{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {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 }{PSTYLE "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "Module 10 : Series" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 18 "1005 Taylor Series " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 17 "O B J E C T I V E" }}{PARA 0 "" 0 "" {TEXT -1 462 "Every function which i s infinitely differentiable can be expressed as a Taylor series. In th is module, we will examine the definition and create Taylor series fro m scratch and automatically.We will also investigate convergence, how \+ the number of terms effect the accuracy of the approximation, Taylor p olynomials, and Taylor's Remainder Theorem. We will also create an ani mation to demonstrate how the increasing terms of a Taylors series eff ect the convergence." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 83 "_______________________________________________________ ____________________________" }}{PARA 4 "" 0 "" {TEXT -1 32 "A. Taylor Series and Polynomials" }}{PARA 0 "" 0 "" {TEXT -1 83 "______________ _____________________________________________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "The \+ Taylor Series for a function near a value x = a, is a power series in \+ ( x-a). We will learn how to find Taylor coefficients, how to expand a function iinto a Taylor series using the formula for Taylor series, a nd how to do it automatically in Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 26 "TAYLOR SERIES FROM SCRATCH" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 184 "The coef ficients of a Taylor series expanded about x = a, are f(n) (a)/n!, and each term is of the from f(n) (a)(x-a)^n/n! We can express the coeffi cient in the following way in Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "(D@@n)(f)(a)/n!;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&---%#@@G6$%\"DG%\"nG6#%\"fG6#%\"aG\"\"\"-%*factorialG 6#F*!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "In general, a Taylor series looks like this - except with an in finite number of terms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sum( (( D@@n)(f)(a) * (x-a)^n)/n!, \+ k = 0..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&---%#@@G6$%\"DG%\" nG6#%\"fG6#%\"aG\"\"\"),&%\"xGF1F0!\"\"F,F1F1-%*factorialG6#F,F5\"\"' " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Here is a more concrete example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := x -> 1/(1- 2*x); a:= 3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%& arrowGF(*&\"\"\"F-,&F-F-*&\"\"#F-9$F-!\"\"F2F(F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"aG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "(D@@10)(f)(a)/10!;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!%C5\")D \"G)[" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sum( (( D@@k)(f)(a ) * (x-a)^k )/k!, k=0..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.#!#6\" #D\"\"\"*&#\"\"#F&F'%\"xGF'F'*&#\"\"%\"$D\"F'*$),&F+F'\"\"$!\"\"F*F'F' F4*&#\"\")\"$D'F')F2F3F'F'*&#\"#;\"%DJF'*$)F2F.F'F'F4*&#\"#K\"&Dc\"F') F2\"\"&F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 130 "This is coefficient for the(x-3)^10 term, and the sum of the f irst six terms of the Taylor series for this function at this value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 40 "AUTOMATICALLY GENERATING A TAYLOR SERIE S" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 402 "As \+ we have sseen in previous modules, we can use the series command to ge nerate a series for a function. By specifying a particular value of x, the series is expanded about that value. When we saw this before, we were using geometric series possibly along with differentiation and i ntegration to find the series. It turns out that these series expansio ns are identical to the Taylor series expansions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "series( 1/(x ^2 - 3*x +2 ), x = 0,20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+M%\"xG# \"\"\"\"\"#\"\"!#\"\"$\"\"%F&#\"\"(\"\")F'#\"#:\"#;F*#\"#J\"#KF+#\"#j \"#k\"\"&#\"$F\"\"$G\"\"\"'#\"$b#\"$c#F-#\"$6&\"$7&F.#\"%B5\"%C5\"\"*# \"%Z?\"%[?\"#5#\"%&4%\"%'4%\"#6#\"%\">)\"%#>)\"#7#\"&$Q;\"&%Q;\"#8#\"& nF$\"&oF$\"#9#\"&Nb'\"&Ob'F0#\"'r58\"'s58F1#\"'V@E\"'W@E\"#<#\"'(GC&\" ')GC&\"#=#\"(v&[5\"(w&[5\"#>-%\"OG6#F&\"#?" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "series( 1/(x^2 - 3*x +2 ), x = 1,20);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#+M,&%\"xG\"\"\"F&!\"\"F'F'F'\"\"!F'F&F'\"\"#F'\" \"$F'\"\"%F'\"\"&F'\"\"'F'\"\"(F'\"\")F'\"\"*F'\"#5F'\"#6F'\"#7F'\"#8F '\"#9F'\"#:F'\"#;F'\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "series( 1/(x^2 - 3*x +2 ), x = 3,20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+M,&%\"xG\"\"\"\"\"$!\"\"#F&\"\"#\"\"!#!\"$\"\" %F&#\"\"(\"\")F*#!#:\"#;F'#\"#J\"#KF.#!#j\"#k\"\"&#\"$F\"\"$G\"\"\"'#! $b#\"$c#F0#\"$6&\"$7&F1#!%B5\"%C5\"\"*#\"%Z?\"%[?\"#5#!%&4%\"%'4%\"#6# \"%\">)\"%#>)\"#7#!&$Q;\"&%Q;\"#8#\"&nF$\"&oF$\"#9#!&Nb'\"&Ob'\"#:#\"' r58\"'s58F4#!'V@E\"'W@E\"#<#\"'(GC&\"')GC&\"#=#!(v&[5\"(w&[5\"#>-%\"OG 6#F&\"#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Note the series is quite different depending on different expansio n values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 28 "CONVERT TO TAYLOR POLYNOMIAL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 462 "A Taylor series is an infinite series. \+ Similarly, the n th degree Taylor polynomial is a polynomial of degree n which coincides with the first (n+1) terms of the Taylor series. Tn (x) = c0 + c1(x-a) + c2(x-a)^2 + ... c n(x-a) n . The series created \+ by the Maple command SERIES have an error terms in \"big Oh \" notatio n. In various situations it is useful to be able to convert the series with error terms to a Taylor polynomial that can be evaluated and gra phed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f := x -> 1/(x+3);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&9$F-\"\"$F-!\"\" F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "series( f(x), x, \+ 7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG#\"\"\"\"\"$\"\"!#!\"\" \"\"*F&#F&\"#F\"\"##F*\"#\")F'#F&\"$V#\"\"%#F*\"$H(\"\"&#F&\"%(=#\"\"' -%\"OG6#F&\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "convert( %, polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0#\"\"\"\"\"$F%*&#F% \"\"*F%%\"xGF%!\"\"*&#F%\"#FF%)F*\"\"#F%F%*&#F%\"#\")F%*$)F*F&F%F%F+*& #F%\"$V#F%)F*\"\"%F%F%*&#F%\"$H(F%*$)F*\"\"&F%F%F+*&#F%\"%(=#F%)F*\"\" 'F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g := unapply( %, x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%& arrowGF(,0#\"\"\"\"\"$F.*&#F.\"\"*F.9$F.!\"\"*&#F.\"#FF.)F3\"\"#F.F.*& #F.\"#\")F.*$)F3F/F.F.F4*&#F.\"$V#F.)F3\"\"%F.F.*&#F.\"$H(F.*$)F3\"\"& F.F.F4*&#F.