Änderungen von Dokument BPE 13.1 Bestandsrekonstruktion und Orientierter Flächeninhalt

Zuletzt geändert von Holger Engels am 2026/10/17 14:38

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Von 28.2 nach 28.3 Von 49.1 nach 50.1

Von Version 28.3

bearbeitet von Holger Engels
am 2026/02/27 14:55

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Auf Version 49.1

bearbeitet von Holger Engels
am 2026/10/09 10:40

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@@ -12,12 +12,13 @@
  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]]
  {{/lernende}}
--Deutung des bestimmten Integrals (Bestandsrekonstruktion, z. B. zurückgelegte Strecke bei veränderlicher Geschwindigkeit, Fläche)
--gekrümmte Randfunktion
--Näherungsweise Berechnung von Integralen (z. B. Kästchenzählen, Ober- und Untersumme)
++--Deutung des bestimmten Integrals (Bestandsrekonstruktion, z. B. zurückgelegte Strecke bei veränderlicher Geschwindigkeit, Fläche)--
++--gekrümmte Randfunktion--
++--Näherungsweise Berechnung von Integralen (z. B. Kästchenzählen, Ober- und Untersumme)--
  Orientierter Flächeninhalt (z. B. Zu- und Abflussmenge)
  Eigenschaften des bestimmten Integrals (Intervalladditivität, Linearität des Integrals, Integralwert ist Null bei ungeraden Funktionen und zu x=0 symmetrischen Integrationsgrenzen)
++== Bestandsrekonstruktion ==
  {{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"}}
  Ein Gefäß sei zu Beginn der Beobachtung mit //10 l// gefüllt. Es wird folgender Zufluss/ Abluss beobachtet:
@@ -26,4 +26,64 @@
  Bestimme den Füllstand nach diesen Veränderungen.
  {{/aufgabe}}
++{{aufgabe id="Fahrstrecke" afb="I" kompetenzen="K4" quelle="[[KMap>>https://kmap.eu/app/browser/Mathematik/Integralrechnung/Rekonstruktion%20einer%20Gr%C3%B6%C3%9Fe]]" zeit="5"}}
++[[image:Fahrstrecke.svg||class="right" width=500]]Das Schaubild zeigt den Geschwindigkeitsverlauf während einer Autofahrt.
++Bestimme die zurückgelegte Strecke.
++{{/aufgabe}}
++
++{{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"}}
++[[image:Photovoltaik.svg||class="right" width=500]]Ein Speicher wird über eine Photovoltaikanlage geladen. Das Schaubild zeigt die Ladeleistung über die Zeit.
++Bestimme näherungsweise die gespeicherte Energiemenge des Speichers um 18:00 Uhr unter der Annahme, dass sie morgens um 6:00 Uhr //2 kWh// betrug.
++{{/aufgabe}}
++
++{{aufgabe id="Test-Rakete" afb="II" kompetenzen="K3,K4" quelle="Holger Engels" zeit="9"}}
++Bei der Erprobung eines neuen Raketenantriebs werden verschiedene Flugprofile getestet. Die Telemetrie wertet dabei aus, wie sich die Geschwindigkeit //v// (in m/s) über die Zeit //t// (in s) verändert (Geschwindigkeits-Zeit-Diagramm). Der Flächeninhalt unter dem Graphen der Geschwindigkeit entspricht dabei dem zurückgelegten Weg //s// (in m).
++
++Bereits eingezeichnet ist ein unrealistisches Flugprofil, bei dem die Rakete schlagartig auf ihre Reisegeschwindigkeit kommt und diese die gesamten 20 Sekunden konstant hält.
++
++[[image:Rakete.svg||width=600]]
++Zeichne für die beiden folgenden Flugprofile jeweils einen möglichen Graphen der Geschwindigkeit //v(t)// in das Koordinatensystem ein. Allen drei Flugprofilen ist die Dauer von 20 Sekunden und die Strecke von 600 Metern gemein.
++(%class=abc%)
++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.
++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.
++{{/aufgabe}}
++
++== Bestimmtes Integral ==
++
++== Eigenschaften ==
++{{aufgabe id="Integraleigenschaften" afb="II" kompetenzen="K1" quelle="Debora Kemm, Benjamin Kaiser" zeit="10"}}
++Ermittle eine aufgsteigende Reihenfolge für den Wert der folgenden Integrale
++(%class=abc%)
++1. {{formula}}\int_{-5}^{5} e^{x+4} \, dx{{/formula}}
++1. {{formula}}\int_{-5}^{5} e^{x-4} \, dx{{/formula}}
++1. {{formula}}\int_{-5}^{5} (-x^5 + 4x^3-2x)\, dx{{/formula}}
++1. {{formula}}\int_{-5}^{5} (\cos(\frac{1}{2}x)-3)\, dx  {{/formula}}
++{{/aufgabe}}
++
++{{aufgabe id="Bilanz null" afb="II" kompetenzen="K2, K5" quelle="Holger Engels" zeit="8"}}
++Es gilt für das bestimmte Integral im Intervall {{formula}}[1; 5]{{/formula}}:
++
++{{formula}}\int_{1}^5 f(x) \, dx = 0{{/formula}}
++
++Gib jeweils eine Funktionsgleichung für {{formula}}f{{/formula}} an, die diese Bedingung erfüllt. Begründe jeweils Deine Wahl.
++(%class=abc%)
++1. {{formula}}f{{/formula}} ist eine trigonometrische Funktion.
++1. {{formula}}f{{/formula}} ist eine Polynomfunktion.
++{{/aufgabe}}
++
++{{aufgabe id="Rechenregeln" afb="III" kompetenzen="K4,K5" quelle="Holger Engels" zeit="9"}}
++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:
++* Der Graph der Funktion {{formula}}f{{/formula}} ist punktsymmetrisch zum Ursprung
++* Der Graph der Funktion {{formula}}g{{/formula}} ist achsensymmetrisch zur y-Achse
++und folgende bestimmte Integralwerte:
++
++(% style="display:flex"%)(((
++((({{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}})))
++)))
++
++Ermittle den exakten Wert des folgenden bestimmten Integrals:
++
++{{formula}}\int\limits_{-5}^{-3}g(x)\ dx{{/formula}}
++{{/aufgabe}}
++
  {{seitenreflexion bildungsplan="" kompetenzen="" anforderungsbereiche="" kriterien="" menge=""/}}

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Fahrstrecke.svg

Author

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	1	+XWiki.nilanurschams

Größe

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	1	+36.3 KB

Inhalt

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Photovoltaik.ggb

Author

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	1	+xwiki:XWiki.holgerengels

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Photovoltaik.png

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	1	+XWiki.nilanurschams

Größe

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	1	+23.8 KB

Inhalt

Photovoltaik.svg

Author

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	1	+xwiki:XWiki.holgerengels

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Rakete.svg

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XWiki.XWikiComments[0]

Autor

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1		-XWiki.kickoff

Kommentar

...	...	@@ -1,1 +1,0 @@
1		-Soll die Aufgabe 1 durch Flächenzerlegung gelöst werden? Oder passt diese Aufgabe eher zu einer anderen BPE?

Datum

...	...	@@ -1,1 +1,0 @@
1		-2023-10-09 15:35:53.47

XWiki.XWikiComments[1]

Autor

...	...	@@ -1,1 +1,0 @@
1		-XWiki.kickoff

Kommentar

@@ -1,2 +1,0 @@
--Laut BP gehört Ober- und Untersumme zu 13.1, daher hab ichs da mal reingestellt.
--Die Aufgabe ist wohl eher als Erarbeitung tauglich.

Datum

...	...	@@ -1,1 +1,0 @@
1		-2023-10-09 16:05:56.242

XWiki.XWikiComments[2]

Autor

...	...	@@ -1,1 +1,0 @@
1		-XWiki.holgerengels

Kommentar

...	...	@@ -1,1 +1,0 @@
1		-Da es eine Erarbeitungsaufgabe ist, habe ich sie hierher verschoben: [[Jahrgangsstufen.BPE_13L.Erarbeitungsaufgabe Untersumme.WebHome]]

Datum

...	...	@@ -1,1 +1,0 @@
1		-2025-12-16 06:21:49.534

Antwort an

...	...	@@ -1,1 +1,0 @@
1		-1

Änderungen von Dokument BPE 13.1 Bestandsrekonstruktion und Orientierter Flächeninhalt

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