Integralrechnung/Integrationsregeln und Australia/LE2 Dangerous Animals: Unterschied zwischen den Seiten

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{{Navigation verstecken|{{Lernpfad Integral}}}}
<!--==Integrationsregeln==-->
Im Folgenden wirst Du einige elementare Integrationsregeln kennenlernen, die Du beim Integrieren ständig benötigen wirst.


{{Box|1=Aufgabe 11|2=
== Interactive exercises ==
Kannst Du eine Regel oder Formel für die Integrale unter folgenden Punkten auf Basis Deines bisherigen Wissens angeben? Die Regel soll so allgemein gehalten sein, dass sie eine Berechnung beliebiger Integrale der folgenden Formen erlauben!
=== Dangerous animals ===
# Welchen Wert hat das Integral einer Summe von Funktionen? Was gilt also für <math>\int\limits_a^b f(x) + g(x) \ \mathrm{d}x</math>?
# Welchen Wert hat das Integral eines Produktes aus einer Zahl und einer Funktion? Was gilt also für <math>\int\limits_a^b c \cdot f(x) \ \mathrm{d}x</math>?
|3=Arbeitsmethode}}


{{Box|1=Aufgabe 12|2=
<div class="memo-quiz" lang="en">
Formuliere selbstständig eine '''allgemeine''' Regel dafür, wie das Integral einer Summe von Funktionen gebildet wird. Benutze dafür wieder die Software Geogebra (Applet oder [http://www.geogebra.org geogebra.org] oder installiert), indem Du die Integrale zweier beliebiger Funktionen <math>f(x)</math> und <math>g(x)</math> in einem beliebigen Intervall <math>[a;b]</math> bestimmst und mit <math>\int\limits_a^b f(x) + g(x) \ \mathrm{d}x</math> vergleichst.
{|  
|3=Arbeitsmethode}}
|-
| [[File:Stonefish in Red Sea (with flash).jpg|130px]] || Stonefish
|-
| [[File:Textile cone (cropped).JPG|130px]] || Textile Cone<br>(killer snail)
|-
| [[File:Casuarius casuarius -Prague Zoo, Czech Republic -head-8a.jpg|85px]] || Cassowary
|-
| [[File:Sydney funnel web spider.jpg|130px]] || Funnel Web Spider
|-
| [[File:Hapalochlaena lunulata.JPG|105px]] || Blue-Ringed Octopus
|-
| [[File:Carybdea branchi9.jpg|130px]] || Box Jelly Fish
|-
| [[File:Great White Shark (14730744390).jpg|130px]] || Shark
|-
| [[File:Brown snake - victoria australia.jpg|130px]] || Brown Snake
|-
| [[File:Saltwater crocodile.jpg|130px]] || Saltwater Crocodile
|}
</div>
 
=== Urban Legends from Australia ===
'''Fill in the gaps. Be careful some of the gaps need [[Participle Constructions]]!'''
 
<div class="lueckentext-quiz" lang="en">
This is a true story a friend of a friend experienced in his Australia holiday:
 
Two American tourists ''were driving (drive)'' on deserted roads through the bush when they accidentally hit a large kangaroo.
 
''Getting (get)'' out of the car, they realized that the kangaroo had died in the accident.
 
''Being (be)'' so drunk, though, they propped up the kangaroo, its lifeless head ''bouncing (bounce)'' from one side to the other,
dressed it in one of their new jackets, and took pictures of themselves with their arms around it.
 
Ater a few minutes of picture-taking, the guys were shocked when it seemed that the kangaroo came back to life! It turns out that the poor kangaroo wasn't dead, but it had passed out, and when it came back to consciousness started to box with the drunken guys! It actually broke one guy's jaw!! It then ''hopped(hop)'' away into the landscape.


<div align="center">
The men couldn't drive their rented car, as the keys were in the green jacket, which was still on the kangaroo, so they ''had to walk (must, walk)'' back to the tournament. A little while later, a pack of kangaroos was seen in the distance, one of them ''wearing(wear)'' the green jacket.  
<ggb_applet height="600" width="600" uselocaljar="true" showmenubar="true" showtoolbar="true" showalgebrainput="true" showreseticon="true" filename="blank.ggb" />
</div>
</div>
{{Lösung versteckt|1=
 
Es gilt die ''Summenregel für Integrale'':<br>
{{Show-Hide|
<math>
This is a true story a friend of a friend experienced in his Australia holiday:
\int\limits_a^b \left( f(x) + g(x) \right) \ \mathrm{d}x = \int\limits_a^b f(x) \ \mathrm{d}x + \int\limits_a^b g(x) \ \mathrm{d}x
 
</math>. <br>
Two American tourists ''were driving (drive)'' on deserted roads through the bush when they accidentally hit a large kangaroo.
Das Integral einer Summe von Funktionen ist gleich der Summe der einzelnen Integrale der jeweiligen Funktionen. Eine Summe wird also gliedweise integriert.
 
''Getting (get)'' out of the car, they realized that the kangaroo had died in the accident.
 
''Being (be)'' so drunk, though, they propped up the kangaroo, its lifeless head ''bouncing (bounce)'' from one side to the other,
dressed it in one of their new jackets, and took pictures of themselves with their arms around it.
 
Ater a few minutes of picture-taking, the guys were shocked when it seemed that the kangaroo came back to life! It turns out that the poor kangaroo wasn't dead, but it had passed out, and when it came back to consciousness started to box with the drunken guys! It actually broke one guy's jaw!! It then ''hopped(hop)'' away into the landscape.
 
The men couldn't drive their rented car, as the keys were in the green jacket, which was still on the kangaroo, so they ''had to walk (must, walk)'' back to the tournament. A little while later, a pack of kangaroos was seen in the distance, one of them ''wearing(wear)'' the green jacket.  
}}
}}
<br>
 
{{Box|1=Aufgabe 13|2=
=== The Whale Museum in Eden ===
Warum ist die Lösung von Aufgabe 12 plausibel?
'''Put in the correct forms - [[Passive|Active or Passive]] - all tenses!'''
# Begründe anschaulich anhand der geometrischen Zusammenhänge!
 
# Begründe anhand der Rechengesetze für Grenzwerte!
<div class="lueckentext-quiz" lang="en">
|3=Arbeitsmethode}}
I <em>arrived (arrive)</em> in Eden, where signs to the Killer Whale Museum <em> had been displayed ()</em>(<abbr title="Die Geschichte ist in simple past, die Schilder wurden noch früher aufgestellt, &rarr; past perfect!">display</abbr>). So I decided to go there because I <em>was tired (to tire)</em> after the long car trip:
{{Lösung versteckt|1=
 
# Die Funktionswerte der Funktionen <math>f(x)</math> und <math>g(x)</math> addieren sich zu den Funktionswerten einer neuen Funktion <math>f(x) + g(x)</math>. Somit addieren sich auch die Flächeninhalte zwischen den Graphen von <math>f(x)</math> und <math>g(x)</math> und der x-Achse.
I <em> toured (tour)</em> the museum, and looked especially at the photos and skeletons of old killer whales. The story of what the museum called "A Modern Jonah" was certainly fascinating - but I found it hard to believe.  
# Der Grenzwert einer Summe ist gleich der Summe der einzelnen Grenzwerte, falls die Grenzwerte existieren: <math>\lim_{\Delta x \to 0} \ \sum_{i=0}^{\infty} \left( f(x_i)+g(x_i) \right) \cdot \Delta x = \lim_{\Delta x \to 0} \ \sum_{i=0}^{\infty} f(x_i) \cdot \Delta x + \lim_{\Delta x \to 0} \ \sum_{i=0}^{\infty} g(x_i) \cdot \Delta x</math>
</div>  
<br>
 
Zur Schreibweise: <math>\sum</math> ist das Summenzeichen (großes griechisches Sigma), es gilt: <math>\sum_{i=0}^n f(x_i) = f(x_0) + f(x_1) + f(x_2) + \dots + f(x_n)</math>, d.h. der Index <math>i</math> durchläuft alle Zahlen von 0 (untere Summengrenze) bis <math>n</math> (obere Summengrenze). Es wird dann die Summe der einzelnen <math>f(x_i)</math> gebildet.
{{Show-Hide|
I <em>arrived (arrive)</em> in Eden, where signs to the Killer Whale Museum <em> had been displayed ()</em>(<abbr title="Die Geschichte ist in simple past, die Schilder wurden noch früher aufgestellt, &rarr; past perfect!">display</abbr>). So I decided to go there because I <em>was tired (to tire)</em> after the long car trip:
 
