lo*sr Flxel Exeu
Ftnsr Spupsrnn A.Y. 2013-20L4 10 Ocronpn 2013
This exam is for two hours only. Use only black or blue ink. Show all necessary solutions and box ali final answers. Calculators and other electronic devices are not allowed. Any form of cheating in examinations or any act of dishonesty in reiation to studies shall be subject to disciplinary action.
I. Let f(*,y) -
2a,
*
Zyz
*
6r'2
* t2rg.
1' Find the rate of change of / at the point (5,0) in the direction of the vector (g,4). {3 poi,ntsl {5 poi,ntsl
2. Determine and classify the critical points of /.
II.
Use the method of Lagrange multipliers to find the point on the plane is nearest to the
origin.
III.
-
4y
z: -
that {s pointsl
fr:
-
tl,
Rewrite
r'/1 r'/i:fr f \F-"'n'
J, J, J**
zd'zd'Yd'x
in spherical coordinates and then evaluate the resulting triple integral.
=-
* z :3
Set up the iterated triple integral in cylindrical coordinates equal to the volume of the solid in the first octant bounded by the paraboloid L y2, the plane x)2 U, the zy-plane and the pointsl
yz-plane.
IV.
2r
[5 points]
'-!.1 :Yg.> \2ry',W:!9osrv) ---
1. Find all possible potential functions for F. [z points] 2. Compute the work done by F in moving a particle from the point (1,0) to the point (0, [2 pitnt !
_tlz).
VI'
Use Green's Theorern to set up the iterated double integral in polar coordinates that is equai
to
f-
f_
(cos(r)
JC
-
a') dr
*
(eo
+ ,u) dy where C is the circle rz + yz :2r.
fi
VII. Let & bethesurfacewithvectorequation Efu,u):uz t+u2 j+uuhwhere
-1
1. Find an equation of the tangent plane to,91 at the point where
2. Set up the iterated rlouble integrai eq,rai
u
ll
u
aS.
u: -l
and
pointsl
-ZSul-l, u:1. [3 poi,ntsJ [9 poi,ntsJ
,Sr
VIII.
Let ,92 be the portion of the plane 2r *5y * z: 10 in the first octant with upward orientation. Compute the flux of. F(r,A,z) : (2n,Zy poi,nts] -2,22) across ,S2.
fi
Page 1 of 2
, -'
i i-t1, :
nl
.
"$ 1-;-i'i'.".,.i?.i.''l r,:':tJ.
IX.,Determine whether the following is convergent or dir/ergent
'.:'-'-
{F}*
.
X.
!
...-:;
t.
i
Consider the
,,
.-
;
,*r*
-,,. -ll, '-
i
,
f".+
,.
,,
,
, :: .*,',*o"ar
ll;
rE
{2
pni;aw;*a$:hf'
,i-'-
:'::iri''
.,.
l
fi'iit"i
.I t.. r-t:..t
..'
i: .J'.:jr
-i
(Y)t/n + z
"*ao
.::; ',2.rDeterinirib'tn.,raliies of r foi whiih the'series is absolutely convergent, conditionally i';;''' I :t"o**rg"dr:.arld divergent. . r
! ,
: ..
I
ii
,,...,r.:, i!4";PoinlsJ
:
1e
: l;x I-n:0 rn, lnl < L,
XI. Given' al=
,]
:,
:" :,:
I .. o9
"'2.itlse the result in (1) to find the '
n. sumofY t_r 4,. n=\
XI!. Give the third-degrm.Tayl.lor,pgly4gmia} for'sini ,it o JZrll. fr
:::::
f,
End of Exam
::::=
il
F*
i-+fi!':1 .:,:-:
i:t.:
'
fl
ll
1;
I
I
i.
ilii
11
:,ti
.:
.r.:
-
.
:i 'i'.,
Page'2.of'2
:.r'::if'.
I
Meru 55 Frual Ex.q.u
SncoNn Spunsrnn A.Y. 2012-201,3 5 Apnrl 2013
This exam is for two hours only. Use only biack or blue iuk. Show all necessary solutions and box all final answers. Calculators and other eiectronic devices are not allowed. Any form of cheating in examinations or any act of dishonesty in relation to studies shall be subject to disciplinary action.
I. Let lb,y) :
-nB
- ZsA.*
1. Find the rate of change of / at the point (1,0) in the direction of the vector 2. Determine and classify the two critical points of /.
II.
Use Lagrange multipliers to find the maximum'and miuimum values of .F'(r, U): Za the constraint *2 +2y2: 18.
III. Let or:
IS pointsJ
fi poi,nts] gr
subject to
[i
points]
Determine whether the sequence {ar,} is convergent or d.ivergent. Is the series
#. Do" convergent?
IS pai,ntsJ
IV, UsethegeometricseriesE*toobtainapowerseriesrepresentationfor n=0
V. Find the third
degree Taylor polynomial
VI. Find the radius of convergence of i
of. e2*
9i9. + 1)5n
+
nz
+ 2 at a :
I**
L,
Consider the vector field "F(r,
il:
ppoi,nts!
[4 pai,ntsJ
Then determine where the power series is absolutely
convergent, eonditionally convergent =o{" and divergent.
VII.
(^,z).
[7 poi,nts]
fuacosn,4Aa +2grsinu). Find all possible potential functions for done by .F in moving a particle from the point ({,1) to ft pointsJ
F then use the result to compute the work the point (0,
VIII.
").
Use Green's theorem
to evaluat"
d,r*6ry f Qo'+2")
d,y whete0 consists of the curve
a:
\/n
from (0,0) to (1,1), the line segment from (1,i).to (0,2), and the line segment from (0,2) to (0,0). [5 poi,nta]
IX. Let ,5r be the solid in the first octant bounded by the coordinate planes and the plane 3u*2y * z : 6. SSI ,p the iterated triple integral in rectangular coordinates equal to the mass of .91 given that the density at any point is equal to the distance of the point from the uy-plane.
X.
[/
points]
Set up the iterated triple integral in sphericai coordinates equal to the volume of the solid in the fust oitrnt that is between the spheres 12 +y2 + 22:4 and n2 +a2 * z2 - 16 and above the cone
[5 poi,nts]
Pagg 1 of 2
.i
ft paintcJ Compute the flqx Ct F(r, U, z) ,',#Ftii:irieif by &1X;"{f6;*. ; '
:
ty +
t.a
:e==*i
,,t * z,a *g)
:
across ,9s gtvtn
')
that the orimtatioq o{ ,93'is
:
End Of EXaf-fl ?=--Total: 60 points i Please return the questionnaire with your bluebooks.
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',,
i'i,
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