Gränsvärden: Skillnad mellan sidversioner

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(33 mellanliggande sidversioner av 2 användare visas inte)
Rad 1: Rad 1:
[[Fil:C Users Lars AppData Local Temp plugtmp-141 plugin-Kvalitetsredovisning 0405 NR.jpg]]
== 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;">
<p>[[Fil:NR.jpg]]   Lars Adiels är lärare på Norra Real och har skapat <b>sidor om gränsvärden</b>
</p>
</div>
<p><br />
</p>
 


Här kommer text om gränsvärden.
Här kommer text om gränsvärden.
Rad 8: Rad 29:


== Omgivningar. ==
== Omgivningar. ==
===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>


Plats för figur
<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 }}                                                           
                               


Ett slutet intervall [a,b] består av alla tal x mellan a och b samt 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|Rita tallinjer i figuren nedan och lägg in intervallen 2<x≤3 ; 4<x<6 ; 1≤x≤1.1}}
{{uppgruta| lägg också in intervallet nedan på en ytterligare tallinje
plats för figur papper
{{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 ===
Om en punkt A finns inne i ett intervall kallas den inre punkt i till intervallet.
Om en punkt A finns inne i ett intervall kallas den inre punkt i till intervallet.


Rad 36: Rad 66:
}}
}}


{{uppgruta|Vilket eller vilka av talen 1 ; 1.414 ; <math>\sqrt{2}</math> ; 3 ; <math>\pi</math> är inre punkter till intervallen  
{{uppgruta|Vilket eller vilka av talen <math>1 ; 1.414 ; \sqrt{2} ; 3 ; \pi</math> är inre punkter till intervallen  
# ] 1.414 , <math>\pi</math> ]
# <math>] 1.414 , \pi ]</math>
# [ <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===
{{defruta|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>A-\epsilon<A<A+\epsilon</math>
där <math>\epsilon</math> är ett godtyckligt positivt tal > 0 (ofta litet) tal. Det kan också skrivas <math>]A-\epsilon, A+\epsilon[</math>.
{{uppgruta|Uppgifter på omgivningar}}
====Punkterade omgivningar====
Ibland undantar man A från själva omgivningen till A då kallas det en punkterad.
{{defruta|De sammanslagna intervallen <math>P_-= \rm{A-a<x<A} </math> och <math>P_+=\rm{A<x<A+b}</math> kallas en '''punkterad omgivning''' ''P'' till A
Det kan också skrivas så här: ''P'' är alla x som uppfyller <math>]a,A[ och ]A,b[</math> där a<A och b>A}}
{{tnkruta|Observera intervallen ovan är öppna}}
plats för figur
{{uppgruta|uppgifter punkterade omgivningar}}
=====Vänster och höger omgivningar=====


== Oegentliga gränsvärden ==
== Oegentliga gränsvärden ==
Rad 48: Rad 108:


== Facit till vissa uppgifter ==
== Facit till vissa uppgifter ==
== GeoGebra ==
=== Tangent och sekant ===
<ggb_applet width="1368" height="621" version="4.0" 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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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Lars Adiels är lärare på Norra Real och har skapat sidor om gränsvärden



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


Facit: (klicka expandera till höger)

Uppgift

Uppgift
lägg också in intervallet nedan på en ytterligare tallinje
[math]\displaystyle{ \pi\leq x }[/math].

Tänk! Detta är ett halvöppet intervall som man också kan skriva ::[math]\displaystyle{ \pi\leq\ x\lt \infty }[/math]


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.


Tänk! Bara punkter A som uppfyller [math]\displaystyle{ a\lt A\lt b }[/math] är inre punkter till intervallet [math]\displaystyle{ a\leq A\leq b }[/math]



Uppgift
Vilket eller vilka av talen [math]\displaystyle{ 1 ; 1.414 ; \sqrt{2} ; 3 ; \pi }[/math] är inre punkter till intervallen
  1. [math]\displaystyle{ ] 1.414 , \pi ] }[/math]
  2. [math]\displaystyle{ [ \sqrt{2} , \sqrt{10} ] }[/math]

Facit: (klicka expandera till höger)

Uppgift
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


Tänk! Observera intervallen ovan är öppna

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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