I <em> toured (tour)</em> the museum, and looked especially at the photos and skeletons of old killer whales. The story of what the museum called "A Modern Jonah" was certainly fascinating - but I found it hard to believe.  
}}
}}
<br>
 
{{Box|1=Aufgabe 14|2=
'''Put in the correct forms - [[Passive|Active or Passive]] - all tenses!'''
Formuliere selbstständig eine '''allgemeine''' Regel dafür, wie das Integral eines Produktes einer Zahl <math>c</math> mit einer Funktion <math>f(x)</math> gebildet wird. Benutze dafür erneut Geogebra (Applet oder [http://www.geogebra.org geogebra.org] oder installiert), indem Du das Integral einer beliebigen Funktion <math>c \cdot f(x)</math> in einem beliebigen Intervall <math>[a;b]</math> bestimmst und mit <math>c \cdot \int\limits_a^b f(x) \ \mathrm{d}x</math> vergleichst, wobei <math>c</math> irgendeine reelle Zahl ist.
 
|3=Arbeitsmethode}}
<div class="lueckentext-quiz" lang="en">
<br>
While they ''were working(work)'' in the bay in 1891, whalers harpooned a huge whale. It ''was badly hurt(<abbr title="hurt ist das Verb, Wer hat hier etwas gemacht? bad ist hier ein Adverb, dass das Verb näher beschreibt!">bad, hurt</abbr>)'', but it was still alive, and it came up out of the water and broke the boat full of whalers to pieces.
<div align="center">
 
<ggb_applet height="600" width="600" uselocaljar="true" showmenubar="true" showtoolbar="true" showalgebrainput="true" showreseticon="true" filename="blank.ggb" />
Afterwards two men <em>couldn't be found (can not, find)</em>.
Hours later, when the sun was going down, the whale came up again and ''died(die)''. The whalers took it and began their work on it. Suddenly they noticed strange movements ' from the whale's body. They cut it open and one of their missing mates, James Bartley, ''was found(find)'' inside the whale. He was unconscious, but he had survived – after fifteen long hours inside the whale's stomach.  
 
James Bartley was seriously ill for two weeks. He lost his hair and was nearly blind.  It was a month before he could speak again. Then he told his mates how it had happened. He had fallen into the whale's mouth, and had felt its huge teeth as he slipped down its throat into its stomach, where fish were swimming around. James Bartley's skin was a deadly white for the rest of his life. He lived another eighteen years.  
</div>
</div>
<br>
 
{{Lösung versteckt|1=
{{Show-Hide|
Es gilt die '''Faktorregel für Integrale''': <br>
While they ''were working(work)'' in the bay in 1891, whalers harpooned a huge whale. It ''was badly hurt(hurt ist das Verb, Wer hat hier etwas gemacht? bad ist hier ein Adverb, dass das Verb näher beschreibt)'', but it was still alive, and it came up out of the water and broke the boat full of whalers to pieces.  
<math>
 
