The Lagrange interpolating polynomial curve through the distinct points p0, p1, p2, … , pn...

The Lagrange interpolating polynomial curve through the distinct points p0, p1, p2, … , pn, is the curve parametrized by the function

where

(that is, αi (t) is the product of (t − j)/(i − j), where j ranges from 0 to n, skipping i).

(a) Verify that αi (i) = 1 for i = 0, 1, 2, … , n. Also, verify that for each i from 0 to n, αi (j) = 0 whenever 0 ≤ j ≤ n and j ≠ i. Determine the range of t that gives the portion of the curve from p0 to pn.

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(b) Calculate the Lagrange interpolating polynomial for p0 = (0, 0), p1 = (2, 1), and p2 = (1, 5). Using a graphics program, obtain a graph of the curve.

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(c) Calculate the Lagrange interpolating polynomial for p0 = (0, 0, 0), p1 = (1, 1, 0), and p2 = (1, 2, 3). Obtain a graph.