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**Unformatted text preview: **Math 41: Calculus Final Exam — December 8, 2008 Name : Section Leader: Bob Joe David Nathan Ian (Circle one) Hough Rabinoff Sher Stiennon Weiner Section Time: 11:00 1:15 (Circle one) • Complete the following problems. In order to receive full credit, please show all of your work and justify your answers. • You do not need to simplify your answers unless specifically instructed to do so. You may use any result from class that you like, but if you cite a theorem be sure to verify the hypotheses are satisfied. • You have 3 hours. This is a closed-book, closed-notes exam. No calculators or other electronic aids will be permitted. If you finish early, you must hand your exam paper to a member of teaching staff. • Please check that your copy of this exam contains 19 pages and is correctly stapled. • If you need extra room, use the back sides of each page. If you must use extra paper, make sure to write your name on it and attach it to this exam. Do not unstaple or detach pages from this exam. • Please sign the following: “On my honor, I have neither given nor received any aid on this examination. I have furthermore abided by all other aspects of the honor code with respect to this examination.” Signature: The following boxes are strictly for grading purposes. Please do not mark. 1 10 8 10 2 13 9 8 3 10 10 13 4 8 11 10 5 10 12 23 6 10 13 12 7 10 14 8 Total 155 Math 41, Autumn 2008 Final Exam — December 8, 2008 Page 2 of 19 1. (10 points) Find each of the following limits, with justification. If there is an infinite limit, then explain whether it is ∞ or-∞ . (a) lim x → + x 2 e (1 /x ) (b) lim x →∞ ln(1 + ln4 x ) ln(2 + 3ln x ) Math 41, Autumn 2008 Final Exam — December 8, 2008 Page 3 of 19 2. (13 points) Differentiate, using the method of your choice. (a) f ( x ) = x 3 csc x- ln | x 2- 1 | + 1 3 e ( x 3 ) (b) g ( t ) = cos t 1 /t (c) F ( z ) = Z z 1 1 + t 2 dt + Z 1 /z 1 1 + t 2 dt Math 41, Autumn 2008 Final Exam — December 8, 2008 Page 4 of 19 3. (10 points) Suppose g is the function g ( x ) = x 2 sin 1 x if x 6 = 0 if x = 0 . (a) Is g continuous at 0? Use the definition of continuity when justifying your answer....

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