This preview shows page 1. Sign up to view the full content.
Unformatted text preview: r = 2 and inside the polar curve r = 4 sin . (You may need the identity 2 sin 2 = 1cos2 .) 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 eld is on the path ~ G ( t ) = t 2 ~ i + 3 t ~ j + 4 t 3 ~ k . At time t = 1 it shuts o 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)?...
View Full
Document
 Fall '09
 ALTHARODIN

Click to edit the document details