{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 260 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 13 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 1 13 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 270 "" 1 13 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 295 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 296 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 300 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 301 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }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 294 43 "An\341 lisis I - Matem\341tica 1 - An\341lisis II (C)" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 15 "TALLER DE MAPLE" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 1 " " } {TEXT 289 13 "LABORATORIO 2" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "En este laboratorio veremos c\363 mo trabajar con sucesiones y aprenderemos a realizar gr\341ficos anim ados.\n" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Sucesione" }{MPLTEXT 0 21 0 "" }{TEXT -1 1 "s" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "El co mando " }{TEXT 296 3 "seq" }}{PARA 0 "" 0 "" {TEXT -1 69 "Para generar sucesiones (num\351ricas o no) podemos utilizar el comando " } {HYPERLNK 17 "seq" 2 "seq" "" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Guardemos en la va riable" }{TEXT 263 6 " pares" }{TEXT -1 50 " la sucesi\363n de los pri meros veinte n\372meros pares:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "pares:=seq (2*i,i=1..10);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 " Podemos luego acceder a la sucesi\363n completa " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "pares;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "o \+ a alg\372n elemento en particular" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "pares[3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "pares[1]+pares[10];" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 264 6 "Pruebe" }{TEXT -1 29 " a generar otras suceciones.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "El comando" }{TEXT 261 1 " " }{TEXT 297 15 "for . ..do... od" }{TEXT 298 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "For...do" 2 "for" "" }{TEXT -1 143 " se emplea para repetir un cierto n\372mero de veces una serie de instrucciones \+ y se puede utilizar para generar sucesiones elemento por elemento." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 0 "" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Generemos los n\372meros pares del 2 al 2 0:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "for k from 1 to 10 \+ do;\n 2*k;\n od;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Asignemos un nombre a cada elemento:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "for k from 1 to 10 do;\na[k]:=2*k;\nod;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "De este modo podemos recuperar los n\372meros generado s. Por ejemplo el quinto par es:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a[5];\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " a[1]+a[10];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "for n from 1 to 10 do\na[n]+a[11-n];\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "Sucesiones...Plot" {TEXT -1 13 "Los c omandos " }{TEXT 260 4 "plot" }{TEXT 299 3 " y " }{TEXT 300 7 "display " }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "Plot" 2 "plot" "" }{TEXT -1 40 " se utiliza para graficar funcione s o " }{TEXT 293 16 "listas de puntos" }{TEXT -1 28 ". Vamos a usar \+ esto \372ltimo.\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 44 "plot ([[1.2,1.8],[2,2],[1,-1]],style=point); " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot ([[ 1.2,1.8],[2,2],[1,-1]],style=line);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plot([[1.2,1.8],[2,2],[1,-1],[1.2,1.8]],sty le=line,color=blue);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Grafiquemos l a sucesi\363n real " }{XPPEDIT 18 0 "a[n] = 1/n;" "6#/&%\"aG6#%\"nG*& \"\"\"F)F'!