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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 }} {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 \+ 7" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 5 "" 0 "" {TEXT 295 7 "Martes " }{TEXT 305 10 "25 de mayo" } {TEXT 306 9 " de 2010 " }{TEXT 303 41 " (Bicentenario de la Rep\372bli ca Argentina)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT 298 16 "Plazo de entrega" }{TEXT 299 39 " : Martes 1 de junio a \+ las 13:00 horas." }}{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 1 de junio a las 13:30 horas." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{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 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 OTRO ALUMNO:" }}{PARA 317 "" 0 "" {TEXT -1 17 "N\372mer o de alumno:" }}{PARA 317 "" 0 "" {TEXT -1 19 "Secci\363n de C\341tedr a:" }}{PARA 315 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 1 "( " }{TEXT 300 61 "El alumno digitador se alterna de laboratorio en labo ratorio)" }}{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 gr upo:" }}{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 53 "Estudiar sumas de Riemann y la definici\363n de integra l" }}{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\341 ficos. Para ello elja la opci\363n Edit->Remove Output->From Worksheet ." }}{PARA 15 "" 0 "" {TEXT -1 345 "Para corregir su laboratorio el ay udante ejecutar\341 primero el laboratorio con Edit->Execute->Worksh eet. El ayudante entregar\341 el laboratorio corregido sin output, per o con los comentarios correspondientes. Los comandos deben estar en s ecuencia l\363gica de modo de que, al ejecutarlos en orden, los c\341l culos sean correctos. Para simplificar use " }{TEXT 307 7 "restart" } {TEXT -1 86 " en cada problema y cargue nuevamente los paquetes que ne cesita (with(plots), etc...) " }}{PARA 15 "" 0 "" {TEXT -1 63 "Todos l os 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 Comandos Ma ple" }}{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 "re store" }{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 q ue 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 ejemplos de ass ume. Para mayor informaci\363n ejecute " }{TEXT 0 8 "?assume;" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 17 "unassign(' var'); " }}{PARA 0 "" 0 "" {TEXT -1 52 "\"borra\" todo lo que se conoc e acerca de la variable " }{TEXT 0 3 "var" }{TEXT -1 56 ", incluyendo \+ las restricciones impuestas con el comando " }{TEXT 335 6 "assume" } {TEXT -1 107 ". De este modo, ella puede ser usada sin restricciones 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 ocu pado \"borrando\" todo lo que se conoce sobre ellas. Es equivalente a \+ aplicar " }{TEXT 0 15 "unassign('var')" }{TEXT -1 17 " a cada variable " }{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\351tica de cimal con N d\355gitos significativos. " }{TEXT 311 32 "Al iniciar Map le 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\355git os sgniticativos, donde N es el valor que tiene la variable Digits." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 14 "evalf(exp r,N) " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Evalua la expresi \363n " }{TEXT 0 4 "expr" }{TEXT -1 52 " en aritm\351tica decimal con \+ N d\355gitos significativos." