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 \(1\). Fordi lineære funktioners vækst ikke afhænger af \(x\), kan lineære funktioner
bruges til at beskrive ting som har en fast vækst.
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Eksempel 3.3.1
Antag at en taxatur koster 40 kr. i starttakst og 15 kr. pr. km.
Vi kan beskrive prisen for en taxatur med en lineær funktion:
\[p(x)=ax+b\]
hvor \(x\) er kørte kilometer og \(p(x)\) 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 funktionen vokser med,
hver gang \(x\) vokser med \(1\). Det må være prisen pr. km, da den jo viser det beløb prisen vokser med, når vi kører en ekstra km. Alt i alt får vi:
\[p(x)=15x+40 \quad , \quad x\geq 0\]
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.
-
a) Hvad er definitionsmængden for \(p\)?
-
b) Tegn grafen for \(p\).
-
c) Hvad er værdimængden for \(p\)?
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Eksempel 3.3.2
Vi bliver ved taxaeksemplet. Vi vil gerne finde ud af, hvad det koster at køre \(5\) km. Vi indsætter \(5\) i forskriften for \(p\):
\[ p(5)=15\cdot 5+40=115 \]
Det koster altså 115 kr. at kører \(5\) km.
Vi vil nu bestemme hvor langt vi kommer for \(70\) kr. Denne gang er det prisen, altså \(p(x)\) vi kender, så vi sætter \(p(x)=70\):
\(\seteqnumber{0}{3.}{0}\)
\begin{align*}
p(x) & = 70 \\ 15x+40 & = 70 && (\text {forskrift indsat})\\ 15x & = 30 \\ x & = 2
\end{align*}
Vi ser, at vi kan køre \(2\) km for \(70\) kr.
Øvelse 3.3.2
Nu er det din tur til at regne på taxaeksemplet.
Øvelse 3.3.3
En maskine koster 400.000 kr. og afskrives med 50.000 kr. om året (dvs. den mister 50.000 kr. i værdi hvert år).
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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 maskinen er afskrevet (har en værdi på 0 kr.)?
Øvelse 3.3.4
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
\[f(x)=2x+3\]
hvor \(x\) er antallet af uger efter 1. maj 1980 og \(f(x)\) er højden.
Øvelse 3.3.5
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 3.3.6
En virksomhed producerer og sælger en vare. Virksomheden har faste omkostninger på 30.000 kr. og variable enhedsomkostninger 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. Hvis du ikke ved hvad
omsætning betyder, så er det det beløb som virksomheden får ind ved salg af \(x\) kg af varen.
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c) Bestem omkostningerne ved en produktion på 100 kg.
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d) Hvor mange kg skal virksomheden sælge, før den begynder at tjene penge?
3.3.6
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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 3.3.7
Antag, at 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.
3.3.7
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a) \(p(x)=-0{,}5x+1550\)
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b) 1300 kr.
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c) 2500 stk.