![Page 1: Math 241 Final examination · Final examination Instructions. Answer the following problems carefully and completely. Show all your work. Do not use a calculator. You may use both](https://reader035.vdocuments.us/reader035/viewer/2022063011/5fc530c584b7020cfd62fdd6/html5/thumbnails/1.jpg)
Instructors D. Krashen and P. Storm December 12, 2007
Math 241Final examination
Instructions. Answer the following problems carefully and completely. Show allyour work. Do not use a calculator. You may use both sides of one 8~ x 11 sheet ofpaper for handwritten notes you wrote yourself. Please turn in your sheet of noteswith your exam. There are 100 points possible. Good luck!
1
Name
Instructors's name
TA's name and time
1. (2)2. (14)3. (6)4. (2)5. (3)6. (8)7. (8)8. (5)9. (5)10. (10)11. (11)12. (6)13. (6)14. (14)
Total (100)
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Here are some integrals you can use:
1. Write whether the following statement is true or false. (You do not need toshow any work.) The product of an odd function f with an odd function 9 isan odd function.
FALSE
2
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2. Use a Fourier transform, a sine transform, or a cosine transform to find thedisplacement u(x, t), for x > 0 and t > 0, of a semi-infinite string if
u(O,t) = 0, u(x,O)= xe-x, and aau
I
= o.t t=o
You may assume the constant a2 of the wave equation is equal to 1. Your finalanswer may contain an integral. t
DO
#Ii W""i)'" ~LL(X,ih(){..:x) Jx ~ ~()
~ ~ 1 uLJ(rt)- 0
1!;1 ~~ ~ - o(J.ll(,{I-I:) t d u~i)_ d- -/ '\__ - ~ U.~o<)-t)
. { tY.-Z _ d~U1s if/;"-~ - ~i -- - ~;lu dU." ~ wM{vr ~c=w-~ fry
~ \ ;}d-[l- cX.1L&)-l:J;:: ~.
q 1[&'<1-1:) - A ClJ~d t) + 13 :5/i1(or'i)~ A ~ 6 w #cbsat ~ ~-
o ==1J 0~ = js { ~/H ~ = ~~ 0 .
o
x..{x
3
![Page 4: Math 241 Final examination · Final examination Instructions. Answer the following problems carefully and completely. Show all your work. Do not use a calculator. You may use both](https://reader035.vdocuments.us/reader035/viewer/2022063011/5fc530c584b7020cfd62fdd6/html5/thumbnails/4.jpg)
Scratch paper
=7 13 0::: 0 ~ 11 &)-t) "" A WS(d -/J
A ;:: 1JL~)O) ::: 'j-;;ilL{X)O)S:: 15ix;x~:: t::~)d-.
4' TT(o< ,-/; \ ::: 4 )~. UJs( rA1:).~ I) [1.+0( 00
~ uLx,-t) c::: ~~i U (,x)t)j = %J J-cX,,~ CftS&<{J sill (o(x) do( ·o
4
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3. Find any two independent solutionsu(x, y) to the followingPDE:
fPu-=ufJxfJy
Neither of your solutions can be the zero function.
e.x+y.IS one S0\U+IO(1.
Olt1a+he\" so \u.b'cn .\3,
- x + Yfa-.::=: e. i:s () ot 0-- ton:s+a1'11: 'Pct .
9 -they P.rr2 fndependen-t .
4. Find a and b real numbers such that
10 - 5i = a + ib.
_:5 r..-Lt-~L
Ii -::.5/,+) b::::- ~ .
5
-
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5. Let
Zl = 2cos('7!-jS)+ 2i sin(7!-jS)
Z2 = 4cos(37r/S) + 4i sin(37r/S)
Find a and b real numbers such that
Zl = a + ib.Z2
6
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6. Show the complex function j(z) = z is not analytic at z = O.
NoY'4 let: ?,; = l X {I;'! y Ie-OLI.n. ~ -:- --LV~ ~ -.:: ~ J..::: J$ -r::: -1.~~ 0 y=70 iy y~~ Lj
1h'~ ~~ow:s. +he \i1Y),"t d()e...s: (101: eXI'~ -1:-
7
hrt \et =l::.X dZ X reCti.
-={ S;::k3-:=1-r. V\1'- x. x - .lO AO o
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7. Find all points z in C satisfying the equation
sinz = 2.
Write the solutions in the form a + ib for a and b real numbers.
