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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 91.1
bearbeitet von Holger Engels
am 2025/05/21 05:13
Änderungskommentar: Neues Bild Ableitungsfunktion 1.svg hochladen
Auf Version 172.1
bearbeitet von Holger Engels
am 2025/07/21 17:10
Änderungskommentar: Neues Bild Schaubild.svg hochladen

Zusammenfassung

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Inhalt
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16 16  * Funktionsterm der Ableitungsfunktion aus Tangentensteigungen aufstellen
17 17  * Beobachtungen bei e^x
18 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.
19 +{{aufgabe id="Tangenten einzeichnen" afb="I" kompetenzen="K4, K5" 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.
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"]]
22 22  {{/aufgabe}}
23 23  
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.
24 +{{aufgabe id="Rauf und runter" afb="I" kompetenzen="K4, K5" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
25 +Markiere zuerst alle Stellen an denen die Kurve die Steigung null hat.
26 +Markiere dann auf der x-Achse Intervalle mit positiver Steigung blau und Intervalle mit negativer Steigung rot.
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="?"}}
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"}}
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}}
32 +(%class=abc%)
33 +1. die Steigung der Tangente in diesem Punkt ist 1
34 +1. die Steigung der Tangente in diesem Punkt ist -1,5
35 +1. die Steigung der Tangente in diesem Punkt ist 0
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=}}
39 +{{aufgabe id="Steigungsfunktion zeichnen" afb="II" kompetenzen="K1, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5" interaktiv=}}
39 39  (% style="float:left; margin-right: 16px" %)
40 40  Skizziere das Schaubild der Steigungsfunktion.
41 41  [[image:Schaubild.svg||width=500]]
42 42  {{/aufgabe}}
43 43  
44 -{{aufgabe id="Zuordnung I" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung"}}
45 +
46 +{{aufgabe id="Beschleunigung" afb="II" kompetenzen="K1, K3, K4, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="6"}}
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 +[[image:Beschleunigung.svg|| width="500px"]]
49 +
50 +(%class=abc%)
51 +1. Zu welchem Zeitpunkt wird die Höchstgeschwindigkeit von {{formula}}180\frac{km}{h}{{/formula}} erreicht?
52 +1. Wann ist die Beschleunigung am höchsten?
53 +1. Skizziere ein Schaubild, aus welchem die Beschleunigung zum Zeitpunkt t hervorgeht.
54 +{{/aufgabe}}
55 +
56 +{{aufgabe id="Zuordnung I" afb="I" kompetenzen="K4, K5" 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 +
45 45  (% style="float:left; margin-right: 16px" %)
46 -| [[image:Polynome zuordnen f.svg||width=200]] | | | | | [[image:Polynome zuordnen A.svg||width=200]]
47 -| [[image:Polynome zuordnen g.svg||width=200]] | | | | | [[image:Polynome zuordnen B.svg||width=200]]
48 -| [[image:Polynome zuordnen h.svg||width=200]] | | | | | [[image:Polynome zuordnen C.svg||width=200]]
49 -| [[image:Polynome zuordnen i.svg||width=200]] | | | | | [[image:Polynome zuordnen D.svg||width=200]]
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]]
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?
66 +{{aufgabe id="algebraischer Zusammenhang I" afb="III" kompetenzen="K1, K2, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="8" interaktiv=}}
67 +Das blaue Schaubild zeigt eine Funktion, das rote Schaubild zeigt ihre Steigungsfunktion.
68 +(%class=abc%)
69 +1. Bestimme die Gleichungen der beiden Schaubilder.
70 +1. Welchen Grad besitzen die beiden Funktionen?
71 +1. Stelle eine Hypothese auf, welchen Grad die Steigungsfunktion einer Funktion 4. Grades hat und überlege dir, wie du die Hypothese überprüfen kannst.
54 54  
55 -b) Skizziere das Schaubild einer möglichen Funktion, welches drei waagrechte Tangenten besitzt. Welchen minimalen Grad hat die Funktion?
73 +[[image:algebra.png||width=300]]
74 +{{/aufgabe}}
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=}}
77 +Das blaue Schaubild zeigt eine Funktion, die roten Schaubilder zeigen ihre möglichen Steigungsfunktionen.
78 +[[image:algebra2.png||width=200]]
56 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.
80 +[[image:algebra3.png||width=200]] [[image:algebra4.png||width=250]]
81 +(%class=abc%)
82 +1. Ordne dem blauen Schaubild seine Steigungsfunktion begründet zu.
83 +1. Welchen (möglichen) Grad besitzen die drei Funktionen?
84 +{{/aufgabe}}
85 +
86 +
87 +
88 +{{aufgabe id="Skizzieren anhand Eigenschaften" afb="III" kompetenzen="K2, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="10"}}
89 +(%class=abc%)
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?
91 +1. Skizziere das Schaubild einer möglichen Funktion, welches drei waagrechte Tangenten besitzt. Welchen Grad hat diese Funktion mindestens?
92 +1. Eine Funktion f hat nur positive Steigungen. Skizziere das Schaubild einer möglichen Funktion.
93 +1. Es ist ein achsensymmetrisches Schaubild einer Funktion 4. Grades gesucht. Folgende Angaben sind bekannt, fülle die Lücken und skizziere das Schaubild der Funktion.
60 60  (% class="border" %)
61 61  |x|-4|-1|0|1 |4
62 62  |Funktionswert|-2,5| |2 |0|
... ... @@ -63,40 +63,14 @@
63 63  |Tangentensteigung|-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!
73 -{{/aufgabe}}
74 -
75 -{{aufgabe id="Aussagen Schaubild" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="?"}}
100 +{{aufgabe id="Aussagen Schaubild" afb="I" kompetenzen="K1, K4, K5, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5"}}
76 76  Gegeben ist das Schaubild einer Funktion. Nimm Stellung zu folgenden Aussagen und begründe deine Antwort.
77 77  [[image:Aussagen.svg|| width="500px"]]
78 -☐ {{formula}}f(-3)=3{{/formula}}
79 -☐ {{formula}}x = 3{{/formula}} ist dreifache Nullstell
80 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}}
104 +☐ die Steigung der Tangente an der Stelle {{formula}}x = 1{{/formula}} ist kleiner als {{formula}}-2{{/formula}}
82 82  ☐ an der Stelle {{formula}}x = 2{{/formula}} liegt eine waagrechte Tangente
83 -☐ die Tangentensteigungen sind negativ für {{formula}}-4 < x < 2{{/formula}}
106 +☐ die Funktionswerte sind positiv für {{formula}}-4 < x < 2{{/formula}}
84 84  ☐ die Tangentensteigungen haben einen Vorzeichenwechsel bei {{formula}}x=-4{{/formula}} von ⊝ ⇾ ⊕
85 85  {{/aufgabe}}
86 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 2.svg
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Ableitungsfunktion 3.svg
Author
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1 -XWiki.wies
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Ableitungsfunktion.ggb
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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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1 +XWiki.wies
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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
XWiki.XWikiComments[1]
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1 +XWiki.holgerengels
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1 +Die Aufgaben "Aussagen Polynomfunktion" und "Aussagen Sattelstelle" wurden nach 12.6 verschoben.
Datum
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1 +2025-06-27 12:51:04.585
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