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epefk - Final Exam Key 2 hours and 45 minutes The average...

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Final Exam Key, 2 hours and 45 minutes April 24, 2003 The average was about 60/100 with the lowest scores on the recent material - power series and polar coordinates. Warning - this key was written days after the exam was graded, so I don’t expect too much interest in it, and I haven’t checked it carefully. The basic methods given are all correct. 1) 1 2 x 4 dx = 32 / 10 = 16 / 5. 2A) From memory, ln(1 + x ) = x - x 2 / 2 + x 3 / 3 - . . . . So, ln(2) 1 - 1 / 2 + 1 / 3 = 5 / 6. Since this is a decreasing alternating series, this answer is too big, but the error is less than the next term, 1/4. 2B) ln(2) = 2 1 dt t 2 - 1 2 · 3 (1 + 4 · 2 3 + 1 / 2) = 25 / 36. Here, for example, y 1 = f ( x 1 ) = f (1 . 5) = 1 / 1 . 5 = 2 / 3. 3a) Set u = tan x and get (tan 3 x ) / 3 + C . 3b) Do IBP twice to get e x ( x 2 - 2 x + 2) + C . 3c) This is improper b/c it has an asymptote at x = 2. A graph is recom- mended. So, it should be split in two (this also helps remove the absolute value signs). The first part is 2 0 | x - 2 | - 1 / 2 = lim t 2 - t 0 (2 - x ) - 1 / 2 = lim t 2 - - 2(2 - x ) 1 / 2 | t 0 = 2 · 2 1 / 2 . You can get the right number without using the ”lim” but you must include that for full credit. The other part gives the same answer, which we add in to get 4
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