Test2_sol - NAME Spring 2011 MAC 2313 Test 2 UFID Section...

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NAME: Spring 2011, MAC 2313, Test 2 UFID: 3/3/2011, Section: 3124 1. Reparametrize the curve r ( t ) = e t cos t i + e t sin t j +2 k with respect to the arc length measured from the point where t = 0 in the direction of increasing t . Then find the curvature of the curve at the point (1 , 0 , 2). (10 points) Solution . Since r ( t ) = e t cos t i + e t sin t j + 2 k , we know r 0 ( t ) = e t (cos t - sin t ) i + e t (sin t + cos t ) j and hence r 00 ( t ) = - 2 sin te t i + 2 cos te t j . Note that k r 0 ( t ) k = 2 e t , we know the arc length parameter starting from t = 0 is s ( t ) = Z t 0 k r 0 ( u ) k du = Z t 0 2 e u du = 2( e t - 1) . At point (1 , 0 , 2), we know t = 0. Therefore r 0 (0) = i + j , and r 00 (0) = 2 j . Hence the curvature at this point is k = k r 0 (0) × r 00 (0) k k r 0 (0) k 3 = 2 ( 2) 3 = 2 2 . 2. Find the limit if exists (5 points each). If not show why the limit does not exist: (1) lim ( x,y ) (0 , 0) xy cos x x 2 + 2 y 2 (2) lim ( x,y ) (0 , 0) e - x
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This note was uploaded on 12/15/2011 for the course MAC 2313 taught by Professor Keeran during the Spring '08 term at University of Florida.

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Test2_sol - NAME Spring 2011 MAC 2313 Test 2 UFID Section...

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