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Name ID MATH 253 Honors Section FINAL EXAM Sections 201-202 1-7 /49 8 /15 9 /10 10 /15 11 /15 Fall 1999 P. Yasskin Multiple Choice: (7 points each) P 4, 4, 4 Its tangent vector is 1. Consider the line through the point plane a. b. c. d. e. x 2y 3z 7. which is perpendicular to the 3, 2, 1 1, 2, 3 7, 6, 5 5, 6, 7 4, 4, 4 x " yz 0 at the point Which of the following points does not lie on this plane? 2. Find the plane tangent to the hyperbolic paraboloid P 6, 3, 2 . a. b. c. d. e. "6, 0, 0 0, 3, 0 0, 0, 2 1, "1, "1 "1, 1, 1 3. Duke Skywater is flying the Millenium Eagle through a polaron field. His galactic measured in lightseconds and his velocity is 2300, 4200, 1600 measured in lightseconds per second. He measures the strength of p 3, 2, 2 the polaron field is p 274 milliwookies and its gradient is milliwookies per lightsecond. Assuming a linear approximation for the polaron field and that his velocity is constant, how many seconds will Duke need to wait until the polaron 286 milliwookies? field has grown to coordinates are v .2, .3, .4 a. b. c. d. e. 2 3 4 6 12 1 4. Consider the surface S parametrized by and 0 t v t 4. Compute dS ;; F u, v u v, u " v, uv R where y, x, y . F for 0 tut2 S a. "32 b. "16 c. 16 d. 32 e. 64 u, v u v, u " v, uv . Find the plane R P R1, 2 3, "1, 2 . Which of the following 5. Consider the surface S parametrized by tangent to this surface at the point points does not lie on this plane? a. b. c. d. e. 3, 0, 0 0, 4, 0 0, 0, "2 1, 1, 0 0, 6, 1 6. Compute area 0 >"x 2 y 2 dx 2xy 3 dy t y t 4 " x2 over the complete boundary of the semicircular traversed counterclockwise. a. 0 b. 16 c. 4 5 d. 80 5 128 e. 5 7. Compute 3 2 5 y3z2 ;; x 3z dy dz 3 dz dx z5 dx dy over the complete surface of the S sphere x2 y2 z2 4 with outward normal. a. 512= 21 2 b. 32= 5 128 = c. 5 d. 16= 3 e. 256= 15 2 8. (15 points) Find the point in the first octant on the surface z 32 x4y2 which is closest to the origin. 3 9. ;; x dA (10 points) Compute 15 R over the region R in the first quadrant 10 y bounded by the curves y x2, y x4 and y 16. 5 0 2 x 4 4 10. (15 points) Find the mass and center of mass of the solid below the paraboloid above the xy-plane, if the density is z 4 " x2 " y2 - x2 y2. (11 points for setting up the integrals and the final formula.) t r 5 11. (15 points) Find the area and centroid 1 of the right leaf of the rose r 2 cos 2 2. -2 -1 0 1 2 (12 points for setting up the integrals and the final formula.) -1 6