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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 "
\n" }{TEXT 275 43 "An\341lisis I - Matem\341tica 1 - An\341lisis II (C
)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 276 15 "T
ALLER DE MAPLE" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 14 " LABORATORIO 3" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Los comandos " }
{TEXT 256 6 "plot3d" }{TEXT 277 3 " y " }{TEXT 278 9 "display3d" }
{TEXT -1 1 "\000" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 " Para graficar
 " }{TEXT 256 27 "funciones de dos variables " }{TEXT -1 24 "utilizare
mos el comando " }{HYPERLNK 17 "plot3d" 2 "plot3d" "" }{TEXT -1 1 "." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "z:=x^2-y
^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot3d(z,x=-4..4,y=-
4..4,axes=boxed,orientation=[76,46]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Grafiquemos d
os funciones juntas;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f:=
ln(x^2*y^4+exp(2))+7;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g:=
9;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot3d(\{f,g\},x=-2..
2,y=-2..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 78 " Si queremos dibujar en distintos colores
 cada figura podemos usar el comando " }{HYPERLNK 17 "display3d" 2 "di
splay3d" "" }{TEXT -1 33 "  que se encuentra en el paquete " }
{HYPERLNK 17 "plots" 2 "plots" "" }{TEXT -1 87 ". Primero cargamos las
 tablas de puntos correpondientes a las figuras en las variables " }
{TEXT 256 4 "fig1" }{TEXT -1 3 " y " }{TEXT 257 5 "fig2 " }{TEXT -1 
17 "respectivamente. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "fi
g1:=plot3d(f,x=-2..2,y=-2..2,color=pink):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 44 "fig2:=plot3d(g,x=-2..2,y=-2..2,color=blue):\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display3d(\{fig1,fig2\});" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 13 "Los comandos " }{TEXT 256 13 "implicitplot " }{TEXT -1 
1 "e" }{TEXT 258 1 " " }{TEXT 279 14 "implicitplot3d" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 13 "Los comandos " }{HYPERLNK 17 "implicitplot" 2 "implicitplot" "
" }{TEXT -1 4 "  e " }{HYPERLNK 17 "implicitplot3d" 2 "implicitplot3d
" "" }{TEXT -1 30 "  se encuentran en el paquete " }{HYPERLNK 17 "plot
s" 2 "plots" "" }{TEXT -1 68 "  y podemos usarlos para dibujar curvas \+
o superficies dadas en forma" }{TEXT 265 10 " impl\355cita" }{TEXT -1 
57 ". Antes debemos \"bajar\" el paquete plots con el comandos " }
{HYPERLNK 17 "with" 2 "with" "" }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 12 "with(plots);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "implicitplot(\{x^2-y^2 =1\},x=-3..3,y=-3..3);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "implicitplot3d(x^2-y^2-z^2=1
,x=-3..3,y=-3..3,z=-3..3,\nscaling=constrained,numpoints=1000);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 11 "El comando " }{TEXT 280 10 "spacecurve" }{TEXT -1 1 "\n
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 11 "El comando " }{HYPERLNK 17 "spacecurve" 
2 "spacecurve" "" }{TEXT -1 30 "   se encuentra en el paquete " }
{HYPERLNK 17 "plots" 2 "plots" "" }{TEXT -1 114 "  y podemos usarlo pa
ra dibujar curvas en el espacio dadas en forma param\351trica. Antes d
ebemos \"bajar\" el paquete " }{TEXT 281 5 "plots" }{TEXT -1 16 " con \+
el comando " }{HYPERLNK 17 "with" 2 "with" "" }{TEXT -1 1 "." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "spacecurve([cos(t),sin(t),t,color=g
reen],t=0..4*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "spacec
urve([x,x^2,x^3],x=-2..2,color=red,thickness=2);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 119 "spacecurve(\n\{[sin(t),0,cos(t),t=0..2*Pi,co
lor=red],\n[cos(t)+1,sin(t),0,numpoints=10,t=-Pi..Pi,color=blue]\},axe
s=FRAME);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 22 "Estudio de Superficies" }}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 16 "Curvas de nivel\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 
"with(plots):" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Dada una funci\363n  " 
}{XPPEDIT 18 0 "z = f(x,y);" "6#/%\"zG-%\"fG6$%\"xG%\"yG" }{TEXT -1 
38 " se llama  \"curva de nivel\" de altura " }{TEXT 270 1 "k" }{TEXT 
-1 15 " , al conjunto " }{TEXT 256 19 "\{(x,y) / f(x,y)=k\}." }}{PARA 
0 "" 0 "" {TEXT -1 52 "Calculemos algunas curvas de nivel de la superf
icie " }{XPPEDIT 18 0 "z = x^2-y^2;" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"*$%
\"yGF(!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:= x^2-y^2;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "g:=implicitplot(\{f =1\},x=-2..2,y=
-2..2, color=BLUE): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "h:=
implicitplot(\{f =-1\},x=-2..2,y=-2..2,color=RED):\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 35 "display([g,h],scaling=constrained);" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "Observar que la \+
curva de nivel de altura k es la proyecci\363n, sobre el plano xy, de \+
la intersecci\363n del gr\341fico de la funci\363n con el plano horizo
ntal z=k:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f1:= plot3d(f,
x=-2..2,y=-2..2,color=GREY):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 42 "f2:= plot3d(1,x=-2..2,y=-2..2,color=PINK):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 30 "display3d([f1,f2],view=-4..4);" }{TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Dibujemos varias curvas de ni
vel:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "g:=seq(implicitplot
(f=i/5,x=-3..3,y=-3..3,numpoints=1000), i=-10..10):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 33 "display([g],scaling=constrained);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "El comando " }{HYPERLNK 17 "contou
rplot" 2 "contourplot" "" }{TEXT -1 54 " permite graficar curvas de ni
vel de manera sencilla.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
100 "contourplot(f,x=-3..3,y=-3..3,contours=[0,-1/2,-1,-2,1/2,1,2],\ns
caling=constrained,numpoints=1000);\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "contourplo
t3d(f,x=-2..2,y=-2..2,contours=[0,-1/2,-1,-2,1/2,1,2],\norientation=[7
0,65],color=black,thickness=2,view=-4..4);\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Intersecci\363n con planos\n
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:\nwith(plots):" 
}}}{PARA 0 "" 0 "" {TEXT -1 112 "Grafiquemos las curvas que se obtiene
n al intersecar la gr\341fica de una funci\363n con plano verticales d
e ecuaci\363n " }{XPPEDIT 18 0 "x = a;" "6#/%\"xG%\"aG" }{TEXT -1 3 " \+
o " }{XPPEDIT 18 0 "y = b;" "6#/%\"yG%\"bG" }{TEXT -1 54 ". Para dibuj
ar curvas en el espacio usamos el comando " }{HYPERLNK 17 "spacecurve
" 2 "spacecurve" "" }{TEXT -1 2 " ." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "f:=(x,y)->x^2-y^2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 70 "c:=spacecurve([0,y,f(0,y)],y=-2..2,color=RE
D,thickness=2,axes=normal):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
2 "c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "g:=plot3d(f(x,y),x
=-2..2,y=-2..2,color=grey):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
58 "p:=implicitplot3d(x=0,x=-2..2,y=-2..2,z=-6..4,color=pink):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display3d(\{c,p,g\},orientat
ion=[54,72],axes=none,view=-6..4);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "c:=seq(
spacecurve([i/2,y,f(i/2,y)],y=-3..3,color=RED,thickness=2),\ni=-6..6):
\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "display3d([c],orienta
tion=[65,58],view=-9..9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "d:=seq(spacecurve([x,i
/2,f(x,i/2)],x=-3..3,color=RED,thickness=2),\ni=-6..6):\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "display3d([d],orientation=[65,58],v
iew=-9..9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "display3d([c
,d],orientation=[65,58],view=-9..9);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "L\355mites
 Direccionales" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Dada una funci\363n de dos varia
bles " }{TEXT 257 7 "f(x,y)," }{TEXT -1 79 " una manera de conjeturar \+
la existencia o no de l\355mite en un punto determinado " }{TEXT 258 
7 "(x0,y0)" }{TEXT -1 124 " es acercarse a ese punto con curvas conten
idas en el dominio de la funci\363n, y calculando la imagen de la curv
a a trav\351s de " }{TEXT 259 1 "f" }{TEXT 282 1 "." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f:=(x,y)->x^4*y^4/(x^2+y^4)^3;" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 59 "Queremos estudiar la existencia de l\355m
ite en un entorno de " }{TEXT 260 5 "(0,0)" }{TEXT -1 35 ". Intentemos
 acercarnos con rectas:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "
p:=plot3d(f(x,y),x=-2..2,y=-2..2,style= PATCHNOGRID):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "l1:=spacecurve([x,x,f(x,x)],x=-2..2
,color=black,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
68 "l2:=spacecurve([x,-x,f(x,-x)],x=-2.2..2.2,color=yellow,thickness=2
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "l3:=spacecurve([x,2*x
,f(x,2*x)],x=-1.6..1.6,color=blue,thickness=2):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 69 "l4:=spacecurve([x,-2*x,f(x,-2*x)],x=-1.6..1.6,
color=red,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "
display3d(p,l1,l2,l3,l4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 10 "Ejercicio:" }{TEXT -1 113 "
  Probar que a lo largo de cualquier recta que pasa por el origen, el \+
l\355mite de f(x,y) restringido a la recta da " }{TEXT 261 1 "0" }
{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Ahora vamos a \+
intentar acercarnos con par\341bolas:\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "c1:=spacecurve([y^2,y,f(y^2,y)],y=-1.6..1.6,color=bla
ck,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "c2:=spa
cecurve([y,y^2,f(y,y^2)],y=-1.5..1.5,color=green,thickness=2):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "c3:=spacecurve([2*y^2,y,f(2*
y^2,y)],y=-1.2..1.2,color=yellow,thickness=2):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 71 "c4:=spacecurve([y,2*y^2,f(y,2*y^2)],y=-1.2..1.
2,color=red,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "display3d(p,c1,c2,c3,c4);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 83 "En la gr\341fica se puede ver que hay algunas p
ar\341blas cuyas im\341genes no pasan por el " }{TEXT 263 5 "(0,0)" }
{TEXT -1 33 ". Esto nos lleva a sospechar que " }{TEXT 265 6 "f(x,y)" 
}{TEXT -1 20 " no tiene l\355mite en " }{TEXT 264 5 "(0,0)" }{TEXT -1 
2 ".\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 10 "Ejercicio:" }{TEXT -1 
62 "  Encontrar cu\341les son las par\341bolas a lo largo de las cuale
s " }{TEXT 267 1 "f" }{TEXT -1 32 " tiende a un l\355mite distinto de \+
" }{TEXT 266 2 "0." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 
268 23 "Observaci\363n importante:" }{TEXT -1 1 " " }{TEXT 256 1 " " }
{TEXT -1 73 "MAPLE une, en forma continua, los puntos del gr\341fico q
ue calcula. Al ser " }{TEXT 256 6 "f(x,y)" }{TEXT -1 16 " discontinua \+
en " }{TEXT 256 7 "(0,0), " }{TEXT -1 67 "el gr\341fico que vemos no s
e corresponde con el real. Un gr\341fico que " }{TEXT 269 2 "no" }
{TEXT -1 52 " una los puntos deja m\341s en claro la discontinuidad." 
