MATHHX B

MATHHX B

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6.2 Eksponentiel vækst

I indledningen til dette kapitel blev det påstået at eksponentielle funktioner vokser med en fast procent. I den næste øvelse vil vi efterprøve den påstand på en konkret funktion.

Øvelse 6.2.1

Betragt sildebenet for \(f(x)=2\cdot 1{,}3^x\):

\[\begin {array}{| l | l | l | l | l |} \hline x & 0 & 1 & 2 & 3 \\ \hline f(x) & 2 & 2{,}6 & 3{,}38 & 4{,}39 \\ \hline \end {array}\]

  • a) Hvor mange procent vokser \(f\) med når \(x\) vokser med 1?

Løsning 6.2.1

  • a) \(f\) vokser med \(30\%\) når \(x\) vokser med \(1\).

Som øvelsen bekræftede vokser eksponentielle funktioner med en fast procent når \(x\) vokser med \(1\). Vi skal nu se på, hvordan man nemt finder væksten. Først definerer vi vækstraten:

  • Definition 6.2.1
    Vækstraten \(r\) bestemmes ved \(r=a-1\).

Vi finder væksten ved hjælp af følgende sætning:

  • Sætning 6.2.1
    For en eksponentiel funktion \(f(x)=ba^x\) med vækstrate \(r=a-1\) gælder at:

    Hver gang \(x\) vokser med 1 vokser \(y\) med: \(r \cdot 100\%\).

    Funktionen skærer \(y\)-aksen i \(b\).

Vækstraten \(r\) er altså den procentvise vækst som decimaltal.

  • Eksempel 6.2.1
    Vi bestemmer vækstraten for funktionen \(f(x)=2\cdot 1{,}3^x\) fra øvelse 1. Vi har

    \[r=a-1=1{,}3-1=0{,}3.\]

    Altså er vækstraten \(r=0{,}3\). Det betyder at funktionen vokser med \(30\%\) hver gang \(x\) vokser med 1.

  • Eksempel 6.2.2
    Vi bestemmer vækstraten for funktionen \(f(x)=7\cdot 0{,}9^x\). Vi har

    \[r=a-1=0{,}9-1=-0{,}1.\]

    Altså er vækstraten \(r=-0{,}1\). At vokse med \(-10\%\) er det samme som at aftage med \(10\%\). Altså aftager \(f\) med \(10\%\) hver gang \(x\) vokser med \(1\).

Øvelse 6.2.2

Bestem vækstraten for følgende funktioner. Hvor mange procent vokser de med, når \(x\) vokser med \(1\)?

  • a) \(f(x)=3\cdot 1{,}2^x\)

  • b) \(f(x)=2000\cdot 4^x\)

  • c) \(f(x)=20\cdot 0{,}8^x\)

Løsning 6.2.2

  • a) \(r=0{,}2\), vokser med \(20\%\).

  • b) \(r=3\), vokser med \(300\%\).

  • c) \(r=-0{,}2\), aftager med \(20\%\).

  • Eksempel 6.2.3
    En vintage guitar kostede \(30.000\) kr. i år 2014 og stiger med \(5\%\) om året. Vi kan beskrive guitarens værdi med funktionen:

    \[f(x)=30000\cdot 1{,}05^x,\]

    hvor \(x\) er antal år efter 2014 og \(f(x)\) er guitarens værdi.

Øvelse 6.2.3

På en fremmet planet var befolkningstallet 1. januar 2014 på \(7{,}15\) milliarder og voksede med ca. \(1{,}2\%\) hvert år.

  • a) Angiv vækstraten for befolkningstallet.

  • b) Opskriv en forskrift for en funktion der beskriver befolkningstallet i milliarder, \(x\) år efter år 1. januar 2014.

  • c) Hvor mange rumvæsener er der ifølge modellen på planeten 1. januar 2020?

  • d) I hvilket årstal kommer vi ifølge modellen første gang over 10 milliarder? Du er nødt til at gøre det grafisk i GeoGebra. Senere vil du lære at gøre det ved beregning.

Løsning 6.2.3

  • a) \(r=0{,}012\).

  • b) \(f(x)=7{,}15\cdot 1{,}012^x\).

  • c) Der er \(7{,}68\) milliarder rumvæsener på planeten i år 2020.

  • d) År 2042.

Øvelse 6.2.4

En dame køber en brugt bil 1. januar 2014. Bilens værdi er givet ved forskriften:

\[f(x)=200000\cdot 0{,}7^x,\]

hvor \(x\) er antal år efter 1. januar 2014 og \(f(x)\) er bilens værdi i kr.

  • a) Hvad kostede bilen, da han købte den?

  • b) Hvor mange procent taber bilen i værdi om året?

  • c) Hvad er den værd 1. januar 2018?

  • d) I hvilket årstal er bilen mindre værd en \(5.000\) kr?

Løsning 6.2.4

  • a) \(200.000\) kr.

  • b) \(30\%\).

  • c) \(48020\) kr.

  • d) År 2024.