State newton's forward interpolation to find f'(X),f"(X),f'"(X) at x=xo
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Consider the equation of the linear interpolation optained in the earlier section :
f1 - f0
f0x1 - f1x0
f(x) @ P1(x) = ax-1b =
x +
x1 - x0
x1 - x0
=
1
[(x1 - x)f0 + (x-x0)f1]
(x1 - x0)
x1 - x
x - x0
x - x0
f0 +
(f1- f0) +
f0
x1 - x0
x1 - x0
x1 - x0
x - x0
= f0 +
(f1- f0)
x1 - x0
= f0 + r Df0 [ r = (x - x0) / (x1 - x0) Df0 = f1 - f0 ]
since x1 - x0 is the step lenght h, r can be written as (x - x0)/h and will be between (0, 1).
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