and whenaretk-misadmaxs.tt they at criticaljohnston/m128s19/9-2-4_02.pdf · ei: fix, y) = 2×4 t...
TRANSCRIPT
Functions of 2- independent variables-
=L = f Cx, y)
whenaretk-misadmaxs.ttand : They on occur at
CRITICAL PTS City I
# •
2¥ HI y'T = O or DateY¥i*i"÷:t
..
±
Partial Derivatives-
Z = fix, y )
IFT C x, y ) = la o
f k tax, y ) - f Cx
, y ) 1stE-.
F partials
fiftyhgyl =
'a'if , oftaHajfCx#Mixed 2nd partial ,
"÷÷÷÷÷÷¥:÷÷÷÷:::::iFOR us they are EQUAL
Exe: z = fix, y ) = cos Hy)
fx = C cos CxyD× = sin . ayy,
¥EEIM)=ftscxDg
-Chain Rule
=- sin Cxyl . y = -
y sincxy )
qtxff.gg#fy=-sinfxyl.Cxyly=-sinCxyl.x
ym
= -Xsincxy)
Fxx = ffx) ×= ( - ysincxyl )×= - y cos Cxy) .
×
xm
= - y'
cos Hy )sin I xy ) ) =
- x cos Cxy) . Cxyly
fyy = y= ( - X
y
=- XZ cos Cxy )
fxy = Cfxly =C - y sin Cxy) )
y'
- (y - coscxy) . x t s inky,
=- xycoscxyl - sin Hy ) =fµ,
EI '. z -
- fix, y ) = y ex Y
Fx ( ye" )
,=yCE
= y . It . Cx y)×= yet ?y
'em.y=f
Fxx =C y
' ex Y) x= y
' Ce " )×= ye
fy =
C.y My =
y.IE?Cxylyt e
". G) y
= y . e' "x text . I = /xle×tteT
Fyy = ( x y Et te" )
*= X Cy EY )
yt (ENE
= X ( y . EY. x t EY . e) t EY . X
= X' ye"
t 2x ex 't findfxyandfyx.ITmake sure they are equal.
EI : =L -
-fix
,y ) = X 't 4xy - 3xy2ty4
Fx = 3×2+4 y- 3yd +0 = 3×444 - 3yd
-
- -
Fxx = GX - 6xy=
Fy = Ot 4x - sexy t 4y3
= 4x - oxy t4y'
fyy = O -6×+12×2 =
-6×+1242 EQUAL
f×y= ( 3544g - 3y' )y= Ot 4 - by = 4 - Gy
fy×= ( 4x-I-xyt4.PL,= 4 - by to =/4-6
⇐ : Find the CRITICAL POINTS of
fix, y ) = X 't Xy t y 2- 3 X t 2
O = fx = 2X t y - 3 ⇒
y=3O = Fy = X t 2 y
-
Fy : Xt 2 y = X t 2 ( 3 - 2x ) = 6 - 3 x = O
⇒ X=6b=2T⇒ 1--3-2121=-171
Exe : Find the CRITICAL values /pts of
fix ,yi= txt Xy - ty = I'
t x y - y"
-xtytz -
- xt¥y=xt¥I××tYIiNot in the domain
x : x¥of AYN IIt x' = O
A.
.
Y⇒Ix ⇒ F-
Er
Linear Approximation
•••R=tf¥Fµfix ⇒ ..÷÷ .
TSuppose
I kn~AX
Tangut u
y - ffxtt ) -
-
f'
HH I xxx )
Linearity
i.IT#tAx)=fCx*)t/ftx*t.AxJJ
'
L'mearApproxinatZ = fix ,y)
F ( X*tAx,y*tAy) - .
•
,•:
* a# pithyx*tAX
-
'
• !:linearttpproximati.no#tAx,y*tAy
-
ffxttax , # ay ) a fHYy* ) t fix't,yHAXtfyfx*y*) . Ay
EI : fix , y) = 2×4 t sins
Estimate Fito, 'T ) using a LINEAR approximation
( x't
, y't ) = ( 0,0 ) fx=4xy tcosx
f*=oAlo,a=otI=±¥¥¥¥×
Fy = 2×2 fykfo) = O
fit ,t ) = f ( Otto ,
Ot 45 )w
W
Ax - Yeo A f- 115
I 0 t Ii AX to . Ay
= ax 4410J