Calc III - 3450:223FINAL Fa1l97

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NAME,ROW

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1.Evaluate Ie (x + y) cis along the portions of the curves x2 + y2 =4 and y =0 as shown in the figure.

y

7x::: I:.

~~G

51. (t.\-O)~I.~O\-

-~ @Jt ~ ~L J

z.. :"1.. -I.

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.1- (-~)-

(!J -0

x

P\ x~<"l

'1::;:U'~t.(!)

J: (~/.., t~ :lJ:,.d J 'ts~ 1{;+ 4(0) '{; jt

CV

- 't [0'" (,t ""cJJt-: 't[<.:t-t."l/:- <-t[l - -/\ = t

0

2. Find the curl and divergence of F =xeYi- ze-Yj+ y Inz k.

-'1~'C

\

~;: (I.. ~ +- ~'1' I' - A/'\) j l0-0) \- [

"

( )(. O-J(-c't

,-(r

1: t -~)" y" /Z''1

, c , - J(t k D

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-+ Y-!t @

I

'1-F =

'r

<:: +-

J

.-:> l' l' \.

V'f...f:: d L Jn .)"

) "< -'-( I'it- -t" ?"

3. Evaluate 1 (7x - Sy)dx + (6x - 7y)dy where C is the circle (x - 3)2+ (y - 3)2=9.s---~

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2

,-.-I'7 ?./ r:.,.-.IJ 11,......

S J (~- -() 014

(9,

/J,~ (;)( - 7'7

ft/< 0 G (f)J I . SfJ4 : /I. 1\T (~r-,' 9ft1"

0 r9

;\It:. ?.. - )'7

fill -:: - ~

4. Findthe fluxof thevectorfieldF =Sxi+ Syj + 4k outward through the surface cut from2 2

the~ottomofz=x +y +Sbyz=7.

~1-

X 1..+ 1 -::.:1-

(2... 13 Pointst L - ,L 11- )( f 'r +.1 - t<.,f- /If-

Y'jj ~ <. Jx) :J.'7; -( '>

K (j) (j) cP<3 y, Lf)" <.J{ ~~ '-') ~ I

x/ ", /) .~;:-~Lf1l~t ~~

J'1:tlJ.!f," fl J<~(I?\-/>..(O))~/~»//'

f ~ r. n JJ : S f,~ 0

@'1.(1' }:

:: S f U~x7.f-J~/'-Y)Y')/jcJD 0 @

: fu~fJ S:u<or)- Lfr)J/,j~

:;)<'1,~ - .;J,,') 1-:'

PI v ,t ~

:: ~Tf' [tio - YJ ~

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5. Let F =(3 + 2xy)i + (x2 - 3y2)j. Is F conservative? Also evaluate Ie F'dr where C

is given by r(t) =(etsin t) i + (etcos t) j, 0::; t::; n.

f -= ~+ ::2"7 Q -' .l(l- J'7 l . /f':) [;]/)" r/) ~;)'" .5'. (" b""'JC,,"v.t.i)~<,:~ .r1::: ...(,.x:. \J/;( - 12POInts

{ f' -.. L. F)..A.J : (lJ-),t,), f("1--1 7t.) ,,,

F~<~ t f- V :: rx ( 'J' J

.r~ -; 3d-d~(

.c::: S><. .>r J'-1,'"( + ,,(d

~~ \, J l

i, 2 ~+j(("):: /' - '7

5' F' J; -: r(0 J -C nJ - {( o~ I)

:>

t(u)~ <0 ,').J

~(iT)" <u)..e")

d«('"1) ~ -J'{ L

5"" 3(7)"'). ~/) t- L

1iT

::: € +-1 CD

:) J::~ - ) t-L

J",. x~ 7 (!j)

6. Calculate the flux ofF =~x2i + sin(xz) j + (xz + eX2Y)kacross the surface S where S is -t~c...

surface of the region bounded by x =0, y = 0, z = 0,z = 1 - x2andy + z = 2.:c1..

...... [;]

y

55 f, A h 0 SJ( y.F JV J, A. J. ~ f/,'"

@l ( I-~'- ~-~ Q)~ ) SSC)( + r) + )( ') /, b cl<0 (:) 0

r ( (i\.. )/

' L

- )C> d< .2 ~- -;:: <:> deL.

)(

:: S ~ ~ ~-~ ~ ;) x.(:J-~) j 1: A~

:;-J,..I :)0<[a-.?-<t - r(I-.;);('- f. x'-1J))"

- ~ - ~ - x" J( ::::

).. '-( Go i:>

C9.5 - J. -J. ~ S/'

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: Jz>' [iX' - ::2x/ - )( s-JJ~

4

7. Findthevectorcompon~ ~f 2i + 3j whicharepar~and perpendicularto -i + 2j.

C

' <31 G-/

[;Jplfr,,/fr-/ <: 2/ ) '> '-- <-1/J'>) ~-0 ~~ .

