IB DP Maths: AI HL复习笔记2.3.3 Exponential Models
Exponential Models
What are the parameters of an exponential model?
- An exponential model is of the form
-
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
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
How can I find the half-life using an exponential model?
- You may need to find the half-life of a substance
- This is the time taken for the mass of a substance to halve
Exam Tip
- Look out for the word "initial" or similar, as a 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
a)State the initial value of the car.
b)Find the age of the car when its value is 17500 NZD.
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