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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 "Normal" -1 319 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 }} {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 \+ 9" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 5 "" 0 "" {TEXT 295 26 "Martes 8 de junio de 2010 " }{TEXT 302 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT 297 16 " Plazo de entrega" }{TEXT 298 42 " : Viernes 18 de junio a las 22:00 h oras." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT 301 31 "Pr\363ximo Laboratorio se sube el" }{TEXT -1 4 " : " }{TEXT 303 27 "No hay pr\363ximo laboratorio." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 4 "" 0 "" {TEXT 367 10 "IMPORTANTE" }{TEXT 370 102 " : Este es e l \372ltimo laboratorio del semestre. F\355jense bien en el plazo de e ntrega. La pr\363xima semana " }{TEXT 369 2 "no" }{TEXT 368 44 " habr \341 atenci\363n en las salas de laboratorio." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 311 "" 0 "" {TEXT -1 0 "" }{TEXT 290 24 "NOMBRE \+ ALUMNO DIGITADOR:" }}{PARA 316 "" 0 "" {TEXT -1 0 "" }{TEXT 291 0 "" } {TEXT 292 27 "N\372mero de alumno digitador:" }}{PARA 0 "" 0 "" {TEXT 293 19 "Secci\363n de C\341tedra:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 289 19 "NOMBRE O TRO ALUMNO:" }}{PARA 317 "" 0 "" {TEXT -1 17 "N\372mero de alumno:" }} {PARA 317 "" 0 "" {TEXT -1 19 "Secci\363n de C\341tedra:" }}{PARA 315 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 300 1 "(" }{TEXT 299 61 "El alumno digitador se alterna de laboratorio en laboratorio)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 310 "" 0 "" {TEXT -1 0 "" }} {PARA 311 "" 0 "" {TEXT 296 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 0 "" 0 "" {TEXT -1 54 "Practicar el integraci\363n por sustituci\363n y por part es." }}{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 sin gr \341ficos. Para ello elja la opci\363n Edit->Remove Output->From Works heet." }}{PARA 15 "" 0 "" {TEXT -1 345 "Para corregir su laboratorio e l ayudante ejecutar\341 primero el laboratorio con Edit->Execute->Wo rksheet. El ayudante entregar\341 el laboratorio corregido sin output, pero con los comentarios correspondientes. Los comandos deben estar \+ en secuencia l\363gica de modo de que, al ejecutarlos en orden, los c \341lculos sean correctos. Para simplificar use " }{TEXT 304 7 "restar t" }{TEXT -1 86 " en cada problema y cargue nuevamente los paquetes qu e necesita (with(plots), etc...) " }}{PARA 15 "" 0 "" {TEXT -1 63 "Tod os los ejercicios a ser realizados por usted est\341n en color " } {TEXT 306 4 "blue" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 62 "La parte verbal de sus repuestas debe ser entregada en color " }{TEXT 305 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 331 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 332 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 304 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 307 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 308 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 309 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 304 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 304 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 311 20 "M\341s sobre gr\341fic os: " }{TEXT 310 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 314 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 312 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 313 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 315 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 317 68 "Estructuras de datos, (ex presi\363n) sucesi\363n, lista, conjunto, tabla: " }{TEXT 316 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 329 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 330 5 "lista" }{TEXT -1 42 " formada por los elementos de la sucesi\363n " }{TEXT 318 1 "s" }{TEXT -1 17 " . En general, si " }{TEXT 319 2 "s " }{TEXT -1 25 "es una sucesi\363n \+ entonces " }{TEXT 320 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 321 1 "s" }{TEXT -1 17 ". En general, si " }{TEXT 322 2 "s " }{TEXT -1 26 "es una sucesi\363n. entonces " }{TEXT 323 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 326 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 324 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 325 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 327 2 "f " }{TEXT -1 39 "a cada elemento de la lista o conjunto " } {TEXT 328 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 333 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 334 9 "f(x)=g(x)" }{TEXT -1 58 " en el intervalo [ a,b], CON LA EXCEPCI\323N del caso en que " }{TEXT 335 10 "f(x), g(x) " }{TEXT -1 34 " sean polinomios, donde encuentra " }{TEXT 336 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 337 3 "x=a" } {TEXT -1 55 ". No necesariamente encuentra la soluci\363n m\341s cerca na a" }{TEXT 338 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 339 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 340 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 342 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 341 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 304 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 304 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 304 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 348 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 354 2 "a " }{TEXT -1 6 "por la" }{TEXT 355 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 352 2 "a " } {TEXT -1 6 "por la" }{TEXT 353 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 356 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 351 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 349 4 "f(a)" }{TEXT -1 39 " y el l\355mite existe. Por ejemplo, para " }{TEXT 350 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 362 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 357 4 "expr" }{TEXT -1 33 " es una expr esi\363n en la variable " }{TEXT 358 1 "x" }{TEXT -1 10 " entonces " } {TEXT 359 12 "diff(expr,x)" }{TEXT -1 19 " es la derivada de " }{TEXT 360 4 "expr" }{TEXT -1 51 " con res pecto a " }{TEXT 361 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 363 7 "funci\363n " }{TEXT -1 16 ", mientras que " }{MPLTEXT 1 0 7 "diff( )" }{TEXT -1 9 " es una " }{TEXT 364 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 "" }}{SECT 1 {PARA 5 "" 0 "Sumas de Riemann" {TEXT 522 18 "Sumas de Riemann: " }{TEXT 524 84 "with(stu dent),rightsum, leftsum, middlesum, trapezoid, leftbox, rightbox, mid dlebox" }{TEXT 525 1 " " }}{PARA 0 "" 0 "" {TEXT 523 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "with(student)" {TEXT 0 13 "with(student)" }{TEXT -1 31 " Activa al paquete de rutinas " }{TEXT 526 7 "student" }{TEXT -1 50 ". Se requiere para ocupar las rutinas que siguen. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 23 "leftsum(f(x), x=a.. b,n)" }{TEXT -1 50 " Calcula la suma de Riemann para una partici\363n \+ de " }{TEXT 520 2 "n " }{TEXT -1 155 "subintervalos de igual longitud \+ en que cada rect\341ngulo tiene como altura el valor de la funci\363n \+ en el extremo izquierdo de cada subintervalo. El valor de " }{TEXT 530 1 "n" }{TEXT -1 40 " es opcional. El valor por omisi\363n es " } {TEXT 531 1 "n" }{TEXT -1 5 "=4 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 0 25 "rightsum( f(x), x=a..b,n)" }{TEXT -1 59 " \+ Idem que el anterior pero para sumas derechas de Riemann." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 26 "middlesum( f(x), x =a..b,n)" }{TEXT -1 64 " Idem que el anterior pero para sumas de punto medio de Riemann." }}{PARA 0 "" 0 "" {TEXT 0 27 "trapezoid( f(x), x=a ..b,n) " }{TEXT -1 8 " Estima " }{XPPEDIT 18 0 "int( f(x),x=a..b) " "6 #-%$intG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 64 " mediante la f \363rmula trapezoidal para una partici\363n uniforme de " }{TEXT 521 1 "n" }{TEXT -1 146 " subintervalos.Esto es, en vez de aproximar cada \+ franja vertical por un rect\341ngulo lo hace por el trapecio que une l os puntos extremos de aquella." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 0 26 "leftbox( f(x), x=a..b, n) " }{TEXT -1 186 "Grafica los rect\341ngulos correspondientes a la suma izquierda de Ri emann de f(x) en el intervalo [a,b] para una partici\363n de n subtint ervalos de igual longitud (valor por omisi\363n es n=4)." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 25 "rightbox( f(x), x=a ..b,n)" }{TEXT -1 61 " Idem que el anterior, pero para la suma derec ha de Riemann" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 26 "middlebox( f(x), x=a..b,n)" }{TEXT -1 69 " Idem que el ant erior, pero para la suma en puntos medios de Riemann." }}{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 34 "resta rt:with(plots):with(student):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Al \341r ea total de los rect\341ngulos verdes es la suma izquierda de Riemann \+ para esta partici\363n y es igual a:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "leftsum(x^2,x=1..3,10);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "lb:=leftbox( x^2,x=1..3,10,shading=GREEN):display(lb) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "El \341rea total de los rect\341ngulos azules es la suma derecha de Riem ann para esta partici\363n y es igual a:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "rightsum(x^2,x=1..3,10);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "rb:=rightbo x( x^2,x=1..3,10,shading=BLUE):display(rb);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Si las superponemos," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(lb,rb);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 528 11 "Integrales:" }{TEXT -1 1 " " } {TEXT 