Matematik 3C
Länkar
- Roger Bengtsson har en sajt på wikidot. Den är CC och innehåller mycket bra förklarande texter, mm.
- Daniel Barker har en sajt med flera kurser på gy 11 och gamla gymnasiet. Här är Ma3c. Daniel är en föregångare på flipped classroom. Det är fritt att läsa och använda men inte full CC (dvs du kan inte själv gå in och ändra).
- Centralt innehåll Ma3C
Trigonometri
Gränsvärden
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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Ser fram emot detta. --Håkan Elderstig 28 augusti 2012 kl. 21.15 (UTC)
Derivator
Provkarta
Denna sida är ett slags provkarta på vad wikiskola har att erbjuda och visar olika typer av delar som kan ingå i en sida. Här finns bilder som ligger på wikimedia, formler som kopierats från wikipedia, filmer, GeoGebra, en quiz och en widget från Wolfram Alpha. Det finns mallar för exempel (blå), definitioner (rosa), länkar, (bruna), uppgifter, (gula), bokhänvisningar (lila), tänkare (orange) samt Khanövningar (gröna).
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Problemlösning med derivatan
Detta är en sammanfattning som introduktion till avsnittet om derivator. Den innehåller ett fysikproblem med en måsjägare.
3.2 Derivator
Använda derivatans definition
Deriveringsregler för polynom
Tillämpningar på derivata
3.3 Derivator och grafer
Rita kurvor med hjälp av derivatan
Största och minsta värde
Derivatans graf
Andraderivatan
Maximi- och minimiproblem
3.4 Merom derivator
Lite Algebra
Derivatan av potensfunktioner
Diskontinuerliga funktioner
Diskreta funktioner
Inflexionspunkt och derivata
Tillämpningar (ej i Liber)
Derivator kommer till användning på många områden inom naturvetenskap, ekonomi, mm. Här kommer ett exempel från fysiken.
| Exempel |
|---|
| Tryck
Antag att [math]\displaystyle{ p(h) }[/math] betyder lufttrycket (i pascal) vid höjden [math]\displaystyle{ h }[/math] (i meter) över havsnivån. Då kommer derivatan [math]\displaystyle{ p'(h) }[/math] att ange hur mycket trycket ökar per meter i höjdled. Derivatan får alltså den fysikaliska enheten pascal per meter. Eftersom trycket i själva verket avtar med höjden, kommer alltså derivatan att bli negativ. Texten i ovanstående avsnitt kommer från Wikipedia.se |
Derivataquiz
Prov
Integraler
Kan man tänka sig någon trevlig frågeställning som ingång till integralerna?