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(MATH100)pme101.pdf
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Math100,L2,MidtermExam,Fall 1999
Giveallthedetails.Nocalculators.
1.Findanequationoftheplanepassingthroughthepoints(1.2.3).(2.3.1)and
(3,1,2).(25pts) 2.Showthatf(x2;y2.2xy)isaharmonicfunctioniff(x.y)ishomonic.(25pts) 3.Reparametrizethecurve
p
22
3.2~~
~r.ti+tcostj+tsint~k 3
inthesameorientationintermsofarclengthmeasuredfromthepointwheret.0(Hint:
t.0).(25pts) 4.Findparametricequationsforthetangentlinetothecurveofintersectionofthe
p
conez.x2+y2andtheplanex+2y+2z.20atthepoint(4.3.5).(25pts)
Math100,L2,MidtermExam,Fall 2001
Giveallthedetails.Nocalculators.6pointsforeachproblem.
1.Finethearclength:
x.3t2+2.y.2t3 ;1.1.t.3:
2.Showthatz.ln(x2+y2)+2tan;1y.xsatis.estheLaplaceequation
@2@2
zz
+.0:
@x2 @y2
3.Findparametricequationsforthetangentlinetothecurveofintersectionofthe
222222
ellipsoidx+2y+z.4andthehyperboloid5x;2y;2z.1atthepoint(1.1.1). 4.Locateallrelativemaxima,relativeminimaandsaddlepointsofthefunction f(x.y).xy2;6x2;3y2 . 5.Findtheabsolutemaximumvalueofthefunctionf(x.y).xysubjecttothe constraint(x+1)2+y2.1andthepointatwhichfobtainsthismaximumvalue.