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SimonFraser University
MATH 251- Summer 2005
Final Exam
Aug 5, 2005,8:30- 11:30am
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Instructions:
- DO NOT OPEN THIS BOOKLET UNTILTOLDTO DO SO.
- Fill in the above box.
- This exam contains 15 pages with a total of 11 questions. Once the exam begins please check to make sure your exam booklet is complete.
- Only complete well-organized solution will receive full credit
- If you run out of space in a problem, use the space on the back of the previous page and clearly indicate where the solution continues.
- Only scientific calculators are allowed.
- No book, paper, or device, other than the usual writing instruments, this booklet and a scientific calculator, shall be within reach of a student during the examination.
- During the examination, speaking to, communicating with, or deliberately exposing written papers to the view of other examinees is forbidden.
Question Marks
Total /
- Suppose that a moving point has initial position vector reO) = 2T , and velocity vector vet) = (1- 2t)T + (3t2 -1)]. Find its position vector r(t) and acceleration vector a(t) 0
sJut;8n.
[6marks]
y'riJ= J V(t)clt
= J { ( H t) t t U{ '=-, );J rit
j
S
: ( {-2 t) tf; ' + (3 -t :./) c1t j .-J ~ -.l = ( i - t 2) i t ( t 3 - tJ j t C oJ ~ 2 ~. 3 ~..j ~ y (0) = (0-0 ) ; + (o-V)J l' C z= 2l -J - t. C:::: 2~ -J 2 ~ 3.1 l .j ... y d):: (t-t ) ~+(t -{)j 1-2"
L ~ J -..
=- (-t +tt2) 1 + (t -t)J ~ --.J. -l 2 ~ (^) J
~(t):;. vIr/)=: r(l-zt) it (3t-I)J l,-l (^) t "2^ /~
:: ((-21) t +(3 -I)J
..J ,...i
== -2~ + bt;
b) Find the equation of the line that is tangent to the curve C and that is parallel to the line: x = -t + 1,y = t + 2,z = 2t + 3. [6 marks]
" Th£ t...jeJ: l:M II th-- Ln.e f~;: - it! /
I
'j::: t+
t;: ztt;,
,. 'trf> II <:-1/ 1,2:>)' ~ /5 I (ft$ t ~ j2: ,- -L I
rJIr »;,1::= W5 t:: E 2.^ I
i (^) - - T7L ( I^5 C{ S tflA;v'l/\l /^ ,..j^ ',l}.,/'I\7 r I ,
-r/u0, iIv<- ~J eJ ~~ : 5 "1k t:n.-e
Tho"f'v -to- p~t ~e i= t.
Y Cf)= ( W4, ~f, [2.;)
- ' -~ JT Jf ... ( 2.- ) ~ I 1f7L,),
:)Q ~ ~~L'()n rrf 176- ~~ ~ , N=1\ 1i 1, - t )
J={- +il i ~ r::7C -t- 2t 1
i~R
SJwL'.' yet)=-^ <^ W5JCj^ t/^ t> /
yi)=: <:- 5h,t/ lJ 12 >,
a) Determine all values of a such that the vector v =< a2,-2a, -1> lies in the plane tangent to the surface z = eX / y at the point (0,1,1). [6 marks]
)M'ffrl I M^ 1t b~ tk^ ~~L^ ~^ of^ -to-^ ~
~M t, w s:wrto.-u i == eX/f, -.! c3-t di eX eX f1 = (- clX ' - ~Y-' 1,/ = z- 'j/ LjL J J>
At thpth.ll (0",1), i/==<-1/1/ I>
" V teh in 1J-R ~~ ~, J ~ ~. V 1 11 \JY
,-.J ~ !. n J-()
)
~ < a (^) / -2 qI -I». < -I (^) I 1 I I >
- (^) -C{-2C1-( ~^ =-(Q+I) 2 =-
l ' () ;:^ -I
3
4. Find the limit (x,y)~(O,O) lim x 2 x + y '), if it exists.
5~~ (
S~j&Y1 2.
[6 marks]
Lei (Y, (p) bJ6 t/J- p~ ~oL~5 of (I)(, ~/
;1= yw&/ y= r~f);
3 J 3
50 1( = Y (ff5 (9 - y, UT5' f) IX "-{- ~ '"L Y 2-~ "L&f vL)1v/'& - ,
)
/ I .'X >
;:;;)-? (~:) 1< '-~ '
=~ y-;> (^) .i? y'ws>e-.
( fOY OIU ~)
" I (pS?OJ ~) ,^ fOY^ C\fj^ ~, /Y} ~
. (9 (^) ~
_f_
.^ , (^). V- c,e.-s.^ - - (^0) J l I^ \ -^ .1. .. L --- 0 Y-7 -Q ( X, 1 ) -? I (), 0 )
1{) (^) - JX1-+~2 -
1- ~ ~ 1- IX 1.+ Ij 1.-
" ~} 1.- -x"t
I
:: (1f. X < (^) {(V I ( .,--^ <.^ / (^) ) / iX ~ ~ 1. ~ ~+!f 1.. "' _1_.. ~~r~-- SKy " - I 1: I ::::. /X '+ 'j 7: ~ / rx I ,
'.. t~ -«K/ :0 t~ /XI =--
( x'~)-) (",':;) ( )[,~) -; 'OJ (/)
.', ~ ~ I)( >
( 'X. ~ ) -) '''. 0 I rx >+ Ij - ~ 0 7
- Find the linear approximation off(x,y) = 4~~ +.JY + 1 at the point (1,4). Use
this to estimate4~ 112 + 13:9 + 1. [7 marks]
S~~, f ( f I 4)::: 4/ {/1- J1 t I ::: 4 j 11- 2.,.-; == 4. 2 =:.8.
