=========================preview======================
(MATH100)fe100.pdf
Back to MATH100 Login to download
======================================================
FinalExamofMATH100 (Fall 2001)
(20pointsforeachproblem)
y2 x2
1.Aparticleismovingonaplaneunderthevector.eldF(x.y).i+j.
(x+y)2 (x+y)2
(a)Findtheworkdonebythe.eldontheparticlewhentheparticlemovesalongthe straightlinefrom(1.3)to(2.6).
.
x.1;cos.
(b)Iftheparticlemovesfrom(2.6)to(1.3)alongthecurve .
y.6;3sin. where.isbetween.and. ,.ndtheworkdonebythesame.eldontheparticle.
2
2.(a)Verifythefollowingequation
@z@z y+x.0 @x@y
forthefunctionz.f(u),whereu.x2;y2 . (b)Afunctionz.f(x.y)isde.nedbytheequationx3y+y3z+z3x.3and f(1.1).1.Please.ndthepartialderivatives@@xzj(x.y).(1.1)and@@x2zj(x.y).(1.1)respectively.
2
3.Evaluatethelineintegral
I
x2
[(e;xy)dx+(xy+siny 6)dy].
C
whereCistheboundarycurveofthe.niteregionboundedbytheparabolay2.2xand thelinex+y.4withcounterclockwiseorientation. 4.Findtheoutward.uxZZ
~
.F.~ndS.
.
where F~(x.y.z).(x 2+cosyz)~i+(y;e z)~j+(z 2+x 2)~k
and.istheboundarysurfaceofthe.nitesolidboundedbythecylinderx2+y2.4,and theplanesx+z.2andz.0. 5.UseLagrangeMultipliersto.ndthemaximumvolumeoftherectangularsolid inthe.rstoctantwithonecornerattheoriginandtheoppositeonelyingintheplane
xyz
++.1,wherea.b.carepositiveconstants.
abc
6.Evaluatethesurfaceintegral
ZZ
~
k.~ndS . z
S
whereSisthesurfaceoftheellipsoidx2.a2+y2.b2+z2.c2.1,~nistheunitoutward ~
normalonthesurface,andkistheunitzdirection.WhycanyounotusetheDivergence theorem.
1