MATH 251H Final Exam Version B Solutions

# Hint find a scalar potential a 70 b 42 correct choice

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Unformatted text preview: /2 0  Þ F  ds 2, 4, 6 to /2 cos 6  6 4 cos 5  sin  d  8  for F  2x, 2y, 2z 0  4 3 2, 4, 6 ttt along the curve  t  r 1, 2, 3 . HINT: Find a scalar potential. a. 70 b. 42 Correct Choice c. 0 d. 42 e. 70  F  f 12. xf  2x yf  2y   Þ F  ds  Þ  f  ds  f 1, 2, 3   Compute Þ F  ds for F  zf  2z Í f x, y, z  x 2  y 2  z 2 f 2, 4, 6  1 2  2 2  3 2 sec x 3 5y, cos y 5  3x 22  42  62  42 counterclockwise around the triangle with vertices 0, 0 , 8, 0 and 0, 4 . Hint: Use Green’s Theorem. a. 12 b. 16 c. 32 d. 64 e. 128 P  sec x 3 Correct Choice 5y Q  cos y 5  3x  Þ F  ds  Þ P dx  Q dy  ÞÞ xQ xQ y P dx dy  yP 3 5 8 ÞÞ 8 dx dy  8Area  8  1  8  4  128 2 4 W ork Out: (Points indicated. Part credit possible. Show all work.) 13. (16 points) Use Lagrange multipliers to find 4 numbers, a, b, c, and d, whose product is and for which a  2b  3c  4d is a minimum. 2 3 Minimize f  a  2b  3c  4d subject to the constraint g  abcd  2 . 3  f  1, 2, 3, 4  g  bcd, acd, abd, abc  f   g: 1  bcd 2  acd 3  abd 4  abc Make the right sides all be abcd and equate: abcd  a  2b  3c  4d a a a So b c d Substitute into the constraint: 2 3 4 4 Í a 4  2  24  16 Í a  2, b  1, c  2 , d  1 abcd  a a a a  a  2 234 24 3 3 3 2 14. (16 points) Find the mass and the y-component of the center of mass of the quarter of the cylinder x2  y2 4 with 0 3 in the first octant z x 0, y 0, z 0 if the mass density is   xyz.   xyz  r cos  r sin  z  r 2 sin  cos  z dV  r dr d dz /2...
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## This note was uploaded on 02/06/2014 for the course MATH 251H taught by Professor Philipb.yasskin during the Spring '11 term at Texas A&M.

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