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{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 51 "Experimental investigation of rad ius of convergence" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nan aimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 78 "Experimental determination of the radius of convergence of a Maclaurin series " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Maclaurin series for \+ " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Sum((-1)^(n+1)*x^ (2*n-1)/(2*n-1),n = 1 .. infinity) = x-x^3/3+x^5/5-x^7/7+x^9/9 + ` . . . `" "6#/-%$SumG6$*(),$\"\"\"!\"\",&%\"nGF*F*F*F*)%\"xG,&*&\"\"#F*F-F *F*F*F+F*,&*&F2F*F-F*F*F*F+F+/F-;F*%)infinityG,.F/F**&F/\"\"$F:F+F+*&F /\"\"&FF+F+*&F/\"\"*F@F+F*%(~.~.~.~GF*" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "This \+ series converges to " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG " }{TEXT -1 5 " for " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$\"\"\"!\"\"%\"xG " }{XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 36 ", so the radi us of convergence is 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "plot([arctan(x),-Pi/2,Pi/2],x=-5..5 ,color=[red,black$2],\n linestyle=[1,4$2]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 489 189 189 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$!\"&\"\"!$! 3!f,Xpw+MP\"!#<7$$!3YLLLe%G?y%F-$!3'zWwsn]YO\"F-7$$!3OmmT&esBf%F-$!3]t 4++4Rc8F-7$$!3ALL$3s%3zVF-$!3GA&=')y(GY8F-7$$!3_LL$e/$QkTF-$!3r1g*)3p7 N8F-7$$!3ommT5=q]RF-$!37,?HYP)GK\"F-7$$!3ILL3_>f_PF-$!3Re3)Q!fO58F-7$$ !3K++vo1YZNF-$!3/];MSU.'H\"F-7$$!3;LL3-OJNLF-$!3yIjn^H]z7F-7$$!3p***\\ P*o%Q7$F-$!3S(fujB')4E\"F-7$$!3Kmmm\"RFj!HF-$!3*Q6Z>'zSR7F-7$$!33LL$e4 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "xx := 0.9;\np(xx),q(xx),arctan(xx); \nxx := 0.98;\np(xx),q(xx),arctan(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+(p\\\"G t!#5$\"+#)3:GtF%$\"+=5:GtF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$ \"#)*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+XtMYx!#5$\"+?vG\\xF%$ \"+o\\(Hv(F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 239 "Even if we take Taylor polynomials of quite high degree, we are not able to determine the radius of convergence with any degre e of accuracy simple by trying to observe where the graphs of the Tayl or polynomials diverge away from the graph " }{XPPEDIT 18 0 "y=arctan( x)" "6#/%\"yG-%'arctanG6#%\"xG" }{TEXT -1 3 ". 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\"\"*&#F(\"$<\"F()F&F2F(F(*&#F(\"$>\"F(*$)F&F6F(F(F/*&#F(\"$,\"F()F&F; F(F(*&#F(\"$.\"F(*$)F&F?F(F(F/*&#F(\"$0\"F()F&FDF(F(*&#F(\"$2\"F(*$)F& FHF(F(F/*&#F(\"$4\"F()F&FMF(F(*&#F(\"$6\"F(*$)F&FQF(F(F/" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "We can use " } {TEXT 0 6 "fsolve" }{TEXT -1 41 " to obtain a numerical estimate for t his " }{TEXT 262 1 "x" }{TEXT -1 12 " coordinate." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "fsolve(p(x)= q(x),x=1..1.02);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\")p:45!\"*" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "For man y Maclaurin series" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[0]+a[1]*x+a[2]*x^2+a[3]*x^3+ ` . . . ` +a[n]*x^n+a[n+1]*x^(n+1)+ ` \+ . . . `" "6#,2&%\"aG6#\"\"!\"\"\"*&&F%6#F(F(%\"xGF(F(*&&F%6#\"\"#F(*$F ,F0F(F(*&&F%6#\"\"$F(*$F,F5F(F(%(~.~.~.~GF(*&&F%6#%\"nGF()F,F;F(F(*&&F %6#,&F;F(F(F(F()F,,&F;F(F(F(F(F(F7F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "the ratio of successive coefficients " }{XPPEDIT 18 0 "a[n+1]/a[n]" "6#*&&%\"aG6#,&%\"nG\"\"\"F)F)F)&F%6#F(!\"\"" }{TEXT -1 22 " approaches the limit " }{XPPEDIT 18 0 "1/R" "6#*&\"\"\"F$%\"RG!\" \"" }{TEXT -1 8 ", where " }{TEXT 309 1 "R" }{TEXT -1 59 " is the radi us of convergence (with the understanding that " }{XPPEDIT 18 0 "R=inf inity" "6#/%\"RG%)infinityG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "1/R=0 " "6#/*&\"\"\"F%%\"RG!\"\"\"\"!" }{TEXT -1 17 " are equivalent)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose t hat " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(x)=a[0]+a[1 ]*x+a[2]*x^2+a[3]*x^3+ ` . . . ` +a[n]*x^n" "6#/-%\"pG6#%\"xG,.&%\"aG6 #\"\"!\"\"\"*&&F*6#F-F-F'F-F-*&&F*6#\"\"#F-*$F'F4F-F-*&&F*6#\"\"$F-*$F 'F9F-F-%(~.~.~.~GF-*&&F*6#%\"nGF-)F'F?F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "q(x) = a[0]+a[1]*x+a[2]*x^2+a[3]*x^3+` . . . `+a[m]*x^m ;" "6#/-%\"qG6#%\"xG,.&%\"aG6#\"\"!\"\"\"*&&F*6#F-F-F'F-F-*&&F*6#\"\"# F-*$F'F4F-F-*&&F*6#\"\"$F-*$F'F9F-F-%(~.~.~.~GF-*&&F*6#%\"mGF-)F'F?F-F -" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "are Taylor polynomia ls, where " }{TEXT 263 1 "m" }{TEXT -1 5 " and " }{TEXT 264 1 "n" } {TEXT -1 17 " are \"large\" and " }{XPPEDIT 18 0 "m>n" "6#2%\"nG%\"mG " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Then \004" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "q(x)-p(x)=a[n+1]*x^(n+1) + a[n+2]*x^(n+2)+` . . . `+a[m]*x^(m)" "6#/,&-%\"qG6#%\"xG\"\"\"-%\"pG6 #F(!\"\",**&&%\"aG6#,&%\"nGF)F)F)F))F(,&F4F)F)F)F)F)*&&F16#,&F4F)\"\"# F)F))F(,&F4F)F;F)F)F)%(~.~.~.~GF)*&&F16#%\"mGF))F(FBF)F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "q(x)-p(x)" "6#,&- %\"qG6#%\"xG\"\"\"-%\"pG6#F'!\"\"" }{TEXT -1 9 " has the " }{TEXT 261 16 "approximate form" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "q(x)-p(x)=a*x^(n+1)+a*r*x^(n+2)+a*r^2*x^(n+3)+` . . . `+a*r^(m-n-1)*x^m" "6#/,&-%\"qG6#%\"xG\"\"\"-%\"pG6#F(!\"\",,*&%\"a GF))F(,&%\"nGF)F)F)F)F)*(F0F)%\"rGF))F(,&F3F)\"\"#F)F)F)*(F0F)*$F5F8F) )F(,&F3F)\"\"$F)F)F)%(~.~.~.~GF)*(F0F))F5,(%\"mGF)F3F-F)F-F))F(FBF)F) " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a=a[n+1]" "6#/%\"aG&F$6#,&%\"nG\"\"\"F)F)" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "r=1/R" "6#/%\"rG*&\"\"\"F&%\"RG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "q(x)-p(x)=0" " 6#/,&-%\"qG6#%\"xG\"\"\"-%\"pG6#F(!\"\"\"\"!" }{TEXT -1 7 ", when " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+r*x+r^2*x^2+` . . . `+r^(m-n-1)*x^(m-n-1);" "6#,,\"\"\"F$*&%\"rGF$%\"xGF$F$*&F&\"\"#F'F)F $%(~.~.~.~GF$*&)F&,(%\"mGF$%\"nG!\"\"F$F0F$)F',(F.F$F/F0F$F0F$F$" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "By inspection, if " } {XPPEDIT 18 0 "m-n" "6#,&%\"mG\"\"\"%\"nG!\"\"" }{TEXT -1 4 " is " } {TEXT 261 4 "even" }{TEXT -1 33 ", this equation has the solution " } {XPPEDIT 18 0 "r*`.`*x=1" "6#/*(%\"rG\"\"\"%\".GF&%\"xGF&F&" }{TEXT -1 10 ", that is," }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x = 1/r" "6#/%\"xG*&\"\"\"F&%\"rG!\"\"" }{XPPEDIT 18 0 "`` =R" "6#/%! G%\"RG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 266 6 "______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 51 "We can obtain a better estimate for the radius of \+ c" }{TEXT 265 10 "onvergence" }{TEXT -1 29 " of the Maclaurin series f or " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 77 " t han we obtained previously if we take higher degree Taylor polynomials for " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "convert( taylor(arctan(x),x,800),polynom):\np := unapply(%,x):\nconvert(taylor( arctan(x),x,820),polynom):\nq := unapply(%,x): \nfsolve(p(x)-q(x)=0,x= 1..1.01);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\"RN7+\"!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "For more rapidly convergent Maclaurin series, we can obtain very accurate esti mates for the radius of convergence by this method." }}{PARA 0 "" 0 " " {TEXT -1 45 "Examples are given in the following sections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 72 "Examples of the experimental \+ determination of the radius of convergence " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT 267 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = cos(x)/(sin(x)+cos( x));" "6#/-%\"fG6#%\"xG*&-%$cosG6#F'\"\"\",&-%$sinG6#F'F,-F*6#F'F,!\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Plot the graph \+ of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 41 " and note t he main features of the graph." }}{PARA 0 "" 0 "" {TEXT -1 29 "(b) Fin d a Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 17 " of degree 9 for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 7 " about " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "(c) Plot the graphs " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 40 " in the same picture over \+ the interval " }{XPPEDIT 18 0 "[-.785, 1];" "6#7$,$-%&FloatG6$\"$&y! \"$!