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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 172.1
bearbeitet von Holger Engels
am 2025/07/21 17:10
Änderungskommentar: Neues Bild Schaubild.svg hochladen
Auf Version 93.1
bearbeitet von Holger Engels
am 2025/05/21 05:13
Änderungskommentar: Neuen Anhang Ableitungsfunktion.ggb hochladen

Zusammenfassung

Details

Seiteneigenschaften
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="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.
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.
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="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.
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.
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"}}
29 +{{aufgabe id="Punkte mit gegebener Steigung finden" afb="?" kompetenzen="" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="?"}}
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 -(%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
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}}
36 36  [[image:Tangentensteigung.svg|| width="700px"]]
37 37  {{/aufgabe}}
38 38  
39 -{{aufgabe id="Steigungsfunktion zeichnen" afb="II" kompetenzen="K1, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5" interaktiv=}}
38 +{{aufgabe id="Steigungsfunktion zeichnen" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="3" interaktiv=}}
40 40  (% style="float:left; margin-right: 16px" %)
41 41  Skizziere das Schaubild der Steigungsfunktion.
42 42  [[image:Schaubild.svg||width=500]]
43 43  {{/aufgabe}}
44 44  
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 -
44 +{{aufgabe id="Zuordnung I" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung"}}
59 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]]
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]]
64 64  {{/aufgabe}}
65 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=}}
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.
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?
72 72  
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]]
55 +b) Skizziere das Schaubild einer möglichen Funktion, welches drei waagrechte Tangenten besitzt. Welchen minimalen Grad hat die Funktion?
79 79  
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.
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.
94 94  (% class="border" %)
95 95  |x|-4|-1|0|1 |4
96 96  |Funktionswert|-2,5| |2 |0|
... ... @@ -97,14 +97,40 @@
97 97  |Tangentensteigung|-2| |0|-1 |
98 98  {{/aufgabe}}
99 99  
100 -{{aufgabe id="Aussagen Schaubild" afb="I" kompetenzen="K1, K4, K5, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5"}}
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="?"}}
101 101  Gegeben ist das Schaubild einer Funktion. Nimm Stellung zu folgenden Aussagen und begründe deine Antwort.
102 102  [[image:Aussagen.svg|| width="500px"]]
78 +☐ {{formula}}f(-3)=3{{/formula}}
79 +☐ {{formula}}x = 3{{/formula}} ist dreifache Nullstell
103 103  ☐ die Tangentensteigungen sind negativ für {{formula}}x \in ]2;5[{{/formula}}
104 -☐ die Steigung der Tangente an der Stelle {{formula}}x = 1{{/formula}} ist kleiner als {{formula}}-2{{/formula}}
81 +☐ die Steigung der Tangente an der Stelle {{formula}}x = 1<-2{{/formula}}
105 105  ☐ an der Stelle {{formula}}x = 2{{/formula}} liegt eine waagrechte Tangente
106 -☐ die Funktionswerte sind positiv für {{formula}}-4 < x < 2{{/formula}}
83 +☐ die Tangentensteigungen sind negativ für {{formula}}-4 < x < 2{{/formula}}
107 107  ☐ die Tangentensteigungen haben einen Vorzeichenwechsel bei {{formula}}x=-4{{/formula}} von ⊝ ⇾ ⊕
108 108  {{/aufgabe}}
109 109  
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 +
110 110  {{seitenreflexion bildungsplan="" kompetenzen="" anforderungsbereiche="" kriterien="" menge=""/}}
Ableitungsfunktion 3.svg
Author
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1 -XWiki.holgerengels
1 +XWiki.wies
Größe
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Beschleunigung.ggb
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Beschleunigung.svg
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Schaubild.svg
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1 -XWiki.holgerengels
1 +XWiki.wies
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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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Ableitungsfunktion.ggb
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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
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1 -Die Aufgaben "Aussagen Polynomfunktion" und "Aussagen Sattelstelle" wurden nach 12.6 verschoben.
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1 -2025-06-27 12:51:04.585
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