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Piecewise Models (DP IB Maths: AI HL)
Revision Note
Linear Piecewise Models
What are the parameters of a piecewise linear model?
- A piecewise linear model is made up of multiple linear models
- For each linear model there will be
- The rate of change for that interval mi
- The value if the independent variable was not present ci
What can be modelled as a piecewise linear model?
- Piecewise linear models can be used when the rate of change of a function changes for different intervals
- These commonly apply when there are different tariffs or levels of charges
- Anything with a constant rate of change for set intervals
- C(d) is the taxi charge for a journey of d km
- The charge might double after midnight
- R(d) is the rental fee for a car used for d days
- The daily fee might triple if the car is rented over bank holidays
- s(t) is the speed of a car travelling for t seconds with constant acceleration
- The car might reach a maximum speed
- C(d) is the taxi charge for a journey of d km
What are possible limitations of a piecewise linear model?
- Piecewise linear models have a constant rate of change (represented by a straight line) in each interval
- In real-life this might not be the case
- The data in some intervals might have a continuously variable rate of change (represented by a curve) rather than a constant rate
- Or the transition from one constant rate of change to another may be gradual- i.e. a curve rather than a sudden change in gradient
Examiner Tip
- Make sure that you know how to plot a piecewise model on your GDC
Worked example
The total monthly charge, , of phone bill can be modelled by the function
,
where is the number of minutes used.
a)
Find the total monthly charge if 80 minutes have been used.
b)
Given that the total monthly charge is £16.59, find the number of minutes that were used.
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Non-Linear Piecewise Models
What are the parameters of non-linear piecewise models?
- A non-linear piecewise model is made up of multiple functions
- Each function will be defined for a range of values of x
- The individual functions can contain any function
- For example: quadratic, cubic, exponential, etc
- When graphed the individual functions should join to make a continuous graph
- This fact can be used to find unknown parameters
- If then
What can be modelled as a non-linear piecewise model?
- Piecewise models can be used when different functions are needed to represent the output for different intervals of the variable
- S(x) is the standardised score on a test with x raw marks
- For small values of x there might be a quadratic model
- For large values of x there might be a linear model
- H(t) is the height of water in a bathtub with after t minutes
- Initially a cubic model might be a appropriate if the bottom of the bathtub is curved
- Then a linear model might be a appropriate if the sides of top of the bathtub has the shape of a prism
What are possible limitations a non-linear piecewise model?
- Piecewise models can be used to model real-life accurately
- Piecewise models can be difficult to analyse or apply mathematical techniques to
Examiner Tip
- Read and re-read the question carefully, try to get involved in the context of the question!
- Pay particular attention to the domain of each section, if it is not given think carefully about any restrictions there may be as a result of the context of the question
- If sketching a piecewise function, make sure to include the coordinates of all key points including the point at which two sections of the piecewise model meet
Worked example
Jamie is running a race. His distance from the start, metres, can be modelled by the function
where is the time, in seconds, elapsed since the start of the race.
a)
Find the value of .
b)
Find the time taken for Jamie to reach 100 metres from the start.
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