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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@@ -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 +26,62 @@
  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. Der orientierte Flächeninhalt unter dem Graphen der Geschwindigkeit entspricht dabei dem zurückgelegten Weg //s// (in m).
++
++Zeichne für jedes der folgenden Flugprofile einen möglichen Graphen der Geschwindigkeit //v(t)// in das Koordinatensystem ein.
++(%class=abc%)
++1. Konstante Geschwindigkeit: Die Rakete legt in den ersten 20 Sekunden eine Strecke von genau 600 Metern zurück. Zur Vereinfachung gehen wir in diesem (unrealistischen) Testfall davon aus, dass die Rakete schlagartig auf ihre Reisegeschwindigkeit kommt und diese die gesamten 20 Sekunden konstant hält.
++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. Auch bei diesem Testflug legt sie in den ersten 20 Sekunden exakt 600 Meter zurück.
++1. Start-, Flug- und Bremsphase: In einem weiteren Testflug soll die Rakete ebenfalls exakt 600 Meter zurücklegen, dafür hat sie aber diesmal 40 Sekunden Zeit. Sie startet aus dem Stand, beschleunigt zunächst mit einer konstanten Rate, fliegt dann eine Zeit lang mit konstanter Geschwindigkeit und bremst am Ende mit einer konstanten negativen Rate wieder bis zum Stillstand (//v(40)=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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XWiki.XWikiComments[0]

Autor

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