PracticeExam2

# E 2 points where does the series converge

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(e) [2 points] Where does the series converge conditionally? Page 5

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Math 222: Exam 2 3. (a) [8 points] Estimate the error in using the first five terms of the power series for e x to approximate e . In particular, find a rational number greater than the error (you may assume that e < 3). (b) [8 points] Recall that X n =0 ( - 1) n 2 n + 1 = π 4 . How many terms of the series would you use to approximate π to within 1 / 100? Justify your answer. Page 6
Math 222: Exam 2 4. [10 points] Set up but do not solve an initial value problem for y ( t ) = pounds of salt in a tank after t minutes. Assume the tank starts with 20 gallons of saltwater at time t = 0 and that each gallon has 1 / 4lb of salt. Pure water is flowing into the tank at a rate of 3 gallons per minute and saltwater is flowing out of the tank at a rate of 4 gallons per minute. Assume the solution in the tank is kept perfectly mixed. Page 7

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Math 222: Exam 2 5. Find the general solution to the following differential equations. (a) [8 points] e y - x dy dx = x . Solve for y ( x ) in your solution. (b) [8 points] xy 0 - 2 y - x 2 = 0 Page 8
Math 222: Exam 2 (c) [8 points] y 00 - 6 y 0 + 13 y = 0 6. (a) [4 points] Solve y 00 - y 0 - 2 y = 0. Page 9

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Math 222: Exam 2 (b) [8 points] Find a particular solution for y 00 - y 0 - 2 y = 12 e 2 x . Page 10
Math 222: Exam 2 (c) [4 points] Solve y 00 - y 0 - 2 y = 12 e 2 x if y (0) = 1 and y 0 (0) = 0. Page 11
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