final265fa2004sol

final265fa2004sol - Math 265 Final Exam 15 Dec 2004 NAME:...

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Math 265 Final Exam 15 Dec 2004 NAME: -iklple A Instructor: %?It A Section: & Instructions: Answer each question completely. Show all work. No credit allowed for mere answers with no work shown. Show the steps of calculations. State the reasons that justify conclusions. 1. (a) Find parametric equations for the line through (2, -3,l) parallel to the vector from (2,0,0) (4,3,1). p: b,-5,() ; = [ , 3, I - &,3,1, (b) Find an equation for the plane through the origin that is perpendicular to the line of part (a). - A - v : 4213~11 2. (a) Compute the gradient of F(x, y, z) = x2 + ex sin y - z at (O,O, 0). (b) Find the equation of the tangent plane to the surface z = + ex sin y at (O,0,0). I bL PIbL. Zlv- - ='I) I. -1 FLW/~, 11 L ,I,-// :" 3
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Math 265 Final Exam Page 2 3. Evaluate 4 (2x2 + 1) dx + (xy) dy where C is the circle of radius 2 with center at the J C origin, oriented counterclockwise. C (5 ~bbL() CUml 1 niby Knd 1 ~k5 I~w~I~ bh~ 1 50 bb~ GQV~\'L 7hw~~. 4. Find a vector in the direction in which f (x, y, z) = x2y3z + 3z2 increases most rapidly at p = (-2,1, -1). What is the rate of change in this direction?
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final265fa2004sol - Math 265 Final Exam 15 Dec 2004 NAME:...

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