{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 14 0 0 
0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 
0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal
" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 
1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 
"Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 
}{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 
2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 
1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 
0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times
" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 
2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 261 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 
1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Normal" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 
2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 264 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 265 1 {CSTYLE "" -1 -1 "Times" 
1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Normal" -1 266 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 
2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 267 1 
{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 
0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 1 "\n" }{TEXT 261 43 "An\341
lisis I - Matem\341tica 1 - An\341lisis II (C)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 15 "TALLER DE MAPLE" }}
{PARA 266 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 14 " LABO
RATORIO 6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 
-1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "En este laboratorio h
allaremos extremos de una funci\363n restringida a un dominio D.\nAnte
s de comenzar es conveniente que repase el m\351todo de " }{TEXT 260 
27 "multiplicadores de Lagrange" }{TEXT -1 3 ".\n\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "with(plots):\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Buscaremo
s los extremos de una funci\363n f(x,y) restringida al dominio D=\{(x,
y) / " }{XPPEDIT 18 0 "g(x,y) <= 1;" "6#1-%\"gG6$%\"xG%\"yG\"\"\"" }
{TEXT -1 7 " donde:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "f:=(
x,y)->.5*x^2+y^2+.5*x*y+.5;\n\ng:=(x,y)->x^2+y^2;\n" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 58 "Primero haremos algunos gr\341ficos para entede
r el problema:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 425 "graf1:=p
lot3d(f(x,y),x=-2..2,y=-2..2,view=0..1.7,numpoints=1000,\nstyle=patchn
ogrid):\ngraf2:=plot3d(f(x,y),x=-1..1,y=-sqrt(1-x^2)..sqrt(1-x^2),nump
oints=1000,\nstyle=patchnogrid):\ncirculo:=spacecurve([cos(t),sin(t),0
],t=0..2*Pi,thickness=2,color=red):\nborde:=spacecurve([cos(t),sin(t),
f(cos(t),sin(t))],t=0..2*Pi,thickness=2,color=blue):\ncilindro:=implic
itplot3d(g(x,y)=1,x=-1..1,y=-1..1,z=0..2,color=yellow,\nstyle=wirefram
e):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "display3d([circulo
,graf1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "display3d([cir
culo,borde,graf1,cilindro]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 33 "display3d([circulo,borde,graf2]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "Hallemos los
 extremos en el borde por el m\351todo de multiplicadores de Lagrange.
 \nPrimero calculamos las derivadas y luego planteamos el sistema de e
cuaciones que da el m\351todo. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 80 "fx:=diff(f(x,y),x);\nfy:=diff(f(x,y),y);\ngx:=diff(g(x,y),x);
\ngy:=diff(g(x,y),y);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "
solve(\{fx=k*gx,fy=k*gy,g(x,y)=1\},\{x,y,k\});\n" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 87 "Dibujemos los puntos cr\355ticos que obtuvimos. En
 el dibujo se ve que todos son extremos:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 447 "extremos1:=pointplot3d([[.9238795325,-.3826834324,0]
,[-.9238795325,.3826834324,0],\n[-.3826834324,-.9238795325,0],\n[.3826
834324,.9238795325,0]],symbol=circle,\ncolor=blue):\nextremos2:=pointp
lot3d([[.9238795325,-.3826834324,f(.9238795325,-.3826834324)],[-.92387
95325,.3826834324,f(-.9238795325,.3826834324)],\n[-.3826834324,-.92387
95325,f(-.3826834324,-.9238795325)],\n[.3826834324,.9238795325,f(.3826
834324,.9238795325)]],symbol=circle,\ncolor=yellow):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 53 "display3d([circulo,borde,graf2,extremos1,
extremos2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 100 "Busquemos (aunque en este ejemplo es evi
edente la respusta) los puntos cr\355ticos en el interior de D:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(\{fx=0,fy=0\},\{x,y\})
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Es claro que (0,0) es m\355n
imo absoluto y esta en D." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
64 "extremos3:=pointplot3d([0,0,f(0,0)],symbol=circle,color=yellow):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "display3d([circulo,borde,
graf2,extremos1,extremos2,extremos3]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 
0 "" }{TEXT 258 13 "Ejercitaci\363n:" }{TEXT 259 0 "" }{TEXT -1 37 "\n
\n1) Encontrar los extremos del plano" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z = 1.5*x+y+2;" "
6#/%\"zG,(*&-%&FloatG6$\"#:!\"\"\"\"\"%\"xGF,F,%\"yGF,\"\"#F," }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "restrigido al dominio " }}
{PARA 263 "" 0 "" {TEXT -1 13 "D=\{(x,y) /   " }{XPPEDIT 18 0 "x^2/2+y
^2 <= 1;" "6#1,&*&%\"xG\"\"#F'!\"\"\"\"\"*$%\"yGF'F)F)" }{TEXT -1 2 "
\}." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 38 "2) Encontrar extremos del paraboloide " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {XPPEDIT 18 0 "z =
 x^2+y^2;" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "restringi
do a la recta " }}{PARA 265 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "x+
y = 1;" "6#/,&%\"xG\"\"\"%\"yGF&F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 53 "  \n3) Hallar m\341ximos y m\355nimos relativos del funci\363n \+
" }}{PARA 267 "" 0 "" {XPPEDIT 18 0 "f(x,y) = .4*(x^4+y^4-x^2-y^2)+1;
" "6#/-%\"fG6$%\"xG%\"yG,&*&-%&FloatG6$\"\"%!\"\"\"\"\",**$F'F.F0*$F(F
.F0*$F'\"\"#F/*$F(F5F/F0F0F0F0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 260 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "restri
ngida al dominio\n" }}{PARA 261 "" 0 "" {TEXT -1 11 "D=\{(x,y) / " }
{XPPEDIT 18 0 "x^2+y^2 <= 2;" "6#1,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F'" }
{TEXT -1 3 "\}.\n" }}{PARA 0 "" 0 "" {TEXT -1 43 "\241No se olvide de \+
analizar el interior de D!" }}}}{MARK "0 0 0" 1 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }