104F09Final

# What is the third derivative f 3 x evaluated at x 0 a

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. What is the third derivative, f (3) ( x ), evaluated at x = 0? (a) - 54 (b) - 9 (c) - 3 / 2 (d) 3 / 2 (e) 9 (f) 54 6

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11. What is the radius of convergence of the series X n =0 2 · 5 · 8 ··· (3 n - 1) 3 · 7 · 11 ··· (4 n - 1) ( x + 1) n ? (a) 0 (b) 1 2 (c) 2 3 (d) 3 4 (e) 4 3 (f) 3 2 (g) 2 (h) 12. Compute X n =1 n 3 n - 1 . (a) 9 4 (b) 3 e - 3 (c) 4 3 (d) r 3 2 (e) 2 (f) 2 3 (g) 3 n 2 / 3 7
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. S 8

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14. Newton’s Law of Cooling states that the rate at which a body changes tempera- ture is proportional to the diﬀerence between its temperature and the tempera- ture of the surrounding medium. Suppose that a body has an initial temperature of 252 F and that after one hour the temperature is 216 F. Assuming that the surrounding air is kept at a constant temperature of 72 F, determine the tem- perature of the body two hours after the starting time, in degrees Fahrenheit. 9
15. Solve y 0 = ln( x ) x y with initial condition y (1) = 2. 10

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16. Compute the ﬁrst three terms of the Maclaurin series for f if the Maclaurin series for f begins 1 + ax + bx 2 + ··· . 11
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What is the third derivative f 3 x evaluated at x 0 a 54 b...

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