quiz03_s107_solns

quiz03_s107_solns - Math 1B Quiz 3 Solutions Section 107...

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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 0 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 0 ( 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. If we want to find n so that | E T | ≤ 10 - 4 , then because of the error formula given above, it is enough to find n that satisfy this equation: K ( b - a ) 3 12 n 2 10 - 4 24(1 - 0) 3 12 n 2 10 - 4 24 12 10 4 n 2 100 2 n 2. (4 points, no partial credit) Circle C if the integral converges and D if it diverges. (a) R 1 1
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