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Änderungen von Dokument BPE 12.7 Monotonie

Zuletzt geändert von Nila Nurschams am 2026/02/27 15:00
Von Version 55.2
bearbeitet von Holger Engels
am 2025/11/21 23:29
Änderungskommentar: Es gibt keinen Kommentar für diese Version
Auf Version 68.1
bearbeitet von Nila Nurschams
am 2026/02/27 14:59
Änderungskommentar: Neues Bild Fahrstrecke.svg hochladen

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Dokument-Autor
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1 -XWiki.holgerengels
1 +XWiki.nilanurschams
Inhalt
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3 3  [[Kompetenzen.K5]] [[Kompetenzen.K4]] Ich kann Funktionen auf strenge Monotonie untersuchen
4 4  [[Kompetenzen.K4]] Ich kann die Wertemenge einer Funktion anhand von Graphen, Funktionstermen und Wertetabellen bestimmen
5 5  
6 +{{aufgabe id="Warum sind einige Aussagen wahr oder falsch?" afb="I" kompetenzen="K5" quelle="Nila Nurschams" zeit="4" tags=""}}
7 +Gegeben ist eine Funktion {{formula}}f{{/formula}}.
8 +Beurteile die folgenden Aussagen und begründe deine Entscheidung:
9 +(%class=abc%)
10 +1. Wenn {{formula}}f^′(x)≥0{{/formula}} gilt, ist {{formula}}f{{/formula}} streng monoton steigend.
11 +1. Eine Funktion mit nur einer Nullstelle der Ableitung ist streng monoton.
12 +1. Ist {{formula}}f^′(x)>0{{/formula}} für alle {{formula}}x{{/formula}}, so besitzt {{formula}}f{{/formula}} keine Extremstellen.
13 +1. Eine streng monoton steigende Funktion kann einen Wendepunkt besitzen.
14 +{{/aufgabe}}
15 +
6 6  {{aufgabe id="Monotoniebereiche bestimmen" afb="II" kompetenzen="K1, K4" quelle="Simone Kanzler" zeit="10" cc="by-sa" tags=""}}
7 7  Gib die Monotoniebereiche der Funktionen {{formula}}f(x){{/formula}} an:
8 8  (%class=abc%)
9 9  1. {{formula}}f(x)=\frac{1}{8}(\frac{1}{3}x^3+\frac{5}{2}x^2-50x+32){{/formula}}
10 10  1. {{formula}}g(x)=e^{(2x+1)}(x-1){{/formula}}
11 -1. {{formula}}h(x)=ae^{(-x-5)}x^2{{/formula}}
21 +1. {{formula}}h(x)=ae^{(-x-5)}x^2;~ a \neq 0{{/formula}}
12 12  {{/aufgabe}}
13 13  
14 -{{aufgabe id="Schaubild skizzieren mithilfe der Monotonie" afb="II" kompetenzen="K1, K4" quelle="Simone Kanzler" zeit="9" cc="by-sa" tags=""}}
24 +{{aufgabe id="Skizzieren mithilfe der Monotonie" afb="II" kompetenzen="K1, K4" quelle="Simone Kanzler" zeit="9" cc="by-sa" tags=""}}
15 15  Gegeben sind folgende Aussagen über die Funktion {{formula}}f(x){{/formula}}:
16 -(%class=abc%)
17 17  1. Für {{formula}}x \in [-\infty;-3]{{/formula}} gilt: {{formula}}f’(x){{/formula}}>0
18 -1. Für {{formula}}x \in [-3;2]{{/formula}} gilt: {{formula}}f’(x){{/formula}}<0
19 -1. Für {{formula}} x \to \infty{{/formula}} gilt:{{formula}} f(x) \to 0{{/formula}}.
27 +1. Für {{formula}}x \in [-3;2]{{/formula}} gilt: {{formula}}f’(x){{/formula}}<=0
28 +1. Für {{formula}}x \to \infty{{/formula}} gilt:{{formula}} f(x) \to 0{{/formula}}.
20 20  
21 -a) Gib für jede Aussage das entsprechende Monotonieverhalten an.
22 -b) Skizziere mithilfe der Aussagen ein mögliches Schaubild der Funktion {{formula}}f(x){{/formula}}.
23 -
30 +(%class=abc%)
31 +1. Gib für jede Aussage das entsprechende Monotonieverhalten an.
32 +1. Skizziere mithilfe der Aussagen ein mögliches Schaubild der Funktion {{formula}}f(x){{/formula}}.
24 24  {{/aufgabe}}
25 25  
26 -{{aufgabe id="Monotonie mit Hilfe des Schaubilds der Ableitung ermitteln" afb="II" kompetenzen="K1, K4" quelle="Ingrid Kolupa, Katharina Justice" zeit="10" cc="by-sa" tags=""}}
27 -Gegeben ist der Graph von {{formula}}f'(x){{/formula}}.
28 -[[image:Ableitungsgraph.svg]]
35 +{{aufgabe id="Aus Schaubild der Ableitung" afb="II" kompetenzen="K1, K4" quelle="Ingrid Kolupa, Katharina Justice" zeit="10" cc="by-sa" tags=""}}
36 +[[image:Ableitungsgraph.svg||class="right" width=350]]Gegeben ist der Graph von {{formula}}f'{{/formula}}.
37 +
29 29  Beurteile die folgenden Aussagen:
30 30  (%class=abc%)
31 -1. Für {{formula}}x \in [2;3]{{/formula}} ist der Graph von f monoton fallend.
32 -1. Zwischen dem Hochpunkt und dem Tiefpunkt des Graphen von {{formula}}f'(x){{/formula}} ist der Graph der Funktion {{formula}}f(x){{/formula}} monoton fallend.
40 +1. Für {{formula}}x \in [2;3]{{/formula}} ist der Graph von {{formula}}f{{/formula}} monoton fallend.
41 +1. Zwischen dem Hochpunkt und dem Tiefpunkt des Graphen von {{formula}}f'{{/formula}} ist der Graph der Funktion {{formula}}f{{/formula}} monoton fallend.
33 33  1. Es gilt: {{formula}}f(-2)<f(0){{/formula}}
34 34  1. Für {{formula}}x<-2{{/formula}} gilt: {{formula}}f''(x) > 0{{/formula}}
35 35  {{/aufgabe}}
36 36  
37 -{{aufgabe id="Monotonie" afb="II" kompetenzen="K2, K1, K5" tags="problemlösen" quelle="Dr. Andreas Dinh" cc="BY-SA" zeit="25"}}
46 +{{aufgabe id="Monotonie" afb="II" kompetenzen="K1, K2, K5" tags="problemlösen" quelle="Dr. Andreas Dinh" cc="BY-SA" zeit="25"}}
38 38  //f// bezeichnet im Folgenden eine im ganzen Definitionsbereich **D** knickfreie Funktion.
39 39  
40 40  Streng steigende Monotonie ist für //f// wie folgt definiert:
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47 47  Eine Funktion kann auch dann streng monoton steigend sein, wenn {{formula}}f'(x)>0{{/formula}} nicht für alle {{formula}}x \in \textbf{D}{{/formula}} gilt.
48 48  {{/aufgabe}}
49 49  
50 -{{seitenreflexion/}}
59 +{{aufgabe id="Freier Fall" afb="II" kompetenzen="K1,K3,K5" quelle="Holger Engels" zeit="6"}}
60 +Die Geschwindigkeit eines Körpers im freien Fall wird in einem vereinfachten Modell durch die Funktion {{formula}}v{{/formula}} mit
61 +
62 +{{formula}}v(t)=\frac{m \cdot g}{\beta}\left(1-e^{-\frac{\beta}{m}\cdot t}\right);~t>=0{{/formula}}
63 +
64 +beschrieben. Zeige, dass die Geschwindigkeit stets zunimmt.
65 +
66 +**Quelle:** Wikipedia [[Fall mit Stokes-Reibung>>https://de.wikipedia.org/wiki/Fall_mit_Luftwiderstand#Fall_mit_Stokes-Reibung]]: Bei kleinen Geschwindigkeit ist die Luftreibung proportional zur Fallgeschwindigkeit.
67 +{{/aufgabe}}
68 +
69 +{{seitenreflexion bildungsplan="" kompetenzen="" anforderungsbereiche="" kriterien="" menge=""/}}
Ableitungsgraph.svg
Author
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1 -XWiki.ingridkolupa
1 +XWiki.holgerengels
Größe
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Größe
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Fahrstrecke.svg
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Größe
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