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT 312 41 "Gr\341fico de funciones y curvas 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 -1 1 " " }{TEXT 412 11 "Ejercicio 1" }{TEXT -1 20 " (Sumas de Riemann)." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Introducci\363n " }}{PARA 0 "" 0 "" {TEXT -1 85 "Los rect\341ngulos que corresponden a la suma de Riemann en puntos izquierdos para una " }{TEXT 466 18 "pa rtici\363n uniforme" }{TEXT -1 5 " con " }{TEXT 460 16 " n subinterval os" }{TEXT -1 23 " en el intervalo [a,b] " }{TEXT 459 1 " " }{TEXT -1 43 "pueden ser graficados mediante el comando " }{MPLTEXT 1 0 23 "lef tbox(f(x),x=a..b,n) " }{TEXT -1 38 "donde n debe ser un entero posit ivo." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 " La correspondiente suma de Riemann representa la suma de las \341reas \+ signadas de estos rect\341ngulos. \311sta se puede realizar usando el comando " }{MPLTEXT 1 0 3 "sum" }{TEXT -1 60 " y definiendo los pun tos de la partici\363n. Por ejemplo, si " }{XPPEDIT 18 0 "f(x)=x^2-x+ 1" "6#/-%\"fG6#%\"xG,(*$F'\"\"#\"\"\"F'!\"\"F+F+" }{TEXT -1 24 ", a=0 , b=1 y n=10," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 14 "f:=x->x^2-x+1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "a:=0; b:=1; n:=10; h:=(b-a)/n;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "for i from 0 to 10 do x[i]:=i/10; od;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "La suma i zquierda es " }{XPPEDIT 18 0 "sum(f(x[i-1])*h,i=1..10)" "6#-%$sumG6$* &-%\"fG6#&%\"xG6#,&%\"iG\"\"\"F/!\"\"F/%\"hGF//F.;F/\"#5" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "unassign('i');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sum(f(x[i-1])*h,i=1..10);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "El comando " }{MPLTEXT 1 0 7 "leftsum" }{TEXT -1 72 " realiza esta suma en forma autom\341tic a. Su sint\341xis es similar a la de " }{MPLTEXT 1 0 7 "leftbox" } {TEXT -1 14 ". Escribimos " }{MPLTEXT 1 0 22 "leftsum(f(x),x=a..b,n) " }{TEXT -1 63 ". Para operar con estos comandos debe cargarse el \+ paquete " }{TEXT 461 7 "student" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "leftsum(f(x),x=0..1,1 0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "#que es lo mismo que obtuvimos def iniendo nosotros los puntos de la partici\363n y formando la correspon diente suma. Geom\351tricamente estamos calculando la suma de las \341 reas de los rect\341ngulos" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "leftbox(f(x),x=0..1,10);# nos muestra los rect\341ngulos cuyas \+ \341reas estamos sumando" }}}{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 86 "Para sumas y rec\341ngulos derechos o de punto medio se tienen las parejas de comandos " }{MPLTEXT 1 0 8 "rig htsum" }{TEXT -1 5 " , " }{MPLTEXT 1 0 8 "rightbox" }{TEXT -1 6 " \+ y " }{MPLTEXT 1 0 9 "middlesum" }{TEXT -1 5 " , " }{MPLTEXT 1 0 9 " middlebox" }{TEXT -1 148 " , las cuales tienen sint\341xis id\351ntic a a las sumas y rect\341ngulos izquierdos que vimos arriba. Para mayor informaci\363n, vea la ayuda online o pinche " }{HYPERLNK 17 "aqu\355 " 1 "" "Sumas de Riemann" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Note tambi\351n que estos comando s pueden actuar para casos generales, no solamente particulares. Por e jemplo," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "L:=value(leftsum(x^2,x=a..b,n));" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 111 "#es el valor de la suma izquierda para un i ntervalo [a,b] cualquiera y un n\372mero n arbitrario