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Zuletzt geändert von Holger Engels am 2026/10/17 14:38
Von Version 33.1
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am 2026/02/27 15:06
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Auf Version 51.4
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12 12  Siehe dazu [[Rekonstruktion einer Größe>>https://kmap.eu/app/browser/Mathematik/Integralrechnung/Rekonstruktion%20einer%20Gr%C3%B6%C3%9Fe]] und [[Obersumme/Untersumme interaktiv>>https://kmap.eu/app/browser/Mathematik/Integralrechnung/Obersumme%20und%20Untersumme#erkunden]]
13 13  {{/lernende}}
14 14  
15 -Deu­tung des be­stimm­ten In­te­grals (Be­stands­re­kon­struk­ti­on, z. B. zu­rück­ge­leg­te Stre­cke bei ver­än­der­li­cher Ge­schwin­dig­keit, Flä­che)
16 -ge­krümm­te Rand­funk­ti­on
17 -Nä­he­rungs­wei­se Be­rech­nung von In­te­gra­len (z. B. Käst­chen­zäh­len, Ober- und Un­ter­sum­me)
15 +--Deu­tung des be­stimm­ten In­te­grals (Be­stands­re­kon­struk­ti­on, z. B. zu­rück­ge­leg­te Stre­cke bei ver­än­der­li­cher Ge­schwin­dig­keit, Flä­che)--
16 +--ge­krümm­te Rand­funk­ti­on--
17 +--Nä­he­rungs­wei­se Be­rech­nung von In­te­gra­len (z. B. Käst­chen­zäh­len, Ober- und Un­ter­sum­me)--
18 18  Ori­en­tier­ter Flä­chen­in­halt (z. B. Zu- und Ab­fluss­men­ge)
19 19  Ei­gen­schaf­ten des be­stimm­ten In­te­grals (In­ter­val­l­ad­di­ti­vi­tät, Li­nea­ri­tät des In­te­grals, In­te­gral­wert ist Null bei un­ge­ra­den Funk­tio­nen und zu x=0 sym­me­tri­schen In­te­gra­ti­ons­gren­zen)
20 20  
21 +== Bestandsrekonstruktion ==
21 21  {{aufgabe id="Gefäß" afb="I" kompetenzen="K4" quelle="[[KMap>>https://kmap.eu/app/browser/Mathematik/Integralrechnung/Rekonstruktion%20einer%20Gr%C3%B6%C3%9Fe]]" zeit="5"}}
22 22  Ein Gefäß sei zu Beginn der Beobachtung mit //10 l// gefüllt. Es wird folgender Zufluss/ Abluss beobachtet:
23 23  
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32 32  {{/aufgabe}}
33 33  
34 34  {{aufgabe id="Photovoltaikanlage" afb="I" kompetenzen="K4" quelle="[[KMap>>https://kmap.eu/app/browser/Mathematik/Integralrechnung/Rekonstruktion%20einer%20Gr%C3%B6%C3%9Fe]]" zeit="5"}}
35 -[[image:Photovoltaik.png||class="right" width=500]]Ein Speicher wird über eine Photovoltaikanlage geladen. Das Schaubild zeigt den Ladeleistung über die Zeit.
36 -Bestimme näherungsweise die Ladung des Speichers um 18:00 Uhr unter der Annahme, dass sie morgens um 6:00 Uhr //2 kWh// betrug.
36 +[[image:Photovoltaik.svg||class="right" width=500]]Ein Speicher wird über eine Photovoltaikanlage geladen. Das Schaubild zeigt die Ladeleistung über die Zeit.
37 +Bestimme näherungsweise die gespeicherte Energiemenge des Speichers um 18:00 Uhr unter der Annahme, dass sie morgens um 6:00 Uhr //2 kWh// betrug.
37 37  {{/aufgabe}}
38 38  
40 +{{aufgabe id="Test-Rakete" afb="II" kompetenzen="K3,K4" quelle="Holger Engels" zeit="9"}}
41 +Bei der Erprobung eines neuen Raketenantriebs werden verschiedene Flugprofile getestet. Bereits eingezeichnet in das Geschwindigkeits-Zeit-Diagramm ist ein unrealistisches Flugprofil, bei dem die Rakete schlagartig auf ihre Reisegeschwindigkeit kommt und diese die gesamten 20 Sekunden konstant hält.
42 +
43 +[[image:Rakete.svg||width=600]]
44 +Der Flächeninhalt unter dem Graphen der Geschwindigkeit entspricht dem zurückgelegten Weg //s// (in m).
45 +
46 +Zeichne für die beiden folgenden Flugprofile jeweils einen möglichen Graphen in das Koordinatensystem ein. Allen drei Flugprofilen ist die Dauer von 20 Sekunden und die Strecke von 600 Metern gemein.
47 +(%class=abc%)
48 +1. Konstante Beschleunigung: Die Rakete startet aus dem Stand (//v(0) = 0//) und beschleunigt für 20 Sekunden mit einer konstanten Rate. Die Geschwindigkeit nimmt also linear zu.
49 +1. Start-, Flug- und Bremsphase: In einem weiteren Testflug startet die Rakete aus dem Stand (//v(0) = 0//), beschleunigt 8 Sekunden lang mit einer konstanten Rate, fliegt dann 4 Sekunden lang mit konstanter Geschwindigkeit und bremst schließlich mit einer konstanten negativen Rate wieder 8 Sekunden lang bis zum Stillstand (//v(20)=0//) ab.
