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Unformatted text preview: Math 1B Quiz 3 Solutions Section 107 September 20, 2009 1. (4 points) For this problem, you will need the following information: Suppose  f 00 ( x )  ≤ K for a ≤ x ≤ b . Then the error E T in the approximation from the Trapezoidal Rule is bounded by  E T  ≤ K ( b a ) 3 12 n 2 . If we wanted to approximate R 1 6 x 6 x +1 dx using the Trapezoidal approximation on n subintervals, how large would n have to be to make the approximation accurate to within . 0001? First we calculate f 00 ( x ). f ( x ) = ( x + 1)6 (6 x 6) ( x + 1) 2 = 6 x + 6 6 x + 6 ( x + 1) 2 = 12 ( x + 1) 2 f 00 ( x ) = 12(2( x + 1)) ( x + 1) 4 = 24 ( x + 1) 3 Now we must find the maximum of  f 00 ( x )  for x ∈ [0 , 1]. On this interval f is increasing, so we only need to check the endpoints. (Careful; f is negative, so just taking the value at the right endpoint is not appropriate here.) Calculate f 00 (0) = 24 and f 00 (1) = 3, so  f 00 ( x )  ≤ 24 on the interval. So we can choose K = 24....
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This note was uploaded on 03/25/2011 for the course MATH 1B taught by Professor Reshetiken during the Fall '08 term at Berkeley.
 Fall '08
 Reshetiken
 Approximation

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