Answer to Question #343209 in Quantitative Methods for Raju

Question

Find Larange’s interpolating polynomial passing through set of points

(0,2) (2,-2),(3,-1),Use it to find

at x = 2

Answer 1

Lagrange Second Order Interpolation Formula  can be written as

$$f(x)=\dfrac{(x-x_1)(x-x_2)}{(x_0-x_1)(x_0-x_2)}y_0$$

$$+\dfrac{(x-x_0)(x-x_2)}{(x_1-x_0)(x_1-x_2)}y_1+\dfrac{(x-x_0)(x-x_1)}{(x_2-x_0)(x_2-x_1)}y_2$$

Substitute

$$f(x)=\dfrac{(x-2)(x-3)}{(0-2)(0-3)}(2)+\dfrac{(x-0)(x-3)}{(2-0)(2-3)}(-2)$$

$$+\dfrac{(x-0)(x-2)}{(3-0)(3-2)}(-1)$$

$$=\dfrac{1}{3}x^2-\dfrac{5}{3}x+2+x^2-3x-\dfrac{1}{3}x^2+\dfrac{2}{3}x$$

$$f(x)=x^2-4x+2$$

$$f(2)=(2)^2-4(2)+2=-2$$