Consider the data in the table
|
n
|
xn
|
f(xn)
|
|
0
|
1.15
|
0.25
|
|
1
|
1.2
|
0.5
|
|
2
|
1.4
|
0.75
|
a. Compute the divided differences f[x0, x1] and f[x0, x1, x2].
b. Find the linear and quadratic interpolating polynomials using the Newton Divided Difference formulation.
c. Suppose we know that the data comes from a function f(x) that is twice differentiable and that |f"(c)| ≤ M for all c in the interval [1, 15, 1.4].
Show that the error satisfies the following inequality
|f(x) - P1(x)| ≤ 0.01 M/ 32, for 1.15 ≤ x ≤ 1.2.