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