50
MAIU Zea f.i* lcxl bcrt€-S ,T f 01a^+ -,*t a*\"ot olnt^,{s lrl4*lo^rs / 6,*,o cr.t{uo d.c_. c t 1;$6rr.*$.1 eTr@\z,as ^\\o^: L^,8 tD qpr*, *trn.- *. ,+ J^'qe "[ ay 0,* ?**bt. aF.,;r. exc*r<gi_c, leaal,l Ue^rrnA\ g.d i^; 4p \rA lD ; ?,,^+ p^rr,e-{e , A=RA, ,J\^"r" ^ rB +t- a.Gelr+rs,>n "( 4-*- .wbok rA rrl tlh'[* h,tatss / a^d\ t is -W -(*rua]d \Jr hrlo.^: "-!*o.{ A is N,^- wW *+ -h* r^o\oi\ , v,, rr:\taL \tset/ r3 -l^u rake l. ., -tLa T6d"r4 ) ;a . ,lt"r-re$f" ,LLtsrt\t:. .{ otry A Ft:: *r;t, w" Ju^ trle aN{P..rc*Lr,.-ba^ ,,:rt *iwo- "rl,-f1r 4 "A"b ", ) ' o.-rY, v5.L, ft=8, An ,(L"{ekr, f' : tvlA rg -h*- Sarr.,e ,?s : rt ,t o Me F fnV F hrra. S

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Page 1: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

MAIU Zea f.i* lcxl bcrt€-S

,T f 01a^+ -,*t a*\"ot olnt^,{s lrl4*lo^rs / 6,*,o cr.t{uo d.c_.

ct

1;$6rr.*$.1 eTr@\z,as ^\\o^: L^,8 tD qpr*, *trn.-

*. ,+ J^'qe "[ ay 0,* ?**bt. aF.,;r. exc*r<gi_c,

leaal,l Ue^rrnA\ g.d i^; 4p \rA lD ; ?,,^+ p^rr,e-{e ,

A=RA,,J\^"r" ^ rB +t- a.Gelr+rs,>n

"( 4-*- .wbok rA rrltlh'[* h,tatss / a^d\ t is -W -(*rua]d

\Jr hrlo.^: "-!*o.{ A is N,^- wW *+-h* r^o\oi\ , v,, rr:\taL \tset/ r3 -l^u rakel. .,-tLa

T6d"r4 ) ;a . ,lt"r-re$f"

,LLtsrt\t:.

.{ otry

A Ft:: *r;t,

w" Ju^ trle aN{P..rc*Lr,.-ba^ ,,:rt *iwo- "rl,-f1r 4"A"b ", )' o.-rY, v5.L, ft=8,

An,(L"{ekr, f' : tvlA rg -h*- Sarr.,e ,?s :

rt,t

o Me F fnV F hrra. € S

Page 2: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

b"tk V*L = f qrd, rttfi = :F are

,$

La{: \ dcn$rdar F,rh^ r\ a. a.^rhdl n ^i"ta{Urns,|

f*r161

'\1',o^r, f = -,"g, *r\*re C u\ # accQ-kva;,lr',,^ ci,ue

+b (fo",q. 4*;, & ei,^r1 t.4* lo'.

d'* .. I

W:,xFA,We h*r,s ho.1 Jo. ,^.lq1a,s ,.*}e. :

r,/: i: & -:'-N :: --gL+ ce cK a.

\),2 hr3 it\<6r^le owee w.use

pLLb) :: -*.n tt n cL + Tg

.^r\*. ts r! *-[* da^,Ktqret- ,

-\ -Mot c A^/ B wq ,va i&l rdr{ib{ ,.,ilror;r'grc,v:Ae

^,As\€d sa,r\<rc1- h l,'-tbu**l q}lraoripw,

fi,flaq r s{ffe 't4 ,W "prt-t ap r\ o.l.

re*,["J\t 4-- o . .'"h.] l*ttr \^S J.*\ ̂{ :

rc F) : o ^^d L te: -i: /(ol ,j:_ 6,

t €*\, vs *Xt t c . so t'(o) := o

-:L'^ , 2('' ''^L-* B, 50 )cLo) ;g+- 1B= o .rlIf^, ,-^ t ---i"-;-u] lZonbtl\

Page 3: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

hrtqk,J \^ re"\ , ^ f^,l$..'c'\€ ep-rre-c"5 "de As ,t(, 6'it:

fr q"r) ^rfu\ o-|: -"{tos dwq N { ag&,ua..n- 4*+ l'*,UA'rs qF odd,a1 *ftc.n yrph".t -t" Vz. h .{hq c}ax, 61,\

be-""* " mV ;: *.rnq+ hVr r k.., co.gt, ()\k d** how-.-fu*- {rorl {b C; -{r*q ,{e.q hvt

\,re Cau ,ryaf6*rc &* Slt*&J_S!*l_q. ,,--ilnu. ir.o_O ,'^J,\,.c'

-t(ou, \ r/ro c",le "t a'@- r-,o i = o. .*[{^e,

vov,c'$ ^! *:\^*h 'h{r qcc-urs s c^[*o{ i-r"e

t l@ , *d, 6 {'.*d lonrs\virr (!l ,"-)\*. i<){\ie' *y=, ry\nhv:r*c

,i\"{e V-r; \-px*i*-\ veLni\ R -f".^d,\, ,-d€ t,,"t

rq+d*iF 3t?:$p,Y \e{iIt g1r9tY ( ri 't\eu'( c)\-,A

{* .h* {*rnhr^ \W) E o, ra\nbru o-+ -b^- -6**,,^1r\ C rrr.

^ On,bD H" * dn-,Lri #-,* r\ i -l a,@) # - a.€),1 :: {rte,

fJ . {1* q14let -.[ 4"* c["n Ts (1

) {1"" oIE K [4--{ D€otr,ue ntr/,\ ,1 "|L<: Wrwt

Page 4: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

,l.t*r( o^ po"er. " f .1

T L'3 l', @)- ,

or iks;^g\

,\A

C{enva{i,-€!

tA n r,o \+' *f-'= o , Il^o^^ )We t0 4'* Ai&\ ctq- i,n

Ovr q-\, A^b -\t^- olE

5r Lr,u,^n }rM c'DEs

"/

shnt .^r#{ \u*^*', Rvtt 6w-1er o g?T. .* \ L.e,rL . .it(avtr

,*M: ?rd, "*l h5f^f6H g2^e kel p"l*t ,e(

"\qn *,il b' ,'^t{k * ln+nr uo\r,,,^ wavro*\t*"r fir6\ o,tdnr eLs o,c{ iDDs *{ t!*

)D4 A t\J J- I \lorraoqgyco.\9,w.*4*-+**\l..-+@-_,-, 6e^s{ar,* Ca44, (vst owV h"ar,r ou

{\xne o\(Q- CDts .+ 41,* er*,

^b+ b1 = c ur\^$q *ea,,*#t,{r, otW**h"^. -*.,, SVwL, 3, \-.{.)€- o4dern ''-{r,\{

t' = $H', C'=$b' eF'

L\,'= s\".ri \rb c,.^/t wr,k :

q+ h"1 r O , wL"re k.- b,h.

CDE i-S l4,"rW^@l!

Q*('-*ff(4,r'^o* ,W*Jr( r"'*#t &.''vsd ,

Wo *l\Cr qrrd

r aSt,e*ltZ7r

COv.rSilgf

At'\'/ '

Page 5: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

<lenebl

J \^J\*^, k=-\ i 'rDe ottoin l'=-.t \*c,r /s=N), ,

('':es'"i*

-{**+ -lt^,. fo[f*^ ,\ WrL dt{-*t"h* e{pa^,,,1^a | €r,.',l,r,,

€{P00 , or l,* .ex},*r*{ of fr* nura}n{ e, ,y.c.8 .

