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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 A" }}{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 53 "Calcula el \341rea de la \+
regi\363n limitada por las curvas " }{XPPEDIT 18 0 "f(x) = x^3-2*x^2+x
-1;" "6#/-%\"fG6#%\"xG,**$F'\"\"$\"\"\"*&\"\"#F+*$F'F-F+!\"\"F'F+F+F/
" }{TEXT -1 3 " y " }{XPPEDIT 18 0 "g(x) = -x^3+3*x-1;" "6#/-%\"gG6#%
\"xG,(*$F'\"\"$!\"\"*&F*\"\"\"F'F-F-F-F+" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solve(x^3-2*x^2+x-1=-
x^3+3*x-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "with(plots)
:\nplot1:=plot(x^3-2*x^2+x-1,x=-2..2,colour=red):\nplot2:=plot(-x^3+3*
x-1,x=-2..2,colour=green):\nplots[display](\{plot1,plot2\});" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "evalf(int(x^3-2*x^2+x-1-(-x
^3+3*x-1),x=1/2-1/2*5^(1/2)..0)+\nint(-(x^3-2*x^2+x-1)+(-x^3+3*x-1),x=
0..1/2+1/2*5^(1/2)),25);" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }{TEXT 259 14 "Ejercicio 2:  " }{TEXT -1 53 " De
termina para qu\351 valores del n\372mero real positivo " }{XPPEDIT 
18 0 "a;" "6#%\"aG" }{TEXT -1 45 " la siguiente funci\363n es continua
 en todo R. " }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "f(x) = ((1+x)^a-1)/x;
" "6#/-%\"fG6#%\"xG*&,&),&\"\"\"F,F'F,%\"aGF,F,!\"\"F,F'F." }{TEXT -1 
17 "             si  " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 7 " >0,
   " }}{PARA 258 "" 0 "" {TEXT -1 2 "f(" }{XPPEDIT 18 0 "x;" "6#%\"xG
" }{TEXT -1 44 ") = 1                                  si   " }
{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 6 "=0,   " }}{PARA 258 "" 0 "" 
{TEXT -1 2 "f(" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 ") =2 " }
{XPPEDIT 18 0 "(1-cos(x)^a)/x^2;" "6#*&,&\"\"\"F%)-%$cosG6#%\"xG%\"aG!
\"\"F%*$F*\"\"#F," }{TEXT -1 21 "                si   " }{XPPEDIT 18 
0 "x < 0;" "6#2%\"xG\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 
22 "Para estos valores de " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 
61 " halla, si existen, sus as\355ntotas verticales y horizontales. " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 42 "restart:\nlimit(((1+x)^a-1)/x, x=0, right);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 37 "limit(2*((1-cos(x)^a)/x^2),x=0,left);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Si x = 0 \+
la funci\363n es continua s\363lo si  " }{XPPEDIT 18 0 "a = 1;" "6#/%
\"aG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 24 "Si x es distinto de 0 y " }{TEXT 276 1 "a" }
{TEXT -1 150 "=1 la funci\363n no tiene as\355ntotas verticales pues e
s continua (es un cociente de funciones compuestas de funciones contin
uas, con denominador no nulo)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a:=1;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 33 "limit(((1+x)^a-1)/x, x=infinity);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "limit((1-cos(x)^a)/x^2,x=-infinity)
;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Las
 rectas  " }{XPPEDIT 18 0 "y = 1;" "6#/%\"yG\"\"\"" }{TEXT -1 3 " e " 
}{XPPEDIT 18 0 "y = 0;" "6#/%\"yG\"\"!" }{TEXT -1 28 " son as\355ntota
s horizontales." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 61 "NOTA: considerando la errata en el enunciado del problema
, f(" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 
"(1-cos(x)^a)/x^2;" "6#*&,&\"\"\"F%)-%$cosG6#%\"xG%\"aG!\"\"F%*$F*\"\"
#F," }{TEXT -1 8 "   si   " }{XPPEDIT 18 0 "x < 0;" "6#2%\"xG\"\"!" }
{TEXT -1 20 ", la soluci\363n ser\355a:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "restart:\nlimit(((1+x)^a-1)/x, x=0, right);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "limit(((1-cos(x)^a)/x^2),x=0
,left);" }}}{PARA 0 "" 0 "" {TEXT -1 113 "La funci\363n no puede ser c
ontinua en x=0 para ning\372n valor de a. Si a=0, se obtiene una disco
ntinuidad eliminable." }}{PARA 0 "" 0 "" {TEXT -1 54 "Considerando a =
 1 si x>0 y a=2 si x<0, se obtiene que" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "limit(((1+x)^1-1)/x, x=infinity);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 36 "limit((1-cos(x)^2)/x^2,x=-infinity);" }}}
{PARA 0 "" 0 "" {TEXT -1 12 "Las rectas  " }{XPPEDIT 18 0 "y = 1;" "6#
/%\"yG\"\"\"" }{TEXT -1 3 " e " }{XPPEDIT 18 0 "y = 0;" "6#/%\"yG\"\"!
" }{TEXT -1 107 " son as\355ntotas horizontales. No hay as\355ntotas v
erticales, siendo la funci\363n continua si x es distinto de 0." }}
{PARA 259 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 256 12 "Ejercicio 3:" }{TEXT -1 1 " " }{TEXT 264 5 "Sean " }
{TEXT 265 1 "z" }{TEXT 266 3 " y " }{TEXT 267 1 "v" }{TEXT 268 101 " l
os n\372meros complejos definidos abajo. Determina el t\351rmino 75 de
 la secuencia de n\372meros complejos " }{TEXT 269 4 "z[n]" }{TEXT 
270 29 ", donde el primer t\351rmino es " }{TEXT 271 1 "z" }{TEXT 272 
16 ", el segundo es " }{TEXT 273 1 "v" }{TEXT 274 83 " y a partir del \+
tercero, cada t\351rmino es el resultado de sumar los dos anteriores.
\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z:=1+2*I;\nv:=1-I;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "z[1]:=z:\nz[2]:=v:\nfor i f
rom 3 to 75 do \nz[i]:=evalc(z[i-1]+z[i-2]):\nod:\nz[75];" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 12 "Ejercicio 4:" }
{TEXT -1 68 " Si una sucesi\363n es contractiva y su constante de cont
ractividad es " }{TEXT 275 7 "c=1/3. " }{TEXT -1 157 "Determinar el pr
imer t\351rmino de la sucesi\363n para el que se puede asegurar con se
guridad que aproxima el l\355mite con un error menor que 10^(-10), sab
iendo que " }{XPPEDIT 18 0 "abs(a[2]-a[1]) = 1/3;" "6#/-%$absG6#,&&%\"
aG6#\"\"#\"\"\"&F)6#F,!\"\"*&F,F,\"\"$F/" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "cota:
=subs(c=1/3,c^(n-1)/(1-c)*(1/3));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "i:=1:\nwhile abs(subs(n=i,cota))>10^(-10) do\n   i:=i
+1:\nod:\ni;" }}}}{MARK "2 0" 58 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }