Question

Let f(x)=xeˣ and x₀=0,x₁=−1,x₂=1.

Find the interpolation polynomial
P2​(x) that interpolates the data points
(x₀,f(x₀)),(x₁,f(x₁)),(x₂,f(x₂)),
using :-
(a) Lagrange form of the interpolation polynomial.
(b) Newton divided differences form of the interpolating polynomial.
(c) Write the interpolation polynomial in standard form.
Show that f[x₀,x₁,x₂]=f[x₂,x₁,x₀]

Answer 1

To find the interpolation polynomial, we can use either the Lagrange form or the Newton divided differences form. In both cases, we evaluate the values of the given points and plug them into the formulas. The interpolation polynomial in standard form is P2(x) = −x*e^(−1) + 0 + x*e^1.

Step-by-step explanation:

To find the interpolation polynomial using Lagrange form, we will use the formula:

P2(x) = L0(x)f(x0) + L1(x)f(x1) + L2(x)f(x2)

where Li(x) = ((x-xj)/(xi-xj)) * ((x-xk)/(xi-xk)) for i ≠ j ≠ k. Evaluating the values, we have P2(x) = −x*e^(−1) + 0 + x*e^1.

To find the interpolation polynomial using Newton divided differences form, we will use the formula:

P2(x) = f(x0) + f[x0,x1](x-x0) + f[x0,x1,x2](x-x0)(x-x1)

Evaluating the values, we have P2(x) = 0 + (e^1 - e^(−1))x + e^(−1)x(x-1).

The interpolation polynomial in standard form is P2(x) = −x*e^(−1) + 0 + x*e^1. To show that f[x0,x1,x2]=f[x2,x1,x0], we can calculate the values using the divided differences formula and verify that they are equal.