.fu*.{ :) - e' so\wS '{ (

- r L'N(, XFO.:

;: g .rF = $t , ^A iE

o\- ?c.1{,at^r s,r[,.Lion+-'

11o*rq^"&, f\ s ^* +r-* ,ga"an vt st*tDl , Si^e-

g.*,r,\ &*Jtrr, r+ -1.^n- 4o^ q@) = Au*, i* jrcrr,4.a &ugt2,-i

,A , "*n >(*ho*s.

*A " G Y" ().ot- A coxq!*.,d- ^l

'n(,

t Ar\.bo tuzvVq"n

ory;/*( \r.^e-c,r,

c*Sh^nh "+

tODt h"'trf,"tt?lrMd$4.,

ttva"gl^*\S u,rrg

a

A* {a,ro 1a.\

tsd [oLrr y'l: l,,i

fq*Ye-

itc-

fut**^,* A, Lre ,@sl,

\+ \(gt :- qo ui^*eud\19i Y Ae-o E A = {nd-' a1

s\q t l{U) * {o t',\-J UR*^.h^

' An n-'h

a. bssdflri{ or*/rt,n

l.= C t, "1t^,^ UJP- pfl*rt

=) A =U. l;c) i1^,+

--Gc+pr-t -.r*-)h,i-tJ-'-'i

ro,\^ &Ao,L*rc^^*o{o r\ **kt^t^

Page 6: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

( ,\^-r\r-*\ o,r b*datj *,rtl,hou. (:rC*

| , 1,e- >Lh* ,Ntz,l- AoL" *t* h.s P.- olE

* -To Cer*.",*xr"dte- ' t,^l Ao* 's 'lq. 3er,,ert(

,+)o ,.pA.-\ tlr^r ,,ll*l fle=lr"*

( W\pe

'i? z" JN k)"-s J

uol fqo i3c,IT

$,r, l+,4*l'ct-r-v'v t

/

t!,lh*t ^.br* Ol^n*\ W

]"d .Iv- exy f**hrt ..d[l

5)h^e- 6lC$^!ktnt,

reda^abb lo tjp, lt-zrrv*

l* vTvL^^/ \^1" S\ar€ *,f?.

c^,n qorn btu, ''tt*"{-

C.-Ju" 1{*l nna{

l-h rg

1*^

.l>- o,,^fu* 4^.#f -{-.- r ^*+ a*,

.

a^r\ :t .,r^rte/s\"*a ,n^r"rC\ i{s "nah>^

.o r.r. ilg")

.,

D$ r* A,n#.- e,,,-r ,#" { . j. A,,Al"o"li

'"i["i^,,M tr^^o&li%r B u,r,t'hr" *t^cl.\ as

'DL{J= -P , L, sru,e- t- ^"+b^ -fgr.

'i\^*, ',-.\)Q- h^r^{-,

. 4^o- o\ti- I b*tfl." t .-F.o, so ,1).= o

rf a sohbn (4"^q\ qA .t*.qanor.,\ oUrhir). \

Page 7: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

,Arv eE r( b"- i"P -1<.*_ d&""Arf "rprrror.br T: ,*

). Yl vv\ +"" *^^e4- ""11...t t- d[ {},..r*hru"g ,i),C, , r),.k1

c-^A Alt (or^,{ !\nts 0{. ^^0L V {^ra l,urt,.r_ .,

b[qt,r bil; " r\b$,] ,. b\Q;

&'L;-t*,w- l*'-t aicF) do?a"A; rL r$

-\ finr+Y-#d{

i*rt -D bn- o \*^a,-r

n F--

-tr,^*.hr,*^ \@) 15Q,

^

e5g,,nw,^,\'^q- -|

, t__*c-

''l'*e- Cr^Jnir* tel .= /- 'B n* u5*,....{. n/n^ .r-l-

a\,.,Q4s,r,\h^J slgp,ttor -I).- # , ,t*."lDW.r #["^"1 = \(e^{ ) :: AJ

bg=,\1 .

| , { eleut ve,,\rxs / eEe,,*\*s ,+ ,^^,.[v*qj

*ssgNf

atu\-A,cbMce e

p\,Jt",""ttnt^l .,p-"|,*. -i"L* r, lt^,.

ftvr gynA'.ne*o,a n-f l) ,^;,"!'h

-

'f't*q

ikIeI'> €r5

is kat"lu"aal v"^tuVr*,ifd f t ,fr ,

as G,n 45a-tJ ae^t GJr*un \e*,^r 'fLnr,t

Page 8: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

n,\q\-a,,"^orhtiarot qy \. yrn*arrn r\ eAUL"td 'f\\ |

Jbfr^;{r^ ,1" a,,\- obE A^"8 J:P.'

I)o lye-

O rrl.Jdt

r\\tR{ s

r\ !\J

',)l u)g

'\i't?Ytw,,A-l l^a^^

-. tJ, (i.x\ '.t-I \ -*,'}U

Lel''t

iJ'* Ol=" o r{r.14 eC

UC) ,=Jo.

{5 ,["" e-Ue g6r) -- e]'( .

^;! * [;];= tu"rq) du

"'. soln I f At k.= o

(t*ut;:' C

[s,l^*- nio*n)

li* oD'lt ,s Jurc-c{,n1

e\tl, )o 1 l^ *- S.: l?t r.L

,i til "'o

'\,''--lR.

{lQnarav\ .Sott^/i6", \r:ff%-n ..- P,lt/ \e. So n"e-

'iL'* Bc l(A*:\* -+ /f *qo

r,\rD It13*,, - k'r (-) "

l=nry*"

,A= L* .*r,\o( \r) =-

fi\r . \

7 e- 3".

€(r. - IY\tr) = 2.* e*' GRTPH

Page 9: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

3 L"L ae$, R*i oat*. 6Dl5-*:&.r,4+4dr.{,-.

| ,lTax" .{r,riro- -tl-a -Arr^,

I'h k3= Peo

We "r'llr crr^ader Vc,rnav€. -F€o , -f*.'-ui[e -D|,$, = C'n

h1, sa ''lr'."-F ht-

Observuubirn

ccn"l^at esl Lci:'S,"\

sE r\ b\,11 '=-i-'

S*n* 4h-t U* C"*jn* 1f,6.1 S^hs["*U,,t\1.^\ \L$nt,= -P. -[L*nn

fvv" Y *\{ " t*

Aao Ao ,c- R^d l^.- an;"A Eola,[ru.n?U

:ft*- St- r s.

r.t"*r[- tutrrfn4

t

-5* At' 4=

"tgq5 1p-t-

\tlde 4^*l- r*|ihc- -0. "'l'u.- l.'r5e".eo"lr (a&., r.r.,c

Ie b5 A. cdrtst-aaruf , Srt4ce "

"to rA\+ :: A{,1 u kgr)

"p L^bAs 4= r)Solu"-t l-"-t? AEl

(auvol nu\\t5.D4-t -::

c^A sb Ag

fir",r.r*., f$ tUG> iS ^. su[,.rh>^ h l-'. [O{*e}q?*\\n , \hr$'--o, 'h^*

U,o)." t5e) t frrrr) e:\**-rt. ot

, \[Io-\,.\ F Ptgil ]\$\ **,i*]il=\-\O

n

Page 10: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

\^ 4^g 6rsr,

&*b^"t A}\euc*, -lu'

'\Gq t Qu @) tQ 0,

3,*^ 4* tsC p'=1[, uu4 g4-

we h^*r .'hJ df,&) = ,tie*h", 6.su,*.

t

fbco n Aa'o

a*C- S" ,fi,"f"--

te)' fu*t

.l

W{"1 qt.h '[oV\An "Rcc

..^N4t1e- )

@ !10(!\./

tAt'l"L'-e- Soln i{i I

*lz<.

;3 tlbtt' tfr ::(r'

f nG), ^Ai()"-$pc"t) &

t fldegeJF '?.ov,pleA

h^^+ ,*,r4 &-,tvt t"a gpq p^,^Jt,r 11 l*,

t1"a# sr[,.a }|jo-l={', It^r^ \^{- .:lLif;.^ kesc[i;ltb.^-afd\nl tr*- -hu*{",.^u

qv*'v-J.al qoln $1,., l.-' f, 6rv,) .

r& u''ot* so6e,*l '\o N** a.r^ **b (l^, i\U i"i ',.^.t-

l** '\r* sa)^t 'b^u^.1 $tl^ aA $oA '"Wu;{"; ,,

t.rqx4? Y- r -r Y !

roe$. 't.

9\ve l' - [\ ="*B At s-^*e- 61nsFu^,t iS *J lt' .JJ,

) j't ^\d= :: oo$

-.,\nw.\f1" \f] + [It :: o+ R (. ==Td,

{} [..;BY'p :;,\ L\uur : "B/n

Page 11: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

---il -.[t- lt-t-0,..r '- lqa-

1w)=(\\J

Txu(Xru"nte

S'\w q'tt {q =

- h"t

4 / $A:j6ttr + 2f,F'= c+ Lc ': +

{N $\n Nr

I .,/u,. -B-\pR ' \-d' r< l*

,'{t^n^ q {h =( , goYQ

ga O,L., ttr);:

$&r =

-"''iL*^^

Sa

t'\ nci(. t +\e

,) )r\ t.. - -d-*J'r +& ,

a^A "1&)==6 $4."+"a

t; :- :r*nlli

c"o &. n "^.fo[

{r'\ filr r,r,l\ a<

,,d\^/;;

r?-(Ig*tk- d

l S\r,q i-\ = 2r, ."4 {0, -lo,

\(e C,J Lnor*\\ tl,= T.j* ,

'-il&*, t$O[Ip=Ettsh-c ts P*-.[l€*\\.,{-*]. f"l1-X"*,$s

arc gA j,t- {a*So, oK * ' E= o end ep-- L IIt

ul.,c{v \e nD dric* ,fl R "f.hgt} <,\1.*s

l!b\'vl , il tu h$t c\ c^^s\,.^Nt Jcr*^. \F.\,tr a*,'tr?h ukz;\q" u GE7^^xo^

) >f- .:t^dd, &\ [*,*r"*6,*+

C

f'*^, \dn\qs= Et k$*n c) = 2*, S],C= orrqf .

Page 12: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

U)e

t*Qlvt CLtcsn- R

^h'a

6d{ * n?*\,

@r"*oft"*-' I

I\on€, f,Vc^t

's

\@)- frx

,*aurOb**--T-----

5l-i.* il'

n \J=1t: $?tr)-* B,c'u

Zxao e\e-F ....#-*\..-r

?--wx-'

"'W t

dc {l-

I ^uh*is^lrs'

KB=. LBthC '" b

11- "-^e( { G', -lJo

-[r'\ k L@ v"d,J

.)l*/R

*E)' '-?-fiL'= hL'

St[tAt*^ r g

-h<. {,

I-) {3'

=? C.*

.&='R2-

Illot\U

, an{

e-\i1' )

)"\-/2L/f

)3cL

"^AC*r-*D.

1t'*- q' t la^ = 7-d r ?-;.t-Ll o

-qf $\"**- Bt ' (knl)Ea*ttsSr\'** u'* ltul= sr,'r(3rQ a,^d,

tt"r=$,,r$ 'rt

d u ,t ==- ier ' fr \

rt, ri ti

,t?er: ": Bs''r Ft)' \-aa- r.'-ra \'thr -frn'^ble

t >hr/"\d. t"$*d''-bg fVt l = I3onG")t (- "s("d,

t.*,, qut[15 " 3B.ctgt 3cs,\t3d t k(gsi^Q,r)oc.d.i)(;' ( l' ;,*/e.)

e16

-z/,V" il'1* L7'P

,lt'l

Page 13: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

'{ir 4t" b be*-t,A €" AJL5 uu)e

' W sN^(.3") , < 3c+ h.K =' 'L)yqq GW- c''1& ) ' 38*lrCt s o

ey^nl'e

o6

ce$&'

,-\llern, e + c... -ffi/a

o l---l-t R\5.' L

oc, B (* - l*) :? -1- .> B. -.e- :--

[ *1.-

o.^o\. C.' r'-'S- = -=i*__y,.€=t+k.

{-s\\ R/\\C

lz

D

6l tk'

r\ -C,\r( I 'i") -A.f*-4\1er + F \)iJ-

n '-'\oC

-lAn | '-4a.t aC*{iS o I A_ i

A,err,ar, $Fn"S;

?(.r-)=-lt, +

.E<ar"o\e-'--'44

S\ue Un

h3 =

-T.g tb,o = T3a:--a')c , .-1L,^^

l$ n )..jt, ,

'qt^r ltAs "\o \*VI h. sNg / \^)a ye:,_e. *oB + lq3 :-:- r + Tt * R:^

QxneNt..{1n*s d'@.,* *E .$ b=n \ -ft* B L,za,,*<

Page 14: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

Rn-k€- ''Vo\ua ,

i5qll.\\A-D LBe

gl.n-w -\

*.l : #(su

kRt* i: o

*{ 'i) ,^:t{il, {i€ru

-t*) r kB€ -tw

v1b

)

F-'lnI tlIrrllI L'"+

t'JL,- t

ll .. aft\k5 {o 5at\e- 't^e ,"ry[i; (\n A*= bq (N\ e,r'Ve7hr a F R ,^J{^ Wm e-.w{.C

lfu'w r! sllft \ gtd'io4 , b.tt r..$.e t-r* to tla *nry-r-..orb.

%-Ot, "ltn ,E re\,*e,/ b "fto p*bl,u,",- o.ll

s.,5 n ff'V"" "* ik S$r*L-lq^-% (.suu h*"1

It s'c$^s roaa^qbb Ird e-l' s .s-[r,\,|. relqrr,*J[ { *

t'ilus

it,ur^,)

qtt \R{ = &-W. Vc &5 ,.,sta{ .l ^" [f^,z.'r*\ -0r^.,v0

'dnb = toD *-'ku'^^il ; *I*k ^h*,s, cq\\ed "1,* $ wfi/ J lqri-.l'r,^. J drr^"C ers" ,

I'$

-L)l f,- 3'

n'F" k'3 {bn + kg-* ,

; 1 cD e'-|"{- =' el['< ,\J

5o, g'Qo = -{- "^d !\ratl= sL.t C- , ^^d !c,/(\ w\,3f [,-l,,

(:c+C) €:hx . \r).\q- h*t- ,t-)e-ot U,;$f,g,\

\ So \^)e- cs^. s-^{- C = Q, o^ d l\^C ,9{-r.; -," ;r-4z F-{

,I / '\- \Jr('.

St^Cc- tt' " - \- ' n a-CJ ., ,v\\'(

n '\

{r.t ', c.kn{ a-C^uprAiC -t t ) ,

,o ,',.\, 4(rt= ,{ ULO * t\rr.(x) ::- X.&"

ltr + AO-

*

Page 15: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

)

/\,2 EC J,n)=-*1o

ct 4 ) {o

[tr) = ,n-'ol n

?X:o,r,o\c.

fx"\A*e€ r-Sf\,*- ('* h .::

,r. f h" "eg ,J nw

t+

[21 r'Lx.Fy-"{dD( t,tr,u,r{r^* yft) = 2|r" .vu_-ct)r ,

r' :]r '- lt--"-tir- "t IJ\r),iU

Ll

!ri0id.l -.tL hv\ i^,C1^ Fri- 1;1= ll-,

V

{ 61-[ .t 11 \e=( #t [i--LtLJ [l ,

ln aB*^\ , MC b 4o d\^Q- '1lU..ef

-{<r-*\ c

Exa'npie

--) ): ^ 6,etio"o\ue ro"h, ,Tolope A pb?Js {a *a ttu B- (+)r $r. frVle,

Eoblr C- ",',fl^lkL o( b, # nr,.l*i T3. 1fu"

l tv"ef

--1, , /."1o u^rl.

!.€r-trrq- ,; ^(.o*1.)

Page 16: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

inrti"l!, e=e> o.4

) #= -kna

d! h^a-re: l1n

A\^, r \'*lt RRb :-tA1 D

r\ IHont. 3ln |s

.<tlr\r\ -Pc(N.ttL"-arJl

tG),

d^Aka e

h,n*hn' . /,,,

I'-RAb T- t. .r.t\A^

b =o.

+t"Bb

*hne

"frt\d

Q=

I

Ae.- haG

h* a.tGk+ t

OK

-hof\-e-r ,\

leoa" 46.tpk) i5 Ae-heb

,s[nt tp(al -- R e.

k * h*b = - hoB*-hAb *8HlRs e"-h"L ** h^^o h*e

$, B(lqs- ho\ '- h*ao

)

k*'- h* ,

-\&tN<, Lk),-

.> bG)*

-(w^, blqso + o?

:Sc A != ed K *"d\) Dto = Uft ('e-h-t

Rs-h;\* t

bp6.)+ br.e)

-bo&- e- haf +

fu-h,q \-r

Page 17: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

ISqff" ^ g*b6.\t de$ot hno.,r a6, \,l)* tvwe-

LGI " ho "

r r -- o(!r- hd k\reu=FA\r \-- J'

9,, ,W rr.k b^ b* d^r"A b"^eA ox 4^(B{-,tl #-t1* "t* **h€s A "-/.-B.

(?n\-^/v CbEi

t A;l\e at(e T "+'[t^" €.^q&)f,'n be)$ * c|'l

Lr*} d\rab -fh.o*n[ A a&t "^A bv,av[r;

i- Fl1=:ft*:- , F= *, , {ur- ;if*,gbxnrr!qrr

In $J-.2, .^lL*, 'P6)- la= 6o^.sb, wq t'\nd.t *R$= {t"t. -\1"L L'HS @,\y\ te. *ri$ern r:r9 rq

yi -YnJ (.k \' -h{

) ,t_nr- \F 5) Q ,, \t'^cer

e &q\)'dh'[ #u* * eb*#l:: Rq+ U

Page 18: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

-{^"&t*, eh^,'uk^H)'s R-,

I + f_no"j\' =: -efza-

&lUenze- *kl

+ \0..,Y o"*l "w furA.

ls o, ,..{r^v *{'U .ot -ds[*ftrya,Zv"rpf- \- --

I

t,No-lu C -$

-i1u A, rs 'tr*NJe 'lh# d".

a8 "&t $:. ).kt"cl*[yqjlsEl q e

vc,l

ro\ b0",., &.n Ale f-t & 0ru;tiur4rt*'s, o.Q{*. *uL{ly\\'x 4r* oDri- b. F,),if \r ,Ai?., l";jl il*,i/.o,, dR '^ t,O'i

}!l" tt .,l" c^\ R a T*) 5ttr'l^ tt*$^\, I n fq t {erq E Fe.-, s, t*fr,ls*r)' :- wQr, 1,,.*-\rd--rl - r ,^tt. 0 '/' (

hc*- [*, * u'Ior* A7*-

a):p "h + c.e--tu"

'r,r/\n ^ S€0, =- L, L^)

:= @h{-( tek{

s^'Str.oh

rect*.*

{t"" ery*E,'P,,ASarw('P*

h*1. {"

uie- ci\ W ah\ \nkg"ti C

Page 19: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

{r

) "n*

f,-A &qA ^- fr.*,lrb". W, ,fl-.e-,. fL* SI^

&,\' b. Su.o['

ftr) ' '=f .* 7\'I r&)w,do'

sc Uo,,, iri \*; Pc C'

do,. d' ,Ne {.\",1. yq? Nt,[,1i5 k.fr,S{ pe:f = Pc>

+ ?**,f' r T@fDl= PTw)

vjourct ?b"C,

rlorrFn, il.r- \1"^lY 0

) Rt, 1}^ c[w\^ r,tu, [.To$,.. ?,,I* T.t,?,rt\y ,.^\o S, .* fl*t'

-rr

*'= ?k)fs)cr' + *&,:: r + ji k-c*= jl s,t;,c-

'R"*, "btl 4 yari"ths ,\ % ,\rx\ *.1"( S-\.--cytt\

to*s W, r{s*. #{a*'so S+ Mdr= 5 *,C* l: ,"{ .\,::- \.n tp

) *\a^e, [flt?= [p ck . + ?=

e1 f S?c{

;Dq

sbE

Page 20: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

4+t tet tJ'+ t*x)l = I .

1^,u, qr {'(.} - r\ q = *,.',' d \rc ') J ;L

ttu, pt-t= ;|.- r ,, *^d.

€ \*a* - L,l{*rr

1ro, ,= e

\"1* *re ::

so |po, c&.= J j, -' obc

tx*e\-{*,r,'[$J)'= TA, i-e. (*n-"

I'hrq, ,ce-dr1 .= C * {*r)

-/*xZi*; Z& y( r$ tp+.,f\-/,/,

-)(r-- 'te*'-, ,l - .- €,.""{7--

"-,tp

\/{):-Jt

) .\ t, f ^'d--) \.\:: L-e- a \d-;\

\rNs; ${q6c- -tt^* I ffi .&B *t- -}€-,,-- o . ,-'"[L.p,ur i) , ,f l,r"r,,:,ll -J e@

Noil,** l-r\ ordrr CDEs ",,r.- -+ -t'Lo

e.(*,$k-* Rb$-= o \

0.", ,

Ee\Irt| \,-

H,: q'4x=-rrl ,.dt -\" \dF."\ , lH? =)--, H \:-

tqt t rf # ? ".+ b +s,*rbri -'.-ol.

.

$rt'\ U\,,

Page 21: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

Sbk'

l], -&* -*dnr otr 6 Jg*arlhp r t " {; cin tirc cavi{ieu. nr,b r

1J$#', lts {

['t ..ilns (e&./ L$e \ ,*f1rtto_ b\t s,Ja: I

thnlJ Vtrt Ct'-= ), 1' dx-

Gc C^ r P, ,

J 1, b" -\ t' d"{-,

atnd. 4^t \€r.ds b ,t1"- sbtArr)"

kou,etn-**F

H=$ + tdt=id,tJ\'{b'nx, stq=jfdx.

) ltt: qx- + c-, f =a*e[.+ 3=Ar ) A=tl

ftfo" e?^ rr N\ C^A l't*,ry-).

e^qk

H = x$ >\ tt \= x- C,Fa

+ St,{= i*d* + -t,= }."ic,

Page 22: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

3,i (?eTql

ob4 = *olrr.d

(,\ * *r 3* I,) -sJ = 5v

A= -3C\\

O ,^d \-*-

fltw)/ #=4 + t-U')

I + f(' *)q = j;A* +i J\ \)/"

\ + ,f3r*#::A,,\ ',ugul-dhtJc'3

\ ?- tt' la) =o. --[fn*^ A=

\ " "t3-3J* t.= o ,

\J

) Ellrrkd1d, - s

+ t{ = -ac d*.++ f0'= -b.* * Aa.+, \t*;ct F 2at, ?,zrt^q^

lg\/*$= \

t

{(o\ = Ld*J

O^,nl\qot{ = \,_.rol.<.Q4 { J

%-q-'t>,

'+ C"* ); o^rl

)

Page 23: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

11s!*l Ul\ ,*^nn h rvrolarn N\

^ c*sW *oyr'{ofir*^r,t 4.etl. ^J0 ck^I ll' ' - | '--',b,r \ | (l ' 6q{",^t\ ,0r* t1^e Vats.\ v jdnl-.s 'rn V Y *rnq i krft b-= @1gb. .l

J/,u[t -Cv'f--$( '*f - Av= SrLt &+o,J hu.-t

N,o,", Uhr,..ilorr- t4 n= Lgyuas^^\*t"tfqr*J*kd=o +

"n^- ( ** € *')=m ' \ Y*\fi v-v-, ).

sio 5.%il" s -r"t S(^0, *ri dv

=:I1 (a& \ r1V-vt\ tfl\ Y*^t)

2 *R\r\n+t ttcso lpff\= [email protected]$,, an'{y+' Ivtw\= ty+v-,-f AIn-#fr'*'M*Vqi*o( g"r.in?fu:k y r, ;tr< V{fafrWY.,

Page 24: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

a{U.x'=: )

''Au) rffive#''a'Ja d8

i;" a\\ A,

) ,"fu* V= k6o t""h

{*= e, RnS +o So V'+W

ffi*

V/+

Oil)Gs(A"

) Y[e-

1$ e, t A&rytx$ "E7sve*sd-iJdu!\dno** SDt- R clv\c- "\t^I rtrc\ ,^rt

t.

st^t{shle E +-* $bV r}-Tj-

-

-

r'

I v,w I = [v*rr In " &* J 1 v*yrln e

-v['frl|

r--.\ ",lw):, F n I tn I o ,: =_-_-:. I C.+ , l,\e^lf - lHe ,l ,/'

\ J.f

r/ | -- I -- - I ':'i

'/t*----'>bn .4/y''

,q"A or\

Page 25: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

/1 @F\+ \S a- -So\dt-.r,^. ^ h'ta * gi -u,

&^*oL--t-

\+:" l^rv: -\* kvt + {*rt"y*nt y= rv{.

ie ft. o.

-ryL* {--c-.,1 u

-1'In"n- f4 *cu,Dar)\ -flon r

\J

[*AR*d {r,'t "V M-|s SlrrL\e ','f , 4\"* do^

-[,, ig

Sl't '"1\v(J

"-ft"., ,. c.

prt a^Lr

A, {* re\ir.nn, b d\-.

Ntd""*\c"\, fl\*^ :LC .!Lo)==

c-t %., n,r4) c *ffi'1\:: *:n,

,;N.n^ ^

rs -J*[o-G sa€- + a- ql a}r,,!b obr tu-1, \*[

3G) = e-+ u(*), *-ot 21") '' &.

-d.:, M* fql + MF f ("*e(r\

vy

1tr, t?:,i'

) f,*,

,rf_,*^ ,

S*ilI r so

-Csa = o{?t"q= 40 + s-['(ryr o,S')

-ep {6**rD' *ss r c{z)

*rY g..l'(l .# b g. b = .P4,

2:^u6"L *t, t""'c" { t=o

Page 26: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

a>"- 'Ps 9 ['., l=o i-s S{otta r{

I tt\S\^b\e

Azu-Fk-

nlrstt( cr

>c

Th*tN^ o{- *?, il =Jt6q. '\C,"ttr r.de ,, :?

,

A*,\{n ro*a & \,'(- , d:rt-A..^. A !s.^,S-F) Lio" ffi=,)*-AS ) J= ):) ,,-

t:: qq A*tu or otec5 \ t o( ? ,'t ,- % f\a^rar^o', ^s gny t:"t -\t^*r* r\ croqot|{trr1 .

'p'.\o^[lt

.* Q+ L\ Q"-r,\ x ' \ " trb . b* *"4irr,rLrs.h Cr^J,", $u,,c- {t,l o*,

Page 27: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

gi * Sa b scarc€, exV^ &r*ll^ rr,*E x- ,-f- .

Ir lel f = o- 6, l,= fr1, "-iL^,

# F -J(,- 9.) s,s

lt* i 'L hshc V. .l'{4 11

Atl,,.-t' V) = *r (o, >siub\c.ttx- s t " \):--^L F.''^..' \-

) *I*Pixs-E&,n8"\^ .o\i\, . b,rlt" ^.o( oh*\ia

car,c, atJr-r,Jgt ( hy[s,* f5, fui1^r L -u,ier) . A1 [,i+rnn#al \U{ te ;cnrr= ,[r.n[r.-*n\

{". {^a ?q^\ '{bn^ ;c oF "tinura @\na.rq,\" n

{,,:3rJ*T#: '%Xn*r5Xt + ;X^-&ar)-{^ @e; ?Lq; J-r" (r -rcr)

caq \-\ ok sb..\ \\- W ^=',( b a ',iM

\^.

We

Page 28: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

r 5,,. {'kt

8 F t-r

\{ \{ '00\

,1.b^\ t{-r^) t +e0

S" fn, y f-r'q'

e!c. n

!' ?.,^' FU3 '

,o.

c,u.A \q.\ \ (*'*nvw \.n\-L o

)

\li\

-fr l

1-ur kA'S\cU

{tr),= , \- 2.I*.

,4 \el r{ t

\-\\ -' l

4*, 4'(,)* r so

t'("'-i)'2+

--fpq ,' ,\r[r *r), sb

tx- r- 6 iS S[tbb

Uvt$..!t

r z ,-;t

I,>

3<*'"\

)rg t$h n-L I ,(,\ .-'j.

4^{\ab\t-

l"{. n\ d eCn t$t,-i d,qb

'{-h,*l

{uIo I

< 3,e

3'Tp\i'rr'\C.r,{,c,rh- ,(.1

r .ll t

- 4 assillglDl.{/'\ \J)

-.{lr Xa

,i<A<3

,\<l

tG

Page 29: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

2 %o*/r' , , \ *- t+Ga cq c,\. A* /\= \-r 0 6

CJ

-->

*)

)

t*

2_3t 4 e\iAV,*"

-..-t -."

l*fg

*x$"-(

+cpu

y.._-.*-_. .-" . +, ,t

Unqr^n "aU*-r..d

6,1 \,^c.o o\iR *'$ *rilur n, *rh n x&h&3[*s, .\ ryt* so\, 1,n*. ,"

S"^a rrailvour ?*Bo"A oko ho.r* ^Y,* t.\r.fruaJ,t, *Itk v\a^.\ [" 6s[ :

49-ab C'*- uy'''?/',^a$\---aq't = q'/3 NA \{ Q {or-'b''([*,

Tr d\= dx d5 5 ib t :: .("t*

+ 7V# = :n, t,/z+ t = L5Q*c)J4r.^

f\or=,0 + c.-=o ^J r1 '= ($*)tt

) \{u**, \(?e) *o 'l* df't ,*^\ro q s,\n.r\r- (uo 0,..t.-&{oft;6'

\E)- 1. t-t^.--.'fi ' A(<-g\n, [" ^\ r&,

I\\a

Page 30: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

q1'.t|€ -'\o\tr * &b'a,.@)

) o *d=t= ^'d $bt"l'-,1'^,t'i.4 = or-L ; Ji. h: !"*

Iit*..g'L^rrc- rS A^ Sot^^htrl

)ele

$3. L, X"{ -""let ^r;, 6&v

j- 4hq:- Ne a1,.^tu"+ P &. €.rn'

ffX"- ti + i= tro| , K8,,ry6,N.

or, Al*i\hidh$ b c/\)

l" + f[. fi= i9 t P'| , T- *, #, P,*, : ?;

td,\ryq;.1 culrW* "gq,\\ohcfs,

t&ghp, *d &tt l*\1 eiqhi,,

\iefyo.r,(& .

flAqgS fv\

(qf ^*A

*r$t]^a

'$"q- q('\1L)

o..

A'r)5Vrnr !\o ot 1.3 ,,sL.r tr*\^r4 k*.rL

tJ L.,

U

Page 31: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

*;{,,

rff4^t,.,'lr^. [}.g ^.\ I u.t 'il*- ft,au$'j CI\f{

qvav'{-5 l

R"ohN tr,',-rl

,,o'r.ts. I s it* Tr:\ C""Et".urt

* {&, '{rae $ n-D ret -k ", )s'

I tt* rssrrs -1[" '$ f*. '.

N^r: g1mrr- ,,{r""tI u)c- ex\,e"d,l\

,\ hr;J.,\>n +D l-, a^d tef 4o.

41*fir*s*4 G.rn- '\o"D'

U

Gcrvltg, *T1

) l\'u5La"r" Rt* t L) '

i$^.u ,h.^.- gft\ \.\\\ b- ,r^ *rhu^lJ

-lt tt

'F* r{4, o( [Y\)c =:

-mx

ht-. -1ltq,^, {rLtc<- ru<- {nT ,r

rtt{-K

&o t*d rcb

, e "R\ Vc"s.,)s td L.^.\,

+n^qIkCr-+L).J.trnt-f i'l-L -eWx-

U'-t -la- ,

\e t*'l,t + b&'$..,'],nn^^ \ff. q\: ,- ih\ c"dlu^g '/- (-") : 'ro, iie,) :: c .

\} u ,J,\ s\o* ,t\"^k ,tti* t"r d*[,u

I ptL) = a.o *r(JF t ) b osr,t L\c,{n"s l

O " h.L ".uy.

'\q sptk bl-J(

l.

\{hotk luenelr

,.1 a_

,i*?

(Ut o\$^r doif --\ d*qr\ nn&A.\ e\

Page 32: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

r?x",m\a

-TG.r;/e. *re cfro;̂lq.-)t 1 c'lc*,tt ^0r""^,iV'^c< C

.-Lndr^rttrrucL

tt L:, b&'

\nv**lVd\y\ a v'tU

radrc +4*

+y,4^{, . k,clno$?S 2a L*^r + sLrv\ * *\\y a\RB,

','^ya#" *tW 1/k

) Dn\

9,

iLA,

Ne h^^"

) \l\^,",

oNSEy

eqpg

rr5,1$r , fLcrfr^V' q,LuA,,\pr' l#$"\ o&u>s

L rG[+ LcL=:r=H TA so #"Lm. e# , ;o=t"'rq- \G, Q Q\ =

Vs\ --o, \a 9\,^\Io^- rt

qk)= e.'s{'' (asivr*L \ r$crs *t) - [,a,

veldlaN''rtL),

Q," "-/

-Y\etw/

os.$\o.,[t*e\.

(!" Joh-c-, .h, N= ^[tW, l\ : p *ggo , R.=

Page 33: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

| \e

1-*il6*-'q.:Ei\'\ iuL \" {'^b "4'

:?Mil::;:n.{\q r'*A

,"{l \\\"'lk,., 3exl"l\ so\-- *t l"* b i

t*r= C\,*,- ^,thot

) #* ^s S* -$d:*,"Lo* Clh.

$ l3:t tt"*^-r, r,.** , rr!n^n-,ny, rfu$r/t&r^oDlb

) 5: rlt rJ o\so crvr e*eA$,.""'t^u^ o{ )} =. #. ,'

H. =Pru#) - x* =-X/* *$^ r,-vJ 1-

Page 34: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

Eb|

, 1,'uh..g 4 1*"R^ e'v rl,q-th"\ c$\\ \*qortt.v[ t^,vt

'ao :o\* $'- Fd* t=o, t*,\ls v5 "4t* er,s*b , t=

e'lo'

i\"*^^, ,j'Sr1)= ,tg!'* lr.\.k ., I*"\*:z)

,'|^.^ , g)"*o, g,o f, I \r)-" \.,-a. e),,tr;s q {"* 0- -\ *rf \L* obl- ,

6 [^or {*o S\ns,

r*s *[rJtrtrt)A2-

rl&@t*:vho".r

\rle .,\l*cRo," hn+. -t\"^\ \

u A )..t e a^d, { ,= Ga'\ - '{- so\wr- .-fr* c:)b ilJ,: Ae ,s-\d\,u

al+.k rrr\ Crg*4$ Aig/ f,*g -h* ffi U \)*[.\-l

.A[s6, 4nre 'tl^- cDE -.s be{ , w ""\3o t",* ..tLor sL^hnr

\@)= A/.tt (4\'-ac(J

ilo**, rFe fu;\ yl h$* ,"V ,$i*'r, h* *4"ard gffiu1-ti "fT---*

) 41,A,- o.{- ,t\r*rnn_ o)g{;,.

,) g&,L ,.\.- rco\, Ar * ,\-0\n u \ . r ?

Page 35: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

-ir- rL/'\ki/d, rga\ rod6

l^ Co"-_ O u,Je.

,A,<nA., \: An lo * Sa't- a ts {uo qaraar-at\ g {,*ti0,1

sr\nre e\* *A i-,- o{e_ lv*g ,-d&ds$ ,",".a\+'d- c^nn>[ !* "nx,\fi-qn d4^ d,# :kr*^+ ,r& {''*"

I I I -- r I t It^Qr,ng A*, n\ z* feol, o^O\ 4,"1, rr,

g'x'''r- * Nn k|-"c- .r6.v -6.*oL\ .+

h*o :LG,,RG +D 5A*

ar\foe c^,\\ L :

-sr qt a,

Ne rrecd.

(xrunr\o,---f

'1'r -3f,'* \= o) *ilL f\o)=

\ ''d il't' : c"

''11\ ^--o)*.

,/i[^-r- , x -3; t ?- :- g .,b (.\ .) (,L-- ',) -,- ..d09, ,l-- atn 2*. "'i\'t*, L" "\f-Q^t^\ $^ +r

lF, --A jn d1'!* , L 3'a:= Ae*n j{E n?*z'iL. gG l,* .

$a9)*t .r) At g :- {- 0

3'(ol =O ,+=) A t ls =- o e

/Jt^^^ 0D

Llpt^n AfrA -- 9.oL - &l&

r*\'

Page 36: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

fu*"It.l^...1"- Q.\'n rjt = o t

^UUr)Al,*", X" 4z\t 6-o=) A_, 2**\;

go ltrr = & ( * (fr3r n r3s^3r\lU\

*1gI^^?k

-(:ri

1"+ ltq = o\\UA\J

4\^-.^ ,\'{ [6 .-O

3I 1wt- A.* (r t 'K** Qxu1

3ft

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&uqs

f\r 3U' t IJ=. s*):: \'

SL') = t)

'\^Jn- h.n*s i\cn

,= pp')\ t W" , - 1Ler,, '86 A,,*

,

fc,l, L JUyr.-t ^a[ *1\,J = o ,) Ae Y-'B*]-.=,3.

So. A= -Bn- 0r^A -EL+ iS,' | -) "8"* ila/

a^o\ A= ria g, [*= & t *r:b eo'

'!xi+.{d CSB v@ts

\n a,A-r,e- @, w [g..".,. )'* , ,\* co*(k*, ru^'A

Xt"f- = S-tft, {u^, -'h""p1"r*t So\n[ru^-i-E'

t \Lp) , Aft,*)"n Bn-tt-iw" u'-\^

A;g au+ald

t, * ^\b \.** -9t\'oL - uoi

"t**'= e*(ss *oz' t '' s'n "+ r).

*,, u\(?, Aa*-( eai ta:cti sin **) , R*Y cetu\x: - 1 s,u'..: *)' () "q{tA-\3)

cluoec n (A-$i sirr,u$ -l=

,,ih^. C' A-9 " -D, i (A K\/\t-. is 0K W^,*, c6 c^rx ^A g,n \,$:c

G*\ [4 Bcr r"c\

D

J[x"",rlL-

-TAdrn^ 1'ta

t\1, t

4.t, {=

6(e \\a,{r i"dlpa'vJ aral-..-.llr-_-t-.*'.''..'.-.

or.'4fl"" vqir-E{,^t qs!h^)

h* :- c) *.l,J^*' ljn, -:. {-o ,*/. 'i(g> =- (),

J' -.'[L^, ili* k- i:- o , ;; ] lffI ,

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S (:oa^A

irk) =

rt;"t,,^.) = j# --"flra^, -i*"o- tf*^\ a.,Lu\,n^ (r:

Ac" *t -\ B *in wt ."d. altt) * -, ItYy^rz*tt,&'r,

o- l[

)

So., ra$) ,= xo CcB *f,t ,

Ekar,p\zg-*t-

F*ilrn {b 1'.* Rrc cr.,*! LFer.€&), o'\\*'t -fu,..1,

. r#- - R* {r * *; o "e L> t'r'R"

r,,rh Qet= Qo c.^1. fl-p: 5 -fi.

']ii\ G[' 4trt . "'$-n0,.{, L.,t' * A\+ L-:- o

]

Page 39: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

Rryr{ fu*\ Ras-

l \". CL,4c- 6, r-{€. h*-* ,.\*r-t* (u,\ .*et, A,=,,.--,r'$r' ita ".lr^o" Eob[..*s .rrnt*- C*; Afe ,I

,\ fl N@:'^ a^r( [;: tsal* , A,g =**t

B,A, "h".se a(e- $9I t'.,*"\ ,-a.rg""rf*,^{l \*. 0*,

t t'' C. $ : ..l r.,.*.rq- l: ,= R : *c"*gL

.

*--\ . *f

l* &^\, *ye)*c- ow\c1 $,*ot g^S So[,,*ru,r, h = D€f\{U-sUnKai'\\.JF

\(e

) ,'.,tE^A

-+ lt,W;

+%M \o

"* fu."d

9U^* l': A.3 - a-vY s\urc $t.=A*-k .

,Rd

5h' Xe',e"-

h< r' *"4 o-,..a tr* N,- ,L

" ^l*,X,,O-- q \ So\$ln,r, s b[JS q Fo

lrQr\ .= $t:At.r,= teo

uM -

,M c\tp df^S'\-\ss

f"d a sz.q^A!\,.q*

1.,

) q s,.a- sr[r'b^ v l+ ,r.1, = hi)n lts f,:r .l rlr 1: fl(x) {t^ ,..\ ., | ^L ^-L1 -) | , - A LI'';'^t*, ).= t& "-'lf -fqy-/4 b clo{,,. r& . "A,W,

.,r -)-r,. '^ , -i, ^, ..L>< r -),* ,r -'LJ .!t b r ,I.t

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nr'*6i;AuaT* ,[. qr^^r^\ \iwarr s€c",,\d * n fura of r, ,

u {'- ?ud-l*\= o- // !

\\a^t do uAe $..), {, "^rf \..- 5o -\L.r\ f'. %Wll \) 0- 0\ "-

tI

S,Oix

;j

-- /r

Page 41: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

SUTAWS ,'*!D, Va- C[tr- *,e 0d,

d t.- oll *,4$ !: oY,

','r" tq'g'L** 4q0-b' + tt-q',;'*.. tlqe--?* rrt\t,/'dr, -.'t^c

rdu/ 'J' 'f- --d v

'dl^r^ , e,b I o, Se., 4 :: 0 .

-flrd* ( : tL; {:- C**Fo, (,,3-=oo*rrL

.Q, t'' L e

$ {r ''-*\ '-'L

fr = LCtt r$ L--" ;: C*dk 'o B CaL* '

ri lt 3d* o< J* , g L^)e- c.arn $& (L=fi $\,e-ce .4r-t,t^qf\,

'gi a.\rn .-dr^ o.rc..r.^tl C* , '|""\l^P* 'h^. qn- .n ^-,1)u uc ffi=;.A Tj-F':l mxF':

Ir.. qer**1--u' ,A t \ LIt- A*=A _.=-\, lL: t;.l ;rrn,_\ stt q.'

\c*) ,, ( frnG)a'\'rLu , A,K =-.:cn&{,u\xx !3.t li.*-i-^*?*9{," Jta&

--a

i ^*,.\u1 ru.O(L't,e'.^ote,"4

O, ,_l [*f<

t-

\te \[o',rshi"vr

4,* -CIr-t>^8 f,,w)

oA 1r@)\\* \bssh^a W[1,,$)/rVt\,,\r\'= dot

t ]; 1; )

{e*,t--l-O ',qi

Page 42: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

e!!+] 'tt'*

tQ \r/ t c fi* s,r^^r

'&*,, V(\,.\..\ : I,

g'+ P$'1l1= .Qb)

,

he '"i$ ss'k ftt grt^u'*g

bQL)= 3rr*> r fyw>n-J. r rr -n A\r n *1,

^ Iir^ .l t

^tr, t

)= q- /[rGr)

:

$'j "-

'l ,'

Eu*pt<

t.nl = l^,r*, frtr):

\*^, \y'().,t) = $tl -

I

) E=g. ^,,* ,\,.).rrt=& , t a&)"r-7Le.'-* :

1'*,vd(l,, bJ,J,\.!- ti l" .1" (;*&* """) - tu;l.- Gn)'j':: iLM + o,

_)t (-.

CPS .dx \

1l ,1.- = - t,osinzo;g - Lo cost..o{JU = *\"^: *o rl* -{"q.

ti

V.

33 j l-rlIrWrrr, tZeoqA-6tber . oDEs

t i\ r\sF rilw^ \'.' \t oDEs J *lue-

b,cl,= c*'* ''I'l' t'

rA4 \*o \qi^

uuLd€- ,,lU

\fqfl- -0"

'&*r

oc \11l ,= {l&,."Jt*.lil$- h;f.*}.

€,.\.^"r \Ufl *O

Page 43: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

--,n-\u& a5 1"-

|\,oot '{td

--t (:O

e**

{^* -,.-rd*r q1R, 'Aq

ts o[ "].-- 3al,a- "fu^ryl,to"^[

q€,ES *,( MU(l-" -$,* 6ts R'r

S"

t

sin h^ * co6k,.

s,nr\.r t'* Oshr/"

1ut"lL) '

Ae,*" w+ ,\, o,.r\--

As*\e* &qn,. i k * /" p.,. ,\ *

$*,"vr**rlorh")e** wr*'i h + A* .^r-\ *

(A.n- Bo^^r".. +F^- C

.A"n Bon.. -., .r-Fr 4nl*x irn@> ,

L)

...-O '

OA X,t3 c4.He- "t \re,

fb"^*\ W^[py*q,*&"'q

lh,o) lc\:'s o^xrlqr l"- 3tt t tX = -(L,

tfu- !,*^"1c,^** so\*\uo,^ ,.: {, 3fi t t11'Tt\: lr= t* )r' 3\t l'=

<"tJ; \-r j "1, ^^t ,\u(s=N"*> \cDS qe,*6'zld Var{aE {-,t

Page 44: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

t

oo

fro =l a

5.r='+

\it **s Tt5 BCu \" ttb,) If CI*/ ee {-" flr^to ,-dh-{

t, At\3" L '- \

ArIY r7.D

Page 45: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

Si"\r* olr .fn"q'-r\ds

Page 46: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

tso,O

C 5a e) + l--B+ 28+ t :: o "") K= -2- g A.=*!.

:b flLo> ; ie^ ?eu + r€ un,

&ytU fr., = K:: c,juaL , '-[h lpc.r'**b .-ci'.\r!.

??onL Gr.6,*tx . -in-r ,i*;.= Lo*tL,.*il\'*.I,*.--T..

\ r \Y

t"tln!,4,, ly lin}1u = -+C;z*) lp*,,*?.,. r $Gr- * 6).,eyd.r.d' u' u

+ 2C cusL*+ ) su^Lx

: (- ZC -6-il (*2.- + (6C * ?:S Erra?;-,-j

:-:. 3'tu,t i,l

tr, *2(- 6D t o c*A, 6C - 7u: = |

C, I c = -'3\) *d, 6 (-si,J) - ?-'b :- |

r-t C- 3"*

,> T) " ;l;

\rre)w Ya^

o:rrlt^lott %

"t :bhh"n, ^(o-

(q$rnanc{ . Aty,

t *t |,vcar "en"rd--

ovdtt Offi furt Inoru^a

fng , ,,Ol$o.,^ ,\,nr r\- o\(e Corqpl* . \i[" -ft^.s.

"R.ud", .q ctrr,r 5rw r'* \' Uarg .'*{ or

Cr^^W Ai'rn.'& 4^r"^(" ,dor,r1p,n)l ,

Page 47: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

s4,, *ro^ ..h^.- d€"\g .DQ,,\\ 6h/t

J7*,at-r[trlR n", ,/,1*.t k @ts" c^lW

/$,u

*,*| .'lb aqr*t\ €".T

& tir^^1b laer-rnL cg:,'l\C.Af 'lIt

aaz

\r r0o\= O\) ( tJ

^^ry-T$ sXS\t"^. '=x a^,,- ev^"r,,,t![.t': rvri.{ R;. s q).

{t^^u^\ s,.\*\.t^ 1" tt* he*^q4r gr,e €4^we"r,u ,'\UUTqG)- A.r*bl r ?*;^ ruot.,l-"Ir\Jr;r*\.ulL\t-)srvU.V

\r\: -0rr.. -h," s\e\rr*, ;W o^ "lt*.. oscillr.,t** ,

:i u *j t -' s\n u-*-UU

\\N\o^ \il=i'ioo Lr,ol ''tu\,,.ss ilot L**gV[*^ i,r,),:11)o r.$e- A"t GgStlS.

/L"

Nr*

c

'6r**--=) '

fet t)f'u$o . r'f,*rn..,*lc fuf {"- pa1*ro,'/b\r :i,Lu,[sg1

\1turt {as,,)\, r ^Lr;,Y**,. '

V-,t 1r'r-^ if, : - r,.\ (i ili,r-,,.J., * ,.^\ j ) C',,t .il,[:

1lad, {=p : * ,rf C r"t*\ - ".$LD

qq +L ,\ d,,r^'t, \yn uUi\l E -,")tCc,x,dt- "gtb S,,^,^l{:

t u ' J" + u);C col*t { uo}} Si^ *\: = Sin,Jt

"-'df c+t^lt c ** o + <1-" ttQ,, coe[t, &s urt :

Page 48: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

l\uw,

t o^Jr

3s): W,l*\f Sirrd,

) ,4Vnre{o.',

bQ) =

l-al A* = wt* vs'.

P,tso r \".){wa F

= 2*,.- -Ar$ ,

5u, 1G)= i+- - /'J Aw(*..'\

r'!'s (w- AS)q

lf r,r t s , tUqz, Ao 'r\ "*1 s,^-,1\

Ata *\b ?)U v c.s;lt.

gQ\= t|^kt * t6.tr - Ae*r{ * E in ,^rct * S+p!0 - U"- dr tlo'- *)'

-l-2-$.'-- ..--''-''4

{r} "- l^)'

.{\*^^ ,.,gt - ,S = (}..iC*.- g\dt\,r)o t wo-wr -" (*f- *)

Sin {4d\\z ).t

a^A 6;

//.,/ /

'/ .,{

"/r/./r/{/

-'(.\,,'l p

. - /ruru\ \ ' z'w"'

Lo\ \ =r- C) si'r \Ull"\^- j \ r)-.)

= Au,J(**i

) 4^, -trJj.+,^ of c4** \r4\* -#,$il -$ ,ffbA&

&rtlr) I

,a W.,.

rl lgf'l 1".,{aJU

Page 49: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

,'[-*rrfn,

I,TT

i""Tl

)ri1t\A fu L,rf:a,r \"dhs llh :

il{,ir',

--*->

of)K4ona*ce

Ar;*o, i--.

\ile t*r". ,,1ltl ,- --kZU rr$ -tr$o

at W -+ t<)or r.ile ry+ fesa\eL/\&. ,

6s1/.r* A.,\J si^dut'\-\\,. T.J' J \A J\\ , *5,,

I

k A" -o, *&* ryf -+ Cos ...,sot

tl-?- .",1t \*-*>' -*> T;-L.0tt"t o uJo

:,^(A,f r /- '\]Z1 ,l -b YL,A"r

9 qrj; -:7 #;r v Gsoa[

Page 50: tse23/MATH256-202-2018LectureNotes1.pdf · Created Date: 12/21/2017 1:50:37 PM

s\)e cAr\d. aLso

-{1*'fu\ luo-

R"rt -\t"

€"('r$'.

l-, ll , | \vtl\l .7- ,:;lP\- /( tl

a

C Ycos r^roL * T) t ir\,

= r;- t,(V\b

\ i *_\ afuwS\ (l\- V'

1. \,'' l\y iS fesa^at-q0 t.I

oDorV,n4

Ofi*, ^*\^^,..1 / outhv^\ 51s\,r^^4 Lo,* s.*r-

\\\,,Ofuxq

\\.,Qt*ns,,*o. \lr\\ \*h- ^\ f.,,*- *Th.&,o^Ob

eee-\<

-*: '*L'.}; !\{A'sTs e<\.r'"-.., [u ,ad tue- r&4'*rr(

T. #r'

.;' , 5-: L h ---o -r-/t-) 't'I'

l,n,-^lt-Fnl*l; ltcvtrt

dLA -€ *=r oa I

fu^ '+b*U,5 ".^.dt