3. Interpolate the function f(x) = 2ª over the interval [−1, 1] by a polynomial(a) Construct the interpolating polynomial p2(x) of degree n = 2 in the Newton form using uniformlyspaced nodes.1(b) Find an upper bound for the error |ƒ(x) — p2(x)| at x = 2(c) Find an upper bound for the error |ƒ(x) — p2(x)| on [−1,1]. Note thatmax f(x) p2(x)| ≤x= [a,b]h³ M39√39max | f(x) — pn(x)| ≤x= [a,b]where h = (b − a)/2 and |ƒ""'(x)| ≤ M3 for x € [a, b].(d) How many equally-spaced nodes are required to interpolate the function f(x) to within € = 0.01?Note that for n > 2,1max | f(n+¹) (§) |(n + 1)! E[a,b]hn + 1n!4where h = (b − a)/n.

We have givenInterval: Step size: Therefore, the nodes are at , and Find the value of f(x) at these…

Interpolate the function f(x) = 2* over the interval [1,1] by a polynomial (a) Construct the interpolating polynomial p2 (x) of degree n = 2 in the Newton form using uniformly spaced nodes. (b) Find an upper bound for the error |ƒ(x) — p₂(x)| at x = ½. (c) Find an upper bound for the error |f(x) - p₂(x)| on [-1,1]. Note that max |f(x) p₂(x)| ≤ x= [a,b] h³ M3 9√3 2 max f(x) pn(x)| ≤ x= [a,b] where h = (ba)/2 and |f"(x)| ≤ M3 for x = [a, b]. (d) How many equally-spaced nodes are required to interpolate the function f(x) to within € = 0.01? Note that for n > 2, 1 max | f(n+¹) (§) (n + 1)! € [a,b] hn+¹n! 4 where h = (b -a)/n.

Interpolate the function f(x) = 2* over the interval [1,1] by a polynomial (a) Construct the interpolating polynomial p2 (x) of degree n = 2 in the Newton form using uniformly spaced nodes. (b) Find an upper bound for the error |ƒ(x) — p₂(x)| at x = ½. (c) Find an upper bound for the error |f(x) - p₂(x)| on [-1,1]. Note that max |f(x) p₂(x)| ≤ x= [a,b] h³ M3 9√3 2 max f(x) pn(x)| ≤ x= [a,b] where h = (ba)/2 and |f"(x)| ≤ M3 for x = [a, b]. (d) How many equally-spaced nodes are required to interpolate the function f(x) to within € = 0.01? Note that for n > 2, 1 max | f(n+¹) (§) (n + 1)! € [a,b] hn+¹n! 4 where h = (b -a)/n.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 93E

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