529 28 "int(), Int(), evalf(Int () )" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 0 19 "int( f(x), x=a..b) " }{TEXT -1 19 " Intenta calcul ar " }{XPPEDIT 18 0 "int(f(x),x = a .. b)" "6#-%$intG6$-%\"fG6#%\"xG/ F);%\"aG%\"bG" }{TEXT -1 29 " mediante m\351todos simb\363licos." }} {PARA 0 "" 0 "" {TEXT 0 18 "Int( f(x), x=a..b)" }{TEXT -1 17 " Es la e xpresi\363n " }{XPPEDIT 18 0 "int(f(x),x = a .. b)" "6#-%$intG6$-%\"fG 6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 99 " NO EVALUADA. Es decir, es la exp resi\363n que al ser evaluada se obtiene la integral de f(x) en [a,b] " }}{PARA 0 "" 0 "" {TEXT 0 25 "evalf ( Int (f(x),x=a..b)" }{TEXT -1 21 " Estima el valor de " }{XPPEDIT 18 0 "int( f(x),x=a..b) " "6#-%$i ntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 10 " mediante " }{TEXT 527 18 "m\351todos num\351ricos " }{TEXT -1 128 "(parecido a sumas de \+ Riemann pero mucho m\341s sofisticado y de alta precisi\363n). Los l \355mities de integraci\363n a,b deben ser n\372meros." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 26 "value( Int( f(x), x=a ..b) " }{TEXT -1 126 ") Intenta evaluar mediante m\351todos simb\363 licos la integral de f(x) en el intervalo [a,b]. Es equivalente a int( f(x), x=a..b)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Ejemplos:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "int( x^2 ,x=a..b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "int(x^2,x=1..2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Int( x^2,x=1..2);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf( Int(x^2,x=1..2));" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "La sig uiente integral no se puede evaluar mediante una f\363rmula expl\355c ita. La \372nica opci\363n es la integraci\363n num\351rica." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot( x^x,x=1..2,color=red); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "int( x^x,x=1..2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf( Int( x^x,x=1..2));" } }}{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 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Problemas" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 " \+ " }{TEXT 371 10 "Ejercicio " }{TEXT -1 35 "1 (Indentificando sumas de \+ Riemann)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Introducci\363n" }}{PARA 0 "" 0 "" {TEXT -1 147 "Ciertos \+ l\355mites de sumas pueden ser evaluados reconociendo la suma en cuest i\363n como una suma de Riemann y calcul\341ndola entonces como una in tegral. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Considere, por ejemplo, el l\355mite, cuando n tiende a infin ity , de la siguiente suma:" }}{PARA 310 "" 0 "" {XPPEDIT 18 0 "n/(n^ 2+1^2)" "6#*&%\"nG\"\"\",&*$F$\"\"#F%*$F%F(F%!\"\"" }{TEXT -1 5 " + \+ " }{XPPEDIT 18 0 "n/(n^2+2^2)" "6#*&%\"nG\"\"\",&*$F$\"\"#F%*$F(F(F%! \"\"" }{TEXT -1 14 " + .... + " }{XPPEDIT 18 0 "n/(n^2+n^2)" "6#*& %\"nG\"\"\",&*$F$\"\"#F%*$F$F(F%!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "O sea, usando la not aci\363n sigma:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 310 "" 0 "" {XPPEDIT 18 0 "limit(sum(n/(n^2+k^2),k=1..n),n=infinity)" "6#-%&limitG 6$-%$sumG6$*&%\"nG\"\"\",&*$F*\"\"#F+*$%\"kGF.F+!\"\"/F0;F+F*/F*%)infi nityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Intentamos hacerlo con Maple:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "S:=Sum(n/(n^2+k^2),k=1..n); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "No es muy claro, que digamos. No obstante, como a ntes, s\355 podemos hallar el l\355mite:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(S,n=infinity); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Otra forma de proceder es indentificar la sua anterior como una suma de Ri emann. Escribimos" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum(n/(n^2+k^2),k=1..n)" "6#-%$sumG6$*&%\"nG\"\"\",&*$F '\"\"#F(*$%\"kGF+F(!\"\"/F-;F(F'" }{TEXT -1 11 " = " } {XPPEDIT 18 0 "[1/n]*sum(n^2/(n^2+k^2),k=1..n)" "6#*&7#*&\"\"\"F&%\"nG !\"\"F&-%$sumG6$*&F'\"\"#,&*$F'F-F&*$%\"kGF-F&F(/F1;F&F'F&" }}{PARA 0 "" 0 "" {TEXT -1 34 " = " }{XPPEDIT 18 0 "[1/n]*sum(1/(1+(k/n)^2),k=1..n)" "6#*&7#*&\"\"\"F&%\"nG!\"\"F&-%$su mG6$*&F&F&,&F&F&*$*&%\"kGF&F'F(\"\"#F&F(/F0;F&F'F&" }{TEXT -1 2 " ." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Vemos qu e los puntos " }{XPPEDIT 18 0 "k/n" "6#*&%\"kG\"\"\"%\"nG!\"\"" } {TEXT -1 4 " , " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 13 "=1, 2, .. . , " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 104 " son los puntos der echos de una partici\363n pareja del intervalo [0, 1] en n subinter valos de largo " }{XPPEDIT 18 0 "h=1/n" "6#/%\"hG*&\"\"\"F&%\"nG!\" \"" }{TEXT -1 35 " cada uno. Esto es, tiene la forma" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 310 "" 0 "" {XPPEDIT 18 0 "[(b-a)/n]*sum(f( a+k*(b-a)/n),k=1..n) " "6#*&7#*&,&%\"bG\"\"\"%\"aG!