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MATH 251H Final Exam Version B Solutions

# MATH 251H Final Exam Version B Solutions - Name Sec 1-12...

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Name Sec MATH 251 Final Exam Spring 2011 Section 200/511 Solutions P. Yasskin Multiple Choice: (4 points each. No part credit.) 1-12 /48 14 /16 13 /16 15 /25 Total /105 1 . Find the projection of the vector u U u 12, u 3,4 U along the vector v u U u 2,1, u 2 U . a . u 12 13 , 3 13 , u 4 13 b . 12 13 , u 3 13 , 4 13 c . u 26 9 , u 13 9 , 26 9 d . 26 9 , 13 9 , u 26 9 Correct Choice e . 108 13 , u 9 13 , 36 13 u ± v u U 24 u 3 u 8 U 13 | v u | U 4 ± 1 ± 4 U 3 proj v u u U u ± v u | v u | 2 v u U 13 9 u u 2 U U 26 9 , 13 9 , u 26 9 2 . If | u | U 4 and | v u | U 3 and the angle between u and v u is u U U 3 then | u ± v u | U a . 6 3 Correct Choice b . 6 c . 3 3 d . 3 e . 12 3 | u ± v u | U | u || v u |sin u U 4 ± 3 ± 3 2 U 6 3 3 . Find the point where the line through the origin perpendicular to the plane 3 x u 3 y ± 4 z U 17 intersects that plane. At this point x ± y ± z U a . 2 Correct Choice b . 3 c . 5 d . 17 e . 1 2 The normal to the plane is: N u U u 3, u U . The line through the origin O in the direction N u is X U O ± tN u or u x , y , z U U u 3 t , u 3 t ,4 t U . Plug the line into the plane and solve for t : 2 x ± 3 y ± 4 z U 3 u 3 t U u 3 u u 3 t U ± 4 u 4 t U U 34 t U 17 . So t U 1 2 . Plug into the line: u x , y , z U U 3 2 , u 3 2 ,2 So x ± y ± z U 2 1

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4 . Find the arc length of the curve r u u t U U u t 2 ,2 t ,ln t U between u 1,2,0 U and u e 2 e ,1 U . a . e 2 Correct Choice b . e 2 ± 1 c . e 2 u 1 d . 2 3 e 3 ± e e . 2 3 e 3 ± e u 5 3 r u u t U U u t 2 t t U v u U 2 t ,2, 1 t | v u | U 4 t 2 ± 4 ± 1 t 2 U 2 t ± 1 t 2 U 2 t ± 1 t L U U u 1,2,0 U e 2 , e ,1 ds U U 1 e | v u | dt U U 1 e 2 t ± 1 t dt U ± t 2 ± ln t ² 1 e U u e 2 ± 1 U u u 1 U U e 2 5 . The pressure, P , density, u , and temperature, T , of a certain ideal gas are related by P U 3 u T . At the point u 1,2,3 U , the density and its gradient are u U 4 and ± u u U u 0.2,0.4,0.1 U , while the temperature and its gradient are T U 300 and ± u T U u 2,1,3 U . Hence the pressure is P U 3 ³ 4 ³ 300 U 3600 and its gradient is ± u P U a . u 1802.4,2704.8,2701.2 U b . u 600.8,901.900.4 U c . u 68,132,42 U d . u 204,372,126 U Correct Choice e . u 1.2,1.2,0.9 U ² P ² x ³ ² P ² u ² u ² x ± ² P ² T ² T ² x U 3 T ² u ² x ± 3 u ² T ² x U 3 ³ 300 ³ 0.2 ± 3 ³ 4 ³ 2 U 204 ² P ² y ³ ² P ² u ² u ² y ± ² P ² T ² T ² y U 3 T ² u ² y ± 3 u ² T ² y U 3 ³ 300 ³ 0.4 ± 3 ³ 4 ³ 1 U 372 ² P ² z ³ ² P ² u ² u ² z ± ² P ² T ² T ² z U 3 T ² u ² z ± 3 u ² T ² z U 3 ³ 300 ³ 0.1 ± 3 ³ 4 ³
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