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{SECT 0 {PARA 257 "" 0 "" {TEXT 259 20 "Bases de Matem\341ticas" }
{TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 261 32 "Soluciones del examen \+
de Maple B" }}{PARA 258 "" 0 "" {TEXT -1 64 "Ingenier\355a T\351cnica \+
en Inform\341tica de Sistemas, de Gesti\363n y LADE" }}{PARA 258 "" 0 
"" {TEXT -1 24 " 1 de Septiembre de 2006" }}{PARA 258 "" 0 "" {TEXT 
-1 0 "" }{TEXT 260 0 "" }{TEXT -1 20 "Duraci\363n: 50 minutos" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
260 "" 0 "" {TEXT 263 13 "Ejercicio 1: " }{TEXT 262 50 "Determinar, si
 es posible, el valor del par\341metro " }{TEXT 265 2 "a " }{TEXT 264 
20 "para que la funci\363n " }{TEXT 266 1 "f" }{TEXT 267 29 " sea deri
vable en su dominio:" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "f(x) = x*ln(x
);" "6#/-%\"fG6#%\"xG*&F'\"\"\"-%#lnG6#F'F)" }{TEXT -1 22 "           \+
  si   0 < " }{XPPEDIT 18 0 "x <= 1;" "6#1%\"xG\"\"\"" }{TEXT -1 4 ", \+
  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 2 "f(
" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 3 ") =" }{XPPEDIT 18 0 "a*(1
-exp(1-x));" "6#*&%\"aG\"\"\",&F%F%-%$expG6#,&F%F%%\"xG!\"\"F,F%" }
{TEXT -1 24 "                si  1 < " }{XPPEDIT 18 0 "x;" "6#%\"xG" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 84 "f:=x->a*(1-exp(1-x)); \ng:=x->x*log(x);\nlimit(f
(x),x=1,right);\nlimit(g(x),x=1,left);\n" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 225 "De esta manera es continua en x=1
, en el resto del dominio es continua por ser composici\363n, diferenc
ia o producto de funciones continuas. Lo mismo sucede con la derivabil
idad. S\363lo hay que considerar, por tanto, el punto x=1." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "diff
(f(x),x);\ndiff(g(x),x);\n" }}}{PARA 0 "" 0 "" {TEXT -1 48 "Siendo est
as funciones continuas en (0,1] y (1, " }{XPPEDIT 18 0 "infinity;" "6#
%)infinityG" }{TEXT -1 31 "), podemos hallar el valor de  " }{XPPEDIT 
18 0 "a;" "6#%\"aG" }{TEXT -1 41 " calculando sus l\355mites laterales
 en x=1:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "limit(diff(f(x)
,x),x=1,right);\nlimit(diff(g(x),x),x=1,left);\na=solve(limit(diff(f(x
),x),x=1)=limit(diff(g(x),x),x=1),a);\n" }}}{PARA 0 "" 0 "" {TEXT -1 
24 "Para hallar el valor de " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 
48 " podemos tambi\351n usar la definici\363n de derivada:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "limit((f(x)-0)/(x-1),x=1,right);\n
limit((g(x)-0)/(x-1),x=1,left);\na=solve(limit((f(x)-0)/(x-1),x=1,righ
t)=limit((g(x)-0)/(x-1),x=1,left),a);\n" }}}{PARA 0 "" 0 "" {TEXT -1 
3 "Si " }{XPPEDIT 18 0 "a = 1;" "6#/%\"aG\"\"\"" }{TEXT -1 45 ", la fu
nci\363n es derivable en todo su dominio." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 14 "Ejercicio 2:  " }
{TEXT -1 21 " Dadas las funciones " }{XPPEDIT 18 0 "f(x) = cos(x);" "6
#/-%\"fG6#%\"xG-%$cosG6#F'" }{TEXT -1 7 "   y   " }{XPPEDIT 18 0 "g(x)
 = sin(2*x);" "6#/-%\"gG6#%\"xG-%$sinG6#*&\"\"#\"\"\"F'F-" }{TEXT -1 
67 ",  halla el \341rea comprendida entre sus gr\341ficas sobre el int
ervalo " }{XPPEDIT 18 0 "[0, Pi/2];" "6#7$\"\"!*&%#PiG\"\"\"\"\"#!\"\"
" }{TEXT -1 6 "      " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 61 "restart:\nf:=x-> cos(x);\ng:=x-> sin(2*x)
;\nsolve(f(x)=g(x),x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
165 "evalf(f(Pi/7)-g(Pi/7));\nevalf(f(2*Pi/6)-g(2*Pi/6));\nArea:=Int(f
(x)-g(x),x=0..Pi/6)+Int(g(x)-f(x),x=Pi/6..Pi/2)=int(f(x)-g(x),x=0..Pi/
6)+int(g(x)-f(x),x=Pi/6..Pi/2);\n" }}}{PARA 0 "" 0 "" {TEXT -1 8 "Tamb
i\351n:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "int(abs(f(x)-g(x)
),x=0..Pi/2);" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 256 12 "Ejercicio 3:" }{TEXT -1 71 "  Estudia los
 intervalos de crecimiento y decrecimiento de la funci\363n  " }
{XPPEDIT 18 0 "f(x) = sin(2*x)^2;" "6#/-%\"fG6#%\"xG*$-%$sinG6#*&\"\"#
\"\"\"F'F.F-" }{TEXT -1 18 "  en el intervalo " }{XPPEDIT 18 0 "[-Pi, \+
Pi];" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 26 "  . Representa la funci\363n.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 64 "restart;\nf:=x->sin(2*x)^2;\ndiff(f(x),x);\nsolve(diff(f(x),x)
=0);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
45 "Al ser la funci\363n seno peri\363dica con periodo " }{XPPEDIT 18 
0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 63 "  y teniendo en cuent
a que se trata del cuadrado de la funci\363n " }{XPPEDIT 18 0 "sin(2*x
);" "6#-%$sinG6#*&\"\"#\"\"\"%\"xGF(" }{TEXT -1 33 " el periodo de est
a funci\363n ser\341 " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"
\"" }{TEXT -1 51 " . Por lo tanto dando valores alrededor de 0 y de  \+
" }{XPPEDIT 18 0 "Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\"\"" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "evalf(sub
s(x=-.5,diff(f(x),x)));\nevalf(subs(x=.5,diff(f(x),x)));\nevalf(subs(x
=.7,diff(f(x),x)));\nevalf(subs(x=.9,diff(f(x),x)));\n \n" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Determinamos que \+
la funci\363n es creciente en " }{XPPEDIT 18 0 "[-Pi, -3*Pi/4];" "6#7$
,$%#PiG!\"\",$*(\"\"$\"\"\"F%F*\"\"%F&F&" }{TEXT -1 3 ",  " }{XPPEDIT 
18 0 "[-Pi/2, -Pi/4];" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F),$*&F&F'\"\"%F)
F)" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "[0, Pi/4];" "6#7$\"\"!*&%#PiG\"\"
\"\"\"%!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "[Pi/2, 3*Pi/4];" "6#7$*
&%#PiG\"\"\"\"\"#!\"\"*(\"\"$F&F%F&\"\"%F(" }{TEXT -1 39 " y decrecien
te en el resto del dominio." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(sin(2*x)^2,x=-Pi..Pi);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 12 "Ejercici
o 4:" }{TEXT -1 5 " Sea " }{TEXT 268 1 "r" }{TEXT -1 22 " un n\372mero
 real y sea " }{XPPEDIT 18 0 "((n+3)/n)^((rn^3+1)/(3*n^2));" "6#)*&,&%
\"nG\"\"\"\"\"$F'F'F&!\"\"*&,&*$%#rnGF(F'F'F'F'*&F(F'*$F&\"\"#F'F)" }
{TEXT -1 21 "  el t\351rmino de orden" }{TEXT 269 3 " n " }{TEXT -1 
15 "de la sucesi\363n " }{XPPEDIT 18 0 "\{a[n]\};" "6#<#&%\"aG6#%\"nG
" }{TEXT -1 10 ". Calcula " }{TEXT 270 1 "r" }{TEXT -1 10 " para que \+
" }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 12 " converja a \+
" }{XPPEDIT 18 0 "exp(3);" "6#-%$expG6#\"\"$" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "restart;\na:=n->(
(n+3)/n)^((r*n^3+1)/(3*n^2));\nlimit(a(n), n=infinity);\nr=solve(exp(3
*r)^(1/3)=exp(3),r);" }}}}{MARK "36 0 0" 131 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }