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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 1 "\n" }{TEXT 270 43 "An\341
lisis I - Matem\341tica 1 - An\341lisis II (C)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 257 15 "TALLER DE MAPLE" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" 
{TEXT -1 14 " LABORATORIO 4" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 183 "En este laboratorio graficaremos \+
distintas rectas  y planos tangentes al gr\341fico de una funci\363n r
eal de dos variables.\nAntes de comenzar es conveniente que repase las
 definiciones de " }{TEXT 267 16 "derivada parcial" }{TEXT -1 2 ", " }
{TEXT 268 20 "derivada direccional" }{TEXT -1 3 " y " }{TEXT 269 16 "p
lano tangente.\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 49 "Ingresamos una funci\363n diferenciable e
n un punto " }{XPPEDIT 18 0 "P = (a, b);" "6#/%\"PG6$%\"aG%\"bG" }
{TEXT -1 5 " con " }{TEXT 258 7 "-2<a <2" }{TEXT -1 4 "  y " }{TEXT 
259 1 " " }{TEXT 256 8 "-2<b<2. " }{TEXT -1 66 "Elegimos, a partir del
 gr\341fico, un rango adecuado para el eje z. \n" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 35 "f:=(x,y)->sin(x*y);\na:=1.2;\nb:=.8;\n" }
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "graf:=plot3d(f(x,y)
,x=-2..2,y=-2..2,style=patchnogrid):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 177 "p:=pointplot3d([a,b,f(a,b)],symbol=circle,color=red,thickness=2
):\n\nminz:=-1;\nmaxz:=2;\n\ndisplay3d(\{graf,p\},orientation=[157,63]
,view=minz..maxz,axes=framed,\nscaling=constrained);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "
\nGraficamos la restricci\363n de " }{TEXT 272 1 "f" }{TEXT -1 28 " a \+
la recta paralela al eje " }{TEXT 273 1 "x" }{TEXT -1 14 " que pasa po
r " }{TEXT 260 6 "(a,b)." }{TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 69 "gx:=spacecurve([a+x,b,f(a+x,b)],x=-a-2..2-a,color=b
lue,thickness=2):\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "display3d(\{
graf,p,gx\},orientation=[165,79],view=minz..maxz,axes=framed,\nscaling
=constrained);" }}{PARA 13 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "\nGra
ficamos la recta tangente a la curva anterior. Para esto calculamos la
 derivada parcial respecto de x en " }{TEXT 261 5 "(a,b)" }{TEXT -1 2 
".\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fx:=diff(f(x,y),x);
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "mx:=eval(fx,\{x=a,y=b\});\n" }
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "lx:=spacecurve([x,b
,mx*(x-a)+f(a,b)],x=-2..2,color=black,thickness=2):\n" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 95 "display3d(\{graf,p,gx,lx\},orientation=[165,79],
view=minz..maxz,axes=framed,scaling=constrained);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "\nAg
reguemos la recta tangente a  la restricci\363n de " }{TEXT 274 1 "f" 
}{TEXT -1 43 " a la recta paralela al eje y que pasa por " }{TEXT 256 
5 "(a,b)" }{TEXT -1 60 ". Para esto calculamos la derivada parcial res
pecto de y en " }{TEXT 256 6 "(a,b)." }{TEXT -1 1 "\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "gy:=spacecurve([a,b+y,f(a,b+y)],y=-
b-2..2-b,color=green,thickness=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "fy:=diff(f(x,y),y);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "my:=eval
(fy,\{x=a,y=b\});" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "ly:=spacecurve
([a,y,my*(y-b)+f(a,b)],y=-2..2,color=black,thickness=2):" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "displa
y3d(\{graf,p,gy,ly,gx,lx\},orientation=[-168,48],view=minz..maxz,axes=
framed,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 40 "mv:=evalf( mx*sqrt(2)/2+my*sqrt(2)/2 );\n" }
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "gv:=spacecurve([a+
t*sqrt(2)/2,b+t*sqrt(2)/2,f(a+t*sqrt(2)/2,b+t*sqrt(2)/2)],t=-2-a*sqrt(
2)/2..2-b*sqrt(2)/2,color=yellow,thickness=2):\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 93 "lv:=spacecurve([a+t*sqrt(2)/2,b+t*sqrt(2)/2,mv*t+f(a,
b)],t=-2..2,\ncolor=black,thickness=2):\n\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 107 "display3d([graf,p,gv,lv,gy,ly,gx,lx],orientation=[17
3,47],view=minz..maxz,axes=framed,scaling=constrained);" }}{PARA 13 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 103 "planotangente:=plot3d(f(a,b)+mx*(x-a)+my*(y-b),x=-3+a..3+a,y=-3
+b..3+b,\ncolor=pink,style=patchnogrid):\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 115 "display3d([graf,p,lv,ly,lx,planotangente],orientatio
n=[170,82],\nview=[-2..2,-2..2,minz..maxz],scaling=constrained);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
262 0 "" }{TEXT 263 13 "Ejercitaci\363n:" }{TEXT 264 0 "" }{TEXT -1 
134 "\n\na) En el ejemplo anterior cambiar el punto P por (0,0).\n   \+
\277Qu\351 peculiaridad tienen algunas de las rectas tangentes? \277Es
 cierto que " }{TEXT 266 18 "en un entorno de P" }{TEXT -1 60 " una re
cta tangente necesariamente s\363lo corta al gr\341fico de " }{TEXT 
275 1 "f" }{TEXT -1 86 " en P?\n   \277C\363mo es el plano tangente? \+
\277Es (0,0) un extremo?\n\nb) Considerar la funci\363n " }{XPPEDIT 
18 0 "f(x,y) = x^2+y^2;" "6#/-%\"fG6$%\"xG%\"yG,&*$F'\"\"#\"\"\"*$F(F+
F," }{TEXT -1 210 "  , P=(0,0), minz=0 y maxz=2.\n   \277C\363mo es el
 plano tangente? \277Es (0,0) un extremo?\n   Probar con otros puntos.
 Observar que la gr\341fica de la funci\363n est\341 siempre por encim
a del plano.\n\nc) Considerar la funci\363n " }{XPPEDIT 18 0 "f(x,y) =
 x^2-y^2;" "6#/-%\"fG6$%\"xG%\"yG,&*$F'\"\"#\"\"\"*$F(F+!\"\"" }{TEXT 
-1 150 "  , P=(0,0), minz=-2 y maxz=2.\n   \277C\363mo es el plano tan
gente? \277Es (0,0) un extremo?\n   Probar con otros puntos. Observar \+
que la gr\341fica de la funci\363n " }{TEXT 265 25 "en cualquier entor
no de P" }{TEXT -1 58 " nunca est\341 totalmente por encima o por deba
jo del plano.\n" }}}}{MARK "0 4 0" 14 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
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