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Warmup i i Is there a function f x y s t Of Sy x
2 Can the vector field below be the gradient vector field ofa function
As youmove in directionof Tf thevalue i Pf fx fy
off Hint use 2nd partial deriincreases By Clairaut's TkmAs move f falongpink y yxoath fromP If f y E fy Xf increases
Pthen fxy fyx I
fCPI Ffp But It lBut that is go no such function fimpossible existso this isnot a gradientvector field
GeometricMethodsfortindingIClassifyingExtremafII Gradient Vector Fields
Local maximum occurs at pts towardswhich all
nearby gradient vectors point sink
Local minimum occurs at pts awayfromwhich all
nearby gradient vectors pointsource
Saddle pts occur at pts towards which some nearbyvectors point away from which others point
n tt r l T f Ie T i
I j d b TMax Min Saddle
teep
gottacrpifi.cat
t saddlePt
Ex a I all vectors same lengthpointing in same
directionno critical pts i.e
no maxima minima orsaddle pts
Contour map for this function
ex I Iitfit
Itv
ye ng
Often the length of a vector is shown by shadingNotedark blue Very very short
EXIT
Pinot aP
acritical w Pipt ie.IEiii i B i
rkI p Di I
f has a local Max at A CED are saddleptsf has a local min at B
Ex 2
No local Maxor min
iiFATT wso c T
gut FAt
saddle2 pairsofvectors onopposite sideswhichpoint inoppositedirection IAisasaddleptAbsoluteRecall Extreme Value Theorem States that if fix is
continuous on a closed interval ai b then f attains
an absolute Max and an absolute min on CaibTMethod a found critical pts on Laib
121 Tested endpts X a x b compared value off at pts in 4 to fca fcb
A closed set in 1122 is one that contains all of its
boundary points A boundary point of a set DE 1122 is
a point Cacb such that every diskcentered at Ca b
contains points in D points not in D bnoofhidnarfz.pt
of3
al t.jyn.inijmsPointb
closeo not dos HInot closed
seDH
Iclosed
a if
theonlyboundary ofD not closed
A boundedset in 1122 is one that is contained in
are bounded but g Isome disk lattenot bounded
b.yinfg.at
ExtremeValue Theorem for a function of 2 variables Itfis continuous on a closed bounded set D in 1122
then f attains an absolute Max an absolute min
on D
ttowtofindabs.extrema.ciFind critical pts on D
2 Find critical points on the boundary of D using singlevariable methods
3 Compare value of f at all points from Cil E Ca
the largest value is the abs max E the smallestvalue is the abs min of f on D
EI Find the absolute extrema of fcx.gl x 2xyt2y onD xy C1122 0 EXE
3 OE y E 2
Find critical pts of f Cin Dfx 2x 2 y O fy 2 2 0
X Ix y
Critical pt Chl in D
Draw Dmy boundary ofD L X o 0EYE22
2 13,2 y L 0 EX E3
4
g
x 3 0 Eye 2Too Ly Ly y O 0 EX E3
Find critical pts of f on boundary of D
L x 0 OE y E 2plug into f flog Ly gCy 0,2
Tiamat the endptsneed to check0,0 E 012
ftp.ffin3 gcxs Hxin YIIYzt4nvcritical pts 2,2 critical pet
endpoints 10,2 already at X 2
3,2 have 0121
Lz X 3 0EYE 2crit pts one
endpts 3,0312
24 y 0 OE XE 3crit pts co o
endpts cool030
Compare values of f at pts found
if i