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.
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