\int\limits_a^b c \cdot f(x) \ \mathrm{d}x = c \cdot \int\limits_a^b f(x) \ \mathrm{d}x
Afterwards two men <em>couldn't be found (can not, find)</em>.
</math>. <br>
Das Integral eines Produktes aus einem konstanten Faktor und einer Funktion ist gleich dem Produkt des konstanten Faktors und des Integrals der Funktion.  
Hours later, when the sun was going down, the whale came up again and died. The whalers took it and began their work on it. Suddenly they noticed strange movements from the whale's body. They cut it open and found one of their missing mates, James Bartley, inside the whale. He was unconscious, but he had survived –after fifteen long hours inside the whale's stomach.  
}}
James Bartley was seriously ill for two weeks. He lost his hair and was nearly blind.  It was a month before he could speak again. Then he told his mates how it had happened. He had fallen into the whale's mouth, and had felt its huge teeth as he slipped down its throat into its stomach, where fish were swimming around. James Bartley's skin was a deadly white for the rest of his life. He lived another eighteen years.  
<br>
{{Box|1=Aufgabe 15|2=
Führe wieder die Plausbilitätsüberlegungen zur Lösung von Aufgabe 14!
# Begründe anschaulich anhand der geometrischen Zusammenhänge!
# Begründe anhand der Rechengesetze für Grenzwerte!
|3=Arbeitsmethode}}
{{Lösung versteckt|1=
# Die Funktionswerte der Funktion <math>f(x)</math> werden mit dem konstanten Faktor <math>c</math> gestreckt. Somit werden auch die Flächeninhalte zwischen dem Graphen von <math>f(x)</math> und der x-Achse mit dem konstanten Faktor <math>c</math> gestreckt.
# Der Grenzwert eines Produkts aus einem konstanten Faktor und einer Funktion ist gleich dem Produkt des Faktors und des Grenzwertes, falls dieser existiert: <math>\lim_{\Delta x \to 0} \ \sum_{i=0}^{\infty} c \cdot f(x_i) \cdot \Delta x = \lim_{\Delta x \to 0} \ c \cdot \sum_{i=0}^{\infty} f(x_i) \cdot \Delta x = c \cdot \lim_{\Delta x \to 0} \ \sum_{i=0}^{\infty} f(x_i) \cdot \Delta x</math>
<br>
Bemerkung: Das erste Gleichheitszeichen gilt aufgrund des Distributivgesetzes, das zweite aufgrund der Grenzwertsätze.
}}
}}
<br>
{{Box|1=Aufgabe 16|2=
# Mache Dich mit der '''Intervalladditivität des Integrals''' (Internet!) vertraut und überzeuge Dich dann von ihrer Gültigkeit mit Hilfe von Geogebra, indem Du Funktionen <math>f(x)</math> und <math>g(x)</math> sowie Grenzen <math>a</math>, <math>b</math> und <math>c</math> so wählst, dass die Zusammenhänge ersichtlich werden!
# Beschreibe Deine Vorgehensweise in 1. Schritt für Schritt in kurzen Stichpunkten!
|3=Arbeitsmethode}}
{{Lösung versteckt|1=
# Ist <math>F</math> eine Stammfunktion von <math>f</math>, dann gilt nach dem 1. Hauptsatz:
  <br>
  <math>\int\limits_{a}^{b} f(x) \ \mathrm{d}x + \int\limits_{b}^{c} f(x) \ \mathrm{d}x = \left[ 
  F(x) \right]_a^b + \left[ F(x) \right]_b^c = \left[ F(b) - F(a) \right] + \left[ F(c) - F(b)
  \right] = F(c) - F(a) = \int\limits_{a}^{c} f(x) \ \mathrm{d}x</math>
# Eine Lösung könnte beispielsweise folgendermaßen aussehen:
## Definiere in Geogebra zwei beliebige Funktionen <math>f</math> und <math>g</math>.
## Definiere beliebige Intervallgrenzen <math>a, \ b \ \mathrm{und} \ c</math>.
## Verschiebe die Intervallgrenzen und beobachte die Werte der Integrale bzw. des Integrals.
## Erkenne, dass ...
}}
{{Fortsetzung|weiter=Aufgaben II|weiterlink=Integral/Aufgaben II}}


[[Kategorie:Integralrechnung]]
{{Australia}}
[[Kategorie:Interaktive Übung]]
[[Kategorie:GeoGebra]]

Version vom 25. April 2022, 18:13 Uhr

Interactive exercises

Dangerous animals

Stonefish in Red Sea (with flash).jpg Stonefish
Textile cone (cropped).JPG Textile Cone
(killer snail)
Casuarius casuarius -Prague Zoo, Czech Republic -head-8a.jpg Cassowary
Sydney funnel web spider.jpg Funnel Web Spider
Hapalochlaena lunulata.JPG Blue-Ringed Octopus
Carybdea branchi9.jpg Box Jelly Fish
Great White Shark (14730744390).jpg Shark
Brown snake - victoria australia.jpg Brown Snake
Saltwater crocodile.jpg Saltwater Crocodile

Urban Legends from Australia

Fill in the gaps. Be careful some of the gaps need Participle Constructions!

This is a true story a friend of a friend experienced in his Australia holiday:

Two American tourists were driving (drive) on deserted roads through the bush when they accidentally hit a large kangaroo.

Getting (get) out of the car, they realized that the kangaroo had died in the accident.

