Gränsvärden: Skillnad mellan sidversioner
Laad (diskussion | bidrag) Ingen redigeringssammanfattning |
Hakan (diskussion | bidrag) Ingen redigeringssammanfattning |
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| (27 mellanliggande sidversioner av 2 användare visas inte) | |||
| Rad 1: | Rad 1: | ||
== Introduktionsföreläsning i två grupper == | |||
Vad händer med uttrycket när x närmar sig 4? | |||
Lösning i WolframAlpha: http://www.wolframalpha.com/input/?i=lim+x-%3E4+f%28x%29+%3D+%28x%5E%280.5%29+-+2%29+%2F+%28x²+-+5x+%2B+4%29 | |||
=== Lösning i tabell-GGB === | |||
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=== Grafen === | |||
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{{clear}} | |||
------ | |||
<div lang="sv" dir="ltr" class="mw-content-ltr"><div class="noprint" style="float:center; border:1px solid blue;width:660px;background-color:#ffcccc;padding:3px;"> | <div lang="sv" dir="ltr" class="mw-content-ltr"><div class="noprint" style="float:center; border:1px solid blue;width:660px;background-color:#ffcccc;padding:3px;"> | ||
| Rad 17: | Rad 31: | ||
===Intervall=== | ===Intervall=== | ||
Om vi tänker oss '''alla''' tal mellan två tal a och b så kallas det ett intervall. | Om vi tänker oss '''alla''' tal mellan två tal a och b så kallas det ett intervall. | ||
Det finns intervall av tre typer. Öppna intervall, slutna intervall och halvöppna intervall (se figurer). | Det finns intervall av tre typer. Öppna intervall, slutna intervall och halvöppna intervall (se figurer). Här gör man så att man ritar öppna cirklar när punkten inte ingår i intervallet och fyllda cirklar när det ingår. Intervallet <math>]1 ,4[</math> är alltså ett öppet intervall dvs 1<x<4. Det indikeras av att hakparenteserna inte sluter om. | ||
<p> På samma sätt är <math>[0.5 , 5]</math> slutet.<math>( 0.5\le x\le5 )</math> och de två sista intervallen halvöppna.</p> | |||
<pdf width="400" height="350">IntervallFig.pdf</pdf> | |||
Alltså | <P>Alltså | ||
{{defruta|Ett öppet intervall ]a,b[ består av alla tal x mellan a och b utom a och b ; a<x<b | {{defruta|Ett öppet intervall ]a,b[ består av alla tal x mellan a och b utom a och b ; a<x<b }} | ||
{{uppgrutaWLink||Rita tallinjer och lägg in intervallen 2<x≤3 ; 4<x<6 ; 1≤x≤1.1</p> | |||
<p>Du kan rita i Geogebra.</p><p>Du kan också rita på eget papper eller trycka ut detta|[http://wikiskola.se/images/Mmpaper.pdf papper]|}} | |||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | |||
'''Facit:''' (klicka expandera till höger) | |||
<div class="mw-collapsible-content">'''Uppgift''' | |||
<pdf>IntervallFacit.pdf</pdf> | |||
</div> | |||
</div> | |||
{{uppgruta| lägg också in intervallet nedan på en ytterligare tallinje | |||
{{uppgruta| lägg också in intervallet på en ytterligare tallinje | |||
::<math>\pi\leq x</math>. | ::<math>\pi\leq x</math>. | ||
{{tnkruta|Detta är ett halvöppet intervall som man också kan skriva ::<math>\pi\leq\ x< \infty</math>}} | {{tnkruta|Detta är ett halvöppet intervall som man också kan skriva ::<math>\pi\leq\ x< \infty</math>}} | ||
}} | }} | ||
Oändlikhetsymbolen <math>\infty</math> kommer att förklaras mer senare. | |||
===Inre punkt i ett intervall === | ===Inre punkt i ett intervall === | ||
| Rad 48: | Rad 70: | ||
# <math>[ \sqrt{2} , \sqrt{10} ]</math> | # <math>[ \sqrt{2} , \sqrt{10} ]</math> | ||
}} | }} | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | |||
'''Facit:''' (klicka expandera till höger) | |||
<div class="mw-collapsible-content">'''Uppgift''' | |||
:1 <math>\sqrt{2}, 3</math> därför att 1.414 ingår inte (öppet) och <math>\pi</math> är inte inre punkt! det är endast en (kant)punkt ett slutet intervall | |||
:2 3 och <math>\pi</math> | |||
</div> | |||
</div> | |||
===Omgivning=== | ===Omgivning=== | ||
| Rad 79: | Rad 108: | ||
== Facit till vissa uppgifter == | == Facit till vissa uppgifter == | ||
== GeoGebra == | |||
=== Tangent och sekant === | |||
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=== Övning gränsvärden === | |||
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=== Arean mellan kurvorna === | |||
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" /> | |||
Nuvarande version från 1 maj 2013 kl. 20.38
Introduktionsföreläsning i två grupper
Vad händer med uttrycket när x närmar sig 4?
