AQA A Level Maths: Pure复习笔记6.3.2 Using Exps & Logs in Modelling

Using Exps & Logs in Modelling

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

6.3.2-Using-Exps-Logs-in-Modelling-Notes-fig2

  •  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)

6.3.2-Using-Exps-Logs-in-Modelling-Notes-fig1

  • 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 = Aekt
    • Decay y = Ae-kt
  • A (constant) is the starting or initial value
    • When t = 0,ekt = e0 = 1

      so   y = A

  • k (constant) determines the rate of growth/decay
  • Many modelling problems are about rates of change –
    • If y = Aektthen     dy/dx = Akekt
    • If y = Ae-ktthen     dy/dx = -Ake-kt

6.3.2-Using-Exps-Logs-in-Modelling-Notes-fig4

6.3.2-Using-Exps-Logs-in-Modelling-Notes-fig5

6.3.2-Using-Exps-Logs-in-Modelling-Notes-fig6

Worked Example

6.3.2-Using-Exps-Logs-in-Modelling-Example-fig1

6.3.2-Using-Exps-Logs-in-Modelling-Example-fig2

6.3.2-Using-Exps-Logs-in-Modelling-Example-fig36.3.2-Using-Exps-Logs-in-Modelling-Example-fig4

6.3.2-Using-Exps-Logs-in-Modelling-Example-fig5

6.3.2-Using-Exps-Logs-in-Modelling-Example-fig6

 

 

 

 

 

 

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