Name Sec MATH 251 Final Sections 511 Spring 2010 1 13 65 15 10 14 21 16 10 Total 106 P Yasskin Multiple Choice 5 points each No part credit 1 2 3 Find the angle between the vectors u 2 2 1 and v 1 2 2 a arccos 8 9 b arccos 8 3 c arccos d arccos 3 8 e arccos 3 8 8 9 At the point x y z where the line r t 1 t t 2 2t intersects the plane x 2y 3z 16 we have x y z a 2 b 2 c 3 d 5 e 16 If a jet flies around the world from West to East directly above the equator in what direction does its unit binormal B point a Down toward the center of the earth b Up away from the center of the earth c North d South e West 1 4 5 Find the z intercept of the plane tangent to the surface a 6 b 1 6 c 5 d 5 e 6 xy 1 at the point 2 3 6 z The temperature in an ideal gas is given by T P where is a constant P is the pressure and is the density At a certain point Q 3 2 1 we have P Q 8 P Q 4 2 4 Q 2 Q 1 4 2 T Q So at the point Q the temperature is T Q 4 and its gradient is a 8 5 6 9 b 4 9 6 c 3 2 2 d 1 2 2 1 2 2 e 2 6 7 If the temperature in a room is T xyz 2 find the rate of change of the temperature as seen by a fly who is located at 3 2 1 and has velocity 1 2 3 a 32 b 36 c 44 d 48 e 52 Find the volume below z xy above the region between the curves y 3x and y x 2 a b c d e 8 81 2 81 4 81 8 243 2 243 8 Compute e x y dx dy 2 2 over the disk enclosed in the circle x 2 y 2 4 C a 1 e 4 2 b 1 e 4 c e 4 2 d e 4 e 2 e 4 3 9 Find the mass of 2 loops of the helical ramp parametrized by r r cos r sin 4 for r 3 R if the density is x2 y2 a 40 b 120 c 200 d 500 3 e 244 3 Correct Choice y x 2 through the helical ramp of problem 9 oriented up 10 Find the flux of F a 4 b 8 3 c 216 d 108 e 1024 3 11 Compute Correct Choice 3 2 2 1 F ds 2xy x 2 along the curve r t 2 t 2 e sin t 1 t 2 e sin 2 t for F HINT Find a scalar potential a 12 b 14 c 22 d 2 e 15 4 2 4 12 Compute 2x sin y 5y dx x 2 cos y 4x dy counterclockwise around the cross shown HINT Use Green s Theorem a 45 y 3 2 1 b 10 0 c 5 0 d 10 1 2 3 x e 45 13 Compute F dS yz xz xyz for F S over the quartic surface z x 2 y 2 2 for z 16 oriented down and out The surface may be parametrized by r r cos r sin r 4 R HINT Use Stokes Theorem a 128 b 64 c 32 d 32 e 64 5 Work Out Points indicated Part credit possible Show all work 14 21 points Verify Gauss Theorem F dV F dS V V 4xz 3 4yz 3 z 4 and the solid V for the vector field F above the cone C given by z x2 y2 or parametrized by R r r cos r sin r below the disk D given by x 2 y 2 9 and z 3 Be sure to check and explain the orientations Use the following steps a 4 pts Compute the volume integral by successively finding dV dV F F V b 8 pts Compute the surface integral over the disk by parametrizing the disk and successively finding r e r e N F R r dS R F D 6 4xz 3 4yz 3 z 4 and C is the cone parametrized by Recall F R r r cos r sin r c 7 pts Compute the surface integral over the cone C by successively finding F R r dS e r e N F C d 2 pts Combine F dS D and F dS C to get F dS V 7 15 10 points Find the average value of the function f x y z x 2 y 2 z 2 within the solid cylinder x 2 y 2 9 for 0 z 4 16 10 points Find the value s of R so that the ellipsoid is tangent to the plane 1 x 4 y 2z 36 2 3 2 x2 y z2 R2 42 32 22 HINT Their normal vectors must be parallel 8
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