Taylor Polynomial vs Interpolated Polynomial

The Interpolated Polynomial is that unique polynomial that passes through the given points. By calculating some points on a function and working out the associated Interpolated Polynomial we can approximate the curve. By imagining being able to calculate with an unlimited number of points and taking a range for those points which tends to zero we can obtain the Taylor Polynomial. Use the interactivity below to explore how the Interpolated Polynomial and the Taylor Polynomial are connected. Things to explore:
  • What can you say when the order of the taylor polynomial is 1?
  • Are you surprised by what happens when you reduce the range of points?
  • Which order of Taylor polynomial is best?
  • What happens when you change the function?