\"\"\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 123 "(d) Co nstruct an animation which you can use to give an estimate for the rad ius of convergence of the Maclaurin series for " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 268 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 3 "(a)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "f := x -> cos(x)/(sin(x)+cos(x));\np1 := plot(f(x),x=-3.. 3,y=-5..5,discont=true):\np2 := plots[implicitplot]( \{x=-Pi/4,x=3*Pi/4\},x=-3..3,y=-5..5,\n color=black,linestyle=3):\n plots[display]([p1,p2]);" }}{PARA 11 "" 1 "" 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$\"3!4w7I=4In\"F17$7$FheoFc`oF\\\\p7$F`\\p7$Fheo$\"3#3w7I=4I2#F17$7$Fh eoF^aoFb\\p7$Ff\\p7$Fheo$\"3ugF,$=4IZ#F17$7$FheoFfaoFh\\p7$F\\]p7$Fheo $\"3kgF,$=4I(GF17$7$FheoFafnF^]p7$Fb]p7$Fheo$\"3cgF,$=4IF$F17$7$Fheo$ \"3M+++++++MF1Fd]p7$Fh]p7$Fheo$\"3!4w7I=4In$F17$7$FheoF_coF\\^p7$F`^p7 $Fheo$\"3QgF,$=4I2%F17$7$FheoFgcoFb^p7$Ff^p7$Fheo$\"3ugF,$=4IZ%F17$7$F heoFbdoFh^p7$F\\_p7$Fheo$\"3)>w7I=4I([F17$7$FheoF]eoF^_pF_eoFaeo-%+AXE SLABELSG6%Q\"x6\"Q\"yFg_p-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fafn;Fcgn$\" \"&F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The \+ graph has vertical asymptotes at " }{XPPEDIT 18 0 "x = -Pi/4, x = 3*Pi /4, x= 7*Pi/4,` . . . `" "6&/%\"xG,$*&%#PiG\"\"\"\"\"%!\"\"F*/F$*(\"\" $F(F'F(F)F*/F$*(\"\"(F(F'F(F)F*%(~.~.~.~G" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(0)=1" "6#/-%\"fG6#\"\"!\" \"\"" }{TEXT -1 6 ", the " }{TEXT 276 1 "y" }{TEXT -1 26 " intercept i s at the point" }{XPPEDIT 18 0 "`` (0,1)" "6#-%!G6$\"\"!\"\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 274 1 "x" } {TEXT -1 21 " intercepts occur at " }{XPPEDIT 18 0 "x = Pi/2+k*Pi;" "6 #/%\"xG,&*&%#PiG\"\"\"\"\"#!\"\"F(*&%\"kGF(F'F(F(" }{TEXT -1 8 ", wher e " }{TEXT 273 1 "k" }{TEXT -1 15 " is an integer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b)\004" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "f := x -> \+ cos(x)/(sin(x)+cos(x));\ntaylor(f(x),x,10):\np := unapply(convert(%,po lynom),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)op eratorG%&arrowGF(*&-%$cosG6#9$\"\"\",&-%$sinGF/F1F-F1!\"\"F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,6\"\"\"F-9$!\"\"*$)F.\"\"#F-F-*&#\"\"%\"\"$F-*$)F.F6F-F-F/*&#\"\"& F6F-*$)F.F5F-F-F-*&#\"#K\"#:F-*$)F.F;F-F-F/*&#\"$A\"\"#XF-*$)F.\"\"'F- 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..1,y=-1..2,\nthickness=2,color=[red,COLOR(HUE,(i+1)/(m+3))]),i=1..m): \nplots[display](frms,insequence=true,title=`animation for radius of c onvergence`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$% )operatorG%&arrowGF(*&-%$cosG6#9$\"\"\",&-%$sinGF/F1F-F1!\"\"F(F(F(" } }{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6$-%(ANIMATEG6,7& -%'CURVESG6%7in7$$!3K++++++]y!#=$\"3[wCe!zliD\"!#97$$!3919%\\FTy$yF.$ \"3\"z(Q`J=#G5$!#:7$$!348G))\\DoDyF.$\"3Y;\")G=p$=x\"F77$$!3#*=U#[#Q_8 yF.$\"3*=%=!)4'[3C\"F77$$!3wCcw*4l8!yF.$\"3'*fZQd'GEb*!#;7$$!3_P%['\\w /xxF.$\"3#))f28B]*[lFG7$$!3I]7`*>IFv(F.$\"3GZ'4-*G.))\\FG7$$!3(e(oH*H& 4/xF.$\"3%4\"ReI8h&Q$FG7$$!3I+D1*Rgal(F.$\"3m')>(eP\"HoDFG7$$!3vC\"G)) \\Dog(F.$\"3'yo#=b2gs?FG7$$!3I]Pf)f!>evF.$\"3ON&\\/_*))RQ#F_p7$$!31](=(zZ`y[F.$\"3v%ey&**Q`I@F_p7$$!3v\\PM0n 1,XF.$\"3$*HJmvW$\\$>F_p7$$!3Z**\\7%R%z7TF.$\"3'oW(oP8`t!)z!#>$\"3-&y6:=Bp3\"F_p7$$!3r'[(o9:QITFhu$\"3h[kS% *)3J/\"F_p7$$!3xU[iS9\\WY!#?$\"3po^5(>mY+\"F_p7$$\"3%>,DJqm0O$Fhu$\"3m 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'eV+PwIo(F.7$F_y$\"3ECAW]z*QX(F.7$Fdy$\"3X#o!eYYx2sF.7$Fiy$\"3uv1s *pF.7$F^z$\"3QEXi]&)*Qw'F.7$Fcz$\"3?3-*3WG]b'F.7$Fhz$\"3%eO?p.>oJ'F.7$ F][l$\"3?R$>6O`B2'F.7$Fb[l$\"3!\\G))R+Syx&F.7$Fg[l$\"3!*H`Gc%4ZFhu7$Fe]l$!3gj,WrN7HAF.7$Fj ]l$!3#*=U^L%p&[cF.7$F_^l$!3wy0[)3\\K2\"F_p7$Facq$!3/>/7Yhr$Q\"F_p7$Fd^ l$!3y!>'=Ej$pu\"F_p7$Ficq$!3PspH')o:0AF_p7$Fi^l$!3[xuG63iWFF_p-F_cl6$F acl#\"#6FcclFf_lFdclFjcl-%&TITLEG6#%Danimation~for~radius~of~convergen ceG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "It looks as though the radius of convergence could be " } {XPPEDIT 18 0 "Pi/4" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 38 ". It ca n certainly be no greater than " }{XPPEDIT 18 0 "Pi/4" "6#*&%#PiG\"\" \"\"\"%!\"\"" }{TEXT -1 32 " because of the singularity at " } {XPPEDIT 18 0 "-Pi/4" "6#,$*&%#PiG\"\"\"\"\"%!\"\"F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The fol lowing commands construct a Taylor polynomials " }{XPPEDIT 18 0 "p(x) " "6#-%\"pG6#%\"xG" }{TEXT -1 19 " of degree 100 and " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 19 " of degree 120 for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = 0 " "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "f := x -> cos(x)/(sin(x)+co s(x));\nconvert(taylor(f(x),x,101),polynom):\np := unapply(%,x):\nconv ert(taylor(f(x),x,121),polynom):\nq := unapply(%,x):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%$cosG6# 9$\"\"\",&-%$sinGF/F1F-F1!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 14 " The values of " }{XPPEDIT 18 0 "p(x),q(x);" "6$-%\"pG6#%\"xG-%\"qG6#F& " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 21 " can be compared for " }{TEXT 275 1 "x" }{TEXT -1 6 " near " } {XPPEDIT 18 0 "Pi/4" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "xx := 0.75;\np(xx);\nq(xx);\nf(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#v!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%odz?&!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ksM*=&!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+zZ1x^!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot([f(x),p(x),q(x)],x=0.76 ..0.79,color=[red,blue,magenta]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"33+++++++w!#=$\"3**oR))H\"=q 7&F*7$$\"3$)***\\i9Rlg(F*$\"3k6-k+luB^F*7$$\"35+vVA)GAh(F*$\"3+D/yf*** 37&F*7$$\"3A+]Peui=wF*$\"3oE3N?))p<^F*7$$\"3I+]i3&o]i(F*$\"3[Nu2flZ9^F *7$$\"3&)*\\(oX*y9j(F*$\"3]ZI,2(p76&F*7$$\"3P+vVTAUPwF*$\"39(zPki'H3^F *7$$\"3;+v$*zhdVwF*$\"3BVMMa#=_5&F*7$$\"3k*\\P>fS*\\wF*$\"3My$e8oM?5&F *7$$\"35+v=$f%GcwF*$\"33r=V+9'))4&F*7$$\"3A++Dy,\"Gm(F*$\"37Vb)RP(f&4& F*7$$\"3o+]7Ytn]F*7$$\"3q+vVb4*\\s(F*$\"3cQga;*)\\ k]F*7$$\"3!**\\7j=_6t(F*$\"3EU)yD\"yTh]F*7$$\"3'***\\P%y!ePxF*$\"3#Q+) \\`I?e]F*7$$\"3p+v=WU[VxF*$\"3q,]cV4Db]F*7$$\"3q+]7B>&)\\xF*$\"3A5KIPn 1_]F*7$$\"3++v$>:mkv(F*$\"3G.*ylGf([]F*7$$\"3A+DcdQAixF*$\"3#GvQeo:k.&F*7$$\"3/+vVVDB(y(F*$\"3h#RQ4/vL.&F* 7$$\"3K+]7TW)Rz(F*$\"3(yRzj&*)**H]F*7$$\"3-++]@80+yF*$\"3s[bm=+&yF*$\"3i35f<<)>+&F*7$$\"3e+ ]i_4QcyF*$\"3=@5piyF*$\"3'*Hu;1`k&*\\F*7$$\" 3?++]:$*[oyF*$\"31;\"[T:YF*\\F*7$$\"3=+]7<[8vyF*$\"3?v%\\ERB%*)\\F*7$$ \"3W+++L'y5)yF*$\"3#3JFtY^k)\\F*7$$\"3#**\\P/)fT()yF*$\"3uJpZkFG$)\\F* 7$$\"3%**\\i0j\"[$*yF*$\"3M>*))*)*)\\-)\\F*7$$\"3M+++++++zF*$\"3/4[ca1 *p(\\F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F_[lF^[l-F$6$7S7$F($\"3m^ %*zVY(RC&F*7$F.$\"3)=%f%Hsi7D&F*7$F3$\"3Am/F:&o$e_F*7$F8$\"3qP+[\"o!Gn _F*7$F=$\"35&4a0c=tF&F*7$FB$\"3#Rk=Y8j%)G&F*7$FG$\"3]o16R5\"**H&F*7$FL $\"3lMoXh$*)HJ&F*7$FQ$\"3#\\Z.]*Q$zK&F*7$FV$\"3RVPG#[(QW`F*7$Fen$\"3uF 4e+M2j`F*7$Fjn$\"3%H1Q1MM6Q&F*7$F_o$\"3E(\\2Of7MS&F*7$Fdo$\"3#Ghz]8H!G aF*7$Fio$\"3#)=p[3X1aaF*7$F^p$\"3i.9-!)[$)zaF*7$Fcp$\"3)[5wGcLL^&F*7$F hp$\"3O(3&4/QLWbF*7$F]q$\"3=5np#GyOe&F*7$Fbq$\"3\\_9J5,p@cF*7$Fgq$\"3u !*y6]m7ncF*7$F\\r$\"3J:Mrx'>Vr&F*7$Far$\"3S))yXS'))zw&F*7$Ffr$\"3')e'3 Dr\"e@eF*7$F[s$\"3_`n,)y%R%)eF*7$F`s$\"3SI\\6L$*fbfF*7$Fes$\"3#fT'Q$3N H-'F*7$Fjs$\"3jZTEbEr,hF*7$F_t$\"3y08K\"fL->'F*7$Fdt$\"3))oePqhV%G'F*7 $Fit$\"3%e&=U(HHLQ'F*7$F^u$\"3&e:5A95G]'F*7$Fcu$\"35@O-FYd>mF*7$Fhu$\" 3ez6$[(\\!\\v'F*7$F]v$\"3Wg,@[s!y)oF*7$Fbv$\"31d86ix>XqF*7$Fgv$\"3we; \"RzZd?(F*7$F\\w$\"3-K%y:ZQvQ(F*7$Faw$\"3M)=D,y\">!e(F*7$Ffw$\"3u)[R9X $**)z(F*7$F[x$\"3nNym%)>jF!)F*7$F`x$\"3#R\"p>pj7\"G)F*7$Fex$\"3gt:Ag%* y`&)F*7$Fjx$\"3;2&)f>y]C))F*7$F_y$\"3;19NbLZg\"*F*7$Fdy$\"3q.%e'Gf1'[* F*7$Fiy$\"35!3$py_Yh)*F*7$F^z$\"3PIbOWX-D5!#<7$Fcz$\"3!fPfOI\"Gq5Fcdl- Fhz6&FjzF^[lF^[lF[[l-F$6$7S7$F($\"3EP)4#f&F*7 $F3$\"3#>)zr!3gW>&F*7$F8$\"35-c#eD(4*>&F*7$F=$\"3>#f1MD,Y?&F*7$FB$\"36 )zfGL**4@&F*7$FG$\"3&f@\"*G2Ny@&F*7$FL$\"3g`@a^;#fA&F*7$FQ$\"3(o<1#=BZ N_F*7$FV$\"3-)y@\\.pg**e_F*7$Fjn$\"3-a#=Bss:F&F*7$F _o$\"3i9yZ%>quG&F*7$Fdo$\"3`8tf*)RZ0`F*7$Fio$\"3Pb]f-.'