Question

Chapter 4 Be able to define an interpolating polynomial for a set of points, and a Cardinal Polynomial. Be able to use Cardinal polynomials to prove the existence of interpolating polynomials, and be able to prove they are unique Be able to state and prove the Recursive Property of divided differences (18. 134) nnd the Invariance Theorem (pg. 135). Problem 7a. Consider the points (1.1),(2,5), (5.41). Find the corresponding Cardinal polynomials and use them to construct the interpolating polynomial of degree no more than 2. You need not simplify this polynomial. Problem 7b. Use the points in find

Problem 7a please.

Answer 1

the general formula is given also

consider the points (1,1),(2,5),(5,41). Find the corou Spondin Cardinal polynomials and use them to construct the interpolating polynomial of degree no more than 2. Ans. The cardinal (Lagrange Cardinal) polynomial corresponding to the point (1,1) is L(a)= (2-2) (2-5) - 02330053 (-2)(a-6) wart) The cardiral polynomial corresponding to the point (2,5) is, L (a) = (2-1) (2-5). tandarts) P - § 9-0 (2-5) (2-0 (2-5) The cardenal polynomial correspouding to the point (5,4) is, L₂ (2) = (2.1) (2-2) = t Card (2-2) (5-1 6-2) The interpotatug polynomial (Lagrange polynomiallie 1. *a-2) (a-5) + 5. (-3) (-1)(a-5)+ 41.js Ca-1) (2-2) ie (2-2)(0-5) - (2-) (8-5) +41 (0-1) (-2) Ingeneral, for three points (2, 4), (22,92) (23,983), the I interpolating polynomial is (4-22) (0-23) 4 (1-2) (9-23) 4 (1-21) (-22) di Ca -2₂) (2, -23) (22-24) (22-23) 1 - 2) (22-22) Look, the polejnonial is of degree 2.