# final 6 answers - cos θ r r dr dθ ANSWER 61 3 ³ 1 π 2...

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MATH 126 – Winter 2007 Final Exam Hints, Answers, and Partial Solutions 1. (a) ANSWER: T 1 ( x ) = 1 5 + 1 25 x (b) HINT: Taylor’s inequality states that the error is bounded by M 2 x 2 , where M is an upper bound for | f 00 ( x ) | . Here, | f 00 ( x ) | = 2 | 5 - x | 3 , which is largest on I when x = 2. So, we can take M to be 2 27 . ANSWER: error 4 27 (c) HINT: Show that f ( n +1) ( x ) = n ! (5 - x ) n +1 and that the error is bounded by 1 3 ± 2 3 ² n +1 . Then solve the system 0 . 04 < 1 3 ± 2 3 ² n +1 < 0 . 05 for n . ANSWER: n = 4 2. (a) HINT: Substitute - 2 x 2 into the Taylor series for 1 1 - x and 3 x into the Taylor series for cos x and subtract. ANSWER: X k =0 ( - 1) k ± 2 k - 3 2 k (2 k )! ² x 2 k (b) HINT: The Taylor series for cos x converges for all x , but the Taylor series for 1 1 - x converges only for x such that | x | < 1. This means that the Taylor series for f ( x ) converges only for x such that | 2 x 2 | < 1. ANSWER: - 1 2 < x < 1 2 (c) ANSWER: T 4 ( x ) = 5 2 x 2 + 5 8 x 4 3. (b) HINT: ZZ R x + p x 2 + y 2 dA = Z π/ 2 0 Z 5 4 ( r
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