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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 173.3
bearbeitet von Holger Engels
am 2025/07/29 10:07
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Auf Version 95.1
bearbeitet von Holger Engels
am 2025/05/21 05:19
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Inhalt
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1 1  {{seiteninhalt/}}
2 2  
3 +[[Kompetenzen.K4]] [[Kompetenzen.K5]] Ich kann Werte der Tangentensteigung graphisch bestimmen
3 3  [[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann aus Werten der Tangentensteigung einen Graphen zeichnen und diesen als Graphen der Ableitungsfunktion deuten
4 4  [[Kompetenzen.K6]] Ich kann Zusammenhänge zwischen den beiden Funktionsgraphen beschreiben
5 5  [[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann erste Hypothesen über einen möglichen algebraischen Zusammenhang zwischen Funktion und Ableitungsfunktion entwickeln
... ... @@ -8,78 +8,52 @@
8 8  **Interaktiv Erkunden:** [[Graphisches Ableiten>>https://kmap.eu/app/browser/Mathematik/Differentialrechnung/Graphisches%20Ableiten#erkunden]]
9 9  {{/lernende}}
10 10  
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.
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="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.
13 13  [[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"]]
14 14  {{/aufgabe}}
15 15  
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.
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.
19 19  [[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"]]
20 20  {{/aufgabe}}
21 21  
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"}}
29 +{{aufgabe id="Punkte mit gegebener Steigung finden" afb="?" kompetenzen="" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="?"}}
23 23  Es ist das Schaubild {{formula}}K_f{{/formula}} einer Funktion {{formula}}f{{/formula}} gegeben. Kennzeichne Punkte auf {{formula}}K_f{{/formula}}, für die gilt:
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
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}}
28 28  [[image:Tangentensteigung.svg|| width="700px"]]
29 29  {{/aufgabe}}
30 30  
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 -
38 +{{aufgabe id="Steigungsfunktion zeichnen" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="3" interaktiv=}}
34 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 -
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" %)
43 43  Skizziere das Schaubild der Steigungsfunktion.
44 44  [[image:Schaubild.svg||width=500]]
45 45  {{/aufgabe}}
46 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.
44 +{{aufgabe id="Zuordnung I" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="4" interaktiv="Interaktiv Zuordnung I"}}
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]]
55 55  {{/aufgabe}}
56 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.
52 +{{aufgabe id="Skizzieren anhand Eigenschaften" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="4"}}
59 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]]
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.
54 +1. Skizziere eine mögliche Parabel 2. Grades, welche eine waagrechte Tangente an der Stelle {{formula}}x = -2{{/formula}} hat. Welche Gemeinsamkeiten haben diese Parabeln?
55 +1. Skizziere das Schaubild einer möglichen Funktion, welches drei waagrechte Tangenten besitzt. Welchen minimalen Grad hat die Funktion?
56 +1. Eine Funktion f hat nur positive Steigungen. Skizziere das Schaubild der Ableitungsfunktion.
57 +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.
83 83  (% class="border" %)
84 84  |x|-4|-1|0|1 |4
85 85  |Funktionswert|-2,5| |2 |0|
... ... @@ -86,14 +86,40 @@
86 86  |Tangentensteigung|-2| |0|-1 |
87 87  {{/aufgabe}}
88 88  
89 -{{aufgabe id="Aussagen Schaubild" afb="I" kompetenzen="K1, K4, K5, K6" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="5"}}
64 +{{aufgabe id="Aussagen Polynomfunktion" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}}
65 +Prüfe die Aussagen! Welche sind wahr? Eine Polynomfunktion 3. Grades ..
66 +☐ hat immer zwei Extrempunkte!
67 +☐ kann auch mal nur einen Extrempunkt haben!
68 +☐ kann auch mal keinen Extrempunkt haben!
69 +☐ hat immer genau einen Wendepunkt!
70 +☐ hat entweder einen Sattelpunkt oder zwei Extrempunkte!
71 +{{/aufgabe}}
72 +
73 +{{aufgabe id="Aussagen Schaubild" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="?"}}
90 90  Gegeben ist das Schaubild einer Funktion. Nimm Stellung zu folgenden Aussagen und begründe deine Antwort.
91 91  [[image:Aussagen.svg|| width="500px"]]
76 +☐ {{formula}}f(-3)=3{{/formula}}
77 +☐ {{formula}}x = 3{{/formula}} ist dreifache Nullstell
92 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}}
79 +☐ die Steigung der Tangente an der Stelle {{formula}}x = 1<-2{{/formula}}
94 94  ☐ an der Stelle {{formula}}x = 2{{/formula}} liegt eine waagrechte Tangente
95 -☐ die Funktionswerte sind positiv für {{formula}}-4 < x < 2{{/formula}}
81 +☐ die Tangentensteigungen sind negativ für {{formula}}-4 < x < 2{{/formula}}
96 96  ☐ die Tangentensteigungen haben einen Vorzeichenwechsel bei {{formula}}x=-4{{/formula}} von ⊝ ⇾ ⊕
97 97  {{/aufgabe}}
98 98  
85 +{{aufgabe id="Aussagen Sattelstelle" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}}
86 +Welche Aussagen treffen auf eine Sattelstelle zu?
87 +☐ Eine Sattelstelle hat eine waagrechte Asymptote
88 +☐ An einer Sattelstelle hat die Steigung ein Maximum oder ein Minimum
89 +☐ An einer Sattelstelle gibt es immer auch einen Krümmungswechsel
90 +☐ Eine Sattelstelle ist auch eine Wendestelle
91 +☐ Eine Sattelstelle kann auch eine Maximalstelle sein
92 +{{/aufgabe}}
93 +
94 +{{aufgabe id="Zuordnung II" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="?"}}
95 +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.
96 +[[image:Ableitungsfunktion.svg|| width="300px"]]
97 +[[image:Ableitungsfunktion 1.svg||width="300px"]] [[image:Ableitungsfunktion 2.svg||width=300]] [[image:Ableitungsfunktion 3.svg||width=300]]
98 +{{/aufgabe}}
99 +
99 99  {{seitenreflexion bildungsplan="" kompetenzen="" anforderungsbereiche="" kriterien="" menge=""/}}
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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 -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 -XWiki.holgerengels
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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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1 -0