mat127practicefinal

# mat127practicefinal - MAT 127 PRACTICE FINAL(1 Consider the...

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MAT 127 PRACTICE FINAL (1) Consider the initial value problem y 00 - y 0 + 3 y = 0 y (0) = 1 , y 0 (0) = - 1 . Assuming the solution to this initial value problem has is the power series y = X n =0 c n x n , find all the coeffiecients c n for n 6. (2) Use the seperation of variables technique to solve the initial value prob- lem y 0 = yln ( x ) y (1) = 2 . (3) (a) Use Euler’s Method with step size 1 to estimate the value y (3), where y denotes the solution to the initial value problem y 0 = y + x 2 y (0) = 1 . (b) Sketch the direction field for the differential equation given in part (a). (4) Determine whether or not each of the following sequences { a n } con- verges. If the sequence converges, then compute the limit. (a) a n = 2 + ( - 2 ) n (b) a n = ( n 3 - n + 2) / ( n 2 - 3 n 3 ) (c) a n = 3 n /n 4 (d) a n = n 2 /n ! (5) Use any method to determine whether or not each of the following series n =1 a n converges. (a) n =1 a n = n =1 (1 + n - 2 ) /n (b) n =1 a n = n =1 ( n 2 + n + 2) / ( n - 8 n 2 ) 1

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2 MAT 127 PRACTICE FINAL (c) n =1 a n = n =1 ( - 1) n +1 (2 + cos ( n )) /n 2 (d) n =1 a n = n =1 ( - 1) n (2 + e - n ) /n (e) n =1 a n = n =1 n 3 / 2 n (6) The differential equation P 0 = 0 . 2 P (1 - P/ 1000)
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