\"\"" }{TEXT -1 14 " en el plano:\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "a:=seq([n,1/n],n=1..40);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "grafa:=plot([a],style=point,color=blue):\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "grafa;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Grafiq uemos las sucesiones " }{XPPEDIT 18 0 "a[n] = 1/n" "6#/&%\"aG6#%\"nG* &\"\"\"F)F'!\"\"" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "b[n] = sin(15*n )/n;" "6#/&%\"bG6#%\"nG*&-%$sinG6#*&\"#:\"\"\"F'F.F.F'!\"\"" }{TEXT -1 7 " y " }{XPPEDIT 18 0 "c[n] = -1/n;" "6#/&%\"cG6#%\"nG,$*&\"\" \"F*F'!\"\"F+" }{TEXT -1 24 " en un mismo gr\341fico." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "b:=seq([n,sin(15*n)/n],n=1..40):\n grafb:=plot([b],style=point,color=red):\n\nc:=seq([n,-1/n],n=1..40):\n grafc:=plot([c],style=point,color=green):\n" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 63 "Si queremos ver los tres gr\341ficos juntos utilizamos \+ el comando " }{HYPERLNK 17 "display" 2 "display" "" }{TEXT -1 43 ". E ste comando se encuentra en el paquete " }{HYPERLNK 17 "plots" 2 "plot s" "" }{TEXT -1 52 " , el cual antes deber\341 ser \"bajado\" con el c omando " }{HYPERLNK 17 "with:" 2 "with" "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "display([grafa,grafb,grafc],ax es=normal,title=\"Lema del sandwich.\",\nxtickmarks=0,symbol=circle); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 7 "Pruebe " }{TEXT -1 41 "a graficar otras suceciones. \+ Por ejemplo " }{XPPEDIT 18 0 "a[n] = (1+1/n)^n;" "6#/&%\"aG6#%\"nG),& \"\"\"F**&F*F*F'!\"\"F*F'" }{TEXT -1 3 " .\n" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Los comandos " }{TEXT 295 6 "taylor" }{TEXT -1 3 " y " }{TEXT 257 7 "convert" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "El comando " }{HYPERLNK 17 "taylor" 2 "ta ylor" "" }{TEXT -1 75 " calcula las aproximaciones de una funci\363n a lrededor de un punto como sigue" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=tan(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "faprx :=taylor(f,x=Pi/4,5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Para des preciar el termino del resto y as\355 obtener el polinomio de taylor u samos el comando " }{HYPERLNK 17 "convert" 2 "convert" "" }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p:=convert(faprx,pol ynom);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Grafiquemos la funci \363n junto con el polinomio." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot([f,p],x=-Pi..Pi,y=-10..10,discont=true,color=[blue,green]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT 272 12 "Animaciones\n" }}{EXCHG {PARA 0 "" 0 "Sucesiones. ..Animaciones" {TEXT -1 53 "Para animar gr\341ficos se pueden utilizar los comandos " }{HYPERLNK 17 "animate" 2 "animate" "" }{TEXT -1 3 " y " }{HYPERLNK 17 "display" 2 "plots,display" "" }{TEXT -1 42 ". Utiliz aremos este \372ltimo con la opci\363n \n" }{TEXT 290 16 "insequence= \+ true" }{TEXT -1 60 ".\nRecuerde que para usar estos comandos antes deb e ejecutar " }{HYPERLNK 17 "with(plots)" 2 "plots" "" }{TEXT -1 2 ":\n " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Ejemplo 1" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 45 "Animemos el gr\341fico de la sucesi\363n compleja \+ " }{XPPEDIT 18 0 "a[n] = (.8+.5*I)^n;" "6#/&%\"aG6#%\"nG),&-%&FloatG6 $\"\")!\"\"\"\"\"*&-F+6$\"\"&F.F/%\"IGF/F/F'" }{TEXT -1 2 ".\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 60 "Primero generemos la sucesi\363n. (Note que debemos separar la " } {TEXT 291 10 "parte real" }{TEXT -1 7 " de la " }{TEXT 292 10 "imagina ria" }{TEXT -1 43 " para as\355 obtener los puntos para dibujar.)