4-~ tL"i. ::: (e.i:j.)~- i( t~)~ II tZ=.t ~ ~Le - 1 =- ().
\ 11;..l:t 4-i.t(-1.<i>+Ltj - '). -+ 1'10
e. = - t?<~ - :a- LJl;Ad-..
~ ~ii d3 ~ t (J-:Hs)_
So oi1e set o~ sol'ns is 'i + a n - l b~(.Hf3){tjl vi A(1Y i(\'h2~e(".
i S;md tlr Iy J -13 > 0 ~ fhe othe, set of oS0/ ~5.~ iC~-(3') ( .r;;)
.
111 I ::'> J -I- ;)1f() - i% J -13 ~ () An;' It'!fecJer:8
L-et :t:::: ~ + lY . 1her\L~ -y ix · / .r;;\e ::: e. t:: :::: LCJ.i~3)
J.+E >0 ~ t1~( i Ld,-d3.J}'" ~ 4> x'" ~ + J1l"n
\i~ \ ;:: \iY \= e.-Y = Ii OA3) \ '" J.t f3- ~ y:c -j.J(J-+f3~
![Page 9: Math 241 Final examination · Final examination Instructions. Answer the following problems carefully and completely. Show all your work. Do not use a calculator. You may use both](https://reader035.vdocuments.us/reader035/viewer/2022063011/5fc530c584b7020cfd62fdd6/html5/thumbnails/9.jpg)
8. Compute the contour integral
where C is the circle Izi = 3.
-The. ~Lt. ~ ~ t is not AI1AlvHc. /1.t -tr.:lJ--_ 1f IT /
o
by t.u,J,/s ttJlh.
f-'~ ;;-j~:=O.C
f Q
9
![Page 10: Math 241 Final examination · Final examination Instructions. Answer the following problems carefully and completely. Show all your work. Do not use a calculator. You may use both](https://reader035.vdocuments.us/reader035/viewer/2022063011/5fc530c584b7020cfd62fdd6/html5/thumbnails/10.jpg)
9. Determine the pole(s) of 5 - 6/Z2. Find the order(s) of the pole(s). Computethe residue(s) at the pole(s).
lhe ~c.J . tli)::: 5 - CP/;gJ- IS
~ Lauren t se\\e.~ C~r1te\e do\dQ,\ ;) ~\th residue
~l-t) h~~ Y\~ o+her poles.
",I,euJr rj"HeY] if) the #0o" Jat 0) rJhere it has t?L poleofO.
10. Determine the pole(s) of1
1 - ez'
Find the order(s) of the pole(s). Compute the residue(s) at the pole(s).
{(~\'" l~e;' :fU\ is d.~i-pec;ojrc.) j,e. .f{;~tJrri)",f(2').
not Maly-lrc. Ai ;>=O. Let's ex.amine.fhj! ~i1:J\JJar.'ty Krst.
10
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Scratch paper
2 1 J.. ~3
Consider- ~~k ~~ =: 1+ ~I+ %" + '-0 +.
By the ,,,-Ilo.fe,;t -j},is V0;1"-" ~~"e.s. o>~"~('~es. (JY\~iI of C.
:9 i'l.) i~ ArPirL tv,,-'y \'Jhe\e.. Seo)=: i .;:'1 1 / 3-.\ =: ~ / :!- \ ::: - i. L i)' /abJ)-J.. f\~l)
Jl ~ i-e) d~l ~l=l)) C LcJ ()
~ 1 l~ ):= 4 ({0):::: j.
it i-E: 0 a-..~.::o
If) By 4e~i J'jr~ \1
(A\\ simile
~ ~ is (!.Y14/11-lu,-- a;I:: 0-1-e I
vole at O. fhe resIdUe." cd; ()l~ I~ )
::: - -;-1 , \ ::: - 11.'00 li-e. ~ -l.=)()
J.-wi-pe.,' oJILi-l:y of ~['l))f-rf1 () AnY in-l-e~ e (" 1 ~nd
\il°-th resldue -1.
.IS
the o-the\ pCJles. are
the~ e. f 0/es ~r.e
11
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11. Compute the integral
111" 1 dBo 5 + 4 cosB
e'fen S ()'Tr
1. r Je _- J. j ~+ 4c.o~.{:1-
-'11
r" J{ti J 5+J7\ Ji'-& --1:
1-:;::::;.JtL
12
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12. Let C be the curve in the complex plane parametrized by C(t) = cos(t)+isin(t),for a :S t :S 71".(Note the 71"!)Compute the value of the contour integral
14
1 dzC z2
CLi) -::: CJ)si: tist'nt = e.lfo tr-
tl Li)::: ie. if'iT' "11
j h _ \ i it d-i: . f -it cd-,"?ie. :=L e tC
00
qJ
:= :z · L e-it] =: - i. (- i -1) = J '0
![Page 14: Math 241 Final examination · Final examination Instructions. Answer the following problems carefully and completely. Show all your work. Do not use a calculator. You may use both](https://reader035.vdocuments.us/reader035/viewer/2022063011/5fc530c584b7020cfd62fdd6/html5/thumbnails/14.jpg)
13. Consider the function
f(x) = edefined on the interval [0,2]. Let
for a ~ x ~ 1for 1 < x ~ 2
00
LBnsin (n;x)n=l
be a sine series for f(x). Using the same values for Bn, for all x in the real linedefine a function
00
g(x) =L Bn sin (n;x) .n=l
Find g( -5/2) and g( -5).
t8 \ ::: i.~
-fil) = - i (0 + i)~(S~) ~ tJtt +~) 0= 6(1) ~
ti-5J::- d(-1) == - 8(1)--to js. odd
15
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14. Solve the Laplace equation Uxx+ Uyy = 0 for a function u(x, y) with 0 ::; x ::; 2,o ::; Y ::; 1 and boundary conditions:
ouu(O,y) = 0, ox(2,y) = 0, u(x,O)= 0,
(7rX
) . (57rX)u(x,l) = 3sin ttIf -2sm &\-*'1 .
uix,y) = ({x\. &Ly1
(IILx) CAy)+ (LX)' C/{y) '" 0q £B~-01ll:::-~
(Lx) &~ ) ,
('{x \+ /\ ~Lx)'" 0 ({()) = ~ '{J)-::: ()'A'> 0~'. Ld 0(= {0\.
(Lx)::: Ac.osCd.XI t 6 ;;1r1&XJ
~L()) -:; 0 q A ~ Q
( '{X)::: 0( 6 uS L.(' J,) ==C) q do(:::: 1+ 11"n ~~ . q IX == ~ + I n.. fIR rl-'" O} l,d) 3)'"
~ ~ 0". f=LX)::: A X-t 13 ~(o) = 0 ~ '~ :: 0i=='&)=: A ::: 0 -
'So tIP 'P 0 ;50 not Ml t1d renil 41j)e .16
Ii()\Jlated
0
0 -Lx
![Page 16: Math 241 Final examination · Final examination Instructions. Answer the following problems carefully and completely. Show all your work. Do not use a calculator. You may use both](https://reader035.vdocuments.us/reader035/viewer/2022063011/5fc530c584b7020cfd62fdd6/html5/thumbnails/16.jpg)
~<. 0 Scratch paper
~'. (LX) '" A Cj)$h(o(X! + g :s,nh{dX) & ct"'v.~Lo) ::: 0 -9 A ~ 0 ,
r::1{~)~ 150( cosh (;1.v{) = 0 q B:::: () .
~ (\0 el6eYJvalue:s~ 0,Sa +he. e.1~rJV ",Iue:s __ tlfe 1\ * (
'It + 1Q2) d-- #tD n:=: 11 1 I'") 3IJ /In 't :A U( v) ) 0\ I ) " ..
wiih e-r6enfd F)x):::: 130s'''cr;i;' x).
Il 9 G~ (y)::: ~o G.~) f GnLO\= 0
q ~)y1:::: fI Cns,nh{~· y) ,
~ uiAr):::: Ans,ohL~+ 1)y) Sin{{~+~)x) fdr n"'Q)j(dr..
~ uixr) '" 1 Unix}J 'i\-::: ()
Lt Lx) i) = 3 sin(1)- d slrJ( ~) ~ ~ An smh(L1 t ¥~ s,n(Lf+~) x)q b~r -the rL:::: 0 Mid IL"" 'J. c-1f ar~ niJYJze.r<J
A j A -~o ::: ~\) J-:::~)
<2 I(Jl} ) . J(jfXJ
'l . I(' 5 rr'11) .
C'.51rX)
uLx,v) :::: ~. sinn If "fO( iF - ~. I{S;
)
. sinh YJ SI(1 if '
! sfr1h{td ~mh Lf17
10...