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Mate m\341tica 3" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 265 15 "TALLER DE MAPLE" }}{PARA 262 "" 0 "" {TEXT -1 0 "" }} {PARA 261 "" 0 "" {TEXT -1 14 " LABORATORIO 1" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "En este laborato rio graficaremos dominios de integraci\363n y calcularemos integrales \+ y vol\372menes. " }}{PARA 0 "" 0 "" {TEXT -1 195 "Veamos algunos ejemp los en los que se usan los distintos comandos que necesitar\341. Ante s, si quiere, puede repasar lo aprendido durante el curso de An\341lis is I. Para ello cliquee en el signo (+). " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 6 "Repaso" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Asignaci\363n" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 224 "\nUno puede pensar que la memoria de una computadora est\341 form ada por buzones cada uno de ellos con un nombre o direcci\363n. En c ada uno de estos buzones o cajitas se puede guardar una informaci\363n .\n\nEn la caja con direcci\363n " }{TEXT 266 1 "a" }{TEXT -1 14 " (l a variable " }{TEXT 267 1 "a" }{TEXT -1 54 ") guardamos el n\372mero e ntero 3 usando el s\355mbolo de " }{TEXT 268 10 "asignaci\363n" } {TEXT -1 92 " \" : = \" y seguido de un punto y coma. Para ejecutar este comando debe apretar la tecla " }{TEXT 269 5 "Enter" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a:=3; " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "MAPLE busca el contenido de la caja con d irecci\363n " }{TEXT 270 1 "a" }{TEXT -1 18 " con el comando " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "a;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Ahora escribimos" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a:=b;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "b:=c;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 5 "c:=4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Contestar " }{TEXT 282 13 "(mentalmente)" }{TEXT -1 27 " las siguientes preguntas:\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "1.- \277Cu\341les son los valores \+ de " }{TEXT 272 1 "a" }{TEXT -1 2 ", " }{TEXT 273 1 "b" }{TEXT -1 2 " \+ y" }{TEXT 274 2 " c" }{TEXT -1 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "2.- Si asigna a " }{TEXT 279 1 "b" } {TEXT -1 33 " el valor 56, \277qu\351 valores toman " }{TEXT 271 1 "a " }{TEXT -1 2 ", " }{TEXT 280 1 "b" }{TEXT -1 3 " y " }{TEXT 275 1 "c " }{TEXT -1 18 "? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "b:=56;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "a;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "b;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "c;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 " Uno puede guardar en una variable una " }{TEXT 283 10 "expresi\363n " }{TEXT -1 25 "aritm\351tica, por ejempl o\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "a:=3*d+e;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Le asigna mos a " }{TEXT 276 1 "d" }{TEXT -1 2 " y" }{TEXT 278 2 " e" }{TEXT -1 13 " los valores " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" } {TEXT -1 6 " y 4" }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" } {TEXT -1 66 ", respectivamente, y luego le preguntamos a MAPLE por el \+ valor de " }{TEXT 277 1 "a" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "d:=sqrt(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "e:=4*sqr t(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "a;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Si uno quiere asignarle a las variables \+ " }{TEXT 281 1 "d" }{TEXT -1 3 " y " }{TEXT 319 1 "e" }{TEXT -1 87 " n uevamente su valor simb\363lico , o sea \"vaciar\" el contenido de las cajas, escribimos\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " e:='e';" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "d:='d';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "a;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "El comando " }{TEXT 284 5 "evalf" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "El comando " } {HYPERLNK 17 "evalf" 2 "evalf" "" }{TEXT -1 155 " da el valor en arit m\351tica en punto flotante de un n\372mero real con la cantidad de d \355gitos que uno indique (si no se indica nada se mostrar\341n 10 d \355gitos)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "4/3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(4/3);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sqrt(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(sqrt(2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "Pi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(Pi,190);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "F unciones y expresiones" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Primero \+ \"limpiemos\" la memoria," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Definimos la " } {TEXT 288 8 "funci\363n:" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:=x->x^3+3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 " \277Funciona como una funci\363n?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(u );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Vemos que funciona precisam ente como se espera que una funci\363n lo haga.\n" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 22 "Consideremos ahora la " }{TEXT 287 10 "expresi\363 n:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "g:=sqrt(3*x-6);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Una " }{TEXT 285 9 "expresi\363n" } {TEXT -1 31 " puede ser transformada en una " }{TEXT 286 7 "funci\363n " }{TEXT -1 16 " con el comando " }{HYPERLNK 17 "unapply" 2 "unapply" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "h:=una pply(g,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(6);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "Si en cambio queremos obtener una expresi\363n a partir de una funci\363n, basta con evaluar la funci \363n en una variable cualquiera. Por ejemplo, a partir de " }{TEXT 289 1 "f" }{TEXT -1 24 " obtenemos la expresi\363n " }{TEXT 290 4 "f(x )" }{TEXT -1 2 ": " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Los comandos " } {TEXT 294 5 "solve" }{TEXT 312 3 " y " }{TEXT 313 6 "fsolve" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Los comandos " }{HYPERLNK 17 "solve" 2 "solve" "" } {TEXT -1 5 " y " }{HYPERLNK 17 "fsolve" 2 "fsolve" "" }{TEXT -1 37 " se usan para resolver ecuaciones. \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Mientras que el comando " }{TEXT 291 5 "solve" }{TEXT -1 19 " resuelve en forma " }{TEXT 292 6 "exacta" }{TEXT -1 53 " una ecua ci\363n o un sistema de ecuaciones, el comando " }{TEXT 293 6 "fsolve " }{TEXT -1 99 " lo hace en forma num\351rica. En este caso es conveni ente indicar el rango donde buscar la soluci\363n. " }}{PARA 0 "" 0 " " {TEXT -1 12 "Por ejemplo:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(cos(x)=1/3,x); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "fsolve(cos(x)=1/3,x=0.. Pi/2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "fs olve(cos(x)=1/3,x=11*Pi/2..6*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "p:=t^4+ 2*t^3-t-2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(p=0,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsolve (p=0,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 295 5 "Solve" }{TEXT -1 42 " permite resolver inecuaciones sencillas.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "inec:=abs(x-1) " 0 "" {MPLTEXT 1 0 14 "solve(inec,x);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "El comando " }{TEXT 300 4 "plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Para graficar utilizamos el comand o " }{HYPERLNK 17 "plot" 2 "plot" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "En el ejemplo que sigue graficaremos la " } {TEXT 300 10 "expresi\363n " }{TEXT -1 31 "que guardaremos en la varia ble " }{TEXT 301 1 "y" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y:= 15*sin(x)/x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(y,x=-20..20,color=blue); " }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 13 "Otro ejemplo:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f:=x->tan(3*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(f(x),x=-4..4,-5..5,discont=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "N\363tese en el \372ltimo ejemplo que en \+ el comando plot el primer argumento es la " }{TEXT 301 9 "expresi\363n " }{TEXT -1 1 " " }{TEXT 300 4 "f(x)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Tambi\351n podemos dibujar dos gr\341ficos junt os" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(\{x^2/10,y\},x=- 10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "El comando " }{TEXT 296 6 "plot3d" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Para graficar " }{TEXT 296 25 "funciones de dos variable " }{TEXT -1 26 "s utilizaremos el comando " }{HYPERLNK 17 "plot3d" 2 " plot3d" "" }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "z:=x^2-y^2; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot3d(z,x=-4..4,y=-4.. 4,axes=boxed,orientation=[76,46]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Grafiquemos d os funciones juntas;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f:= ln(x^2*y^4+exp(2))+7;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g:= 9;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot3d(\{f,g\},x=-2.. 2,y=-2..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Los com andos " }{TEXT 297 12 "implicitplot" }{TEXT 314 1 " " }{TEXT -1 1 "e" }{TEXT 298 1 " " }{TEXT 315 14 "implicitplot3d" }{TEXT 316 1 "\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 13 "Los comandos " }{HYPERLNK 17 "implicitplot" 2 "imp licitplot" "" }{TEXT -1 4 " e " }{HYPERLNK 17 "implicitplot3d" 2 "imp licitplot3d" "" }{TEXT -1 30 " se encuentran en el paquete " } {HYPERLNK 17 "plots" 2 "plots" "" }{TEXT -1 68 " y podemos usarlos pa ra dibujar curvas o superficies dadas en forma" }{TEXT 299 10 " impl \355cita" }{TEXT -1 35 ". Antes debemos \"bajar\" el paquete " }{TEXT 320 5 "plots" }{TEXT -1 16 " con el comando " }{HYPERLNK 17 "with" 2 " with" "" }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "implici tplot(\{x^2-y^2 =1\},x=-3..3,y=-3..3);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "implicitplot3d(x^2-y^2-z^2=1,x=-3..3,y=-3..3,z=-3..3, \nscaling=constrained,numpoints=1000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 302 11 "El comando " }{TEXT 317 7 "display" }{TEXT 318 1 "\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "El comando " }{HYPERLNK 17 "display" 2 "plots,display" "" }{TEXT -1 123 " permite mostrar var ios gr\341ficos simult\341neamente. Antes de usar este comando por pr imera vez se debe ''bajar'' el paquete" }{TEXT 303 6 " plots" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "graf1:=plot(x,x=-1.2..1.2,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "graf2:=implicitplot(x^2+y^2=1,x=-1..1,y=-1..1,colo r=blue,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display([graf1,graf2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 " \nSi los gr\341ficos que quiere dibujar son de tres dimensiones hay qu e usar el comando " }{HYPERLNK 17 "display3d" 2 "plots,display3d" "" } {TEXT -1 1 "." }}}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "El comando " } {TEXT 256 3 "int" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart; \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "El comando " }{HYPERLNK 17 "int" 2 "int" "" }{TEXT -1 38 " calcula una funci\363n primitiva de una " } {TEXT 258 9 "expresi\363n" }{TEXT -1 13 ". Por ejemplo" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f:=1/x/(1+x^2)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "F:=int(f,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Tambi\351n podemos calcular una integral definida" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "INT:= int(f,x=1..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Vam os a calcular la integral " }{XPPEDIT 18 0 "int(int(x^2/(y^2),x),y);" "6#-%$intG6$-F$6$*&%\"xG\"\"#*$%\"yGF*!\"\"F)F," }{TEXT -1 41 " en la regi\363n delimitada por las curvas " }{XPPEDIT 18 0 "y = x;" "6#/%\" yG%\"xG" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "x*y = 1;" "6#/*&%\"xG\"\"\"%\"yGF &F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 186 "Para poder calcul ar integrales dobles debemos entregar la regi\363n en la forma adecuad a. Procedamos entonces a dibujar primero la regi\363n en el plano so bre la que queremos integrar. Como " }{XPPEDIT 18 0 "x = 2" "6#/%\"xG \"\"#" }{TEXT -1 65 " es una curva definida en forma impl\355cita util izaremos el comando" }{TEXT 307 13 " implicitplot" }{TEXT -1 67 ". Par a las otras dos, como son gr\341ficos de funciones, utilizaremos " } {TEXT 308 4 "plot" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "y1:=1/x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "ecua:=x=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "y2:=x;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "fig1:=plot(y1,x=.2..3,y=0..3 ,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "fig2:=impli citplot(ecua,x=0..2,y=0..4,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "fig3:=plot(y2,x=.2..3,y=0..3,color=green):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "display(\{fig1,fig2,fig3\}); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 " Encontremos los puntos de in tersecci\363n entre las curvas " }{XPPEDIT 18 0 "y[1];" "6#&%\"yG6# \"\"\"" }{TEXT -1 6 " e " }{XPPEDIT 18 0 "y[2];" "6#&%\"yG6#\"\"#" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(y1 =y2,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "La soluci\363n -1 no n os interesa y la regi\363n queda descripta como " }{TEXT 309 11 "1 < x <2, " }{XPPEDIT 18 0 "1/x;" "6#*&\"\"\"F$%\"xG!\"\"" }{TEXT 310 7 " \+ < y " 0 " " {MPLTEXT 1 0 11 "f:=x^2/y^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "y la integral la calculamos como sigue" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "INT:=int(int(f,y=1/x..x),x=1..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Vo l\372menes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "with(plots,display3d);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Gra fiquemos el volumen comprendido por los paraboloides " }{XPPEDIT 18 0 "z = x^2+y^2;" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)" }{TEXT -1 15 " y " }{XPPEDIT 18 0 "z = (x^2+y^2)/2+1;" "6#/%\"zG,&*& ,&*$%\"xG\"\"#\"\"\"*$%\"yGF*F+F+F*!\"\"F+F+F+" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "p1:=x^2+y^2;\np2:=1/2*(x^2+y ^2)+1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot3d(\{p1,p2\}, x=-2..2,y=-2..2,orientation=[30,92]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "\nVeamos en qu\351 curva se cortan las dos gr\341ficas.\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(p1=p2,y);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "\nObservamos que se cortan en el c \355rculo " }{XPPEDIT 18 0 "x^2+y^2 = 2;" "6#/,&*$%\"xG\"\"#\"\"\"*$ %\"yGF'F(F'" }{TEXT -1 8 " , " }{XPPEDIT 18 0 "z = 2;" "6#/%\"zG \"\"#" }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 " g1:=plot3d(p1,x=-sqrt(2)..sqrt(2),y=-sqrt(2-x^2)..sqrt(2-x^2),color=bl ue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "g2:=plot3d(p2,x=-sq rt(2)..sqrt(2),y=-sqrt(2-x^2)..sqrt(2-x^2),color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "display3d([g1,g2],view=0..2,style=w ireframe,orientation=[43,76]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "\nCalculemos el volumen,\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "int(int(p2-p1,y=-sqrt(2-x^2)..sqrt(2-x^2)),x=-sqrt(2)..sqrt(2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 263 12 "Ejercitaci\363n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Abrir una nueva hoja de trabajo (w orksheet) cliqueando debajo del men\372 " }{TEXT 256 4 "File" }{TEXT -1 11 " la opci\363n " }{TEXT 257 3 "New" }{TEXT -1 101 " y realizar l os siguientes ejercicios. Para guardar el contenido de la misma, elegi r debajo del men\372 " }{TEXT 260 4 "File" }{TEXT -1 16 " la alternati va " }{TEXT 261 7 "Save as" }{TEXT -1 74 ". A medida que avance el la boratorio se recomienda elegir la alternativa " }{TEXT 262 4 "Save" } {TEXT -1 41 " a fin de guardar lo que ya est\341 hecho. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 304 1 "1" }{TEXT -1 22 ".-Ejercicio de repaso:" }}{PARA 0 "" 0 "" {TEXT -1 26 "(a) Resolve r la ecuaci\363n " }{XPPEDIT 18 0 "sqrt(1-x^2) = -x;" "6#/-%%sqrtG6#, &\"\"\"F(*$%\"xG\"\"#!\"\",$F*F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "(b) Graficar las funciones " }{XPPEDIT 18 0 "f(x) = sqrt( 1-x^2);" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&\"\"\"F,*$F'\"\"#!\"\"" }{TEXT -1 9 " y " }{XPPEDIT 18 0 "h(x) = -x;" "6#/-%\"hG6#%\"xG,$F'!\" \"" }{TEXT -1 118 " en un mismo gr\341fico. \n(c) Comparar las res puestas obtenidas en los puntos (a) y (b). \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 1 "2" } {TEXT -1 23 ".- Calcule la integral " }{XPPEDIT 18 0 "int(int(x^2/(y^2 ),x),y);" "6#-%$intG6$-F$6$*&%\"xG\"\"#*$%\"yGF*!\"\"F)F," }{TEXT -1 45 " en la regi\363n delimitada por las curvas " }{XPPEDIT 18 0 " y = x^2/8;" "6#/%\"yG*&%\"xG\"\"#\"\")!\"\"" }{TEXT -1 7 " , " } {XPPEDIT 18 0 "y = 8*x^2;" "6#/%\"yG*&\"\")\"\"\"*$%\"xG\"\"#F'" } {TEXT -1 6 " , " }{XPPEDIT 18 0 "x*y = 8.;" "6#/*&%\"xG\"\"\"%\"yGF &-%&FloatG6$\"\")\"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 138 "................................................................. ...................................................................... ..." }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 306 1 " 3" }{TEXT -1 79 ".- Dibuje el cuerpo limitado por las siguientes super ficies y halle su volumen:" }}{PARA 0 "" 0 "" {TEXT -1 16 "(a) El cili ndro " }{XPPEDIT 18 0 "x^2+y^2 = 4;" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF' F(\"\"%" }{TEXT -1 18 " , el paraboloide " }{XPPEDIT 18 0 "z = x^2+y^2 ;" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)" }{TEXT -1 13 " y el plano " }{XPPEDIT 18 0 "z = 0.;" "6#/%\"zG-%&FloatG6$\"\"!F(" }}{PARA 0 " " 0 "" {TEXT 258 5 "Ayuda" }{TEXT -1 62 ": cada dibujo gu\341rdelo en \+ una variable y luego con el comando " }{TEXT 259 10 "display3d " } {TEXT -1 75 "dib\372jelos en un mismo gr\341fico. El cilindro lo puede dibujar con el comando " }{TEXT 260 14 "implicitplot3d" }{TEXT -1 26 ". Elija distintos colores." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "(b) El paraboloide " }{XPPEDIT 18 0 "z = x^2+2 *y^2;" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"*&F(F)*$%\"yGF(F)F)" }{TEXT -1 15 " y el cilindro " }{XPPEDIT 18 0 "z = 4-2*y^2;" "6#/%\"zG,&\"\"%\"\"\" *&\"\"#F'*$%\"yGF)F'!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 146 ".............................................................. ...................................................................... .............." }}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{MARK "1 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }