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fx166f2009 - Math 166 Final Exam Page 5 5(14 points...

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Math 166 Final Exam December 16, 2009 Full Name: Instructor Section: Instructions: Answer each question completely. Show all necessary work. No credit is allowed for mere answers with no work shown. State the reasons that justify your conclusions. However, “I did it on my calculator” is not sufficient justification for any result. 1. (14 points) Find the indicated limit, or state (and justify) that it does not exist. (a) lim r →- 3 3 r 2 - 10 r 2 + 2 r - 3 (b) lim x 0 sin(2 x 2 ) e - x 2 - 1
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Math 166 Final Exam Page 2 2. (14 points) Evaluate each improper integral or show that it diverges. (a) Z 2 1 x 2 - 1 dx (b) Z π sin 2 x dx
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Math 166 Final Exam Page 3 3. (14 points) Find the convergence set for the power series X k =1 k 2 3 k ( x + 1) k
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Math 166 Final Exam Page 4 4. (14 points) (a) Given the power series 1 1 - t = 1 + t + t 2 + t 3 + · · · , - 1 < t < 1 find a power series for x 2 x 2 + 3 . (b) Write a geometric series that converges to 5.
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Unformatted text preview: Math 166 Final Exam Page 5 5. (14 points) Determine whether each infinite series converges or diverges. Indicate the test(s) you use. (a) ∞ X k =1 (-1) k 2 k k 2 + 1 (b) ∞ X n =2 ln n n 2 Math 166 Final Exam Page 6 6. (16 points) (a) Find P 2 ( x ), the Taylor polynomial of order 2 based at 1 for f ( x ) = ln x . (b) Bound the error | R 2 ( x ) | = | ln x-P 2 ( x ) | if 1 2 ≤ x ≤ 3 2 . (You my use the formula R n ( x ) = f n +1 ( c ) ( n + 1)! ( x-a ) n +1 for some number c between a and x .) Math 166 Final Exam Page 7 7. (14 points) (a) Name the curve with polar equation r = 2 sin θ , and sketch its graph. (b) Set up and evaluate an integral in polar coordinates for the area inside of the graph of part (a)....
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