de subdivisiones ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(\{a=0,b=1,n=10\}, L);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "#corresponde al cas o del intervalo [0,1] dividido en 10 subintervalos. Es lo msmo que hab er calculado" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "value(lefts um(x^2,x=0..1,10));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 6 "S obre " }{TEXT 464 5 "value" }{TEXT -1 2 ", " }{TEXT 465 5 "evalf" }} {PARA 0 "" 0 "" {TEXT -1 11 "El comando " }{TEXT 0 11 "value(expr)" } {TEXT -1 42 " intenta evaluar la expresi\363n no evaluada " }{TEXT 0 4 "expr" }{TEXT -1 39 " mediante m\351todos simb\363licos o exactos." }}{PARA 0 "" 0 "" {TEXT -1 11 "El comando " }{TEXT 0 13 "evalf( expr) \+ " }{TEXT -1 41 "intenta evaluar la expresi\363n no evaluada " }{TEXT 0 4 "expr" }{TEXT -1 40 " mediante m\351todos num\351ricos aproximados ." }}{PARA 0 "" 0 "" {TEXT -1 4 "La " }{TEXT 462 1 "f" }{TEXT -1 41 " final se refiere a \"punto flotante\") ( " }{TEXT 463 14 "floating p oint" }{TEXT -1 3 " )." }}{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 32 "EJECUTE LOS SIGUIENTES COMANDOS:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:= x -> exp(-x^2/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student);" }}}{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 387 42 " Grafique f(x) en el intervalo [0,1] y " } {TEXT 408 24 "s\363lo en base al gr\341fico," }{TEXT 409 31 " encuentr e una cota inferior ( " }{TEXT 404 1 "m" }{TEXT 405 26 " ) y una cota \+ superior ( " }{TEXT 406 1 "M" }{TEXT 407 60 " ) para el \341rea bajo el gr\341fico de f(x) en dicho intervalo." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 5 "Ayuda" }}{PARA 0 "" 0 " " {TEXT -1 126 "Busque un rect\341ngulo inscrito y uno circunscrito a \+ la regi\363n encerrada entre el gr\341fico y el eje X en el intervalo \+ pertinente." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 384 9 "Respuesta" }{TEXT 385 1 ":" } {TEXT 383 0 "" }{TEXT 386 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 "1ii)" }}{PARA 0 "" 0 "" {TEXT 389 104 "Obtenga ahora la suma derecha correspondiente a una partici\363n \+ uniforme de [0,1] en n=15 subintervalos , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 473 2 "a)" }{TEXT -1 1 " " }{TEXT 469 109 "definiendo los puntos distinguidos de la partici\363n, planteando la correspondiente suma de Riemann y usando " }{TEXT 410 3 "sum" } {TEXT 411 15 " para evaluarla" }}{PARA 0 "" 0 "" {TEXT 390 0 "" }} {PARA 0 "" 0 "" {TEXT 474 2 "b)" }{TEXT -1 1 " " }{TEXT 470 10 "median te " }{TEXT 471 8 "rightsum" }{TEXT 472 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 388 9 "Respuesta" }{TEXT 391 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 "1i ii)" }}{PARA 0 "" 0 "" {TEXT 393 93 "Repita el ejercicio anterior, est a vez para las suma izquierda asociada a la misma partici\363n." }} {PARA 0 "" 0 "" {TEXT 394 0 "" }}{PARA 0 "" 0 "" {TEXT 392 9 "Respuest a" }{TEXT 395 1 ":" }}{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 397 36 "Revise ahora sus estimaciones para " }{TEXT 400 2 "m " }{TEXT 401 4 " y " }{TEXT 402 1 "M" }{TEXT 403 58 " , obte nidas en la primera parte de este mismo ejercicio. " }{TEXT 468 10 "Ju stifique" }}{PARA 0 "" 0 "" {TEXT 398 0 "" }}{PARA 0 "" 0 "" {TEXT 396 9 "Respuesta" }{TEXT 399 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 419 11 "Ejercicio 2" }{TEXT -1 45 " (Integrales son