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
~