Unit 14: Copying a function with a polynomial
About this unit
A Taylor polynomial copies a function's value and derivatives at one point, each term being f⁽ⁿ⁾(a)/n! times (x − a)ⁿ, so hard functions become sums you can add up.
By the end, you can: Build polynomials that copy a function's value, slope, bend and more at one point, see why each coefficient is the nth derivative over n!, and write the series for eˣ and sin x.
Modules
5 modulesLearn
- Match the value, then the slope, then the bendArticle · A polynomial can be tuned to share a function's value, slope and curvature at a point; for eˣ at 0 that gives 1, then 1 + x, then 1 + x + x²/2.Not started
- Where the n! comes fromArticle · Differentiating cₙxⁿ n times leaves n!·cₙ, so cₙ = f⁽ⁿ⁾(a)/n!; this gives the series for eˣ and, from the derivative cycle, for sin x.Not started
- Add terms, widen the fitExperiment · Raise the degree of the eˣ polynomial until it stays within 0.05 of the curve over a band more than 4 units wide.Not started
- The recipeReview · Match derivatives at a point, divide by n!, add the terms; more terms fit a wider stretch, and a calculator only needs a few once x is small.Not started