50 +{{/aufgabe}}
51 +
52 +{{aufgabe id="Streifenmethode" afb="II" kompetenzen="" quelle="Stefanie Walz" zeit=""}}
53 +[[Freier Fall.svg||width=300 class=right]]Das Schaubild zeigt die Höhe über der Zeit für einen Fall aus 5 Metern Höhe, modelliert durch die Funktion {{formula}}h(t)=-5\cdot t^2+5{{/formula}}. Für //n// Streifen und einer Streifenbreite von {{formula}}\frac{b}{n}{{/formula}} ergibt sich die Untersumme folgendermaßen:
54 +
55 +{{formula}}U_n = \Delta x \cdot h(1\Delta x) + \Delta x \cdot h(2\Delta x) + ... + \Delta x \cdot h(n\Delta x){{/formula}}
56 +{{formula}}U_n = \Delta x \cdot (-5 \cdot (1\cdot Deltax)^2 +5) + \Delta x \cdot (-5 \cdot (2\cdot Deltax)^2 +5) + ... + \Delta x \cdot (-5 \cdot (n\cdot Deltax)^2 +5){{/formula}}
57 +{{formula}}U_n = -5\cdot b^3\frac16(2+\frac3n+\frac1n^2+5b{{/formula}}
58 +
59 +{{/aufgabe}}
60 +
61 +== Bestimmtes Integral ==
62 +
63 +== Eigenschaften ==
64 +{{aufgabe id="Integraleigenschaften" afb="II" kompetenzen="K1" quelle="Debora Kemm, Benjamin Kaiser" zeit="10"}}
65 +Ermittle eine aufgsteigende Reihenfolge für den Wert der folgenden Integrale
66 +(%class=abc%)
67 +1. {{formula}}\int_{-5}^{5} e^{x+4} \, dx{{/formula}}
68 +1. {{formula}}\int_{-5}^{5} e^{x-4} \, dx{{/formula}}
69 +1. {{formula}}\int_{-5}^{5} (-x^5 + 4x^3-2x)\, dx{{/formula}}
70 +1. {{formula}}\int_{-5}^{5} (\cos(\frac{1}{2}x)-3)\, dx {{/formula}}
71 +{{/aufgabe}}
72 +
73 +{{aufgabe id="Bilanz null" afb="II" kompetenzen="K2, K5" quelle="Holger Engels" zeit="8"}}
74 +Es gilt für das bestimmte Integral im Intervall {{formula}}[1; 5]{{/formula}}:
75 +
76 +{{formula}}\int_{1}^5 f(x) \, dx = 0{{/formula}}
77 +
78 +Gib jeweils eine Funktionsgleichung für {{formula}}f{{/formula}} an, die diese Bedingung erfüllt. Begründe jeweils Deine Wahl.
79 +(%class=abc%)
80 +1. {{formula}}f{{/formula}} ist eine trigonometrische Funktion.
81 +1. {{formula}}f{{/formula}} ist eine Polynomfunktion.
82 +{{/aufgabe}}
83 +
84 +{{aufgabe id="Rechenregeln" afb="III" kompetenzen="K4,K5" quelle="Holger Engels" zeit="9"}}
85 +Gegeben sind zwei im Intervall {{formula}}[-5; 5]{{/formula}} stetige Funktionen {{formula}}f{{/formula}} und {{formula}}g{{/formula}}. Es sind keine Funktionsterme bekannt, lediglich die Symmetrieeigenschaften:
86 +* Der Graph der Funktion {{formula}}f{{/formula}} ist punktsymmetrisch zum Ursprung
87 +* Der Graph der Funktion {{formula}}g{{/formula}} ist achsensymmetrisch zur y-Achse
88 +und folgende bestimmte Integralwerte:
89 +
90 +(% style="display:flex"%)(((
91 +((({{formula}}\int\limits_{0}^{3}f(x)\ dx=5{{/formula}}))) ((({{formula}}\int\limits_{0}^{5}g(x)\ dx=14{{/formula}}))) ((({{formula}}\int\limits_{-3}^{3}(f(x)+g(x))\ dx=18{{/formula}})))
92 +)))
93 +
94 +Ermittle den exakten Wert des folgenden bestimmten Integrals:
95 +
96 +{{formula}}\int\limits_{-5}^{-3}g(x)\ dx{{/formula}}
97 +{{/aufgabe}}
98 +
39 39  {{seitenreflexion bildungsplan="" kompetenzen="" anforderungsbereiche="" kriterien="" menge=""/}}
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Photovoltaik.ggb
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Photovoltaik.svg
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Rakete.ggb
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Rakete.svg
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XWiki.XWikiComments[0]
Autor
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1 -XWiki.kickoff
Kommentar
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1 -Soll die Aufgabe 1 durch Flächenzerlegung gelöst werden? Oder passt diese Aufgabe eher zu einer anderen BPE?
Datum
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1 -2023-10-09 15:35:53.47
XWiki.XWikiComments[1]
Autor
... ... @@ -1,1 +1,0 @@
1 -XWiki.kickoff
Kommentar
... ... @@ -1,2 +1,0 @@
1 -Laut BP gehört Ober- und Untersumme zu 13.1, daher hab ichs da mal reingestellt.
2 -Die Aufgabe ist wohl eher als Erarbeitung tauglich.
Datum
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1 -2023-10-09 16:05:56.242
XWiki.XWikiComments[2]
Autor
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1 -XWiki.holgerengels
Kommentar
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1 -Da es eine Erarbeitungsaufgabe ist, habe ich sie hierher verschoben: [[Jahrgangsstufen.BPE_13L.Erarbeitungsaufgabe Untersumme.WebHome]]
Datum
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1 -2025-12-16 06:21:49.534
Antwort an
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1 -1