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

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Exponential Models

What are the parameters of an exponential model?

  • An exponential model is of the form

    •  space f left parenthesis x right parenthesis equals k a to the power of x plus c or space f left parenthesis x right parenthesis equals k a to the power of negative x end exponent plus c for space a greater than 0

    • space f open parentheses x close parentheses equals k straight e to the power of r x end exponent plus c

      • 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

      • a to the power of x where a greater than 1

      • a to the power of negative x end exponent where 0 less than a less than 1

      • straight e to the power of r x end exponent where r greater than 0

    • Exponential decay is represented by

      • a to the power of x where 0 less than a less than 1

      • a to the power of negative x end exponent where a greater than 1 

      • straight e to the power of r x end exponent where r less than 0

  • 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

Examiner Tips and Tricks

  • Look out for the word "initial" or similar, as way of asking you to make the power equal to zero to simplify the equation

  • Questions regarding the boundary of the exponential model are also frequently asked

Worked Example

The value of a car, V (NZD), can be modelled by the function

space V open parentheses t close parentheses equals 25125 cross times 0.8 to the power of t plus 8500 comma blank t greater or equal than 0

where t is the age of the car in years.

a) State the initial value of the car.

2-3-3-ib-ai-sl-exponential-models-a-we-solution

b) Find the age of the car when its value is 17500 NZD.

2-3-3-ib-ai-sl-exponential-models-b-we-solution
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Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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