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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 98.1
bearbeitet von Stephanie Wietzorek
am 2025/05/21 07:45
Änderungskommentar: Neuen Anhang Beschleunigung.ggb hochladen
Auf Version 173.5
bearbeitet von Holger Engels
am 2025/08/02 07:35
Änderungskommentar: Es gibt keinen Kommentar für diese Version

Zusammenfassung

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Titel
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1 -BPE 6.3 Momentane Änderungsrate und graphisches Ableiten
1 +BPE 6.3 Graphisches Ableiten
Dokument-Autor
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1 -XWiki.wies
1 +XWiki.holgerengels
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
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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 -{{aufgabe id="Beschleunigung" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="?"}}
20 -Ein Auto soll auf freier Autobahn auf {{formula}}180/frac{km}{h}{{/formula}} beschleunigen. Die Geschwindigkeit wird durch {{formula}}v(t)=180\cdot(1-\exp{-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.
21 -[[image:Beschleunigung.svg|| width="500px"]]
22 -
23 -(%class=abc%)
24 -1. Zu welchem Zeitpunkt wird die Höchstgeschwindigkeit von {{formula}}180/frac{km}{h}{{/formula}} erreicht?
25 -1. Wann ist die Beschleunigung am höchsten?
26 -1. Skizziere ein Schaubild, aus welchem die Beschleunigung zum Zeitpunkt t hervorgeht.
27 -{{/aufgabe}}
28 -
29 -{{aufgabe id="Tangenten einzeichnen" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
30 -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.
31 31  [[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"]]
32 32  {{/aufgabe}}
33 33  
34 -{{aufgabe id="Rauf und runter" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
35 -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.
36 36  [[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"]]
37 37  {{/aufgabe}}
38 38  
39 -{{aufgabe id="Punkte mit gegebener Steigung finden" afb="?" kompetenzen="" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="?"}}
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"}}
40 40  Es ist das Schaubild {{formula}}K_f{{/formula}} einer Funktion {{formula}}f{{/formula}} gegeben. Kennzeichne Punkte auf {{formula}}K_f{{/formula}}, für die gilt:
41 - die Steigung der Tangente in diesem Punkt ist 1
42 - die Steigung der Tangente in diesem Punkt ist 1,5
43 - die Steigung der Tangente in diesem Punkt ist 0
44 - die Steigung der Tangente in diesem Punkt ist {{formula}}-\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
45 45  [[image:Tangentensteigung.svg|| width="700px"]]
46 46  {{/aufgabe}}
47 47  
48 -{{aufgabe id="Steigungsfunktion zeichnen" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="3" interaktiv=}}
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 +
49 49  (% 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 +
41 +{{aufgabe id="Steigungsfunktion zeichnen" afb="II" kompetenzen="K1, K4, K5" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5" interaktiv=}}
42 +(% style="float:left; margin-right: 16px" %)
50 50  Skizziere das Schaubild der Steigungsfunktion.
51 51  [[image:Schaubild.svg||width=500]]
52 52  {{/aufgabe}}
53 53  
54 -{{aufgabe id="Zuordnung I" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung I"}}
55 -(% style="float:left; margin-right: 16px" %)
56 -| [[image:Polynome zuordnen f.svg||width=200]] | | | | | [[image:Polynome zuordnen A.svg||width=200]]
57 -| [[image:Polynome zuordnen g.svg||width=200]] | | | | | [[image:Polynome zuordnen B.svg||width=200]]
58 -| [[image:Polynome zuordnen h.svg||width=200]] | | | | | [[image:Polynome zuordnen C.svg||width=200]]
59 -| [[image:Polynome zuordnen i.svg||width=200]] | | | | | [[image:Polynome zuordnen D.svg||width=200]]
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.
60 60  {{/aufgabe}}
61 61  
62 -{{aufgabe id="Skizzieren anhand Eigenschaften" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="4"}}
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.
