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When Is Simpson's Rule Exact? (a) Show that if f is a polynomial of degree 3 or lower, then Simpson's Rule gives the exact value off f(x) dx. (b) Find the approximation S4 for f(x³ - 6x² + 4x) dx and verify that S4 is the exact value of the integral. (c) Use the error bound given in (4) to explain why the statement in part (a) must be true.

When Is Simpson's Rule Exact? (a) Show that if f is a polynomial of degree 3 or lower, then Simpson's Rule gives the exact value off f(x) dx. (b) Find the approximation S4 for f(x³ - 6x² + 4x) dx and verify that S4 is the exact value of the integral. (c) Use the error bound given in (4) to explain why the statement in part (a) must be true.

Advanced Engineering Mathematics
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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This is from James Stewart Calculus Textbook

 

When Is Simpson's Rule Exact? (a) Show that if f is a polynomial of degree 3 or lower, then Simpson's Rule gives the exact value of f(x) dx. (b) Find the approximation S4 for f(x³ - 6x² + 4x) dx and verify that S4 is the exact value of the integral. (c) Use the error bound given in (4) to explain why the statement in part (a) must be true.
Transcribed Image Text:When Is Simpson's Rule Exact? (a) Show that if f is a polynomial of degree 3 or lower, then Simpson's Rule gives the exact value of f(x) dx. (b) Find the approximation S4 for f(x³ - 6x² + 4x) dx and verify that S4 is the exact value of the integral. (c) Use the error bound given in (4) to explain why the statement in part (a) must be true.
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