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Final_S13

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(6) Show your work; correct answers only will receive only partial credit (unless noted otherwise). (7) Be careful to avoid making grievous errors that are subject to heavy penalties. (8) If you need more blank paper, ask a proctor. Out of fairness to others, please stop working and close the exam as soon as the time is called. A signiﬁcant number of points will be taken oﬀ your exam score if you continue working after the time is called. You will be given a two-minute warning before the end. 1 2 3 4 5 6 7 8 9 10 11 12 Total 1

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1. (20 points) Solve the diﬀerential equation: a) (10 points) x 3 e y - y 0 = e x + y . b) (10 points) yy 0 = cos x,y (0) = 9 . 2
2. (20 points) Solve the diﬀerential equation (Find the general solution): a) (10 points) y 00 + 7 y 0 = 0. b) (10 points) y 00 + 6 y 0 + 10 y = 0. 3

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3. (10 points) Solve the following initial value problem: y 00 + 10 y 0 + 25 = 0 , y (0) = 3 , y 0 (0) = 0 . 4
4. (10 points) A radioactive sample decayed to 80% of its original mass in 1 year. Find the half-life time for this sample. 5

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5. (10 points) A cup of iced coﬀee with temperature 10 C is placed in a room with temperature 25 C . After 10 minutes the temperature of the cup was equal to 15 C . Find the temperature of the cup after an hour. 6
Test the following series for convergence: 6. (10 points) X n =1 3 n n + 3 7. (10 points) X n =1 n + 5 n - 10 7

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8. (10 points) X n =1 sin n - 1 n 3 + 1 9. (10 points) X n =2 ( - 1) n ln n 8
10. (15 points) Consider the series X n =1 ( - 1) n ( x + 1) n n · 7 n . a) (10 points) Find the radius of convergence b) (5 points) Find the interval of convergence 9

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11. a) (10 points) Find the Taylor series for the function f ( x ) = 1 (1+ x ) 2 with center at a = 5. b) (5 points) Find the radius of convergence for this series 10
12. (10 points) Present the integral as a sum of inﬁnite series and estimate it with error within 0 . 001: Z 0 . 1 0 sin x - x x 3 dt. 11
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