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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
32~~
~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,SolutionsofMidtermExam,Fall 1999
1.Findanequationoftheplanepassingthroughthepoints(1.2.3).(2.3.1)and (3,1,2).(25pts) Solution.LetP1.(1.2.3).P2.(2.3.1)andP3.(3.1.2).Then
~~~~~~~~
P1P2.(2i+3j+k);(i+2j+3k).i+j;2~k. ~~~~~~
~~~
P1P3.(3i+j+2k);(i+2j+3k).2i;j;k: Thus .
. ~~~
ijk
.
. ~~
~n.P1P2.P1P3.11;2.;3i;3j;3~k
.

2;1;1 isanormalvectoroftheplane.Thustheequationoftheplaneis: ;3(x;1);3(y;2);3(z;3).)x+y+z.6: 2.Showthatf(x2;y2.2xy)isaharmonicfunctioniff(x.y)isharmonic.(25pts) Solution.Letz.f(x2;y2.2xy).Then @z .2xf1(x 2;y 2.2xy)+2yf2(x 2;y 2.2xy). @x @2
z
2222222222
.2f1(x;y.2xy)+4xf11(x;y.2xy)+8xyf12(x;y.2xy)+4yf22(x;y.2xy). @x2
@z
2222
.;2yf1(x;y.2xy)+2xf2(x;y.2xy). @y
@2z
2222222222
.;2f1(x;y.2xy)+4yf11(x;y.2xy);8xyf12(x;y.2xy)+4xf22(x;y.2xy): @y2
f
Since 11(x 2;y 2.2xy)+f22(x 2;y 2.2xy).0.
wehave
@2 @2 zz
+.4x 2(f11(x 2;y 2.2xy)+f22(x 2;y 2.2xy))@x2 @y2
+4y 2(f22(x 2;y 2.2xy)+f11(x 2;y 2.2xy)).0:
Thusz.f(x2;y2.2xy)isharmonic. 3.Reparametrizethecurve
p
22
32~~
~r.ti+tcostj+tsint~k 3
inthesameorientationintermsofarclengthmeasuredfromthepointwheret.0(Hint: t.0).(25pts) Solution.Note
pp
~~
~r0(t).2ti+(cost;tsint)j+(sint+tcost)~k:
p
jj~r0(t)jj.2t+(cost;tsint)2+(sint+tcost)2
p
.2t+cos2t;2tcostsint+t2sin2t+sin 2t+2tsintcost+t2cos2t
p
.2t+1+t2
p
.(1+t)2
.1+t:
So
Z
t t2
s.(1+.)d..t+. 0 2
equivalently,
pt2;2t;2s.0.)t.1+1+2s:
Therefore,
pq22ppppp
~
~r.(1+1+2s)3i+(1+1+2s)cos(1+1+2s)~j+(1+1+2s)sin(1+1+2s)~k: 3
4.Findparametricequationsforthetangentlinetothecurveofintersectionofthe
p
conez.x2+y2andtheplanex+2y+2z.20atthepoint(4.3.5).(25pts)
p
Solution.Letf(x.y.z).x2+y2;z.Then
xyfx.p.fy.p.fz.;1:
22 22
x+yx+y
Thus 43
~
~~
rf(4.3.5). i+j;k: 55
Moreover,
~
~~
~n.i+2j+2k
isanormalvectoroftheplanex+2y+2z.20.Thus
.
. ~~~
ijk
1613
.
43 . ~~~
~v.rf(4.3.5).~n.;1.i;j+k

5555

122
isadirectionvectorofthetangentlineoftheintersectioncurve.Hencethetangentline
is: 1613 x.4+t.y.3;t.z.5+t: 55