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "z:=.8+.5*I;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a:=seq([Re(z^n),Im(z^n)],n=1..50): \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Ahora generaremos una suces i\363n de gr\341ficos. Cada elemento de la sucesi\363n es un " }{TEXT 273 8 "\"cuadro\"" }{TEXT -1 3 " o " }{TEXT 274 7 "\"frame\"" }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "cuadros:=seq(p lot([a[n]],color=blue),n=1..50):\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Para obtener la animaci\363n usamos el comando " }{TEXT 275 7 " display" }{TEXT -1 15 " con la opci\363n " }{TEXT 286 15 "insequence=t rue" }{TEXT -1 182 ".\n\nNota: para comenzar la animaci\363n haga \"cl ick\" con el bot\363n izquierdo del \"mouse\" sobre el gr\341fico y lu ego sobre el bot\363n \"play\" que aparecer\341 en la parte superior d e la pantalla.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "display ([cuadros],style=point,insequence=true,scaling=constrained,\nsymbol=c ircle);" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 16 " Insequence=false" }{TEXT -1 37 " muestra todos los \"cuadros\" juntos: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "display ([cuadros],st yle=point,insequence=false,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 279 10 "Ejemplo 2\n" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Dada una funci\363n" }{TEXT 265 1 " " }{TEXT -1 14 "f der ivable en" }{TEXT 266 1 " " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" } {TEXT -1 54 " la dibujaremos junto con la recta secante a f por (" } {XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 3 ",f(" }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 5 ")) y " }}{PARA 0 "" 0 "" {TEXT -1 1 "(" }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 7 "+ h, f (" }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 2 "+h" } {TEXT 267 3 ")) " }{TEXT -1 24 "para algunos valores de " }{TEXT 280 1 "h" }{TEXT -1 1 "." }{TEXT 268 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart;\nwith(plots):\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "f:=x->-3*x^2+x+3;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "x0:=-0.5;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "\nDefinamos una funci\363n, que llamaremos" }{TEXT 281 1 " " }{TEXT 276 1 "m" }{TEXT -1 23 ", que dada la variable " }{TEXT 277 1 "h" }{TEXT -1 64 ", nos d evuelva la pendiente de la recta que pasa por los puntos " }{TEXT 269 1 "(" }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 2 ",f" } {TEXT 275 1 "(" }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 4 ") ) y" }{TEXT 270 2 " (" }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" } {TEXT -1 2 "+h" }{TEXT 272 1 "," }{TEXT -1 1 "f" }{TEXT 274 1 "(" } {XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 5 "+h)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "m:=h->(f(x0+h)-f(x0))/h;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 0 "" }{TEXT -1 74 "Gua rdamos los gr\341ficos de algunas secantes; para esto usamos la estruc tura" }{TEXT 273 1 " " }{HYPERLNK 17 "for...do...od; " 2 "for" "" } {TEXT 260 1 "." }}{PARA 0 "" 0 "" {TEXT -1 60 "\nRecordar que la ecuac i\363n de la recta secante al gr\341fico de " }{TEXT 301 2 "f " } {TEXT -1 25 "que pasa por los puntos (" }{XPPEDIT 18 0 "x[0];" "6#&%\" xG6#\"\"!" }{TEXT -1 2 ",f" }{TEXT 262 1 "(" }{XPPEDIT 18 0 "x[0]" "6# &%\"xG6#\"\"!" }{TEXT -1 4 ")) y" }{TEXT 259 2 " (" }{XPPEDIT 18 0 "x[ 0];" "6#&%\"xG6#\"\"!" }{TEXT -1 2 "+h" }{TEXT 258 1 "," }{TEXT -1 1 " f" }{TEXT 261 1 "(" }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 9 "+h)) es\n " }{XPPEDIT 18 0 "y = m(h)*(x-x[0])+f(x[0]);" "6#/%\"yG,& *&-%\"mG6#%\"hG\"\"\",&%\"xGF+&F-6#\"\"!!\"\"F+F+-%\"fG6#&F-6#F0F+" } {TEXT -1 38 ".