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c'- rJ~l.jj E PCXJdX 1 2) Let -::""'-'~: ~ r, cJ{'('('f, ,j.yti" "' 0nJ~r< \/().r,;"Jf,fIv-?, ryr I1 (J) c c., ;t:<..CI.. rl e /rr 11/- f,t·er·)<: Qt r <. --. --- ------ Ro..rto(Or¥L ~·cJ2.eQ-? (S:S~~ c.~ \ ~---- ::::----- 1" G fR .~ '2.ClrcciofO?..- VO.fLta.i!Le, '- cAei€ .. run/necJ ~ y'i-s ~tL,L~ .. :6JCi(_ ~ p?.s::lb .. i./eJ~ to -.frecl /1:. /n -SO/77·€- I n -i.oz.-vo..J! . 0<1 !L~~ ~: ~. rY1"Q ~"ve- 0r:'~~ 11k 5~l ~_/~ 11~fKfw;HL ~~ Ip~) ,tp,,:=.L\ IJpflO~J [12 rf~ 1K ) = p;J /iP( rE (0.,6') )2:, p,,_'J'- '.. "-.- _,K} 1J(E(q g) ---,~'-- t)' c...on-t./nUoU:J:J c~e' J () _ , ---------------- ------... ' l:..h~ 1-3 P (X ) ~cIr. -ihcvt I Jp (f e (c;, g )) - .! -----. LU1 r' f~"Y7 dir'yt v ~:~d;,c>re OIL [ Cl I -1 J JJ ex) .-.1. X6 C~ 1] / 110 ( 1 E(qJ;)) ,.- g~~ A/o(fY(~ ,~,. Jt'~b.;U;on. .2/, [- X/'L ," q, 2(X)·~ ~2-/; e IP(r8(q)S»)~ ~ JJcx)ofx ~ -e ~f (g) -, e<.9i"') -£ - - xl2. ._] ~ le?? c f. '): -:::: $/ 2-- J y ~oo "0(77 }Xpg.dsJ;7'cn (m€().J'-. ~ -of CL 2n.nd<Y?<..lA\ru~ Of j f~( CL Ji~~ ~e... lE [1] =- 2,"-: ik -. P K,------?]

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Page 1: )2:,personal.maths.surrey.ac.uk/st/S.Zelik/teach/... · c'-rJ~l.jjE PCXJdX 1 2) Let -::""'-'~: ~r, cJ{'('('f,,j.yti" "'0nJ~r< \/().r,;"Jf,fIv-?, ry r I1 (J) c--.c.,;t:

c'- rJ~l.jj

E PCXJdX 1

2)

Let -::""'-'~: ~r, cJ{'('('f, ,j.yti" "'0nJ~r< \/().r,;"Jf,fIv-?, ryr I1 (J)c c., ;t:<..CI.. rl e/rr11/- f,t·er·)<: Qt r <.--. --- ------

Ro..rto(Or¥L ~·cJ2.eQ-? (S:S~~ c.~ \~---- ::::-----

1" G fR .~ '2.ClrcciofO?..- VO.fLta.i!Le, '- cAei€ ..run/necJ ~ y'i-s

~tL,L~ ..:6JCi(_ ~ p?.s::lb ..i./eJ~ to -.frecl /1:. /n -SO/77·€-I n -i.oz.-vo..J! .

0<1

!L~~ ~: ~. rY1"Q ~"ve- 0r:'~~ 11k 5~l ~_/~11~fKfw;HL ~~ Ip~) ,tp,,:=.L\ IJpflO~J

[12 rf~1K ) = p;J

/iP( rE (0.,6') )2:, p,,_'J'-'.."-.- _,K} 1J(E(q g)---,~'--t)' c...on-t./nUoU:J:J c~e' J () _ ,---------------- ------... ' l:..h~ 1-3

P (X ) ~cIr. -ihcvt

I Jp (fe (c;, g )) -.! -----.

LU1r'f~"Y7 dir'yt v ~:~d;,c>re OIL [ ClI -1 JJJ ex) .-.1. X6 C~ 1]/

110 ( 1E(qJ;)) ,.-g~~A/o(fY(~ ,~,.Jt'~b.;U;on.

.2/,[- X/'L ," q,

2(X)·~ ~2-/; e IP(r8(q)S»)~ ~ JJcx)ofx ~

- e~f (g) -, e<.9i"') -£ - - xl2. ._]~ le?? c f. '): -:::: $/ 2-- J y~oo "0(77

}Xpg.dsJ;7'cn (m€().J'-. ~ -of CL 2n.nd<Y?<..lA\ru~ Of jf~( CL Ji~~ ~e...

lE [1] =- 2,"-:ik-.P K,------?]

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17·:::: i '-e,X rd-c-ciw~I' 'o-rz..------.g2.. ~ E [ 1<-~ - cer?cuteo{ vv/ fs f2z

IvCV?.. [ 11 = 1=[ 1'-] - (E [1])~

/VWL [r1:= f[(1- £[13)"- }~

/rr;~Qn-t

VQJt.c'<.V? cc:...----~

Pe.mfl\

-- ~ enea.ru 'i!;jE [c1'"- l:TnY-::z, = I; [ T '" 2. 1 ELf] + E [)'32..] '-

= r::: [ JL l ~'2. 1::[ j J, . k f~:3 c ET" 3, ~ = E [ i '-J - EL 'f"3 '<-.0'1!?f..VY/02.

~ i.h~ ciEMioJsi'C<;n of! C\ z=ndor--rc vo.rJcJZ,~ ~

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L e;t l' o.r.J, re- ~ {wo 7...Qf7c1orrr. . va.rua ..i,,{:flY.;,,

(n1-<::J?.tL QQJ1eJzcJl. &;t -:r . . .. 1" e~ 17.-. _ L a..r; 0107/'7 VOl7_/~ )CJ / // '7

. Le,(l ':<;J(.l~ {he... d(;'S~ c.e~e.

r~--=---1JwitJ? ~-e-tcJ Pi<

1~= 1. cc: I wlL~ ~-e~ Cj, e.

IN:'-t to j!,nJ I::k j =il71; p =U.etJ

!!< 1=J~; 2'=2c)- ?J

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-- d~·~~.

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W ~d CCv<... \1\/ e... 3~) i><" EfV;YJerco£ ? :

PK= EL T~1k] = £["1=11<;, 'Z.E/RJ = r P;<.e

OYlcA )~ to· :t PI<; ~7in ·l!1e. cent/'n £40 c..e:o ~e..

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Imp.o-uC.o..nt Pwd}~L ca5':>.e.: incl-E.?"-"Ci~-t ~VorU~ 1" Of/iJ tL .' .- no CcJ1~~U2-fo--Z,I00 ~iween de.event 3~eoZ vv / t.Pc T QJ'( c;f . rz..

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E f1 '21 = s s X ':'i pcx) '0;) clX "'21 = .'{S' X ':'i j\ GX), J''1J'::l) o!Xcllj =- (S' xJVx)oCx) (S'~JQG~)oI~)= '£[1J £[2].

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I~c/~-e. o;ldemeat: /Cei7-tLc~ ~n7/"·t ·£.h-e.oC-Q-i'l(~

L e"'t XK k:= f) ,"/00 k l'nd€p~oc!Ko-t (/I~?eY2_.iOYl t / ))

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e;~ /./~)

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,'nd/epend enee.

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Page 10: )2:,personal.maths.surrey.ac.uk/st/S.Zelik/teach/... · c'-rJ~l.jjE PCXJdX 1 2) Let -::""'-'~: ~r, cJ{'('('f,,j.yti" "'0nJ~r< \/().r,;"Jf,fIv-?, ry r I1 (J) c--.c.,;t:

oflA (-0 = el"~ e~j, WiQ{)-eJL--~----J p?-O~

/-/0\1\/ to F'nd {~e cAj~l,w'U;CJn

J;~?/~J. '?

2'-O@6un;~e /f ~ ;'s ne LI"} K'vtC;-.t~~

An ,,>W€J?,.: tG<3 - n,,,,..rno.k-. cl; .). vU/on., ..Let X- = e l'. ----- --7 ~

jJr C_X.. < ~ ') _ ~<~~_~J)~~)J~" 7~-.- - , -

..S> JX ..('.1) = oli IF" (x< ~ » - 2=ew£.,/ - - ~ ----In OWL CC~:

Page 11: )2:,personal.maths.surrey.ac.uk/st/S.Zelik/teach/... · c'-rJ~l.jjE PCXJdX 1 2) Let -::""'-'~: ~r, cJ{'('('f,,j.yti" "'0nJ~r< \/().r,;"Jf,fIv-?, ry r I1 (J) c--.c.,;t:

L c.=CiL0<-S-: eE' n , cc", ~ O-nc( J eO<;i"f"l i"El L,nL 2 (!). ~~-ve.."~j~~ Y-t2JI/~ie,J. /_.- ,'.------ ,-,' , __ (06(nt Con

f(y~~---- ~~~) ik- en<=-..

- f (c1L X +j?:, IJ J , _.po/nt on -Lfie~fd'k fex)

/

I'ta.nd( ~ jJ.) /3 ~ (J J

A l~&-~C __ic ellef : .f G C ~( !R') /.~ c.on Vexi. ·f j.-f 1(( x)~ 0 v XI .

T~eo?J2.rY( (f'LOI'YL CLJ~) Le-c )-f E C'L(IR) J

~n,}Et-?./c-_. ole.(? ': .D: If).~ 10 I Q----= t If', ", --J can Ve- Xl

f~ eV"";j X, ~ E It<.

'3uch 1:AoX j?- +0~~J

-nd...X-+(?o ~) ~ d... l(x) +fi fell)

Page 12: )2:,personal.maths.surrey.ac.uk/st/S.Zelik/teach/... · c'-rJ~l.jjE PCXJdX 1 2) Let -::""'-'~: ~r, cJ{'('('f,,j.yti" "'0nJ~r< \/().r,;"Jf,fIv-?, ry r I1 (J) c--.c.,;t:

"f (~X +f1~") - ~f(X)-fb f{~) =: f!(x+j3(tj-X)--.f(x)1- CID

f6 L f(':\)~ pon1 = Sf' (x+y. S (~-X»)O<~? ('0-><)

- f2, S -J?( x+ S('cJ-XJ) cA '3 (\l~:)= - p, (~-":<) k rf '( X+ '3( ':\-x»-o 0 Ci I-r(x+~ (\:1- x) ) 1of s<, =- p, ( '!:, - x) SJ3:{ re x t-~ '3+cL?:i) ('j-x~ Xo 0

XJ..<s, ~y,-x) J.-'3 d'e" =-d..f> (", - X) 2.. S Ss-f"( X+ (f-> -'3-td..'Z:"'l)(Y,~x~ oft:

if If 11c~')~0)0V=crw£1eJl.C:> ~ a./-~

Now &t /fiE'- Ine~ -~ I?ofc{ anc4 &t ~7f2,er<- - 11

-f ( cLx + /3> ~) ==- ·f (X' -r{?J 6 X ) ~ f ex) +f(x) j6L'1X + .f~K) l~j(\"f (x -16. ><.) -=- f( x) + flex'')~)< + .~ F..\ I< L-{ " ,

f? (cLX+Jb '-.;) -.J.. Fex)- (b pc'!» = - @iI'» {'ex') 6.X":-,.-, -~-----

6X~O "~ Lfl/(x')~oJ ~---,----,..-Q.

yv ~e re d(J~~tJZi5 /0 eJ~ a.1> ~ 6'e.C01Y7 ~ §o/u.u.£i;j fi) ;'f t:.Ae fiAJ?d/on .,.,f r':3 E:J?f?J:'uc

P2.0[O>si'£Jor(: Let f &e... con ve X G\.J(cA -£f?t e. ?r-cy;yf2c

of ·f of ~ no ·t Con t,OA >( e/17EC>.-: "L 'j(<''I~'':::. le, cl '

Gut J P 11(x) > Q l VX , Tg eIt

_¥ (~X+~ v, h <k f(Xh;4?(~~c> 0:-3Y oR E '0I~f : E-E.Q...-I:.fJJe- pie t:Lu?.f.L.o.:tt!5e ~ /7/7$ a.p -Lae.. e~<kL

TR;~ l'ne'J4'tc{..t/'(J c.eve ..Be- exiendecA -to n - (JC-:>/n -£'5,.

j e.n se n '-.!.2-12-9 v\.vJ!/lg: Le { ·f tk. c.on ve..x. TR-eYL.,

[f (6" X, +H+d..~ X;, ) ~ d,f(X')+,:,:,+d..~-f?(:{j], ~ E5J~)(f+"'+d-n 0

Page 13: )2:,personal.maths.surrey.ac.uk/st/S.Zelik/teach/... · c'-rJ~l.jjE PCXJdX 1 2) Let -::""'-'~: ~r, cJ{'('('f,,j.yti" "'0nJ~r< \/().r,;"Jf,fIv-?, ry r I1 (J) c--.c.,;t:

Pu:cf.. / IhcJuve.t/<3n ~ n. .. ~e- C-~f~ ~ c-~

i (dc, x, i"d-. •..X2 +d-.3, X3') =f ("",'ch) L cvx,"/' d~ 1<2-J T "''' Y;, ') ~d\-1+d.2.. I I Ii

~ ; I~ ~

~ ~F()()il",f(lj)"=. (d,Tek-j !(cAl. +~-0=---...j)). f(.)" _i J X I ..:l. --l X 2.. + cA? X3 <."'1i"v<~ U)) T(ll 2.- ~ ••••••

n Ii t, J I-, 01, ';( F' j

~ ( d,"/' <I.,:) [01~~ f ('X / ') -I ::;. '- -Pex2-1] + .1, f ('10)""" 0( I fCx,) ~2fr1,hJ/q)~1(i,n, ~ ik ~-t . J • . Cld . + Cl! / -'3:,. C '-~jn verA)

L et ilk ~f'h 01 t do noet =ntOJn eneaJL sl!o?,,-'5(3d) ..ptICX) >0 vrX!) . T H€rL.~._~~---

f (J" X,f' f.~n. xn)= d.., f(x, )+d..,f(~)'+,L"f(Xn) '~:x.: X2.== ~+ ,-:AI- > 0 V i ~ ~~~

~, c:.o..n & agis::v'necJ f=rn.fhe- =8 ~i"d"d'6r,ROJ1JOh( v'eD,I<:Ji(: 1" e- ?J:l,nolorr, VClrtia.i~

CL D 'vv/·t Fe. pea Go. e/CI;c:! de;. ~ 0 . 62d-<.-r(.d'':>(j~ / ~_~_yLet ..f2 ~ COnv'~X. rRerc..

: (ITJ'.oo ~~3) r£ii(i)J-~{~JBI7S-V( -5>. yjl-:-:-fC-:--E-[T]-) -·~-E-[I-(j)-J }

fC5L COn tirJiA~ ""'-Oe- i t i'3> ub,o -Lwe-. (~A-e. i{)t~

OU<...L -LA €.- t irn/tJ3 0.(2 CCY2JwhfoncA/nG ,on i.~o..L. ~ ~!)\"/k-E" E'CjltAQ.(>"Cl ? - ,'I m pcs-da-.r,t l

'-------

Lei ,p 8e- Convex a.nd do not conf:.cv'n -I:Ae.. 81781'L 3e()~

( "";;3, Ire'X)J' aJ) ~ ~ t.f2uX. -tJ,e. "l.aJ?o1crT? vo.n.iolk<1' ~GL>::. nL ~ 'fv'vo -of;ff~t \/~f..fl:6 (w/~h nOl1-~erw

p20~;el~) . S> I~(E[{]') < t: [Fe n] J

[; q«c<£ ~ ,n :TVlsv-? in~o.f;~ is Q. v'~exc.w;:::16_""~~ .'.T·f ~ClA. CUlQ..... ~tI ~ un E-)<0JY}1~ w, ih ::t 4 ut I 'fU ~- Clen'?~1:: ~ ?and(1n~ vC(j{,/'o •..l~e~~· \/v/f..l. & 01« !,'

l cA i sbJ IJd/c-o'S ")

