104F09Final

# According to taylors remainder estimate the maximum

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8. According to Taylor’s remainder estimate, the maximum possible error in the use of 4 n =0 x n n ! to approximate e x on the interval [ - 1 , 1] is which value? (a) e 240 (b) e 120 (c) 1 48 (d) 1 24 (e) 1 24 e (f) 1 12 e 5

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9. Evaluate n =0 ( - 1) n π 2 n +1 3 2 n (2 n + 1)! . 10. Let f ( x ) = n =0 ( - 1) n n 2 x n . What is the third derivative, f (3) ( x ), evaluated at x = 0? 6
11. What is the radius of convergence of the series n =0 2 · 5 · 8 · · · (3 n - 1) 3 · 7 · 11 · · · (4 n - 1) ( x + 1) n ? 12. Compute n =1 n 3 n - 1 . (a) 9 4 (b) 3 e - 3 (c) 4 3 (d) 3 2 (e) 2 (f) 2 3 (g) 3 n 2 / 3 7

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Free answer questions Each of these questions is worth 8 points. Please show all work that you wish to be considered for partial credit and put a box around the final answer to each (part of the) problem. 13. A standard US Mailbox has a cross-section shaped like a square of side S with a semicircle attached. Where is the center of mass of this cross-section? State your answer in coordinates with the origin at the center of the square.
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