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Mathematics 3150 2, Summer 2006
Departmentof Mathemaiics,Universityof Urah
June 26, 2006
Student ID Number:
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This is a closed-book, clGed nor6 exadination_
This exam besins at ll:00 a n. and ends ai 12:00 E.n sharD.
Th6 elm is mad" up ot 2 qu6r @N lo, totat ot 20 Doints.
Write your a]t$€.s clearly lf tDu show " merety .
answer, rhen you are tikeiy to receive zero
""-u"i".t
petial
credit. So show and explain your work.
* Confine yolr wo.k to
tbis worksheet- you may use both sids of the paper. ,jfhere is ds e €xrra shei ot
paper p€r problem to. yob b write on ifyou wish.
" There is a fo.mula sheet in the very b&k of rhis worksheer which you may ure ar w,lt.
1. (10 poinhs) Solve, explicirly, the bear equation,
r ( r , o ) : rf.f r " . o r = 0 ,
& { 0 , t ): L ( r , t ) : 0 ,
=
u^=z .ft v vi^'-(^r<r-\d*
k
.!Q\--x- --+
- @ --)
ft@\
vr!x, t I --
.
2
5'.
z--J
t+ --o
/ ---{ \
(nrtz)
s
rffit
(-')^'''
2 . f l 0 p o i n t s ) . o n s r d c r r h F l o l l o w i n 8t h c w a r F
:0,
a',0;:",fir',or
u ( 0 ,t ) : u ( l , t ) : 0 ,
(er<:=-t@\z
o ----
wtz t)-- ?*,
bn-- 2-
r'(T(
(-r)
d =o
of
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c'ot(s+'rtt)