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

    •  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 value of c represents the boundary for the function

      • It can never be this value

    • The value of 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 is a constant percentage increase or decrease

    • Such as functions generated by geometric sequences

  • For example, suppose V open parentheses t close parentheses equals 24000 open parentheses 0.95 close parentheses to the power of t plus 6000 is the value of a vehicle in dollars t years after it was purchased

    • The value of the car decreases by 5% each year

    • The initial value of the car is $24000+$6000 = $30000

    • The boundary is $6000

      • The car will never reach this value

  • 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

These models are different to quadratic and cubic models, the constant term is not the initial value.

  • The initial value of f open parentheses x close parentheses equals 100 straight e to the power of 2 x end exponent is 100

  • The initial value of f open parentheses x close parentheses equals 100 straight e to the power of 2 x end exponent plus 50 is 100 plus 50 equals 150

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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