M408D_Sample Final - r = 2 and inside the polar curve r = 4...

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Final Exam Math 408D Name . Fall 1996 SSN . You must show your work in order to insure that you receive full credit for a problem. Please write legibly and label the problems clearly. Circle your answers. 1. (10 points) Find the interval of convergence of the series X n =0 ( - 1) n nx n 3 n ( n 2 + 1) . 2. (10 points) Estimate Z 1 2 0 dx 1 + x 4 accurate to 2 decimal places. 3. (10 points) Two surfaces, x 2 + y 2 + z = 4 and x 2 + 3 y 2 = z , intersect in a curve in 3-space. Find parametric equations for the tangent line to the curve at the point (0, -1, 3). 4. (10 points) Find the distance from the point (1, 0, 2) to the line given by parametric equations x = 2 + 2 t , y = 1 - t , and z = 2 t . 5. (10 points) Let f ( x, y ) = xe xy . Find the directional derivative of f at the point (2,0) in the direction of the vector ~u which makes an angle of 60 with the positive x -axis. 6. (10 points) Show that the points (1, 1, 1), (0, 2, 2) and (-1, 0, 1) do not lie on a straight line and find the equation of the plane containing these three points. 7. (10 points) Find the area of the region which lies outside the polar curve
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Unformatted text preview: r = 2 and inside the polar curve r = 4 sin θ . (You may need the identity 2 sin 2 θ = 1-cos2 θ .) 8. (10 points) Find the volume of the solid bounded above by the surface z = 9-x 2-y 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 2-1 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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