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Mathematics202Examination2March12,2004NameDoallofyourworkinthebluebooklets.
Pleaselabelyouranswersclearly,asIwillnothavetimetoperformextensivesearchesforanswers.
Whenyouarenishedwiththeexam,pleaseputthisexaminationinsideofthebluebooklet.
Nocreditwillbegivenforanswerswithoutexplanations.
Cheatingwillresultinafailinggrade.
Theproblemsarenotarrangedinorderofincreasingdiculty,soyoumightwanttoreadallofthembeforebeginning.
Calculatorsarenotpermittedonthisexamination.
1.
(10points)LetF:R3→R3bedenedbyF(x,y,z)=(xy2,x2z,yz).
Compute*F.
2.
(10points)Letf(x,y)=exsiny.
Compute·(f).
3.
(10points)Thefunctionf(x,y)=x3+y33x3yhasacriticalpointat(1,1).
Usethesecondderivativetesttoseeifthiscriticalpointisalocalmaximum,alocalminimum,orasaddlepoint.
4.
(10points)Findanequationforthetangentplanetothegraphofx2y3z4+xyz=2atthepoint(2,1,1).
5.
(10points)Ifz=yf(x2y2),wheref(t)isaC∞function,showthatyzx+xzy=x.
Hint:Lett=x2y2,andusethechainrule.
6.
(15points)Considerthepathdenedbyx=sin2ty=cos2t0≤t≤π2Findthearclengthofthispathusingtheintegralformulaforarclength.
7.
(15points)Supposethatv(t):R→R2andw(t):R→R2aretwosmoothpathsinthexy-plane.
Showthatddt(v(t)·w(t))=ddt(v(t))·w(t)+v(t)·ddt(w(t)).
Hint:Writev(t)=(v1(t),v2(t))andw(t)=(w1(t),w2(t)),andexpandthedotproduct.
8.
(20points)Acertainshippingcompanywillacceptanypackageintheshapeofarectangularparallelepipedsolongasthesumoftheheightandtwicethesumofthewidthandthedepthisnomorethan72inches.
WhatisthemaximumvolumeofsuchapackageSetthisproblemupasamaximizationproblemofafunctionoftwovariables,andndthecriticalpoint(thereisonlyonephysicallyreasonableone).