IBMaths: AI HLDPRevision Notes2. Functions2.6 Further Modelling with Functions2.6.4 Piecewise Models

Piecewise Models (DP IB Maths: AI HL)

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Linear Piecewise Models

What are the parameters of a piecewise linear model?

  • A piecewise linear model is made up of multiple linear models space f subscript i open parentheses x close parentheses equals m subscript i x plus c subscript i
  • 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

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

Exam Tip

  • Make sure that you know how to plot a piecewise model on your GDC 

Worked example

The total monthly charge, £ C, of phone bill can be modelled by the function

space C left parenthesis m right parenthesis equals open curly brackets table row cell 10 plus 0.02 m end cell row cell 9 plus 0.03 m end cell end table blank table row cell 0 less or equal than m less or equal than 100 end cell row cell m greater than 100 end cell end table close,

wherespace m is the number of minutes used.

a)
Find the total monthly charge if 80 minutes have been used.

2-3-1-ib-ai-sl-piecewise-models-a-we-solution

b)
Given that the total monthly charge is £16.59, find the number of minutes that were used.

2-3-1-ib-ai-sl-piecewise-models-b-we-solution

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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 space f subscript i left parenthesis x right parenthesis
    • 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 f open parentheses x close parentheses equals open curly brackets table row cell f subscript 1 left parenthesis x right parenthesis end cell row cell f subscript 2 left parenthesis x right parenthesis end cell end table blank table row cell a less or equal than x less than b end cell row cell b less or equal than x less than c end cell end table close then f subscript 1 open parentheses b close parentheses equals f subscript 2 open parentheses b close parentheses

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

Exam 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, x metres, can be modelled by the function

x open parentheses t close parentheses equals open curly brackets table row cell 3 t end cell row cell 125 minus a left parenthesis t minus 15 right parenthesis ² end cell end table blank table row cell 0 less or equal than t less than 5 end cell row cell 5 less or equal than t less than 15 end cell end table close

where t is the time, in seconds, elapsed since the start of the race.

a)
Find the value of a.

2-6-4-ib-ai-hl-nonlinear-piecewise-model-a-we-solution

b)
Find the time taken for Jamie to reach 100 metres from the start.

2-6-4-ib-ai-hl-nonlinear-piecewise-model-b-we-solution

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Dan

Author: Dan

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.