IB DP Maths: AI HL复习笔记2.6.4 Piecewise Models

What are the parameters of a piecewise linear model?

• For each linear model there will be
  • The rate of change for that interval m­_i
  • The value if the independent variable was not present c_i

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?

• Linear models have a constant rate of change
  • In real-life this might not be the case
  • A function might increase (or decrease) gradually rather than at a constant rate

What are the parameters of non-linear piecewise models?

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