126_Au07_Sol2 - Math 126, Sections C and D, Fall 2007,...

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Unformatted text preview: Math 126, Sections C and D, Fall 2007, Solutions to Midterm II 1. Consider the vector function r ( t ) = < 4 t + 1 , t 2- 3 t, cos( t ) > . (a) Find the unit tangent vector T at (1 , , 1). (4 points) T (0) = r prime (0) | r prime (0) | r prime ( t ) = < 4 , 2 t- 3 ,- sin( t ) > r prime (0) = < 4 ,- 3 , > T (0) = < 4 ,- 3 , > | < 4 ,- 3 , > | = < 4 ,- 3 , > 16 + 9 + 0 = < 4 5 ,- 3 5 , > (b) Find the normal plane to the curve at (1 , , 1). (3 points) 4 5 ( x- 1)- 3 5 ( y- 0) + 0( z- 1) = 0 4 5 ( x- 1)- 3 5 y = 0 (c) Find the curvature at (1 , , 1). (4 points) = | r prime r primeprime | | r prime | 3 So we need r primeprime ( t ) = < , 2 ,- cos( t ) > r primeprime (0) = < , 2 ,- 1 > r prime r primeprime = < 3 , 4 , 8 > So = 9 + 16 + 64 5 3 = 89 125 1 2. Consider the two lines r 1 ( t ) = < 2 t- 1 , 3 t, 5 t- 2 > and r 2 ( s ) = < 4 s- 1 , 6 s + 1 , 2 s + 2 > . (a) Show that they are skew. That is, show that they do not intersect, and that they are not parallel....
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126_Au07_Sol2 - Math 126, Sections C and D, Fall 2007,...

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