\\K&F*7$F^p$\"3 lEu\">!GlW`F*7$Fcp$\"3Q-N!)*y#zq`F*7$Fhp$\"33#G5)[\"yaR&F*7$F]q$\"3EHG cM&GuU&F*7$Fbq$\"3mSxXZ%)*)eaF*7$Fgq$\"33y(RF,Bs\\&F*7$F\\r$\"3())*H;, sxPbF*7$Far$\"3+I,!pgXZe&F*7$Ffr$\"38eSaZ%yCj&F*7$F[s$\"3]Ud$[m(R*o&F* 7$F`s$\"3sr4-()[4bdF*7$Fes$\"32F,$oM%G=eF*7$Fjs$\"3=q!fZpDM*eF*7$F_t$ \"3?Rnvw@JzfF*7$Fdt$\"3_:)>[!pFsgF*7$Fit$\"3#p(e41_[rhF*7$F^u$\"3oy?#[ \"oR$H'F*7$Fcu$\"35s*o7EaXT'F*7$Fhu$\"3A9zw1rHdlF*7$F]v$\"33-:_\")Ru*p 'F*7$Fbv$\"3[xW%[8Z6(oF*7$Fgv$\"3mVxlI@#)[qF*7$F\\w$\"3!4b-/2GKD(F*7$F aw$\"3kP*=llWLZ(F*7$Ffw$\"3Cq09v[TFxF*7$F[x$\"3g%ynYkSs*zF*7$F`x$\"3/V @?\"3R7I)F*7$Fex$\"3\"*og4?)RNj)F*7$Fjx$\"3Un3i!Qy&o*)F*7$F_y$\"3Ac!R` R%*4R*F*7$Fdy$\"3%[2m'z\"Qp!)*F*7$Fiy$\"3_Wk/TESH5Fcdl7$F^z$\"3%4on*HS k!3\"Fcdl7$Fcz$\"3gql2L([79\"Fcdl-Fhz6&FjzF[[lF^[lF[[l-%+AXESLABELSG6$ Q\"x6\"Q!Fe^m-%%VIEWG6$;$\"#w!\"#$\"#zF]_m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "Th e values of these two Taylor polynomials start to diverge away from ea ch other as well as diverging away from " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 12 " as soon as " }{TEXT 277 1 "x" }{TEXT -1 17 " is greater than " }{XPPEDIT 18 0 "Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "In fact they inters ect very close to " }{XPPEDIT 18 0 "x = Pi/4" "6#/%\"xG*&%#PiG\"\"\"\" \"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "evalf[55](fsolve(p(x)-q(x)=0,x=0.78 ..0.79));\nevalf[55](Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"XS&= !Q%)\\BH\\5sv)>e%3m:'4$[uRj\")R&y!#b" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"X_XwP%)\\BH\\5sv)>e%3m:'4$[uRj\")R&y!#b" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "This suggests that the ra dius of converge is " }{XPPEDIT 18 0 "R = Pi/4;" "6#/%\"RG*&%#PiG\"\" \"\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT 269 8 "Question" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x ) = 1/(1+exp(x));" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)-%$expG6#F'F)!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Plot the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 41 " and note the main features of the graph." }}{PARA 0 "" 0 "" {TEXT -1 29 "(b) Find \+ a Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 18 " of degree 13 for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 7 " about " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "(c) Plot the graphs " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p(x)" " 6#-%\"pG6#%\"xG" }{TEXT -1 40 " in the same picture over the interval \+ " }{XPPEDIT 18 0 "[-3.5,3.5]" "6#7$,$-%&FloatG6$\"#N!\"\"F)-F&6$F(F) " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 123 "(d) Construct an ani mation which you can use to give an estimate for the radius of converg ence of the Maclaurin series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 270 8 "Solution" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "(a)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f := x -> 1/(1+exp(x));\nplot(f(x),x=-6..6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&\"\"\"F-,&F-F--%$expG6#9$F-!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"'\"\"!$\"3W`O VoPFv**!#=7$$!3z******\\TVQd!#<$\"3#36Y\"y_!z'**F-7$$!3l****\\-r%3^&F1 $\"3%3w65'ftf**F-7$$!3A+++l;!\\D&F1$\"3]m'>Tm_![**F-7$$!3o*****\\lfs* \\F1$\"3&[AtX1*)G$**F-7$$!3%)****\\s@%3u%F1$\"3ls59dGW8**F-7$$!3J++]U. 6.XF1$\"3%f+-r.o/*)*F-7$$!3')****\\-G&pD%F1$\"37M//WXKg)*F-7$$!3(***** \\AjP-SF1$\"3Zr[$G9d0#)*F-7$$!33++]sih[PF1$\"3IVe5da\"*p(*F-7$$!3%)*** ***pGf([$F1$\"3w'3gvzELq*F-7$$!3)******\\J$odKF1$\"3ZQFACA[H'*F-7$$!3y ******4'f))*HF1$\"3dFx87eAD&*F-7$$!33+++]J(*QFF1$\"3IwI(\\lvGR*F-7$$!3 #)*******QC&)[#F1$\"3#*zz#=6MLB*F-7$$!3/++]AH4hAF1$\"3/JU$=-Jg0*F-7$$! 3%*******4\\l!*>F1$\"35,3/mS7)z)F-7$$!3'*******R%e:w\"F1$\"3v)e$zjt/M& )F-7$$!33++]#yk]\\\"F1$\"3gF$*e?DPo\")F-7$$!3M+++SFam%**F-$\"3m@6i`b\"**p#F-7$$\"3k*****\\J igC\"F1$\"37i!o;:DQB#F-7$$\"3%*****\\P7$$\"3Y****\\P/&f\\#F1$\"3US`xXdD9wF_v7 $$\"3q+++5zj_FF1$\"3wPRDpVy$*fF_v7$$\"3=****\\<3;%*HF1$\"3A,TF:m.pZF_v 7$$\"3;++]Z=iYKF1$\"3oRoQSm%[u$F_v7$$\"3[******\\'[M\\$F1$\"3K]!o%Q7#* \\HF_v7$$\"3W****\\PM&=v$F1$\"3=^Dy4)zNH#F_v7$$\"3v+++gzs+SF1$\"3[Rt!H nNtz\"F_v7$$\"35+++0\"Q_D%F1$\"3e;i*y3=\"*R\"F_v7$$\"3q++]x2k2XF1$\"3+ odklDU!4\"F_v7$$\"3d+++?EdRZF1$\"3oyn8^[hm')F_s7$$\"3M+++&o#R0]F1$\"3g 95Za_4dmF_s7$$\"3++++?`9V_F1$\"3gqPS\">UeD&F_s7$$\"3G++]<#Rm\\&F1$\"33 ZPm>Oy$3%F_s7$$\"3F++]A_ERdF1$\"3aqnORX\"o?$F_s7$$\"\"'F*$\"3MuZjcJisC F_s-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fb[l-%+AXESLABELSG6$Q\"x6\"Q!Fg[l- %%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "f(x) -> 0" "6#f*6#-%\" fG6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"!F-F-F-" }{TEXT -1 4 " as " } {XPPEDIT 18 0 "x -> infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"% )infinityGF*F*F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x) -> 1" "6#f* 6#-%\"fG6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"\"F-F-F-" }{TEXT -1 4 " \+ as " }{XPPEDIT 18 0 "x -> -infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arr owG6\",$%)infinityG!\"\"F*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 36 " \+ has no real number discontinuities." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 3 "(b)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "f := x -> 1/(1+exp(x));\ntay lor(f(x),x,14):\np := unapply(convert(%,polynom),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-, &F-F--%$expG6#9$F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"p Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,2#\"\"\"\"\"#F.*&#F.\"\"%F.9$F.! \"\"*&#F.\"#[F.*$)F3\"\"$F.F.F.*&#F.\"$![F.*$)F3\"\"&F.F.F4*&#\"#<\"&S 1)F.*$)F3\"\"(F.F.F.*&#\"#J\"(?:X\"F.*$)F3\"\"*F.F.F4*&#\"$\"p\"*+WL>$ F.*$)F3\"#6F.F.F.*&#\"%ha\",+K33\\#F.*$)F3\"#8F.F.F4F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 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*45F17$Fax$\"3?$Gy)HlN;*)Fex7$Fgx$\"3gt[P]g#)*)zFex7$F\\y$\"3c#3,#>se= wFex7$Fay$\"3)o[Rk6AuP)Fex7$F]jp$\"3#*etYrj+%y*Fex7$Ffy$\"3[d=s)*[)fC \"F17$Fc_o$\"3)*GKeqO'[l\"F17$F[z$\"3P!yGVL]+J#F17$F[`o$\"37M?C\"z$\\? MF17$F`z$\"3y&\\3T-Qg<&F17$F_gm$\"35h+,>$few(F17$Fez$\"3wu/1wI]n6F.7$F cao$\"374Q1_7U_9F.7$Fggm$\"3))4a)QkH\\!=F.7$Fc\\q$\"3%z$3@%\\i5,#F.7$F [bo$\"3+*fU+U\"*)RAF.7$F[]q$\"3kA/M#*Gv$\\#F.7$Fjz$\"3)=uY0W3_x#F.-Fb_ l6$Fd_l#\"#@Ff_lFg[lFg_lF]`l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "It looks as though the radius o f convergence is about 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 54 "The following commands construct a Taylor polynomi als " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 19 " of degree 100 and " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 19 " of d egree 120 for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " \+ at " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "f : = x -> 1/(1+exp(x));\nconvert(taylor(f(x),x,101),polynom):\np := unapp ly(%,x):\nconvert(taylor(f(x),x,121),polynom):\nq := unapply(%,x):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&\"\"\"F-,&F-F--%$expG6#9$F-!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The values of " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "q(x)" "6#-% \"qG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 21 " can be compared for " }{TEXT 278 1 "x" }{TEXT -1 8 " near 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "xx := 3.0;\np(xx);\nq(xx);\nf(xx);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#xxG$\"#I!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+q0Ze]!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]q " 0 "" {MPLTEXT 1 0 61 "plot([f(x),p(x),q(x)],x =3.05..3.18,color=[red,blue,magenta]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"3#)************\\I!#<$\" 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$\"3qU0q4kw^DFa^l7$F_u$\"3nE4lB-i.GFa^l7$Fdu$\"3sJ\"G*3_=bIFa^l7$Fiu$ \"3mbL'H#*)4`LFa^l7$F^v$\"3Io:r*eZ>l$Fa^l7$Fcv$\"3I\"*\\kh8Z8SFa^l7$Fh v$\"3e>!RGe+.R%Fa^l7$F]w$\"3+Eq&*\\&fi#[Fa^l7$Fbw$\"3o$=-%y&Q%)H&Fa^l7 $Fgw$\"3IQgghZpYeFa^l7$F\\x$\"32u.`!QsCV'Fa^l7$Fax$\"3qjpgWOX'4(Fa^l7$ Ffx$\"3#zwem9Zn#yFa^l7$F[y$\"3?24$4$3Xn&)Fa^l7$F`y$\"3e.1@$fDr]*Fa^l7$ Fey$\"3\\!*=xzK#Q/\"F*7$Fjy$\"3+,b?(eON:\"F*7$$\"3;++0\\6'e<$F*$\"3Q)* 4xWz<57F*7$F_z$\"3#Hv!y:0op7F*7$$\"3=v=i'o(eyJF*$\"3Dx=p;t(pL\"F*7$Fdz $\"3nZ/r30$zS\"F*-Fiz6&F[[lF\\[lF_[lF\\[l-%+AXESLABELSG6$Q\"x6\"Q!F`_m -%%VIEWG6$;$\"$0$!\"#$\"$=$Fh_m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The valu es of these two Taylor polynomials start to diverge away from each oth er as well as diverging away from " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 12 " as soon as " }{TEXT 279 1 "x" }{TEXT -1 17 " is g reater than " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 "This begins to suggest that the radius of converg e is " }{XPPEDIT 18 0 "R=Pi" "6#/%\"RG%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "In fact they intersect very close to " } {XPPEDIT 18 0 "x = Pi;" "6#/%\"xG%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "evalf [60](fsolve(p(x)-q(x)=0,x=3.1..3.16));\nevalf[60](Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"gnU*3yn;v$*Rpr>%)G]zKQVEYQKz*e`EfTJ!#f" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"gn%\\(4#e5v$*Rpr>%)G]zKQVEYQKz*e`Ef TJ!#f" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "If we extend the domain of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 29 " to allow complex values for " }{TEXT 293 1 "x" }{TEXT -1 14 ", we see that " }{XPPEDIT 18 0 "1+ exp(x) = 0" "6#/,&\"\"\"F%-% $expG6#%\"xGF%\"\"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x = Pi*i" "6 #/%\"xG*&%#PiG\"\"\"%\"iGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "Note that by Euler's formula " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "exp(i*theta) = cos*theta+i*sin*theta;" "6#/-%$expG 6#*&%\"iG\"\"\"%&thetaGF),&*&%$cosGF)F*F)F)*(F(F)%$sinGF)F*F)F)" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 8 "we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(i*Pi) = cos*Pi+i*sin*Pi;" "6#/ -%$expG6#*&%\"iG\"\"\"%#PiGF),&*&%$cosGF)F*F)F)*(F(F)%$sinGF)F*F)F)" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "1+exp(i*Pi) = 0" "6#/,&\"\"\"F%-%$expG6#*&%\"iGF%%# PiGF%F%\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "subs(x=Pi*I,1+exp(x));\neval (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$-%$expG6#*&%#PiGF$^# F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " } {XPPEDIT 18 0 "f(x)=1/(1+exp(x))" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)-%$ expG6#F'F)!\"\"" }{TEXT -1 36 " has a singularity at a distance of " } {XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 44 " units from the origin in t he complex plane." }}{PARA 0 "" 0 "" {TEXT -1 183 "If you imagine that the notion of Taylor series can be extended to functions of a complex variable, then the location of this singularity might suggest that th e radius of convergence " }{TEXT 261 22 "can be no greater than" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT 285 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 " Let " }{XPPEDIT 18 0 "f(x) = 1/(x+exp(x))" "6#/-%\"fG6#%\"xG*&\"\"\"F) ,&F'F)-%$expG6#F'F)!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Plot the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 41 " and note the main features of the graph." }}{PARA 0 "" 0 "" {TEXT -1 29 "(b) Find a Taylor polynomial " }{XPPEDIT 18 0 "p(x) " "6#-%\"pG6#%\"xG" }{TEXT -1 17 " of degree 9 for " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " about " }{XPPEDIT 18 0 "x=0" "6#/ %\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "(c) Plot th e graphs " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 40 " in the same p icture over the interval " }{XPPEDIT 18 0 "[-.567, 1];" "6#7$,$-%&Flo atG6$\"$n&!\"$!\"\"\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 123 "(d) Construct an animation which you can use to give an estimate \+ for the radius of convergence of the Maclaurin series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "solve(x+exp(x)=0);\nevalf(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%)LambertWG6#\"\"\"!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!+/HVrc!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 " f(x) = 1/(x+exp(x))" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F'F)-%$expG6#F'F)!\" \"" }{TEXT -1 56 " has a vertical asymptote where the denominator is z ero." }}{PARA 0 "" 0 "" {TEXT -1 18 "This occurs where " }{TEXT 287 1 "x" }{TEXT -1 1 " " }{TEXT 281 1 "~" }{TEXT -1 16 " -0.5671432904. " } }{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 33 ": With the option \" discont=true\" " }{TEXT 0 4 "plot" }{TEXT -1 39 " does not discover th is discontinuity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 229 "f := x -> 1/(x+exp(x));\np1 := plot(f(x),x= -2.. -.5671432905,y=-5..5):\np2 := plot(f(x),x=-.5671432903..3,y=-5..5 ):\np3 := plots[implicitplot](x=-.5671432904,x=-1..0,y=-5..5,\n col or=black,linestyle=3):\nplots[display]([p1,p2,p3]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,& 9$F--%$expG6#F/F-!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7gn7$$!\"#\"\"!$!3gp(yuT%*GO&!#=7$$!3w& G/9!ywo>!#<$!3?j^yvI0naF-7$$!3>8QSKGfT>F1$!3Ok3SC:PhbF-7$$!3e(\\@wtJ5 \">F1$!3iv/'R*\\\"=n&F-7$$!3wL]1k!o-)=F1$!3UpZIL(yzy&F-7$$!3%=$*y[g]' \\=F1$!3&\\#)y&QN\"*3fF-7$$!3pIe!4Ik7#=F1$!3D2'*\\Au6EgF-7$$!3Uz0MF>(= z\"F1$!3k3([n(z*H:'F-7$$!3orJb#Hu9w\"F1$!3_6:7#yi0H'F-7$$!39UgOVTGWbs;F1$!3L:b+ #)*p\\t'F-7$$!3aa(pYl\\;k\"F1$!3:I3$)[\"Gh!pF-7$$!3_([wJ)zh5;F1$!3=*R' )eL1z3(F-7$$!3cCQmWx/5R\\\"F1$!3qSI7,FjxyF-7 $$!3G!>XcW*3i9F1$!37*eFKR[y7)F-7$$!3cTS*4#G#RV\"F1$!31rFIS\"F1$!3H(Rw8t,=k)F-7$$!3OM^DmBft8F1$!3sH;'Hw5a#*)F-7$$!3IG+qm \"))GM\"F1$!3sXuBCG(QC*F-7$$!3%4o%pE@p98F1$!3W5W_,X**e&*F-7$$!3S=iR))) yUG\"F1$!3(4E)>B/EE**F-7$$!3?A!*G\"3)o_7F1$!3'QgB=2xT.\"F17$$!3$>LN3>) =D7F1$!39z7,3Pbt5F17$$!3wG7wbv[&>\"F1$!3US6F17$$!3+Z/_$*Q![;\"F 1$!3%G#f;/.es6F17$$!3b&43i\"fyM6F1$!3GNX]$=m&H7F17$$!3*)fFmz>u06F1$!3- vMbqVp!H\"F17$$!3w'y%)*RO\\t5F1$!3'=Q?J6:nO\"F17$$!3By7[sq^W5F1$!3/)Qh gS>PW\"F17$$!3P)yYb]?F17$$!3]?[HrOPk')F-$!3Rh))y'z*>UAF17$$ !3U1zMVh#eN)F-$!37\"QA'GX!y[#F17$$!3!*f9>-#e'e!)F-$!3#*>\"eG-PUy#F17$$ !3'oBU91hZv(F-$!35\"fKKm(ouJF17$$!3OtFXU.Q`uF-$!3/$f..m]Kp$F17$$!3'y3# *=wUk<(F-$!3CL:^?0n_VF17$$!3u2'z@/T!foF-$!3ixMHuQQ'[&F17$$!3?(R&HgJ:vl F-$!3Y4/)edCZ<(F17$$!3)=\"p3n%pCF'F-$!3mtvTJD6t5!#;7$$!3@\\@54ghFhF-$! 3-Tt!oja-T\"F_z7$$!3_'Q<6biF)fF-$!35$zsrc*4h?F_z7$$!3ZS!)eR,$\\!fF-$!3 f4m]bfJWFF_z7$$!3J$pe!Gx4FeF-$!3kk!pp+N26%F_z7$$!3G?SHA:=)y&F-$!3))[F. 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}{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 19 " of degree 1 20 for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " at " } {XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "f := x -> 1/(x+exp(x));\nconvert(taylor(f(x),x,101),polynom):\np := unapply(%,x ):\nconvert(taylor(f(x),x,121),polynom):\nq := unapply(%,x):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*& \"\"\"F-,&9$F--%$expG6#F/F-!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The values of " }{XPPEDIT 18 0 "p(x) " "6#-%\"pG6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6# %\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 21 " can be compared for " }{TEXT 292 1 "x" }{TEXT -1 18 " ne ar LambertW(1) " }{TEXT 284 1 "~" }{TEXT -1 14 " 0.5671432904." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "xx := 0.53;\np(xx);\nq(xx);\nf(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#`!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+y!pE\\%!# 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)*o0)[%!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+!)HX'[%!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot([f(x),p(x),q(x)],x=0.55 ..0.57,color=[red,blue,magenta]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"3U+++++++b!#=$\"3Ina!))4;(zV F*7$$\"3BLL$3VfV]&F*$\"3?;yOo8VxVF*7$$\"37n;H[D:3bF*$\"3S,%>='[WvVF*7$ $\"3BMLe0$=C^&F*$\"3iH015C@tVF*7$$\"3#RL$3RBr;bF*$\"3uCRW]o'4P%F*7$$\" 3&pm\"zjf)4_&F*$\"3[;QYXOtoVF*7$$\"3'RLe4;[\\_&F*$\"3SJS%enkmO%F*7$$\" 3\\+]i'y]!HbF*$\"3[D58%*Q_kVF*7$$\"3&QLezs$HLbF*$\"33*z2e^6BO%F*7$$\"3 X+]7iI_PbF*$\"3%)G9Jry5gVF*7$$\"3#omm@Xt=a&F*$\"3Omk8dG%yN%F*7$$\"3!QL $3y_qXbF*$\"3%>A!3i#\\eN%F*7$$\"3!*****\\1!>+b&F*$\"3=u=$ya1ON%F*7$$\" 3]++]Z/NabF*$\"3I;]sHjN^VF*7$$\"3]++]$fC&ebF*$\"3)[H(HU%*=\\VF*7$$\"3+ 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8.