final265sp2006

Final265sp2006 - not required to evaluate the integral 6(14 points Find the area of the part of the surface z = 2 x y 2 that is directly over the

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Math 265 Final Exam 2006/05/03 Instructor: Answer each question completely. Show all work. No credit is allowed for mere answers with no work shown. Show the steps of calculations. State the reasons that justify conclusions. 1. (12 points) Find parametric equations of the line of intersection of the two planes 2 x - 3 y - 5 z = 15 and x - y - 2 z = 5. 2. (14 points) Find all local extrema of f ( x, y ) = 2 x 2 + 3 xy + 4 y 2 - 5 x + 2 y . Classify each as a local maximum, local minimum, or saddle.
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3. (12 points) On the curve r ( t ) = t i + t 2 j + 2 3 t 3 k fnd the velocity vector v ( t ) and the acceleration vector a ( t ) at any t . Find a T and a N , the tangential and normal components o± acceleration, at t = 1.
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4. (12 points) Given the function g ( x, y, z ) = x 2 ln y + yz , Fnd an equation for the tangent plane to the surface g ( x, y, z ) = 2 at the point (2 , 1 , 2). 5. (12 points) Convert the cylindrical coordinate integral Z π 0 Z b 0 Z r 0 r 3 z dz dr dθ to an equivalent iterated integral in Cartesian coordinates. [It is
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Unformatted text preview: not required to evaluate the integral.] 6. (14 points) Find the area of the part of the surface z = 2 x + y 2 that is directly over the triangle in the x-y plane with vertices (0 , 0), (0 , 1) and (1 , 1). 7. (12 points) Find the work done by the force ±eld F = ± cos(ln( x-y + 1)) x-y + 1 + 2 x sin x 2 ,-cos(ln( x-y + 1)) x-y + 1 + 3 y 2 e y 3 ² . to move an object counterclockwise around the boundary of the region in the ±rst quad-rant lying between the curves y = x 2 and y = √ x . 8. (12 points) Use Stokes’s Theorem to evaluate Z Z G curl v · n dS , where v = yz i + x j + ( y + z ) k and G is the part of the paraboloid z = x 2 + y 2 below the plane z = 4, with the upward normal....
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This note was uploaded on 04/07/2008 for the course MATH 265 taught by Professor Gregorac during the Spring '08 term at Iowa State.

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Final265sp2006 - not required to evaluate the integral 6(14 points Find the area of the part of the surface z = 2 x y 2 that is directly over the

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