Unformatted text preview: r = 2 and inside the polar curve r = 4 sin θ . (You may need the identity 2 sin 2 θ = 1cos2 θ .) 8. (10 points) Find the volume of the solid bounded above by the surface z = 9x 2y 2 and below by the region in the xy plane enclosed by the curves y = x , x = 1, and y = 0. 9. (10 points) Find all points on the paraboloid z = 9 x 2 + 4 y 2 at which the normal vector is parallel to the line through the points (4, 2, 5) and (2, 6, 4). 10. (10 points) Let f ( x,y ) = 1 4 x 21 2 y 2 . Find the maximum and minimum values of f on the disk of radius 1 centered at the origin. Bonus: (5 points) A spaceship outside any gravitational ﬁeld is on the path ~ G ( t ) = t 2 ~ i + 3 t ~ j + 4 t 3 ~ k . At time t = 1 it shuts oﬀ its rockets and coasts along the tangent line to the curve at that point. How close does it come to the point (9, 15, 50)?...
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 Fall '09
 ALTHARODIN
 Euclidean geometry, 3space, 60◦

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