Fall 2012 Prelim #3 Soln - Math 1910 Prelim 3 7:30 PM to...

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Math 1910, Prelim 3 November 27, 2012, 7:30 PM to 9:00 PM You are NOT allowed calculators, the text or any other book or notes. SHOW ALL WORK! Writing clearly and legibly will improve your chances of receiving the maximum credit that your solution deserves. Please label the questions in your answer booklet clearly. There are 5 questions . Write your name and section number on each booklet you use. You may leave when you have finished, but if you have not handed in your exam booklet and left the room by 8:45pm, please remain in your seat so as not to disturb others who are still working. 1. (20 points) Suppose f ( x ) = 6 x 2 + 2 x + 1. (a) Calculate R 1 0 f ( x ) dx exactly. Solution Z 1 0 f ( x ) dx = Z 1 0 (6 x 2 + 2 x + 1) dx = 2 x 3 + x 2 + x 1 0 = 2 + 1 + 1 = 4 . (b) Calculate T 1 the estimate for R 1 0 f ( x ) dx for one interval for the Trapezoidal Rule. Solution T 1 = 1 2 (1 - 0)[ f (0) + f (1)] = 1 2 [1 + 9] = 5 . (c) Calculate the error estimate for T 1 , which is K 2 ( b - a ) 3 / (12 N 2 ), where K 2 is the maximum of the absolute value of second derivative for f ( x ) on the interval [ a, b ], and N is the number of subintervals. Solution f 0 ( x ) = 12 x + 2 , f 00 ( x ) = 12 . The f 00 ( x ) is constant, hence its maximum value is same as that, K 2 = 12. With b = 1, a = 0 and N = 1, the error estimate is: E = 12(1 - 0) 3 / (12) = 1 (d) Calculate the actual error | R 1 0 f ( x ) dx - T 1 | . How does this compare to the error estimate in Part (c)?

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