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

Exponential Models

What are the parameters of an exponential model?

• An exponential model is of the form

  •   or for 

  • 

    • Where e is the mathematical constant 2.718…

  • The c represents the boundary for the function

    • It can never be this value

  • The 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

    • where 

    • where 

    • where 

  • Exponential decay is represented by

    • where 

    • where  

    • where 

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

  • Such as functions generated by geometric sequences

• 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  or  the half-life is the value of t such that:

  • 

  • You can solve for t using your GDC

• For  the half-life is given by 

  • 

  • 

  • 

• For the half-life is given by 

  • 

  • 

  •