2-11 - pair Add up the areas A = Δ x 3 f x i-1 4 f x i f x...

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February 11, 2005 Announcements Review L’Hospital Rule Today § 7.7 Approximate Integration What can you do when you are asked to estimate 1 0 e x 2 dx or 1 0 cos( x 3 ) dx ? 1
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Let f be a continuous function on [ a, b ]. Divide [ a, b ] into n subintervals of width Δ x . Take n to be EVEN. Here Δ x = b - a n and x 0 = a, x n = b, x i = x i - 1 + Δ x = a + i Δ x for 1 i n. Recall b a f ( x ) dx = lim n →∞ n i =1 f ( x * i x where x * i is any sample point in the i th -interval [ x i - 1 , x i ].
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Midpoint Rule b a f ( x ) dx M n = Δ x [ f x 1 ) + f x 2 ) + · · · + f x n )] where ¯ x i = 1 2 ( x i - 1 + x i ) Trapezoidal Rule Instead of using Rectangles use Trapezoids whose top side is given by the cord between the end- points x i - 1 and x i of the i th -interval. b a f ( x ) dx T n = Δ x 2 n i =1 [ f ( x i - 1 ) + f ( x i )] = Δ x 2 [ f ( x 0 ) + 2 f ( x 1 ) + · · · +2 f ( x n - 1 ) + f ( x n )]
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Simpson’s rule Replace the linear approximation by a quadratic approximation. Use each subsequent PAIR of intervals to de- termine the best fitting quadratic equation ( y = Ax 2 + Bx + C ) between the three points deter-
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Unformatted text preview: pair. Add up the areas A = Δ x 3 [ f ( x i-1 ) + 4 f ( x i ) + f ( x i +1 )] under the graph of this quadratic equation (parabola). Z b a f ( x ) dx ∼ S n S n = Δ x 3 [ f ( x ) + 4 f ( x 1 ) + 2 f ( x 2 ) + ··· +2 f ( x n-2 ) + 4 f ( x n-1 ) + f ( x n )] Error bounds Suppose | f 00 ( x ) | ≤ K for a ≤ x ≤ b . If E T is the error in the Trapezoidal Rule and E M is the error in the Midpoint Rule, then | E T | ≤ K ( b-a ) 3 12 n 2 and | E M | ≤ K ( b-a ) 3 24 n 2 . Error bound for Simpson’s rule Suppose | f (4) ( x ) | ≤ K for a ≤ x ≤ b . If E S is the error in Simpson’s Rule, then | E S | ≤ K ( b-a ) 5 180 n 4 ....
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