Page 14: )2:,personal.maths.surrey.ac.uk/st/S.Zelik/teach/... · c'-rJ~l.jjE PCXJdX 1 2) Let -::""'-'~: ~r, cJ{'('('f,,j.yti" "'0nJ~r< \/().r,;"Jf,fIv-?, ry r I1 (J) c--.c.,;t:

SeVfG~ .fo..rn~ l:n~£/e51,. (exc<Yn?~ ') :

I) Ine.tp~·t.(J &Lween OJLI'-6hmvtje.. CV7e'i ~'p~c..

x- .~0) L

Page 15: )2:,personal.maths.surrey.ac.uk/st/S.Zelik/teach/... · c'-rJ~l.jjE PCXJdX 1 2) Let -::""'-'~: ~r, cJ{'('('f,,j.yti" "'0nJ~r< \/().r,;"Jf,fIv-?, ry r I1 (J) c--.c.,;t:

CL) R.o.nd0rY<. =rr1""LVJd'~(J OJ?O( d! ~C<>.<n L'NIn. cde..tefl.JY7,n/-d/e- ~ :

':5cvne.. ·fez. o.J2t 8~~.1 )

''C.cV7dolYJ vaJGJo..ie~

f" (L ) . 4.(L ) = .i]eut CD iJe 7£v?cAorr-c (·the~

A-t ( i..) anJ. fJ-t. (i..) '- J!.,:>o

A~ U) = (f + '- )"- 32r>?j'GJurdlr/(j ) :Ut. CL) = Cl f L ) -t_ of! ~=t;,'2)O-ncA

L e n'r iJ'I, Cl. .

{ETAlU)j E[:r1t: CL)] ;>~ ,I L' is; not ~;n'.,/,iC

A-oof, Foc&'IA/'S fZDrrL m<5Z.f:L fr'U7~ ;;7~tJ

etLr~~.EI[~j - ~en.en ;n~cd;tJj IUL ~(X)= ~ '® .' ,x, o{o not CCfl~/? el7~ s~ ~ ;l7eCfl-(Ci~

6~ / ~ '3L:z/d. 0~~~4 u~

Exam!".&...: Let ~ & o€?6?,:cA 1:0 P"?j Y?R.. £' pound~ a..i:

-l",ne. T o..noi r?d: fJ2,e i.,..,t~ w.Xe.. (!) f5~ 2s:v7cAorn

~d!.,oil.< 4u:t/o0 ; ", /R:, nown l ). !-IovV m ucf.c.. rno/?'a' e.c:vc VV E"_~o vv fli>,--,.,if!, e. eCU7 ~ aL t = O! .

~_o_eJ_'_lo_n_ (na../ve): T/2;.€...."... fy~t; vc~~ afJ p _-L. T eve -if ~ e..,.moment t=o i", Q ~dorrc vonicJ~

/!V(L) 0 (/r<S"]

B l~t,W EO- c.o..nno t. .lSc"fWYI( Q. '1.r:J1ci"'t"rf a..n-, mO«n i oF men ~f'l..orr< the tfCV1 ~ 1I -0 need !>Oh']etf5/'(j r:A~f/}/"ti c..

-- (<;,'m~ ~oeulio0)~i.~e. ihe.. eXf>.'!:..da:tior, 77 _!in~werz.. ~ _

IPo ::: P £L( 1+iy I> 1J IN fi<?:J<£ ; '},$e fP'=glJu-,.-, :

eo..cJ. CRC7./c:.Q.. / .1 ndeecJ f.fie- fuiun_L \;~)

,'3 CL2jwn G~ 4'V1c.tOrh VGv'?.'~!F V ( i).:=:;. .Po (It ~)n !

