e304k - Exam III and Key MAC 2312 Aug 5, 2004 S Hudson 1)...

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Exam III and Key Aug 5, 2004 MAC 2312 S Hudson 1) (5pts) Find the Taylor polynomial of order n=2 centered at a = 1 for f ( x ) = x . 2) (5pts) Find the McLaurin Series (a=0) for f ( x ) = sin( x 2 ) by substituting into a known series. 3) (10pts) Find the sum of the series (or write ”Diverges”): a) k =0 π +1 π k b) k =1 1 k +2 - 1 k +3 4) (40pts) Compute the integrals (or write ”diverges”). Show all your work. a) R 1 ln( x ) x 2 dx b) R +1 - 1 dx x 2 c) R x 2 ( x +2) 3 dx d) R dx 9 - 4 x 2 e) R sec 4 x tan 2 x dx f) R 1 x 3 +4 x dx g) R 2 x +5 x 2 +4 x +5 dx h) R ln(1 + x ) dx 1
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5) (10pts) For each series, answer either Converges (C) or Diverges (D) and state which “test” you are using. a) k =1 2 k k 2 +1 b) k =1 ( - 1) k k 2 k +3 6) (10pts) Choose ONE proof, explain thoroughly: a) Calculate R sec( x ) dx , explaining each step. b) State and prove the Divergence Test. 7) (20pts) Answer True or False:
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e304k - Exam III and Key MAC 2312 Aug 5, 2004 S Hudson 1)...

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