How this calculator works
Taylor's theorem bounds the remainder when the function has enough derivatives throughout the interval joining the centre and evaluation point. Supply a valid upper bound M for the magnitude of the derivative of order n + 1 on that entire interval. The calculator combines M, distance and factorial size; it cannot verify your derivative bound or the function's differentiability.
Formula
|Rₙ(x)| ≤ M|x-a|^(n+1)/(n+1)!, provided |f⁽ⁿ⁺¹⁾(t)| ≤ M throughout the interval between a and x.
Worked example
For a degree-2 approximation about 0 at x = 0.5 with M = 1, the bound is 0.5³/3! = 1/48 ≈ 0.0208333333. With n = 3, x = 2, a = 0 and M = 3, it is 3 × 16/24 = 2.
Advertisement
Frequently asked questions
Is M just the derivative at the centre?
Not necessarily. It must bound the derivative magnitude everywhere between the centre and evaluation point.
Does the bound equal the actual error?
Usually it is only a guaranteed ceiling, provided the assumptions hold. The actual error may be much smaller.
Why is the answer zero at the centre?
The distance factor vanishes. A Taylor polynomial agrees with the function at its centre when its coefficients come from that function.