$y%[F*7$F^p$\"3%4_jFxb&))[F*7$Fcp$\"3fT$R$))GsT\\F*7$Fhp$\"39]@Z>N: \"*\\F*7$F]q$\"3Sxm<#z+U0&F*7$Fbq$\"3\\y)eB`Na6&F*7$Fgq$\"39&GL0=M!*=& F*7$F\\r$\"3'f!42ie#fE&F*7$Far$\"3!R5Q(p,!RN&F*7$Ffr$\"3u&*QD%\\*GUaF* 7$F[s$\"3o)ox**eQla&F*7$F`s$\"3yIX'3`Cbm&F*7$Fes$\"3'3fAT\"fzydF*7$Fjs $\"3.0h\"[$4>7fF*7$F_t$\"3[,d_h/:jgF*7$Fdt$\"3$4/>&*4n\\A'F*7$Fit$\"3_ U%)Q!QhgR'F*7$F^u$\"33ip!*=:M/mF*7$Fcu$\"3S*Q(p;MW4oF*7$Fhu$\"3EtAtAE' *[qF*7$F]v$\"3)4px`WdfG(F*7$Fbv$\"3AW]vb>vovF*7$Fgv$\"3)>.ZuQf%fyF*7$F \\w$\"3aB%efq\"=\">)F*7$Faw$\"3%fQ)3%G,ba)F*7$Ffw$\"3S2=J,H:^*)F*7$F[x $\"3S%yw+p([y$*F*7$F`x$\"3)H4P82Fh&)*F*7$Fex$\"3Tq$zHu5u.\"Fdcl7$Fjx$ \"3Cf364ZC*3\"Fdcl7$F_y$\"3%G=JGb'4a6Fdcl7$Fdy$\"3pw!ym\\lu@\"Fdcl7$Fi y$\"3cN6Px37\"H\"Fdcl7$F^z$\"3-jN5sy-o8Fdcl7$Fcz$\"3#\\s>)[CIe9Fdcl-Fh z6&FjzF[[lF^[lF[[l-%+AXESLABELSG6$Q\"x6\"Q!Fe^m-%%VIEWG6$;$\"#b!\"#$\" #dF]_m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 112 "The values of these two Taylor polynomia ls start to diverge away from each other as well as diverging away fro m " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 12 " as soon as \+ " }{TEXT 288 1 "x" }{TEXT -1 17 " is greater than " }{XPPEDIT 18 0 "La mbertW(1);" "6#-%)LambertWG6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 54 "This begins to suggest that the radius of converge is \+ " }{XPPEDIT 18 0 "R = LambertW(1);" "6#/%\"RG-%)LambertWG6#\"\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "In fact they intersect v ery close to " }{XPPEDIT 18 0 "x = LambertW(1);" "6#/%\"xG-%)LambertWG 6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "evalf[100](fsolve(p(x)-q(x)=0,x=0.5 6..0.57));\nevalf[100](LambertW(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"_qU&zvhpW>KpmXo'3$zXIAz58N\"3D^'=(y:Qv\\bN5Amo***H(Qy4/HVrc!$+\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qYH_<'pW>KpmXo'3$zXIAz58N\"3D^ '=(y:Qv\\bN5Amo***H(Qy4/HVrc!$+\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT 298 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f( x) = 1/(x^2+cos(x));" "6#/-%\"fG6#%\"xG*&\"\"\"F),&*$F'\"\"#F)-%$cosG6 #F'F)!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Plot th e graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 41 " an d note the main features of the graph." }}{PARA 0 "" 0 "" {TEXT -1 29 "(b) Find a Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG " }{TEXT -1 18 " of degree 16 for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 7 " about " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "(c) Plot the graphs " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 40 " in the same picture over \+ the interval " }{XPPEDIT 18 0 "[-2, 2];" "6#7$,$\"\"#!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 123 "(d) Construct an animation wh ich you can use to give an estimate for the radius of convergence of t he Maclaurin series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 299 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 " (a) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "f := x -> 1/(x^2+cos(x));\nplot(f(x),x=-5..5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&\"\"\"F-,&*$)9$\"\"#F-F--%$cosG6#F1F-!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 555 202 202 {PLOTDATA 2 "6%-%'CURVESG6$7bo7$$!\"&\"\"!$ \"3wWY(QCB^&R!#>7$$!3YLLLe%G?y%!#<$\"3))G')e]fpfVF-7$$!3OmmT&esBf%F1$ \"3Gk)zNS\"ooZF-7$$!3ALL$3s%3zVF1$\"3k,Sg10E0`F-7$$!3_LL$e/$QkTF1$\"3p \"o*=SW\"\\%fF-7$$!3ommT5=q]RF1$\"3*RiystbLq'F-7$$!3ILL3_>f_PF1$\"3z.% 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YJF-$\"35fK2,40&***Ffn7$$!3vDMLLe*e$\\!#?$\"3+!=)*e=y)****Ffn7$$\"3+l; a)3RBE#F-$\"3=D\"flXTu***Ffn7$$\"3bsmTgxE=]F-$\"3g!*eV$p@u)**Ffn7$$\"3 7!o\"HKk>uxF-$\"3Ys]v`o&)p**Ffn7$$\"3womT5D,`5Ffn$\"3h4\\NpK\"[%**Ffn7 $$\"3Gq;zW#)>/;Ffn$\"3#=jw1H$ps)*Ffn7$$\"3!=nm\"zRQb@Ffn$\"3;=VM+;8s(* Ffn7$$\"3mOLL$e,]6$Ffn$\"3`<>?qY2%Ffn$\"3[oMVv^xB #*Ffn7$$\"36QLe*[K56&Ffn$\"3,\\TH$)=wA))Ffn7$$\"3summ\"zXu9'Ffn$\"3q,# 4!e'z$p$)Ffn7$$\"3#yLLe9i\"=sFfn$\"3O%4#G!=]R'yFfn7$$\"3#4+++]y))G)Ffn $\"3LoU(G<(4QtFfn7$$\"3%>++DcljL*Ffn$\"3hN3z!RG&=oFfn7$$\"3H++]i_QQ5F1 $\"3O#H$fUBu0jFfn7$$\"3U+](=-N(R6F1$\"3tRYFIXoCeFfn7$$\"3b++D\"y%3T7F1 $\"3O1Jz)pHYO&Ffn7$$\"3G+]P4kh`8F1$\"3[vEb0!*R$)[Ffn7$$\"3+++]P![hY\"F 1$\"3!GDgY#QYOWFfn7$$\"3iKLL$Qx$o;F1$\"3x>x!eIL\"Ffn7$$\"3a** *\\7`Wl7$F1$\"3S'G&*H)*\\&R6Ffn7$$\"3enmmm*RRL$F1$\"3_ 0" "6#f*6#-%\"fG6#%\"xG7\"6$%)operatorG%&arrowG 6\"\"\"!F-F-F-" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "x -> infinity" "6#f *6#%\"xG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "f(x) -> 0" "6#f*6#-%\"fG6#%\"xG7\"6$%)operatorG%&a rrowG6\"\"\"!F-F-F-" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "x -> -infinity " "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\",$%)infinityG!\"\"F*F*F*" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 36 " has no real number discontinuities." }}{PARA 0 "" 0 "" {TEXT -1 18 "The derivative of " }{XPPEDIT 18 0 "f(x )" "6#-%\"fG6#%\"xG" }{TEXT -1 8 " is f '(" }{TEXT 301 1 "x" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "`` = (sin(x)-2*x)/((x^2+cos(x))^2);" "6#/%!G* &,&-%$sinG6#%\"xG\"\"\"*&\"\"#F+F*F+!\"\"F+*$,&*$F*F-F+-%$cosG6#F*F+F- F." }{TEXT -1 35 ", which is equal to zero only when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 " The function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 30 " \+ has the maximum value 1 when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "Note also that " } {XPPEDIT 18 0 "f(x)=1/(x^2+cos(x))" "6#/-%\"fG6#%\"xG*&\"\"\"F),&*$F' \"\"#F)-%$cosG6#F'F)!\"\"" }{TEXT -1 39 " is an even function, so the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 26 " is \+ symmetrical about the " }{TEXT 302 1 "y" }{TEXT -1 7 " axis. 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" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 137 "f := x -> 1/(x^2+cos(x));\nconvert(taylor(f(x ),x,141),polynom):\np := unapply(%,x):\nconvert(taylor(f(x),x,161),pol ynom):\nq := unapply(%,x):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf* 6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&*$)9$\"\"#F-F--%$cosG6#F1 F-!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The values of " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 21 " can be co mpared for " }{TEXT 303 1 "x" }{TEXT -1 11 " near 1.5. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "xx := 1 .5;\np(xx);\nq(xx);\nf(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$ \"#:!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*91\"4V!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*:.!4V!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5c(*3V!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot([f(x),p(x),q(x)],x=1.6..1.63,color=[ red,blue,magenta]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"33+++++++;!#<$\"3-@g'\\**=8&R!#=7$$ \"31+]i9Rl+;F*$\"3%=Xtb1t!\\RF-7$$\"3)*\\PC#)GA,;F*$\"3I\\dUq)>r%RF-7$ $\"35+v$eui=g\"F*$\"3$)4e1$QC\\%RF-7$$\"3++D'3&o]-;F*$\"3rvGO-brURF-7$ $\"31](oX*y9.;F*$\"3?*HR_F=0%RF-7$$\"3,]P9CAu.;F*$\"3wM>y$>#[QRF-7$$\" 3)*\\P*zhdVg\"F*$\"3\")GZ#H*\\PORF-7$$\"3;]P>fS*\\g\"F*$\"3\"pOX%Ro>MR F-7$$\"3)*\\(=$f%Gcg\"F*$\"3rq)[6!o-KRF-7$$\"3****\\#y,\"G1;F*$\"33D) \\n#ezHRF-7$$\"3:+Dr\"zbog\"F*$\"30l$y:xJy#RF-7$$\"3,+](4&G]2;F*$\"3h$ Ry!==iDRF-7$$\"35+]7nD:3;F*$\"3!*)*RJqRSBRF-7$$\"3****\\-*oy(3;F*$\"3d T7n\"yn7#RF-7$$\"3#*\\PpnsM4;F*$\"3(3T?)o)G$>RF-7$$\"3-+]siL-5;F*$\"3) *p/(>\\Cq\"RF-7$$\"33++!R5'f5;F*$\"3E83o$Qt]\"RF-7$$\"33]P/QBE6;F*$\"3 L!RFh$\\!G\"RF-7$$\"3$****\\\"o?&=h\"F*$\"3@;]57!)z5RF-7$$\"3$*\\Pa&4* \\7;F*$\"3PDS%z@(f3RF-7$$\"32]7j=_68;F*$\"3%*oO9IE]1RF-7$$\"33+vVy!ePh \"F*$\"3SOD^z#=V!RF-7$$\"3/](=WU[Vh\"F*$\"3\\kV%=Q8B!RF-7$$\"3/+DJ#>&) \\h\"F*$\"37m?T@>:+RF-7$$\"33]P>:mk:;F*$\"3%R2#*=(z!z*QF-7$$\"3))\\iv& QAih\"F*$\"3!QiOHgbf*QF-7$$\"3(**\\PPBWoh\"F*$\"33zrzJ![Q*QF-7$$\"3/++ bjm[<;F*$\"3'HrC&R=n\"*QF-7$$\"3)**\\(yb^6=;F*$\"3Y>3#f(Ra*)QF-7$$\"3( *\\PMaKs=;F*$\"3A,l*)3i[()QF-7$$\"3++D6W%)R>;F*$\"31u&='>E?&)QF-7$$\"3 (****\\@80+i\"F*$\"3%RC)4*z^J)QF-7$$\"3%***\\7,Hl?;F*$\"3q!)*4]Cj4)QF- 7$$\"3()\\P4w)R7i\"F*$\"3)46\"[,6)*yQF-7$$\"30+]x%f\")=i\"F*$\"3)[;pb> :o(QF-7$$\"3.]P/-a[A;F*$\"3IanT'HyZ(QF-7$$\"3/](=Yb;Ji\"F*$\"3%>*yXP-l sQF-7$$\"3,+]i@OtB;F*$\"39K7QX2dqQF-7$$\"3$*\\PfL'zVi\"F*$\"3i@g-[[RoQ F-7$$\"3/++!*>=+D;F*$\"3GfSn+.ImQF-7$$\"3.+DE&4Qci\"F*$\"3'p!Q)*R%fT'Q F-7$$\"3&)\\P%>5pii\"F*$\"3-rZ_@u.iQF-7$$\"3))***\\:$*[oi\"F*$\"3eK+c& [)3gQF-7$$\"3))*\\7<[8vi\"F*$\"3ILR[Pf&y&QF-7$$\"3!*****Hjy5G;F*$\"3wT t`i,'e&QF-7$$\"3&)\\P/)fT(G;F*$\"3)*=2[MLt`QF-7$$\"3')\\i0j\"[$H;F*$\" 3#Hz$eO()p^QF-7$$\"3!*************H;F*$\"3cn-mDM^\\QF--%'COLOURG6&%$RG BG$\"*++++\"!\")$\"\"!F`[lF_[l-F$6$7S7$F($\"3f#pvXpzQ7&F-7$F/$\"3=L\\ \\l3=\">&F-7$F4$\"3SN!f]diID&F-7$F9$\"3ILRfhSbE`F-7$F>$\"3^ts_k6$\\S&F -7$FC$\"3sJO\"p5ov[&F-7$FH$\"3UAubI>aobF-7$FM$\"3YC4V&*y/dcF-7$FR$\"3% =o'[qP%Qv&F-7$FW$\"3WHi(H$H&f&eF-7$Ffn$\"3/Yunhr jqgF-7$F`o$\"3YPj(o.!f$>'F-7$Feo$\"3TzE;odNCjF-7$Fjo$\"3A'\\Vy1\"pdkF- 7$F_p$\"3W)*Q*zmM`e'F-7$Fdp$\"3'=J^'*)\\pXnF-7$Fip$\"3S\\$p5ED#*)oF-7$ F^q$\"3gXJE+^blqF-7$Fcq$\"3=*RK!e@uF-7$F]r$\" 3HG>$>+3Rh(F-7$Fbr$\"3-ia'>(f(f#yF-7$Fgr$\"3&*em&=gG:.)F-7$F\\s$\"3c7F Y-[[l#)F-7$Fas$\"3Ylmad!*pA&)F-7$Ffs$\"3y7w]&)[.f()F-7$F[t$\"3iJ9=;1*z -*F-7$F`t$\"3JO]I9F*7$Fax$\"3Kak%3UG&*[\"F*7$Ffx$\"3/6p*Q8f8b\"F*7$F[y$\"3G0i6. SA6;F*7$F`y$\"3y%GQ<7COo\"F*7$Fey$\"3u4'y+v\"*>v\"F*7$Fjy$\"3)30-8v])G =F*7$F_z$\"3\\%zOy\\_k!>F*7$Fdz$\"37OtRk@\\%*>F*-Fiz6&F[[lF_[lF_[lF\\[ l-F$6$7S7$F($\"3stbem#Q3&[F-7$F/$\"35*4(['yj(4\\F-7$F4$\"3+,KADyQk\\F- 7$F9$\"3?s\")R]IyH]F-7$F>$\"34'\\X81>,5&F-7$FC$\"3AAnWDf!\\<&F-7$FH$\" 3%3V`+!3y[_F-7$FM$\"3=1Q-P_;I`F-7$FR$\"3i(3[\")4(*)>aF-7$FW$\"31sdB5ML :bF-7$Ffn$\"3%\\(Q!Q\\_,i&F-7$F[o$\"3THq#o&yT=dF-7$F`o$\"3d%*p\"GxBh$e F-7$Feo$\"3uMT)[PTB'fF-7$Fjo$\"3=Cbri_3#4'F-7$F_p$\"3t/GE\\IBIQ7F*7$F]w$\"3/WbZ\"eoIH\"F*7$Fbw$\"3#f;aF MB+N\"F*7$Fgw$\"3UIzz**H\\89F*7$F\\x$\"35xK!**>u&y9F*7$Fax$\"35LAg.qR \\:F*7$Ffx$\"3z;F89k;C;F*7$F[y$\"3I6E$=X)4(p\"F*7$F`y$\"3_pBf\\!zfy\"F *7$Fey$\"3)[qS,Hf0(=F*7$Fjy$\"3ct-N&yjj'>F*7$F_z$\"3!=U(*o,KQ1#F*7$Fdz $\"3;zBEKoEv@F*-Fiz6&F[[lF\\[lF_[lF\\[l-%+AXESLABELSG6$Q\"x6\"Q!Fe^m-% %VIEWG6$;$\"#;!\"\"$\"$j\"!\"#%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The valu es of these two Taylor polynomials start to diverge away from each oth er as well as diverging away from " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 12 " as soon as " }{TEXT 300 1 "x" }{TEXT -1 29 " is g reater than about 1.62. " }}{PARA 0 "" 0 "" {TEXT -1 74 "This begins t o suggest that the radius of converge is approximately 1.62. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "evalf[35](fsolve(p(x)-q(x)=0,x=1.61..1.63));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Doyt'3m/X*\\BD.h%zM@;!#M" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "For a real number " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cos*i*xi = cosh* xi;" "6#/*(%$cosG\"\"\"%\"iGF&%#xiGF&*&%%coshGF&F(F&" }{TEXT -1 8 ", w here " }{TEXT 304 1 "i" }{TEXT -1 24 " is the imaginary unit. " }} {PARA 0 "" 0 "" {TEXT -1 19 "This follows since " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*i*xi = 1-i^2*xi^2/2!+i^4*xi^4/4!-i^ 6*xi^6/6!+` . . . `;" "6#/*(%$cosG\"\"\"%\"iGF&%#xiGF&,,F&F&*(F'\"\"#F (F+-%*factorialG6#F+!\"\"F/*(F'\"\"%F(F1-F-6#F1F/F&*(F'\"\"'F(F5-F-6#F 5F/F/%(~.~.~.~GF&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1+xi^2/2!+xi^4/4!+xi^6/6!+` . . . `;" "6#/%!G,,\" \"\"F&*&%#xiG\"\"#-%*factorialG6#F)!\"\"F&*&F(\"\"%-F+6#F/F-F&*&F(\"\" '-F+6#F3F-F&%(~.~.~.~GF&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = cosh*xi;" "6#/%!G*&%%coshG\"\"\"%#xiGF' " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 28 " is a re al number such that " }{XPPEDIT 18 0 "x = xi;" "6#/%\"xG%#xiG" }{TEXT -1 31 " is a solution of the equation " }{XPPEDIT 18 0 "x^2=cosh*x" "6 #/*$%\"xG\"\"#*&%%coshG\"\"\"F%F)" }{TEXT -1 14 ", then taking " } {XPPEDIT 18 0 "zeta = i*xi;" "6#/%%zetaG*&%\"iG\"\"\"%#xiGF'" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "x = zeta;" "6#/%\"xG%%zetaG" }{TEXT -1 15 " is a zero of " }{XPPEDIT 18 0 "x^2+cos*x" "6#,&*$%\"xG\"\"#\"\" \"*&%$cosGF'F%F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "Th is follows since " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " zeta^2+cos*zeta = i^2*xi^2+cos*i*xi;" "6#/,&*$%%zetaG\"\"#\"\"\"*&%$co sGF(F&F(F(,&*&%\"iGF'%#xiGF'F(*(F*F(F-F(F.F(F(" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -xi^2+cosh*xi;" "6#/%!G,&*$%#xiG\"\"#!\"\"*&%%coshG\"\"\"F'F,F," }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "xi = evalf[35](fsolve(x^2=cosh(x),x=1.61..1.63)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#xiG$\"D*4Un3m/X*\\BD.h%zM@;!#M " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "If we extend the domain o f " }{XPPEDIT 18 0 "f(x) = 1/(x^2+cos(x));" "6#/-%\"fG6#%\"xG*&\"\"\"F ),&*$F'\"\"#F)-%$cosG6#F'F)!\"\"" }{TEXT -1 29 " to allow complex valu es for " }{TEXT 305 1 "x" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 36 " has a singularity at a distance of " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 44 " units from the origin in the complex plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 183 "If you imagine that the notion of Taylor series can be extended to functions of a complex variable, then the location of thi s singularity might suggest that the radius of convergence " }{TEXT 261 22 "can be no greater than" }{TEXT -1 1 " " }{XPPEDIT 18 0 "xi;" " 6#%#xiG" }{TEXT -1 1 " " }{TEXT 306 1 "~" }{TEXT -1 37 " 1.62134794610 325234994504660867421. " }}{PARA 0 "" 0 "" {TEXT -1 62 "The calculatio ns above suggest that the radius of convergence " }{TEXT 307 1 "R" } {TEXT -1 16 " is this number " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "We can easily get a more accu rate value for " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "xi = evalf[100](evalf[110](fsolve(x^2=cosh(x),x=1.61..1.63)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%#xiG$\"_q[KX\"[uyDL#ypg+_%459bj4Zjj& pdqO='f_(*)4Un3m/X*\\BD.h%zM@;!#**" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 36 "We can also obtain the complex zero " } {XPPEDIT 18 0 "zeta = i*xi" "6#/%%zetaG*&%\"iG\"\"\"%#xiGF'" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "x^2+cos(x)" "6#,&*$%\"xG\"\"#\"\"\"-%$cosG 6#F%F'" }{TEXT -1 11 " directly. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "zeta = evalf[100](evalf[110] (fsolve(x^2+cos(x),x=1.62*I,complex)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%%zetaG^#$\"_q[KX\"[uyDL#ypg+_%459bj4Zjj&pdqO='f_(*)4Un3m/X*\\B D.h%zM@;!#**" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exampl e 5" }}{PARA 0 "" 0 "" {TEXT 271 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = 1/(1+x*arctan (x));" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)*&F'F)-%'arctanG6#F'F)F)!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Plot the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 41 " and note the main features of the graph." }}{PARA 0 "" 0 "" {TEXT -1 29 "(b) Find \+ a Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 18 " of degree 16 for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 7 " about " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "(c) Plot the graphs " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p(x)" " 6#-%\"pG6#%\"xG" }{TEXT -1 40 " in the same picture over the interval \+ " }{XPPEDIT 18 0 "[-3/2, 3/2];" "6#7$,$*&\"\"$\"\"\"\"\"#!\"\"F)*&F&F 'F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 123 "(d) Construct a n animation which you can use to give an estimate for the radius of co nvergence of the Maclaurin series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 272 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 48 "f := x -> 1/(1+x*arctan(x));\nplot(f(x),x=-5.. 5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG %&arrowGF(*&\"\"\"F-,&F-F-*&9$F--%'arctanG6#F0F-F-!\"\"F(F(F(" }} {PARA 13 "" 1 "" {GLPLOT2D 555 202 202 {PLOTDATA 2 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0" "6#f*6#-%\"fG6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"!F-F- F-" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "x -> -infinity" "6#f*6#%\"xG7\" 6$%)operatorG%&arrowG6\",$%)infinityG!\"\"F*F*F*" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 36 " has no real number discontinuities." }}{PARA 0 "" 0 " " {TEXT -1 18 "The derivative of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 8 " is f '(" }{TEXT 289 1 "x" }{TEXT -1 1 ")" } {XPPEDIT 18 0 "`` = -(arctan(x)+x/(1+x^2))/((1+x*arctan(x))^2);" "6#/% !G,$*&,&-%'arctanG6#%\"xG\"\"\"*&F+F,,&F,F,*$F+\"\"#F,!\"\"F,F,*$,&F,F ,*&F+F,-F)6#F+F,F,F0F1F1" }{TEXT -1 35 ", which is equal to zero only \+ when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6 #%\"xG" }{TEXT -1 30 " has the maximum value 1 when " }{XPPEDIT 18 0 " x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "N ote also that " }{XPPEDIT 18 0 "f(x) = 1/(1+x*arctan(x));" "6#/-%\"fG6 #%\"xG*&\"\"\"F),&F)F)*&F'F)-%'arctanG6#F'F)F)!\"\"" }{TEXT -1 39 " i s an even function, so the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6 #%\"xG" }{TEXT -1 26 " is symmetrical about the " }{TEXT 290 1 "y" } {TEXT -1 7 " axis. 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0 "" 0 "" {TEXT -1 54 "The following commands construct a Taylor polynomials " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 19 " of degree 140 and " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" } {TEXT -1 19 " of degree 160 for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\" xG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "f := x -> 1/(1+x*arctan(x));\nconvert(taylor(f(x),x, 141),polynom):\np := unapply(%,x):\nconvert(taylor(f(x),x,161),polynom ):\nq := unapply(%,x):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#% \"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&F-F-*&9$F--%'arctanG6#F0F-F- !\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The values of " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 21 " can be compared for " }{TEXT 291 1 "x" }{TEXT -1 11 " near 0.8. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "xx := 0.75; \np(xx);\nq(xx);\nf(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"# v!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#3\"zWn!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+s,zWn!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+x+zWn!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot([f(x),p(x),q(x)],x=0.82..0.84,color=[red,bl ue,magenta]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6 '-%'CURVESG6$7S7$$\"3]*************>)!#=$\"3yz\"HE2trR'F*7$$\"3IKL$3Vf V?)F*$\"3%zgj%GM2&R'F*7$$\"3=m;H[D:3#)F*$\"3y`1x3sC$R'F*7$$\"3ILLe0$=C @)F*$\"3=X`9yT>\"R'F*7$$\"3)HL$3RBr;#)F*$\"37o5#[LG\"*Q'F*7$$\"3-m;zjf )4A)F*$\"3[a1N.J2(Q'F*7$$\"3.L$e4;[\\A)F*$\"3!QVGyMo^Q'F*7$$\"3c**\\i' y]!H#)F*$\"3a!Hz4!o>$Q'F*7$$\"3#HLezs$HL#)F*$\"3!zx)4#fe6Q'F*7$$\"3_** \\7iI_P#)F*$\"3=s'4opF\"zjF*7$$\"3*emm@Xt=C)F*$\"3IZ_(4[RqP'F*7$$\"3)G L$3y_qX#)F*$\"3W)RV&f3?vjF*7$$\"33++]1!>+D)F*$\"3!\\+(yp<8tjF*7$$\"3e* ***\\Z/Na#)F*$\"3KML#p*\\0rjF*7$$\"3e****\\$fC&e#)F*$\"3I[9C7W0pjF*7$$ \"3=L$ez6:BE)F*$\"3m5/FX$QsO'F*7$$\"33mm;=C#oE)F*$\"3Y-:Ky'z]O'F*7$$\" 3Ymmm#pS1F)F*$\"3#yDj[p^KO'F*7$$\"31**\\i`A3v#)F*$\"3+Eou\"e#=djF*7$$ \"37+](3zMuG)F*$\"3-Q%)z6\">_N'F*7$$\"3slm\"H_?7<`jF*7 $$\"3%em\"zihl&H)F*$\"3GCPE=8H^jF*7$$\"3)GL$3#G,**H)F*$\"3;-KtYVE\\jF* 7$$\"3yK$ezw5VI)F*$\"3InoUB(frM'F*7$$\"3(***\\PQ#\\\"3$)F*$\"3]PysJ$G` M'F*7$$\"3zKLe\"*[H7$)F*$\"3>n\"y!*4^LM'F*7$$\"3!******pvxlJ)F*$\"3r7= F5#48M'F*7$$\"3a***\\_qn2K)F*$\"3#\\lpNR7$RjF*7$$\"3&)**\\i&p@[K)F*$\" 3Cd)=()4\"QPjF*7$$\"3n***\\2'HKH$)F*$\"3q*[`ScP_L'F*7$$\"3xlmmZvOL$)F* $\"3)f\"\\+cAJLjF*7$$\"3A****\\2goP$)F*$\"3$ee^eOd7L'F*7$$\"3%GLeR<*fT $)F*$\"3IA*R@.'RHjF*7$$\"3g****\\)HxeM)F*$\"3C(=.o'=OFjF*7$$\"3]m;H!o- *\\$)F*$\"3j!>TYg[aK'F*7$$\"3?**\\7k.6a$)F*$\"3g(ey*\\%\\MK'F*7$$\"3/m m;WTAe$)F*$\"3%p%Rmdc\\@jF*7$$\"3a**\\i!*3`i$)F*$\"3>:!Q%45X>jF*7$$\"3 [KLL*zymO)F*$\"3pne[ND[i**z`w'F*7$F=$\"3K>\"[$4'Q>z'F*7$FB$\"3Ld7(H'*G0# oF*7$FG$\"3M\\[(ou*4\\oF*7$FL$\"3OU.Gbj#4)oF*7$FQ$\"3(Q2@l\\9k\"pF*7$F V$\"31K`K([$eapF*7$Fen$\"31.K!p$Q(p*pF*7$Fjn$\"3@\\KJ4$Hr.(F*7$F_o$\"3 EG!\\+zAd3(F*7$Fdo$\"3T&)\\!ob)QQrF*7$Fio$\"33#Hy1b(3$>(F*7$F^p$\"3qH \\5+UNYsF*7$Fcp$\"3$*pSvR%pWJ(F*7$Fhp$\"3-X\"fpr;lP(F*7$F]q$\"3u;MqS#y SX(F*7$Fbq$\"3:GM(=_2z_(F*7$Fgq$\"3G$*Q_88\"\\h(F*7$F\\r$\"3r*3\"Q_Q,/ xF*7$Far$\"3d#pTd7mR!yF*7$Ffr$\"3!*y7hDG[-zF*7$F[s$\"3:gmGR)*[;!)F*7$F `s$\"3CdvAr-/W\")F*7$Fes$\"3I\\u\"QO$>j#)F*7$Fjs$\"31Ac!*o'y4S)F*7$F_t $\"3-]TSCy%Rb)F*7$Fdt$\"3:)4*f:p![r)F*7$Fit$\"3y9#=eYg<)))F*7$F^u$\"3a 'yJPdH63*F*7$Fcu$\"3El7Z0cut#*F*7$Fhu$\"3=qY%znEX\\*F*7$F]v$\"3@z3\"y1 F!4(*F*7$Fbv$\"3#)y*3'fBPg**F*7$Fgv$\"3&)yi\"[!eS@5!#<7$F\\w$\"3%G;TU) G$)\\5Fbbl7$Faw$\"313/&4PY'z5Fbbl7$Ffw$\"3+_A1&RaJ6\"Fbbl7$F[x$\"3)Qg& [.r!y9\"Fbbl7$F`x$\"3m*=y6=Oe=\"Fbbl7$Fex$\"3iN'>#>bKE7Fbbl7$Fjx$\"37n UT6#RhE\"Fbbl7$F_y$\"3SE[%GGi]J\"Fbbl7$Fdy$\"3sAaDPu+i8Fbbl7$Fiy$\"3(o yK[q:cT\"Fbbl7$F^z$\"3KU6-p?gq9Fbbl7$Fcz$\"37+tIa**)R`\"Fbbl-Fhz6&FjzF ^[lF^[lF[[l-F$6$7S7$F($\"31U_9Pq$\\h'F*7$F.$\"3RL&o-@%HKmF*7$F3$\"3E#H odw'z[mF*7$F8$\"3)>/Wn1U!pmF*7$F=$\"3Ks0N]WP\"p'F*7$FB$\"3(o(=J^Rr:nF* 7$FG$\"3<5`Y4:KSnF*7$FL$\"3mU^saE/onF*7$FQ$\"3!GPzHn3$*z'F*7$FV$\"3E2R OePKLoF*7$Fen$\"3%*H%*)H\"e`roF*7$Fjn$\"3s2m\">UD\"3pF*7$F_o$\"3/dKE>+(F*7$Fio$\"3%*G\")*)*)oU`qF*7$F^p$\"3`yQ[U% *3/rF*7$Fcp$\"3+U_/l'f&prF*7$Fhp$\"3vq:B2P#)HsF*7$F]q$\"3<,5`R&RfI(F*7 $Fbq$\"3!pc')G9]\"ztF*7$Fgq$\"372GD:8JmuF*7$F\\r$\"3'4Gb0R1lb(F*7$Far$ \"3i'3qfqL(ewF*7$Ffr$\"3ag)HHO60w(F*7$F[s$\"3Go#)*3nv%zyF*7$F`s$\"3#[i ]1<$)R,)F*7$Fes$\"3t;an$Q)*39)F*7$Fjs$\"3%**R$[?\")3*G)F*7$F_t$\"3[^\" Qa+&Hb%)F*7$Fdt$\"3=qf/\"Qr=j)F*7$Fit$\"3(>JoVDnp\"))F*7$F^u$\"3$49/S0 #HS!*F*7$Fcu$\"3iM2+JoFe#*F*7$Fhu$\"3*H0rl:f1^*F*7$F]v$\"3YPX?04He(*F* 7$Fbv$\"3q%3AxDK^+\"Fbbl7$Fgv$\"3axhU&Q.].\"Fbbl7$F\\w$\"3)f3ay82)o5Fb bl7$Faw$\"3tBqY%e8Y5\"Fbbl7$Ffw$\"3Uh7I!)REX6Fbbl7$F[x$\"3O>xi@*Gx=\"F bbl7$F`x$\"3a!\\?(**)*zM7Fbbl7$Fex$\"3!=Lr+$HU&G\"Fbbl7$Fjx$\"3nZ_1q8o N8Fbbl7$F_y$\"3Koi9gD0)R\"Fbbl7$Fdy$\"3)*fC&=f,&e9Fbbl7$Fiy$\"3KBb!)3f ?G:Fbbl7$F^z$\"3mQKbXuS+;Fbbl7$Fcz$\"3gpm=S[Z%o\"Fbbl-Fhz6&FjzF[[lF^[l F[[l-%+AXESLABELSG6$Q\"x6\"Q!Fe^m-%%VIEWG6$;$\"##)!\"#$\"#%)F]_m%(DEFA ULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The values of these two Taylor polynomials start to diverge away from each other as well as diverging away from " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 12 " as soon as " } {TEXT 280 1 "x" }{TEXT -1 30 " is greater than about 0.833. " }}{PARA 0 "" 0 "" {TEXT -1 75 "This begins to suggest that the radius of conve rge is approximately 0.833. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "evalf[20](fsolve(p(x)-q(x)=0 ,x=0.82..0.84));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5K2_'4gflbL)!#? " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "For a real number " } {XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "arctan* i*xi = i*arctanh*xi;" "6#/*(%'arctanG\"\"\"%\"iGF&%#xiGF&*(F'F&%(arcta nhGF&F(F&" }{TEXT -1 8 ", where " }{TEXT 294 1 "i" }{TEXT -1 24 " is t he imaginary unit. " }}{PARA 0 "" 0 "" {TEXT -1 19 "This follows since " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*i*xi = i* xi-i^3*xi^3/3+i^5*xi^5/5-i^7*xi^7/7+` . . . `;" "6#/*(%'arctanG\"\"\"% \"iGF&%#xiGF&,,*&F'F&F(F&F&*(F'\"\"$F(F,F,!\"\"F-*(F'\"\"&F(F/F/F-F&*( F'\"\"(F(F1F1F-F-%(~.~.~.~GF&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = i*xi+i*xi^3/3+i*xi^5/5+i*xi^7/7+` \+ . . . `" "6#/%!G,,*&%\"iG\"\"\"%#xiGF(F(*(F'F(*$F)\"\"$F(F,!\"\"F(*(F' F(*$F)\"\"&F(F0F-F(*(F'F(*$F)\"\"(F(F3F-F(%(~.~.~.~GF(" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=i*( xi+xi^3/3+ xi^5/5+xi^7/7+` . . . `)" "6#/%!G*&%\"iG\"\"\",,%#xiGF'*&F)\"\"$F+!\" \"F'*&F)\"\"&F.F,F'*&F)\"\"(F0F,F'%(~.~.~.~GF'F'" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = i*arctanh*xi;" " 6#/%!G*(%\"iG\"\"\"%(arctanhGF'%#xiGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 " xi;" "6#%#xiG" }{TEXT -1 28 " is a real number such that " }{XPPEDIT 18 0 "x = xi;" "6#/%\"xG%#xiG" }{TEXT -1 32 " is a solution of the equ ation " }{XPPEDIT 18 0 "x*arctanh*x = 1;" "6#/*(%\"xG\"\"\"%(arctanhG F&F%F&F&" }{TEXT -1 14 ", then taking " }{XPPEDIT 18 0 "zeta = i*xi;" "6#/%%zetaG*&%\"iG\"\"\"%#xiGF'" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "x = zeta;" "6#/%\"xG%%zetaG" }{TEXT -1 15 " is a zero of " } {XPPEDIT 18 0 "1+x*arctan*x;" "6#,&\"\"\"F$*(%\"xGF$%'arctanGF$F&F$F$ " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "This follows since \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+zeta*arctan*zet a = 1+i*xi*arctan*i*xi;" "6#/,&\"\"\"F%*(%%zetaGF%%'arctanGF%F'F%F%,&F %F%*,%\"iGF%%#xiGF%F(F%F+F%F,F%F%" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1+i^2*xi*arctanh*xi;" "6#/%!G,& \"\"\"F&**%\"iG\"\"#%#xiGF&%(arctanhGF&F*F&F&" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1-xi*arctanh*xi" " 6#/%!G,&\"\"\"F&*(%#xiGF&%(arctanhGF&F(F&!\"\"" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "xi = evalf[20](fsolve(x*arctanh(x)=1,x=0.82..0.8 4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#xiG$\"5W)pk4gflbL)!#?" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "If we extend the domain of " } {XPPEDIT 18 0 "f(x) = 1/(1+x*arctan*x);" "6#/-%\"fG6#%\"xG*&\"\"\"F),& F)F)*(F'F)%'arctanGF)F'F)F)!\"\"" }{TEXT -1 29 " to allow complex valu es for " }{TEXT 295 1 "x" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 36 " has a singularity at a distance of " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 44 " units from the origin in the complex plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 183 "If you imagine that the notion of Taylor series can be extended to functions of a complex variable, then the location of thi s singularity might suggest that the radius of convergence " }{TEXT 261 22 "can be no greater than" }{TEXT -1 1 " " }{XPPEDIT 18 0 "xi;" " 6#%#xiG" }{TEXT -1 1 " " }{TEXT 296 1 "~" }{TEXT -1 25 " 0.83355655960 096469844. " }}{PARA 0 "" 0 "" {TEXT -1 62 "The calculations above sug gest that the radius of convergence " }{TEXT 297 1 "R" }{TEXT -1 16 " \+ is this number " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 44 "We can easily get a more accurate value f or " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "xi = evalf [100](evalf[110](fsolve(x*arctanh(x)=1,x=0.82..0.84)));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%#xiG$\"_qRkYQc#>7E*fsB1,Nybo?^Z#oz)*\\.A#z\"RS \"\\$>UAq9;O%)pk4gflbL)!$+\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "We can also obtain the complex zero " } {XPPEDIT 18 0 "zeta = i*xi" "6#/%%zetaG*&%\"iG\"\"\"%#xiGF'" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "1+x*arctan*x;" "6#,&\"\"\"F$*(%\"xGF$%'arc tanGF$F&F$F$" }{TEXT -1 11 " directly. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "zeta = evalf[100](evalf [110](fsolve(1+x*arctan(x),x=0.83*I,complex)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%%zetaG^#$\"_qRkYQc#>7E*fsB1,Nybo?^Z#oz)*\\.A#z\"RS\" \\$>UAq9;O%)pk4gflbL)!$+\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = ln(x+1)/x;" "6#/-%\"fG6#%\"xG *&-%#lnG6#,&F'\"\"\"F-F-F-F'!\"\"" }{TEXT -1 2 ".\004" }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Plot the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 41 " and note the main features of the graph." }} {PARA 0 "" 0 "" {TEXT -1 29 "(b) Find a Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 17 " of degree 9 for " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " about " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "(c) Plot the graphs " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 40 " in the same picture over the interval " }{XPPEDIT 18 0 "[-1, 1];" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 123 "(d) Construct an animation which you can use to give an \+ estimate for the radius of convergence of the Maclaurin series for " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = 1/(x+cos(x)); " "6#/-%\"fG6#%\"xG*&\"\"\"F),&F'F)-%$cosG6#F'F)!\"\"" }{TEXT -1 2 ". \004" }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Plot the graph of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 41 " and note the mai n features of the graph." }}{PARA 0 "" 0 "" {TEXT -1 29 "(b) Find a Ta ylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 17 " of degree 9 for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " about " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 20 "(c) Plot the graphs " }{XPPEDIT 18 0 "f(x )" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p(x)" "6#-%\" pG6#%\"xG" }{TEXT -1 40 " in the same picture over the interval " } {XPPEDIT 18 0 "[-.739, 1];" "6#7$,$-%&FloatG6$\"$R(!\"$!\"\"\"\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 123 "(d) Construct an animat ion which you can use to give an estimate for the radius of convergenc e of the Maclaurin series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "_____________________ _____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "_______________________________ ___" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 31 "(a) Find the \+ Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 25 " of degree 20 centred at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"! " }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = 1/(1+x^2+x* sin(x));" "6#/-%\"fG6#%\"xG*&\"\"\"F),(F)F)*$F'\"\"#F)*&F'F)-%$sinG6#F 'F)F)!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "(b) Plot th e graph of the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#% \"xG" }{TEXT -1 39 " found in (a) along with the graph of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " over the interva l " }{XPPEDIT 18 0 "-1<=x" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``< =1" "6#1%!G\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "(c) Plot a graph to show the absolute error when the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 37 " found in (a) is used to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 18 " on the interval " }{XPPEDIT 18 0 "-1/2 <= x;" "6#1,$*&\"\"\"F &\"\"#!\"\"F(%\"xG" }{XPPEDIT 18 0 "`` <= 1/2;" "6#1%!G*&\"\"\"F&\"\"# !\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "What is the maxi mum absolute error when " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 24 " is used to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 18 " on the interval " }{XPPEDIT 18 0 "-1/2 <= x; " "6#1,$*&\"\"\"F&\"\"#!\"\"F(%\"xG" }{XPPEDIT 18 0 "`` <= 1/2;" "6#1% !G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 3 " ? " }}{PARA 0 "" 0 "" {TEXT -1 31 "(d) Find the Taylor polynomial " }{XPPEDIT 18 0 "q(x);" "6#-%\"qG6 #%\"xG" }{TEXT -1 25 " of degree 24 centred at " }{XPPEDIT 18 0 "x = 0 ;" "6#/%\"xG\"\"!" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f (x) = 1/(1+x^2+x*sin(x));" "6#/-%\"fG6#%\"xG*&\"\"\"F),(F)F)*$F'\"\"#F )*&F'F)-%$sinG6#F'F)F)!\"\"" }{TEXT -1 49 ", and find a numerical solu tion for the equation " }{XPPEDIT 18 0 "p(x)=q(x)" "6#/-%\"pG6#%\"xG-% \"qG6#F'" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "0 " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 31 "(a) Find the Taylor polynomial " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 25 " of degree 20 cen tred at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 18 " for t he function " }{XPPEDIT 18 0 "f(x) = 2/(x^2+2*cosh(x));" "6#/-%\"fG6#% \"xG*&\"\"#\"\"\",&*$F'F)F**&F)F*-%%coshG6#F'F*F*!\"\"" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "(b) Plot the graph of the Taylor poly nomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 39 " found \+ in (a) along with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\" xG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "-2<=x" "6#1,$\" \"#!\"\"%\"xG" }{XPPEDIT 18 0 "``<=2" "6#1%!G\"\"#" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 71 "(c) Plot a graph to show the absolute er ror when the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"x G" }{TEXT -1 37 " found in (a) is used to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "-1/2 <= x;" "6#1,$*&\"\"\"F&\"\"#!\"\"F(%\"xG" }{XPPEDIT 18 0 "` ` <= 1/2;" "6#1%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "What is the maximum absolute error when " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 24 " is used to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "-1/2 <= x;" "6#1,$*&\"\"\"F&\"\"#!\"\"F(%\"xG" } {XPPEDIT 18 0 "`` <= 1/2;" "6#1%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 "? " }}{PARA 0 "" 0 "" {TEXT -1 31 "(d) Find the Taylor polynomial " } {XPPEDIT 18 0 "q(x);" "6#-%\"qG6#%\"xG" }{TEXT -1 25 " of degree 24 ce ntred at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 18 " for \+ the function " }{XPPEDIT 18 0 "f(x) = 2/(x^2+2*cosh(x));" "6#/-%\"fG6# %\"xG*&\"\"#\"\"\",&*$F'F)F**&F)F*-%%coshG6#F'F*F*!\"\"" }{TEXT -1 49 ", and find a numerical solution for the equation " }{XPPEDIT 18 0 "p( x)=q(x)" "6#/-%\"pG6#%\"xG-%\"qG6#F'" }{TEXT -1 17 " in the interval \+ " }{XPPEDIT 18 0 "0 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }