,
a /
"Iru*
Midterm #3
Nlathenatics 3150 2, Summer 2006
Deparimeri of I\4athonatics.Uni\'€rsityoI Utah
July 17.2006
Student ID Number:
'
1 , , , : . d , l o - d b o o l ,. l . - , 1n r - . F . 1 m ' n 1 r i o . .
+ Tlis exam besirs at 1l:00 a.m. and eDds ai t2:00 p.'n. sharD.
* This exan is maile up ot 2 qx€stions for a tota.l ot 20 points.
* WriLe your flswers clcelY. If you show merely a rurnerical answ€r, then you
a.e tik€ly to receiw z€ro
parrial .redi'. So \1
J . , l , . 1, , . , , ' . \ o r .
"\
'l herc js
" Confine you work to ll,is workshe€t. You mar usc botl sides of the paper.
atso an dtra sheet of
paper per prohlem for vo! ro *rite on if rox wish
+ 'fhere is a fomuld sn€t nr tne reN bacl ol tlris $lrlsheet \rhich you
nray Ne al. will.
A=b=4
1. (10 points) Comprle -8,., lor all m.n > l. {lrere
",.
'+s:
!
I
!
r-,,
"i'r('l''')si,,(nr?)
A
I P '
3r,n =r1fl fJ
0
0<:r<1.0<g<1.
v"ra(P'rtrtr a")
(r*1)
,o^-(*'1t dS
\0
Ailw I ,
t
..t-
*-
"to'
fk E+ L+nra!. ;
1L=
y'-
.1L'- 1
't' q J ^- M ^ (r"rtccla(t.
fr- ar,r^( mX*) dt
r'l
tr^X''n/ 4
.v'= fu
-rfu?
,i")h
1
cs3(mrrrc)
!nz.o)
v = J
nT(
Pan*t:
sdlJ^ird, paazz-.
$r.a;1-',,)dn= - q'.$.l.=-q$_
ha
(-t\
|
=
ff ,rr"(,^-n-uo.
(t t)-
|-
R.,5 U"a. h"'t! tsr"1^:
--*.fit# *g-|-9^l-^t""jr5
? B,,n
* r [r-!ry) fj5^. t^"Xr9
= J ,lr"-'.fJ
^*(yy\3
m{
gl-'
-+
+G,f'Ir-c'f]. 1!1|s
lnnr(,
(kt "l
('"6**)
-)
-t-'r*)f
(-'f.'\r
crl. u f C-,f.'['rn^'(:)
\t
=
'
( A
\ iZ
t{
'i.1:
*r-a'" a'^J ln
.-
\y_te/
't"l'
_
'{y\/t(
,t
- oll ' nrvr^
Yv\r/
t
i <)r
* o$'
!n
o}I '
t {P,^ ,, f\
lt
2. (10 points) Solve the two-4imensional nav€ equation.
€;tn=o*-ra^
with bour:tdary cond;tions
z ( 0 , y . i ) : u ( 1 , qt ) = 0
u ( " , 0 . , = u ( r ,r . t ) : 0
{ l < r < 1 .l > 0 ,
0<r<1,t>0,
6l-.J^.
and initial condition
?l(r, !.0) : 100
:)
0<u<1,0<y<1.
3.,^-- Po fola^;1-r.tra* { r;t ^(y'19
l-
= t00
a
=
\
l€ao
tnrtd-
if
1 6 0 0 \*
(ryo)
s r.-A n -- </1,
I
\ o
n{15,t) =
l- (-r)
nT
I
o
(-r)h
1-
------=
-{(-
biJ( 6-T
6lt.--'"i^
'
r- g^gt*tq'