lo*srFlxel Exeu

Ftnsr Spupsrnn A.Y. 2013-20L410 Ocronpn 2013

This exam is for two hours only. Use only black or blue ink. Show all necessary solutions and boxali final answers. Calculators and other electronic devices are not allowed. Any form of cheating inexaminations 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).

2. Determine and classify the critical points of /.{3 poi,ntsl

{5 poi,ntsl

II. Use the method of Lagrange multipliers to find the point on the plane 2r - 4y * z :3 thatis nearest to the origin. {s pointsl

III. Set up the iterated triple integral in cylindrical coordinates equal to the volume of the solidin the first octant bounded by the paraboloid z: L - x)2 - y2, the plane fr: U, the zy-planeand the yz-plane.

tl, pointsl

IV. Rewriter'/1 r'/i:fr f \F-"'n'

J, J, J** zd'zd'Yd'x

in spherical coordinates and then evaluate the resulting triple integral. [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, _tlz).

[2 pitnt !VI' Use Green's Theorern to set up the iterated double integral in polar coordinates that is equaif-

to f_ (cos(r) - a') dr * (eo + ,u) dy where C is the circle rz + yz :2r. fi pointslJC

VII. Let & bethesurfacewithvectorequation Efu,u):uz t+u2 j+uuhwhere -ZSul-l,-1<u12.

1. Find an equation of the tangent plane to,91 at the point where u: -l and u:1.

2. Set up the iterated rlouble integrai eq,rai u ll u aS.

,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 -2,22) across ,S2. fi poi,nts]

[3 poi,ntsJ

[9 poi,ntsJ

Page 1 of 2

-nl

i i-t1, : , -' . "$ 1-;-i'i'.".,.i?.i.''l

IX.,Determine whether the following is convergent or dir/ergent

'.:'-'- t. {F}* ,, .-

-ll, -,,.

, . ,, , , :: .*,',*o"ar .,.

. i ; '-

X. Consider the ,*r* i , f".+ ll;"*ao

(Y)t/n + z

r,:':tJ. rE l

...-:; ! ,i-'-

:'::iri'' fi'iit"i

. t..I r-t:..t ..'

fr

f,

il

F*

fl

ll

1;

I

I

i.

I I ..

,]

o9

umofY n."'2.itlse the result in (1) to find the s t_r 4,.' n=\

XI!. Give the third-degrm.Tayl.lor,pgly4gmia} for'sini ,it o JZrll.

::::: End of Exam ::::=

i-+fi!':1

.:,:-: i:t.: '

ilii :.r'::if'.

{2 pni;aw;*a$:hf'

i: .J'.:jr -i

.::;',2.rDeterinirib'tn.,raliies of r foi whiih the'series is absolutely convergent, conditionallyi';;''' r I :t"o**rg"dr:.arld divergent. ! , : .. . :, ii ,,...,r.:, i!4";PoinlsJ

1eXI. Given' al= : I rn, lnl < L, : :" I

l;x - :,:n:0

11 :,ti .:

.r.: - . :i'i'.,

Page'2.of'2

Meru 55

Frual Ex.q.uSncoNn 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 finalanswers. Calculators and other eiectronic devices are not allowed. Any form of cheating in examinationsor 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 (^,z). IS pointsJ

2. Determine and classify the two critical points of /. fi poi,nts]

II. Use Lagrange multipliers to find the maximum'and miuimum values of .F'(r, U): Za - gr subject tothe constraint *2 +2y2: 18. [i points]

III. Let or: #. Determine whether the sequence {ar,} is convergent or d.ivergent. Is the series

Do" convergent? IS pai,ntsJ

IV, UsethegeometricseriesE*toobtainapowerseriesrepresentationfor I**

ppoi,nts!n=0

degree Taylor polynomial of. e2* + nz + 2 at a : L, [4 pai,ntsJV. Find the third

VI. Find the radius of convergence of i 9i9. Then determine where the power series is absolutely

=o{" + 1)5n

convergent, eonditionally convergent and divergent. [7 poi,nts]

VII. Consider the vector field "F(r, il: fuacosn,4Aa +2grsinu). Find all possible potential functions forF then use the result to compute the work done by .F in moving a particle from the point ({,1) tothe point (0,

"). ft pointsJ

VIII. Use Green's theorem to evaluat" f Qo'+2") d,r*6ry d,y whete0 consists of the curve a: \/nfrom (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 thedensity at any point is equal to the distance of the point from the uy-plane. [/ points]

X. Set up the iterated triple integral in sphericai coordinates equal to the volume of the solid in thefust oitrnt that is between the spheres 12 +y2 + 22:4 and n2 +a2 * z2 - 16 and above the cone

Pagg 1 of 2

[5 poi,nts]

.i

ft paintcJ

Compute the flqx Ct F(r, U, z) : ty + ,,t * z,a *g) across ,9s gtvtn that the orimtatioq o{ ,93'is,',#Ftii:irieif by &1X;"{f6;*. ; '

t.a : ') :

:e==*i End Of EXaf-fl ?=---i Total: 60 pointsPlease return the questionnaire with your bluebooks.

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