M116ExFF11Solutions

# M116ExFF11Solutions - Math 116 Final Exam Name EXAM...

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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 0 < 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 . True False Solution: Since cos( t ) changes concavity over the interval - π t π , the trapezoid rule is guaranteed to give neither an under- nor an overestimate over that interval. c . [2 points] If the area A under the graph of a positive continuous function f ( x ) is infinite, then the volume of the solid generated by rotating A around the x -axis could be either infinite or finite depending on the function f ( x ) .

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