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5lJ -=- (x c, " :;: I_J ( ) -::= (( ( " .. ) E ·let r fJ I .; c J " . (.\ L \V f C\,h, X ( - . s. - \ ( ') - -- ) J J JJ or J .\.. J Pe.l ) ::- J (f I' A, ) )( c, , l,v \ \ Jx ", \ ' p ( '- , 0\ x I \ / -::- J (f X -\ eLl< r t t X j d X = .x'= [. (p I, ,j " J (x \ = . .. 'IN . \ pi> p (\. ) \ I I \ '"-- ------------_-J. \ \ i E't-v \ f P (h, ) q c.,

5lJ - George Mason Universitymath.gmu.edu/~igriva/HW1.pdf · Pro . pe~~l. U-,MPGt\')Ot\ \ ~ V. I ~ F. I (p) ~ rb -\=6 . r (/) ~ fJ '=-P ~o.x /9 "b",~e ~ p~ 1», Property . l ,n~1>

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5lJ -=- (x c, " :;: I_J

( ) -::= (( ( " .. )E ·let r fJ I .; c J "

. L~~ (.\ L \V ~ f C\,h, X

( - . s. - ~('. \ ( ') - -- )r· J J JJ or J .\.. ~x

J Pe.l ) ::- J ( f I' A, )

)(

c, , l,v \ \ ~o Jx",

\ ' p ('-

, 0\ x ~ I \ /

-::-

J (f X -\ eLl< r • t

t X~X-f

j ~. d X = . x'=

[. (p I, ,j "

J (x \

=. .. 'IN ..r~ \ pi> ~ p(\. )~ \ I I \

'"-- ~-q ------------_-J.

\

\ i E't-v \

f ~ P(h, )

q c.,

§ L�\ J � 3 Cx,t) � •

f: f (x i+) i

" � \o."' \.R!_ lo.w �

::: -

J J c:,r cA X

J�

,,

----

- ----

o.i A.\\ouJ it'IreP-0eiOnds. be~.J\ \/eh,des; - ­

kt i be, -the, \e()~th 0 ct be_ e. ch~ ',(IS- be.t>-eBl OJ r _

_ e, ass. e ttf\ rl1\ e ~ _ _ _ d:: .3 (O('.~' \j VJYt~ \lIS (Y1~/~_ d _ __ ~__ u.e 0: "no p': ~ -so d;< .J- -J.,

-__ t -J., ~ DV-t ­

''J~~(7)-~):: ~p- a. \iJ~ (\ct~ 1ha.t L u:(\ \oe. ~ r iA ,:.>rl'f~ f.) .. ~ whee d,:::.O\ , r~A - ­

- __ So PM:'~ Mel ';- _tlu{..l:fr.- ­\jJe -th~ V-: "$ ----\ ::-l Cr --lit'-) _-t - ---- ­

-f :f

\\ P oK-f

'e - \I . 1_ ~ -I ­_ __ t',\\J \: r p z. ~ \ i)'",~d <. _,.. 1'} P -_ _ _ __ \_ 'J~" FCfM,) =' ~~ (r,..- f, J~o i'/ldeed. ¥- ,f-- ­

N 3: 2Ft) f t

l

);-=-~i.~p(~"2. ~O_-~,. a)lf-o~--f--~-(

_.~-----

(V\i\e~

d =) 4- ,:>ec.o"J,<:> I~ V L SQ:> hf'

6 se<..ot'\J \ \' SCb ~ v

"",\eS• tLet Q.,~ ~f v== f'

• ,{ L 50 fW\\ -t • cGl\'le(~ f'\ '\y V If S~c. = ~ ::;> V~

r if ~ec.. ) \\1'

d .\+ f(I\ 3)66Q se.~J f\ •\/:: ==:> Y= If- ~~c. If ~ S.Uo(/) f-t \. f

• So \1'\ {V\ i \es V= ( ~~) . ~ hf'

. ~t{\ua i,,) l.\.l'\' (Of"~ J t => d - \

fd f: l c! f p"

. ® M\\~) ~ V£ 5(/)-:::So) V .(~ - ~. J '-' hrB&

~(j) ~ s ­. If V > ("

®\) = -

W . ( ~ t-.J)

cunes @• a ')5 e \

S<b = ~ ( p - ft.) ~ \

S(/) - ~If ( ~ - I') ~

5¢ _.

.... ... _---®

(C.Ot\~ )

V

5. � C.. ( C.o(\\j

• Si"'-"'- pro \.,.,_ ,\.-,.\e, .\-\-. \- V= fY1) """"''\ be co"-tl"u.•u. ''

®

\J -

®

v�

�s

� (; - l'J

s

( � -1)-

= 4-'+

= �� C �� Y.Y-CI> -t >

_?_ ( { T � -}__') �

'+tf �

= f 5 lU.

)\ ,-

100 ""i

� "'

Pro pe~~l U-,MPGt\')Ot\

\~ V I ~ FI

(p) ~ rb -\=6 r (/) ~ fJ '=- P~o.x

/9 "b",~e ~ p~ 1», Property l ,n~1> ~,.JJ a~ :>\<>p c- 0.[ 'f"-p

IS t\~~o.\\~.

L') 'f (p~to.'I.) : (/;)

V:: F (p~~~) :

I

~.J 2 F(p) P f 1/ ( pJ ~ (/) ~{ (/)'=P ~ p" v ""'"

j ~r,"\~.J Proper~r ~ ~(A. ~U.\I j rp-=td>~

~ t5. ') ( ~ ~ (b")• () ~~Q\"'a~ ~

Vasia
Text Box
More accurately, F''(rho) does not exist when rho = 1/(l+440) or rho = 1/(l + 880/3), so we cannot say if the flux is concave in the whole region, but it is definitely concave in each subregion.

1:" L~. (-

[I] @] . f\~ure S. j ~ Cl.PPeQ~ +0 be ()f' \~ver~e-j ?O-\Q.ho\o wIth

\1e{~ x ~ (SdJJ e <bJ ~ )( - ",Jer C.€P~~ 0.\ ()) ~ 1~ ¢.

~ y -:: - 'I- ( X _ S¢) 1. t 8 cD ''2.S"

v ~ - '+ pI. + li ? - ~~ * e0 l'Z.s 5

- Lf (J"'l. +fV ~ ¥- p J1'2.. 5 +­

lliJ • CS~\,l (l\\ t\ C I (p') -=I d:> -tor ¢ ~ P f f>P1O<X

J (p') == P \J = P f CpJ c (p'):- J' (p) ( 'Cq. 5..\ 5 • ? 1. I ~)

. ~o) c't.p)= J"(p):: 2 f'(pJ ~ PF/(p) f:(j) ~or cb '=PI::Pf<'~)/ J

\kJ~ ~~ \ ~ ~ f1)~ \y ~ ( p.'2.\ 6)

L ftp)' f f" (p') = 2 CI~5) f' ~2. + fC-,~~

- lj.

I'2.S

/.

V:fplee.- 6~ le\cl mIte

a R -Y.. m o.klc <)-~~~", ~ _ _ __

W~be3·\n\M"\h "=-"~(I-~) __ _

lhe {~ "J '= p" \~ -\t'efe1otf. f "111\ (\ - f;.) ~ VJ f - ~ p" 6~": ~Ulili(\(j fmin =- 6) _ W~ \<.MJJ q,oodro:tiC- hw ron"";ri\ _ ct-r_

~t~ ~(~~~-~~~~~-~~~-- ~ }:. ) ).. - PN- -"\ ~ -1 <Il\

~. . _ _ . . S\fI.te1.o)-:;o 0Jtd 'J(f~)::-

-A

p~/2. PM- P DenO\t'd: -.;':, bt w~n Flv.x I.) iY\~i1'lVO

b) PM'· jOffi rl.ofVi" .,*- \) deJ\')\~ ~e..-.e..11t).~~

• "J It\: rv ~~13S . ~ uJr\et\ ~~; ~~ I h}~h-weut i.> \.\ r -e '\ ct. O\.lJ ICDJ"S •

'" ~ p"" ~\s <\te m. f\\lJ\'\l C1 p:. flAk ~ \,.\~itI.e.f'e~re. co\\e -+\\. \\

ir(\ffir~ra'\L

5. g a.) ti- l ti .f CX,u) - .!"()()

�t ::.

d,X

- ., z. � 1- (t d

l . y 1-r; .)- { .,.r)

cl 'I -

� ..... -

- ?

v -v-

'(' -

� u(s)

j = -t u(x-2.t) +

j(x,c.)-:.. -({x)= u{r)

J"':: t 1 f{x- 2 i)

b\/ ~

Jl.

/~

- ((j.

J..r. -= .r .c) ~

2l-d

~ d)<.-ar­

~ -2)

+ rj-t

-ell

$ J-t

=--) :: - 6.,-4

-..,,J-:-Or::() =/ L

1. (5) .....

/ V - r 4 )

~.> X ­ {" >

==::> X=' bi

( = \ 6..1- ....

tidf' :;. y

-> .::dr f

:='? .Ad)=-:­ r + u(~) I

{,­

,,­

~ '3 c) :?J + 2. 6va) - (.)bi

~A ....(:::- f7­

t (~.r)

::: I Ji ~. -= I

('rJ:=:--? 5" - r)

~) ~ x- t / or

f (x/e) - f( ) := -I +-./

U ( ) t>

J= -- ( -:. -I

­i-f1/6 t) I

­

t -.fz;.tJ

:J­J f-

I

- (