Lecture13bc - Here a = , b = , and f ( x ) = (2 x – 3) 3...

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Instructor: Quanlei Fang Dept. of Math, University at Buffalo, Spring 2009 Welcome to Math 122 Lecture 13 Survey of Calculus and its Applications I1 § 9.4 Approximation of Definite Integrals ! The Midpoint Rule ! The Trapezoidal Rule ! Simpson’s Rule ! Error Analysis
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Approximate the following integral by the midpoint rule. We have ! x = ( b a )/ n = . The endpoints of the four subintervals begin at a = 1 and are spaced 1 unit apart. The first midpoint is at a + ! x /2 = . The midpoints are also spaced unit apart. According to the midpoint rule, the integral is approximately equal to
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Approximate the following integral by the trapezoidal rule. 2 x " 3 ( ) 3 1 4 # dx ; n = 3 As in the last example, ! x = and the endpoints of the subintervals are a 0 = , a 1 = , a 2 = , and a 3 = . The trapezoidal rule gives
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Approximate the following integral by Simpson’s rule. 2 x " 3 ( ) 3 1 4 # dx ; n = 3
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Obtain a bound on the error of using the midpoint rule with n = 3 to approximate
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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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This note was uploaded on 02/20/2009 for the course MTH 122 taught by Professor Buettgens during the Spring '08 term at SUNY Buffalo.

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Lecture13bc - Here a = , b = , and f ( x ) = (2 x – 3) 3...

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