f
I I I
x=4- .---
2/JX-,-JY+12JX - rfi"t~-t-I'JX
, ~ I -L- } 11" +zfrv:TjIf-t-I '2[i - {fii-f¥i-I T'i
{,
' t J
l. X I r Sf'1-%+/ }1 2 I
-f (f.CI-):: ~I --:=-L ~ / IT ~f?i f I I (4 4
5'", {ex, if) ~ f (fA) T Ix (1,4). ('X-/)+fl/,4) (~- 4)
=: g -/- -f (rx -I) + i (Ij- 4)
4 fi,,~+0it1 ~ f (/,2/ 3. fJ
~ f? f -1. (/. 2 - I ) T ~ ( ). 7- 4-)
.:::: g 7- l. x o. Z t- i" (- o. I)
= y 0 T~
a) Evaluate (^) f
1 0^ fY 1 SIn x^ x^ dxdy^ by^ first reversing the order of integration.
s~~ r
~ II, (^) /~::j
(?
[7 marks]
O~ Y:Sf, ~~ X~I J
(?~t,X.~11 ()~ ~~XJ ,
R:: {(X,~) I
= {(X,~)
( I J1) J.^ ~ I^.^ ~y /Y^ drx clJ
~ _5 J ~;r d A
fZ tj(
:: J
I
0 J^ V IX^ -YM'Xely 1/^ cI)(
IX
= J / ».;1 " tI /X 0 ?.
f
:: - Cff5~ /0::- W5f-1-
b) Find the volume of the solid region that lies below the surface z = l+x x~ y
and above the plane region R. R is bounded by the graphs of xy = 1, xy = 4,
x = 1and x = 4. (Hint: Let u = x, v = xy .)
<)'-- elvtl \9r1 J
~J.A
4 ~^ "^ \J^ 'X~=^4
'
\ t/ I~ -
1)(-4-
[/ i 1'11"-\ 4
[8 marks]
S (^) J
' .- I)( (,J v= ",,<w2 of A
rz
[vt u::/x) v~ x~.
S =-? (U, It) I I S U ~ 1-,
dLI (};\ dV
.j;;:: I, ';Ij =AJ1dX::: 1;
. (^) --,..) {,( - J. 1. ::/dX dj ;;v ('JI./
J"X J ~f
':X
I~V~4J dV '
d1::'l.'.
I
/o
~ X (^) j :: rx =- if
J{U/V)
d (IX. LJ)
d(XU (^) )
j
'
.' , 'J -:: (^) .I ;;) atexL~ if~62 :: -L U
J- J J
rx ~ -
~
' (
V -
/
J rx, ~) I
; - .- (^) I f X L~ L ~. d A ~ I t V. L' (^) J (U J v)'0/ [( C{ v
fZ 5
= J
"4-(4-<~.-L dudv {J{ /1Vt ~i'
'-
J
4--Lcf~ f '
)
'1 /-Jv
- (^) I Vi (^) / /iV-z..
- [L I (I{ J 11. f-} t~ ( It V Z)J 11=
).,4. -{(£"IT-i,J),^11
- Find the moment of inertia Iz = JJJT (x2 + y2)5(x,y,z) dV of the solid T with 5(x,y,z) = z , where the solid Tlies between the spheres x2 + y2 + Z2 = 1 and x2 + y2 + Z2= 9 in the first octant. [8 marks] 1 z ~ J j S (IX 1-+ ~ L). z d V J T
'In Sf~I' ffi-! ~ ci.;nY-fu I
'X~ f$h,f ~ <9 / If' P5h.1$i1l~ / z= f C»5t,
IX L -1- Y <=: ( f 5h,cp Cir5 ~ /-+ C f n:.,f ;;,'"()/ :: f~~7-+.
T= {(f,1>, OJI t ~(d, o~4~ ~/ O~ (9s ~)
A 3:J
I~ ~ f 02 J 0 L (^) J ' (^) I f~>t. fW5tf~chdpJcfJ@T
:: (~7'-' 1'-^3 J
); Jo ) .),(^ f..^ " 5,}+^ (hS^ +^ d^ P^ df d 6i
:: (f:d~)(f%,3~ ~+'f)-~/(j-df)
~ f. (i %01 f ) :. {-fp b ) / J:: I J (^) ( }^ b-^ )
=2'"2('7; ::; I
a) Given f (x, y, z) = tan x + Z2 In y , find div (\1 f) and curl (\1 f). [5 marks] 1- :c't
5Mv'{)n vf= <fx,f~1 fl>== <~1<1 3; 2~£n1>,
cL'v(vf ) :: ~ ( ~IX ) -I fg (f) -I ~ (21! lj )
:::: 2 SUI)(. 5M/Xt... X - + T 2 t, J
- -f) Zl ()
::: 2 J4/X £P,., t - - IF T 2u, fj.
ewrl (V f) ~
iJr Uw~(Vf):
) )^ .-.:>\ ~ L
'~ J K .) ..sL J JY Jj dJ
~~ f 2~£"'j
0= t( fg(n )- ~I ;"))- J ((zfi))-J~(~'X))
,..j. z :r: 2 1- ..j ..j
+ t (~(f)-~ 13-U~<) = ~('f - --g )-rO+I<. D
b) Determine if the vector field F(x,y) = (x2 + yeXY)z+(y2 +xeXY)] is
conservative. [4 marks]
S;Uvm r f ~ ;;{ \ 1 ~ I)(ij, Q :: ~ ~ 'X f 'X ~
~ - ~ ) X ~ J~ - 'X ~ l' ~
d~ - e -f ~f, .iX, ~'~ -.' iKt .j
.: - J J dt::: ~ JX!" . ~!I s Ct [en ~v",-t'\r.e
v~ j,'~. 14
