Name SecMATH 251 Final Spring 2010Sections 511 P. YasskinMultiple Choice: (5 points each. No part credit.)1-13 /65 15 /1014 /21 16 /10Total /1061.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 8 /9d.arccos 3/ 8e.arccos(3/8)2.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.−2b.2c.3d.5e.163.If a jet flies around the world from West to East, directly above the equator,in what direction does its unit binormal Bpoint?a.Down (toward the center of the earth)b.Up (away from the center of the earth)c.Northd.Southe.West14.Find the z-intercept of the plane tangent to the surfacexyz= 1 at the point(2,3,6).a.6b.16c.5d.−5e.−65.The temperature in an ideal gas is given by T = κPρwhere κ is a constant,P is the pressure and ρ is the density. At a certain point Q =(3,2,1), we haveP(Q)= 8 ∇P(Q)=(4,−2,−4)ρ(Q)= 2 ∇ρ(Q)=(−1,4,2)So at the point Q, the temperature is T(Q)= 4κ and its gradient is ∇T(Q)=a.κ(−8.5,6,9)b.κ(4,−9,−6)c.κ(3,2,−2)d.κ12,2e.κ −12,226.If the temperature in a room is T=xyz2, find the rate of change of the temperatureas seen by a fly who is located at(3,2,1)and has velocity(1,2,3).a.32b.36c.44d.48e.527.Find the volume below z = xy above the region between the curves y = 3x and y = x2.a.812b.814c.818d.2432e.24388.Compute∫∫Ce−x2−y2dxdy over the disk enclosed in the circle x2+ y2= 4.a.π2(1 − e−4)b.π(1 − e−4)c.π2e−4d.πe−4e.2πe−439.Find the mass of 2 loops of the helical rampparametrized byR(r,θ)=(rcosθ,rsinθ,4θ)for r ≤ 3if the density is ρ = x2+ y2.a. 40πb. 120πc. 200πd.5003πe.2443π Correct Choice10.Find the flux of F=(y,−x,2)through the helical ramp of problem 9 oriented up.a.4πb.83πc.216πd.108π Correct Choicee.10243π11.Compute∫(2,1)(3,2)F⋅ dsfor F=(2xy,x2)along the curve r(t)=((2 + t2)esinπt,(1 + t2)esin2πt).HINT: Find a scalar potential.a.12b.14c.22d.2e.15 − 4 2412.Compute∮(2xsiny − 5y)dx +(x2cosy − 4x)dycounterclockwise around the cross shown.HINT: Use Green’s Theorem.a. −45b. −10c. 5d. 10e. 450 1 2 30123xy13.Compute∫∫S∇× F⋅ dSfor F=(−yz,xz,xyz)over the quartic surface z =(x2+ y2)2for z ≤ 16oriented down and out. The surface may be parametrized byR(r,θ)=(rcosθ,rsinθ,r4)HINT: Use Stokes’ Theorem.a. −128πb. −64πc. −32πd. 32πe. 64π5Work Out: (Points indicated. Part credit possible. Show all work.)14.(21 points) Verify Gauss’ Theorem∫∫∫V∇⋅ FdV =∫∫∂VF⋅ dSfor the vector field F=(4xz3,4yz3,z4)and the solid Vabove the cone C given by z = x2+ y2or parametrized by R(r,θ)=(rcosθ,rsinθ,r),below the disk D given by x2+ y2≤ 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:∇⋅ F, dV,∫∫∫V∇⋅ FdVb.(8 pts) Compute the surface integral over the disk by parametrizing the disk andsuccessively finding:R(r,θ), er, eθ, N, FR(r,θ),∫∫DF⋅ dS6Recall: F=(4xz3,4yz3,z4)and C is the cone parametrized byR(r,θ)=(rcosθ,rsinθ,r).c.(7 pts) Compute the surface integral over the cone C by successively finding:er, eθ, N, FR(r,θ),∫∫CF⋅ dSd.(2 pts) Combine∫∫DF⋅ dSand∫∫CF⋅ dSto get∫∫∂VF⋅ dS715.(10 points) Find the average value of the function f(x,y,z)=x2+y2+z2within the solid cylinder x2+ y2≤ 9 for 0 ≤ z ≤ 4.16.(10 points) Find the value(s) of R so that the ellipsoidx242+y232+z222= R2is tangent to the plane12x +43y + 2z = 36.HINT: Their normal vectors must be
View Full Document
Unlocking...