Exponential Models (DP IB Applications & Interpretation (AI): HL): Revision Note

Exponential models

What are the parameters of an exponential model?

• An exponential model is of the form

  •  $f\left(x\right)=k{a}^x+c$ or $f\left(x\right)=k{a}^{-x}+c$ for $a>0$

  • $f\left(x\right)=k{\mathrm{e}}^{rx}+c$

    • Where e is the mathematical constant 2.718…

  • The value of $c$ represents the boundary for the function

    • It can never be this value

  • The value of $a$ or $r$ describes the rate of growth or decay

    • The bigger the value of $a$ or the absolute value of $r$ the faster the function increases/decreases

What can be modelled as an exponential model?

• Exponential growth or decay

  • Exponential growth is represented by

    • $a^x$ where $a>1$

    • $a^{-x}$ where $0<a<1$

    • ${\mathrm{e}}^{rx}$ where $r>0$

  • Exponential decay is represented by

    • $a^x$ where $0<a<1$

    • $a^{-x}$ where $a>1$ 

    • ${\mathrm{e}}^{rx}$ where $r<0$

• They can be used when there is a constant percentage increase or decrease

  • Such as functions generated by geometric sequences

• For example, suppose $V\left(t\right)=24000{\left(0.95\right)}^t+6000$ is the value of a vehicle in dollars $t$ years after it was purchased

  • The value of the car decreases by 5% each year

  • The initial value of the car is $24000+$6000 = $30000

  • The boundary is $6000

    • The car will never reach this value

• Examples include:

  • V(t) is the value of car after t years

  • S(t) is the amount in a savings account after t years

  • B(t) is the amount of bacteria on a surface after t seconds

  • T(t) is the temperature of a kettle t minutes after being boiled

What are possible limitations of an exponential model?

• An exponential growth model does not have a maximum

  • In real-life this might not be the case

    • The function might reach a maximum and stay at this value

• Exponential models are monotonic

  • In real-life this might not be the case

    • The function might fluctuate

How can I find the half-life using an exponential model?

• You may need to find the half-life of a substance

  • This is the time taken for the mass of a substance to halve

• Given an exponential model $f\left(t\right)=k{a}^{-t}$ or $f\left(t\right)=k{\mathrm{e}}^{-rt}$ the half-life is the value of t such that:

  • $f\left(t\right)=\frac{k}{2}$

  • You can solve for t using your GDC

• For $f\left(t\right)=k{a}^{-t}$ the half-life is given by $t=\frac{\ln 2}{\ln a}$

  • $\frac{k}{2}=k{a}^{-t}$

  • $a^t=2$

  • $t\ln a=\ln 2$

• For $f\left(t\right)=k{\mathrm{e}}^{-rt}$ the half-life is given by $t=\frac{\ln 2}{r}$

  • $\frac{k}{2}=k{\mathrm{e}}^{-rt}$

  • ${\mathrm{e}}^{rt}=2$

  • $rt=\ln 2$

• For example, suppose$M\left(t\right)=100{\mathrm{e}}^{-3t}$ is the mass of a substance after $t$ hours

  • Solve $100{\mathrm{e}}^{-3t}=50$ to find the half-life

    • ${\mathrm{e}}^{-3t}=\frac{1}{2}$

    • ${\mathrm{e}}^{3t}=2$

    • $3t=\ln 2$

    • $t=\frac{1}{3}\ln 2=0.2310...$