Lösning i WolframAlpha: http://www.wolframalpha.com/input/?i=lim+x-%3E4+f%28x%29+%3D+%28x%5E%280.5%29+-+2%29+%2F+%28x²+-+5x+%2B+4%29
Lösning i tabell-GGB
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Grafen
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Här kommer text om gränsvärden.
Upplägget.
Motivering.
Omgivningar.
Intervall
Om vi tänker oss alla tal mellan två tal a och b så kallas det ett intervall. Det finns intervall av tre typer. Öppna intervall, slutna intervall och halvöppna intervall (se figurer). Här gör man så att man ritar öppna cirklar när punkten inte ingår i intervallet och fyllda cirklar när det ingår. Intervallet [math]\displaystyle{ ]1 ,4[ }[/math] är alltså ett öppet intervall dvs 1<x<4. Det indikeras av att hakparenteserna inte sluter om.
På samma sätt är [math]\displaystyle{ [0.5 , 5] }[/math] slutet.[math]\displaystyle{ ( 0.5\le x\le5 ) }[/math] och de två sista intervallen halvöppna.
Alltså
| Definition |
|---|
| Ett öppet intervall ]a,b[ består av alla tal x mellan a och b utom a och b ; a<x<b |
| Uppgift
|
Rita tallinjer och lägg in intervallen 2<x≤3 ; 4<x<6 ; 1≤x≤1.1
Du kan rita i Geogebra. Du kan också rita på eget papper eller trycka ut detta papper |
Facit: (klicka expandera till höger)
| Uppgift |
|---|
lägg också in intervallet nedan på en ytterligare tallinje
|
Oändlikhetsymbolen [math]\displaystyle{ \infty }[/math] kommer att förklaras mer senare.
Inre punkt i ett intervall
Om en punkt A finns inne i ett intervall kallas den inre punkt i till intervallet.
plats för figur
| Definition |
|---|
| En punkt A som ligger ligger helt inne i ett intervall kallas inre punkt till intervallet.
|
| Uppgift |
|---|
Vilket eller vilka av talen [math]\displaystyle{ 1 ; 1.414 ; \sqrt{2} ; 3 ; \pi }[/math] är inre punkter till intervallen
|
Facit: (klicka expandera till höger)
- 1 [math]\displaystyle{ \sqrt{2}, 3 }[/math] därför att 1.414 ingår inte (öppet) och [math]\displaystyle{ \pi }[/math] är inte inre punkt! det är endast en (kant)punkt ett slutet intervall
- 2 3 och [math]\displaystyle{ \pi }[/math]
Omgivning
| Definition |
|---|
| Om en punkt A är inre punkt till ett öppet intervall U kallas U en omgivning till A |
Ofta kommer vi att använda symmetriska omgivningar till en punkt som [math]\displaystyle{ A-\epsilon\lt A\lt A+\epsilon }[/math]
där [math]\displaystyle{ \epsilon }[/math] är ett godtyckligt positivt tal > 0 (ofta litet) tal. Det kan också skrivas [math]\displaystyle{ ]A-\epsilon, A+\epsilon[ }[/math].
| Uppgift |
|---|
| Uppgifter på omgivningar |
Punkterade omgivningar
Ibland undantar man A från själva omgivningen till A då kallas det en punkterad.
| Definition |
|---|
| De sammanslagna intervallen [math]\displaystyle{ P_-= \rm{A-a\lt x\lt A} }[/math] och [math]\displaystyle{ P_+=\rm{A\lt x\lt A+b} }[/math] kallas en punkterad omgivning P till A
Det kan också skrivas så här: P är alla x som uppfyller [math]\displaystyle{ ]a,A[ och ]A,b[ }[/math] där a<A och b>A |
plats för figur
| Uppgift |
|---|
| uppgifter punkterade omgivningar |
Vänster och höger omgivningar
Oegentliga gränsvärden
Gränsvärden.
Alternativa definitioner.
Facit till vissa uppgifter
GeoGebra
Tangent och sekant
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Övning gränsvärden
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Arean mellan kurvorna
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