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}{PSTYLE "Nor mal" -1 310 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" -1 311 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 312 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "_pstyle24 " -1 313 1 {CSTYLE "" -1 -1 "Times" 1 14 255 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Normal" -1 314 1 {CSTYLE "" -1 -1 "Times" 1 14 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" -1 315 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 1 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 316 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 317 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 318 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "" 5 319 1 {CSTYLE "" -1 -1 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 255 "" 0 "" {TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT -1 17 "C\341lculo I-MAT1610" }}{PARA 18 "" 0 "" {TEXT -1 13 "Laboratorio \+ 5" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 5 "" 0 "" {TEXT 295 7 "Martes " }{TEXT 305 9 "4 de mayo" }{TEXT 306 9 " de 2010 " }{TEXT 303 51 " (en 1675 se inaugura el Observatorio de Greenwich)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 319 "" 0 "" {TEXT 298 16 "Plazo de entrega" }{TEXT 299 41 " : Martes 11 de mayo a \+ las 13:00 horas. " }{TEXT 415 56 "(EN 1904, NACE SALVADOR DALI, PRECU RSOR DEL SURREALISMO)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT 302 30 "Pr\363ximo Laboratorio se sube el" }{TEXT -1 4 " : \+ " }{TEXT 304 36 "Martes 11 de mayo a las 13:30 horas." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 311 "" 0 "" {TEXT -1 0 "" }{TEXT 290 23 "NOMBRE ALUMNO DIGITADOR" }{TEXT 296 1 ": " }}{PARA 316 "" 0 "" {TEXT -1 0 "" }{TEXT 291 0 "" }{TEXT 292 27 "N \372mero de alumno digitador:" }}{PARA 0 "" 0 "" {TEXT 293 21 "Secci \363n de C\341tedra: 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 289 19 "NOMBRE OTRO ALUMNO: " }}{PARA 317 "" 0 "" {TEXT -1 17 "N\372mero de alumno:" }}{PARA 317 " " 0 "" {TEXT -1 21 "Secci\363n de C\341tedra: 1" }}{PARA 315 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 1 "(" }{TEXT 300 61 "El alum no digitador se alterna de laboratorio en laboratorio)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 310 "" 0 "" {TEXT -1 0 "" }}{PARA 311 "" 0 "" {TEXT 297 16 "NOMBRE DEL GRUPO" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT 294 33 "Secci\363n de Laboratorio del grupo:" }}{PARA 255 "" 0 " " {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 9 "Objetivos" }}{PARA 15 "" 0 "" {TEXT -1 70 "Presenta ci\363n del m\351todo de Newton para hallar soluciones de ecuaciones" }}{PARA 15 "" 0 "" {TEXT -1 45 "Problemas de razones de cambio relacio nadas \030" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Instrucciones" }}{PARA 15 "" 0 "" {TEXT -1 119 "Entregue el laboratorio sin output alguno y s in gr\341ficos. Para ello elja la opci\363n Edit->Remove Output->From \+ Worksheet." }}{PARA 15 "" 0 "" {TEXT -1 345 "Para corregir su laborato rio el ayudante ejecutar\341 primero el laboratorio con Edit->Execut e->Worksheet. El ayudante entregar\341 el laboratorio corregido sin ou tput, pero con los comentarios correspondientes. Los comandos deben es tar en secuencia l\363gica de modo de que, al ejecutarlos en orden, l os c\341lculos sean correctos. Para simplificar use " }{TEXT 307 7 "re start" }{TEXT -1 86 " en cada problema y cargue nuevamente los paquete s que necesita (with(plots), etc...) " }}{PARA 15 "" 0 "" {TEXT -1 63 "Todos los ejercicios a ser realizados por usted est\341n en color " } {TEXT 309 4 "blue" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 62 "La parte verbal de sus repuestas debe ser entregada en color " }{TEXT 308 9 "dark red." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "Glosario de Coman dos Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 334 23 "Asignaci\363n a variables:" }{TEXT -1 1 " " }{TEXT 0 3 " := " }{TEXT -1 2 ", " }{TEXT 0 8 "unassign" }{TEXT -1 3 " , " }{TEXT 0 7 "restore" }{TEXT -1 2 ", " }{TEXT 0 6 "assume" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 22 "var := comando maple; " }} {PARA 0 "" 0 "" {TEXT -1 81 "Se evalua el comando maple y la expresi \363n que resulta es asignada a la variable " }{TEXT 0 3 "var" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 12 "assume(x>a );" }{TEXT -1 1 " " }{TEXT 0 19 "assume(n,integer); " }{TEXT -1 1 " " }{TEXT 0 14 "assume(z,real)" }}{PARA 0 "" 0 "" {TEXT -1 55 "Son ejempl os de assume. Para mayor informaci\363n ejecute " }{TEXT 0 8 "?assume; " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 17 "unassi gn('var'); " }}{PARA 0 "" 0 "" {TEXT -1 52 "\"borra\" todo lo que se c onoce acerca de la variable " }{TEXT 0 3 "var" }{TEXT -1 56 ", incluye ndo las restricciones impuestas con el comando " }{TEXT 335 6 "assume " }{TEXT -1 107 ". De este modo, ella puede ser usada sin restriccione s m\341s adelante. Las comillas ' ' son fundamentales. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 8 "restore;" }}{PARA 0 "" 0 "" {TEXT -1 125 "reinicializa todas las variables que se hayan oc upado \"borrando\" todo lo que se conoce sobre ellas. Es equivalente a aplicar " }{TEXT 0 15 "unassign('var')" }{TEXT -1 17 " a cada variabl e " }{TEXT 307 3 "var" }{TEXT -1 21 " que se haya ocupado." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "?assume" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 310 25 "Aproximaciones decimales:" }{TEXT -1 1 " " }{TEXT 0 5 "evalf" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 11 "Digits:= N;" }}{PARA 0 "" 0 "" {TEXT -1 90 "Define que de ahora en adelante se ocupa aritm\351ti ca decimal con N d\355gitos significativos. " }{TEXT 311 32 "Al inicia r Maple Digits vale 10" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 13 "evalf( expr) " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 120 "Eval\372a la expresi\363n en artim\351tica decimal con N d\355gitos sgniticativos, donde N es el valor que tiene la variable D igits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 14 " evalf(expr,N) " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Evalua l a expresi\363n " }{TEXT 0 4 "expr" }{TEXT -1 52 " en aritm\351tica dec imal con N d\355gitos significativos." }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 5 "" 0 "" {TEXT 312 41 "Gr\341fico de funciones y cu rvas en el plano" }{TEXT -1 2 ", " }{TEXT 0 4 "plot" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 21 "plot( f(x), x=a..b); " } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 94 "Grafica a y=f(x) en el intervalo [a,b]. El rango y escala del eje Y se ajusta autom\341ticam ente" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 43 "pl ot ( f(x) , x=a..b, scaling=constrained);" }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 78 "Idem que el anterior pero las escalas en los ejes \+ X a Y est\341n en relaci\363n 1:1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 0 30 "plot ( f(x) , x=a..b, y=c..d);" }}{PARA 0 "" 0 "" {TEXT -1 96 " Muestra la porci\363n del gr\341fico de y=f(x) \+ que yace en el rect\341ngulo a <= x <= b, c<= y <= d." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 41 "plot( f(x), x=a..b, view=[x1..x2,y1..y2])" }}{PARA 0 "" 0 "" {TEXT -1 121 "Muestra la por ci\363n del gr\341fico de y=f(x) para a <= x <= b pero enmarcado e n una ventana con x1 < x < x2, y1 < y < y2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 44 "plot ( f(x) , x=a..b, y=c..d, di scont=true);" }}{PARA 0 "" 0 "" {TEXT -1 91 " Idem que el anterior, pe ro se usa cuando f es discont\355nua (y f no toma valores complejos) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 0 55 "plot( f(x), x=a..b, y=c..d, discont=true, \+ color=COLOR);" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 123 "Idem qu e el anterior donde y=f(x) se grafica con color COLOR, donde COLOR pue de ser red, blue, green, cyan, magenta, etc..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 0 78 "plot( [f(x) ,g(x), h(x)] , x=a..b, y=c..d,discont=true, color=[red,blue,cyan]);" } }{PARA 0 "" 0 "" {TEXT -1 124 "Idem que en el anterior pero se grafica n las tres funciones f,g,h al mismo tiempo en colores red,blue, cyan r espectivamente." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 307 42 "plot( [x(t), y(t) , t=a..b], color=red ); " }}{PARA 0 " " 0 "" {TEXT -1 79 "grafica los puntos (x(t),y(t)) en el plano XY cu ando t var\355a desde a hasta b." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 307 63 "plot( [x(t), y(t) , t=a..b], color=red, \+ scaling=constrained) );" }}{PARA 0 "" 0 "" {TEXT -1 149 "Idem que el a nterior, pero las escalas en los ejes X,Y est\341n en relaci\363n 1:1. (equivalente a presionar el bot\363n 1:1 en la barra de men\372 del g r\341fico)." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 314 20 "M\341s sobre gr\341fic os: " }{TEXT 313 29 "with(plots), animate, display" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT 0 12 "with(plots) " }{TEXT -1 97 "Activa al paqu ete plots donde se encuentran las rutinas, display, implicitplot, text plot, etc...." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 55 "animate( \{ f(x), g(x,a)\}, x=x1..x2, a=a1..a2, nframes); " }{TEXT -1 163 " Crea una animaci\363n donde en cada cuadro se graf ica a las funciones f(x), g(x,a) en el intervalo [x1,x2], con a varia ndo desde a1 hasta a2 con un incremento de " }{XPPEDIT 18 0 "(a2-a1)/( nframes-1);" "6#*&,&%#a2G\"\"\"%#a1G!\"\"F&,&%(nframesGF&F&F(F(" } {TEXT -1 16 ". En total hay " }{TEXT 317 7 "nframes" }{TEXT -1 10 " c uadros. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 26 "p1:= plot( f(x), x=a..b): " }{TEXT -1 23 " Guarda en la variable \+ " }{TEXT 0 2 "p1" }{TEXT -1 72 " el gr\341fico de y=f(x), el cual pued e ser desplegado mediante el comando " }{TEXT 315 13 "display(p1). " } {TEXT -1 87 "Note que la \372nica manera en que el gr\341fico puede se r desplegado es mediante el comando " }{TEXT 316 9 "display. " }{TEXT -1 39 "Es conveniente terminar el comando con " }{TEXT 0 1 ":" }{TEXT -1 11 " en vez de " }{TEXT 0 1 ";" }{TEXT -1 30 " para evitar ouput in deseable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 19 "display(p1,p2,p3); " }{TEXT -1 49 " Grafica los gr\341ficos guarda dos en las variables " }{TEXT 318 9 "p1,p2,p3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 74 "display( sucesi\363n o lis ta de de variables con gr\341ficos, insequence=true); " }{TEXT -1 72 " Muestra cuadro a cuadro los gr\341ficos en la lista creando una animac i\363n. " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 320 68 "Estructuras de datos, (ex presi\363n) sucesi\363n, lista, conjunto, tabla: " }{TEXT 319 53 "[ s] , \{s\}, nops(s) , op(s) , seq(s) , s[n] , map(f,s)" }}{PARA 0 "" 0 " " {TEXT -1 153 "Para agrupar varios datos en una misma variable, se pu ede usar una de las siguientes estructuras de datos: (expresi\363n) s ucesi\363n, lista, conjunto, tabla." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 0 22 "exp1, exp2, ... , expn" }{TEXT -1 12 " \+ Es la " }{TEXT 332 8 "sucesi\363n" }{TEXT -1 229 " formada por las expresiones exp1, exp2, etc. En general, expresiones separadas por co mas definen a un sucesi\363n. Las expresiones pueden ser de distintos \+ tipos. Sucesiones respetan el orden de sus elementos y aceptan repetic iones" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 4 "[s ] " }{TEXT -1 8 " Es la " }{TEXT 333 5 "lista" }{TEXT -1 42 " formada por los elementos de la sucesi\363n " }{TEXT 321 1 "s" }{TEXT -1 17 " . En general, si " }{TEXT 322 2 "s " }{TEXT -1 25 "es una sucesi\363n \+ entonces " }{TEXT 323 3 "[s]" }{TEXT -1 79 " es una lista. Listas resp etan el orden de sus elementos y aceptan repeticiones" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 4 "\{s\} " }{TEXT -1 56 "Es el conjunto formado por los elementos de la sucesi\363n " }{TEXT 324 1 "s" }{TEXT -1 17 ". En general, si " }{TEXT 325 2 "s " }{TEXT -1 26 "es una sucesi\363n. entonces " }{TEXT 326 3 "\{s\}" }{TEXT -1 153 " e s un conjunto. En maple un conjunto emula al concepto matem\341tico de conjunto: elementos repetidos se eliminan y el orden de los elementos no importa. ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 15 "T[algo]:=valor;" }{TEXT -1 146 " Si T no es una lista , c rea la tabla T con una entrada. Con asignaciones adicionales del tipo \+ T[indice]=valor se agregan elementos a la tabla T." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 5 "s[k] " }{TEXT -1 43 "el ele mento k-\351simo de la lista o conjunto " }{TEXT 329 2 "s," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 7 "T[algo]" }{TEXT -1 59 " Si T es una tabla, valor asociado a \"algo\" en la tabla." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 8 "nops(s) " } {TEXT -1 46 "el n\372mero de elementos de la lista o conjunto " } {TEXT 327 1 "s" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 0 6 "op(s) " }{TEXT -1 55 " la sucesi\363n co n los elementos del conjunto o lista " }{TEXT 328 31 "s (le quita los \+ par\351ntesis a s)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 21 "seq( a(n), n=n1..n2 )" }{TEXT -1 52 " la sucesi\363n a(n1 ), a(n1+1), a(n1+2), .... a(n2). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 0 28 "[ seq ( (a(n), n=n1..n2 ) ] " }{TEXT -1 54 "la lista con los elementos a(n1), a(n1+1), ... , a(n2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 28 "\{ seq ( (a(n), n= n1..n2 ) \} " }{TEXT -1 93 " el conjunto con los elementos a(n1), a(n1 +1), ... , a(n2). Elementos repetidos se eliminan. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 9 "map(f,s) " }{TEXT -1 58 " la lista o conjunto que se obtiene de aplicar una funci\363n " }{TEXT 330 2 "f " }{TEXT -1 39 "a cada elemento de la lista o conjunto " } {TEXT 331 2 "s " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Ejemplos:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sucesion:=1,4,2,10,9,1,-4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "lista:= [sucesion];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "conjunto:= \{sucesion\};" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "op(conjunto); op(lista);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "nops(lista) , nops(conjunto );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f:= x-> x^2; map(f,li sta);map(f,conjunto);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "se q( k^2,k=1..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "datos:= \+ [ [1,2], [3,4], [5,6]];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "d atos[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "datos[2][1];" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "Una ta bla consiste en una asociaci\363n entre valores de un \355ndice y expr esiones. Las tablas se definen en forma din\341mica. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "T[1]:= 3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "T[4]:= 2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "T[cabeza_de_pescado]:= cola_de_pescado;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "op(T);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 336 12 "Los comandos" }{TEXT -1 1 " " }{TEXT 0 12 "fsolve,solve" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 28 "fsolve( f(x)=g(x), x=a..b); " }} {PARA 0 "" 0 "" {TEXT -1 46 "Calcula por metodos num\351ricos UNA solu ci\363n de " }{TEXT 337 9 "f(x)=g(x)" }{TEXT -1 58 " en el intervalo [ a,b], CON LA EXCEPCI\323N del caso en que " }{TEXT 338 10 "f(x), g(x) " }{TEXT -1 34 " sean polinomios, donde encuentra " }{TEXT 339 24 "TOD AS las raices reales" }{TEXT -1 22 " en el intervalo [a,b]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 23 "fsolve( f(x)=g(x ), x=a)" }{TEXT -1 47 " busca por m\351todos num\351ricos UNA soluci on de " }{XPPEDIT 18 0 "f(x)=g(x)" "6#/-%\"fG6#%\"xG-%\"gG6#F'" } {TEXT -1 27 " comenzando la b\372squeda en " }{TEXT 340 3 "x=a" } {TEXT -1 55 ". No necesariamente encuentra la soluci\363n m\341s cerca na a" }{TEXT 341 4 " x=a" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 20 "solve( f(x)=g(x), x)" }{TEXT -1 6 " . Si " }{TEXT 342 5 "f, g " }{TEXT -1 248 "son polinomios, encuentra \+ todas las raices, en otros casos intenta encontrar tantas soluciones c omo pueda mediante reducciones algebraicas. Si no encuentra soluciones ya sea por que no hay o por que no pudo hallarlas, retorna NULL, es d ecir \"nada\"." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Ejemplo" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eq := x^4-5*x^2+6*x=2;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solv e(eq,x);" }}}{PARA 0 "" 0 "" {TEXT -1 62 "N\363tese que el 1 aparece d os veces: esto se debe a que es una " }{TEXT 343 10 "ra\355z doble" } {TEXT -1 73 " de la ecuaci\363n: esto es, que el polonomio en cuesti \363n es divisible por " }{XPPEDIT 18 0 "(x-1)^2" "6#*$,&%\"xG\"\"\"F &!\"\"\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 44 "Podemos formar una lista c on las soluciones:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sols : = [solve(eq,x)];" }}}{PARA 0 "" 0 "" {TEXT -1 73 "Y ahora referirnos a una cualquiera de ellas por su posici\363n en la lista:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sols[3];" }}}{PARA 0 "" 0 "" {TEXT -1 44 "Nos entrega el tercer elemento de la lista. " }}{PARA 0 "" 0 " " {TEXT -1 54 "Podemos tambi\351n verificar que la soluci\363n es corr ecta:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(x=sols[3],eq);" }}}{PARA 0 "" 0 "" {TEXT -1 52 " No parece tan evidente que ambos lados sean iguales:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(s ubs(x=sols[3],eq));" }}}{PARA 0 "" 0 "" {TEXT -1 14 " Y s\355 lo eran! " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 345 21 " Algebra: los comandos" }{TEXT -1 1 " " }{TEXT 0 7 "expand," }{TEXT -1 1 " " }{TEXT 0 51 "combine, simplify, factor, normal, coeff, quo, rem. " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 19 "expand ( expresion);" }}{PARA 0 "" 0 "" {TEXT -1 139 "Expande la expresi\363n desarrollando potencias de binomios, distribuyendo productos con suma s, o expandiendo f\363rmulas trigonom\351tricas, etc.. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 20 "combine( expresion); " }}{PARA 0 "" 0 "" {TEXT -1 64 "aplicado a ciertas expresiones reali za la operaci\363n inversa de " }{TEXT 344 6 "expand" }{TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 20 "simplify( expresion);" }}{PARA 0 "" 0 "" {TEXT -1 294 "aplica las reglas de simp lificaci\363n que cumplen las funciones t\355picas trigonom\351tricas , exponenciaci\363n, logaritmos, polinomios, etc.. La noci\363n que Ma ple tiene de una expresi\363n simplificada es con toda probabilidad di ferente de la que Ud. tiene, la que es a su vez diferente de la de su \+ vecino." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 18 "factor(polinomio);" }}{PARA 0 "" 0 "" {TEXT -1 74 "factoriza el polin omio en factores con coeficientes enteros , racionales. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 23 "factor(polinomio,re al);" }}{PARA 0 "" 0 "" {TEXT -1 96 "factoriza el polinomio en factore s con coeficeintes reales usando aritm\351tica de punto flotante. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 26 "factor(pol inomio,complex);" }}{PARA 0 "" 0 "" {TEXT -1 90 "factoriza el polinomi o en factores lineales complejos usando aritm\351tica de punto flotant e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 307 18 "no rmal(expresion);" }}{PARA 0 "" 0 "" {TEXT -1 92 "obtiene una expresi \363n en la forma denominador/numerador, con t\351rminos comunes simpl ificados." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 23 "coeff( expresion ,x^k);" }}{PARA 0 "" 0 "" {TEXT -1 97 "obtiene el coeficiente de x^k en la expresion, siempre que \351sta sea una suma \+ de pontencias de x. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 0 32 "q:= quo(a,b,x); r:= rem(a,b,x);" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "Si " } {TEXT 0 3 "a,b" }{TEXT -1 17 " son polinomios, " }{TEXT 0 1 "q" } {TEXT -1 20 " es el cuociente de " }{TEXT 0 1 "a" }{TEXT -1 14 " divid ido por " }{TEXT 307 1 "b" }{TEXT -1 3 " y " }{TEXT 0 1 "r" }{TEXT -1 16 " es el resto de " }{TEXT 0 1 "a" }{TEXT -1 14 " dividido por " } {TEXT 307 1 "b" }{TEXT -1 37 ". El resto y el couciente satisfacen " } {TEXT 0 8 "a= b*q+r" }{TEXT -1 28 ", donde grado(r) < grado (b)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "limite" {TEXT 351 51 "L\355mites, l\355mites por la izquierda y por la derecha:" } {TEXT -1 1 " " }{TEXT 0 7 "limit()" }{TEXT -1 1 ":" }}{PARA 15 "" 0 " " {TEXT -1 2 " " }{TEXT 0 21 "limit(f(x),x=a,left) " }{TEXT -1 7 "cal cula" }{XPPEDIT 18 0 "limit(f(x),x = a,left);" "6#-%&limitG6%-%\"fG6#% \"xG/F)%\"aG%%leftG" }{TEXT -1 46 " , que es el l\355mite de f(x) cuan do x tiende a " }{TEXT 357 2 "a " }{TEXT -1 6 "por la" }{TEXT 358 11 " izquierda." }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 0 22 "limit(f(x) ,x=a,right) " }{TEXT -1 7 "calcula" }{XPPEDIT 18 0 "limit(f(x),x = a,r ight);" "6#-%&limitG6%-%\"fG6#%\"xG/F)%\"aG%&rightG" }{TEXT -1 46 " , \+ que es el l\355mite de f(x) cuando x tiende a " }{TEXT 355 2 "a " } {TEXT -1 6 "por la" }{TEXT 356 9 " derecha." }}{PARA 15 "" 0 "" {TEXT -1 0 "" }{TEXT 0 16 "limit(f(x),x=a) " }{TEXT -1 8 "calcula " } {XPPEDIT 18 0 "limit(f(x),x = a);" "6#-%&limitG6$-%\"fG6#%\"xG/F)%\"aG " }{TEXT -1 33 " , el cual existe s\363lo cuando " }{XPPEDIT 18 0 " limit(f(x),x = a,left);" "6#-%&limitG6%-%\"fG6#%\"xG/F)%\"aG%%leftG" } {TEXT -1 2 "= " }{XPPEDIT 18 0 "limit(f(x),x = a,right);" "6#-%&limitG 6%-%\"fG6#%\"xG/F)%\"aG%&rightG" }{TEXT -1 132 "= L. Si los l\355mites por la izquierda o por derecha no existieran o no fueran iguales ent onces el l\355mite no existe y maple responde " }{TEXT 359 9 "undefine d" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Ejemplos:" }}{PARA 0 "" 0 "" {TEXT -1 42 "Considere a la f unci\363n definida por tramos" }{TEXT 354 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f:= x-> piecewise( x<0, x+2, x>=0, x^2-x+1);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "plot(f(x), x=-2..2, discont=true,color=red);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 10 "Vemos que " }}{PARA 15 "" 0 "" {TEXT -1 48 "cuando x tien de a cero con x <0, f(x) tiende a 2" }}{PARA 15 "" 0 "" {TEXT -1 47 "c uando x tiende a cero con x>0, f(x) tiende a 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 8 "Es decir" }} {PARA 310 "" 0 "" {XPPEDIT 18 0 "limit(f(x),x = 0,left) = 2;" "6#/-%&l imitG6%-%\"fG6#%\"xG/F*\"\"!%%leftG\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "limit(f(x),x = 0,right) = 1;" "6#/-%&limitG6%-%\"fG6#%\"xG/F*\" \"!%&rightG\"\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Maple puede calcular estos l\355mites" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "limit(f(x), x=0);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit(f(x), x=0,right);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(f(x), x=0,left);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "Para nuestra funci\363n, para " } {XPPEDIT 18 0 "a <>0" "6#0%\"aG\"\"!" }{TEXT -1 63 " los l\355mites \+ por la izquierda y por la derecha son iguales a " }{TEXT 352 4 "f(a)" }{TEXT -1 39 " y el l\355mite existe. Por ejemplo, para " }{TEXT 353 3 "a=2" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "L1:=limit( f(x),x=2,left);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "L2:= limit( f(x),x=2,right); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "L:= limit(f(x),x=2);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(2);" }}}{PARA 0 "" 0 "" {TEXT -1 21 "En general para a>0, " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "assume(a,positive);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(f(x),x=a,left);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(f(x),x=a,right);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "limit(f(x),x=a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(a);" }}}{PARA 0 "" 0 "" {TEXT -1 21 "En general para a<0, " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "assume(a,negative) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(f(x),x=a,left); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(f(x),x=a,right); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "limit(f(x),x=a);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(a);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "unsassign('a ');" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "derivada" {TEXT 365 10 "Derivadas:" }{TEXT -1 1 " " }{TEXT 0 12 " D(), diff()" }} {PARA 0 "" 0 "" {TEXT 0 5 "D(f) " }{TEXT -1 82 "Si f es una funci\363n entonces D(f) es la funci\363n f ' y entonces D(f)(x) es f ' (x)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Por ejemp lo, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(sin); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Sin \+ intentamos " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "D(sin(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "con el prop\363sito de obtener cos(x) vemos que no funciona." }} {PARA 0 "" 0 "" {TEXT -1 18 "Para ello usamos:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "D(sin)(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 11 "(D @@2) (f)" }{TEXT -1 95 " Si f es una funci\363n entonces (D@@2)(f) es la funci\363n f '', qu e es la segunda derivada de f . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 14 "diff(expr, x) " }{TEXT -1 3 "Si " }{TEXT 360 4 "expr" }{TEXT -1 33 " es una expr esi\363n en la variable " }{TEXT 361 1 "x" }{TEXT -1 10 " entonces " } {TEXT 362 12 "diff(expr,x)" }{TEXT -1 19 " es la derivada de " }{TEXT 363 4 "expr" }{TEXT -1 51 " con res pecto a " }{TEXT 364 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Por ejemplo," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(sin(x), x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 15 "diff( expr,x$2)" }{TEXT -1 62 " Es la segunda derivada de expr con respecto a la variable x. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 15 "diff( \+ expr,x$3)" }{TEXT -1 62 " Es la tercera derivada de expr con respecto a la variable x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 15 "diff( expr,x$n)" }{TEXT -1 62 " Es la n-\351sima derivada de expr con respecto a la variable x." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "Una diferencia importante es que " } {MPLTEXT 1 0 5 "D(f) " }{TEXT -1 8 " es una " }{TEXT 366 7 "funci\363n " }{TEXT -1 16 ", mientras que " }{MPLTEXT 1 0 7 "diff( )" }{TEXT -1 9 " es una " }{TEXT 367 9 "expresi\363n" }{TEXT -1 103 ". As\355, por ejemplo, si deseamos evaluar la derivada de la funci\363n f(x)=sin(x )/(1+x) en el punto x=0," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=x->sin(x)/(1+x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(x);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "#Es la derivada en un punto x" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "#Es la derivada evaluada en el punto x=0. Usando dif f es as\355:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f(x), x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(x=0,%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Problemas" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 400 10 "Problema 1" } {TEXT -1 31 " (El m\351todo de Newton-Raphson )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Introducci\363n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 396 26 "LA DE SCRIPCION DEL METODO" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 141 "Es un algortimo para aproximarse, en forma iterativa, \+ a una soluci\363n de una ecuaci\363n del tipo f(x)=0, donde f es una funci\363n diferenciable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Se parte de una aproximaci\363n inicial " } {XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 92 " (tan cercana a l a ra\355z buscada como se pueda ) a partir de la cual se genera una su cesi\363n " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 4 " , \+ " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 4 " , " } {XPPEDIT 18 0 "x[2]" "6#&%\"xG6#\"\"#" }{TEXT -1 60 " , ... que, en c ondiciones apropiadas, converge a la ra\355z." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "La idea es que, teniendo \+ ya el valor " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT -1 99 " (el cual se espera est\351 cerca de la ra\355z) se \"reemplaza\" la \+ curva y=f(x) por su tangente en ( " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG 6#%\"nG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 " f(x[n])" "6#-%\"fG6#&%\"xG 6#%\"nG" }{TEXT -1 93 " ) (la cual debe parecerse a la curva en una \+ vecindad del punto de contacto) y se define " }{XPPEDIT 18 0 "x[n+1] " "6#&%\"xG6#,&%\"nG\"\"\"F(F(" }{TEXT -1 128 " como la abcisa del pu nto en que dicha tangente intersecta al eje X . En otras palabras, s e reemplaza la ecuaci\363n original " }{XPPEDIT 18 0 "f(x)=0" "6#/-% \"fG6#%\"xG\"\"!" }{TEXT -1 6 " por " }{XPPEDIT 18 0 "g[n](x)=0" "6#/ -&%\"gG6#%\"nG6#%\"xG\"\"!" }{TEXT -1 11 " donde " }{XPPEDIT 18 0 "g[n](x)" "6#-&%\"gG6#%\"nG6#%\"xG" }{TEXT -1 41 " es la ecuaci\363n \+ de la tangente en ( " }{XPPEDIT 18 0 "x[n] , f(x[n])" "6$&%\"xG6#% \"nG-%\"fG6#&F$6#F&" }{TEXT -1 3 " )." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "Se espera que " }{XPPEDIT 18 0 "x[n +1]" "6#&%\"xG6#,&%\"nG\"\"\"F(F(" }{TEXT -1 42 " constituya una mejo r aproximaci\363n que " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" } {TEXT -1 79 " y as\355, iterando, obtener la sucesi\363n que converja al valor exacto de la ra\355z." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 68 "Por ejemplo, si intentamos buscar la ra \355z postiva de la ecuaci\363n " }{XPPEDIT 18 0 "x^2-2=0" "6#/,&*$% \"xG\"\"#\"\"\"F'!\"\"\"\"!" }{TEXT -1 23 " ( o sea, el n\372mero \+ " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 40 " ) partie ndo de la aproximaci\363n incial " }{XPPEDIT 18 0 "x[0]=2" "6#/&%\"xG 6#\"\"!\"\"#" }{TEXT -1 58 " , graficamos la curva y su tangente en e l punto inicial:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=x->x^2-2;x0:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "g0:=x->f(x0)+D(f)(x0)*(x-x0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(\{f(x),g0(x)\},x=0..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 57 "La intersecci\363n de la tangente con el eje x \+ es x1:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "x1:=solve(g0(x)=0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "y ahora iteramos partiendo de x1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 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Podr\355a alejarse de cualquier ra\355z o bie n converger a otra ra\355z cercana (por eso es importante que la aprox imaci\363n inicial sea lo m\341s cercana posible)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Por ejemplo: Deseamos obt ener una ra\355z de la ecuaci\363n " }{XPPEDIT 18 0 "x^3=x+1" "6#/*$ %\"xG\"\"$,&F%\"\"\"F(F(" }{TEXT -1 22 " ( esto es, h(x) = " } {XPPEDIT 18 0 "x^3-x-1=0" "6#/,(*$%\"xG\"\"$\"\"\"F&!\"\"F(F)\"\"!" } {TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "Por el teorema del valor intermedio sabemos que hay una \+ ra\355z entre 0 y 2 ( h(0)<0 , h(2)>0 y h continua en [0,2] \+ )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Empe zamos con valor incial x0=0.6 :" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "h:=x->x^3-x-1; x0:=0.6;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "H:=x->h(x0)+D(h)(x0)*(x-x0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(\{h(x),H(x)\},x=0..2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "La tangente no es horiz ontal. Ella corta al eje X, pero muy lejos de la ra\355z :" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(H(x)=0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 205 "Esto es , si en alg\372n punto de la iteraci\363n llegamos a una valor en el c ual la tangente es casi horizonatal, nos vamos a ir alejando de la ra \355z (peor a\372n, podr\355amos caer en un punto en el cual la tangen te " }{TEXT 381 3 " es" }{TEXT -1 41 " horizontal y all\355 el m\351t odo colapsar\355a)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 399 12 "EL ALGORITMO" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "En general, la \+ tangente a la curva " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG " }{TEXT -1 17 " en el punto ( " }{XPPEDIT 18 0 " x[n] , f(x[n]" "6$ &%\"xG6#%\"nG-%\"fG6#&F$6#F&" }{TEXT -1 22 " ) tiene ecuaci\363n \+ " }{XPPEDIT 18 0 "g[n](x)=f(x[n])+D(f)(x[n])*(x-x[n])" "6#/-&%\"gG6#% \"nG6#%\"xG,&-%\"fG6#&F*6#F(\"\"\"*&--%\"DG6#F-6#&F*6#F(F1,&F*F1&F*6#F (!\"\"F1F1" }{TEXT -1 56 " y si denotamos su intersecci\363n con el eje X por " }{XPPEDIT 18 0 "x[n+1]" "6#&%\"xG6#,&%\"nG\"\"\"F(F( " }{TEXT -1 17 " tenemos que " }{XPPEDIT 18 0 "0=g[n](x[n+1])" "6# /\"\"!-&%\"gG6#%\"nG6#&%\"xG6#,&F)\"\"\"F/F/" }{XPPEDIT 18 0 "``=f(x[n ])+D(f)(x[n])*(x[n+1]-x[n])" "6#/%!G,&-%\"fG6#&%\"xG6#%\"nG\"\"\"*&--% \"DG6#F'6#&F*6#F,F-,&&F*6#,&F,F-F-F-F-&F*6#F,!\"\"F-F-" }{TEXT -1 23 " , de lo cual obtenemos" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 310 " " 0 "" {XPPEDIT 18 0 "x[n+1]=x[n]-f(x[n])/D(f)(x[n])" "6#/&%\"xG6#,&% \"nG\"\"\"F)F),&&F%6#F(F)*&-%\"fG6#&F%6#F(F)--%\"DG6#F/6#&F%6#F(!\"\"F :" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "E sta recurrencia, junto con " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"! " }{TEXT -1 92 " = aproximaci\363n inicial , nos define la sucesi\363 n que esperamos converja a la ra\355z buscada." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Notemos de paso, que si \+ " }{XPPEDIT 18 0 "D(f)(x[n])=0" "6#/--%\"DG6#%\"fG6#&%\"xG6#%\"nG\"\"! " }{TEXT -1 76 " el m\351todo colapsa. A\372n si no es cero pero muy c ercana a cero, la fracci\363n " }{XPPEDIT 18 0 "f(x[n])/ D(f)(x[n])" "6#*&-%\"fG6#&%\"xG6#%\"nG\"\"\"--%\"DG6#F%6#&F(6#F*!\"\"" }}{PARA 0 " " 0 "" {TEXT -1 101 "puede ser enorme y as\355 alejarnos de la ra\355z perseguida (tal como hab\355amos apreciado geom\351tricamente)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 150 "Finalmen te hacemos notar que, si bien existen m\351todos num\351ricos para est imar el error cometido en la en\351sima iteraci\363n (la diferecia ab soluta entre " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT -1 76 " y la ra\355z buscada) por el momento podemos contentarnos con asumir que si " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT -1 5 " y \+ " }{XPPEDIT 18 0 "x[n+1]" "6#&%\"xG6#,&%\"nG\"\"\"F(F(" }{TEXT -1 123 " tienen los primeros k decimales en com\372n, entonces tenemos una p aroximaci\363n que es buena hasta la k-\351sima cifra decimal. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 370 0 "" } {TEXT -1 1 " " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "1i)" }}{PARA 0 " " 0 "" {TEXT -1 2 " " }{TEXT 379 107 "Los babilonios usaban la siguie nte recursi\363n para aproximarse a la ra\355z cuadrada de un n\372mer o positivo a :" }}{PARA 0 "" 0 "" {TEXT 382 0 "" }}{PARA 310 "" 0 "" {XPPEDIT 287 0 "x[n+1]=[1/2]*(x[n]+a/x[n])" "6#/&%\"xG6#,&%\"nG\"\"\"F )F)*&7#*&F)F)\"\"#!\"\"F),&&F%6#F(F)*&%\"aGF)&F%6#F(F.F)F)" }}{PARA 0 "" 0 "" {TEXT 383 0 "" }}{PARA 0 "" 0 "" {TEXT 384 45 "Comente sobre l a eficacia de dicho algoritmo." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 371 9 "Respuesta" }{TEXT 372 1 ":" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 380 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "1ii)" }}{PARA 0 "" 0 "" {TEXT 374 59 "Usando el m\351to do de Newton encuentre, con una precisi\363n de " }{XPPEDIT 281 0 "10^ (-4)" "6#)\"#5,$\"\"%!\"\"" }{TEXT 385 27 " , la ra\355z de la ecuaci \363n " }{XPPEDIT 282 0 "x^3-2*x^2-5=0" "6#/,(*$%\"xG\"\"$\"\"\"*&\" \"#F(*$F&F*F(!\"\"\"\"&F,\"\"!" }{TEXT 386 101 " que se encuentra en e l intervalo [ 0, 4 ]. Compare su respuestas con las que entrega Maple usando " }{MPLTEXT 1 0 6 "fsolve" }}{PARA 0 "" 0 "" {TEXT 387 0 "" } }{PARA 0 "" 0 "" {TEXT 373 9 "Respuesta" }{TEXT 375 1 ":" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 5 "1iii)" }}{PARA 0 "" 0 "" {TEXT 377 49 "Indique cu\341ntas ra\355ces tiene la ecuaci \363n f(x) = " }{XPPEDIT 284 0 "2*x*cos(2*x)-(x-2)^2=0" "6#/,&*(\"\"# \"\"\"%\"xGF'-%$cosG6#*&F&F'F(F'F'F'*$,&F(F'F&!\"\"F&F/\"\"!" }{TEXT 393 3 " , " }{TEXT 394 25 "justificando su respuesta" }{TEXT 395 1 ". " }}{PARA 0 "" 0 "" {TEXT 378 0 "" }}{PARA 0 "" 0 "" {TEXT 376 9 "Resp uesta" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "1iv)" }}{PARA 0 "" 0 "" {TEXT 390 37 "Usando el m\351 todo de Newton encuentre " }{TEXT 391 5 "todas" }{TEXT 392 156 " las r a\355ces de la ecuaci\363n de la parte anterior con una precisi\363n d e 4 cifras decimales correctas. Compare sus respuestas con las que ent rega Maple usando " }{MPLTEXT 1 0 6 "fsolve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 388 9 "Respuesta" }{TEXT 389 1 ":" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 413 10 " Problema 2" }{TEXT -1 31 " (Tasas de cambio relacionadas)" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Introducci \363n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 297 "Un punto (x(t), y(t)) se mueve a rapidez constante por una curva C \+ definida impl\355citamente por F(x,y) = C en el plano XY (en una posi ble interpretaci\363n puede pensarse que dichas ecuaciones determinan \+ las coordenadas del punto en funci\363n del tiempo de modo que el par \+ de ecuaciones describen el " }{TEXT 414 11 "itinierario" }{TEXT -1 13 " del punto). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "El problema es determinar las tasas de cambio de x(t) e y(t) en un punto dado de la curva (las velocidades de cada coordenada )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 " El siguiente ejemplo muestra como hacerlo." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Consideramos la curva definida \+ implicitamente por F(x,y)=3, donde " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "F:= (x,y)-> x*y+x^3+y^2; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "implicitplot( F(x,y)=3, x=-4..4,y=-4..4,grid=[30,30]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 217 "Se sabe que el punto (x(t),y(t)) viaja a velocidad constante e igual a 3 a lo largo de la curva y se pide dete rminar las tasas de cambio x'(t), y'(t) en el instante en que el punto pasa por el punto (1,1) de la curva." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "La ec uaci\363n que relaciona x(t) con y(t) es la ecuaci\363n de la curva : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eq:= F(x(t),y(t))=3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " Las " }{TEXT 408 49 "tasas de cambio de x(t) e y(t) est\341n relaciona das" }{TEXT -1 89 " por la siguiente ecuaci\363n, que se obtiene deriv ando implicitamente la ecuaci\363n anterior." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "diff(%,t); colle ct(%,diff(x(t),t));eq1:=collect(%,diff(y(t),t));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 72 "La velocidad con que se mueve un punto (x(t),y(t)) en el plano XY es " }{XPPEDIT 18 0 " s(t) = sqrt(diff(x(t),t)^2+diff(y(t),t)^2);" "6#/-%\"sG6#%\"tG-%%sqrtG 6#,&*$-%%diffG6$-%\"xG6#F'F'\"\"#\"\"\"*$-F.6$-%\"yG6#F'F'F3F4" } {TEXT -1 3 " =3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Entonces " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eq2 :=9 = diff(x(t),t)^2+diff(y(t),t)^2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 187 "Para determinar x'(t), y'(t) debemos reemplazar x(t)=1, y(t)=1 en la ecuaci\363n deq1, y resolver el sistema de dos ecuaciones para las dos incognitas x'(t), y'(t) definidas por deq1, deq2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Renombramos x'(t) y'(t) ...." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eq3:=subs(\{diff(x(t) ,t)=xp,diff(y(t),t)=yp\},eq1);" }}}{PARA 0 "" 0 "" {TEXT -1 48 "Usamos que estamos en el punto x(t)=1, y(t)=1..." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "eq4:=subs( \{x(t)=1, y(t)=1\},eq3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "eq5:=subs( \{diff(x(t),t)=xp, diff( y(t),t)=yp\},eq2);" }}}{PARA 0 "" 0 "" {TEXT -1 110 "Resolvemos el sis tema de 2x2 (es la intersecci\363n de una recta por el or\355gen con u na circunferencia de radio 3)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sols:=solve(\{eq4,eq5\},\{xp ,yp\});" }}}{PARA 0 "" 0 "" {TEXT -1 139 "Tenemos 2 soluciones: xp =9/5, yp =-12/5 corresponde al caso en que el punto se mueve a la der echa pasando por (1,1) (pues x' >0) y" }}{PARA 0 "" 0 "" {TEXT -1 129 "xp = - 9/5, yp = 12/5 corresponde al caso en el punto se \+ mueve de derecha a izquierda pasando por (1,1) (pues x' < 0 ). ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 6 "2i) " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 403 163 "Conside re el mismo punto P en la curva F(x,y) definido en la introducci\363 n y otro punto Q que se mueve a velocidad constante e igual a 6 por \+ la curva G(x,y) = 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "G:= (x,y) -> -2*x*y-3*x^3+y^2; G(-1,-1); \+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "implicitplot( G(x,y)=2, x=-4..4,y=-4..4,grid=[70,70]) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 401 329 "Se pide determinar la \+ velocidad con que cambia la distancia entre los puntos P en la curva \+ F(x,y) =3, y Q en la curva G(x,y) = 2, cuando ellos viajan a velocid ades constantes e iguales a 3 y a 6 respectivamente, en un instante en que P pasa por (1,1) de derecha a izquierda y Q pasa por (-1,.- 1) de arriba hacia abajo" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 316 "" 0 "" {TEXT 402 9 "Respuesta" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "2ii)" }}{PARA 0 " " 0 "" {TEXT 404 157 "Un minutero de un reloj mide 8 mm de longitud y \+ el horario 4 mm. \277Con qu\351 velocidad cambia la distancia entre la s puntas de las menecillas a la una en punto?" }}{PARA 0 "" 0 "" {TEXT 405 0 "" }}{PARA 316 "" 0 "" {TEXT 406 9 "Respuesta" }}{PARA 0 " " 0 "" {TEXT 407 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 5 "2iii)" }}{PARA 0 "" 0 "" {TEXT 409 155 "Una bola de 10 cms de radio se hunde, a raz\363n de 1 cm/se g, al interior de un hemisferio de 50 cent\355metros de radio que est \341 parcialmente lleno con agua." }}{PARA 0 "" 0 "" {TEXT 410 106 " \277 Con qu\351 rapidez sube el el nivel del agua en el preciso instan te en que la bola se sumerge por completo?" }}{PARA 0 "" 0 "" {TEXT 411 0 "" }}{PARA 316 "" 0 "" {TEXT 412 9 "Respuesta" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "34" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }