Quiz 3 Solution - MAC2313 Quiz 3 Solution 1. Reparametrize...

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Unformatted text preview: MAC2313 Quiz 3 Solution 1. Reparametrize ~ r ( t ) = e t ,e t sin( t ) ,e t cos( t ) with respect to the arclength starting at the point (1 , , 1). Solution: We need to find the arclength variable s = Z t || ~ r ( u ) || du . ~ r ( t ) = D e t ,e t sin( t ) + e t cos( t ) ,e t cos( t )- e t sin( t ) E || ~ r ( t ) || = q e 2 t + e 2 t (sin 2 ( t ) + 2cos( t )sin( t ) + cos 2 ( t )) + e 2 t (cos 2 ( t )- 2cos( t )sin( t ) + cos 2 ( t )) = 3 e 2 t = 3 e t s = Z t || ~ r ( u ) || du = Z t 3 e u du = 3 e t- 3 Solving for t , we obtain t = ln( s 3 + 1) and can rewrite ~ r ( s ) = s 3 + 1 , ( s 3 + 1)sin(ln( s 3 + 1)) , ( s 3 + 1)cos(ln( s 3 + 1)) . 2. For the function ~ r ( t ) = h sin(2 t ) ,t, cos(2 t ) i find the curvature, the equation of the normal plane, and the equation of the osculating plane at the point (0 ,, 1). Solution: First to find ~ T and ~ N ....
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Quiz 3 Solution - MAC2313 Quiz 3 Solution 1. Reparametrize...

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