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Unformatted text preview: Here a = , b = , and f ( x ) = (2 x – 3) 3 . Differentiating twice, we find that " " f x ( ) = How large could  f "" ( x ) be if x satisfies 1 # x # 4? Since the function 48 x – 72 is clearly increasing on the interval from 1 to 4 (in fact, it’s increasing everywhere), its greatest value occurs at x = . Therefore, its greatest value is so we may take A = in the preceding theorem. The error of approximation using the midpoint rule is at most NOTE: We have hitherto determined that the exact value of this integral is 78 and that the midpoint approximation for it (using n = 3) is 72. Therefore, this approximation was in error by 78 – 72 = 6. Our result in this exercise says that our midpoint approximation error should be no greater than 15. Since 6 is no greater than 15, this result suggests that our midpoint approximation was done correctly....
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 Spring '08
 BUETTGENS
 Numerical Analysis, Approximation, Definite Integrals, Integrals, Midpoint Rule, Approximation theory

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