Winter 2006 - Hall's Class - Practice Exam 2

# Winter 2006 - Hall's Class - Practice Exam 2 - Math 20C...

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Unformatted text preview: Math 20C Calculus Practice Midterm II Solutions 1. Let r ( t ) = h 2 t, 1 3 t 3 ,t 2 i . (a) (5 points) Compute the curvature of r ( t ) at the point (2 , 1 3 , 1). At this point, t = 1. Thus, r ( t ) = h 2 ,t 2 , 2 t i r 00 ( t ) = h , 2 t, 2 i | r ( t ) | = √ 4 + t 4 + 4 t 2 = 2 + t 2 r ( t ) × r 00 ( t ) = h- 2 t 2 ,- 4 , 4 t i | r ( t ) × r 00 ( t ) | = √ 4 t 2 + 16 + 16 t 2 r (1) = h 2 , 1 , 2 i | r (1) | = 3 | r (1) × r 00 (1) | = √ 4 + 16 + 16 = 6 Using the curvature formula above, we get: κ = 6 3 3 = 2 9 (b) (5 points) Compute the unit tangent and unit normal vectors at the point (2 , 1 3 , 1). r ( t ) = h 2 ,t 2 , 2 t i r (1) = h 2 , 1 , 2 i | r ( t ) | = √ 4 + t 4 + 4 t 2 = 2 + t 2 | r (1) | = 3 T ( t ) = 1 2+ t 2 h 2 ,t 2 , 2 t i T (1) = r (1) | r (1) | = h 2 3 , 1 3 , 2 3 i T ( t ) = 1 2+ t 2 h , 2 t, 2 i +- 2 t (2+ t 2 ) 2 h 2 ,t 2 , 2 t i = 1 (2+ t 2 ) 2 h- 4 t, 4 t, 4- 2 t 2 i T (1) = h- 4 9 , 4 9 , 2 9 i So N (1) = T (1) | T (1) | = h- 2 3 , 2 3 , 1 3 i (c) (5 points) Find the equation of the osculating plane at the point (2 , 1 3 , 1)....
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Winter 2006 - Hall's Class - Practice Exam 2 - Math 20C...

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