Tangent Planes andDirectioualideuvative.si/2eviewW

Assume you are given a function Z -

- floe , y ) and

a poult - Coco, yo ,

to ).

=P.

How do we found the Tangent plane to f at P ?

you have seen in class thai the following vectors

belong to the The Tangent plane i

J = L I,0

,f. else , ya to ) )

D= LO,

I, fylxo.yo.to ) )

so if you want thee equation of the plane you

find the normal vector

it =T x T = C f se i fy ,

- I )

and after some algebra you geti

Z = Zo t fsc Coco, go ) ( x - xo ) t fy Go

, yo) ( y - go )

huearap.pro#eatieu thee idea is that thee value of a point on

a surface can be approximated by a point on thee Tg plane .

Save thug youdid in Cole I :

Doc = x - Ko by e y - yo Dz -

- Z - Zo

( xo.yo.to ) is a poceit on thee surface while be, y ,

't ) ou the Tg plane .

so thee eq . of thee Tg plane tells you that for small changes

of doc and by ,

the changes ou z,

Da, is given by :

DE a fsc Coco, yo ) Doc t fyfxqyo ) By

Deet given e function z = flag ) and a unit vector -b ).

then the direceiaealoativeoffcx.ee ) air the et direction

at a point Coca, yo

) is given by

D-ufcxqyot-fcxo.yd.outfyfxo.ge

g¥ieEr of fix , y ) at Coco, yo ) is defined by

J-flxo.yot-Lfxcxo.ydifylxo.ge

with this notation D= f Coco , yo ) = Of Geo, to) ;

wi

scalar product

Exilegiven the feueiioee fix ,yI= e

" F-

y'e

"

a find the equation of the tangent plane at Coco, go) -

. he,

I )

Cu, find all posable partial derivatives of f

Cue) find the dewoitine of f in the detection of si,is at the point

C o,I )

C V ) at thee point Can, find the direction where wax of dewoitine occurs .

Solutionil let 's feud thee partial olezeroetiuesn

f-a( x. g) = ¥ fcxiy ) -

- ¥ ( e" T

- g3e" ) = rye

" 'T-

y' ex

.

fycx.gl --

Ig ( e" 'T

- y'e" ) -

- jet safe ".

in our craze Coco, yo ) = Co

, , ) ,So Zo = f Coco

, yo ) -- I - I -

- O

f- xC o

,D= I - I =D f y Co , it =

- 3

so thee equation of the e tangent plane at Coca, go ) =

( oil )

is given by i

Z = Zo t fxlxeyo ) C a - do ) t Ey Coco, yo ) Cy - yo )

Z = - 3 ( y - i )

Cm) we have alzeeeocy fooled foe and fy .

So now we need

fax , fyy , fog and f-you

. fax = Is ( rye'T

- y'ex ) = ye

'T- gse

"

fyy -

- If ( -

zaget F

- sy'e" ) = -

E Ey(g- I .

e' %) - age"

=

= - z ( - Ey- I

e'T

+ j÷

. aged% ) - aye

"

.

fay = fyac = Ig ( rye'T

- jet ) --

Lyfe'Ttry .

ed % - zg2e"

.

Cues we have a forcer be for thee olirettoeeal deductive only for

unit vectors ! Lc,IS

.is not eerie -

First step there is to make it eerie. it =

Ill, , ) I

Since Ici, , > I - F

,

it =L fz , rt )

remember that Dei f Coco, yo ) =Ff Coco .ge ) . it

.

Ttfcxoyo ) = Tf Co, 1) = L face , . ) ; tycoon ) = do,-3 )

so we get : Dei f Co,D= 20

,-3 ) . L 'E , fz S = - ¥

( N ) we want to find thee direction,

i. e . a vector ei,

where the neat directional dewoitine occurs

Dei f Con ) = Efcon ) . it =to flan

t.lu/eosoEfco.n#but it is a cent vector !

so what we have found is Die f loci ) =I Effort I eoso

Pthink of this as a feuetiou of O that you want to maximize

you know that the cosine is maximum when 0=0 ( cos 0--1 )

0--0 eeeeoeuo that it and Tf can have the same detection,

C and detestation )

So et is a cent vector wi thee save olerecetoee as Tf Con ).

we know the ai Jfc -0 a) = Eo, -3 ) so it = L0,-3)_ = LO

,- I )

I Lo,-331

So the eeeaxiceu of the detection at derivative of f at Co, , )

occurs along the direction it = do,

- is.