\"\"F(%\"nGF*F(-%$s umG6$-%\"fG6#,&F)F(*(%\"kGF(,&F'F(F)F*F(F+F*F(/F4;F(F+F(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "dond e " }{XPPEDIT 18 0 "a=0 " "6#/%\"aG\"\"!" }{TEXT -1 7 " y " } {XPPEDIT 18 0 "b=1" "6#/%\"bG\"\"\"" }{TEXT -1 48 ". Por tanto, el l \355mite de la suma anterior es " }{XPPEDIT 18 0 "int(f(x),x=a..b)" " 6#-%$intG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 6 " = " } {XPPEDIT 18 0 "int(f(x),x=0..1)" "6#-%$intG6$-%\"fG6#%\"xG/F);\"\"!\" \"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "P ara \"descubirir\" la funci\363n " }{XPPEDIT 18 0 "f" "6#%\"fG" } {TEXT -1 13 " notamos que" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 310 "" 0 "" {XPPEDIT 18 0 "f(a+k*(b-a)/n)" "6#-%\"fG6#,&%\"aG\"\"\"*(% \"kGF(,&%\"bGF(F'!\"\"F(%\"nGF-F(" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "f(k/n)" "6#-%\"fG6#*&%\"kG\"\"\"%\"nG!\"\"" }{TEXT -1 7 " = " } {XPPEDIT 18 0 "1/(1+(k/n)^2)" "6#*&\"\"\"F$,&F$F$*$*&%\"kGF$%\"nG!\"\" \"\"#F$F*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "de donde tenemos que " }{XPPEDIT 18 0 "f(x)=1/(1+x^2)" "6#/-%\"f G6#%\"xG*&\"\"\"F),&F)F)*$F'\"\"#F)!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "As\355, el l\355mite de la suma es " }{XPPEDIT 18 0 "int(1/(1+x^2),x = 0 .. 1)" "6#-%$intG6$*&\"\"\"F' ,&F'F'*$%\"xG\"\"#F'!\"\"/F*;\"\"!F'" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Calcul\341ndola con \+ Maple obtenemos:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "int(1/(1+x^2),x = 0 .. 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 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "1i)" }}{PARA 0 "" 0 "" {TEXT 373 39 "Repita lo hecho en el ejemplo para " }{XPPEDIT 411 0 "Limit(Sum((n^3*k+k^4)/(n^5),k \+ = 1 .. n),n = infinity)" "6#-%&LimitG6$-%$SumG6$*&,&*&%\"nG\"\"$%\"kG \"\"\"F/*$F.\"\"%F/F/*$F,\"\"&!\"\"/F.;F/F,/F,%)infinityG" }}{PARA 0 " " 0 "" {TEXT 374 0 "" }}{PARA 0 "" 0 "" {TEXT 375 111 "(esto es, calcu le el l\355mite como tal y luego represent\341ndolo como una integral \+ adecuada y evaluado esta \372ltima)" }}{PARA 0 "" 0 "" {TEXT 376 0 "" }}{PARA 0 "" 0 "" {TEXT 372 9 "Respuesta" }{TEXT 377 1 ":" }}{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 379 14 "Repita par a " }{XPPEDIT 412 0 "Limit(Sum(k/(4*n^4-k^2*n^2),k = 1 .. n),n = inf inity)" "6#-%&LimitG6$-%$SumG6$*&%\"kG\"\"\",&*&\"\"%F+*$%\"nGF.F+F+*& F*\"\"#F0F2!\"\"F3/F*;F+F0/F0%)infinityG" }}{PARA 0 "" 0 "" {TEXT 380 0 "" }}{PARA 0 "" 0 "" {TEXT 381 121 "(es posible que Maple no pueda e valuar el l\355mite de la suma. No obstante s\355 es posible hacerlo l lev\341ndolo a una integral)" }}{PARA 0 "" 0 "" {TEXT 382 0 "" }} {PARA 0 "" 0 "" {TEXT 378 9 "Respuesta" }{TEXT 383 1 ":" }}{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 385 27 "Calcule el \+ l\355mite, cuando " }{XPPEDIT 413 0 "n" "6#%\"nG" }{TEXT 386 12 " ti ende a " }{XPPEDIT 414 0 "infinity" "6#%)infinityG" }{TEXT 387 31 " \+ , de la siguiente expresi\363n:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 319 "" 0 "" {XPPEDIT 18 0 "[1/n]*``" "6#*&7#*&\"\"\"F&%\"nG!\"\" F&%!GF&" }{TEXT 388 2 "( " }{XPPEDIT 18 0 "[sec(Pi/(4*n))]^2" "6#*$7#- %$secG6#*&%#PiG\"\"\"*&\"\"%F*%\"nGF*!\"\"\"\"#" }{TEXT 389 5 " + " }{XPPEDIT 18 0 "[sec((2*Pi)/(4*n))]^2" "6#*$7#-%$secG6#*(\"\"#\"\"\"%# PiGF**&\"\"%F*%\"nGF*!\"\"F)" }{TEXT 390 5 " + " }{XPPEDIT 18 0 "[se c((3*Pi)/(4*n))]^2 " "6#*$7#-%$secG6#*(\"\"$\"\"\"%#PiGF**&\"\"%F*%\"n GF*!\"\"\"\"#" }{TEXT 391 14 " + ..... + " }{XPPEDIT 18 0 "[sec((n* Pi)/(4*n))]^2" "6#*$7#-%$secG6#*(%\"nG\"\"\"%#PiGF**&\"\"%F*F)F*!\"\" \"\"#" }{TEXT 392 1 ")" }}{PARA 0 "" 0 "" {TEXT 393 0 "" }}{PARA 0 "" 0 "" {TEXT 394 27 " a) Mediante los comandos " }{TEXT 404 5 "limit" } {TEXT 405 1 "/" }{TEXT 406 5 "Limit" }{TEXT 407 4 " y " }{TEXT 408 3 "sum" }{TEXT 409 1 "/" }{TEXT 410 3 "Sum" }}{PARA 0 "" 0 "" {TEXT 395 0 "" }}{PARA 0 "" 0 "" {TEXT 396 83 " b) Expres\341ndolo como una int egral sobre el intervalo [0, 1] (y evaluando \351sta)." }}{PARA 0 " " 0 "" {TEXT 397 1 " " }}{PARA 0 "" 0 "" {TEXT 398 60 " c) Expres\341 ndolo como una integral sobre el intervalo [0, " }{XPPEDIT 415 0 "Pi/ 4" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 399 22 "] (y evaluando \351sta) ." }}{PARA 0 "" 0 "" {TEXT 400 0 "" }}{PARA 0 "" 0 "" {TEXT 401 55 "Es criba la igualdad entre las dos integrales obtenidas." }}{PARA 0 "" 0 "" {TEXT 402 0 "" }}{PARA 0 "" 0 "" {TEXT 384 9 "Respuesta" }{TEXT 403 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 " " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 416 11 "Ejercicio 2 " }{TEXT -1 38 " (Integraci\363n por cambio de variables)" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Introducci\363n" }}{PARA 0 "" 0 "" {TEXT -1 55 "La f\363rmula de cambio de variables para antiderivadas es" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Int(f(g(x))*d iff(g(x),x),x)" "6#-%$IntG6$*&-%\"fG6#-%\"gG6#%\"xG\"\"\"-%%diffG6$-F+ 6#F-F-F.F-" }{TEXT -1 7 " = " }{XPPEDIT 18 0 "Int(f(u),u)" "6#-%$I ntG6$-%\"fG6#%\"uGF)" }{TEXT -1 14 " , donde " }{XPPEDIT 18 0 "u= g(x)" "6#/%\"uG-%\"gG6#%\"xG" }{TEXT -1 13 " (y luego, " }{XPPEDIT 18 0 "du=diff(g(x),x)*dx" "6#/%#duG*&-%%diffG6$-%\"gG6#%\"xGF,\"\"\"%# dxGF-" }{TEXT -1 3 " )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "El \"cambio de variables\" " }{XPPEDIT 18 0 "u=g( x) " "6#/%\"uG-%\"gG6#%\"xG" }{TEXT -1 88 " puede, en forma m\341s ge neral, venir dado por cualquier relaci\363n entre ambas variables: " } }{PARA 0 "" 0 "" {XPPEDIT 18 0 "h(x)=g(u)" "6#/-%\"hG6#%\"xG-%\"gG6#% \"uG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 68 "Cuando se aplica a integrales (definidas) la f\363rmula de arriba reza:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 310 "" 0 "" {XPPEDIT 18 0 "Int(f(g(x))*diff(g(x),x),x=a..b)" "6#-%$IntG6$*&-%\"fG6 #-%\"gG6#%\"xG\"\"\"-%%diffG6$-F+6#F-F-F./F-;%\"aG%\"bG" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "Int(f(u),u=alpha..beta)" "6#-%$IntG6$-%\"fG6#% \"uG/F);%&alphaG%%betaG" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "donde " }{XPPEDIT 18 0 "alpha " "6 #%&alphaG" }{TEXT -1 5 " y " }{XPPEDIT 18 0 "beta" "6#%%betaG" } {TEXT -1 23 " son los valores de " }{TEXT 436 1 "u" }{TEXT -1 25 " \+ que corresponden a " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 6 " y " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 88 ", re spectivamente. En este caso los intervalos deben ser tales que la tra nformaci\363n " }{TEXT 437 1 "x" }{TEXT -1 5 " --> " }{TEXT 438 2 " \+ u" }{TEXT -1 16 " sea uno-a-uno." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 27 "Por ejemplo, en la integral" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 310 "" 0 "" {XPPEDIT 18 0 "Int(x^2*exp(x^ 3),x)" "6#-%$IntG6$*&%\"xG\"\"#-%$expG6#*$F'\"\"$\"\"\"F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 " hacemos el cambio de variables " }{XPPEDIT 18 0 "x^3=u" "6#/*$%\"xG \"\"$%\"uG" }{TEXT -1 17 ", con lo cual " }{XPPEDIT 18 0 "du=3*x^2* dx" "6#/%#duG*(\"\"$\"\"\"*$%\"xG\"\"#F'%#dxGF'" }{TEXT -1 9 " <---> \+ " }{XPPEDIT 18 0 "(du/3)=x^2*dx" "6#/*&%#duG\"\"\"\"\"$!\"\"*&%\"xG\" \"#%#dxGF&" }{TEXT -1 53 ". De este modo la integral original se con vierte en" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 310 "" 0 "" {XPPEDIT 18 0 "[1/3]*Int(exp(u),u)" "6#*&7#*&\"\"\"F&\"\"$!\"\"F&-%$In tG6$-%$expG6#%\"uGF/F&" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "la cual es inmediata (de las conteni das en la tabla dada al comienzo) y vale " }{XPPEDIT 18 0 "exp(u)+ C" "6#,&-%$expG6#%\"uG\"\"\"%\"CGF(" }{TEXT -1 17 ". Reemplazando " }{TEXT 439 1 "u" }{TEXT -1 7 " por " }{XPPEDIT 18 0 "x^3" "6#*$%\"xG \"\"$" }{TEXT -1 11 " obtenemos" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 310 "" 0 "" {XPPEDIT 18 0 "Int(x^2*exp(x^3),x)" "6#-%$IntG6$*&% \"xG\"\"#-%$expG6#*$F'\"\"$\"\"\"F'" }{TEXT -1 7 " = " }{XPPEDIT 18 0 "[1/3]*exp(x^3)+C " "6#,&*&7#*&\"\"\"F'\"\"$!\"\"F'-%$expG6#*$%\" xGF(F'F'%\"CGF'" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "la cual puede ser verificada f\341cilment e derivando el lado derecho para obtener el integrando." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 73 "Maple puede hacer \+ la transformaci\363n de la integral mediante el comando " }{TEXT 440 9 "changevar" }{TEXT -1 94 " (el que requiere que se cargue previame nte el paquete student) . Su sint\341xis es simple: " }{TEXT 441 10 "changevar(" }{XPPEDIT 18 0 "h(x)=g(u)" "6#/-%\"hG6#%\"xG-%\"gG6#% \"uG" }{TEXT 442 2 ", " }{XPPEDIT 18 0 "expr" "6#%%exprG" }{TEXT 443 2 ", " }{XPPEDIT 18 0 "u" "6#%\"uG" }{TEXT 444 1 ")" }{TEXT -1 10 " , \+ donde " }{XPPEDIT 18 0 "h(x)=g(u)" "6#/-%\"hG6#%\"xG-%\"gG6#%\"uG" } {TEXT -1 67 " es la f\363rmula que expresa la relaci\363n entre la an tigua variable, " }{TEXT 445 1 "x" }{TEXT -1 26 " , y la nueva variab le, " }{TEXT 446 1 "u" }{TEXT -1 3 ", " }{TEXT 447 4 "expr" }{TEXT -1 34 " denota la integral (en la forma " }{TEXT 448 3 "Int" }{TEXT -1 6 ") y " }{XPPEDIT 18 0 "u" "6#%\"uG" }{TEXT -1 88 " denota la n ueva variable, en t\351rminos de la cual debe ser expresada la nueva i ntegral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Para el ejemplo anterior lo har\355amos as\355:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart: w ith(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "I1:=Int(x^ 2*exp(x^3),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "I2:=chang evar(x^3=u,I1,u);" }}}{PARA 0 "" 0 "" {TEXT -1 24 " Su valor es, sabem os, " }{XPPEDIT 18 0 "[1/3]*exp(u)" "6#*&7#*&\"\"\"F&\"\"$!\"\"F&-%$e xpG6#%\"uGF&" }{TEXT -1 39 ". Igual le pedimos a Maple que lo haga:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "value(I2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " y entonces sutitu\355mos:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(u=x^3,%);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "Finalmen te verificamos el resultado derivando esta \372ltima expresi\363n. Deb emos obtener x^2*exp(x^3) :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Si hubi\351semos tenid o que calcular la integral " }{XPPEDIT 18 0 " Int(x^2*exp(x^3),x=2..3 )" "6#-%$IntG6$*&%\"xG\"\"#-%$expG6#*$F'\"\"$\"\"\"/F';F(F-" }{TEXT -1 29 " ,simplemente habr\355amos usado" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "I3:=changevar(u=x^3,Int (x^2*exp(x^3),x=2..3),u);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "N\363tese que changevar inmediatamente coloc\363 los l\355mites de integraci\363n que corresponden a la nueva variable de integraci\363n, " }{TEXT 451 1 "u" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Vemo s que la integral ya est\341 resuelta en la variable " }{TEXT 449 1 " u" }{TEXT -1 48 ". No se hace necesario traer de vuelta al viejo " } {TEXT 450 1 "x" }{TEXT -1 69 " puesto que los l\355mites de la segund a integral ya han sido ajustados" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "A modo de segundo ejemplo consideremos:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "I1:=Int(x*sqrt(1+x^2),x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 28 "Con el cambio de variable : " }{XPPEDIT 18 0 "u=1+x^2" "6#/%\"uG,&\"\"\"F&*$%\"xG\"\"#F&" }{TEXT -1 31 " la i ntegral se transforma en:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "I2:=changevar(u=1+x^2,I1,u);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "que es de las integrales de la tabla. Su valor es:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "value(I2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "y sustituyendo \+ obtenemos:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "subs(u=1+x^2,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "nos muestra que el resultado era correcto." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "Otra est rategia que funciona a veces es reemplazar toda la expresi\363n bajo l a ra\355z por un cuadrado, de modo de eliminar la ra\355z. En este cas o, " }{XPPEDIT 18 0 "1+x^2=t^2" "6#/,&\"\"\"F%*$%\"xG\"\"#F%*$%\"tGF( " }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "I3:=changevar(t^2=1+x^2,I1,t);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "value (I3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(t=sqrt(1+x^2) ,%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 " que es el mismo resultado obtenido antes." }}{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 3 "2i)" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 453 12 "Desarrolle, " }{TEXT 452 14 "sin usar Maple" } {TEXT 454 16 ", la integral " }{XPPEDIT 18 0 "Int(x*sqrt(1+x^2),x)" "6#-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,&F(F(*$F'\"\"#F(F(F'" }{TEXT 455 25 " usando la sustituci\363n " }{XPPEDIT 18 0 "t^2=1+x^2" "6#/*$%\" tG\"\"#,&\"\"\"F(*$%\"xGF&F(" }{TEXT 456 47 ". Confirme que la integra l se transforma en " }{XPPEDIT 18 0 "Int(t^2,t)" "6#-%$IntG6$*$%\"t G\"\"#F'" }{TEXT 457 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 417 9 "Respuesta" }{TEXT 418 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 4 "2ii)" }}{PARA 0 " " 0 "" {TEXT 429 100 "Para cada una de las siguientes integrales busqu e una o varias sustituciones que la transforme (use " }{TEXT 458 9 "ch angevar" }{TEXT 459 260 " si lo desea) directamente en una de las inte grales inmediatas de la tabla dada al comienzo o bien en una que pueda expresarse como combinaci\363n de aquellas usando simplificaciones al gebraicas/trigonom\351tricas en el integrando . S\363lo entonces resu \351lvala (usando " }{TEXT 460 5 "value" }{TEXT 461 135 " ). Si se tra ta de una primitiva (integral indefinida) traiga de vuelta la variable original y verifique su respuesta por derivaci\363n : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 462 4 "Nota" }{TEXT 463 79 ": No es preciso que haga todo con Maple. Varios pasos los puede hacer \"a mano\"." }}{PARA 0 "" 0 "" {TEXT 464 0 "" }}{PARA 0 "" 0 "" {TEXT 465 5 "a) " }{XPPEDIT 18 0 "Int(x/(4+x^4),x = 0 .. 1);" "6#-%$ IntG6$*&%\"xG\"\"\",&\"\"%F(*$F'F*F(!\"\"/F';\"\"!F(" }}{PARA 0 "" 0 " " {TEXT 466 0 "" }}{PARA 0 "" 0 "" {TEXT 510 9 "Respuesta" }{TEXT 511 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 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 467 5 "b) " }{XPPEDIT 18 0 "In t(x^5/(x+1),x = 0 .. 1);" "6#-%$IntG6$*&%\"xG\"\"&,&F'\"\"\"F*F*!\"\"/ F';\"\"!F*" }}{PARA 0 "" 0 "" {TEXT 468 0 "" }}{PARA 0 "" 0 "" {TEXT 512 9 "Respuesta" }{TEXT 513 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 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 469 5 "c) " }{XPPEDIT 18 0 "Int((exp(3*x)+ 1)/(exp(x)+1),x)" "6#-%$IntG6$*&,&-%$expG6#*&\"\"$\"\"\"%\"xGF-F-F-F-F -,&-F)6#F.F-F-F-!\"\"F." }{TEXT 470 0 "" }}{PARA 0 "" 0 "" {TEXT 471 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 514 9 "Re spuesta" }{TEXT 515 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 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 472 5 "d) " } {XPPEDIT 18 0 "Int(exp(x)/((exp(x)+1)*ln(exp(x)+1)),x)" "6#-%$IntG6$*& -%$expG6#%\"xG\"\"\"*&,&-F(6#F*F+F+F+F+-%#lnG6#,&-F(6#F*F+F+F+F+!\"\"F *" }}{PARA 0 "" 0 "" {TEXT 473 0 "" }}{PARA 0 "" 0 "" {TEXT 516 9 "Res puesta" }{TEXT 517 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 474 5 "e) " }{XPPEDIT 18 0 "Int(1/((1+x)*sqrt( x)),x)" "6#-%$IntG6$*&\"\"\"F'*&,&F'F'%\"xGF'F'-%%sqrtG6#F*F'!\"\"F*" }}{PARA 0 "" 0 "" {TEXT 475 0 "" }}{PARA 0 "" 0 "" {TEXT 518 9 "Respue sta" }{TEXT 519 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 476 5 "f) " } {XPPEDIT 18 0 "Int(1/(x*sqrt(x^6-4)),x)" "6#-%$IntG6$*&\"\"\"F'*&%\"xG F'-%%sqrtG6#,&*$F)\"\"'F'\"\"%!\"\"F'F1F)" }}{PARA 0 "" 0 "" {TEXT 477 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 420 0 "" }}{PARA 0 " " 0 "" {TEXT 419 9 "Respuesta" }{TEXT 421 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 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 6 "2iii) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 423 22 "Transforme la integral" }}{PARA 310 "" 0 "" {XPPEDIT 329 0 "Int(1/sqrt(x*(x-1)*(x-2)),x=3..5)" "6#-%$IntG6$*& \"\"\"F'-%%sqrtG6#*(%\"xGF',&F,F'F'!\"\"F',&F,F'\"\"#F.F'F./F,;\"\"$\" \"&" }}{PARA 0 "" 0 "" {TEXT 478 0 "" }}{PARA 0 "" 0 "" {TEXT 479 2 "e n" }}{PARA 310 "" 0 "" {XPPEDIT 330 0 "2*Int(1/sqrt((2-sin(u)^2)),u = \+ Pi/4 .. Pi/3)" "6#*&\"\"#\"\"\"-%$IntG6$*&F%F%-%%sqrtG6#,&F$F%*$-%$sin G6#%\"uGF$!\"\"F3/F2;*&%#PiGF%\"\"%F3*&F7F%\"\"$F3F%" }{TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 20 "" 0 "" {TEXT -1 5 "Ayuda" }}{PARA 0 "" 0 "" {TEXT -1 55 "En un primer paso transforme la integral original en " } {XPPEDIT 18 0 "Int(2/sqrt((2+t^2)*(1+t^2)),t = 1 .. sqrt(3))" "6#-%$In tG6$*&\"\"#\"\"\"-%%sqrtG6#*&,&F'F(*$%\"tGF'F(F(,&F(F(*$F/F'F(F(!\"\"/ F/;F(-F*6#\"\"$" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 422 9 "Respuesta" }{TEXT 424 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 " \+ " }{TEXT 425 11 "Ejercicio 3" }{TEXT -1 25 " (Integraci\363n por parte s)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Introducci\363n" }}{PARA 0 "" 0 "" {TEXT -1 39 "La f\363rmula d e integraci\363n por partes es" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 310 "" 0 "" {XPPEDIT 18 0 "Int(u(x)*diff(v(x),x),x)" "6#-%$IntG6 $*&-%\"uG6#%\"xG\"\"\"-%%diffG6$-%\"vG6#F*F*F+F*" }{TEXT -1 7 " = \+ " }{XPPEDIT 18 0 "u(x)*v(x)" "6#*&-%\"uG6#%\"xG\"\"\"-%\"vG6#F'F(" } {TEXT -1 5 " - " }{XPPEDIT 18 0 "Int(v(x)*diff(u(x),x),x)" "6#-%$Int G6$*&-%\"vG6#%\"xG\"\"\"-%%diffG6$-%\"uG6#F*F*F+F*" }{TEXT -1 2 " ." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "La trans formaci\363n del lado izquierdo al lado derecho se puede llevar a cabo usando Maple: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 480 45 "intparts( Int(u(x)*diff(v(x),x),x) , u(x) )" }{TEXT -1 49 " dar\341 como respuesta el lado derecho. Tal como " }{TEXT 481 9 "changevar" }{TEXT -1 13 ", el comando " }{TEXT 482 8 "intparts " }{TEXT -1 36 " requiere que se cargue el paquete " }{TEXT 483 7 "st udent" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Por ejemplo:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart: with(student):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "In:=Int(x^2*sin(x),x);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 46 "como sabemos, \351sta se integra por par tes con " }{XPPEDIT 18 0 "u(x)=x^2 " "6#/-%\"uG6#%\"xG*$F'\"\"#" } {TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "intparts(In,x^2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "N\363tese que Mple no realiza lo \+ que cualquiera de ustedes habr\355a hecho: cancelar los dos signos men os en el segundo t\351rmino del lado derecho. Hay que forzarlo a hacer lo usando " }{TEXT 484 6 "expand" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "In2:=expand( %);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 208 " La nueva integral requiere, otra vez, integraci\363n por partes. Afort unadamente no es necesario separar la integral del resto del lado dere cho. Se puede usar intparts directamente sobre la expresi\363n compl eta:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "In3:=expand(intparts(In2,x));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Finalmente hemos arribado a una integral conocida: Usando " }{TEXT 485 5 "value" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "value(In3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 26 "y lo chequeamos derivando:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%, x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "q ue era el integrando original." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 123 "Podemos integrar por partes usando alg \372n par\341metro, lo cual es conveniente para obtener f\363rmulas de reducci\363n. Por ejemplo:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "i[n]:=Int(x^n*exp(x),x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "expand(intparts(i[n],x^n)); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Nuev amente Maple nos sorprende no realizando una simplificaci\363n evident e en el integrando : " }{XPPEDIT 18 0 "x^n/x=x^(n-1)" "6#/*&)%\"xG%\" nG\"\"\"F&!\"\")F&,&F'F(F(F)" }{TEXT -1 22 ". Lo forzamos usando " } {TEXT 486 9 "simplify " }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Como la integra l del lado derecho es " }{XPPEDIT 18 0 "i[n-1]" "6#&%\"iG6#,&%\"nG\" \"\"F(!\"\"" }{TEXT -1 41 " hemos obtenido la f\363rmula de reducci \363n:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 310 "" 0 "" {XPPEDIT 18 0 "i[n]=x^n*exp(x)-n*i[n-1]" "6#/&%\"iG6#%\"nG,&*&)%\"xGF'\"\"\"-%$ expG6#F+F,F,*&F'F,&F%6#,&F'F,F,!\"\"F,F4" }{TEXT -1 2 " ." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 487 4 "Nota" }{TEXT -1 15 " : El comando " }{TEXT 488 8 "simplify" }{TEXT -1 73 " puede no \+ ser de mucha ayuda en integrales trigonom\351tricas: Por ejemplo:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "i:=Int((sec(x))^2*(tan(x))^3,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(i);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Hemos visto tam bi\351n en clases que integrales como " }{XPPEDIT 18 0 "Int(exp(x)*si n(x),x)" "6#-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%$sinG6#F*F+F*" }{TEXT -1 166 " pueden obtenerse integrando dos veces por partes, obteniendo as \355 una ecuaci\363n en la integral original, y despejando \351sta de \+ tal ecuaci\363n: Con Maple lo har\355amos as\355:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart: wit h(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "e1:=Int(exp( x)*sin(x),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "e2:=intpar ts(e1,exp(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "e3:=intpa rts(e2,exp(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve(e1 =expand(e3),e1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Que es el resultado de la integral original. Verificando: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}}{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 3 "3i)" }}{PARA 0 "" 0 "" {TEXT 491 53 "Reduzca, usando \+ integraci\363n por partes, la integral " }{XPPEDIT 348 0 "Int(cos(x)* ln(cot(x)),x);" "6#-%$IntG6$*&-%$cosG6#%\"xG\"\"\"-%#lnG6#-%$cotG6#F*F +F*" }{TEXT 494 79 " a una integral inmediata, y luego eval\372ela. C ompruebe su respuesta derivando." }}{PARA 0 "" 0 "" {TEXT 496 0 "" }} {PARA 0 "" 0 "" {TEXT 495 4 "Nota" }{TEXT 497 100 ": No se obsesione c on hacerlo 100% v\355a Maple. Ay\372dese de \351l y use tambi\351n sus propios conocimientos." }}{PARA 0 "" 0 "" {TEXT 492 0 "" }}{PARA 0 " " 0 "" {TEXT 490 9 "Respuesta" }{TEXT 493 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 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "3ii)" }}{PARA 0 "" 0 "" {TEXT 431 53 "Reduzca, usando inte graci\363n por partes, la integral " }{XPPEDIT 18 0 "Int(sin(ln(x)),x )" "6#-%$IntG6$-%$sinG6#-%#lnG6#%\"xGF," }{TEXT 489 79 " a una integr al inmediata, y luego eval\372ela. Compruebe su respuesta derivando." }}{PARA 0 "" 0 "" {TEXT 432 0 "" }}{PARA 0 "" 0 "" {TEXT 430 9 "Respue sta" }{TEXT 433 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 5 "3iii)" }}{PARA 0 " " 0 "" {TEXT 498 53 "Reduzca, usando integraci\363n por partes, la int egral " }{XPPEDIT 18 0 "Int(exp(arcsin(x)),x)" "6#-%$IntG6$-%$expG6#- %'arcsinG6#%\"xGF," }{TEXT 499 79 " a una integral inmediata, y luego eval\372ela. Compruebe su respuesta derivando." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 434 9 "Respuesta" }{TEXT 435 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 4 "3i v)" }}{PARA 0 "" 0 "" {TEXT 502 42 "Establezca una f\363rmula de reduc ci\363n para " }{XPPEDIT 18 0 "I[n] = Int(x^n*sin(x),x);" "6#/&%\"IG6 #%\"nG-%$IntG6$*&)%\"xGF'\"\"\"-%$sinG6#F-F.F-" }{TEXT 503 55 " y lu ego \372sela iter\341ndola repetidamente para calcular" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Int(x^8*sin(x),x)" "6#-%$IntG6$*&%\"xG\"\")-%$sinG 6#F'\"\"\"F'" }{TEXT 504 40 ". Verifique su respuesta por derivaci\363 n." }}{PARA 0 "" 0 "" {TEXT 505 0 "" }}{PARA 0 "" 0 "" {TEXT 501 4 "No ta" }{TEXT 506 41 " : Puede usar un loop para la iteraci\363n." }} {PARA 0 "" 0 "" {TEXT 507 0 "" }}{PARA 0 "" 0 "" {TEXT 500 9 "Respuest a" }{TEXT 508 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "3v)" }}{PARA 0 "" 0 "" {TEXT 427 62 "Usando integraci \363n por partes y cambio de variable calcule " }{XPPEDIT 18 0 "Int (arcsin(x)/x^2,x)" "6#-%$IntG6$*&-%'arcsinG6#%\"xG\"\"\"*$F*\"\"#!\"\" F*" }{TEXT 509 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 426 9 "Respuesta" }{TEXT 428 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 "" }} {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 "35" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }