Änderungen von Dokument BPE 6.3 Graphisches Ableiten

Zusammenfassung

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  • Ableitungsfunktion 1.svg
  • Ableitungsfunktion 2.svg
  • Ableitungsfunktion 3.svg
  • Ableitungsfunktion.svg
  • Aussagen.ggb
  • Aussagen.svg
  • Beschleunigung.ggb
  • Beschleunigung.svg
  • Schaubild.ggb
  • Schaubild.svg
  • algebra II.ggb
  • algebra.ggb
  • algebra.png
  • algebra2.png
  • algebra3.png
  • algebra4.png
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Titel

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1		-BPE 6.3 Momentane Änderungsrate und graphisches Ableiten
	1	+BPE 6.3 Graphisches Ableiten

Inhalt

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1	1	{{seiteninhalt/}}
2	2
3		-=== Kompetenzen ===
4		-[[Kompetenzen.K4]] [[Kompetenzen.K5]] Ich kann Werte der Tangentensteigung graphisch bestimmen
5	5	[[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann aus Werten der Tangentensteigung einen Graphen zeichnen und diesen als Graphen der Ableitungsfunktion deuten
6	6	[[Kompetenzen.K6]] Ich kann Zusammenhänge zwischen den beiden Funktionsgraphen beschreiben
7	7	[[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann erste Hypothesen über einen möglichen algebraischen Zusammenhang zwischen Funktion und Ableitungsfunktion entwickeln
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10	10	**Interaktiv Erkunden:** [[Graphisches Ableiten>>https://kmap.eu/app/browser/Mathematik/Differentialrechnung/Graphisches%20Ableiten#erkunden]]
11	11	{{/lernende}}
12	12
13		-* Kann eine Tangente den Funktionsgraphen schneiden?
14		-
15		-* Bereiche mit positiver/ negativer Steigung schraffieren
16		-* Punktweise graphisch ableiten
17		-* Qualitativ graphisch ableiten
18		-* Zusammenhänge HP, TP, SP vorwärts und rückwärts
19		-
20		-* Funktionsterm der Ableitungsfunktion aus Tangentensteigungen aufstellen
21		-* Beobachtungen bei e^x
22		-
23		-{{aufgabe id="Tangenten einzeichnen" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
24		-Zeichne jeweils die Tangenten an den Stellen {{formula}}x\in\{-1, 0, 1\}{{/formula}} ein und bestimme deren Steigungen.
	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.
25	25	[[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"]]
26	26	{{/aufgabe}}
27	27
28		-{{aufgabe id="Rauf und runter" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
29		-Markiere jeweils auf der x-Achse Intervalle mit positiver Steigung blau und negativer Steigung rot. Markiere die Stellen mit Steigung Null.
	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.
30	30	[[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"]]
31	31	{{/aufgabe}}
32	32
33		-{{aufgabe id="Punkte mit gegebener Steigung finden" afb="?" kompetenzen="" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="3"}}
	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"}}
34	34	Es ist das Schaubild {{formula}}K_f{{/formula}} einer Funktion {{formula}}f{{/formula}} gegeben. Kennzeichne Punkte auf {{formula}}K_f{{/formula}}, für die gilt:
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}}
	24	+(%class=abc%)
	25	+1. die Steigung der Tangente in diesem Punkt ist 1
	26	+1. die Steigung der Tangente in diesem Punkt ist -1,5
	27	+1. die Steigung der Tangente in diesem Punkt ist 0
39	39	[[image:Tangentensteigung.svg|| width="700px"]]
40	40	{{/aufgabe}}
41	41
42		-{{aufgabe id="Zuordnung" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung"}}
	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	+
43	43	(% style="float:left; margin-right: 16px" %)
44		-| [[image:Polynome zuordnen f.svg||width=200]] | | | | | [[image:Polynome zuordnen A.svg||width=200]]
45		-| [[image:Polynome zuordnen g.svg||width=200]] | | | | | [[image:Polynome zuordnen B.svg||width=200]]
46		-| [[image:Polynome zuordnen h.svg||width=200]] | | | | | [[image:Polynome zuordnen C.svg||width=200]]
47		-| [[image:Polynome zuordnen i.svg||width=200]] | | | | | [[image:Polynome zuordnen D.svg||width=200]]
	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]]
48	48	{{/aufgabe}}
49	49
50		-{{aufgabe id="Skizzieren anhand Eigenschaften" afb="?" kompetenzen="" quelle=" Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung"}}
	41	+{{aufgabe id="Steigungsfunktion zeichnen" afb="II" kompetenzen="K1, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5" interaktiv=}}
51	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:
	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]]
53	53
54		-(% class="border" %)
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 |
	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?
58	58	{{/aufgabe}}
59	59
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!
	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.
	83	+(% class="border" %)
	84	+|x|-4|-1|0|1 |4
	85	+|Funktionswert|-2,5| |2 |0|
	86	+|Tangentensteigung|-2| |0|-1 |
67	67	{{/aufgabe}}
68	68
69		-{{aufgabe id="Aussagen Sattelstelle" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}}
70		-Welche Aussagen treffen auf eine Sattelstelle zu?
71		-☐ Eine Sattelstelle hat eine waagrechte Asymptote
72		-☐ An einer Sattelstelle hat die Steigung ein Maximum oder ein Minimum
73		-☐ An einer Sattelstelle gibt es immer auch einen Krümmungswechsel
74		-☐ Eine Sattelstelle ist auch eine Wendestelle
75		-☐ Eine Sattelstelle kann auch eine Maximalstelle sein
	89	+{{aufgabe id="Aussagen Schaubild" afb="I" kompetenzen="K1, K4, K5, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5"}}
	90	+Gegeben ist das Schaubild einer Funktion. Nimm Stellung zu folgenden Aussagen und begründe deine Antwort.
	91	+[[image:Aussagen.svg|| width="500px"]]
	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}}
	94	+☐ an der Stelle {{formula}}x = 2{{/formula}} liegt eine waagrechte Tangente
	95	+☐ die Funktionswerte sind positiv für {{formula}}-4 < x < 2{{/formula}}
	96	+☐ die Tangentensteigungen haben einen Vorzeichenwechsel bei {{formula}}x=-4{{/formula}} von ⊝ ⇾ ⊕
76	76	{{/aufgabe}}
77	77
78		-{{seitenreflexion bildungsplan="" kompetenzen="" anforderungsbereiche="" kriterien="" menge=""/}}
	99	+{{seitenreflexion bildungsplan="5" kompetenzen="4" anforderungsbereiche="4" kriterien="5" menge="3"/}}

Ableitungsfunktion 1.svg

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	1	+XWiki.holgerengels

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Inhalt

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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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algebra.png

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algebra2.png

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algebra3.png

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algebra4.png

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XWiki.XWikiComments[0]

Datum

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

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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.

XWiki.XWikiComments[1]

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

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	1	+XWiki.holgerengels

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