Question

12. Given the data set: We want to find the interpolating polynomial of degree 2 through these points. a) Write the interpolating polynomial in Lagrange form b) Write the interpolating polynomial in Newton form.

Answer 1

Interpolating polynomial language polynomial f(x) = (x - x_1) (x - x_2)/(x_0 - x_1) (x_0 - x_2) f(x_0) + (x - x_0) (x - x_2)/(x_1 - x_0) (x_1 - x_0) f(xi) + (x - x_0) (x - x_1)/(x_2 - x_0) (x_2 - x_1) f(x) f(x) = (x - 1) (x - 2)/(0 - 1) (0 - 2) times 4 + (x - 0) (x - 2)/(1 - 0) (1 - 2) (0) + (x - 0) (x - 1)/(2 - 1) (2 - 0) (0) f(x) = 4/2 (x^2 - 3x + 2) = 2(x^2 - 3x + 2) rightarrow f(x) 2(x^2 - 3x + 2) language from 2(x^2 - 3x + 2) newton's form the divided deference table f(x) = f(x_0) + (x - x_0) [x_0 x_1] (x - x_0) (x - x_1) [x, x_0, x_1] = 4 + (x - 0) x - 4 + (x - 0) (x - 1) 2 = 4 - 4x + 2x^2 - 2x f(x) = 2x^2 - 6x + 4