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e111k - Exam I MAC 2312 and Key July 6 2011 S Hudson 1[30pt...

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Exam I July 6, 2011 MAC 2312 and Key S Hudson 1) [30pt] Short calculations, 5 pts each, not much partial credit likely. Simplify terms like sin( π ) and e 0 , but not like 121 / 44. a) d dx [ R x 0 te t dt ] b) R 4 0 x + e x dx c) R 2 1 1 t dt f) R π 0 sin 2 x dx e) R 1 0 dx 2 - x f) R 3 0 | x - 2 | dx = 2) [15pts] State and prove the F.T.C., Part Two, about computing the derivative of a definite integral. You can use other definitions and theorems in your proof - but point out when you are doing so. Partial credit is likely, even with small gaps, if you are making a good effort to explain your reasoning. 3) [15pts] Find the exact area under f ( x ) = x 2 , on [0,2] using a limit of sums (as done in class and in Ch.5.4). Use a regular partition and the Right Endpoint Rule. This is roughly equivalent to asking for the definition of R 2 0 x 2 dx and a calculation of that. To make this long problem a bit easier, you get an outline and some formulas to choose from (but not all are needed, or even correct). A) The interval [0 , 2] gets split into n parts and x * k is the right endpoint of the k th part. Which two of these formulas are correct (choose and explain) ? Δ x = 1 /n , Δ x = 2 /n , x * k = k/n , x * k = 2 k/n , x * k = 2 k - 1 /n . B) Now, we add up the areas of n rectangles to get an estimate of the total area, called E n . Which two of these are correct formulas for E n (choose and explain) ? n 1 k/n , 2 n 1 ( x * k ) 2 /n , n ( n + 1) / 2, n ( n + 1) / 3 n 2 , 8 n ( n + 1)(2 n + 1) / 6 n 3 .

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