=========================preview======================
(MATH100)examin20.pdf
Back to MATH100 Login to download
======================================================
Math100-IntroductiontoMultivariableCalculus
FINALEXAMINATION

FallSemester,1999
TimeAllowed:2.5Hours. TotalMarks:100

StudentName:
StudentNumber:
1.(10marks)Locateallrelativemaxima,relativeminimaandsaddle pointsofthefunction
33
f(x.y).x+y;3x;3y:
2.(10marks)Evaluatethedoubleintegral
ZZ
1
dxdy.
D(1+x2+y2)2
where
n.o
.
22 R2
D.x+y..x.0.y.0:
3.(15marks)Evaluatethelineintegral
I
22
(e x+y 2)dx+(ey+x 2)dy
C
whereCistheboundaryoftheregionbetweeny.2x2andy.2x andisorientedcounterclockwise.
4.(15marks)Evaluatethelineintegral
Z
(sinx+siny)dx+(1+xcosy)dy.
C
whereCisthelinesegmentfrom(0.0)to(...).
1
Math100-IntroductiontoMultivariableCalculus 2
~
5.(15marks)Findthe.uxofFacrossthesurfaceofconicalsolidbounded by
q
z.x2+y2andz.1.
where
2
~~2~2~
F(x.y.z).xi+yj+zk:
6.(15marks)Findthevolumelyinginsideboththespherex 2+y 2+z 2.a 2 andthecylinderx2+y2.ax,whereaisapositiveconstant.
7.(20marks)UsetheDivergenceTheoremtoevaluatethesurfaceintegral
ZZ
~
F(x.y.z).n~dS.
.
where
~2~~~
F(x.y.z).2xi+(y+z;4xy)j+k
and.isthesurfaceoftheupper-hemisphere

q
z.1;x2 ;y2.z.0
withoutthebottomdisk(i.e.,notincludingthediskx 2+y 2 .1on thexy-plane)andisorientedupward.
Math100-IntroductiontoMultivariableCalculus 3
1.Locateallrelativemaxima,relativeminimaandsaddlepointsofthe function
33
f(x.y).x+y;3x;3y:
SolutionThecriticalpoints:
2
fx.3x;3.0.)x..1.
2
fx.3y;3.0.)y..1:
Hence,therearefourcriticalpoints:
(1.1).(1.;1).(;1.1).(;1.;1).
Also,
.6x.f.6y..0.)D.f;f2.36xy:
fxxyyfxyfxxyyxy
Thus,
at(1.1).D.0&fxx.0.)relativeminimum, at(1.;1).D.0.)saddlepoint, at(;1.1).D.0.)saddlepoint, at(;1.;1).D.0&.0.)relativemaximum.
fxx
Math100-IntroductiontoMultivariableCalculus 4
2.Evaluatethedoubleintegral
ZZ
1
dxdy. D(1+x2+y2)2
where
n.o
.
22 R2
D.x+y..x.0.y.0:

Solution
ZZ
1
dxdy D(1+x2+y2)2
Z.Z
2 R1 . rdrd.
(1+r2)2
00
.
..r.R
.
.11
.
..;.
.
221+r2
r.0
.R2 . : 4(1+R2)
Math100-IntroductiontoMultivariableCalculus 5
3.Evaluatethelineintegral
I
22
(e x+y 2)dx+(ey+x 2)dy
C
whereCistheboundaryoftheregionbetweeny.2x2andy.2x andisorientedcounterclockwise. SolutionByGreen'sTheorem,wehave
I
22
(e x+y 2)dx+(ey+x 2)dy
C
ZZ
.(2x;2y)dxdy
R
ZZ
12x 1 .(2x;2y)dydx.;: 02x2 5
Math100-IntroductiontoMultivariableCalculus 6
4.Evaluatethelineintegral
Z
(sinx+siny)dx+(1+xcosy)dy.
C
whereCisthelinesegmentfrom(0.0)to(...).

SolutionSince
@@ (sinx+siny).(1+xcosy). @y @x
weknowthatthevector.eld
~~

(sinx+siny)i+(1+xcosy)j
isconservative.Thepotentialfunction.satis.es
(
@.
@x .sinx+siny.
@..1+xcosy:
@y

Solvingthe.rstequation,wehave
..;cosx+xsiny+h(y):

Substitutingthisintothesecondequationfor.tohave
xcosy+h0(y).1+xcosy:

Henceh(y).y+C,whereCisaconstant.Thus
.(x.y).;cosx+xsiny+y+C:
Finally,thelineintegralis
.
(...)
.
.

[;cosx+xsiny+y+C]..2+.:
.
(0.0)

Math100-IntroductiontoMultivariable