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A10_2011 - University of Toronto at Scarborough Department...

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University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT A37H Summer 2011 Assignment # 10 You are expected to work on this assignment prior to your tutorial during the week of July 25th. You may ask questions about this assignment in that tutorial. At the beginning of your TUTORIAL during the week of August 1st you need to submit the following homework problems. STUDY: Improper Integral material - from LEC (this material is also found in the Chap- ter 14 end of chapter exercises), Chapter 22 (up to pg 457). HOMEWORK PROBLEMS: 1. Determine whether the following improper integrals converge or diverge. Find the value of the improper integral if it converges. (a) Z -∞ x 3 e - x 4 dx (b) Z 1 0 x ln( x ) dx (c) Z π 0 sec 2 ( x ) dx (d) Z 0 x arctan( x ) (1 + x 2 ) 2 dx Warning: You will need to use more than one integration technique to find the appropriate primitive in part (d). 2. If the sequence { a n } converges, then prove that its limit is unique.
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