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Unformatted text preview: Math 116 — Final Exam December 15, 2011 Name: EXAM SOLUTIONS Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 14 pages including this cover. There are 10 problems. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck. 3. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor when you hand in the exam. 4. Please read the instructions for each individual problem carefully. One of the skills being tested on this exam is your ability to interpret mathematical questions, so instructors will not answer questions about exam problems during the exam. 5. Show an appropriate amount of work (including appropriate explanation) for each problem, so that graders can see not only your answer but how you obtained it. Include units in your answer where that is appropriate. 6. You may use any calculator except a TI92 (or other calculator with a full alphanumeric keypad). However, you must show work for any calculation which we have learned how to do in this course. You are also allowed two sides of a 3 00 × 5 00 note card. 7. If you use graphs or tables to find an answer, be sure to include an explanation and sketch of the graph, and to write out the entries of the table that you use. 8. Turn off all cell phones and pagers , and remove all headphones. Problem Points Score 1 10 2 12 3 10 4 9 5 10 6 7 7 12 8 12 9 11 10 7 Total 100 Math 116 / Final (December 15, 2011) page 2 You may find the following expressions useful. And you may not. But you may use them if they prove useful. “Known” Taylor series (all around x = 0 ): sin( x ) = x x 3 3! + ··· + ( 1) n x 2 n +1 (2 n + 1)! + ······ for all values of x cos( x ) = 1 x 2 2! + ··· + ( 1) n x 2 n (2 n )! + ··· for all values of x e x = 1 + x + x 2 2! + ··· + x n n ! + ··· for all values of x ln(1 + x ) = x x 2 2 + x 3 3 x 4 4 + ··· + ( 1) n +1 x n n + ··· for 1 < x < 1 (1 + x ) p = 1 + px + p ( p 1) 2! x 2 + p ( p 1)( p 2) 3! x 3 + ··· for 1 < x < 1 Math 116 / Final (December 15, 2011) page 3 1 . [10 points] Indicate if each of the following is true or false by circling the correct answer. No justification is required. a . [2 points] Let a n be a sequence of positive numbers. If a n ≤ 7 n 2 3 n 1 for all values of n ≥ 1 , then a n must converge. True False Solution: Since lim n →∞ 7 n 2 3 n 1 = lim n →∞ 7 n 8 n 1 = 0 and < a n ≤ 7 n 2 3 n 1 , lim n →∞ a n = 0 . Therefore, the sequence a n converges (to zero). b . [2 points] The trapezoid rule is guaranteed to give an underestimate of R π π cos tdt ....
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This note was uploaded on 01/28/2012 for the course MATH 116 taught by Professor Irena during the Fall '07 term at University of Michigan.
 Fall '07
 Irena
 Math, Calculus

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