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fx265s2010 - ρ = 2 below the cone φ = π 3 and above the...

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Math 265 Final Exam S2010 Name & Section: Instructor: Answer each question completely. Show all work. No credit is allowed for mere answers with no work shown. Give exact answers unless decimal approximations are requested. 1. (12 points) Find parametric equations for the tangent line to the curve x = 2 t 2 + t , y = 2 t 2 , z = t 3 - t , at t = 2. 2. (12 points) Find parametric equations of the line of intersection of the planes 2 x + 3 y + 2 z = 2 and x + y + z = 3 .
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3. (12 points) Find all points ( x, y ) at which the tangent plane to the graph of z = 3 x 2 + 2 xy + 2 y 2 - 4 x + 5 y is parallel to the plane 4 x - 6 y - 2 z = 8. 4. (12 points) Find all critical points of the function f ( x, y ) = x 2 + y 4 - 4 xy , and classify each critical point as a local maximum, a local minimum, or a saddle point.
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5. (14 points) Find the center of mass of the solid inside the sphere
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Unformatted text preview: ρ = 2, below the cone φ = π/ 3 and above the plane z = 0, if the density is proportional to the distance from the origin. Use symmetry where possible. 6. (12 points) Apply Green’s Theorem to evaluate I C ( e x 2 +2 y ) dx +(sin y 3 + x 2-x ) dy , where C is the triangle with vertices (0 , 0), (1 , 0) and (1 , 2), traversed in the counterclockwise direction. 7. (14 points) Evaluate ZZ R ( x + y ) sin( x-y ) dA , where R is the triangle with vertices (0 , 0), ( π, π ) and ( π,-π ). 8. (12 points) Compute the flux of the vector field F ( x, y, z ) = x i + ( y-2 x ) j + ( x-z ) k across the surface of the solid determined by x 2 + y 2 + z 2 ≤ 9, with the outward normal....
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