zum Zeitpunkt t hervorgeht.
54 54  {{/aufgabe}}
55 55  
56 -{{aufgabe id="Zuordnung I" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung I"}}
57 - Ordne jedem Funktionsgraph (grün) den Graphen ihrer Steigungsfunktion (blau) zu. Begründe deine Zuordnung.
58 -
59 -(% style="float:left; margin-right: 16px" %)
60 -| [[image:Polynome zuordnen f.svg||width=200]] | | | | | [[image:Polynome zuordnen C.svg||width=200]]
61 -| [[image:Polynome zuordnen g.svg||width=200]] | | | | | [[image:Polynome zuordnen D.svg||width=200]]
62 -| [[image:Polynome zuordnen h.svg||width=200]] | | | | | [[image:Polynome zuordnen B.svg||width=200]]
63 -| [[image:Polynome zuordnen i.svg||width=200]] | | | | | [[image:Polynome zuordnen A.svg||width=200]]
64 -{{/aufgabe}}
65 -
66 -{{aufgabe id="algebraischer Zusammenhang I" afb="III" kompetenzen="K1, K2, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="8" interaktiv=}}
57 +{{aufgabe id="Algebraischer Zusammenhang I" afb="III" kompetenzen="K1, K2, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="8" interaktiv=}}
67 67  Das blaue Schaubild zeigt eine Funktion, das rote Schaubild zeigt ihre Steigungsfunktion.
68 68  (%class=abc%)
69 69  1. Bestimme die Gleichungen der beiden Schaubilder.
70 70  1. Welchen Grad besitzen die beiden Funktionen?
71 71  1. Stelle eine Hypothese auf, welchen Grad die Steigungsfunktion einer Funktion 4. Grades hat und überlege dir, wie du die Hypothese überprüfen kannst.
72 -
63 +
73 73  [[image:algebra.png||width=300]]
74 74  {{/aufgabe}}
75 75  
76 -{{aufgabe id="algebraischer Zusammenhang II" afb="II" kompetenzen="K1, K2, K4, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5" interaktiv=}}
67 +{{aufgabe id="Algebraischer Zusammenhang II" afb="II" kompetenzen="K1, K2, K4, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5" interaktiv=}}
77 77  Das blaue Schaubild zeigt eine Funktion, die roten Schaubilder zeigen ihre möglichen Steigungsfunktionen.
78 78  [[image:algebra2.png||width=200]]
79 79  
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83 83  1. Welchen (möglichen) Grad besitzen die drei Funktionen?
84 84  {{/aufgabe}}
85 85  
86 -
87 -
88 88  {{aufgabe id="Skizzieren anhand Eigenschaften" afb="III" kompetenzen="K2, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="10"}}
89 89  (%class=abc%)
90 90  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?
Ableitungsfunktion.ggb
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