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Änderungen von Dokument BPE 12.7 Monotonie

Zuletzt geändert von Nila Nurschams am 2026/02/27 15:00
Von Version 68.1
bearbeitet von Nila Nurschams
am 2026/02/27 14:59
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Auf Version 9.2
bearbeitet von Ingrid Kolupa
am 2025/10/13 17:12
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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}}
6 +{{aufgabe id="Monotonie mit Hilfe des Schaubilds der Ableitung ermitteln" afb="I" kompetenzen="K1, K4" quelle="Ingrid Kolupa, Katharina Justice" zeit="5" cc="by-sa" tags=""}}
7 +Gegeben ist der Graph von {{formula}}f'(x){{/formula}}.
8 +[[image:Ableitungsgraph.svg]]
9 +1. In welchen Bereichen ist {{formula}}f(x){{/formula}} monoton steigend?
10 +1.
11 +1. Listenpunkt
15 15  
16 -{{aufgabe id="Monotoniebereiche bestimmen" afb="II" kompetenzen="K1, K4" quelle="Simone Kanzler" zeit="10" cc="by-sa" tags=""}}
17 -Gib die Monotoniebereiche der Funktionen {{formula}}f(x){{/formula}} an:
18 -(%class=abc%)
19 -1. {{formula}}f(x)=\frac{1}{8}(\frac{1}{3}x^3+\frac{5}{2}x^2-50x+32){{/formula}}
20 -1. {{formula}}g(x)=e^{(2x+1)}(x-1){{/formula}}
21 -1. {{formula}}h(x)=ae^{(-x-5)}x^2;~ a \neq 0{{/formula}}
22 -{{/aufgabe}}
23 23  
24 -{{aufgabe id="Skizzieren mithilfe der Monotonie" afb="II" kompetenzen="K1, K4" quelle="Simone Kanzler" zeit="9" cc="by-sa" tags=""}}
25 -Gegeben sind folgende Aussagen über die Funktion {{formula}}f(x){{/formula}}:
26 -1. Für {{formula}}x \in [-\infty;-3]{{/formula}} gilt: {{formula}}f’(x){{/formula}}>0
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}}.
29 29  
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}}.
33 33  {{/aufgabe}}
34 34  
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 -
38 -Beurteile die folgenden Aussagen:
39 -(%class=abc%)
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.
42 -1. Es gilt: {{formula}}f(-2)<f(0){{/formula}}
43 -1. Für {{formula}}x<-2{{/formula}} gilt: {{formula}}f''(x) > 0{{/formula}}
44 -{{/aufgabe}}
45 -
46 -{{aufgabe id="Monotonie" afb="II" kompetenzen="K1, K2, K5" tags="problemlösen" quelle="Dr. Andreas Dinh" cc="BY-SA" zeit="25"}}
17 +{{aufgabe id="Monotonie" afb="II" kompetenzen="K2, K1, K5" tags="problemlösen" quelle="Dr. Andreas Dinh" cc="BY-SA" zeit="25"}}
47 47  //f// bezeichnet im Folgenden eine im ganzen Definitionsbereich **D** knickfreie Funktion.
48 48  
49 49  Streng steigende Monotonie ist für //f// wie folgt definiert:
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56 56  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.
57 57  {{/aufgabe}}
58 58  
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=""/}}
30 +{{seitenreflexion/}}
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