‘____.__._._1.'

-QgéJ;_.‘

I

'.

1.:~

:-.‘i \

;,<

F.

5~1

~_,<,.~...,.._..A_..___.<?‘<4.

-__,____'¢_;.<.~.;-;~'

'..t-v1.¢-{'4-‘~.-?.:.-'.'1-c(_;=1,

.; 4

I

v

I W.»~>

Magg§§3gg,ot2 Ei><@m*S@ #=“1;‘1Merging W am’ ““*“‘ _ No \\/\2A1“M1\+<;Q

‘ 7 ' “ "" 1

_/u

5% P 'Sin¥guIé§rY Saiuiioii 1 V‘

.. .,,. -._~4

» H -( \ 'N4 u

;,;=“. .:-v 1:-V:-. 1-» '_:!;‘:g.¢:?74.Z Q ."?"?n:: ’

» -1‘ '" ~ -'-,.»'-.- . .

¥~3?.'.»~.?;;-'~?

‘\.~-V. -31-‘. 1-‘ ,-- =¢=~~-‘ . -Y

,1:-“.1,~

.1

. Z’

a=sa\;sa¢11-

i x ‘ Q Y

* - ‘we=z@sem*~a~\~§.~:~v , , 15- ‘Y? W 5?a . < -M ' :_;.L;.;-,_‘,2~. ;_- .{_. 5'5 3;

'-v~=j<»:w@~ 1 f'7i:‘?Y§'-‘.1 1 J Y ' l "

)'f;;g-_ E -)@_G\ r. . s

@vMs MP Q» »\I\ r’ ‘ ‘ - . 4 ‘Z '1 t

" In- ‘A A. ‘,. 4, _. ,,,_

u A <iiIf' eq- §(1,9,P) = 0 mag possess o solution; which1

4'2"

I‘Li1-

E-3

'4

docs not. involve om5 arbitrorg constant and, Ln gevfwerol

is mot obi".-oir1e0L _fron"1- ihe geerol 5olukior\ U5 9ivim9 ’°"H

I)(_‘}rf_|:CU‘|Qf- ’v¢|UQ 'Lo the O1'bi£rO_1‘§_] cons!-.0r'1k5 , is ca'|l‘e.c1. ‘_

iimgqlor Solutio,V‘ ' ' i ’

P-aiiscrimilganf -

J)iffer<:nuqL‘img_@ Pccrkiog w.,.t']>A, 5_¢ 9_1£(Y.9, P) -e F3 --‘~)-?--Q5 i

-~ ~.~- "‘ - 1.:>» . av

0 Q)C?"\'-Sizdr the orj-I~lir\<?_or ,dif§L-_- eq- 9} first Ofdfv f(x_,y,p} =. -_ _.-..-__

If we; elim"irwoLe' P‘ j-rorn @ and ®. {Hem ti-{Q »;;__|¢',»n;mo,-‘L ,-l._‘_ -

(1-e_suIun_g eq-) , £3 called 1>-; d1$¢,;mL,,m,,,t 5.0,: Qq. @' ____W_T .

' -

.1-I-'..4-.*.'.7-.. ' ‘- |

,v'.\L‘?;_-"{"¥1!f-;fF3'_.7“.-\“.‘{~§x§‘"_ ' V ‘

111

\ .

I

-~

8

|I

I

\

’ Remark% ’ ‘ % ‘

- “r 7Com * !*‘ _~ -.$LCi€r' (HE \1fWQO§Y' of flfj (1.,-(;l€1‘, I-(){_’LJ)!D7 2; (-~_

If ihe‘ eq- 6) is quaolrokig Lrw P i_-<1 is‘ of the shape i\'f2+B? +(_Z =bUirw ihé P-disc. is givgm by f]3Z_i;[\Q __-_O

I 4, ,

n%

Q miscrnunnamamfr'""" 5 "M ~-'——~——~ _--.-cf. ___1p-.:_.;- '

Lat the 9'9-“QFOI Solubiom o[»(D, IDQ ‘(P(‘)L’~l:;'C_\ 2

])if__f¢renL'@ PClT'£!:'()HLij W.r.L_ C‘ I-_r€ O d:)(x ‘

' ’ J ’ ; Z "“*‘_"“‘“" \. .' Dc

.

0 ~--~-»~-~ V» Q;

.1»

l

1

.¢‘

' ” ‘ “ -~ . we Qt the €_\£rr'u»"*-L"-'- -\ ’» ,

- Ii, we elamcnot (1 Pom O’ 'C"‘@- \ Sk 2 ' ,,=: .

'v

1':3-'1

-1...-,_-,_.

,,|

L.‘.-.

.¢'

-in-:-.1,

, 2

J

-'.' r

.w.

.-..._‘-.~1.

:‘;\»

,.:,.

2: '

J.<.-.-“-‘W.

¢~-r:-::i-5

..,,1

-4

vi

A

1

\

1 9 I

' ' " - both we c1i'.$crirT\ir'\ont5;U\O“- ~

I 153 ¢Qmmom porblf‘ (

i F1'

: | I ;

_/

i

_l - ‘

gerwerol sol “is 3 = cx—cz =37 19 = KP 3r LIX?I

% 11/ l >

-"r" ' 112, 'K; >%:;€;;'-»->. . | - ' ' ' ' ’~ f

Y€.'$ulL(F\9: ail‘) Coiled C"d'i5c'-‘CFC? . -_-1' .. f .

.!

"1 3> , ' -

r..l

€ Vemarii % ? ."'"'"' . ‘ . '1. ‘

. . - . _ - - ' _-_. . -|~,@_ shape AC -\-BC/* L- "* ’eq, @ ¢5 quQdrOlLC L1": C 1 " e‘ OJ t -

' > . " - I ~ -._-

Uwen U-13, 4;-Adi5¢._i5 given D3 B-LiF\(, ._ O‘

L

1 D¢:€ermin'ati;0n ""0?-*~'Sin*g;-"~S0§; " .~ -

I Y ! . 'H

5“ 'ose, we wOr\L to H <1 the §i~9=-*'°-L 50" °f °“ff'e‘*'.§"‘*”P*= ° ¢

11 . F 1 ~

6 Fimol ‘the.’9ér\_é,r‘O\ olukior‘ O§®

6 Find the. c-c>iLsc;ri»"i"‘Or‘t

Ft-fwd the -1>__dL5Cr|'.n'\1'OfWt

" ‘ sotcffcs the diff-'eq~@, '6 We--,>"m9~'°”, 5°'““°»”‘ °5@

T511619; drsc. ?IWé'€imd » A

. \ . -I n

1. ll “ \ ' iv-\"' ' '

We _caF§'¢"‘ob£oir\ the. singular 5O'\uEi0F\ Of {he dig‘ QC!‘ §(X-9*P\=°'-‘@;§

<>Lire.c:{.l‘:J }\'Of‘" eq-@', Q5 » ' u V Q A

1 1 1.-1.1. Q ' 1; 1-," ' ,g,mth0%‘Find P-<>u$¢r¢m"‘<>1"=* §?“@ “*1 POT °§ “Li *6" ‘O

5OLi5fie5 the cu“, e_q_@, is tfxe 5>in_gulor :sol- bag @ '

Example’ J % L V EX3mP5@L,%solve onot-‘find singular so|- 0} 50"/Q <>"d fmo‘ >'*'"‘9u"-" 55"‘ OF

2 .Z

P—>'.P+y = Q ---@ XP-—1'=1P+‘i‘»¢ =05oI=_V 9-__xP_P1 S0l=~ . il -_ -’ .

IL is C/(vD~L')’OUfL)5 €q- 5Q dis ‘ .251) = 'x'P+, “X ‘ w

‘ V: -I or C‘}_ C7‘_,rH :0 .'J)z'U(ir*(=_~ v'\Lz'QM:f‘\§’} \\'1‘ \ '1'.

~- . cw __ . 1P _ ~=Jlfjgular 50! 2 31 §_'_L(1_i1.p; z;(xP 6“ ,@ A;_

we fimok the §t.'f\9ulOv‘ 1501- 01(1), O5; . ' -

‘ . IY

'-wziia;~_'|l~L£-u-'ru-

F

--P~:

1‘?

I

Ma!§@%,‘§@v org %

'» | ' W ' ' n V

1 fw V '» 1‘ 1'1 " '

. . LnL . . -" i

5O|\/€; ogrwdg find igulor $011 o§ I @ ! ~ I ‘ '*‘1 @ 0 Q .,

’ (_x‘+1>.P - 2xsP -ac’ =

~_SO'|- " ‘iv

2x9? = (X3-|)P2~ xz

1 I; 2>c9P _= XZP

=:

'__.'..._-.__;..._"_.'_.»._.'

\

-1-1

IN

._"><

>1!-"

“Fvb‘

\"/

.5;

-'1.|.1

Ii-_.~_.;.

, .

______--._,..~._,

_____\,

.

vl:1

;_=.._-._'_

.1:.'\

_.

1

I

I (15 )

rI1

Il

1 Exampla: ' ‘_,L%__p_

O.

> .

'

ii V

‘ ,‘ ‘ {T|2U)z——'-\X-I-nL —-= o ' “ “L f’

..§ . 92.: 4 - I

V 5if\CQ' C—oLisc_- 0-“id P-—d~1'5¢.; {-1 .

, $_ornQ _' -I ‘

g ’ " ' '1' 'j2=l-IX is the Singular _S0[.AQ‘@-

‘

' l

U16. Sir‘-.3§.nl0r jQ(. c,_{ 4:

U v;D'i{f- W-r-E 1, é. get‘ u -~ I '3 l 3 3

GU P I -1 -z -2 -1 XPV "*1->51P~\‘(/‘ = O - ,—iT'>.. X‘p>-(Xp %L_[;_‘_Pd . S .5

.'_ - n

df — Q "-

gip -- .d_E 3+; dp _*-I 'P+ discriminant '

:7 2? = X d1~*P7'v'Z 011*

':>P+~[15-—£ '-_-_-, ( _.

! ‘ :

N

P1311 T5

Cl? '

1 ' .4 P-cli5c- o{ (D, is _9i~/eh as\ . -

~31 52.4/-\c-=0 .-’ .

| *2 ' 7 9 l H 1 1 3 3 '.;..:>»P(l.+_|'51 /.)%‘).__:_x<\'_;_L_i,/+ %13_%LE=O :3? 7., ‘j .-.LiO_"7L O

.:-__7' 3 '1. 3_- ,_ ¢ ~ " X (x-§;|—L|a) .=- 0

Z” %('*""é*“i= (‘P-X 9‘~"- ~ 0 %)_i~oax3' !_,'3' ' 1"»_ ‘- ’ T =0 Or 131'-Llcn ~-_-. ¢

_ ‘. - \ |

TComsioL€r , ' H‘

-if |-1---—-.,_-———?_=O 0r.P-7(_d__.:E.._=o‘P omV . ":7 TC "0 or 70:12.1 1]

?'\"' --- -|( .-:_ (:._

% 'P—X9£;£='o_ .1~Ql;e 1:.)

_ ‘<>m"€

=7’ |r\P

1’?A P ZQX

- -lr\>L'+lrwc -

I

I ;> >

=‘|i! ‘ I? !.__‘p Sincé X=° ‘o ‘

-0]‘

We} fir wriké‘. Q), O5

- 7(-3—\~ '-' : .__;_._ ®

"U.-

Jr.

Q“f-\

11'\/N

PU{h.7"\9 (YT C0, we gei . ' 5 eq ".

Z 2 2 I '4 X? -1- ;>Z\~_] - ~'»- -\ 63(-@ P, ~l Cl —"ZC.X.‘;] ._.)L2 2 Q-—~—

‘VF’

\_/

_!~I

(Tat? gérjer-0| 501-0? (Qv T: O -1- O + o -+ 0¢ Q -;R.p_1-5. of eqq . __(

. ' ’

I ‘ } )0»|

‘ x

I

I = O i5- @- Jirgulorr 501- @§@

' ,,.i

_L-_-,__7,‘;-,.—_.,..—_-..__-‘-=7_'sr-‘=1-.-Lu-<.'*"

_____._,__,,...__....-..~._--v.-_.-'___.----\--4--*‘

.v -

:;--nae',v||,-.‘-n.Iu'v:--In\

~:, ,

I

K =

>.; ___7 .' or .'_-.. O ‘ 1

; P *1 § x (

I

R1.

-/aw ,4]. V ‘

.‘

1.,it 1‘r

X :"' ‘

-‘.- ;' ‘

Q ‘

1* .1» , ‘

:11-'=‘-1%;-,-.'2wr?

._\.. -.~.‘.-‘-.' ~-.:*‘.-' --A.»

\?7'.';-Q.-

.t "I._{;;-)2; . _

'1 ‘\-' V

..-,-¢. .

..

-. - - 1~,~ 13$3-£23‘. I . ‘~

$15-j;*:~ 2.." 7, 1 * Oil

2, P;" < ‘ '

,~,,_. .-,!_.\ 1». -.#:. ‘ . ' ' ~ 7L _

*5gr -/-5£rc'e. ';c.-@4¢s¢- Od P“ d‘5°' -7 v.

‘W0 -* ~¥{ 1 ~ 1 ' |

33 i‘ ‘ré: T k '1 -:;¥..-11»-. .

R0 ’-.

>;';§%:~;-1,-,- ;»;

=.-1!>'~.-4,.‘

)4 ‘_\ .. ,.<4.‘,',-r;.., I

>,. ,'_.;}.'

‘ -’~7~-. _:

!g;.; -2 . {;.>52-M

5@;._ , % ~ we {ml We sw-sU¥@'" ~\.r_»,,*§1;=ae<1> ' ' -

1;}:-‘.-" ‘V

U4.-'~;» -. M1»

1-: I‘ I

.! .r

r~

\,,- _

..,., ,~:.'-7.‘; 5.‘

>4 -

¢ w§:.~.-”.v 4-.‘

‘

.-__.;.: -I

—.. ,.

V

->

JEP. III-~1' 1.

1ea:.‘.§i‘.~:' '!54% 1 If-*"f»’¥";' 1»

Tic. .-f .v

» * V 7, % B“ QM = O. ‘ Q19 “

dgql».-;_r.) _.

3%

.'/Vif‘,_‘_<_>'§’_l-5__Y

‘:i.',-:-,-,,<I_L.''_

" V ’ ‘? * "X d1 + P’ dx ' P

-.7!“ I H" i ' 4 b I‘ I-| I P~* ' I ~". . .1 olEU 3 , fl? \_ § F , 7 P P 1 \ 1 .5 ):1 P 3 \ 4 ~‘>34~ 1 P '

§ :7 J I H ‘B O I l

Lvgg

'.~'..-._.~.§~.

..

~¢,r_-" ' ‘

if | ‘I > .1‘-v . .,. ‘

kv ~

. »

1'~i,'\.-L..-

~

n-...;_;_.\.'..-

lr*.*1:.:~1,,»,:§» - ='7_ (-70-‘W 7' O

$7" —— ‘, I '

s

A ' ‘[3 ‘

i

j M -_.% j ~"~"'"~"~11¢; *LLLLL

Q_\rn1mor'\t 2P - V '"1. .4 '

.

' =>1> \-- -w--w- =0 J

=>% * K-1¢_% =0, s . ‘ % Q“ A

::7',.- _L2.‘=_ qg . 3?. -.: O

P_oLisc.riminor\*1 - :7 ‘g_ ;o of ._X <1? =0‘ K . -~‘_p?-_' ~P_a1

"»v;‘:._ 3

II AP-’olisc- for @, "5 9We’~" bi Qomscler 7 l M

O V ’ Va‘ = O

Ore. 5an'>»'e_-"J ~ ' ""- ~ ’

" V

~~ % " “ V => WP = '""**“‘F. '

_ ‘. . (0 I _. , _

§ Y xL_.'q"d;T 5'T9TQr5o.\f{! ‘ => ' P=cx '

;r_ by; nnokin the 5°‘b5k‘L'*‘l“°n5

»_ P — u dv -v oluaft :

- putting in @, we 99-.\Example ?

% 2 - * . »

1 1.C)|LZ___'Z%-C,X-F 4-UL = O_2 .

Solve. )cP +3P('12(+5§ -r 9'=°_'%@ V ‘ 2 1 ,-1» c7L—'z‘jC‘+Ll = 0"" ‘*1’\

) .

9

- 8 A (9.@»f\er-oi b'O\- 0‘

H _’ ' Sirwgulor joh- Qnd imd the 5ir\9ulOY‘.5°\'

\¢:;»y':=;'.} ' ' I ' ‘;1 J~/ere. H = u , I "—f 5%; (',—d~|Lsc.ri nqirwont

xgp Q: ._ d du . C__O(Yi5C_ o'f® £5 9‘-vem 05dg=;dcL~ d =3-‘1:L‘§_-_'—* r -,,_, A

‘ _____.__ udu ' 2 z

- ~<>w1 di ~- => €7-‘d)—'-DL-H =<>%

<$1

we

N.

N

-uolv-—

.__ Qdu :7 H =1-{X. U

._/ __ _€______.

P-- oi£:>c.rm’n'»’W<Jt

, ,//1*‘ F155 eq-Q7 bec.0n'\@.§, O5 V

. N .

‘ET\1":1:' ~ "~

,~_'v")v.'-' .

s;.._-5. ~

»I ‘ .3):," K33:’

rM

A»

31

3 | *

¥ 1

7. . “ __U LL Uqbk X O _ -— [.1 AL _

\2+d"-vd*>+’8 <‘*°“""dMJr '§'

* .

(‘N ‘ 5

P» disc of (D, is gm/¢>_»f=. 0-J-v:‘{,~ . .‘. ‘ ‘ ‘L

$5?‘ 2/ d 2 L P11 O

' ,

_-._

. 4'»'1"- 1 \

- xx .l.I§- 1 | - . . .1- :. >-- »- . -'1~' , -' "xii '11’. - 71‘-T 1'51’

, 1" .- . - ,1. . . 1. .4 -- ~ _ . - ' » ‘ ' . -1 .' ' . V . ‘ _ ' -_-I" " ' " - . "-'.""'\""'<,". 7}’-':‘.1;:j5, - -. _ -'- ' '- =. ' '. ~,'_'-5 ‘V - " ~ _ 3 ~ f § - »’- - I ’ ' _--_~._~__, . ,;. _-".'-._':~’\ h-=,'_=',-:5,'.E!‘ _ ,‘ ' 1': , . : 4- _ . "‘§' 1; -_ ' 4 — . ' ' 1 ' ',’..; ' .’ 5 I 5, . 'U'I' 7‘ 1 5‘.',,j.':_' _ ‘V

"_ '- I 1'-' ‘ "?',1‘~T '."I|;\» ‘ .> ; U 1 '. - , -' of -~ . .',‘ -'. ‘ _.,§‘-‘ ,1!" '- ..-.'I-4'n‘ ‘, , _ -_. 0 J ' / <\ .1, 5

.64.

1 *3 ,

' '> ' 2 L 1 o ‘ / J "

1;‘ I ‘ : R O O ¥l é

1; '

--‘.1.-":_“-x".'___i<;_:<_._:':...-.-

n

"4;;.;.~.;~..»-;....;:5»;,,.‘‘§-av?-‘

F

1

1!, ' ' ¥

I ‘L .. |9l-- 5.7-‘.: ‘2.; | . ,_ .

1‘ t K.

1 , . .

.I

y .

I

x} .I " F

1! I

~: ,I

t I - \-»$ I x" "

\

. ' . . . ‘Ii;"'1 ;‘f_j;,,f,;

~ F‘ |, .,~ \.'. . _

. ' '¥

-.., _ _~ . . ' -

51-T'1§'~1,lOr.-..§OL"1-_' -§|'.*.-‘_3",-._.:“I§§','7~_-~v ._ __ '- , _-, . . }-" '-

""c:- dis’¢ramvm_<>'r;t ; - ' V __-bi

'< C._—"d,i5c-’.0[ ®,A"£-5’ given Q3 .4

Ag ._ _

.

?<p.|'

/\-‘

=7!" a’5+>5.—i\»-4. % ML" ,1:.+&1;'=-* - -

* 54 ~ \ - 5

O I I I I ' v ' .'1%-2'1g j? A %Hf%w@ I H,,wW@

\ I ' 1. . ~

X 3 I I _ | J I 1 * M5

- P-Ad.isci-._oj@('z5',9ive.n b5 3, .' ~ - ' 3 3‘=*~-‘| Y

; ' ’ 2.~;_ - ‘ r_ ., =~_6qc_1._ mac?» sac» »

-

. _ .;»-~-~ ._.f<‘t,§"_-.1}Y»/a ‘ .__.____-_ __,____ _,,,_ __.. -7, T,_ _ _ w ..., . -=_—.,.....-.~.».,... ‘ ' '1 -.x--...w~$

.1 . - . '. > "- ~ *g.IE._: - . 11 ~.\. ‘ | I . f V _._‘ ---- '- 1 ' fum-

- 1 ~ 1 ’ I '

'3, .' y_ac;— q_o= = * - . - .- -.-.a@,».»,.-;->

i 5 f ~ . >_ 9 ~ "I ® 5;= I. ":7 X’ .-.= - qQ_97_‘_. . .‘_

~_‘ -. », ._ IF.

