Assignment:
Q(i) Use the tabulated data and Newton's method of divided differences to (a) calculate the degree three interpolating polynomial based on x_0, x_1, x_2, x_3 and (b) calculate the degree four interpolating polynomial based on x_0, x_1, x_2, x_3, x_4.
x_k || 0 | -1 | 1 | 3 | 2
y_k || 5 | 15 | 3 | 47 | 9
(Use exact arithmetic (with fractions, if necessary); do not re-order the given points.)
Q(ii) Check whether the function f(x) is a cubic spline, where
f(x) = {0.5x^3 - 2x^2 + 1.3x - 8 if 0 <= x <= 1
{0.5(x-1)^3 - 0.5(x-1)^2 - 1.2(x-1) - 8.2 if 1 <= x <= 3
{0.5(x-3)^3 + 2.4(x-3)^2 + 2.8(x-3) - 8.6 if 3 <= x <= 4
Provide complete and step by step solution for the question and show calculations and use formulas.