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[lFh[l" 1 5 0 1 10 0 2 1 1 1 2 1.000000 78.000000 -97.000000 0 0 "Curv
e 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curv
e 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" 
"Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Cur
ve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 2
7" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "
Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curv
e 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46
" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "C
urve 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve
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"" }{TEXT 260 0 "" }}{SECT 1 {PARA 257 "" 0 "" {TEXT 269 14 "Ejercitac
i\363n 1" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 261 1 "1" }{TEXT -1 25 ".- Reemplazar en el item
 " }{TEXT 273 22 "Estudio de superficies" }{TEXT -1 41 " el ejemplo da
do por estos otros:\n      (" }{TEXT 274 5 "Nota:" }{TEXT -1 137 " es \+
probable que tambi\351n haya que modificar distintas opciones como num
points, view, rangos para x e y, etc, dependiendo de cada ejemplo)" }}
{PARA 0 "" 0 "" {TEXT -1 5 " a)  " }{XPPEDIT 18 0 "f(x,y) = x^2*y;" "6
#/-%\"fG6$%\"xG%\"yG*&F'\"\"#F(\"\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " b) " }{XPPEDIT 18 0 "f(
x,y) = x^2+y^2;" "6#/-%\"fG6$%\"xG%\"yG,&*$F'\"\"#\"\"\"*$F(F+F," }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 4 " c) " }{XPPEDIT 18 0 "f(x,y) = 1/(x^2+y^2)" "6#/-%\"fG6$%\"xG%\"
yG*&\"\"\"F*,&*$F'\"\"#F**$F(F-F*!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 3 " d)" }{XPPEDIT 18 0 "f(x,y) = x^2*y^2+
1" "6#/-%\"fG6$%\"xG%\"yG,&*&F'\"\"#F(F+\"\"\"F,F," }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " e) " }
{XPPEDIT 18 0 "f(x,y) = y/sqrt(x);" "6#/-%\"fG6$%\"xG%\"yG*&F(\"\"\"-%
%sqrtG6#F'!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 1 "2
" }{TEXT -1 120 ".- Para las siguientes superficies, hallar algunas cu
rvas de nivel y cortes con planos. Tratar de imaginar la gr\341fica y \+
" }{TEXT 265 5 "luego" }{TEXT -1 39 " verificar graficando con los com
andos " }{TEXT 263 7 "plot3d " }{TEXT -1 2 "o " }{TEXT 264 14 "implici
tplot3d" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 4 "a)  " }{XPPEDIT 18 0 "z = sqrt(x^2+y^2);" "6#/%\"zG-%
%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F," }{TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "b)  " }{XPPEDIT 18 0 
"z^2 = x^2+y^2;" "6#/*$%\"zG\"\"#,&*$%\"xGF&\"\"\"*$%\"yGF&F*" }{TEXT 
-1 6 "\n\nc)  " }{XPPEDIT 18 0 "z^2+x^2+y^2 = 1;" "6#/,(*$%\"zG\"\"#\"
\"\"*$%\"xGF'F(*$%\"yGF'F(F(" }{TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }{TEXT 266 1 "3" }{TEXT -1 23 ".- Graficar la funci
\363n " }{XPPEDIT 18 0 "z = x^2+y;" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"%\"yG
F)" }{TEXT -1 28 ",\n\na) utilizando el comando " }{TEXT 267 6 "plot3d
" }{TEXT -1 28 ".\n\nb) utilizando el comando " }{TEXT 268 14 "implici
tplot3d" }{TEXT -1 1 "." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }
{TEXT 260 14 "Ejercitaci\363n 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 1 "1
" }{TEXT -1 100 ".- Estudiar los l\355mites restringidos a curvas que \+
pasen por el origen, de las siguientes funciones:\n" }}{PARA 0 "" 0 "
" {TEXT -1 3 " a)" }{TEXT 32 2 "  " }{XPPEDIT 18 0 "(x^2-y^2)/(sqrt(x^
2+y^2+1)-1);" "6#*&,&*$%\"xG\"\"#\"\"\"*$%\"yGF'!\"\"F(,&-%%sqrtG6#,(*
$F&F'F(*$F*F'F(F(F(F(F(F+F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 5 " b)  " }{XPPEDIT 18 0 "x^3*y^3/(x^2+y^3);" "6#*(
%\"xG\"\"$%\"yGF%,&*$F$\"\"#\"\"\"*$F&F%F*!\"\"" }{TEXT -1 1 "\n" }}
{PARA 0 "" 0 "" {TEXT -1 5 " c)  " }{XPPEDIT 18 0 "x*y/abs(x-y);" "6#*
(%\"xG\"\"\"%\"yGF%-%$absG6#,&F$F%F&!\"\"F+" }{TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " d) " }{XPPEDIT 
18 0 "sin(x*y)/(abs(x)+abs(y))" "6#*&-%$sinG6#*&%\"xG\"\"\"%\"yGF)F),&
-%$absG6#F(F)-F-6#F*F)!\"\"" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT 
-1 4 " e) " }{XPPEDIT 18 0 "(x^2+y^2)/(x*y-y+x);" "6#*&,&*$%\"xG\"\"#
\"\"\"*$%\"yGF'F(F(,(*&F&F(F*F(F(F*!\"\"F&F(F-" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 1 "2" }{TEXT -1 41 ".- Anali
zar la existencia del l\355mite en (" }{TEXT 263 4 "0,0)" }{TEXT -1 
38 " para las funciones del item anterior." }}}}{MARK "1 0" 0 }
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