MATHHX B
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3.3 Anvendelser af lineære funktioner
En lineær funktion har forskriften \(f(x)=ax+b\). Vi husker, at den skærer \(y\)-aksen i \(b\) og vokser med \(a\), hver gang \(x\) vokser med \(1\). Derfor kan lineære funktioner bruges til at beskrive situationer, hvor vi starter
på en bestemt værdi og så efterfølgende har en fast vækst eller fald.
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Eksempel 3.3.1
Antag at en taxatur koster 40 kr. i starttakst og 15 kr. pr. km (jaja, jeg ved godt, man også skal betale en tidstakst, men den ser vi bort for her).
Vi kan beskrive prisen for en taxatur vha. en lineær funktion
\[f(x)=ax+b\]
hvor \(x\) er kørte kilometer og \(f(x)\) er prisen. Vi ved at \(b\) er skæringen med \(y\)-aksen, hvilket svarer til \(x\)-værdien \(0\). Altså må \(b\) være starttaksten da den jo svarer til at taxaen har kørt \(0\) km. Så
mangler vi bare \(a\), som er det, vi skal gå op, hver gang \(x\) vokser med \(1\). Altså må \(a\) være prisen pr. km, da det jo svarer til det, prisen vokser, hver gang vi kører en km. Alt i alt får vi:
\[f(x)=15x+40 \quad , \quad x\geq 0\]
Læg mærke til, at der står \(x\geq 0\). Det er fordi, man ikke kan køre et negativt antal km.
Øvelse 3.3.1
Vi tager udgangspunkt i ovenstående eksempel.
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a) Hvad er definitionsmængden for \(f\)?
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b) Tegn grafen for funktionen \(f(x)\).
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c) Hvad er værdimængden for \(f\)?
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d) Beregn prisen, når man kører 10 km.
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e) Beregn, hvor langt kan man komme for 100 kr.?
Øvelse 3.3.2
En maskine koster 800.000 kr. og afskrives med 50.000 kr. om året.
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a) Bestem forskriften for funktionen \(v(x)\), der beskriver maskinens værdi.
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b) Hvor mange år går der, før maskine er afskrevet?
Øvelse 3.3.3
En person besluttede sig for at holde øje med sin græsplæne i en periode. Personen fandt ud af, at græsset højde, målt i cm, kunne beskrives med funktionen
\[f(x)=1{,}5x+5\]
hvor \(x\) er antallet af uger efter i 1. maj 1980 og \(f(x)\) er højden.
Øvelse 3.3.4
En virksomhed sælger en vare.
Prisen \(E(x)\), som funktion af efterspørgslen \(x\), er givet ved:
\[E(x)=-2x+400\quad ,\quad x\geq 0\]
Prisen \(U(x)\), som funktion af udbuddet \(x\), er givet ved:
\[U(x)=2x+200\quad ,\quad x\geq 0\]
Prisen er i kr. og mængden er i stk.
Ligevægtmængden er den mængde \(x\), hvor udbud og efterspørgsel er lige store (surprise). Den tilhørende pris, kaldes ligevægtsprisen (suprise surprise).
Øvelse 3.3.5
En virksomhed producerer en vare. De faste omkostninger er 30.000 kr., og derefter er der en enhedsomkostning på 140 kr. pr. kg. Virksomheden kan sælge varen for 200 kr. pr. kg.
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a) Bestem en forskrift for funktionen \(C\) som beskriver omkostningerne \(C(x)\) i kr. som funktion af vægten \(x\) i kg.
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b) Bestem en forskrift for funktionen \(R\) som beskriver omsætningen \(R(x)\) i kr. som funktion af vægten \(x\) i kg.
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c) Bestem omkostningerne ved en produktion på 100 kg.
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d) Hvor mange kg. skal virksomheden producere, før det giver overskud?
Løsning 3.3.5
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a) \(C(x)=140x+30000\)
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b) \(R(x)=200x\)
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c) 44000 kr.
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d) 500 kg.
Øvelse 3.3.6
Prisen på en vare kan beskrives ved en lineær funktion \(p(x)=ax+b\), hvor \(x\) er afsætningen i stk. Ved en pris på 800 kr. afsættes 1500 stk. og ved en pris på 1000 kr. afsættes 1100 stk.
\(\begin {array}{ | c | c | c | c | c |c |} \hline x & 1500 & 1100 \\ \hline p(x) & 800 & 1000 \\ \hline \end {array}\)
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a) Bestem en forskrift for \(p\).
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b) Bestem prisen ved en afsætning på 500 stk.
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c) Bestem afsætningen ved en pris på 300 kr.