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:=.Lf <.~I; .;i>.r

-.(-f-, f, '> (!)

f{r/e_"l: (~/.- L 2, 3) !:t- /-t.:>.);:J- <. / <$'

.2.\-7-; J-J->

CD :: < '~-/ /'r'>

8. Let V be the solid that lies above z =7 and inside x2 + l + i =64. Let the density of

this solid be given by xyz. SET UP BUT DO NOT EVALUATE INTEGRALS

a) in spherical coordinates to find the mas~.o~ V.

[;j-ziT ("}"i r

MC!.\) ~

S J S . 8 Points

- - ?1' U 0 7 (fs.,d C')6-J(fj'~df":~)~

'"", CD @ r;;~ fi) (fi")/,~,"'f' fh~ -OJ p,;2; @ @ Jlh,ftf-

r

Ov~ t:' t(ot...A\." f=-8 -:::) -1 -= p'. 4

h. -I '9.\.f-: C'"j Ii'

b) in cylindrical coordinates to find the volume ofV.

Vol~-~.......

):~(JJ

ser., ~0 1 r JtJ, de-

@ ~ Q

[;j

<. \.X -+ '1 t- y~ -:: (,,'-(

1.. '\.. rf.. r '7 :: 'o<-t- '11-: tj

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9. ThepointP(2, 1,-1) lies onbothF(x, y, z) = -x + y2+ z2= 0 and G(x,y, z) =

X2 + 2y2 + 3z2 - 9 =O. Find an equation of the line through P that is tangent to both

surfaces F and G. Hint: The line is perpendicular to both normals at P.

X - :L 4- t (~'1) h.t)'i ~- \ +- t C:/I.f) W

~ t tl~)~ ~ -,

~ <to f': vi":: L.-I } ~~ / ,:)-c'> ::

1\ ~ &-:: \7 ~ -= < ~ ><, ) '-1'7/ (.. 'to> :

<~II ~) -J) (9z.. <-J) 1-(1 -(.. '> GJ

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-:. (-Il + Y) /' - J ( '- f1) +'L C It - J)

:::- -L(t

10. Let w = f(x,y) wherex =u - 2v and y =u+ 2v.

a) Find ~: in terms of fx and fy.

d,l.-v ,d,j,o,..> ~J( ~t,..,.. ~>-'- -r--,J LA.. - ~ -'" .}LA.. .J '<' J u.

:::.. fx. 4 I

CD

+ r . )I

C9

b) Find ::~ in terms of fxx' fyy, and fxy.

~(

.lV"\ .l \~N 7v-) ~ y~ L t" ~ f, J

[r ~x f" J '7~ -- +-t;< - +- >()(, ,JAJ ) JAr

.cV f[)

, [-~r:<~

- I YJ - /1 k

@

r j;(. + ~ d. '-I \f]x. ~ ~) ;;- J

eJ+- ~C/) )

V+ J- (x, --.1 C7~

-- ;L ~x<><..t J. ~'rl

I

~

~

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:l... -:>...

c.r -""

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11. Fmd::" Of~",g, off(x,y)=," - 3xy" th, po;n!(2,3)mth, <!IT"tionOf~-i + 2j.

~ ~ vi-. V'Dv\ f 12Points

- / x~'f 1. X\.'I

~- ~)(.7 ~ - 2'-1) )( e. -)~ ..

.

.

@ 2;.s)

/ . p.. n \:: ,,--I~e - 9J l..f~ -(. . {-II;). ')tJ?-

It.. I\.- ,- Il-e.. t"'<11-'ie -IL

<'~I/ .1 ')

~\---:=--.V ..>

(f)

":..._Ye.il._.1

JF @JJi-12. Find the absolute maximum and minimum values obtained by the function

f(x, y) =2xy - 2x - 2y + 1 on the triangular region R in the xy-plane with vertices at

'-(the points (0, 0), (0, 3), and (3, 0).

i~;;)) v ~x~)

.{ )(>:>! ::- :;).'"+-\

::.;, D::'~ ~j

Lc)))~ ""J')~I

c... ~"I:<.. c.. f~f.;)c..

C (I.; I): ';l - .:}- ~ +- I ': -/

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0f-x : ::27 -.l ':0-:> =-) '1:: I

p) ~ Q.x. -..1.::..:> ;:;.,> X~Ix

+:'(0,.\ : \ (3)f ( ), u) :: ..r

P(o;o) :: I 0P-(O)l) ~ -(- V

"\ '"'p. ~.. x.

~ x) ~"x) :. .)xU-x) -;;).-x @ ~ (i-x) +-/L

:: Jx ~(.,.x- I-

I().(I ~..>() ~ - '1><+-L ::--:> :::~ X ~ ~...) ~I -:. 112..

(V1/...( : I /°.-1 \\)6..f

"I::'~ -) 'Yt ( J) ~ ) CD( OJ1)

: Cx - :J-'iL-~:;{-(.~ +I

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