Page 16: )2:,personal.maths.surrey.ac.uk/st/S.Zelik/teach/... · c'-rJ~l.jjE PCXJdX 1 2) Let -::""'-'~: ~r, cJ{'('('f,,j.yti" "'0nJ~r< \/().r,;"Jf,fIv-?, ry r I1 (J) c--.c.,;t:

Wt·£g ~><~cxo.:i-.Jor;" _._ __ OD

L~[ FV(i)] -__Po E( \If" l") =E ,-F ((It-i) n) _[( (jft>") '> p]go; /n me...o..-n ~o~ w/ et.. 6:.~

}P. ( E[ (I-ri)0] -.£-L-~(-l+~-)-:-]_- -1-)-:>-0-7

,'./2 ~o.u- ~ f~;'3 ~i:(~~, And.. ~he- C07.J~cJC 0J1~'C...

/~J ogv/00l--~Yo 1::> () IPo _ E[( I+<-)"J . c.JZec.f<; ,

fn=e.. oIe1.cv& 6n.. expedo.:l/<:Jn8 of C-OI??fX'«nc/,I?!J o.no(o1/~0:>un6'n8 -f(A~.

l~iu!- '/fU e%/on iRe.. ffAnJJo r't If' (x)=- O+itl =- If (x) ~ X :] >( ~ C?

,'~ /-/:. c:.onVeJX CJL con c:,c,-I\/e ?

o i(X) ~ --t (It x) -t:-;L. f /1( X ) =- I:. (l:.- ij (/+X 2t -L.t ) ~o

An'JweJC..: Convex; if! g~ ~o] ~~ E§J 0<. ~

ConcQve: ;,f /t:_(-L-/)<O! z~ tE(o;r)

Lef'Y<fYlQ' Let ([)-tk«vxlorrc (the Same /o<-o.i1..t,~)

and. I) ~~) ~)

~ lE[ (I~'T'].> E U~TL)I-1oJ /f[(l+iytJ

2) Let ~3. TgefL

[ E [ (/+<. )<1.+,9. J ? E[ (If I. )"J ' f [ ClTi. /' ~]

( ,1 0,0 Me fu..nc.i.Jond e(,~Q>t;'oy( )Ro.ndorn =~C H~ '--t--(:-~-/ f<n.. t1>e. e.XFda:L'OIJf$., ~i) = Of-i)· IfG~

---.::.------ T~_-t ' X>P2..oo.(Z: S;/nc..f2- T;7--t > .i-Lfi-e.. ·funut/o'( X '? X I~ ccnVe-. --

J€-ns-en ~ T,' " J----~ , . ~ ", T-'tJ) /-i,-'t} EL (liLl} =E[~(I'irt') 1-10] ~(E'l(l'L) ,

Page 17: )2:,personal.maths.surrey.ac.uk/st/S.Zelik/teach/... · c'-rJ~l.jjE PCXJdX 1 2) Let -::""'-'~: ~r, cJ{'('('f,,j.yti" "'0nJ~r< \/().r,;"Jf,fIv-?, ry r I1 (J) c--.c.,;t:

AnCLfo3~)

)E [ (I t i ,)0> ]

a..nd

IEr (I-I'[)t?> ]

c.on...ve'x

CiD Ro.nctoCD... ~n-t and j?lJ0tUfUL VG'e,~ .Let n. '.

( A K 'f..) - (Je CL net c.~ "S tULc:;ur'0 .~ ?>oTn-e-.) K. K=o '

~'n~ pi-o-jed:: anJ &;;t © !k- t/J;e- ':?.D-ndorn

In·teJ"lJ2/.Jc 'Z-G\..:te.. .T~eT'L) t!Je p'V o.n..al .~V CV?SL ~')o zo.-ncio/'V}

!P V ( i ') = i: Ak (I-tL )' ·tkl IF VT (i) =- i..~1(I-n' )T--t~ Ak.J,,,~ol) ~o I

wf!,d olx-~ t~e .ex~da.i,,·on~?we. RGWE:..- tf,e... ··fcJ(JryJu [CL

.D- Vr (L)-= (1t-L)'. PV(L) J -- not. 'hw1'!- foG.. tf-O---1---=-----L..:.=--=-----=--=- __ 'j e.Xfdai,on ~ .

Tg eO{j2j'Y?: Let ~ co.:rJ2L,feovv--s Ai<. > o} K~ I }._ ) n f3e­

1~"V·t,'V'e. (e )(C~ ol ·/;he /r)lt.I'a.i!..- ;'11 ve.>kt7?ef7·6 Ae) o'/nd. 0I.;;t:

~. T~ert

;-E-[ -F -V(-I. )-] -<-1.:-- -[ (-If-I- -{ J-'-. -E -[ -P-V-( L-Y-] ].

Page 18: )2:,personal.maths.surrey.ac.uk/st/S.Zelik/teach/... · c'-rJ~l.jjE PCXJdX 1 2) Let -::""'-'~: ~r, cJ{'('('f,,j.yti" "'0nJ~r< \/().r,;"Jf,fIv-?, ry r I1 (J) c--.c.,;t:

o

Pu::o..f2; I nrJeecA1 ~>;o + eemfflC( 9 @n .~ '------..- n

f[FVJ == Ao' f[Clti,rJ+ £'AK f[Ofi)'-tkJ < lio' E[{/ii)'] +Z(A/('EI(/-Iz;rJH{it~}~K"'-i K"i

.= E[ (Ifi] J ( Aot ~' JlK f[ (Ito)-~"]) = f [(Ni}'J . £Lpv] 0

.Con~/,&~: /f the. ~'/7~ rwj~ /-3 ~c.h ~t EL PV] z:.O

(er~06r6-n of vc~ ~i;/!x:xl} -/;,4€~ £ [i:"V] < 0 ono( WE:-. e~i/1-e.. rnc~ n i'n rn-eo...n 1/.

® rhe ~ of 2.arlcle>rr, ,nt~ <o.XE'_ Mt{!, ,nde("encknti n CJt..J:!.177 -E:f7 -6' .

;Vow, d,ff.QJLQn t UlT.0?-=-.i:. z.o.;t,€5', I'=.- ev=cJ ~.LJ.. I L2..) L~) ._'? Lt( OJ(cJ.t.h~ ctn.e..- /ndE/;'iYldE. ..i1't 'w/tJr i ~~~'O(//

(AI< I iK. ).20 - net c.c~ d?.e.s.-IJYY( I~"L somE f1u..v/l~:> [>'2.0) ed.

IFV =: 40 (if",) (1+'0) +/1, (/+'2)(Jf'0)+:A~) ) PV=. 40 + A, ;+,,);~(Jt,5~Ot,,)~-:;";;,;,)'(,j~"

Iwut_tEr1 u:'dvL udJ.'LonoJ ~(:£ion .G!::- 5J~f=cJYYI""£'" ,oci- th~ 1Le/I " cl Llr77~ GUll cs (n' o.~ f&-w~ 'r7>vde. oi !,1rY/- LAni -k . )~ 'll

FV O-nd PV o.l1J2- 2.OJ£donr VCVliC~ ~cJ". wMcxX aku.i: expet::fu.-6c:s."

T~e~ t)Ld /tK=~I T~en...

/E[J:v] = FV(HLl) = i;, A" (I+EfL})T-~

2) Let ~ and I,AK ~'2J fo'-.- ~?~TRe"-. -------

}E[FV] < .(/..t£[i.])n.EL PV] 1

p~co.[Z: )'---:- i) ~i'l?~ (,; Po LI( 0Jl..JL InJeP!E1deni.- 1=[ (/ti,).l.ooCftLk,)]= (HE(L) /"_

Ref'Y>01.,~: J) /~ ~t ~ i·1 "So/Y7e of tK. CN'Lc2.- /lo-i, ,'/7 t~, "Slnc..e...

-to?- ()<t<.:L ~E [0+£) t:1:F 11lIZ1J (1+f[ i '1 ) i: }

2.) ~i°tfC12.- LJ "'Lf( OJUL /ndependen't onc{ E [ ~]., ~ [ y~1~J..

. ~ 'n-~Q -I-ED-J)n. E [(ifLIY" (Jt~"y J::: C f[ (IN:,)J· E[Ufi, y'}...(E[Ufc;~')l· E:[(lh~5iJ); ~l-tE[£J)'>i. :::.,L

> ( 'O)n-R. -['." J/tELLl .= 1=- (1-t}.x-tl)·(J+f0t'2. ) .. (J-tin\