l\355mite s de sumas de Riemann)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Introducci\363n" }}{PARA 0 "" 0 "" {TEXT -1 25 "Como ud. vi\363 en clases, " }{XPPEDIT 362 0 "int(f(x),x = a . . b);" "6#-%$intG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 6 " es e" } {TEXT 413 0 "" }{TEXT -1 36 "l l\355mite de las sumas de Riemann de " }{TEXT 414 4 "f(x)" }{TEXT 467 1 " " }{TEXT -1 9 "en [a,b]." }}{PARA 0 "" 0 "" {TEXT -1 26 "Por ejemplo, para calcular" }}{PARA 310 "" 0 " " {XPPEDIT 257 0 "int(x^3,x = 0 .. 1);" "6#-%$intG6$*$%\"xG\"\"$/F';\" \"!\"\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "usamos partic iones uniformes y sumas derechas:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart: with(student):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->x^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "h:=1/n;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "S_n:=sum(f(k*h)*h,k=1..n);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Este es el valor de la suma cor respondiente a una partici\363n en " }{TEXT 415 2 "n " }{TEXT -1 126 " subintervalos. Podemos calcular los valores correspondientes a cualq uier n que deseemos usando el comando subs. Por ejemplo" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs( n=12,S_n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "Es el va lor correspondiente a " }{TEXT 417 1 "n" }{TEXT -1 20 " = 12 subinter valos." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Podemos verificar lo anterior usando el comando " }{TEXT 418 8 "right sum" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf(rightsum(f(x),x=0..1,12));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Ahora calculamos el valor de la integral tomando el \+ l\355mite, cuendo " }{TEXT 416 1 "n" }{TEXT -1 20 " tiende a infinit o:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "value(S_n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(S_n,n=infinity);" }}}{PARA 0 "" 0 "" {TEXT -1 75 " Que es \+ el valor de la integral, como puede confirmarse usando el comando " } {HYPERLNK 17 "int" 1 "" "Integrales" }{TEXT -1 3 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "int(f(x), x=0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "o, alternativamente," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Int(f(x),x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{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 45 "Si intentamos hacer lo mi smo con la funci\363n " }{XPPEDIT 18 0 "g(x)=sqrt(x)" "6#/-%\"gG6#%\" xG-%%sqrtG6#F'" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "S:=sum(sqrt(k*h)*h,k=1..n); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(S);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "Maple no es capaz de hallar una f\363rmula cerrada para el valor de la suma. No obstant e, es capaz de calcular el l\355mite de \351sta cuando " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 12 " tiende a " }{XPPEDIT 18 0 "infinit y" "6#%)infinityG" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(S,n=infinity);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Que es el valor correcto de " }{XPPEDIT 18 0 "int(sqrt(x),x=0..1)" "6#-%$intG6$-%%sqrtG6#%\"xG/F);\" \"!\"\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Podemos usarlo para calcular integrales sobre int ervalos generales:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "int(exp(x),x=a..b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "2i)" }}{PARA 0 "" 0 "" {TEXT 423 27 "De manera similar calcule " }{XPPEDIT 309 0 " int(x^5,x = 0 .. 1);" "6#-%$intG6$*$%\"xG\"\"&/F';\"\"!\"\"\"" }{TEXT 420 91 " tomando esta vez el l\355mite de sumas izquierdas. Verifique su respuesta usando el comando " }{TEXT 453 3 "int" }{TEXT 454 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 421 9 "Respues ta" }{TEXT 422 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 425 71 "Maple no pudo e valuar la suma de Riemann correspondiente a la funci\363n " } {XPPEDIT 312 0 "sqrt(x)" "6#-%%sqrtG6#%\"xG" }{TEXT 426 80 " en el in tervalo [0, 1] cuando usamos una partici\363n uniforme del intervalo e n " }{TEXT 475 1 "n" }{TEXT 476 90 " subintervalos. No obstante la s uma s\355 puede ser evaluada usando otro tipo de partici\363n. " }} {PARA 0 "" 0 "" {TEXT 427 8 "Eval\372e, " }{TEXT 455 14 "sin usar Mapl e" }{TEXT 456 78 ", la suma de Riemann correspondiente a la partici \363n del intervalo [0, 1] en " }{XPPEDIT 315 0 "n" "6#%\"nG" }{TEXT 428 34 " subintervalos mediante puntos " }{XPPEDIT 317 0 "x[i]=i^2/ n^2" "6#/&%\"xG6#%\"iG*&F'\"\"#*$%\"nGF)!\"\"" }{TEXT 429 5 " , " } {XPPEDIT 319 0 "i = 0;" "6#/%\"iG\"\"!" }{TEXT 430 8 ", ...., " } {XPPEDIT 321 0 "n" "6#%\"nG" }{TEXT 431 103 ". (Elija los puntos disti nguidos en cada subintervalo como prefiera). Luego calcule el l\355mit e (cuando " }{XPPEDIT 323 0 "n" "6#%\"nG" }{TEXT 432 12 " tiende a \+ " }{XPPEDIT 325 0 "infinity" "6#%)infinityG" }{TEXT 433 65 " ) de la \+ suma y verifique que obtiene el valor de la integral " }{XPPEDIT 327 0 "int(sqrt(x),x = 0 .. 1)" "6#-%$intG6$-%%sqrtG6#%\"xG/F);\"\"!\" \"\"" }{TEXT 434 2 " ." }}{PARA 0 "" 0 "" {TEXT 435 0 "" }}{PARA 0 "" 0 "" {TEXT 436 0 "" }}{PARA 0 "" 0 "" {TEXT 424 9 "Respuesta" }{TEXT 437 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 "2iii)" }}{PARA 0 "" 0 "" {TEXT 439 61 "Haga los mismos c\341lcul os de la parte anterior, pero esta vez " }{TEXT 457 21 "usando comando s Maple" }{TEXT 458 35 " tan intensamente como sea posible," }}{PARA 0 "" 0 "" {TEXT 440 0 "" }}{PARA 0 "" 0 "" {TEXT 438 9 "Respuesta" } {TEXT 441 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 38 "Comentarios sobre sum/Sum, limit/Limit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "El comando " }{TEXT 373 3 "sum" }{TEXT -1 55 " intent a calcular simb\363licamente la suma. Por ejemplo," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "j:='j': sum( j^3,j=1..n);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 51 "Por contraste, la forma inerte de este comando es " } {TEXT 374 3 "Sum" }{TEXT -1 48 ". \311ste deja la suma \"sin evaluar \". El comando " }{TEXT 375 5 "value" }{TEXT -1 63 " convierte esta \+ forma inerte a la forma anterior. Por ejemplo," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "j:='j': Sum(j^3,j=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 67 "Lamentab lemente, muchas sumas no pueden ser evaluadas. Por ejemplo," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "j: ='j': S:=sum((sec(j))^2,j=1..n);" }}}{PARA 0 "" 0 "" {TEXT -1 40 "obvi amente, lo mismo acontecer\341 usando " }{TEXT 376 3 "Sum" }{TEXT -1 11 " y luego " }{TEXT 377 7 "value. " }{TEXT 378 0 "" }{TEXT -1 65 " Eso s\355, podemos evaluar para valores espec\355ficos de la variable \+ " }{TEXT 379 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "n:=10: evalf(S);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "nos d\341 el va lor (nm\351rico, por el uso del comando " }{TEXT 380 6 "evalf " } {TEXT -1 8 " ) de " }{XPPEDIT 18 0 "sum(sec(j)^2,j = 1 .. 10);" "6#- %$sumG6$*$-%$secG6#%\"jG\"\"#/F*;\"\"\"\"#5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "En otras ocasi ones, sum recurre recurre a funciones ad-hoc creadas justamente para carcterizar dichas sumas (o expresiones similares). Por ejemplo," }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "j:='j': n:='n': sum((1/j),j=1..n);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Algo enteramente an\341logo ocurre con los comandos " }{TEXT 381 5 "limit " }{TEXT -1 5 " y " }{TEXT 382 5 "Limit" }{TEXT -1 2 " ." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "2iv)" }} {PARA 0 "" 0 "" {TEXT 442 23 "Intente evaluar ahora " }{XPPEDIT 338 0 "int(tan(x^2),x = 0 .. 1);" "6#-%$intG6$-%$tanG6#*$%\"xG\"\"#/F*;\" \"!\"\"\"" }{TEXT 444 91 ". Primero como l\355mites de sumas de Riem ann y, si ello no es posible, usando los comandos " }{TEXT 447 3 "int " }{TEXT 448 2 ", " }{TEXT 449 3 "Int" }{TEXT 450 2 ", " }{TEXT 451 5 "evalf" }{TEXT 452 7 " , etc." }}{PARA 0 "" 0 "" {TEXT 445 0 "" }} {PARA 0 "" 0 "" {TEXT 443 9 "Respuesta" }{TEXT 446 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 506 11 "Ejercicio 3" }{TEXT -1 34 " (Aproxima ci\363n mediante trapecios)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Introducci\363n" }}{PARA 0 "" 0 "" {TEXT -1 202 "Los m\351todos de sumas de punto izquierdo, medio o dere cho se basan, todas ellas, en aproximar \"la regi\363n bajo la curva\" por rect\341ngulos, obtenidos a partir de una partici\363n del interv alo de integraci\363n. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Otra opci\363n es levantar en cada punto " } {XPPEDIT 18 0 "x[i]" "6#&%\"xG6#%\"iG" }{TEXT -1 89 " de la partici \363n una vertical hasta tocar la curva y luego unir \"los puntos de arriba\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "De este modo se obtiene un trapecio con base en [ " }{XPPEDIT 18 0 "x[i-1]" "6#&%\"xG6#,&%\"iG\"\"\"F(!\"\"" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "x[i]" "6#&%\"xG6#%\"iG" }{TEXT -1 15 " ] y alturas " }{XPPEDIT 18 0 "f(x[i-1] )" "6#-%\"fG6#&%\"xG6#,&%\"iG\"\"\"F+!\"\"" } {TEXT -1 5 " y " }{XPPEDIT 18 0 "f(x[i])" "6#-%\"fG6#&%\"xG6#%\"iG" }{TEXT -1 20 ". Su \341rea es, pues," }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 310 "" 0 "" {XPPEDIT 18 0 "[(f(x[i-1] )+f(x[i]))/2]*(x[i]-x[ i-1])=[(f(x[i-1] )+f(x[i]))/2]*Delta*x[i]" "6#/*&7#*&,&-%\"fG6#&%\"xG6 #,&%\"iG\"\"\"F0!\"\"F0-F)6#&F,6#F/F0F0\"\"#F1F0,&&F,6#F/F0&F,6#,&F/F0 F0F1F1F0*(7#*&,&-F)6#&F,6#,&F/F0F0F1F0-F)6#&F,6#F/F0F0F6F1F0%&DeltaGF0 &F,6#F/F0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Si usamos una partici\363n uniforme en " }{TEXT 477 1 "n" } {TEXT -1 18 " subintervalos ( " }{XPPEDIT 18 0 "Delta*x[i]=(b-a)/n" " 6#/*&%&DeltaG\"\"\"&%\"xG6#%\"iGF&*&,&%\"bGF&%\"aG!\"\"F&%\"nGF/" } {XPPEDIT 18 0 "``=h" "6#/%!G%\"hG" }{TEXT -1 14 " para todo " } {TEXT 478 1 "i" }{TEXT -1 51 " ) la suma de las \341reas de los trapec ios ser\341, pues" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 310 "" 0 " " {XPPEDIT 18 0 "h*Sum([(f(x[i-1] )+f(x[i]))/2],i=1..n)" "6#*&%\"hG\" \"\"-%$SumG6$7#*&,&-%\"fG6#&%\"xG6#,&%\"iGF%F%!\"\"F%-F-6#&F06#F3F%F% \"\"#F4/F3;F%%\"nGF%" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "[h/2]*``" "6 #*&7#*&%\"hG\"\"\"\"\"#!