prac2 - x 2 4 + y 2 9 + z 2 25 = 1 does not intersect the...

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Math 192 , Second Prelim, Thursday, October 28, 7:30 - 9:00 PM Fall 2004 No calculators. An 8 . 5 × 11 inches sheet with information on both sides is allowed. Recommendations: Write your name and section number on the exam book- let. Carefully read every problem. If you don’t know how to solve a problem, move on. Draw accurate pictures and explain clearly each of your steps. Write neatly. 1. (a)(8 points) Find the point(s) of intersection of the following pair of curves in polar coordinates: ± r = 1 - cos( θ ) r = cos( θ ) . (b)(8 points) Find the point(s) of intersection of the following pair of curves in polar coordinates: ± r 2 = 5 cos(2 θ ) r 2 = 5 sin(2 θ ) . 2. (14 points) Find the critical points of f ( x, y ) = x 3 - 3 x +2 y 2 - y 4 and classify them as local maxima, local minima or saddle points. 3. (20 points) The ellipsoid
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Unformatted text preview: x 2 4 + y 2 9 + z 2 25 = 1 does not intersect the plane 15 x-10 y +3 z = 90. Find the point on the ellipsoid closest to the plane and ±nd the point on the ellipsoid farthest from the plane. 4. (15 points) Find the mass of the thin plate covering the region R between the curves y = √ 2-x 2 , y = √ 1-x 2 and y = 0 if the density is δ ( x, y ) = e x 2 + y 2 . 5. (15 points) Compute the volume of the region bounded by the following inequalities: ≤ x ≤ 2 ≤ y ≤ p 4-x 2 x 2 + y 2 ≤ z ≤ 4 . 6. (20 points) Consider the region R bounded by the following inequalities: ≤ y 2 cos( θ ) ≤ r ≤ 4 ≤ z ≤ k-y, such that k ≥ 4 . For what k is the volume of R equal to 100?...
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This note was uploaded on 06/01/2008 for the course MATH 1920 taught by Professor Pantano during the Spring '06 term at Cornell.

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