\"%(=#F.)F3\"\"'F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$Z&\"%(=# " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 242 "Her e we define a function f(x), expand it into a series with 7 terms, con vert it to polynomial, then use the UNAPPLY command to change the poly nomial expression into an actual function, g(x). Finally we evaluate t he Taylor polynomial at x =1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________________________ ____________________________________" }}{PARA 4 "" 0 "" {TEXT -1 23 "B . Animated Convergence" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________ ____________________________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 242 "The Taylor Polynomials gradually converg e to the Taylor Serires which is a representation of the original func tion in some interval of convergence. In this section, we'll see with \+ our own eyes how this convergence takes place in an animation." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "First we define a function and a generic Taylor polynomial. Then we define som e constant so that our graph desplays for a =< x =< b, and c = " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "a := -6: b := 6: c := -3: d:= 4:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := x -> sin(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$sinG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "g := (x,k) -> convert(series(f(x), x, k), polyno m):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "on := x -> piecewise ( x <0, 0, x < 1, 1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#onGR6#% \"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6'29$\"\"!F12F0\"\"\"F3F3F( F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 523 "display( plot( f(x ), x = a..b, y = c..d, thickness = 3, color = blue), animate( on(t-1)* g(x,2) , x = a..b, t = 0..7, view = c..d, color = cyan), animate( on(t -2)*g(x,4) , x = a..b, t = 0..7, view = c..d, color = coral),animate( \+ on(t-3)*g(x,6) , x = a..b, t = 0..7, view = c..d, color = green), ani mate( on(t-4)*g(x,8) , x = a..b, t = 0..7, view = c..d, color = violet ), animate( on(t-5)*g(x,10) , x = a..b, t = 0..7, view = c..d, color = red), animate( on(t-6)*g(x,12) , x = a..b, t = 0..7, view = c..d, col or = coral) ); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%(ANIMATEG627)-%'CURVESG6$7S7$$!\"'\"\"!$F.F.7$$!++++]d!\"*F/7$$!+ ++++bF3F/7$$!++++]_F3F/7$$!+++++]F3F/7$$!++++]ZF3F/7$$!+++++XF3F/7$$!+ +++]UF3F/7$$!+++++SF3F/7$$!++++]PF3F/7$$!+++++NF3F/7$$!++++]KF3F/7$$!+ ++++IF3F/7$$!++++]FF3F/7$$!+++++DF3F/7$$!++++]AF3F/7$$!+++++?F3F/7$$!+ +++])\\:%z#!#=7$$!3!******\\2<#pe!#<$\"3Bo'[PGbC-%F [w7$$!3z******\\TVQdF_w$\"3]i#*o@G0#=&F[w7$$!3s***\\iiSYi&F_w$\"32H)GM 7l'>hF[w7$$!3l****\\-r%3^&F_w$\"3UN>Ii&>\"ypF[w7$$!3A+++l;!\\D&F_w$\"3 c-L?N69k&)F[w7$$!3^******f13E^F_w$\"3w%eduQIk:*F[w7$$!3o*****\\lfs*\\F _w$\"3M*\\wU0!)pf*F[w7$$!3G**\\P%GbJ$\\F_w$\"3uY)=eK*Hd(*F[w7$$!3w*** 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$\"3@!o![Y)4ZT)F[w7$F^q$\"3gM2ckh%)*[*F[w7$Faq$\"35`8#ob\\\\(**F[w7$Fd q$\"3'>bNO=d)R)*F[w7$Fgq$\"3wE!G'f8'H4*F[w7$Fjq$\"3a7a*=es1y(F[w7$F]r$ \"3q*G8R5*[%)fF[w7$F`r$\"3UD\\hQF\"e\"QF[w7$Fcr$\"3_S%e:%fu39F[w7$Ffr$ !3=pzF$G[))3\"F[w7$Fir$!37&=\"RafwDNF[w7$F\\s$!3_vM&oZ@#fdF[w7$F_s$!3k &>7Z`X!owF[w7$Fcs$!3/eR(pn2y;*F[w7$Ffs$!3\"oln)><*G-\"F_w7$Fis$!3?j)RR n`**3\"F_w7$F\\t$!3dz=)*)H " 0 "" {MPLTEXT 1 0 22 "resta rt; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name ch angecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1330 "tay_plot := proc( F, c1, c2, k1, k2, a, b, eps)\nlocal x1, x 2, y1, y2, delta, dely, m, M, A, B, i, n, g1,g2;\nn := 100: delta := \+ (b-a)/n: x2 := a:\nM := maximize( F(x), \{x\}, \{x=a..b\}): m := min imize( F(x), \{x\},\{x=a..b\}):\ng1 := unapply( convert(series( F(x), \+ x=c1, k1) , polynom), x);\ng2 := unapply( convert(series( F(x), x= c2, k2) , polynom), x); for i from 1 to n do\nx1 := x2: x2:=x1 + delta: \ny1 := evalf( F(x1)): y2 := evalf( F(x2)):\ndely := abs( evalf( F(x1 )- g2(x1)));\nif( dely < eps )\nthen B||i := polygonplot([[x1, M],[x1, y1],[x2,y2],[x2,M]], color = maroon, style = patchnogrid):\nelse B||i \+ := polygonplot([[x1, M],[x1,y1],[x2,y2],[x2,M]], color = khaki, style \+ = patchnogrid): fi:\ndely := abs( evalf( F(x1) - g1(x1)));\nif (dely < eps)\nthen A||i := polygonplot([[x1, M],[x1,y1],[x2,y2],[x2,m]], co lor = violet, style = patchnogrid):\nelse B||i := polygonplot([[x1, M] ,[x1,y1],[x2,y2],[x2,m]], color = khaki, style = patchnogrid): fi:\n od:\ndisplay([ plot( F(x), x = a..b, thickness = 4, color = red),\nplo t( g1(x), x = a..b, y = m..M, thickness = 2, color= blue),\nplot( g2(x ), x = a..b, y = m..M, thickness = 2, color= green),\nseq( A||i, i=1.. n), seq( B||i, i=1..n),\nplot( [[c1,m],[c1,F(c1)]], x = a..b, color= b lue, linestyle = 3),\nplot( [[c2,F(c2)],[c2,M]], x = a..b, color= gree n, linestyle = 3)],\naxes = framed );\nend:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 ".....Here is another form.....( delete on or the two)......" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}{PARA 7 " " 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1071 "tay_plot := proc( F , c1, c2, k1, k2, a, b, eps)\nlocal x1, x2, y1, y2, delta, m, M, A, B, i, n, g1,g2, CL, png;\nn := 100: delta := (b-a)/n: x2 := a: png := 'patchnogrid';\nM := maximize( F(x), x=a..b): m := minimize( F(x), x =a..b):\ng1 := unapply( convert(series( F(x), x=c1, k1) , polynom), x) ;\ng2 := unapply( convert(series( F(x), x= c2, k2) , polynom), x); for i from 1 to n do\nx1 := x2: x2:= x1 + delta: y1 := evalf( F(x1)): \+ y2 := evalf( F(x2)):\nif( abs( evalf( F(x1) - g2(x1))) < eps)\nthen CL := maroon; else CL := khaki; fi:\nA||i := polygonplot( [[x1,M],[x1, y1],[x2,y2],[x2,M]], color = CL, style = png):\nif( abs( evalf( F(x1) \+ - g1(x1))) < eps)\nthen CL := navy ; else CL := khaki; fi;\nB||i := \+ polygonplot( [[x1,m],[x1,y1],[x2,y2],[x2,m]], color = CL, style = png) :\nod:\ndisplay([ plot( [F(x), g1(x), g2(x)], x = a..b, y = m..M, thic kness = [4,2,2], color = [red, coral, green] ),\n plot( [[c 1,m],[c1,F(c1)],[c2,f(c2)],[c2,M] ],x = a..b, thickness = 3, color = [ coral, green] ),\nseq( A||i, i=1..n), seq( B||i, i = 1..n) ], axes \+ = framed ); \nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f : = x -> ln(1+x); c1 := 3; c2 := 3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%#lnG6#,&\"\"\"F09$F0F(F(F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1G\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(f(11));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]m!\\[#!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "tay_plot( f, c1, c2, 3, 6 , -0.6, 10, .06);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6[x-%'CURVESG6%7U7$$!3w**************f!#=$!3'*\\:u=t!H;*F*7$$!3Lmm; 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