{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 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0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 260 1 " " }{TEXT 262 25 "An\341lisis Matem\341tico I (B)" }}{PARA 257 "" 0 " " {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 261 16 "TALLER DE MAPLE " } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 14 " LABORATORIO 1" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "El comando " }{TEXT 256 5 "evalf " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "El comando " }{HYPERLNK 17 "ev alf" 2 "evalf" "" }{TEXT -1 133 " da el valor aproximadode un n\372me ro real con la cantidad de d\355gitos que uno indique (si no se indic a nada se mostraran 10 d\355gitos)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "4/3;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(4/3);" }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sqrt(2);" }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(sqrt(2));" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "Pi;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(Pi,190);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 10 "Funciones " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Definimos la " }{TEXT 263 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 7 "f(u +1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Vemos que funciona precis amente como se espera que una funci\363n lo haga." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "El comando " }{TEXT 256 5 "limit" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 " Para calcular l\355mites utilizamos el comando " }{HYPERLNK 17 "limit" 2 "limit" " " }{TEXT -1 22 " .\nCalculemos el " }{XPPEDIT 18 0 "limit((x-1)/( 3*x+2),x = infinity);" "6#-%&limitG6$*&,&%\"xG\"\"\"F)!\"\"F),&*&\"\"$ F)F(F)F)\"\"#F)F*/F(%)infinityG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Q:=(x-1)/(3*x+ 2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(Q,x=infinity);" }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Asignamos a " }{TEXT 285 1 "f" }{TEXT -1 6 " la " }{TEXT 258 7 "funci\363n" }{TEXT -1 3 " " }{XPPEDIT 18 0 "proc (x) options operat or, arrow; sin(x)/x end;" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"*&-%$ sinG6#F%\"\"\"F%!\"\"F*F*F*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f:=x->sin(x)/x;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "y calculamos " }{XPPEDIT 18 0 "limit(f(x),x = 0);" "6#-%&limitG 6$-%\"fG6#%\"xG/F)\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "limit(f(x),x=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "\nObserve que el argumento del comando " } {TEXT 259 5 "limit" }{TEXT -1 18 " es la expresi\363n " }{TEXT 264 4 "f(x)" }{TEXT -1 17 " y no la funci\363n " }{TEXT 265 1 "f" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "\nPodemos tambi\351n cal cular el l\355mite por la izquierda o por la derecha:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(Q, x=-2/3,right);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "El comando " }{TEXT 256 4 "plot" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Para graficar utilizamos el coma ndo " }{HYPERLNK 17 "plot" 2 "plot" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "y:= sin(x)/x;" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "p lot(y,x=-20..20,color=blue);\n " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Otro ejemplo:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f:=x ->tan(3*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(f(x),x= -4..4,-5..5,discont=true);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "N \363tese en el \372ltimo ejemplo que en el comando " }{TEXT 277 4 "plo t" }{TEXT -1 27 " el primer argumento es la " }{TEXT 257 9 "expresi \363n" }{TEXT -1 1 " " }{TEXT 256 5 "f(x) " }{TEXT -1 16 "y no la func i\363n " }{TEXT 266 1 "f" }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Tambi\351n podemos dibujar dos gr\341ficos juntos" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot([x^2/10,y],x=-10..10); \n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot([x^2,-x^2,x^2* sin(1/x)],x=-.1..0.1,color=[black,black,red]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Fun ci\363n cuadr\341tica" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "restart:\nwith(plots):\nwith(plottools): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Graficaremos par\341bolas, ju nto con el v\351rtice y el eje de simetr\355a. Tamabi\351n hallaremos \+ las ra\355ces.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a:=-1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "b:=4;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 6 "c:=-3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=x->a*x^2+b*x+c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "parabola:=plot(f(x),x=-b/2/a-5..-b/2/a+5):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 68 "vertice:=pointplot([-b/(2*a),f(-b/(2*a))],colo r=blue,symbol=circle):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "e je:= display(line([-b/(2*a),f(-b/(2*a))],[-b/(2*a),f(-b/(2*a)+5)],line style=3)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display([para bola,vertice,eje],scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(f(x)=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "En el ejemplo anterior cambiar los valores de " }{TEXT 267 8 "a, \+ b y c" }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "En el \+ pr\363ximo ejemplo tambi\351n graficaremos par\341bolas pero esta vez \+ las definiremos a partir del " }{TEXT 268 7 "v\351rtice" }{TEXT -1 6 " y de " }{TEXT 269 1 "a" }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "restart:\nwith(plots):\nwith(plottools):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "xvertice:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "yvertice:=3;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "a:=-1.5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f:=x->a*(x-xvertice)^2+yvertice;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "parabola:=plot(f(x),x=xvertice-5..xvertice+5):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "vertice:=pointplot([xvertice ,yvertice],color=blue,symbol=circle):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "eje:= display(line([xvertice,f(xvertice)],[xvertice,f (xvertice+5)],linestyle=3)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "dis play([parabola,vertice,eje],scaling=constrained);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve (f(x)=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "En el ejemplo anteri or cambiar los valores de " }{TEXT 256 12 "a, xvertice " }{TEXT -1 2 "e " }{TEXT 270 8 "yvertice" }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Lo que sigue es un gr\341fico " }{TEXT 274 7 "animado " }{TEXT -1 47 " en el que se muestra como var\355a la gr\341fica de \+ " }{XPPEDIT 18 0 "y = a*x^2;" "6#/%\"yG*&%\"aG\"\"\"*$%\"xG\"\"#F'" } {TEXT -1 16 " para valores de" }{TEXT 272 1 " " }{TEXT 271 2 "a " } {TEXT 273 0 "" }{TEXT -1 18 "desde -8 hasta 8. " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " for k from -40 to 40 do\na:=k/5:\np[k]:=plot (a*x^2,x=-2..2):\nod:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "display ([seq(p[k],k=-40..4 0)],insequence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Haga '' click'' con el mouse sobre el gr\341fico de arriba. Aparecer\341 una b otonera como de video. Apriete el bot\363n " }{TEXT 282 5 "play " } {TEXT 283 0 "" }{TEXT -1 30 " y haga sonreir a su par\341bola." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 35 "Funciones exponencial y logar\355tmica" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "restart:\nwith(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Graficaremos funciones exponenciales." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k:=0.5;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 5 "a:=3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " f:=x->k*a^x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(f(x),x=-5..5,scaling=constrain ed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "En el comando " }{TEXT 276 4 "plot" }{TEXT -1 32 " anterior, se utiliz\363 la opci\363n " } {TEXT 275 19 "scaling=constrained" }{TEXT -1 218 ". Esto hace que MAPL E elija la misma escala en los ejes x e y. Como los valores que toma l a funci\363n exponencial son muy grandes es conveniente dejar que MAPL E elija dos esacalas distintas, como en el ejemplo que sigue:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(f(x),x=-5..5);\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Grafiquemos la inversa de " } {TEXT 286 1 "f" }{TEXT -1 12 ", junto con " }{TEXT 287 1 "f" }{TEXT -1 44 " y la recta bisectriz del primer cuadrante.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "invf:=x->log[a](x/k);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "plot([f(x),invf(x),x],x=-2..5,y=-4..6,color =[blue,red,black],scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Notar la simetr\355a respecto de la recta." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Funciones a trozos " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Par a definir funciones a trozos utilizamos el comando " }{HYPERLNK 17 "pi ecewise" 2 "piecewise" "" }{TEXT -1 3 " . " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "f:=piecewise(x>0,x^2-1,x<0,-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f,x=-1..1);\n\n\n" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 92 "g:= piecewise(x<-1,cos(Pi*(x+1)),-11,cos(3*Pi*(x-1)));" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 18 "plot(g,x=-5..5);\n\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 73 "h:= piecewise(x<-1,abs(x),-1 9,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(h,x=-2..2);\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f:=piecewise(x<0,exp(1/x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(f,x=-30..10,axes=framed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Discontinui dades" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Daremos ejemplos de funciones disc ontinuas." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "g:=pi ecewise(x>0,sqrt(x+9)/sqrt(x)-3/sqrt(x),x<0,(1-cos(x))/(x^2+x));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "limit(g,x=0,left);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "limit(g,x=0,right);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "limit(g,x=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 " Como el l\355mite existe en " } {XPPEDIT 18 0 "x[0] = 0" "6#/&%\"xG6#\"\"!F'" }{TEXT -1 18 ", se trata de una " }{TEXT 278 23 "discontinuidad evitable" }{TEXT -1 32 ", pues el \372nico problema es que " }{TEXT 288 1 "g" }{TEXT -1 21 " no est \341 definida en " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 79 " .\nVeamos el gr\341fico. Observar que MAPLE no hace caso de esta discontinuidad.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot( g,x=-0.2..0.5,thickness=3,color=blue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "f:= pie cewise(x>1,(sqrt(3*x+1)-sqrt(x+3))/(x^2-x),x<1,sin(-2*x+2)/(x^2+x-2)); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "limit(f,x=1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "limit(f,x=1,left);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "limit(f,x=1,right);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Conclu imos que " }{TEXT 289 1 "f" }{TEXT -1 19 " no es continua en " } {XPPEDIT 18 0 "x[0] = 0;" "6#/&%\"xG6#\"\"!F'" }{TEXT -1 82 ". Como el l\355mite no existe pero s\355 los limites laterales entonces se trat a de una " }{TEXT 279 33 "discontinuidad de primera especie" }{TEXT -1 21 ".\nVeamos el gr\341fico.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(f,x=0..2,discont=true,color=grey);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "h:=sin(1/x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "limit(h,x=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "MAPLE cont esta que el l\355mite es todo el intevalo [-1,1]. O sea que este l\355 mite " }{TEXT 280 10 "no existe " }{TEXT -1 85 "seg\372n la definici \363n vista en la te\363rica.\nLo mismo sucede por derecha y por izqui erda:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(h,x=0, right );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "limit(h,x=0,left);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x[0] = 0" "6#/& %\"xG6#\"\"!F'" }{TEXT -1 8 " es una " }{TEXT 281 33 "discontinuidad d e segunda especie" }{TEXT -1 40 ".\nVeamos el gr\341fico en un entorno del 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot(h,x=-0.1..0.1,color=yellow,numpoints=1000); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "\277Comprende ahora la respue sta que dio MAPLE al calcular el l\355mite?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "z:=x *sin(1/x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Observar que esta f unci\363n est\341 entre las funciones y=-x e y=x, luego el l\355mite \+ en " }{XPPEDIT 18 0 "x[0] = 0" "6#/&%\"xG6#\"\"!F'" }{TEXT -1 66 " deb e ser 0. Por lo tanto se trata de un discontinuidad evitable.\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "limit(z,x=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "plot([x,-x,x*sin(1/x)],x=-.1..0.1,c olor=[black,black,blue],numpoints=1000); " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}}{MARK "1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }