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