Using Exps & Logs in Modelling (CIE A Level Maths: Pure 3)

What is exponential modelling?

• Exponential modelling is where real-life situations are modelled using exponential growth and decay 

• Exponential growth is typically used for …
  • … population (animals and humans) increase
  • … the value of an investment under compound interest

•  Exponential decay is typically used for …
  • … levels of radioactivity
  • … quantity of a drug in a person’s bloodstream
  • … depreciation (eg value of a standard car)

• It is important to consider if a model is only relevant for a certain period of time
• • Can a population increase forever?
  • Will a very old car still be worth a small amount of money?
• For example: Consider a substance that initially contains 2000 radioactive atoms where the number of radioactive atoms, N, at time, t hours, is modelled by
  • y is N
  • A = 2000 – the initial number of radioactive atoms
  • k > 0 so the negative indicates this must be exponential decay
  • t is used (rather than x) as the situation is time-dependent

How do I solve modelling problems? 

• Recognise the form of an exponential model
  • Growth y = Ae^kt
  • Decay y = Ae^-kt

• A (constant) is the starting or initial value
  • When t = 0,

    e^kt = e⁰ = 1

    so   y = A

• k (constant) determines the rate of growth/decay
• Many modelling problems are about rates of change –
  • If y = Ae^kt

    then     dy/dt = Ake^kt

  • If y = Ae^-kt

    then     dy/dt = -Ake^-kt