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pm1bSolns - Math 265(Butler Practice Midterm I B(Solutions...

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Math 265 (Butler) Practice Midterm I — B (Solutions) 1. Find the projection of h 2 s, 1 , s - 1 i onto the vector h- 2 t, 5 - t 2 , 4 t i . The formula for projection of a vector u onto a vector v is u · v v · v v . Applying it to our case with u = h 2 s, 1 , s - 1 i and v = h- 2 t, 5 - t 2 , 4 t i we have that the projection is h 2 s, 1 , s - 1 i · h- 2 t, 5 - t 2 , 4 t i h- 2 t, 5 - t 2 , 4 t i · h- 2 t, 5 - t 2 , 4 t i h- 2 t, 5 - t 2 , 4 t i = (2 s )( - 2 t ) + (1)(5 - t 2 ) + ( s - 1)(4 t ) ( - 2 t ) 2 + (5 - t 2 ) 2 + (4 t ) 2 h- 2 t, 5 - t 2 , 4 t i = - 4 st + 5 - t 2 + 4 st - 4 t 4 t 2 + 25 - 10 t 2 + t 4 + 16 t 2 h- 2 t, 5 - t 2 , 4 t i = 5 - 4 t - t 2 t 4 + 10 t 2 + 25 h- 2 t, 5 - t 2 , 4 t i = 5 - 4 t - t 2 ( t 2 + 5) 2 h- 2 t, 5 - t 2 , 4 t i . (On a side note we see that this projection only depends on t , even though the vector we were projecting involved s .)
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2. A particle travels along the parametric curve h e - t cos t, e - t sin t i starts at (1 , 0) at time t = 0 and then spirals into the origin (0 , 0) as t → ∞ . How far will the particle have traveled when it reaches the origin? (In other words, what is the arc length of the parametric curve for 0 t < .)
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