FRIDAY OCTOBER 23 EASTPRESSRECORD

LASTTIMEOLDefied open theopen disc opensetsand open neighborhoods of points

r Bthedisc H

of radius gmopenneighborhoodof P

centeredat Q iit

The limit as Cky cash of f ix yis if as lay approachcab far yapproachesL

A function fixy is continuous95 forevery point ca btElR

liar fix g LHey Cab

theorem Thecomposition ofcontinuous functions is continuous

Anexample d a functionwithno limit as cx.gl Co07

fix g XIX2ty2

FLEE.LYfHigh fix and XIL 2fa2X2

IE itI Fix y a constant along L

22 But fordifferent slopes thereare different constantEI Alo y y X

f1 1 1 1 2

Along 4 2 xfIX2 7 1 5

Socoo 7,2 does not exist

I 2.3 o DIFFERENTIATION f

SLOGAN Differential functionsare the ones that can be approximated

by linear functions

This begs thequestion

Q What is a linear function

A linear functionff RE IRf ixy

atb Xtc ya b care scalars

Don g 1123 01.12

gCXyZ l atbxtcyt.dz

A function tix y is

DIFFERENTIABLE at Xoyo E 1122

If there is a linear function

lcx.gl so that

lion fcx.yilcx.gl oyt lxo.ydycx.gl exoyo Il

If tcx.gs a differentiable

we say lex yis the LINEAR

APPROXIMATION

If lay a tbxtey isthe linear approximation

then

thepartial derivative GoyoI b

thepartial derivative gytc o yoko

How DO YOU COMPUTEPARTIAL DERIVATIVES

If fcx.gl 55 differentiable at Xoydthen

No yo Lhamoflkoothiyol fix y

T

Note flxoth yo a

moth o1 variable find

MotoTreaty as constant anddifferentiate with respect to

E fcx.gl ex7

ofKyl ye

Exampleio gcx yz sinCx e It y 2

01,2 3 1 cosC1 3 t 22

4 0 10

Exercise The function

gcx.y.tt ltX2txytyz

is differentiable on all of 1123Compute the linear approximation

lex y Z at bxtcytdzof guy 7 at Cbl1 c

1123

Ansi 0 2xty b f 11,411 2 IH 3

Hoy Xt Z c doggie1,17 111 2

day y ct ffz.ci1,11 1O 2

Icky z at 3xt2ytZ

What is a We know

Iim glay zl at3xt2ytzO oXyid d flexy.tl 441111

This can only happen if

4 914411at3tt2 lt at6

Soa2fT o

t wThe linear approximationof the function guy 2at Hoyo 7d is

lcxiy.zl glxgyo.to t0Kay x Xo

toddyexo yo zo ly yo tf CXoyo 7 Iz Zo

Let fu y or gun yallbe a

differentiable function The derivative

of fury at Ho yoo is the1 2 matrix

Dt Wo yo cxoyo Italoyd

The gradient of fcx.ysatcxo.y.si'sThao yo ftexoyo yCxoyet

Example fix g x2sityD fcxo.yoj IIZxosicydxocosly.IT

1754,1 25inch Cosell

Except gcw x y 2 w2txyt 22

Dg waxy 2 I2W y X 27