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mid2o - parallel to the line with parametric equations x =...

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Math 126 C, D - Spring 2006 Mid-Term Exam Number Two May 11, 2006 Name: Section: 1 10 2 10 3 10 4 10 5 10 6 10 Total 60 Complete all questions. You may use a scientific, non-graphing calculator during this examination. Other elec- tronic devices are not allowed, and should be turned off for the duration of the exam. If you use a trial-and-error or guess-and-check method, or read a numerical solution from a graph on your calculator, when an algebraic method is available, you will not receive full credit. You may use one hand-written 8.5 by 11 inch page of notes. Show all work for full credit. You have 50 minutes to complete the exam. .
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1. Find the slope of the tangent line to the polar curve r = 1 θ , θ > 0 at the point where it intersects the cartesian curve x 2 + y 2 = 1 9 .
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2. At what point(s) is the tangent line to the curve x = t 3 3 t, y = t 2 + 2 t parallel to the line with parametric equations
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Unformatted text preview: parallel to the line with parametric equations x = 3 t + 5 , y = t − 6 ? 3. For any m > , the helix determined by the position function v r ( t ) = a cos t, sin t, mt A has constant curvature that depends on m . Find the value of m such that the radius of curvature at any point on the curve is 3. 4. A particle is moving so that its position is given by the vector function v r ( t ) = a t 2 , t, 5 t A Find the tangent and normal components of the particle’s acceleration vector. 5. Reparametrize the curve v r ( t ) = a 5 t − 1 , 2 t, 3 t + 2 A with respect to arc length measured from the point where t = 0 in the direction of increas-ing t . 6. Let f ( x, y ) = x 2 y + x sin y − ln( x − y 2 ) . (a) Find f y ( x, y ) . (b) Find f xy ( x, y ) ....
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