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^ktthen?????dy/dx = Ake^kt
  • If?y = Ae^-ktthen?????dy/dx = -Ake^-kt