Being (be) so drunk, though, they propped up the kangaroo, its lifeless head bouncing (bounce) from one side to the other, dressed it in one of their new jackets, and took pictures of themselves with their arms around it.

Ater a few minutes of picture-taking, the guys were shocked when it seemed that the kangaroo came back to life! It turns out that the poor kangaroo wasn't dead, but it had passed out, and when it came back to consciousness started to box with the drunken guys! It actually broke one guy's jaw!! It then hopped(hop) away into the landscape.

The men couldn't drive their rented car, as the keys were in the green jacket, which was still on the kangaroo, so they had to walk (must, walk) back to the tournament. A little while later, a pack of kangaroos was seen in the distance, one of them wearing(wear) the green jacket.

This is a true story a friend of a friend experienced in his Australia holiday:

Two American tourists were driving (drive) on deserted roads through the bush when they accidentally hit a large kangaroo.

Getting (get) out of the car, they realized that the kangaroo had died in the accident.

Being (be) so drunk, though, they propped up the kangaroo, its lifeless head bouncing (bounce) from one side to the other, dressed it in one of their new jackets, and took pictures of themselves with their arms around it.

Ater a few minutes of picture-taking, the guys were shocked when it seemed that the kangaroo came back to life! It turns out that the poor kangaroo wasn't dead, but it had passed out, and when it came back to consciousness started to box with the drunken guys! It actually broke one guy's jaw!! It then hopped(hop) away into the landscape.

The men couldn't drive their rented car, as the keys were in the green jacket, which was still on the kangaroo, so they had to walk (must, walk) back to the tournament. A little while later, a pack of kangaroos was seen in the distance, one of them wearing(wear) the green jacket.


The Whale Museum in Eden

Put in the correct forms - Active or Passive - all tenses!

I arrived (arrive) in Eden, where signs to the Killer Whale Museum had been displayed ()(display). So I decided to go there because I was tired (to tire) after the long car trip:

I toured (tour) the museum, and looked especially at the photos and skeletons of old killer whales. The story of what the museum called "A Modern Jonah" was certainly fascinating - but I found it hard to believe.

{{{1}}}


Put in the correct forms - Active or Passive - all tenses!

While they were working(work) in the bay in 1891, whalers harpooned a huge whale. It was badly hurt(bad, hurt), but it was still alive, and it came up out of the water and broke the boat full of whalers to pieces.

Afterwards two men couldn't be found (can not, find).

Hours later, when the sun was going down, the whale came up again and died(die). The whalers took it and began their work on it. Suddenly they noticed strange movements ' from the whale's body. They cut it open and one of their missing mates, James Bartley, was found(find) inside the whale. He was unconscious, but he had survived – after fifteen long hours inside the whale's stomach.

James Bartley was seriously ill for two weeks. He lost his hair and was nearly blind. It was a month before he could speak again. Then he told his mates how it had happened. He had fallen into the whale's mouth, and had felt its huge teeth as he slipped down its throat into its stomach, where fish were swimming around. James Bartley's skin was a deadly white for the rest of his life. He lived another eighteen years.

While they were working(work) in the bay in 1891, whalers harpooned a huge whale. It was badly hurt(hurt ist das Verb, Wer hat hier etwas gemacht? bad ist hier ein Adverb, dass das Verb näher beschreibt), but it was still alive, and it came up out of the water and broke the boat full of whalers to pieces.

Afterwards two men couldn't be found (can not, find).

Hours later, when the sun was going down, the whale came up again and died. The whalers took it and began their work on it. Suddenly they noticed strange movements from the whale's body. They cut it open and found one of their missing mates, James Bartley, inside the whale. He was unconscious, but he had survived –after fifteen long hours inside the whale's stomach. James Bartley was seriously ill for two weeks. He lost his hair and was nearly blind. It was a month before he could speak again. Then he told his mates how it had happened. He had fallen into the whale's mouth, and had felt its huge teeth as he slipped down its throat into its stomach, where fish were swimming around. James Bartley's skin was a deadly white for the rest of his life. He lived another eighteen years.


Map of Australia

Australia - the continent Down UNDER

  1. Fact File
  2. Dangerous Animals
  3. Exploring the Outback
  4. Dream Time
  5. Convicts
  6. Work and Travel
  7. Planning a Holiday


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