Is a solution to the differential equation y 00 2 y 2

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) is a solution to the differential equation y 00 - 2 y - 2 y 3 = 0. 11. (7 pts) Solve the differential equation y 0 = xe - y x 2 + 1 . Find the particular solution such that y (0) = 1.
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Final Exam, Page 10 of 15 May 8, 2010 12. Let f ( x ) = 10 9 x 2 be a probability density function on [1 , 10]. Find the following: (a) (2 pts) P (2 x 3) (b) (2 pts) The mean (expected value) of x (c) (2 pts) The variance of x (d) (2 pts) The standard deviation of x
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Final Exam, Page 11 of 15 May 8, 2010 13. (5 pts) Let f ( x, y ) = 1 2 ye - x be a joint probability density function on the sample space { 0 x < , 0 y 2 } . Find P (0 x < , 1 y 2) 14. (6 pts) Consider the sequence n ! n n . Write the first four terms of the sequence (without simplifying). Determine whether this sequence converges or diverges. If it converges, find its limit. If it diverges, explain why.
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Final Exam, Page 12 of 15 May 8, 2010 15. Determine if each of the following series converges or diverges. If it converges, find its sum . (a) (4 pts) X n =0 6 n - 2(3 n ) 10 n (b) (4 pts) X n =0 1 n + 3 - 1 n + 4 (c) (4 pts) 3 - 3 4 + 3 16 - 3 64 + 3 256 + . . .
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Final Exam, Page 13 of 15 May 8, 2010 16. Determine if each of the following series converges or diverges. For each series,
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