\n\nConsideraremos h=2/n con n=1..10. \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "for n from 1 to 10 do\n\nsecante:= m(2/n)*(x-x0)+f(x0):\n\ncuadro[n]:=plot([secante,f(x)],x=-1..2,color=[ blue,black]):\n\nod:\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Creamos una secuencia con todos los \"cu adros\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "s:=seq (cuadro[ n],n=1..10):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 262 1 "\n" }{TEXT -1 35 "Mostramos el gr\341fic o con el comado " }{TEXT 278 7 "display" }{TEXT 282 1 " " }{TEXT -1 14 " y la opci\363n " }{TEXT 263 15 "insequence=true" }{TEXT -1 77 " \+ (para ver la animaci\363n haga \"click\" sobre el gr\341fico y luego s obre \"play\"):\n" }}{PARA 0 "" 0 "" {TEXT 264 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display([s],insequence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 16 "Insequence=false" }{TEXT -1 36 " muestra todos los \"cuadros\" juntos:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "display([s],insequence=false);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Ejemplo 3\n " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Animemos distintas aproximacio nes de Taylor a la funci\363n " }{XPPEDIT 18 0 "f(x) = sin(x);" "6#/-% \"fG6#%\"xG-%$sinG6#F'" }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart;\nwith(plots):\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "f:=sin(x);\na:=0;\nk:=30;" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 84 "Hallamos los gr\341ficos de f y sus polinomios de Taylo r desde el orden 0 hasta el k-1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "for n from 1 to k do\nt:=taylor(f,x=a,n):\npol:=conv ert(t,polynom):\ngrafpol[n]:=plot([pol,f],x=-16..16,y=-2..2,color=[blu e,green],\nthickness=3,numpoints=500):\nod:" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 74 "Creamos una secuencia con los gr\341ficos y luego la mo stramos con animaci\363n:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "cuadros:=seq(grafpol[n],n=1..k):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display([cuadros],insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 13 "Ejercitaci\363n:" }}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 283 3 "1.-" }{TEXT -1 68 "Utilice el ejemplo1 para graficar la sucesi\363n de n\372meros complejos " }{XPPEDIT 18 0 "a[n] = (sqrt(2)/2+sqrt(2)*I/2)^n;" "6#/&%\"aG6#%\"nG),&*&-%%sqrtG6# \"\"#\"\"\"F.!\"\"F/*(-F,6#F.F/%\"IGF/F.F0F/F'" }{TEXT -1 3 ".\n\n" } {TEXT 284 3 "2.-" }{TEXT -1 75 "En el ejemplo 2 agregar en cada cuadro el gr\341fico de la recta tangente en (" }{XPPEDIT 18 0 "x[0],f(x[0]) ;" "6$&%\"xG6#\"\"!-%\"fG6#&F$6#F&" }{TEXT -1 4 ").\n\n" }{TEXT 287 3 "3.-" }{TEXT -1 24 "En el ejemplo 3 cambiar " }{TEXT 302 1 "f" }{TEXT -1 10 " por \n\na) " }{XPPEDIT 18 0 "f(x) = cos(x);" "6#/-%\"fG6#%\"xG -%$cosG6#F'" }{TEXT -1 19 " b) " }{XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 24 " \+ c) " }{XPPEDIT 18 0 "f(x) = 1/(1+x);" "6#/-%\"fG6#%\"xG*&\"\"\"F) ,&F)F)F'F)!\"\"" }{TEXT -1 14 " \n\nd) " }{XPPEDIT 18 0 "f(x) \+ = log(1+x);" "6#/-%\"fG6#%\"xG-%$logG6#,&\"\"\"F,F'F," }{TEXT -1 13 " \+ e) " }{XPPEDIT 18 0 "f(x) = 1/(1+x^2);" "6#/-%\"fG6#%\"xG*&\" \"\"F),&F)F)*$F'\"\"#F)!\"\"" }{TEXT -1 14 " f) " }{XPPEDIT 18 0 "f(x) = x^7-x^5+x^3-x;" "6#/-%\"fG6#%\"xG,**$F'\"\"(\"\"\"*$F'\" \"&!\"\"*$F'\"\"$F+F'F." }{TEXT -1 1 "\n" }{TEXT 257 4 "Nota" }{TEXT -1 68 ": para una mejor visualizaci\363n restrinja el dominio y el eje de las " }{TEXT 256 2 "y." }}{PARA 0 "" 0 "" {TEXT 271 2 "4." }{TEXT -1 22 "-Considere la funci\363n " }{XPPEDIT 18 0 "f(x) = x*sin(x);" "6 #/-%\"fG6#%\"xG*&F'\"\"\"-%$sinG6#F'F)" }{TEXT -1 62 " para x entre 0 \+ y 10. Hallar y graficar la recta tangente en (" }{XPPEDIT 18 0 "x[i],f (x[i]);" "6$&%\"xG6#%\"iG-%\"fG6#&F$6#F&" }{TEXT -1 7 ") para " } {XPPEDIT 18 0 "x[i] = i/5;" "6#/&%\"xG6#%\"iG*&F'\"\"\"\"\"&!\"\"" } {TEXT -1 73 ", i=0..50. Generar una animaci\363n en la que cada cuadro sea el gr\341fico de " }{TEXT 303 1 "f" }{TEXT -1 33 " junto con la \+ recta tangente en " }{XPPEDIT 18 0 "x[i];" "6#&%\"xG6#%\"iG" }{TEXT -1 2 ".\n" }{TEXT 288 4 "Nota" }{TEXT -1 69 ": para una mejor visualiz aci\363n del gr\341fico, restringir el eje de las " }{TEXT 285 2 "y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }}}}{MARK "4 0 0" 12 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }