S09fnlansl

S09fnlansl - is used in the spherical coordinate sense, and...

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Answers for Sample Final Exam BEWARE: The following are just the numerical answers. Since the justi±cation is missing, they would not receive full credit on the real exam. 1. (a) r ( t )= ° 1 , 4 , 6 ± + t ° 3 , 2 , 1 ± ; (b) 5 / 14. 2. R ( u )= j +2 k + u ( i + j + k ); at ( 2 , 1 , 0). 3. (a)11; (b) 2 x 6 y +9 z =19 . 4. Maximum value is 3 3 / 2 / 5 5 / 2 , achieved at the four points ( 1 5 , 1 5 , 3 5 ), ( 1 5 , 1 5 , 3 5 ), ( 1 5 , 1 5 , 3 5 ) and ( 1 5 , 1 5 , 3 5 ). 5. ° 1 / 3 1 / 9 ° y 1 / 3 f ( x, y ) dx dy + ° 1 1 / 3 ° y y f ( x, y ) dx dy . 6. (8 2 2) / 15. 7. 8 πb 4 / 5. 8. (a) f ( x, y )= x 2 y + xy 3 +3 y 2 / 2; (b) 105.5. 9. -2/3. 10. div F ( x, y, z )= xy +2 y + z 3 ; curl F ( x, y, z )= xz i +(3 xz 2 yz ) j ; grad(div F )( x, y, z )= y i +(2+ x ) j +3 z 2 k . 11. (9 3 8 2+1) / 15. 12. 5 π/ 2. ******************** Details for question 7. [Beware: The notation is a bit confusing here because the symbol ρ
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Unformatted text preview: is used in the spherical coordinate sense, and the density at the point ( , , ) is given by the function f ( , , ) = .] The region is described in spherical polars by 0 2 b cos , 0 2 , 0 / 2. Then E dV = / 2 2 2 b cos 2 sin ddd = / 2 2 4 4 sin 2 b cos dd = 4 b 4 / 2 2 (cos ) 4 sin d d = 8 b 4 / 2 (cos ) 4 sin d = 8 b 4 (cos ) 5 5 / 2 = 8 b 4 1 5 = 8 b 4 5 ....
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This note was uploaded on 09/07/2011 for the course MATH 39578 taught by Professor Penner during the Spring '09 term at USC.

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