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{SECT 0 {PARA 257 "" 0 "" {TEXT 260 20 "Bases de Matem\341ticas" }
{TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 262 32 "Soluciones del examen \+
de Maple B" }}{PARA 258 "" 0 "" {TEXT -1 58 "Ingenier\355a T\351cnica \+
en Inform\341tica de Sistemas, de Gesti\363n " }}{PARA 258 "" 0 "" 
{TEXT -1 21 " 8 de Febrero de 2006" }}{PARA 258 "" 0 "" {TEXT -1 0 "" 
}{TEXT 261 0 "" }{TEXT -1 20 "Duraci\363n: 50 minutos" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 258 14 "Ejercicio 1:  " }{TEXT 263 81 "Determinar con 27 decimal
es exactos el \341rea de la regi\363n limitada por las curvas " }
{XPPEDIT 18 0 "h(x) = x*(x^2-3*x+3);" "6#/-%\"hG6#%\"xG*&F'\"\"\",(*$F
'\"\"#F)*&\"\"$F)F'F)!\"\"F.F)F)" }}{PARA 0 "" 0 "" {TEXT -1 2 "y " }
{XPPEDIT 18 0 "g(x) = x^2;" "6#/-%\"gG6#%\"xG*$F'\"\"#" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(x
*(x^2-3*x+3)=x^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "with
(plots):\nplot1:=plot(x*(x^2-3*x+3),x=0..3,colour=green):\nplot2:=plot
(x^2,x=0..3,colour=red):\nplots[display](\{plot1,plot2\});\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "evalf(int(x*(x^2-3*x+3)-x^2,
x=0..1)+int(-x*(x^2-3*x+3)+x^2,x=1..3),27);" }}}{PARA 11 "" 1 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 14 "Ejercicio
 2:  " }{TEXT -1 53 " Determina para qu\351 valores del n\372mero real
 positivo " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 45 " la siguiente \+
funci\363n es continua en todo R. " }}{PARA 258 "" 0 "" {XPPEDIT 18 0 
"f(x) = ln(1+x)^a/x;" "6#/-%\"fG6#%\"xG*&)-%#lnG6#,&\"\"\"F.F'F.%\"aGF
.F'!\"\"" }{TEXT -1 29 "                         si  " }{XPPEDIT 18 0 
"x;" "6#%\"xG" }{TEXT -1 7 " >0,   " }}{PARA 258 "" 0 "" {TEXT -1 2 "f
(" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 51 ") = 2                  \+
                       si   " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 
6 "=0,   " }}{PARA 258 "" 0 "" {TEXT -1 2 "f(" }{XPPEDIT 18 0 "x;" "6#
%\"xG" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "(1-cos(x)^(2*a))/(x^2);" "6#
*&,&\"\"\"F%)-%$cosG6#%\"xG*&\"\"#F%%\"aGF%!\"\"F%*$F*F,F." }{TEXT -1 
22 "                 si   " }{XPPEDIT 18 0 "x < 0;" "6#2%\"xG\"\"!" }
{TEXT -1 0 "" }{TEXT -1 0 "" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT 
-1 22 "Para estos valores de " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT 
-1 60 " halla, si existen, sus as\355ntotas verticales y horizontales.
" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "restart:limit(a*log(1+x)/x, x=0, right);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "limit(((1-cos(x)^(2*a))/x^2),x=0,le
ft);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "
Si x = 0 la funci\363n es continua s\363lo si  " }{XPPEDIT 18 0 "a = 2
;" "6#/%\"aG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 24 "Si x es distinto de 0 y " }{XPPEDIT 18 0 
"a = 2;" "6#/%\"aG\"\"#" }{TEXT -1 149 ", la funci\363n no tiene as
\355ntotas verticales pues es continua (es un cociente de funciones co
mpuestas de funciones continuas, con denominador no nulo)." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a:=2;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "limit(a*log(1+x)/x, x=i
nfinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "limit(((1-cos(
x)^(2*a))/x^2),x=-infinity);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 10 "La recta  " }{XPPEDIT 18 0 "y = 0;" "6#/%
\"yG\"\"!" }{TEXT -1 27 " es un as\355ntota horizontal." }}{PARA 11 "
" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 12 "Ej
ercicio 3:" }{TEXT -1 6 " Sea \{" }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%
\"nG" }{TEXT -1 41 "\} la sucesi\363n definida recursivamente por" }}
{PARA 258 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"
\"" }{TEXT -1 5 " = 45" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 
18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 5 " =   " }{XPPEDIT 18 0 "abs
(2*a[n-1])/3+2;" "6#,&*&-%$absG6#*&\"\"#\"\"\"&%\"aG6#,&%\"nGF*F*!\"\"
F*F*\"\"$F0F*F)F*" }{TEXT -1 13 "             " }{XPPEDIT 18 0 "2 <= n
;" "6#1\"\"#%\"nG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 82 "Calcular el t\351rmino 24 de la sucesi
\363n. Construir el primer t\351rmino de la sucesion " }{XPPEDIT 18 0 
"\{b[n] = a[n+1]-a[n]\};" "6#<#/&%\"bG6#%\"nG,&&%\"aG6#,&F(\"\"\"F.F.F
.&F+6#F(!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 45 "que, en valor absoluto, \+
sea menor que 10^\{-3\}" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "a:=proc(n)\nif n=1 then 45\nelse 2*
abs(a(n-1))/3+2 fi:\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "evalf(a(24));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "b:=pro
c(n)\nif n=1 then a(2)-a(1)\nelse -a(n)/3+2 fi:\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "n:=1:\nwhile abs(b(n))>=10^(-3) do \+
\n  n:=n+1: \nod:\nn;" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 257 12 "Ejercicio 4:" }{TEXT -1 92 " Determina \+
el conjunto de n\372meros complejos cuyo cuadrado es igual a dos veces
 su conjugado:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "z:=a+b*I;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalc(z^2)=2*(a-b*I);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ecuacion1:=a^2-b^2=2*a;\necu
acion2:=2*a*b=-2*b;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 3 "Si " }{TEXT 264 4 "b=0 " }{TEXT -1 9 "entonces " }{TEXT 
265 2 "a " }{TEXT -1 15 "tiene que ser :" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(a^2=2*a);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "Si " }
{TEXT 266 2 "b " }{TEXT -1 20 "no es cero entonces:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "solve(subs
(a=-1,ecuacion1));" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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