Suppose you're given the data points
(t1,f1) = (0.0, 0.1)
(t2,f2) = (0.4, 0.6)
(t3,f3) = (1.0, 0.2)
Compute the eight coefficients of the natural cubic spline f(t) that interpolates this data. You should assume that f(t) is made up of two cubic pieces
f1(t) = a1 t^3 + b1 t^2 + c1 t + d1, 0 <= t < 0.4,
f2(t) = a2 t^3 + b2 t^2 + c2 t + d2, 0.4 <= t <= 1.
You do not have to do all the work by hand, but you must explain how you obtained the solution (and provide any code you used to obtain it, if applicable).