f(x) = x^(1/3)....?

Question

The function
f(x) = x^(1/3)
is to be tabulated on the interval [1, 10] with constant mesh spacing.
(a) How many points are required for piecewise linear interpolation to yield five decimal places accuracy?
(b) What convenient step size would you choose in practise?

Answer 1

f"(x) = -2/9 x^(-5/3)
So if the mesh spacing is Δx then the maximum error is
|(Δx/2)^2 / 2! (-2/9) (1)^(-5/3)| = 1/9 (Delta;x/2)^2.
So we want 1/9 (Delta;x/2)^2 < 5×10^-6
=> Δx < 0.0134 (3 d.p.)
=> we need at least (10-1)/0.0134 = 670 points for accuracy to 5 d.p.

In practise, we'd use a step size of 0.01.