Let f(x) = sin 2x. (a) Find the Hermite interpolating polynomial of degree at most three using xo = 0 and x₁ = □, then use it to approximate ƒ (²). (b) Use the error formula below to find the error bound, then compare it with the actual error. f(x) − H₂n+1(x) - ƒ (²n+2) (5(x))(x − x₁) ². = (2n + 2)! i=0
Let f(x) = sin 2x. (a) Find the Hermite interpolating polynomial of degree at most three using xo = 0 and x₁ = □, then use it to approximate ƒ (²). (b) Use the error formula below to find the error bound, then compare it with the actual error. f(x) − H₂n+1(x) - ƒ (²n+2) (5(x))(x − x₁) ². = (2n + 2)! i=0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let \( f(x) = \sin 2x \).
(a) Find the Hermite interpolating polynomial of degree at most three using \( x_0 = 0 \) and \( x_1 = \pi \), then use it to approximate \( f\left(\frac{\pi}{5}\right) \).
(b) Use the error formula below to find the error bound, then compare it with the actual error.
\[
f(x) - H_{2n+1}(x) = \frac{f^{(2n+2)}(\xi(x))}{(2n+2)!} \prod_{i=0}^{n}(x-x_i)^2.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2f2c4bd-bf4c-4a3b-a0a2-6333c3306a45%2Fd1bfa6c0-c800-4b6c-942b-d0d67d219ba4%2F5an3lzk_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( f(x) = \sin 2x \).
(a) Find the Hermite interpolating polynomial of degree at most three using \( x_0 = 0 \) and \( x_1 = \pi \), then use it to approximate \( f\left(\frac{\pi}{5}\right) \).
(b) Use the error formula below to find the error bound, then compare it with the actual error.
\[
f(x) - H_{2n+1}(x) = \frac{f^{(2n+2)}(\xi(x))}{(2n+2)!} \prod_{i=0}^{n}(x-x_i)^2.
\]
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