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(MATH100)review.pdf
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MATH100L2:ReviewProblems
1.Findtheequationofthelinepassingthroughthepoints(1.;1.2)and(2.1.3). 2.Findtheequationoftheplanepassingthroughthepoints(1.;1.1),(1.2.3)and (1.1.0). 3.Forwhatvalueofkdothefourpoints(1.1.;1).(0.3.;2).(;2.1.0)and(k.0.2) laalieonaplane. 4.Findtheequationofthelinepassingthroughthepoint(1.1.1)andparalleltothe intersectionlineoftheplanesx+y+z.0and2x;y;z.1. 5.Whatisthedistancefromthepoint(2.0.0)totheintersectionlineoftheplanes x+y+z.0and2x;y;z.1.
6.Findanarcparametrizationoftheintersectioncurveofx2+y2.1andx+y+z.2 with(1.0.1)thereferencepointandcounterclockwiseorientationwithrespecttothexy-plane.Whatisthelengthofthewholecurve.
7.Findthefollowinglimits:
x2(y;1)2 xy
(1)lim.(2)lim: (xy)!(01)x2+(y;1)2(xy)!(00)x+y
8.Findtheequationofthetangentplaneandthenormallineofthesurfacez.
p
1+x3y3atthepoint(2.1).
9.Iff(x.y)isharmonic,showthat

.
xyf.;
2222
x+yx+y
isalsoharmonic(cf.Example9.2). 10.Byapproximatelywhatpercentagewillthevalueofw.x3y2.z5increaseor decreaseifxincreasesby2%,yincreasesby1%andzdecreasesby;3%. 11.Findthedirectionalderivativeoff(x.y.z).x2y;yz3+zatthepointP(1.;2.0) ~
inthedirectionofthevector~u.2i+ ~j;2~k,and.ndthemaximumrateofincreaseof fatP. 12.Findanequationofthetangentplanetothelevelsurfaceoff(x.y.z).x2y+ y2z+z2xat(1.;1.1). 13.Findanequationofthetangentlineoftheintersectioncurveofthesurfaces
x 2+2y 2+z 2.1andx 3+2y+z.4:
14.Locateallrelativemaxima,relativeminimaandsaddlepointsoffifany:
24
f(x.y).xy++: xy
15.UseLagrangemultiplierto.ndthemaximumandminimumvaluesoffsubject tothegivenconstraint.Also,.ndthepointsatwhichtheseextremevaluesoccur.
f(x.y.z).xyz.x 2+y 2+z 2.1: