TurmaB

B

G A~AK \ TVTurm~A .Q.. B

F3Qc.<:~estã~ (3,5) Seja G = G(x, y) uma função de classe C2 em IR2 e considere

. .!l". 1.!l". dada por

F(s, t) = sG(st, -s).

8F 8F 82F(a) Calcule -(s t ) - ( t ) - ( ) .

8s ',

8ts, e

8s8ts, t em termos das denvadas parciais de G.

(b~ Sabendo que o plano tangen~e ao gráfico de G no ponto (-2, -2, G( -2, -2)) temequaçao x - y + 2z + 1 = O, determme o vetar \7G( -2, -2) e o valor de G( -2, -2).

(c) Determine o vetar unitário 11= (a, b) tal que a derivada direcional 8F(2 _1) .

, . ~_, sejamaxnna.

uU .

1='0 0:-) ~ A G (~\.""k,\~V,,-t~

I ) X (-1 ,-10) ~ IX ~ ~ ~ -r ~ ~,A

~~(~I~~ -A~ ~::-~~~~OU 8A 'd t

2-

f ~e~ -fi,L r tWv CD~r~ ot..c-~@.

~ ~ (A,~~ i. G0:0;ê> ,l\(I",.t))+ Ar~(ÂJ/"f),~l<itJ~ + ~~'\-t\IAô~ ~) C7!>

G Ln. 0.1':, 'd~\ 1)

<0 J

~(~,-0 ~ G(At -A) -1-j\, It ~ (M,-/»- ~ v.-t\-i;1 .?lI;.

I l(7)L_ ~~ . J I. ~--, ""-"

~" -- -- -- -

~(A -l) =' A) ~(~l'" ,10\~

(A ,-t))~ -t _3G. (X\il,~,~(t>I~~~ lôt I L 'Ox 'Ot d':). ,. otJ

~\I>-t\ -:o /;,2. ~ (Á1::, -i» .'O

\v )

8Ã--

9L-F rh 0- ~ 2." dG (4 -h)-ttt~l;;tI-~~+iG(~-~~

_ _ \ ,./) /'"_ I 2. tl A <dtJ dX.-- 3~

of:,dt ~IL 2'f. un --j -

iF _ 0 ,-t) :o 2-;.~(;;t!-;l)+tt4 (~,;~-l ~(~I-~oA ot . 'dI-

ô7C ~d~~ ~ ~._" ~" ~_.------~

._.

;. I-

b).

~ T o ~ ~ OJJ ~ G /vIN ht, (-2 -2 (;/- --2~\

Á

r - I I. c2, éJ).

o- -7»:

-Z,-2"-

b ~ . ~021~2)) ~6W,

~ 111) 6::C b(-2, i) + R(X-'l-2-) "bC~:I::~ .,'u '.

~

~ CL-~~~-~~~~--=- c; f2,--2) ~

r_ ..1)" -t'-,,-

~ 6= iL

V0~vJV~->- .e. J,l- )~uJC D~

(À ~ - :i (0

L LO- âJ" + G (-21- 2-) "

--~

.' 2 t,;}- 2. t~),.G:t2/-2)~ ~

l~:G(-21-2) =- -

Xz iZ ~ b tZ,-2) " ti ( ~,}

. ~

2) G ck G~~z- ~ ç cL ~

6;2- JL ,', ç=- ~I

oL~~ ~ ' r:~ ~ elo. ~) -kAMfr>

~. (62,__

~ ~ '\J~ l~\ -I)' It = \\<;J~ (QI--~ 1\ llft 1\

~G-

~~

'

(J~

'.

"'i2.h l02\-\) CÀA-i4{- ~

~lciMuirU)40d-

olJ- .\?OJW-- !vL "'- ~ .

OL\. s~ ~\\ 'T ~ loLI --D 1I

VW\<L A":2 ~ -te -( ~ ~,~ ') = lAtf-~J= t21-~

R.-cVo\ ~ (À.

9F l~ 1-1)= G02,-2) --~ H,~2)- 2ôG (-2,-2)

9pôX' ô~

.;"-1-2l~+i)"-1

]i (d,-~ ~ 4- .~(-21-~ ~ lt(:i\ = -2 .

~,r' À;1t

.27 X

2.)

~ \J F(2,-1)" ti I -,) 1I'\fr: (2,-D I) ~ If1T ~ ~ ~-1 f-J..

-2\H\- iR \2

I

~