# mid2u - (a) and (b). 3 3. (12pts) Consider the surface...

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Name Quiz Section MIDTERM II Math 126, Section A February 22, 2007 Problem Total Points Score 1 12 2 12 3 12 4 14 Total 50 5(Bonus) 3 - You may use a scientiﬁc calculator and one one-sided sheet of handwritten notes. No other notes, books or calculators are allowed. Please turn oﬀ your cell phone. - Show all your work to get full credit. - Read instructions for each problem CAREFULLY. - Leave all your answers in EXACT form. - Check your work! 1

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1. (12pts) Consider the curve given by the equation in polar coordinates r = 4cos θ + sin θ. (a)(6pts) Find the Cartesian equation of the curve. Sketch the curve. (b)(6pts) Find the equation of the tangent line to the curve at the point θ = π/ 4. 2
2. (12pts) Consider the parametric curve given by the vector function ~ r ( t ) = ( t,t 2 ,t 3 ). (a)(4pts) Find the equation of the normal plane to the curve at the point when t = 1. Hint. The normal plane is the plane perpendicular to the tangent line. (b)(4pts) Find the equation of the normal plane to the curve at the point ( - 1 , 1 , - 1). (c)(4pts) Find the parametric equations of the line of intersection of the planes from

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Unformatted text preview: (a) and (b). 3 3. (12pts) Consider the surface deﬁned by the equation f ( x,y ) = x 2 y + y 3 + x . (a)(6pts) Find the tangent plane to the surface at the point (-2 , 1 , 3). (b)(6pts) Find all second partial derivatives of f ( x,y ). 4 4. (14pts) (a)(5pts) Find the velocity and position vectors of a particle that has the acceleration vector ~a ( t ) = (2 , cos t, sin t ) , the initial velocity ~v (0) = h , ,-1 i and the initial position ~ r (0) = h 1 , 1 , i . (b)(1pt) Find the position vector at the time t = 1. 5 Answer the following two questions in any order. Simplify your answers as much as possible. (c)(4pts) Find the curvature at t = 1. (d)(4pts) Find the length of the projection of the acceleration vector at t = 1 on the unit normal vector at t = 1. 6 5. (3pts) ( Bonus, full credit only) . Show that if a particle moves with the constant speed, then the velocity and acceleration vectors are orthogonal. 7...
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## This note was uploaded on 05/07/2008 for the course MATH 126 taught by Professor Smith during the Spring '07 term at University of Washington.

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mid2u - (a) and (b). 3 3. (12pts) Consider the surface...

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