\"\"F'%!GF'" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "f (x[0])+2*f(x[1])+2*f(x[2])" "6#,(-%\"fG6#&%\"xG6#\"\"!\"\"\"*&\"\"#F+- F%6#&F(6#F+F+F+*&F-F+-F%6#&F(6#F-F+F+" }{TEXT -1 8 " +....+ " } {XPPEDIT 18 0 "2*f(x[n-1])+f(x[n])" "6#,&*&\"\"#\"\"\"-%\"fG6#&%\"xG6# ,&%\"nGF&F&!\"\"F&F&-F(6#&F+6#F.F&" }{TEXT -1 2 ") " }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "3i)" }}{PARA 0 "" 0 "" {TEXT 492 20 "Grafique la funci\363n " }{XPPEDIT 370 0 "f(x)=exp( -x^2/2)" "6#/-%\"fG6#%\"xG-%$expG6#,$*&F'\"\"#F-!\"\"F." }{TEXT 493 239 " en el intervalo [0,1] junto a los trapecios de aproximaci \363n correspondientes a una partici\363n uniforme del intervalo en \+ n=4 subintervalos. Despliegue tambi\351n en el gr\341fico los rect \341ngulos de punto izquierdo y los de punto derecho." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 10 "Sugerencia" } }{PARA 0 "" 0 "" {TEXT -1 4 "Use " }{MPLTEXT 1 0 7 "leftbox" }{TEXT -1 4 " y " }{MPLTEXT 1 0 9 " rightbox" }{TEXT -1 48 " para los rect \341ngulos. Para los trapecios use " }{MPLTEXT 1 0 9 "pointplot" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 479 9 "Respuesta" }{TEXT 480 1 " :" }}{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 491 90 "Para los datos de la parte anterior calcule las sumas izquierdas, \+ derechas y de trapecios." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 481 9 "Respuesta" }{TEXT 482 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 "3iii)" }}{PARA 0 " " 0 "" {TEXT 487 27 "Maple dispone del comando " }{MPLTEXT 1 0 9 "tra pezoid" }{TEXT 488 95 " para producir en forma autom\341tica la suma de las \341reas de los trapecios. Inf\363rmese sobre \351l " } {HYPERLNK 17 "aqu\355" 2 "student[trapezoid]" "" }{TEXT 489 76 " y \+ \372selo para confirmar la suma de trapecios obtenida en la parte ante rior." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 483 9 " Respuesta" }{TEXT 484 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 "3iv)" }}{PARA 0 "" 0 "" {TEXT 490 118 "Dem uestre que, en cualquier situaci\363n, la suma de los trapecios produ ce el promedio de las sumas izquierda y derecha." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 485 9 "Respuesta" }{TEXT 486 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 3 "3v )" }}{PARA 0 "" 0 "" {TEXT 496 142 "Sean SM y ST las sumas de pun to medio y de trapecios correspondientes a una cierta funci\363n f(x) y una partici\363n de un intervalo [a,b]. " }{TEXT 503 8 "Explique" }{TEXT 504 129 " (basado en argumentos gr\341ficos) por qu\351 si f \+ no cambia su concavidad en [a,b] entonces SM y ST son cotas de \+ la integral" }}{PARA 0 "" 0 "" {XPPEDIT 371 0 "Int(f(x),x=a..b)" "6#-% $IntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT 497 72 ". Establezca cu\341 l es cota superior y cu\341l es cota inferior en cada caso." }}{PARA 0 "" 0 "" {TEXT 498 0 "" }}{PARA 0 "" 0 "" {TEXT 499 44 "Use lo anteri or para aproximar el valor de " }{XPPEDIT 372 0 "Int(exp(-x^2/2),x=0. .1)" "6#-%$IntG6$-%$expG6#,$*&%\"xG\"\"#F,!\"\"F-/F+;\"\"!\"\"\"" } {TEXT 500 34 " con un error no mayor que 1/10." }}{PARA 0 "" 0 "" {TEXT 501 0 "" }}{PARA 0 "" 0 "" {TEXT 502 91 "Compare con el resultad o de la pregunta 1iv) y con el valor obtenido mediante el comando " }{MPLTEXT 1 0 3 "int" }{TEXT 505 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 494 9 "Respuesta" }{TEXT 495 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 "" }} {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 }