. - » ‘ - ,=>~-.:t7;r';.;-.

- »r¢ H; we 9éF~§ ' ~ ,j$%Ei§- * , .. .s.‘!|- ,.~.!§§

. 3 -3 V <- "i-=<-£13’: - - 80 5' ~

x l%<=‘~~.:-;:'>r-=-* . ‘ >‘ ,. ~| -. Q‘;

. Y _- - . ;_.-j. 2 1-;;<‘.;»i~~='é13- . B2-~4A_c~=_'o',, . H _ »§L ‘r3 -‘=7 ---l- = - — .3-91‘ =

*-‘M-x‘-3 =-‘ "o V V * F ' P ' -'~7‘5". 2, J ';\ .

. , . - ~ . “-..-'j'.~'-"3 Ii‘;.'.§~‘.','.;!

' H ' ' ‘,2 ftE"~‘:?;'-'I§;'I ' . -.»--

" , _ :-.; ii, , V ‘ 7' ‘Hr *

I ' - I ‘Q3 " '2 ii V .. . . . -_~_-~-.-+ X SJ ‘F "' O‘ ('P' .. _ ,> _ ‘ -, .1, --;~d!.'i.R;H-'1>f§‘<§£,

__ ._ X

V .. ., L ,. ~_:'_._A_=;:_~»v 7..

. N4? 1 . __-__—-_ . . ._\-~;J,--_,:.J-'; :‘£~- -_ \ ",-. . ,4

' ,...,\f‘¢. , .‘. - 15 . A° . ,. ~ " H" 3‘ - ~ ' s‘=- "

' ' “ ' -' -" »" ; ' , - ' , ' " " J ’ > ' I-':>\.4» f' 3&2?“

1 z-1‘1£a>;L|?%(>¢1+\1¢;§»a ;-_ 1.<§§:2;j.'1?1<*-~>6<%<>’v» ‘ '

';' H'*'”7L ‘ A’ Q Q A’

7(.'+\_-'3 ._-A ’.. Ir-I -V; -H

@- I------- »- ,-' . - ' ._-=.:_=-¢=-!.-5~ -2'

» » ;'§~_‘~:‘.-';‘-F’ \;. .,{ a<

‘ _-1 -.3.-‘.-:55:-‘(B .

-_ ¢.:'- \;.=.i_..=,,.

.-5. J eq; 2 .

5_£r'\¢e. '0-_-"d/~_is¢.. "g/jd. 1'?-’<_>u5¢..' U . i-e_ e'q__ @~ ‘.5 A5b‘t‘_s"f1;_'ed 4_. > -

. 3 - ._ ~ ; - - , . -. » _ '\j_;_:j_,.*-:L~"1#",

- ~ bk! 2-Jzvc 3 o " ' **

I

11414.1 I k‘|--nm~¢"‘s¢,»2~.&a - 1 i ~* L}

*’_*1_+H.‘=|»_'1 is Qin mau . ' ' '

, ‘ I , =; W

2-! ' ‘

§ 1‘ L’;

iI‘ll V I

ll 4

A ‘

r

I. .

r ,I

n

5 !

1I 1' Vv u ‘:1.

»_,._ » ~_'_-2:’.- . . ;.;-,,.--;_~.-

-~@= ; , : .:;¢uv@$§#.. _-

V9 r_5Q\ 05¢) 41 "_._ ‘U1_L_-_qa:_'"_ O is» -

~ _ - .-5¢",“9_“-‘-'°’. 5-°L"31?-‘ .\ ( .;‘1.-‘--W4‘

\1| 1 -| % 1 ~| - ~-I - ; =~ ,=

1 - .’;~.‘-74 2~:.: "»..'

. ;.-Z I; »» '3'.‘-._-fr

. " '. E?;"1-. ‘

. ;_r"- ",‘.l'e". .._,;_

. 1;.‘-,.-"vs. "' ' , V-‘J5:

v.34.7.K*?».»-.-;.w

;;~u~»“ >>;_,.'/:;§§

L .’‘Xi _; . _ . ,_S

‘I ' ‘ \ ; F

'2 ~.,:- ’. .-' -‘. . Q,

.i __

. .~‘. ‘ "\ . _¢___. , »

1!

:-, 11 I

0

‘:-7"-,-:-\--_3.-_;.=\_=;=_-2.

.%‘

v»:.

I Avallable at % ‘ ;

~1‘ ~

'. ‘- = _ ?www.mathc|'£Y°|'9 " r I/__+-<

,r_ r. . , --

J;$1

:¥;-°§».i.',-*.-.-_‘ Y - .7 -

.s~4-

.

Malm;ily.org i //5">1: M in Manand maths = ".7 w . 1

‘?:"£;§§4&=. @:=.§I15;vv. ' P - ' ' _ - “'6 - ~' §m¥“\“'?-~ , é Q ~ ' = ' , ' »:§':';r kn. \¥‘1r"/-==d-."~i'* ‘ '

’ ' Q» 1. . .

ii 1lv§r< - 1 ‘ I-

<;§\':' .3»-~.-,Y~-

~,>..

..<17i“

H11.’

4\.'.":9" .&¢

,

% [L I

N: -("'xv . .1 £-.1 U

,. ‘ 1

4 4‘ ‘

~ 5,:. _.,

, #:-,-_ K"

1 ;. ,,.

W r‘ 'gt .

‘:i!1{$§A@

.-V

»_ {,1 . . \1i:r- 1;’! ,

‘ 4

2

~‘,

‘ -1,. , ~ '_ .. . .-, ,;

.5\4__

.1.

~¢

.._.',' ’..__,

,3».

». . » _‘ , » ~ '

owe Ond _l"~‘L4_‘5[f\9ulQf ~ \ Solve. amok §ir'x_d 5lmrJ.>o\ <3

% L ~ V ¢Y= 1 1L‘O O "8-é=,_ P _,_ ‘_[ j 1>(_)(-_L1§ --1'PX.‘:|-r§;|-b -_-.v»‘ ._'-

D '- H: ‘I15 Cloiirouk-5 QC‘.’ ' S02. 1.’ - V l‘ 2 1

’ ” . V @.,PvLz~<>P ",1,P1%1+%4"\?.=°, 50 its vggnrol 5pl- 1:>_. , I

.

_ . n '“ ’ Li; L5 ngt $olvob\e §-OF 73* E5

IJ

5 = ¢1+¢»_. %@% » j V - "

~ T ~ 1 1 “ M 1 = 1; * _ ,Sigulcwr SO|, ‘ P-X __2P3gLj -1- E] -»?O~ b - O

‘1 Sin ulor 50!. O§ (9)65 " ’- -~ 1 1 . 1 .- 1 " '*

J».-‘

aid 1“/@_ {ud - 3 4 =7 Px__7_P1_\5+‘;j _-— OP+\:>_ ~

'\‘1}"1'rY;';'_:15l - ;H . Y » - - , - '

;;.~;- ( ' . . ~ - _-_=~ 1- 55) --. c1_P -r b -*~CZ-—cll5crir~niir\oi,_ - ‘ ’ (P .- '

.14

44. %P.}>e-' \- -. >. . - . ' , : I . " .

..j_;; .~ pk tiring valug,-_ of Q weysétv ;- 50 if; '3€f\e’f"Ol 59!. 15,

<=*;§s;,’: =V ~ . I -.1 -- - '

'“f"-.‘. '4 1 1».

n4‘ .

1.1 ;_.\,.‘,

, .

> _

Li

;D@1r';@r=@*£w"s“-~-*°»We9° %”L 1

gr-~'-..’ .

1.\_‘~, ..2' 11%.‘.

‘EV?

'1!“/I-~ '2.

1- ..,\-.>: - ~

-$59-=7:;<4..

§::;§..J -

. |;-1.>=. .- .‘L . .

\1»* ‘;ii(l»:~;’f"é.;»‘?'P—§0EiSC.rzirS;\'*r\ontvV I t - l 7- ‘D ‘ ' I C. .1 o.I.cZ_‘_b7~

§;;;~=,=¢i;"_ Y=->1 3 +c‘>_L--zcxy -_-_ ac:-b

' » ' . '1-\ - » " -A V - " ‘ '

- . 51.1

I I

\

,

'1.

.'75*‘

_;-:;:;-_-q1:I;;§-:-=

\'\

\

/"\

-

V ~ L 2:; ‘PX- .-:.i/2}‘-P'\b

r’ . .-,;,'~;rI_»__., ‘.~ | - ‘ ., _ ‘ __ - ' :1 ‘ 1

[= WO ’= Imi L I ‘"’» 2+ =!»%-RF 1 /‘:""’*‘=.* ,_ 4, _' '- - - » - V .

-|;:\“.¢.-.‘=.'.'J'- . ' < ‘ ~ ~" -

~ I » ' ’"" 7 ’ , ’ - 1 . - - ' ..1 T”? ° * <-‘M - .1 ~% 1* I“ <>’<%*r°“*" 13°-v ~

J ,

_§

G-\-

(Tu

n- ==~C)LdZlC)-1', '~ I l ! 1+‘?

1' H,» '1' ' . S. '

. *4 (-7-O X + ('6 - - " -jMn‘

‘ “J ' V . =5. L-1-C1 £5/ci*c’+B ‘

'_@ °'\*H°"H N-r-L we. Sek 2 2 1 1 1 1“ A

‘wgxwr1.¢.:1»l-{¢_?..‘17y\_-'_»-¢~‘s:??'.*/‘¢~.'r%~.~

~-L,¢_J:4».:.-._

—.....'2.:;=.':._.--.=

\

c_'I_,

~

,-\

_' ' =X+rTp 11

‘:\’~%2_.,.‘.:’:bl V ._ —_-v -P = __lL‘ . , . ,_

; ’ -' , ;‘ L1 H) ' E => gxz-c1-2 C -_~2x~_;jc 1-(UL L§ ) "—' U ——-_--43,)

?i*;'§'§'i-%.- ' ' . ‘ V

gr‘: ‘|:!k P 9 v ’4 H I, '\‘ , .,~ '_

,1? P. 5

A "ue °¥P in @ -we 991' ' 5i"‘\9u|of Sol-» ’ V V

K. ' . » -‘ _.,_~, - _. ._ -

1

A- ’"v‘2-72'.-':I_t:

41

in1:; - --

1 51 |

»

-'-., ..

3544,.-.~:;?.Yr: '

' ‘,:?I4”.

.-.1. 3 Z

Lf,‘ I, " '1}IQ.

ii.» I];-‘I,,4/" _ ' I

;[r\¢_e_ ¢.T;T(L5¢! Tnd P_di5c . Q-Qlxscrzn/~urwC>&rt.I \ I ,

! H 1

"'F‘T-'\ 2

Q-L) / B-4/\C=re? sen»/>e% _'

% ¢—<>“S<=< Gf P-Q/<13‘ is swam ha~ ’ ’ -1- 1- ' »

'/\

VJ"\./3-

>1-

MD

N

N

~U0

N,

__ _.

4 ~ =2 (—2><a)¢q (>L~a)\&1--D3 = Q, V15 a Szrwgulav’ 50!» o§ @ ‘ '1 I 1/

‘:2. . =‘.?.:] ’ LDQH __ L!\(~,(|.Jl_b)(-—CJ\\_j-r O *3) ‘Z OW 1

I

|

Q’ A

.1 ,_;_. __,, - , , _ r ' ‘.,_... ' —""'_'—_ .

I ~ - > ~ A -» . 1 ’~. ‘ *\

\-.'¢»q\~~wn-an-_..1

|

\

I

2.1. u‘. '- = O 2 “ 3' -L ' ‘

7 ‘ AI; 4.. - =* c>¢‘—<1¢ ~1-c><-‘J +5 ~b = Q»

* ’

. n—l V . . . ,4-:1-~~..... .' -. -- '

2

. V °‘ "'\ » \ Y

‘ " ~ ~ =~ %M +‘ ~L\=V we. .»~.Qw ;1.~.d the S-$o!- ac) M, '

<.‘‘I._-

'.. 'I.'_-—-

1' 'X'-"w‘

N‘L__

1.-‘ .

r

1

LC

I

.n-4.. . .

NID

v""1 é\

1-‘\

3‘-

VJ ;r-

;-'-1'.='-‘=-,.:'.

.v

-..-..".<,

'2

._..'.'..':..‘4.'>3;=;'_‘_-.-4:-_-|....-.1

1

-x

xl...

.~;

--.2

- 1

.-':\

i.<..'.‘.‘,'bvj_-;_-=_;;.;._

_..

._:

7

I

Y

X

=> (X—s)’- L4 1 cl-'-an = O 1 » -:15

. u = '

D \ +f,(XA-

,1 /

-»-_-- L . .... ._ .( _ - Zféwl. L<4_'

LC.

\/--_-—

'-a_-q______._

\-5

. .4 ..__.._._.-_-§',_

! ' . 1

' E ‘

A » . V . — 1. + + c>~1~: ¢,(._\x~._-< “._\ —- - » .

J O 1' ¢- 1 I L 1 , 1%Y ‘Job;

‘ I" _ ‘ii |'l>¥ -1- iii!‘ =- b ' -

+1?-—JP | |-_ ,1 -'_-.._‘_°__'Q”‘§':‘L__

.1.\.

‘IL _£5' »"joL.$oI\'/oble for x,g,1f ‘P-FJ-L-LSQ‘. cf eq_ @ is 9‘-wed bi]

SQ, W3 Wfig if, Q3, ,_: O

-7 Q; -.2)L _ x'._. Q. 8 - b ) '= O')(P-('P"|'|§_H(’P'f'|3-1-|='Q .__ ( 8)"; q(;_ z)( 1, -1

=7 XP ‘"v':’=-;‘."!'?""'* _L" ='o - -. 2 1 I 2.1 1 1‘ 1 i~;

:7’ W E1 + g =7‘ k§)Lz-régz-I. Ii 1'5 c_/O1.'r.'0iué% e_q.' 50 ‘I-{:5 -> 2 2 ' 1 2 '

gerwc-:-'1-Q'_| 501- L5 ‘_ Cx_,‘__.l__

P“ ‘+7’-~ qw-q(><_&1-b1~Q%1+

>§t‘ I

9~CTN

' . "' b)L-1-OE}:

Q~0;0»-

0'...

‘J

H _ I ‘ C—r| _>l'r"\C E. " C — Ol l'5C‘ Or 91- P- <I1(ij(_‘

I ' Y ' a . ' . ' - 'I | __- =7 v'j(C.+O =' c(c+|3x+\ »'

9w

_5Qf‘|’TQ}__ .

UU

Uw

KN

N

1-I

I“

~

=7 C':l-r-3 =-cz>L+cx+l_ -. 'I'O~\_-j = Ob I-S lhe. 2 ~ ~. » A

=> _cxA+,<_:x_- cg+|-3 =0 9 Sirgulonr so:- o[ eq-(1)._=» c2'>L .-'+ cil-ED -rL\—&i) _=‘o .7

' - b ‘

. ____®Sigulou-501;- 4' - .~ ' LIP =

. . 1 1 ;We. fu’WOLA .$zr19u.IO:-V50! o_.f (D, 05 , ipz iv

..._/Hc .' 'l'_ 9

- ’. . , $Oi:-

!__ - 1scnV'm-_nOn.t_ '1 _1>ifj.,'g w.,},t 9 We eut, 9

=v 1‘ E A => 90¢ =

"-4 ' ~-=.i - =W 8. ‘C1-‘P Q

= - P C-)-P __,

1+5 11%-ux+q13 =0 . _. H 6PoLP .

--.1 1 1. =7 X+'i+21E\ —LnCA= o => §aP“OLP = jqow

:7 (_1+E§§1-LIX = o‘ H‘ :7 8/3?: =" ~

3P- C(£5Crz'rY1zff\OrL :9 ;28_!'P —; 3'-y C79

P——CL£3'C~ of @, £5 Qive by :7 -.-1 L;-“C2 ~ \/

B— 4:/\C = O =7 \ (U _. Q13

:7 -. 2 -L : . " .‘i 1 (X ';j)—‘-iX’(l H) O , 2-‘, |v,:._L‘ _=g_&U_‘, L53‘

I1

. 1 '

. ,

!

2?-“P=::.."F-P,

=0

O

C—<>‘i5<:-» of ® Ls, gzvah by '9:-3Lg'= 1‘;--21> §I " I ' H1|I I 1

B2 ‘Ac! P ‘ 1 ' 1 I ,

Avaiiqbleétathy.Qrgcitwww.m

':z'C‘"!‘“"7..F‘ww-

:1:-.

;sE

I

E

I 1rl

I

5

n

I r

1

1

r

151» ,

,i,-.'_"»1 .

.. ,,--.-

A ‘.. .

2:v.-.\

_ ,1“.-'~,5 , ,-_ -

. 4” »

= E ' 1 l ' | ‘L ‘ 1 1'7 §j1-|- Q1 + '2 Q *5] -.= 13 11

I ‘ _ 1 1ri \ . I i Q)

$23

-.L'-, V.

@“?;;"-';\,.

..'.: ,.T1 \

.

1 La x PL -'=~ (5 x - \)L ___ @'q"!"»‘V,,'>' ,' 3 ' ' I.$1; 'I’!

"'-if % ' L‘|_,::!_‘.!_;iT_1‘._-EV . E

>

. #1

,5 I :1 ;='i; 5?. 5- ' 2 Z I ' ‘ '1 ' .4; » 71> -ZXH -141-P »'= O 1

>

4,~,;‘.

. - ‘/5»=.>;~1;;".,_-

-V ‘ I‘

. ._ 5i"\gu.lOr 501- :7 (“J*'c) == X1' " ' 1

%@..'.\é!§§i'-L'5’39£@55w9~;,'

''.=_.‘_:_'f_'

'""''’ g'v.<:

N

.-.-._r>

-‘Z

r-—\

\s=N

,0DuJ

Q.-d

H

F ’>

'*'Z§':- 2.’ 1 1 '€x{g,.\;:

1

~

L

'.7§‘*".‘ '“"' ""' '

//,2.

.._,. , .. 71- . . 7-0-=.. ‘ '-|»._ ;}-w.-1;-,:- ;- . ‘ I

Q55

"l1;.';‘.‘{'.

-.~»-.‘-

~,-'____‘_-_-_=‘»1_‘_‘-',_'__-'v~§:..~¢:.-'.'i\..-n’\";i'..\l-\

"‘-\"M't;.J-J

=v' x2+31+2xu - L11 i-.=‘> (wag).-qx=Q

Puurlhfj ffw @ , wa geL _

(51+) _.= 91

’ :. .2, =_4 Sihce c'-disc-' end I’=- ou'¢¢. ‘ 7 L‘ q L3*c{" 9*

H?’ are §'€5’9{'>"Y‘*_F--I

- H1? 5=(:nc'—13

/’-\-/ .S0|$— I

.,»>»:*~i I

--

‘°F@ "5 ;L7L+3)—-1-UL:-_Q =7 ( 7" 3‘I’-\-C) :21 ‘

"13 + 23¢ .\.5_13 =0 -_ .... .--'

Smsuror SO\- ~i’ WE rw OW fimi U\Q Sirgulfxr" §o| us

P‘ W- 1°-O"-'w"~~O~*= V

1‘

u" ‘nu

u;zm.?»*~ ‘ » "’ '.’ '33,: -_

. . J PL L ‘ d‘ ',

' = /l...__.l5:"‘ C“ ‘5¢' °f _@,'-5 give b3U ‘~ .

P . ’= 1 ----‘"=lf;" '13:-AA; = 0<1- -- " - ‘J x I \ 2 '1

rf; .‘r<-- 'A G

j§1:i§.'.l

““ K-‘a1-I

55 I - ' i ‘ -v:_ I‘. -1-—)~=.r;1». .- -

:

’ => I3. = '5 _.I3/a.'#ILC11-J /5-*6") Z1 .. A .

<

'1;|{3{;;‘__“.'71

. .‘\ 1-5'?’-)"~5, :1 .Q ‘r;-‘X!-‘"_L'_:

, '1~\"§-‘;1":5.'.' 3]I;,:'.;\"_‘.g,5";1:.€.=;.‘,. Q‘-:2 ; * I»:

*-’/l$='1?-.~.‘

_-;_.'_‘..'..:_....'.._'..2_-'_.'_='_..'-'..-=..=.-.........

:_ ""‘ """";] '-'--— -Mix -—)111 1-f__ P_J§ _ (7-EJ)"‘-i(U.—-X3.) =OA

djv =1’-["7;“""-'/>.‘i“_\d-7L l :9 uyji-iL:(__%J'*—x?')_=o ._

d'_J : — _‘/> 72,1] '/~ . [=7_ d. 30 ‘

V ‘ 3 \ I ' P?” oLLS<l1’i-'Y\iv'\or'\L».-' ~’ ' \_ - /L IL - .- *‘“_ " - .

i =7 t =_ X — 7L i'('. : | .

?T}5:§;=}-I:v§jA‘Z;. \ 3: " ' , ' P"d‘>'<'~' °[ @. as givem \;'.-‘-JF>‘r=;‘..=‘;F'-i. " ' 3 . I’ 5/1. /.1]

,¥ :4; A= 1 _ 1 - ,4 — Q 1

‘ %A ~ 15--I: Ac = Qn -,r'£-fk, _. ' '

P ..Y;-£37;-‘.‘

- .'.‘¢-.,,».- ‘xi-_-V

._ PM ._

W". 525 ’

E’$‘-~~’;-.-Z'-

'

~. 1‘ n ; 2.

- ,;§;l 1 U +c +13: xu‘-~ W .

‘ \1i'l’;' ‘ 1' -1 L. -- ..;‘.

‘4 ..r

_

=:~2 .

~~\:.'~> =_,=,=._» _

.-_‘,. V

~51 -‘.,

\

!53’¢"Y,-.:!‘-4:2-vj -_ I1"

F5?‘,1 4 1

R1,; 4-:_:|,. -.-

~,!

_i.

;__~

,~

FL

¢'g3.";:.7.'-'€_- 1 . » . ‘ _ . " "Y Jt \;u wvuacl. my Q, 7_ 1 y

,1‘ . ‘ I P~r2Ty_-_L{)(_U :0-~_ CLT GU I(J¥'“»Z ‘Hf: new [

V § ......___ _ " _5U!:~ .

A ) X _ \' 11

. 1 “a I ‘

,.¢

>11.

1‘->3/A._ £11.]

F'_.'

.5-r¢)' '=_ \'_ 15/‘...I] .,_-,3 =1> "x(5'*'c-) IL -r )\_.-'2_'.u

(J+¢) =.~. 7§(_1+\-111

r

\-1

3'

»

D.__-_-1

, 1 -=17 O ~ u um-9>q =0

"-1 Q

C0‘ - . .mmon ‘n C‘ d~.‘5C: 2 P. dasc1:5 1':Q

Bub @ £5 not $QtLS1Qd b =

1QD__

LOC

incl '\h€: :>'1r_wc3uL0: ->0 . (11

~

,,, V; ,.1_ - ,1‘ ' , - ' D l X'_ C! -r UK,-.'AL1 -I) = 0 ' J ~

no ulaggbg. :_ rap] |~ U L ‘ ' W“ ' L I -I. 1' I, 5 .

;'-:""=’_\‘ 34 2'.-? W » ' O\ - O (I) € I

1 1 3

L,“ E5 = P+'2?k1: 1

‘ I

i 1:

2

,

\

Y

K

I u;.\j~ _l --2- ;.

J31

4//a 1 \ . ' Y = _' ' . ' 11 .

v .. ____¢_.. . _. . -,

- . C [3-\$c_f\Y‘(\\\':\C!V_r_\th .. /" 7.

w».a....,..._..~_..._.._"-

__.|.

.<.....:.v\"-:.'-_.:_..:uL...-"‘

-_-A'

-I

I“-<.

":-

5;

1

X

~;

‘ X

.]'\.-<1

jg

. I - * ,.§ “ +l'\1<.'.--H ~~° . I ‘_.~ 3 -:3 1 -~ - M‘ T --| H3 *1!‘ - 1' I 2'vP (*LP\»QiM

I . H 0 ,

A 1 " - L15: 1>1+'2PvL A =

- :,y_v 1- .- . - ‘rs K '\\)lL'v'| ~ ‘ - ' -

P .' ‘ §L'\-_~,¢r'u'n\n<\n\- ° L) " :3.4 " " y | 1. ~ ‘.\_ »

. , 7 . _17iffe.renl;r¢t£r~g W-1'-L X WE (letBL; MAC '= ° 1

" % 1 Y-3' - _ F A “

u \-/.-\H=o L‘ ="'2X P1+7'i1P dp ZLPHLQE

.-1 I

_..- alv-V --.-'

_ _/L(_~.‘_,.)~_

<15?» <1[\Ii*¥ Q =V ,"3§<~*

~ 015*- ¢

I ~- _ =-_->.1>».-v-\-. "—_-7. P

\ —- ' " '. , ==<> M , -1» .

$3-_ <_3.\b'C..r_"Y\.\r_.cx‘.,rE—=- .

'——~""’_"‘__;“’_-‘E-,‘ - ' '| L‘. I b‘ I ‘ll. ‘ - I P _ - .-... ' > '

. \3’- d35crimYn¢=“‘¢ °5- '5_ av r J " 6: ' f /X?) < P 1 “ O

‘ "L; AC = O. 1 \- _,__- ._:u

. -__ B A '__ ‘A’ » '

.. E":___? O_.:_l._|U-\'H(?>w.-\'.'L=O —; I1- 1776 :0 or 'P—'x'5~T£--0 ’

- ==> XkZ’>*--1),-= O ‘Co‘rw5£oleAr» '- ',, X

.,> x=o or (32.-'\)=0W ._ P__XQ_E__ SOA

. _<,"-‘\r\c.e.‘ ,'

l

; -

. , . . A ._ Q1’ ~-=. '

; ' boih Y’-oL1sc..] and. <_L-d-=~<'-- - ‘"7 X OUQ '- P -

1__O -‘lb/)o‘é_ :.'\ngu\.cu"’:7:

. .

€~ :5-o\u\:.\or\ ‘oil ¢°yl ’ . ‘ I? .J L

” .= j¢w@ M %:% '~»=A ' =7 P = _cx i -

Putt-_1'rw_g in éq. @, wg get

- c;vL+2cw.—qx\_§~=o —'-—

‘i V .

’ _-1 1 ,w<-1 now find the S-'sol- oi (9, as6P5 +3.PX—';J = 0 - ..

S0! =~

. > \

\ » ‘ _C—- d~i5c:ri'n’w£r*\orwt

11] -~“H\. 1-L. ' 3P1 =

=731=

<11 -2 - =>3* = '3? 9-§%+P'_;’5(zPgr

3 - 6P3 C-°Li$_<1-016) is giver-w bf} -

P.

'Q

I’

//3

"Tl

>4-~.J_.-

\-/'

.1

_»"\

I.;__;

*2»

2'3.\./

‘s~_!<>~

.*4;

. . . CM1:0 \b C.0-~‘\N\on \r‘\ -

1

,=

____._ . . __-—: I--—--'—~~ ~, 9.,

2 I

& ‘*J' '

...,-.._-

i

2

'l

. 76 V ‘@ A :

’ Sirgulor 591- . < 1'

, . 1 1 H 7. ‘

I ‘

\

I

-2 |

I 2

515-6P5 I ‘ B"""A€' l

8 I

Lax 1» nsx'~3 = o

-4-

.5“

5%’-‘E

1}?

F4?

'+L{\j :0'1? 3-4P5 ---_%zQ(P \ 2_,_ -,_ __— ow P—oL'5 " " , zdam, I29? wag 1 ¢"”"_£.Q‘?_1_~

.‘v P—dis<:- Of CD i5 S"”@"'°‘;:

1

I

'11 lI»;1'.~1=’ '1‘ . .12:/f N-1“-;lZ‘j,.Wé: QQL, | :2’ (1x‘>’---‘*1-1-"‘I1-H3 ~%

I

Q?

\'

r

54.

'1

'1

w1

‘I

E

\

\

'

»

~.._-_..-,_......__.._..,__._-...--._...

=5.35;‘;-3'-';='~.~:‘-.--;'..'.‘.'..

~

.

.=_.1...‘..1-.—\

,1'1‘:

'___‘-~¢_.__'-___-4-_;-_.

='_'r;;:"'.§_-;.‘ . , . . ' ‘. ' , 7 ' ”%+~|29P = -5(_L_ 5 '<_-Lg ‘ 1515 {(A6 -_ 0

,2,‘

-<

'.

1‘

_.

U

L--\

—¢1,9-ULC

I‘

'\\-—~.

LC

ON

(J

1?;

)' -3

,1;

I ' 1 ' ~ ’=’ (‘"I'=*—1)~"(@P+>@°‘ll)-O.n

-z,

»

,..

-J.-

-2

.4

.1

4

P'1}+ 50145 .

//7

, - ' .

Q ! - ::) ~ 3x1 L I _._ _

=->1P<~'~-2-6‘.1m<~r6w-*-*3 *0 W '" '<"****° ~‘"P ,' Pl‘ L. E ::v- 6 ‘ 1

:7 ("l}51*5H) (1?-r%%3 =O ._ LDL -1- lévbj =Q

=7 vC'+u~j =<> A _'“?

. _ are sonne-COf'15id€r, "

' 2P"+39lé% ,=

;;>,3=gf_,__‘ ~32: .E

%__' H111

'¥s‘u~‘~

1

.._-Q--r‘

. ‘ L '_ _ ..-__" ~ 7@‘_‘. qt] =; O 75 b-§O\- O‘ £,§1_’_:

\

2-? '1_;“‘*63-“O °". 2P+3cZ%’=-0 ' 51'»’1C@- c- oLis<;- and PI; F1-1'50

.- \ ~QL’*\ T 2P J )LP+)L£J_P +1 = O ----5--:1}

» .—-—-'—--2§‘_¥. '“"

.\_-’ 5.,_*_

7:1; : .

| 1 1

' :7 Ir‘?-' ".-= -2 lng‘ + Inc - -7 ' ' -2 -I=7

C. 81-

50 I 1-~ '

. 11_7L&5P = —XP.—\ ,

- ‘J = -M’—1F'%1.>@ff;.W-*'t”w,w@-sat %

» u.. . I V A

f’“°L""‘-:8?-'51-TF1 ® we. 9e.£ d5 P '2 "4? '3 -‘.;.g:L- 5 -‘L";L = '-(7L¢—i-\—'P)--(-X, ‘P -a—;L-‘IX. Y9}

H '3" 3' /d1'.X ‘"3 =0 - ’

:7 ~1:~ 3 <>¢* P30?’ 011* x‘P“ '2

5‘? +.-3cx—3 =0--§@ “~ 2- ' ' '

“'0'” SOVI :-. 0

W@- qow find the s~5Q|.,Q¥@,a5

C-<1i,sc.rim5¢,~,Q,-,t ,

-> 'P~—.-*=--" —-—'-—— dp2 x"‘P *7‘ x’P1)};Y,L

i , ' :7 "" 4'“ \4- ._ -2P(I 78?,’ 1Q FIN) %.Fi=O

, P ot ",1

1'7 |-_;_.'_-:_Q or 7_'p+X%\_;=O‘C-ToLi5¢-;§_E~§¢f @, - . . | V - ! 7@pZ_ . .

2-.3_ 1 _4

35»—”.1~!AC,

L3 L\/Q1-,1

_ z__ .

—7 v9X. H(67( 33) -o 2_P+X£i_

=9’ 976‘-t-ZHH3 =0 ‘ Om. ' " '4 .. - P

C0f\3£oLe,r,;

-" gm

P-discrirnimor-wk I :7 Ki; -_- -2117.4_( X

‘ p'°u5c' bf Q); £5 Qivérw by‘ ’ :7 Imp7 15 ‘MW1- -

\

T‘ “2!!’W7(_+.lr’TC

I1

I

1

»_-._;.._.._-1

5

v.. ‘

E?‘ . ~

.:;! .':

i‘ ' .-

59‘ ‘Q Y

-. I. '. F A x 1- KI I ! ‘v mi“ _ I

I

I

_ .,..._._.-..,_._.'_.___,_._- __. . _\

,._--~_~.___...-- ——-—-' "'\ .

.‘..‘_».:e_.-._ . '

. rr. |._v~.: --;', ~_ 2 1 \

..§.9_x_-LIea(-~.n =9 ‘"7 *°=‘l/1*“ ' “1-=> I '“<9'>C‘+2;u~f'=_o- I

P“*l“'“‘3 in Qq-@,W@9e*/IJ 1. 1 - " __ - L

-_-_-v I X,."£l-" "I')(‘8"€"."'I'I -0 §

> \ '

Since ' c.-disc-‘is e,qQ'ol £0 I ~ ' ’ A

.

2 =%>- ci + cm ‘F x = A '

p__di5¢_. .2 A31.-1-833:6 1'5 -4 . I

a \§1;rj9'.%l~I0Ii1‘.',$OI- cIj3 @, \ _“l’i'_q_;9_‘._*Ii1L_~gI_' I ' I

I, Q I I . wé how find ih 5-I_<C5l- Ufa) OJ‘F! _ ~ "2 * (‘VI » . .- , . .3. 3 . ,

- . ,>_‘.1i3 17 ’ = 0_ -———-® " - =_ - . I I

I HPHI Y C—ci¢5crLrnin0n"tI" S I‘ '

.'_

V 2E:I'P3 -4= xPH+ me <1-"d"5¢' °I @ I-‘~9“’-6" bu I I. {

i =>~ 2:1 = M».,. mt? LB--'4"\‘l=°. , . . 7 .

._ )_ _-

i ,p¢._ 1-.¢:w-1-£ X , we get . =3 x [IX --O| , '1.

Q ; dP 3 -‘~ P I-1 =9 X(“*‘1 “I 2°1% on - law" * ‘?("*?“" %C‘3“° I 0 1i ~':. I, --

P 1-1. P ' 2'3 . I X o».~1>XEI LI--~ ':7 2}: =1 -;L__¢i- +P-36X,P ‘ii +361?- ._-.I.<>*>¢. _ I C“ I‘

I P- okiscrimrhoht1'1 3

I =v P _ 1¢3 -561 QIP I I é

P3.-,-‘:?§'f.-'l’;;§»:‘£.’1_l _ °‘* -Ta oi?» - ~ "P—<>‘i5¢» of @/ 15 9%?" b %‘~4-2‘;‘¢;:.=.§;=,,-. .

Ir => "_* 36v'5’=?5i:i?=L‘='~‘ " 115$ QLP

‘IE 1

»~

;-_l_.\-

-.1.

,i

I 2. 21 .

E-:37 7L;:O,)(.=O’ ‘KEI-1-|=OiIj| 2) [-- 2% -=0 OI’ P—X :O -I I

2

U-I

U3

8!!

‘£1‘lI#,,

‘J

Q9_71‘U

. _ ' =7 1_(X3_qw:O

;1‘§}.:?..Q~ )<P_XQRw=O —;on 7 1-»? Lv<.~f- Q = 9

OH, Commom in both (1-discHIP,-"X.--"=0 ‘- "'

_ Qnd P~ disc .

. g ' 1 I

Eg 1 :»;_l_ won?! wj IP I‘ \ -» 7L==_0 , 1a—L| o Ore_/d. _ .

7k

... Q13 = Q1. *$\§°.""’9"'°' ‘I”"I°‘ G3 ,‘-7 P X . t _ '

I I

Available at I0

I~ www.mathc|ty.org

ii?’ ’~1~~~* *' *’** ’ ‘ '» I-5 4 5"“/I if ? i

I .-.\

"B-1-|Ac =0 .-. =7’ ""P == Ir‘-vC2+ lrwc ‘I . "ii§ _. _' ' -

6,n

.*.

‘.-

iv

>2

-.

jz;.__

i

I A A .wmj

.. .. - ‘i 2 3' =7 ')L§I -LIX" =0 I

I' -— -—»— __,, "— f‘ __ O=7 P(i Pq 1(|

§z'r\c_e._ 7(_=0, 1\j1_-L; = 0 dz-e,/ QM‘-_ ‘ ~ I I

'1':

"“-/‘..\'..__.,.,-...__..:

\

._»

~

\I

I\

I o

W:

-J—A"*4“-4'"~‘-—-'~*1~l-L-=:;."»mn¢>'1*¥"'=~r:s:<,-z:<';~:‘:>;-L‘...

.--».;~.»-_<'-_-.,.,___,_.,_,_,‘____,,_______;

w‘_q .V

.-.=;_-_;;<_‘-.'..'=_::_f'.;-'.'___.=1'."»;.'-_

-"111.=.-'~_-:

..'.‘".-J‘,-V. V.>_‘__'._:

.=-j_:‘-

I

.===_:L_

‘\ , r.

.=

1- \

*3 ;

v.

-=<>;~.r~_11;;,~:1-—-_

,~‘....-.~.v..

_;...:'..._J.;'.'_'.-.;L:‘_~..;.,-_

\

7»; :

“ .': vi

. 1‘ ‘.2 I1 - B

1>,¢,____33;g-_-r~-1'an"_-‘_~..~=~_.-u,-.._...\_u._::'"-*="/"‘_>:.*.:~.P.'.-:1;1:;:_m;;;;;::_.-._..‘---

‘\.'.__.._§..‘____.i'_-.-.¢=_;»..;___..~-_~-......._;_-;..-.._.<..,._--~»-~

5 ‘ii J)3 \

I

\

:~

-'=-'7\P=-_ '_ ‘

Matiéiéémrg' ‘L/ Mering Man and maths

'-_§

8P.->&~ ~ 11?putting if-1 e_q. @, WQ get’ 1 §O|:

F 1. 1‘ cl?»-2¢:x3HI + |?.v:L3=-_Q _

"lP :1 =-2"/X-\-5P3~,g .

:7 C 7L"~.7-C51 + :1 =Q

5-'27X»=Q'--

Singular 5Q|',_ ' “=7 '19 =_8Px-, "2TIPXWE. HON, find 5£r19ulOr jQ[. Q3

VV-1‘-t, X», WE. geL -

'1 (lg "3(P+XdP‘) '27 15321“-J 1'““ I - - .__C— dL§Crim[nQnt ' GL1 di ( Q9‘)

.~ =7!2P=8P-8X93)-31 §_‘1‘$g-LPSihce @ L5 nab quadratic Y* 00k. P1 *. P3 Obi

-'- cl-o£i5¢j'..§'_o Q7 :' o QL as, §=7 HP + ll =g>< q 21 4"“S f ‘W - P‘ ( “L P3)'§¢’.,| 1,.-_ .' -

4 ‘

.> ,.v

.~;» :;= ®’-;P!~_'of_bio[]H W.-,».£ Q we 93%.

uM ' =7 P(~+"%3~1X<‘*+%%>%§z-=¢ I1?: '1," ‘S 3

< P/( gga O

H (IX. 9 -'-17"'z_c:X -39 =.

‘1 - '39 .2’ 3 4. ' - : __ P<51)»-2(%%3‘***” =0 P “%“@;=<>

_:_

H :7 21 ._____

=7 El lg‘ rz_ o-I516 H 1° ;

.

. .-__|.... _ . -

=" 8":-'%?.;—:1."o8a“ + I911 =°

% —° ='> q =1-\+v __ '3 P =

~ Lu: " P__' ' ‘

‘- ‘ 3::I I? LH"P1 O °' P‘2°‘~%<?¢""°§ £1 1 1 I

75* "> C = ‘E13 ‘PM’ 1""@/ we§et-C-orxscole»-_ ,

=>'S.@L_P _.1.g<£¢_P "1 1

v

\

P .

6 ~ =v 'nP = L 1if ,0 . 2 r\x_ + Inc' ' ~\\'~" ."}'>'?1-':!’- . " -:' V: .____" -5 ___O -P —C-W -

~v - 9~a¢;;=»~1%6~>¢ - ~"f7 99.?-5L*7Lf

P— discrirriimont [ ' =Y _ V

1 ' 1

3

8 (cqri-\I X _‘

:\,_'

~/

Q)

mr<.,

_!~

'><

- q{- @, pcrluaug W-r-tP, we get-_ __ 1 /-LDLP3-— ayp = 0

F3

N/(V?>41

\/N

LC.

l’*J

J

C3

_

’-:1 Q13 —2"zx = o ___

_. -7 acx3°"—~:1<%g -2-; =0’ M Q;1

fifwgulor 591,. l l

"=7 TAP--35 = O L=, p _3_§_'- he. .. t We ow %"m< H"*@» 5-sol-om) Qiax P“‘m3""@ S’

/1

, II >_.\

Y , x

,1 ' r 111',’

=5 |F“|P == lr1X+ Inc "- K

| C1 1I 3 1

1

I

.2“

I

I

1

I

Z C. . 51:.-X1 ‘X ‘:1.

n\

6 _ 32__ ==v -- a1a“- |oa:_3“+ mu = o - —v OK’ 5 V °-~ A '

.

I .

.1

=1

1 1

M.-;~.=.-'a;J . .

V.

___./%'u____

‘C....'

\)_‘_:_ ~

- 9 HZ’-',-:_;:*_'.:~:i6L‘ X6

3 - 3 -disc» 'rY1zIr\Qf\b ' »' -.. §H- 21. =-"O I L ' ' . V

15 <2» + @ V

.._;,

‘..

~..<-....()

g §l:§*‘*€- ' I ’ ' ‘‘ .s

J 0 \'

, - 123

-Diff-® paw-L£Vol\j' w-~r-£ C. \*/Q QQL

%;w.";% -_..1w“ ».»m6=<>% 23 " * W

:61“ HX3 zqc x/Z-zqcg = OWI

.2, .

~ 9 <=;=_k5u>¢ *.Y 3" »- ~ __*

. H __\7_ £3? ___7_'I __ Q

A - _ L‘ ~-'z'lX.‘=O-ore some ‘ ' *9‘ 89 '18

. - x ‘ ”_;‘-Z-"7/H 5z'nc'&;='. ~§I;-:olisc- and P-—ol1'sc- X - 3

4!. 1, . 6- . » . 3___ 3 __ ,

1 jg = aux LS the 5-501. =7 j¢l3, 27* -0- X ,

‘f| \ W \ ‘ -.1‘; 2713-}. L.[ E13 -= O

Ivesigoté §¢.- Sigl-1|0~“ ~ 1 _

' . P.~- oi '§¢»Y Y» '__’1Q__§_ ' A ;__0 5;» ha wdiqs 1=-<>#=§¢~ -1—~—?—-‘-1‘ " % 7

~ \

; __'z>£ff- ® p<1\"l:_CC\|l:| w-_r_-t I5, _we,9eL

'6-—éP+ZlJ-1'1’=<> --—@ ‘

-e~:g.:,?_¢_1_r+f‘{'-2' > vv “

P-c:liS:v'ci'r-*1‘rw'\ir'\¢:1r'\t . :7 'P‘L.=- H i l I

.‘I-I

GP-'\ I: O “ 33 '- —~ ' L‘ 5’<9 /45 X I1 3'1-8'-27x =" 'L -

\-'3) == ‘ ~ l I ' ;

- -P =>8J-_.-:2-1 ¢-2'1'>£==;_:'1|f\8.== lnl - ' "

1 = * '

‘=7 P|r'_\¢'=-.\f“\\ =7 333-;1\J3--Q71? -_‘;;-.

1 :7? P==<> |_e=\, \r\| =-0*; :5 ('3

\ 3 '3 ‘ ‘ ‘

2. 2, .1

.,;' ./H - ~_ ~ \ "’>_ 3. 3__/ )(_-( 4 2: (.1M ,N ._ 5imCe eq§@ 3Oh_5fL_eO,\ 1/x(x 1)(Y 1)1 x3 > )

, , >LfJ,~'_z"!vL3- =0puL£in9- in eq- (Q, WQ get ' = ’

.7 H ‘ 6 ’

- "8 X:/Z‘-I2".('7%'z5g _-2-I :0 A I

!f!.;..

\

I

\.

!

‘_I\.

1

- =7 27 3 3:.-' . ‘, XTHB O

=5 XLH-O =0 J 51'r‘\QQ_ ¢"d"§¢.- <>*"\<>( F-—d"$¢-

are Jan’-\e_ ~

'2? O’ _H-l ‘=0.; o;@_

Take vb =0 ‘ ' ‘ -

I Since I O %£ é O1mv¢~>-ug5Le_' jor 5,~m§ul0,-

' 1 H

I 501» 05 firdirwfj P-.-;n.sc.-

O'1

O0

w

. '15‘ ‘Z ' I, ,.

- _. ' I

I

E

._.______.__...-_.____,._.____

\v '\ \‘ "i

3 = 2',-'1

. K 3 L

"...~“.a;>.

1,

‘I'_.:v.

',;..~'

. I

.<:=wa;,,‘—,~.~:-

5‘-4&2:

‘Q1 1

E113.

.» »:v':'\l‘zi

'r‘%.1.\

:31$9‘:~ u‘fa?

/:>‘,

354:

Y451‘ .

4:15.‘"I

122%

-;.. ,2

1..

',l I .

5-.’-_~,._‘,».

M11‘x

v.3;

=7‘.1

$4

» 1":;, .=

f“! »:' -

ti.‘

r1

4

,-, g -1

a'._$.<

-A3,

._,

91¢

,1

3!»'-'-U.

,,\P.6';

v

4

.._‘¢

» /2.15_f~ _-~.__.__,_ .. - ..,__.._._i_,. _ _» 'f='.";,' '-= .- ‘ _-- _ , -__»... 4

€‘,f2"l§,._'~I’:T_-"1".‘5'

~_.__.__.._...-_.........'-

'..;-:.;.-._--_

.' ’ 5 ‘-=| ord. dy '= '

1

.J= »’ - ’.. » _ ' 37L—6X+2=3 I O

~

-'\'.

-.l:‘ '-\.=I J, . .

.,v.

:'.==‘:i.;-‘.1

.1

..;:;L~~L'L1

-,

,=4,

up._.1-‘~..-:

.,|

,.f.'~\/

._i..|

1

4

, .4

,,

_...‘-'L~;'l..-.

hx‘

.

:1‘ wP’—— HP = o X =

.4‘;_

F \.,~; .

I5‘...~

.2"-1‘?-

/,‘;_:,_

1

1-,4-=4»

1K

1'.‘-.in.;_xM» 1

_\_.-

.~..

~e(.

X.,,.;,' I.

/{I1r->-.-.-

1*-él;

si 7

* I

2;}. .

2 :70 PH = O, 3-1-3 =9 3 TO|(€ .X_2

i [+81-_—_o" I '

<Li>1;.':.:~..:.—_T.-2...

1-“".."‘..

\

<-

3I

I

,4

1

'1

.:i

- -? % . ""114

,:bv X =0 am ‘/P :01».- X = O £5 not 5in_9u!- Sol. o1£@ ‘ P- di$Crir"r\irWo.r\b

T°‘k€-4 3'1 '=='0_ P—disc- of G] ‘L5 give by

' » ' B‘--4/\c=@511 O - - 1,:> 0-114-LIX(x~n(x-2)[—(3x%~¢;~.<»2) )=

:;1~<"e@ 1-, 5O(:1'§‘fL'&.O(.b3 3:. I => M-»>.<:~-1»<31‘-@X1~%@>‘@n<>cP'-_-.0 ' , . ’*”.°“=°'*"'¢=°, X-1'=‘?.

I - . I 2

"I = O 1'5 0 SLSOL of @ 1 . - .I , .’____(. , . -

/:1 7 ’ F‘r5t1 W€ w.r£h€_ Q7, O5 .

Q’ ;'_.. "

q H><(x-7’ — = ,__P-'21=1+1_y‘*= 0 _-_@% '~ “L 2) (“Q 6"”) §>='*""@

I 5% ‘ v%

Take X=0

:P~(il5CI’L'r'Y1irOf\77L = 0 _-. <_;L_lL __;) .e _._

<13 ‘ " P *1>m-@ Poll .@ Y” . V" 'Port: 3 wrt.P, wegef: 5lnCQ_@ £5 Sousjied.

0 amok Q93 = Q ‘

I ‘--I ._ .. )(_=o_ L5 Q f,'*SO|- 0f@I? P =0 .

' >

Pym;-»3 C0, We. get 1 = 1 -2% =0 ,-_Q 3% "=0

. » .

""5& -: '9 ',,u‘1=~o 5/-'F\Cl€_® £5 satised by "___

°" (|~31)(|+'5:)=o " 7<=l orwok = 0 ' .

or ("”'In(\.+ED(l+9z)_'—=o' 1 3 =0. H x_] Z O £5’ O S'5ol' 6)__._L

TH. \-5 ejected)‘ X: 1 _-, Q =0, ._ I ‘._

U 5'; Lrnagiforg ; - -GL3 I e -15 =0‘

; I l I ) X 2 _§'g=| -'-.‘g~_7%=o ,_'_QP=O "D -83 ‘

' ' ‘ ' 3L'"2' :2 O ‘ '

5_"‘¢Q @ ¢5;$Otz'sf:'eo( by ' / L5 O 5 5°" °§@-= :1: =1 ck ow. _; ' I - ' .g Gr“. O i

. > I

. -- ~ _,._~. 4 > 7 -...-._->\'__ Q’ »-- -—- -~~ 77:3’-_-'.?f'..‘.¢j_f.‘H v - . - - ‘ I R

\ ii

.._-___.___..._."_

1-.4;-‘_%1-53-if-‘.

-,~_'.'-‘...\

‘......

fl

,, .

~:i

<2?

..--.--1-V"‘___>\_\___..__.._._

.__'.._-_.a-.i.-u

i

i

n

_i‘. 1

.r,|'

i

_V 4. it _ ::..A__

' '_-= O i5 S'SOl- 0%’ ' 4’ ‘I25

- Ei='—l -'-?._3-"=<>|i-e_.7>= . ~‘ -, 4 .

| § n

I

|

O ' s

5ince._:® is sohl-sffed bg D

us X: "14’:-3 -9‘-X-=0 ile y=

“ks. _3x-?¢63><.+2H=o I . ';

T°‘."~B 1+5"-'-° _X=., 6:J3e—2q '= 61-{T5_'._s--_¢"_5 .

.. " . ' - ' I ’ 6 ‘_ . ._. -JV: 2‘. . . GL5 4 6

. <15 P. I

y'=-1 and ii‘:-L=o * 5'3"_\_Ce Z-'5 “Qt jOh.5Fl.ed _~

si ti+i=0 is s-sol-o{(13, _' “*1 X‘~=|i'f'-3'_or'>'d g-"g=oZ

' ‘

' T‘o|<e. y = 0 ' Q‘

3X,-6x~-|-7- ==|<;~. is not s._\-..-1|-<»f(_'j.

= .'. Q5. _ - f\LV H O cix "3? "Q P = O

» 13--H¥_g1>_+351: 0 __-..__-4)

D“,-‘Ce @ 55 "Of" Sokisfzfeol by _‘SQ|1— V » _‘I*1’./Hy=O',L€:§l.f'\""'Ci=g.“i -_-'-Q ‘ ' ' ‘ I ""“"

- dx - ~ . 13-d.i$cr('r~nir\onL _

3=Q is not 0 S~‘So[.Q§_® A . 7 V

, POrhLd|\3~_ W-r-L. pi wg Qg

In I I .

; P~'di5<-_rirncmamt '

_ VI >2

‘Diff-(D. portgiolly »‘v-v-t P, weget :7 P ti (%X5)2 " PUt.EiFW9 ¢jr"\ Eq-(D, we

P IZP -f-_3)(_ = ' .O |

' 3/2 ' ' ‘/.1 ' LL §

I ‘::$ qP7' -= "51. » 10%‘ 15]) -— 8;]

- =‘* *(%***7'I%***"L\*‘1\ +8u=<>% -I=" P = Ix/-M, ~ L %

. ' => =+(i1g)"*-;(_ aw) +831 »- 0P““"’“S1} in _eq~ ®,'we get . ' 3 3 “ ’

| , Y 1|+ 1,2 I P1 :>;_2[_.t(_q_-Ky 2»£LX Y z_‘*(j_(*/q))+aX(r(—3/Q)-\j= 3 §'(3 9.n*83 ."°_,__- I i1‘

1 I ' ’ '3 ‘

. '

Y» :<-1/L.) (—2~':1“=§> ‘Z’ '-* (_%w')'/*= my’

é / .

::> I'_1('—[7y;1)}& (Q1. tnlkx 9 != O 1?? It 2- IL{ ‘= O

‘/3_ 3 >

H = 114-I/15’»< => ~<w~_»>“ /<7=7 32 = q» (-1/4)-XL :> %‘_,i 1333- 163“ = c:.

7- 3

Iv 5 1 '”“ ,.=> _’¥'_ v@3':i3— 3“ =07L3_’_Hz 27 I .

- - 1_/ l .

y \1/ I \

’ % M Availab\e atwww.mathci*ty-Org

I

\

I 1 I ' 9 & ‘5 3 § I L I 5 =>P’—"~w = o ‘ %? ' g.

,3: ¢ LIP +376-P -3 == 0 '__@ - 1 A ’ 1‘Ari ' . > . 4 =7 3P = I-I13 ‘I SDI: ' I,

I ' =7 P' 3

S.>2,-

»

$1

._?-I

:5 .’<;,!\-;,4.

r 1|

'='.1v...‘

-, 1

.‘_

-.: ff

.g\-

."~

ILi,.,.___,?,_-¢ ,-_:___.._ ., >-»~..»-_-_.--. ., .. , _.__ _ <- -- - _ .___ '1-..._.

. V . ; ‘ 1 7; _ .- -- \ - \.__....___ . ,__P. IT. _*.-._ |

=- ' .

I .-7,|- _. ‘ ' I v " - l-

..._..,-.9;

>.\ . __‘(an

‘ mziiI V .—.,__.\

)-..-,.=\~ 14 I

‘ ___».:.

J",-;=‘.-y

;w . ,1

3.‘.-.‘

|

A

\

L \ ‘

‘r

‘\ ‘ .7u

.\.,-X if{ix_'/=7

,1,

_,;

-3»

;f'(,.;- 1",

;\x=—’ _-

I

.>._,' .' \

1*

7

.'v'_v

> »

\ 1

J ,

.|11

\ . 11

. . V

- ' -

.,s .,~11*

'-'}':~.-'‘J ¢

51' rf ." -'W » 1

:V.l '1.»

L», 1_§\ -.

$1‘:

1"

1'

,1.1

~ ,1-:

._1 {Z.3.‘ 1

wig; E

'12.?

>15.-; - -

.~,-F,‘<: -Y

§'~,§.v:.;"Z

N:‘A5-‘T , = 1 :1 99 _. . ‘ - “

I! | I if it ' E I ? = (@912 )3“ L4x(~§E,$(“§ %X2§ + 8(-$1")-1

», L ' 6 5 6

R

H‘.'2 1.v-|

»

1». ‘f

‘H’

J ..4,54‘

,-'¢‘1§:'.~§_,: 1;

0 -_,

'" '1

'.&__‘f-.~':_~ -'1 "§q5"x*. -; .".

.5.>,:->,-

gw. .~ .s.1F§";- ~17);.,_.\.,

Qlyz

‘F “ "=1* A

I

vw-'‘v.-1‘...¢'-.'2.‘..n-.....;_..';'__'-.-'...-_-~.f-L_.

'3=:

.;'.'

: 1

.6¢

..L'.....'._'Ll.ll...=_._""~"--"$5.;-.

=,;|.

-'-..-L;.1’.-_

_._._’&\._

9.1.1‘

1*

_~.nv.L~

O

__»

{KI 7

E1 ‘. H

i; _\.‘1 I4 - 1;; " 2 2--1 , 8 out 37L

' " 2

I-1», a “’v"> 2

. 3 I' _ ‘ 3 3 H . V26 ,

%~:=3

}t To-ke. 7L3+57'= 0 I ' t. ,uX S I-27 ‘j : O . "

' => s“(H_>@ -1.197 = O4, ‘3='TX .

3 3Y

* _.-’__. =7 3:0 , .L4)L-'213_L-;_.~)

' _:> ':: _~3_2HP=_3x 5 Q , L44-2jg_-.,

=v _P‘=—_L 'F<>r<eg=Q

P?-1' ' -H- _. ' __ . .LLL"19LT\L S0f€q @1WQ9&L I.H_Q .. C§§:_L___o L_e_p=O

.l_-In-S pf eq, Q) I jince. @_¢f=>-QtL'5f¢eu{ b3

-— ‘:!=O amok d.9AI uP3— ‘-r aw-3 ,- -.O_L_x,=¢>’ _.

.= q(i-

=l4("?

, is O. @;@>, H - - ' ~

1l>.cf'_ 3 LW) .-‘5-§;- —w ‘{T¢>~'_<§“-.».{ ~>¢_1w =<>

6 - 3 L{‘;(3--‘Z1’ 5] 1; Q .

=-_-.__aL--—'2)_L_;.L '

' -) xi1'9 i’ 119 --=‘ L113 A V

7- 3‘Z 27(>(.3)__97(- ' ' ‘:7 H __. _t_._(_X3 ‘_ ;_ .

——-——*- - .. ____= _L_{_

‘ 7.33 23 ~" 21 Q11‘ BX

Q; 17" g(-Q) *~ :=_ J - or P = *4/912_ 27 H)__233 15 V. F Pu£E<'*’W9' rm L-H~5 of eq.@ .

F '@+=+‘":,1;:%_§, O}

18 '1 2233

= P—- 1-(X8?-9-8'5 “‘

Z, 2?*§+9t-3-13A1

. .\.', _Lk “ 1 .1 2

~' 1

= /75:1 =Fo='R."H-5 5uX_ 6_Li_.L' + |_1_§_-_. '

729 Eqjl ' W’

' I ‘ = * ‘6L17(6—l9'Z7L6-r 128%’ 'R. 3 . . : Q ,= . .E .P"f,l‘»ZP—4>¢&J =o___@ .729 . 5

. 50!‘:-~ ,

. §:'F7QQ, Eq. ® is soft-sf-{ed by

P—- O‘-i5c:‘rif1’\£f\Or*\h fj = $.78 Gad GL3 ix:‘ »-' 2'! 3; --

‘/3PS

1

» ‘

1

is

’ "=> P

POrh'o.Ilg W-r-l: P, .v\/Q gQ£ t

2 - ax -77;; =0 1.5 Q 5_SO(_ o}@ .

—2>@.=o *

'1.I :> 3P = 27} \__

I 1

‘Tr

_/Sc

74...

.

=i__

/\UN

' | I | (

‘.1'.

I V , 1 , . "

1‘ . l 4'11?‘ (J§\)1— L119 "= ‘O ; ‘P3; '

.-»

~

;

‘

»\

ii

i4

W

>1

,.‘-

J

1:J!

'1

a

5, _,

J

‘14‘

r

1

»

~.‘1---A-__.,_._-¢,..___.....

:-

1

~

. . -~u- -qua-_ ...~'"I"!§“'-. ____‘_4 ‘nu;-_ -_ uv».,<(--A .‘ ..".'7_r.,,\ 1- - . . ri .,‘_‘,q_ -_ ,_' ‘X,‘__,v\ -- . Y; . vb-..* -. \ - _ , . ,- \ pf“:

_, -W Fl -'~:- - ---- _-,....- 4 .1 »..-_/'~ X?’ ‘L-, ;

»I¢£', "'. \ 1 : = --~|- ' ' ' .v~. ‘~ . . “- "5: 1- . r~|_ . ‘/ .~ .

_- ~ -~ "- -1

-Y1

_-._,..._.....\.___.

-:~

.?Lr;

\

->|

.__;-In ‘Ix_."_i "4. ..

1

.11

.' !V

I2.;_.-;_--

I‘

»

4

~r

; ,

--9

\~.

, .P¢¢'=““9 W Q9‘ 53: W?’-§‘J‘W. ‘ ‘Ma‘n§i‘y§a0rg‘i1 /" » 2 ' Mer in "an an ma s

41¢<%+J-2;2i%§1> M * Q 9

.1‘ ' '

L-‘<4--<0-+3

V

if.,:_-Y

.’\uiJ\J/'\

x/\,(JR,

N

>4U-7

I

(Z-.-2-\—1-\\‘;§ =0 " 7! I "

\ 3-1 3 ~_L\ = "

:1‘; ~:(-?5} x("=/Q 5 .=\“" ! M

‘/11-.) 1-_ X '= v | '

at 11-

3" ‘ 17 ‘f(7¢7LL'-—2131\ =0 .

I

7.

:27 2X'6’_;~2-')(_g7' =. O

. 1.. 7_ = '2. ‘

E; .i’/ 1'-*7 “K -.:.Q,"; 21% -213 =0A _ 1, 2 -

* -__-7 1-.-_o" 7_>g~219.=o, _ \I

1 ‘

i

~:

q *

1;~\..1:’;.X, .

....

"._(\:“\"

.1.

1

l>

1

Z +akéQ x=Q (M "“ 1”__,__i_Z.__-i——-——-

. ' Y '.' % => P’=:~.+7-=-2. 4 ’ -. "-T 4Puu¢m9 in 1_-H-5 O§ Qq-(D '

‘X _ .1. .oLX_ | 1('21\*i_ Vi‘-O ' GL9-9 L-611. P-.0‘ifsf, we wrike. @, Q5,‘ i " !__'H,S C4 A

-J . 3 . 71 . ‘/ '_

2 Y-“W1”--@' ?"’ “Oh ‘““‘5‘e'°k: =[~:2(:iLi§2‘q-2vZ%1("'?:\“\'11i(’£1><‘3O' b5*)L-_—_Q‘OfWd‘_Ci1*.-.:.

. A _-i_21X"- 2'! ‘-

‘ Y - - § . '|

Toké 2 L‘ ' 2

21“

11 i =1 121:1‘ =-' j 1 1 .

, * .f1>_' 52 = g-qx . 1 ' ‘

1 , K-1w=<> =:~»@(—%<;)/1W%w>~*-\} A

Ni/H ‘I \ | *F'.\ * "

7-'2_'|H2='O_ ' - '

_i~>.3:

I.

zyll *5,“=* 5'-‘i('\*\ ‘ .‘ _

‘ -,. \.-f1~f'Z‘f:.i

I . “ Avai\ab\¢ atmg.

I ’ ? ' \~wW-'“a‘hd-Aty

§)1)?@'f‘__'Z <1-(2/33213» “*9 =0 '

J , 1

I

F

. __\.’1

u

:

1

4 . °“3 O _. v 1

i=0? (-3 ~<>@<>~ S-S<>\-@%@ I - B(2"33/23 {+“§L)y21‘\‘Yi“(%*Yv@]'