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Unformatted text preview: MATH 241 FINAL EXAM May 13, 2006 Instructions: Number the answer sheets from 1 to 9. Fill out all the information at the top of
each sheet. Answer problem n on page n, n 2 1,    ,9. Do not answer one question on more than one sheet. If you need more space use the back of the correct sheet. Please write out and sign the
Honor Pledge on page 1 only. SHOW ALL WORK The Use of Calculators Is Not Permitted On This Exam 1. (20 points)
(a) Find an equation for the plane through P = (3,3,1) that is perpendicular to the planes
x+y=2z and 2x+z=10. (b) Find parametric and symmetric equations for the line through P perpendicular to the plane
found in part (a). 2. (25 points) The position vector of a particle at any time t is given by t3
r(t) = 2ti + t2j + 3k (a) Find the velocity, acceleration, and speed of the particle at any time t. (b) Find the tangential and normal components of the acceleration vector at any time t.
(c) Find the curvature of the trajectory at any time t. (d) Find the total distance travelled by the particle in the time interval 0 _<_ t g 1. 3. (15 points) Find the point on the hyperbolic paraboloid z : x2 — 33/2 at which the tangent
plane is parallel to the plane 830 + 3y —— z = 4 4. (20 points) Suppose that the gravitational potential due' to a point mass at the origin at the
point (x, y, z) in space is given by the formula 1
E : ——~——
($2 + y2 + Z2)1/2
(a) Find the rate of change of potential with respect to distance at the point P z: (2, 1, 2) in
the direction of the vector v 2 7i — 4j + k.
(b) Find the maximal directional derivative DUE at the point P and the direction u in which
that maximum occurs. 5. (25 points) A rectangular parallelepiped lies in the ﬁrst octant with three sides on the coordinate
planes and one vertex on the plane 2:1: + y + 4.2 2: 12. Find the maximum possible volume of
the parallelepiped. 6. (30 points)
(a) Write a double integral in rectangular coordinates (as, y) which gives the volume V of the region bounded below by the my plane, above by the paraboloid z = 3:2 + y2 and on the sides by the parabolic cylinder 12 y2
— —=1.
9+4 You may, if you wish, use symmetry to simplify your answer.
(b) Find V by making the change of variable a: 2: 3u cos 1;, y = 2a sin 11. 7. (20 points) Find the mass M of an object which occupies the solid region bounded above by the
sphere m2 + y2 + z2 = 1 and below by the my plane if its density is 6(13, y, z) = 1 + 2:. 8. (25 points) Let E be that part of the plane 356 + 2y — z = 4 that lies inside the cylinder
:52 + y2 = 9. Let C be the boundary of Z oriented counterclockwise as viewed from above. Use Stokes’s theorem to compute f0 F  dr where F(x, 3/, z) = ——yzi + :czj + 22k. 9 (20 Pomts) Compute f IS F  ndS where E is the boundary of the solid region
D = {($,y,z) : x2+y2 g4, ~1 g z 33}, F(:1:, y, z) = $3i + y3j + 22k, and n points outward. ...
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This note was uploaded on 09/23/2008 for the course MATH 241 taught by Professor Wolfe during the Spring '08 term at Maryland.
 Spring '08
 Wolfe
 Calculus

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