63 63  (%class=abc%)
64 -1. Skizziere eine mögliche Parabel 2. Grades, welche eine waagrechte Tangente an der Stelle {{formula}}x = -2{{/formula}} hat. Welche Gemeinsamkeiten haben diese Parabeln?
65 -1. Skizziere das Schaubild einer möglichen Funktion, welches drei waagrechte Tangenten besitzt. Welchen minimalen Grad hat die Funktion?
66 -1. Eine Funktion f hat nur positive Steigungen. Skizziere das Schaubild der Ableitungsfunktion.
67 -1. 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 +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]]
70 +
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?
75 +{{/aufgabe}}
76 +
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.
68 68  (% class="border" %)
69 69  |x|-4|-1|0|1 |4
70 70  |Funktionswert|-2,5| |2 |0|
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71 71  |Tangentensteigung|-2| |0|-1 |
72 72  {{/aufgabe}}
73 73  
74 -{{aufgabe id="Aussagen Polynomfunktion" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}}
75 -Prüfe die Aussagen! Welche sind wahr? Eine Polynomfunktion 3. Grades ..
76 -☐ hat immer zwei Extrempunkte!
77 -☐ kann auch mal nur einen Extrempunkt haben!
78 -☐ kann auch mal keinen Extrempunkt haben!
79 -☐ hat immer genau einen Wendepunkt!
80 -☐ hat entweder einen Sattelpunkt oder zwei Extrempunkte!
81 -{{/aufgabe}}
82 -
83 -{{aufgabe id="Aussagen Schaubild" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="?"}}
89 +{{aufgabe id="Aussagen Schaubild" afb="I" kompetenzen="K1, K4, K5, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5"}}
84 84  Gegeben ist das Schaubild einer Funktion. Nimm Stellung zu folgenden Aussagen und begründe deine Antwort.
85 85  [[image:Aussagen.svg|| width="500px"]]
86 -☐ {{formula}}f(-3)=3{{/formula}}
87 -☐ {{formula}}x = 3{{/formula}} ist dreifache Nullstell
88 88  ☐ die Tangentensteigungen sind negativ für {{formula}}x \in ]2;5[{{/formula}}
89 -☐ die Steigung der Tangente an der Stelle {{formula}}x = 1<-2{{/formula}}
93 +☐ die Steigung der Tangente an der Stelle {{formula}}x = 1{{/formula}} ist kleiner als {{formula}}-2{{/formula}}
90 90  ☐ an der Stelle {{formula}}x = 2{{/formula}} liegt eine waagrechte Tangente
91 -☐ die Tangentensteigungen sind negativ für {{formula}}-4 < x < 2{{/formula}}
95 +☐ die Funktionswerte sind positiv für {{formula}}-4 < x < 2{{/formula}}
92 92  ☐ die Tangentensteigungen haben einen Vorzeichenwechsel bei {{formula}}x=-4{{/formula}} von ⊝ ⇾ ⊕
93 93  {{/aufgabe}}
94 94  
95 -{{aufgabe id="Aussagen Sattelstelle" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}}
96 -Welche Aussagen treffen auf eine Sattelstelle zu?
97 -☐ Eine Sattelstelle hat eine waagrechte Asymptote
98 -☐ An einer Sattelstelle hat die Steigung ein Maximum oder ein Minimum
99 -☐ An einer Sattelstelle gibt es immer auch einen Krümmungswechsel
100 -☐ Eine Sattelstelle ist auch eine Wendestelle
101 -☐ Eine Sattelstelle kann auch eine Maximalstelle sein
102 -{{/aufgabe}}
103 -
104 -
105 -
106 -{{seitenreflexion bildungsplan="" kompetenzen="" anforderungsbereiche="" kriterien="" menge=""/}}
99 +{{seitenreflexion bildungsplan="5" kompetenzen="4" anforderungsbereiche="4" kriterien="5" menge="3"/}}
Ableitungsfunktion.ggb
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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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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.
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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.