{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 260 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 262 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 262 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Brown E mphasis" -1 281 "Times" 1 12 128 42 42 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Navy Emphasis" -1 282 "Times" 1 12 35 35 142 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{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 0 128 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 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 "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 "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 } {PSTYLE "Normal" -1 256 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 "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT 259 40 "Comparison of polynomial approxi mations " }}{PARA 0 "" 0 "" {TEXT -1 65 "by Peter Stone, Dept. of Appl ied and Environmental Sciences, RMIT" }}{PARA 0 "" 0 "" {TEXT -1 61 "p eter.stone@rmit.edu.au . . or . . peterstone@optusnet.com.au" }}{PARA 0 "" 0 "" {TEXT -1 56 "This worksheet is based on a Maple worksheet wr itten by " }}{PARA 0 "" 0 "" {TEXT -1 77 "Jim Herod, School of Mathema tics, Georgia Tech. (Retired) jherod@mail.tds.net" }}{PARA 0 "" 0 "" {URLLINK 17 "http://www.mapleapps.com/categories/mathematics/calculus/ html/polyApprox.html" 4 "http://www.mapleapps.com/categories/mathemati cs/calculus/html/polyApprox.html" "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 9.10.2003" }} {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 12 "Introduction" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 16 "Given a funtion \+ " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 16 " on an interva l " }{XPPEDIT 18 0 "[a, b]" "6#7$%\"aG%\"bG" }{TEXT -1 61 ", we are in terested in making a polynomial approximation for " }{XPPEDIT 18 0 "g( x);" "6#-%\"gG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 " We are faced with two questions: " }}{PARA 15 "" 0 "" {TEXT -1 52 "How large do we choose the degree of the polynomial." }}{PARA 15 "" 0 "" {TEXT -1 55 "How do we choose the coefficients for the polynomial. \+ \004" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 " Three choices of classical polynomials are considered in this workshee t.. If the function is sufficiently " }{TEXT 260 14 "differentiable" } {TEXT -1 28 ", we can try the well known " }{TEXT 260 18 "Taylor polyn omials" }{TEXT -1 123 ". The Taylor polynomials would likely be the fi rst choice for most students to make in getting a polynomial approxima tion. " }}{PARA 0 "" 0 "" {TEXT -1 35 "\nAlternatively, if the functio n is " }{TEXT 260 10 "integrable" }{TEXT -1 192 ", we can choose the c oefficients so that area between the graphs of function and the polyn omial approximation is as small as possible. This scheme will give ris e to an approximation with the " }{TEXT 260 20 "Legendre polynomials" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "A less familiar sequence of polynomials which can be used to approximate a continuous function " }{XPPEDIT 18 0 "g(x);" "6#-%\" gG6#%\"xG" }{TEXT -1 40 ", uniformly over a closed interval, are " } {TEXT 260 21 "Bernstein polynomials" }{TEXT -1 56 ". \nThe Bernstein p olynomials are constructed as follows:" }}{PARA 0 "" 0 "" {TEXT -1 4 " For " }{TEXT 266 1 "x" }{TEXT -1 12 " in [0, 1], " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "B(n,x) = Sum(g(k/n)*matrix([[n], [k]] )*x^k*(1-x)^(n-k),k = 0 .. n);" "6#/-%\"BG6$%\"nG%\"xG-%$SumG6$**-%\"g G6#*&%\"kG\"\"\"F'!\"\"F2-%'matrixG6#7$7#F'7#F1F2)F(F1F2),&F2F2F(F3,&F 'F2F1F3F2/F1;\"\"!F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "w here " }{XPPEDIT 18 0 "matrix([[n], [k]])= n!/ (k!*(n-k!))" "6#/-%'mat rixG6#7$7#%\"nG7#%\"kG*&-%*factorialG6#F)\"\"\"*&-F.6#F+F0,&F)F0-F.6#F +!\"\"F0F7" }{TEXT -1 30 " is the binomial coefficient. " }}{PARA 0 " " 0 "" {TEXT -1 47 "Note that no differentiabiliby is required for " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 45 ", and no integrab ility is used. The function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 66 " is simply evaluated at an evenly spaced grid across the interval." }}{PARA 0 "" 0 "" {TEXT -1 89 "Bernstein polynomials are h ardly known by undergraduate science and engineering students." }} {PARA 0 "" 0 "" {TEXT -1 162 "This worksheet suggest a good reason: wh ile the technique for making the approximations with Bernstein polynom ials is simple, the rate of convergence can be slow." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 206 "In this worksheet, we \+ provide an example for which the the function is infinitely differenti able at zero, the Taylor series converges, and yet the Taylor series f ails to make the expected approximation for " }{XPPEDIT 18 0 "g(x);" " 6#-%\"gG6#%\"xG" }{TEXT -1 191 ". For this example the Legendre polyno mials and the Bernstein polynomials can be used to make an approximati on, but the first of these makes an approximation with a smaller degre e polynomial." }}{PARA 0 "" 0 "" {TEXT -1 35 "We agree that a series o f functions" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "S = S um(u[n](x),n);" "6#/%\"SG-%$SumG6$-&%\"uG6#%\"nG6#%\"xGF," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "represents a function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 51 " on a certain set A if and only if for every point " }{TEXT 267 1 "x" }{TEXT -1 67 " of the set A, the series S converges to the value of the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 14 " at that point" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Taylor Polynomials " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 348 "Immediately after the question of convergence of the Tay lor series of a function comes the question, \"Does the Taylor series \+ of a given function represent the function throughout the interval of \+ convergence?\" The clue to the answer lies in Taylor's formula with a \+ remainder. To clarify the situation we recall the situation for the Ta ylor's series." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n](x)=g(a)+g*`'`(a)*(x-a)+g*`''`(a) /2!*(x-a)^2+ `@@`(g,3)*``(a)/3!*(x-a)^3+` . . . `+`@@`(g,n)*``(a)/n!*( x-a)^n" "6#/-&%\"SG6#%\"nG6#%\"xG,.-%\"gG6#%\"aG\"\"\"*(F-F0-%\"'G6#F/ F0,&F*F0F/!\"\"F0F0**F-F0-%#''G6#F/F0-%*factorialG6#\"\"#F6,&F*F0F/F6F >F0**-%#@@G6$F-\"\"$F0-%!G6#F/F0-F<6#FDF6,&F*F0F/F6FDF0%(~.~.~.~GF0**- FB6$F-F(F0-FF6#F/F0-F<6#F(F6),&F*F0F/F6F(F0F0" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 7 "be the " }{TEXT 268 1 "n" }{TEXT -1 36 " t h Taylor polynomial of a function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#% \"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = S[n](x) + R[n](x)" " 6#/-%\"fG6#%\"xG,&-&%\"SG6#%\"nG6#F'\"\"\"-&%\"RG6#F-6#F'F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "R[n] (a)=`@@`(g,n+1)*``(c)/n" "6#/-&%\"RG6#%\"nG6#%\"aG*(-%#@@G6$%\"gG,&F( \"\"\"F1F1F1-%!G6#%\"cGF1F(!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 21 "for some real number " }{TEXT 269 1 "c" }{TEXT -1 9 " bet ween " }{TEXT 270 1 "a" }{TEXT -1 5 " and " }{TEXT 271 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 83 "This form of the remainder is \+ obtained by an application of the Mean Value Theorem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Suppose that we have a n interval containing 0 on which " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#% \"xG" }{TEXT -1 38 " and all of its derivatives exist. If " }{TEXT 272 1 "x" }{TEXT -1 71 " is a point of the interval, then the limit of the polynomial sequence " }{XPPEDIT 18 0 "S[n](x)" "6#-&%\"SG6#%\"nG6 #%\"xG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" } {TEXT -1 294 " if and only if the remainder has limit zero. Techniques for showing the limit of the remainder is zero vary with the function concerned. To make any general statement specifying conditions under \+ which a given function is represented as a limit of the Taylor polynom ials is extremely difficult." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" } {TEXT 263 7 "Example" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "We consider the following function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 51 ", where the Tayl or series about zero determined by " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6# %\"xG" }{TEXT -1 22 " does not converge to " }{XPPEDIT 18 0 "g(x)" "6# -%\"gG6#%\"xG" }{TEXT -1 14 ", even though " }{XPPEDIT 18 0 "g(x)" "6# -%\"gG6#%\"xG" }{TEXT -1 94 " is infinitely differentiable on any inte rval containing zero. Here is the definition of g(x)." }}{PARA 256 "" 0 "" {TEXT -1 2 "\n " }{XPPEDIT 18 0 "g(x) = PIECEWISE([0, x <= 0],[ex p(-1/(x^2)), 0 < x]);" "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$\"\"!1F'F,7$- %$expG6#,$*&\"\"\"F4*$F'\"\"#!\"\"F72F,F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "We draw the graph of " }{XPPEDIT 18 0 "g(x)" "6#- %\"gG6#%\"xG" }{TEXT -1 4 " on " }{XPPEDIT 18 0 "[-1, 1]" "6#7$,$\"\" \"!\"\"F%" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "g := x -> piecewise(x<=0,0,exp(-1/x^2)):\n'g(x)'=g(x); \nplot(g(x) ,x=-1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%*PIECEWI SEG6$7$\"\"!1F'F,7$-%$expG6#,$*&\"\"\"F4*$)F'\"\"#F4!\"\"F8%*otherwise G" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6 $7X7$$!\"\"\"\"!$F*F*7$$!3ommm;p0k&*!#=F+7$$!3wKL$3s%HaF/F+7$$!3Q+++]$*4)*\\F/F+7$$!39+++]_&\\c%F /F+7$$!31+++]1aZTF/F+7$$!3umm;/#)[oPF/F+7$$!3hLLL$=exJ$F/F+7$$!3*RLLLt If$HF/F+7$$!3]++]PYx\"\\#F/F+7$$!3EMLLL7i)4#F/F+7$$!3c****\\P'psm\"F/F +7$$!3')****\\74_c7F/F+7$$!3)3LLL3x%z#)!#>F+7$$!3KMLL3s$QM%FioF+7$$!3] ^omm;zr)*!#@F+7$$\"3%pJL$ezw5VFio$\"3#yRkN8_`&>!$^#7$$\"3s*)***\\PQ#\\ \")Fio$\"3N8S_Kf=>S!#$)7$$\"3GKLLe\"*[H7F/$\"3#4W$4K=Hi=!#Y7$$\"3I**** ***pvxl\"F/$\"3W%*oq)fv[d\"!#L7$$\"3#z****\\_qn2#F/$\"3DC?m'4m9_)!#G7$ $\"3U)***\\i&p@[#F/$\"3S 3%*))!#B7$$\"3ElmmmZvOLF/$\"3-7>jR^2d7F`p7$$\"3i******\\2goPF/$\"3w%[F \"zJu^()F`p7$$\"3-mm\"H2fU'RF/$\"3df0vqA$Qs\"!#?7$$\"3UKL$eR<*fTF/$\"3 98blV`r#4$Fjs7$$\"3/mm\"HiBQP%F/$\"3G(QwJe'*zO&Fjs7$$\"3m******\\)Hxe% F/$\"3A+tT\\MAT')Fjs7$$\"3ckm;H!o-*\\F/$\"3$\\$oeHI=.=Fio7$$\"3y)***\\ 7k.6aF/$\"3Y%y'p^,R'G$Fio7$$\"3#emmmT9C#eF/$\"3?MoGqZ(\\B&Fio7$$\"33** **\\i!*3`iF/$\"3sZcJ:n/]xFio7$$\"3%QLLL$*zym'F/$\"3kz`hNb&[0\"F/7$$\"3 wKLL3N1#4(F/$\"3A#34QA]%p8F/7$$\"3Nmm;HYt7vF/$\"31!G#pI:M+#F/7$$\"3]mmmT6KU$)F/$\"3!>>f2_OmP#F/7$$\"3 a****\\P$[/a)F/$\"3,l@WO,_QDF/7$$\"3fKLLLbdQ()F/$\"3C " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Is it not clear that the function " }{XPPEDIT 18 0 "g(x) " "6#-%\"gG6#%\"xG" }{TEXT -1 32 " has derivatives of all orders? " }} {PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "x<=0" "6#1%\"xG\"\"! " }{TEXT -1 18 ", all derivatives " }{XPPEDIT 18 0 "`@@`(g,n)*``(x)" " 6#*&-%#@@G6$%\"gG%\"nG\"\"\"-%!G6#%\"xGF)" }{TEXT -1 11 " are zero. " }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "x>0" "6#2\"\"!%\"x G" }{TEXT -1 107 ", the derivative of any order can be found by succes sive applications of standard differentiation rules to " }{XPPEDIT 18 0 "exp(-1/x^2)" "6#-%$expG6#,$*&\"\"\"F(*$%\"xG\"\"#!\"\"F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 25 "\nWe establish below that " } {XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 4 " is " }{TEXT 260 33 "infinitely differentiable at zero" }{TEXT -1 11 ", and that " } {TEXT 260 32 "all derivatives at zero are zero" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 49 "With this result the Taylor series determ ined by " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 20 " about zero would be" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "0 + 0*`.`*x+ 0*`.`*x^2 + 0*`.`*x^3 + ` . . . `" "6#,,\"\"!\"\"\"*(F$F%%\" .GF%%\"xGF%F%*(F$F%F'F%F(\"\"#F%*(F$F%F'F%F(\"\"$F%%(~.~.~.~GF%" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 79 "This series represents t he function identically 0 everywhere, but the function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 62 " is not identically zero on a ny open interval containing zero." }}{PARA 0 "" 0 "" {TEXT -1 95 "\nTo ward making a proof that all the derivatives are zero at zero, we esta blish the following:\n\240" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 273 1 "n" }{TEXT -1 29 " is a positive integer then " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Limit(g(x)/(x^n),x = 0,right) = 0 ;" "6#/-%&LimitG6%*&-%\"gG6#%\"xG\"\"\")F+%\"nG!\"\"/F+\"\"!%&rightGF1 " }{TEXT -1 14 " ------- (i). " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 264 14 "______________" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Justific ation of (i) " }}{PARA 0 "" 0 "" {TEXT -1 37 "Here is Maple's verific ation of (i).\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Limit(exp (-1/x^2)/x^n,x=0,right);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%&LimitG6%*&-%$expG6#,$*&\"\"\"F,*$)%\"xG\"\"#F,!\"\"F1F,)F/%\"nGF1/ F/\"\"!%&rightG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "To make a formal proof we use l'Hospita l's rule." }}{PARA 0 "" 0 "" {TEXT -1 6 "Write:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-1/(x^2))/(x^n)=x^(-n)/exp(1/(x^2)) " "6#/*&-%$expG6#,$*&\"\"\"F**$%\"xG\"\"#!\"\"F.F*)F,%\"nGF.*&)F,,$F0F .F*-F&6#*&F*F**$F,F-F.F." }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Then . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 219 "xi := x -> x^(-n):\n'xi(x)'=xi(x);\neta := x -> ex p(1/x^2);\n'eta'(x)=eta(x);\nDiff('xi(x)',x)=diff(xi(x),x);\nnm := rhs (%):\nDiff('eta(x)',x)=diff(eta(x),x);\ndn := rhs(%):\nnm/dn;\nDiff('x i(x)',x)/Diff('eta(x)',x)=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#xiG6#%\"xG)F',$%\"nG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$etaGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#*&\"\"\"F0*$)9$\" \"#F0!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$etaG6#%\"xG-%$ expG6#*&\"\"\"F,*$)F'\"\"#F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%%DiffG6$-%#xiG6#%\"xGF*,$*()F*,$%\"nG!\"\"\"\"\"F/F1F*F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%$etaG6#%\"xGF*,$*(\"\"#\"\"\"F* !\"$-%$expG6#*&F.F.*$)F*F-F.!\"\"F.F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&**)%\"xG,$%\"nG!\"\"F&F,F&F*F'-%$expG6#*&F&F&*$)F *F'F&F-F-F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%%DiffG6$-%#xiG6# %\"xGF+\"\"\"-F&6$-%$etaGF*F+!\"\",$*&#F,\"\"#F,*()F+,&%\"nGF1F5F,F,F9 F,-%$expG6#,$*&F,F,*$)F+F5F,F1F1F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Note that the last expression is " }{XPPEDIT 18 0 "n/2" " 6#*&%\"nG\"\"\"\"\"#!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "``(g(x)/(x ^(n-2)));" "6#-%!G6#*&-%\"gG6#%\"xG\"\"\")F*,&%\"nGF+\"\"#!\"\"F0" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "From l'Hospital's rule we have " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Limit(g (x)/(x^n),x = 0,right) = n/2;" "6#/-%&LimitG6%*&-%\"gG6#%\"xG\"\"\")F+ %\"nG!\"\"/F+\"\"!%&rightG*&F.F,\"\"#F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(g(x)/(x^(n-2)),x = 0,right);" "6#-%&LimitG6%*&-%\"gG6#%\"xG\" \"\")F*,&%\"nGF+\"\"#!\"\"F0/F*\"\"!%&rightG" }{TEXT -1 15 " ------- ( ii). " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 265 20 "_____________ _______" }{TEXT -1 16 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "n=1" "6#/%\"nG\"\"\"" }{TEXT -1 14 ", (ii) give s " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(g(x)/x,x = 0,right) = 1/2;" "6#/-%&LimitG6%*&-%\"gG6#%\"xG\"\"\"F+!\"\"/F+\"\" !%&rightG*&F,F,\"\"#F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(x*exp(-1 /(x^2)),x = 0,right) = 0;" "6#/-%&LimitG6%*&%\"xG\"\"\"-%$expG6#,$*&F) F)*$F(\"\"#!\"\"F1F)/F(\"\"!%&rightGF3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "n=2" "6#/%\"nG\"\"#" }{TEXT -1 13 ", (ii) gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Limit(g(x)/(x^2),x = 0) = Limit(exp(-1/(x^2)),x = 0);" "6#/-%&LimitG6$ *&-%\"gG6#%\"xG\"\"\"*$F+\"\"#!\"\"/F+\"\"!-F%6$-%$expG6#,$*&F,F,*$F+F .F/F//F+F1" }{TEXT -1 5 " = 0." }}{PARA 0 "" 0 "" {TEXT -1 4 "For " } {XPPEDIT 18 0 "n = 3;" "6#/%\"nG\"\"$" }{TEXT -1 13 ", (ii) gives " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(g(x)/(x^3),x = \+ 0,right) = 3/2;" "6#/-%&LimitG6%*&-%\"gG6#%\"xG\"\"\"*$F+\"\"$!\"\"/F+ \"\"!%&rightG*&F.F,\"\"#F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(g(x) /x,x = 0,right) = 0;" "6#/-%&LimitG6%*&-%\"gG6#%\"xG\"\"\"F+!\"\"/F+\" \"!%&rightGF/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "For " } {XPPEDIT 18 0 "n = 4;" "6#/%\"nG\"\"%" }{TEXT -1 13 ", (ii) gives " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(g(x)/(x^4),x = \+ 0,right) = 4/2;" "6#/-%&LimitG6%*&-%\"gG6#%\"xG\"\"\"*$F+\"\"%!\"\"/F+ \"\"!%&rightG*&F.F,\"\"#F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(g(x) /(x^2),x = 0,right) = 0;" "6#/-%&LimitG6%*&-%\"gG6#%\"xG\"\"\"*$F+\"\" #!\"\"/F+\"\"!%&rightGF1" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 10 "and so on." }}{PARA 0 "" 0 "" {TEXT -1 37 "The formula (i) follows by induction." }}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We n ow see how (i) helps to show that all the derivatives of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 18 " at zero are zero." }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 262 14 "1st derivative" } {TEXT -1 4 " of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 12 " at zero is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L imit(g(x)/x,x = 0);" "6#-%&LimitG6$*&-%\"gG6#%\"xG\"\"\"F*!\"\"/F*\"\" !" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 18 "which is 0 by (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 262 14 "2nd derivative" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "g(x) " "6#-%\"gG6#%\"xG" }{TEXT -1 12 " at zero is " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Limit(g*`'`(x)/x,x = 0);" "6#-%&LimitG 6$*(%\"gG\"\"\"-%\"'G6#%\"xGF(F,!\"\"/F,\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "We can ob tain a formula for " }{XPPEDIT 18 0 "g*`'`(x)" "6#*&%\"gG\"\"\"-%\"'G6 #%\"xGF%" }{TEXT -1 7 ", when " }{TEXT 274 1 "x" }{TEXT -1 95 " is pos itive, by applying the standard differentiation rules. We may as well \+ let Maple do this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 51 "g := x -> exp(-1/x^2);\nDiff('g(x)',x)=diff( g(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)oper atorG%&arrowGF(-%$expG6#,$*&\"\"\"F1*$)9$\"\"#F1!\"\"F6F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"gG6#%\"xGF*,$*(\"\"#\" \"\"F*!\"$-%$expG6#,$*&F.F.*$)F*F-F.!\"\"F7F.F." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(g*`'`(x)/x,x = 0,right) = 2;" "6#/-%&LimitG6%*(% \"gG\"\"\"-%\"'G6#%\"xGF)F-!\"\"/F-\"\"!%&rightG\"\"#" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Limit(g(x)/(x^4),x = 0,right) = 0;" "6#/-%&LimitG6%* &-%\"gG6#%\"xG\"\"\"*$F+\"\"%!\"\"/F+\"\"!%&rightGF1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Also " }{XPPEDIT 18 0 "Limit(g*`'`(x)/x, x = 0,left) = 0;" "6#/-%&LimitG6%*(%\"gG\"\"\"-%\"'G6#%\"xGF)F-!\"\"/F -\"\"!%%leftGF0" }{TEXT -1 10 ", because " }{XPPEDIT 18 0 "g*`'`(x) = \+ 0;" "6#/*&%\"gG\"\"\"-%\"'G6#%\"xGF&\"\"!" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "x<=0" "6#1%\"xG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 262 14 "3rd derivative" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG " }{TEXT -1 12 " at zero is " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Limit(g*`''`(x)/x,x = 0);" "6#-%&LimitG6$*(%\"gG\"\"\"- %#''G6#%\"xGF(F,!\"\"/F,\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "We can obtain a formula f or " }{XPPEDIT 18 0 "g*`''`(x)" "6#*&%\"gG\"\"\"-%#''G6#%\"xGF%" } {TEXT -1 7 ", when " }{TEXT 275 1 "x" }{TEXT -1 61 " is positive, by a pplying the standard differentiation rules." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "g := x -> exp(-1/x ^2);\nDiff('g(x)',x$2)=diff(g(x),x$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#,$*&\"\"\"F1*$ )9$\"\"#F1!\"\"F6F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$ -%\"gG6#%\"xG-%\"$G6$F*\"\"#,&*(\"\"'\"\"\"F*!\"%-%$expG6#,$*&F2F2*$)F *F.F2!\"\"F;F2F;*(\"\"%F2F*!\"'F4F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(g*`''`(x)/x,x = 0,right) = -6*``(Limit(g(x)/(x^5),x = 0,right)) +4*``(Limit(g(x)/(x^7),x = 0,right));" "6#/-%&LimitG6%*(%\"gG\"\"\"-%# ''G6#%\"xGF)F-!\"\"/F-\"\"!%&rightG,&*&\"\"'F)-%!G6#-F%6%*&-F(6#F-F)*$ F-\"\"&F./F-F0F1F)F.*&\"\"%F)-F66#-F%6%*&-F(6#F-F)*$F-\"\"(F./F-F0F1F) F)" }{TEXT -1 5 " = 0." }}{PARA 0 "" 0 "" {TEXT -1 5 "Also " } {XPPEDIT 18 0 "Limit(g*`''`(x)/x,x = 0,left) = 0;" "6#/-%&LimitG6%*(% \"gG\"\"\"-%#''G6#%\"xGF)F-!\"\"/F-\"\"!%%leftGF0" }{TEXT -1 10 ", bec ause " }{XPPEDIT 18 0 "g*`''`(x) = 0" "6#/*&%\"gG\"\"\"-%#''G6#%\"xGF& \"\"!" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "x<=0" "6#1%\"xG\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "It will be possible to show, in a similar way, that all h igher derivatives of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 16 " are zero, when " }{TEXT 276 1 "x" }{TEXT -1 82 " is zero, if w e can establish that the formulas for the successive derivatives of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 5 " for " }{TEXT 277 1 "x" }{TEXT -1 28 " positive all have the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`@@`(g,n)*``(x)=Sum(a[n,p]*``(f(x) /(x^p)),p = n+2 .. 3*n)" "6#/*&-%#@@G6$%\"gG%\"nG\"\"\"-%!G6#%\"xGF*-% $SumG6$*&&%\"aG6$F)%\"pGF*-F,6#*&-%\"fG6#F.F*)F.F6!\"\"F*/F6;,&F)F*\" \"#F**&\"\"$F*F)F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 23 "wh ere the coefficients " }{XPPEDIT 18 0 "a[n,p]" "6#&%\"aG6$%\"nG%\"pG" }{TEXT -1 14 " are integers." }}{PARA 0 "" 0 "" {TEXT -1 77 "This cert ainly seems to be the case from the following experimental evidence." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "g := x -> exp(-1/x^2):\n'g(x)'=g(x);\nDiff('g(x)',x)=diff(g(x ),x);\nDiff('g(x)',x$2)=diff(g(x),x$2);\nDiff('g(x)',x$3)=diff(g(x),x$ 3);\nDiff('g(x)',x$4)=diff(g(x),x$4);\nDiff('g(x)',x$5)=diff(g(x),x$5) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%$expG6#,$*&\"\"\" F-*$)F'\"\"#F-!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-% \"gG6#%\"xGF*,$*(\"\"#\"\"\"F*!\"$-%$expG6#,$*&F.F.*$)F*F-F.!\"\"F7F.F ." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"gG6#%\"xG-%\"$G6$F *\"\"#,&*(\"\"'\"\"\"F*!\"%-%$expG6#,$*&F2F2*$)F*F.F2!\"\"F;F2F;*(\"\" %F2F*!\"'F4F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"gG6# %\"xG-%\"$G6$F*\"\"$,(*(\"#C\"\"\"F*!\"&-%$expG6#,$*&F2F2*$)F*\"\"#F2! \"\"F " 0 "" {MPLTEXT 1 0 70 "g := x->exp(-1/x^2):\n'g (x)'=g(x);\nDiff('g(x)'/x^p,x)=diff(g(x)/x^p,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%$expG6#,$*&\"\"\"F-*$)F'\"\"#F-!\"\"F1 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&-%\"gG6#%\"xG\"\"\")F +%\"pG!\"\"F+,&**\"\"#F,F+!\"$-%$expG6#,$*&F,F,*$)F+F2F,F/F/F,F-F/F,** F4F,F-F/F.F,F+F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "At each s tage, the new powers of " }{TEXT 278 1 "x" }{TEXT -1 74 " in the denom inators must be obtained by adding 1 or 3 to previous powers." }} {PARA 0 "" 0 "" {TEXT -1 109 "We can conjecture the following formulas for the first and last coefficients in each sum for the derivatives. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[n, n+2] =(-1)^(n+1)*(n+1)!" "6#/&%\"aG6$%\"nG,&F'\"\" \"\"\"#F)*&),$F)!\"\",&F'F)F)F)F)-%*factorialG6#,&F'F)F)F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[n,3n]=2^n" "6#/&%\"aG6$%\"nG*&\"\"$\" \"\"F'F*)\"\"#F'" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 231 "Although it would be preferable to have \+ formulas for all the coefficients, we shall resist the impulse to prov e even these two formulas.\nThe original author, Jim Herod, did not at tempt to find general formulas for the coefficients." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We have now established that the Taylor series for " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 30 " about 0 does not converge to " }{XPPEDIT 18 0 "g(x)" "6 #-%\"gG6#%\"xG" }{TEXT -1 83 " on any open interval containing zero, b ut instead converges to the zero function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Legendre Polynomials " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 51 "We now loo k at a different set of polynomials: the " }{TEXT 260 20 "Legendre Pol ynomials" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "These polynom ials " }{XPPEDIT 18 0 "P[0](x), P[1](x), P[2](x), `. . .`" "6&-&%\"PG6 #\"\"!6#%\"xG-&F%6#\"\"\"6#F)-&F%6#\"\"#6#F)%&.~.~.G" }{TEXT -1 31 " h ave the following properties:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(P[i](x)^2,x = -1 .. 1) <> 0;" "6#0-%$IntG6$*$-&%\"P G6#%\"iG6#%\"xG\"\"#/F.;,$\"\"\"!\"\"F3\"\"!" }{TEXT -1 14 " ------- \+ (i)," }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(P[i](x)*P[j](x),x = -1 .. 1) = 0" "6#/-% $IntG6$*&-&%\"PG6#%\"iG6#%\"xG\"\"\"-&F*6#%\"jG6#F.F//F.;,$F/!\"\"F/\" \"!" }{TEXT -1 5 ", if " }{XPPEDIT 18 0 "i<>j" "6#0%\"iG%\"jG" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "From (i) and (ii) it follows that, if " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 22 " is a polynomial, and " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x) = Sum(a[i]*P[i]( x),i = 0 .. n);" "6#/-%\"gG6#%\"xG-%$SumG6$*&&%\"aG6#%\"iG\"\"\"-&%\"P G6#F/6#F'F0/F/;\"\"!%\"nG" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[i] = \+ int(g(x)*P[i](x),x = -1 .. 1)/Int(P[i](x)^2,x = -1 .. 1);" "6#/&%\"aG6 #%\"iG*&-%$intG6$*&-%\"gG6#%\"xG\"\"\"-&%\"PG6#F'6#F0F1/F0;,$F1!\"\"F1 F1-%$IntG6$*$-&F46#F'6#F0\"\"#/F0;,$F1F:F1F:" }{TEXT -1 16 " ------- ( iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Moreover, if f is any function such that " }{XPPEDIT 18 0 "Int(f(x)^2 ,x = -1 .. 1)" "6#-%$IntG6$*$-%\"fG6#%\"xG\"\"#/F*;,$\"\"\"!\"\"F/" } {TEXT -1 42 " is finite, then the polynomial of degree " }{TEXT 279 1 "n" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "S(x)" "6#-%\"SG6#%\"xG" }{TEXT -1 7 ", with " }{XPPEDIT 18 0 "Int((g(x)-S(x))^2,x = -1 .. 1);" "6#-%$ IntG6$*$,&-%\"gG6#%\"xG\"\"\"-%\"SG6#F+!\"\"\"\"#/F+;,$F,F0F," }{TEXT -1 41 " as small as possible, is the polynomial " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S(x) =sum(a[i]*P[i](x),n = 0 .. n)" "6# /-%\"SG6#%\"xG-%$sumG6$*&&%\"aG6#%\"iG\"\"\"-&%\"PG6#F/6#F'F0/%\"nG;\" \"!F7" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "with\240" }{XPPEDIT 18 0 "a[i]" "6#&%\"aG6#%\"iG" } {TEXT -1 16 " given by (iii)." }}{PARA 0 "" 0 "" {TEXT -1 43 "Maple kn ows about the Legendre polynomials." }}{PARA 0 "" 0 "" {TEXT -1 25 "He re are the first four. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "with(orthopoly,P):\nP(0,x);P(1,x);P (2,x);P(3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\" \"\"\"#!\"\"*(\"\"$F%F&F'%\"xGF&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,&*&#\"\"&\"\"#\"\"\"*$)%\"xG\"\"$F(F(F(*&#F,F'F(F+F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "As before, let" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "g(x) = PIECEWISE([0, x <= 0],[exp(-1/( x^2)), 0 < x]);" "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$\"\"!1F'F,7$-%$expG 6#,$*&\"\"\"F4*$F'\"\"#!\"\"F72F,F'" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 70 "In order to get a Len gendre Polynomial approximation for the function " }{XPPEDIT 18 0 "g(x )" "6#-%\"gG6#%\"xG" }{TEXT -1 51 ", we compute the coefficients by th e formula (iii)." }}{PARA 0 "" 0 "" {TEXT -1 21 "Because the function \+ " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 25 " is zero on th e interval " }{XPPEDIT 18 0 "[-1, 0]" "6#7$,$\"\"\"!\"\"\"\"!" }{TEXT -1 28 ", it sufficies to integrate " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6# %\"xG" }{TEXT -1 47 " from 0 to 1 in the numerator of formula (iii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Example 1: n = 5" }} {PARA 0 "" 0 "" {TEXT -1 33 "First calculate the coefficients " } {XPPEDIT 18 0 "a[i]" "6#&%\"aG6#%\"iG" }{TEXT -1 12 " in the sum " } {XPPEDIT 18 0 "S(x) = sum(a[i]*P[i](x),n = 0 .. n)" "6#/-%\"SG6#%\"xG- %$sumG6$*&&%\"aG6#%\"iG\"\"\"-&%\"PG6#F/6#F'F0/%\"nG;\"\"!F7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "with(orthopoly,P):\nn := 5;\nfor i from 0 to n do\n \+ int(P(i,x)*exp(-1/x^2),x=0..1)/int(P(i,x)^2,x=-1..1);\n a[i] := ev alf(%);\nend do;\ni := 'i':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG \"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"#F&-%$expG6#! \"\"F&F&*&F%F&*&%#PiGF%-%$erfG6#F&F&F&F&*&F'F+F.F%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"!$\"*\"Gp`W!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"$\"\"%\"\"\"-%#EiG6$F(F(F(!\"\"*&#F&F'F(-%$exp G6#F,F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"\"$\"+,jr86! #5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"&\"\"#\"\"\"-%$expG6#! \"\"F(F,*&#\"#:\"\"%F(*&%#PiG#F(F'-%$erfG6#F(F(F(F,*(F/F(F0F,F2F3F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"#$\"*SB#e7!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"#@\"\")\"\"\"-%$expG6#!\"\"F(F,*&#\"# x\"#;F(-%#EiG6$F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\" \"$$\"*3l,,*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"$6\"\"\")\" \"\"-%$expG6#!\"\"F(F(*&#\"$\"H\"#;F(*&%#PiG#F(\"\"#-%$erfG6#F(F(F(F(* (F/F(F0F,F2F3F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"%$\"(n ]N$!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"%J8\"#k\"\"\"-%#EiG6 $F(F(F(!\"\"*&#\"#**\"\")F(-%$expG6#F,F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"&$!(w@***!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "g := x -> piecewise(x<=0,0,exp(-1/x^2)):\n'g(x)'=g(x ); \nS := x->Sum(a[i]*P(i,x),i=0..n);\nevalf(plot([g(x),S(x)],x=-1..1, color=[red,green],thickness=2),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"gG6#%\"xG-%*PIECEWISEG6$7$\"\"!1F'F,7$-%$expG6#,$*&\"\"\"F4*$)F' \"\"#F4!\"\"F8%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SGf* 6#%\"xG6\"6$%)operatorG%&arrowGF(-%$SumG6$*&&%\"aG6#%\"iG\"\"\"-%\"PG6 $F39$F4/F3;\"\"!%\"nGF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7X7$$!\"\"\"\"!$F*F*7$$!5nmmmm;p0k&*!#?F+7$ $!5LLLL$3s%HaF/F+7$$!5++ +++]$*4)*\\F/F+7$$!5+++++]_&\\c%F/F+7$$!5+++++]1aZTF/F+7$$!5mmmm;/#)[o PF/F+7$$!5LLLLL$=exJ$F/F+7$$!5LLLLLL2$f$HF/F+7$$!5********\\PYx\"\\#F/ F+7$$!5LLLLLL7i)4#F/F+7$$!5********\\P'psm\"F/F+7$$!5********\\74_c7F/ F+7$$!4LLLLL3x%z#)F/F+7$$!4LLLL$3s$QM%F/F+7$$!2mmmmm\"zr)*F/F+7$$\"3LL LLezw5V!#>$\"5&=OasN8_`&>!$`#7$$\"3++++v$Q#\\\")Fbp$\"5%oB_DD$f=>S!#&) 7$$\"4LLLL$e\"*[H7Fbp$\"55=bO4K=Hi=!#[7$$\"4+++++qvxl\"Fbp$\"5dq[pq)fv [d\"!#N7$$\"4++++]_qn2#Fbp$\"5#Q0Uim4m9_)!#I7$$\"4++++Dcp@[#Fbp$\"5'3< \\8<_)3N*)!#F7$$\"4++++]2'HKHFbp$\"5%[<#))=!*>3%*))!#D7$$\"4nmmmmwanL$ Fbp$\"5!3/#>jR^2d7!#B7$$\"4+++++v+'oPFbp$\"5Oy(\\F\"zJu^()F_s7$$\"5lmm m\"H2fU'RF/$\"5Bgi0vqA$Qs\"!#A7$$\"4LLLLeR<*fTFbp$\"5%3L_bOM:F4$Fjs7$$ \"5lmmm\"HiBQP%F/$\"5)4nRwJe'*zO&Fjs7$$\"4+++++&)Hxe%Fbp$\"5#RII<%\\MA T')Fjs7$$\"4nmmm\"H!o-*\\Fbp$\"5H3SoeHI=.=!#@7$$\"4++++DTO5T&Fbp$\"5yN (y'p^,R'G$F_u7$$\"4nmmmmT9C#eFbp$\"5$*))QoGqZ(\\B&F_u7$$\"4++++D1*3`iF bp$\"57\"el:`rY+v(F_u7$$\"4LLLLL$*zym'Fbp$\"55wz`hNb&[0\"F/7$$\"4LLLL$ 3N1#4(Fbp$\"5Rj#34QA]%p8F/7$$\"4nmmm\"HYt7vFbp$\"5G8!G#pI:M+#F/7$$\"4nmmm;9@BM)Fbp$\"5%o8>f2_OmP#F/7$ $\"5++++]P$[/a)F/$\"5$y`;Uk8?&QDF/7$$\"4LLLLL`v&Q()Fbp$\"5%3sJx&*[U%*p #F/7$$\"4++++DOl5;*Fbp$\"5A;'HLM:Ov.$F/7$$\"4++++v.Uac*Fbp$\"5bfZ6Gi([ BN$F/7$$\"\"\"F*$\"5g@BWr6WzyOF/-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-F $6$7W7$F($\"*K$)GC\"!#57$F-$\"5RD@\"3*)=Afl%Fjs7$F1$!5JSau`RduRZ!#C7$F 4$!5mom(=oo:RW$Fjs7$F7$!5yRZSWwS)GA&Fjs7$F:$!5tgh85mMW=dFjs7$F=$!52)Go JoNW7K&Fjs7$F@$!5x*R1X)4KW*G%Fjs7$FC$!5dHi\\C>_:\"z#Fjs7$FF$!5nD\\Oaq$ 3/2\"Fjs7$FI$\"5W&GB+C%R\"QZ(F_s7$FL$\"5KrFIKoYYgAFjs7$FO$\"5iZgIgYT>[ PFjs7$FR$\"5W'\\QW=R6g\"\\Fjs7$FU$\"5YlC^2rCfkcFjs7$FX$\"5f)p?Epdu)))f Fjs7$Fen$\"50U()eC(*pL9fFjs7$Fhn$\"5u+-%\\R&pDiaFjs7$F[o$\"5*f6X(RO=r6 XFjs7$F^o$\"50CPT@_f_FLFjs7$Fao$\"5(*[>M\\/PA>e6?Fjs7$Fjo$!5_([1*\\(z!G_QFjs7$F]p$!5%z>Hw=b32v&Fjs7 $F`p$!5kE$)pgp:0$[(Fjs7$Fgp$!5\\@A****H#HUm)Fjs7$F]q$!5>A*3zWx#[c%*Fjs 7$Fcq$!5KY\\_fjcC'e*Fjs7$Fiq$!5CsVU\"evC`'))Fjs7$F_r$!5IV.+5**Q\\/sFjs 7$Fer$!5qmy'fzXX90%Fjs7$F[s$\"5\\'3)eT&Q:sO\"F_s7$Fas$\"5/y48rAS(*=iFj s7$F\\t$\"5]epbM1LNL8F_u7$Fft$\"5H:Q\"4f:WSI#F_u7$F[u$\"5UGjjh7WX=MF_u 7$Fau$\"5m/mJ6'yE'4[F_u7$Ffu$\"5G0zJ)zJ![5kF_u7$F[v$\"5ypU)=ub![f$)F_u 7$F`v$\"5L^I4^g'R<0\"F/7$Fev$\"5F$ocNJOaDI\"F/7$Fjv$\"5keBINfw#He\"F/7 $Fdw$\"5bSAp\\wQdp=F/7$F^x$\"5+?%*))zT3kLAF/7$Fcx$\"5!*GOsZJR44CF/7$Fh x$\"5Fn:\")=/]_#f#F/7$$\"5lmmm\"zW?)\\*)F/$\"5iOx;A&f5qz#F/7$F]y$\"5_0 M&f.?63,$F/7$$\"5++++++PDj$*F/$\"5_D/&RYAuUA$F/7$Fby$\"5q0At#e!QYYMF/7 $$\"5++++v=5s#y*F/$\"5%**ege:6d]p$F/7$Fgy$\"+I/\"R&RFhz-F\\z6&F^zF+F_z F+-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Ffel-%%VIEWG6$;F(Fgy%(D EFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "This approximation using Legendre polynomials is rather \+ crude, but it is certainly better than any Taylor polynomial at 0." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Example 2: n = 40 " }} {PARA 0 "" 0 "" {TEXT -1 33 "First calculate the coefficients " } {XPPEDIT 18 0 "a[i]" "6#&%\"aG6#%\"iG" }{TEXT -1 12 " in the sum " } {XPPEDIT 18 0 "S(x) = sum(a[i]*P[i](x),n = 0 .. n)" "6#/-%\"SG6#%\"xG- %$sumG6$*&&%\"aG6#%\"iG\"\"\"-&%\"PG6#F/6#F'F0/%\"nG;\"\"!F7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "with(orthopoly,P):\nn := 40;\nfor i from 0 to n do\n int(P(i,x)*exp(-1/x^2),x=0..1)/int(P(i,x)^2,x=-1..1);\n a[i] := e valf(%,20);\nend do:\ni := 'i':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"nG\"#S" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "There is no discernable difference in the graphs of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "S(x) " "6#-%\"SG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "g := x -> piecewise(x<=0,0, exp(-1/x^2)):\n'g(x)'=g(x); \nS := x->Sum(a[i]*P(i,x),i=0..n);\nevalf( plot([g(x),S(x)],x=-1..1,color=[red,green],thickness=[1,2]),20);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$\"\"!1F 'F,7$-%$expG6#,$*&\"\"\"F4*$)F'\"\"#F4!\"\"F8%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$Sum G6$*&&%\"aG6#%\"iG\"\"\"-%\"PG6$F39$F4/F3;\"\"!%\"nGF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7X7$$!\"\" \"\"!$F*F*7$$!5nmmmm;p0k&*!#?F+7$$!5LLLL$3s%HaF/F+7$$!5+++++]$*4)*\\F/F+7$$!5+++++]_&\\c%F/F+7 $$!5+++++]1aZTF/F+7$$!5mmmm;/#)[oPF/F+7$$!5LLLLL$=exJ$F/F+7$$!5LLLLLL2 $f$HF/F+7$$!5********\\PYx\"\\#F/F+7$$!5LLLLLL7i)4#F/F+7$$!5********\\ P'psm\"F/F+7$$!5********\\74_c7F/F+7$$!4LLLLL3x%z#)F/F+7$$!4LLLL$3s$QM %F/F+7$$!2mmmmm\"zr)*F/F+7$$\"3LLLLezw5V!#>$\"5&=OasN8_`&>!$`#7$$\"3++ ++v$Q#\\\")Fbp$\"5%oB_DD$f=>S!#&)7$$\"4LLLL$e\"*[H7Fbp$\"55=bO4K=Hi=!# [7$$\"4+++++qvxl\"Fbp$\"5dq[pq)fv[d\"!#N7$$\"4++++]_qn2#Fbp$\"5#Q0Uim4 m9_)!#I7$$\"4++++Dcp@[#Fbp$\"5'3<\\8<_)3N*)!#F7$$\"4++++]2'HKHFbp$\"5% [<#))=!*>3%*))!#D7$$\"4nmmmmwanL$Fbp$\"5!3/#>jR^2d7!#B7$$\"4+++++v+'oP Fbp$\"5Oy(\\F\"zJu^()F_s7$$\"5lmmm\"H2fU'RF/$\"5Bgi0vqA$Qs\"!#A7$$\"4L LLLeR<*fTFbp$\"5%3L_bOM:F4$Fjs7$$\"5lmmm\"HiBQP%F/$\"5)4nRwJe'*zO&Fjs7 $$\"4+++++&)Hxe%Fbp$\"5#RII<%\\MAT')Fjs7$$\"4nmmm\"H!o-*\\Fbp$\"5H3Soe HI=.=!#@7$$\"4++++DTO5T&Fbp$\"5yN(y'p^,R'G$F_u7$$\"4nmmmmT9C#eFbp$\"5$ *))QoGqZ(\\B&F_u7$$\"4++++D1*3`iFbp$\"57\"el:`rY+v(F_u7$$\"4LLLLL$*zym 'Fbp$\"55wz`hNb&[0\"F/7$$\"4LLLL$3N1#4(Fbp$\"5Rj#34QA]%p8F/7$$\"4nmmm \"HYt7vFbp$\"5G8!G#pI:M+#F/7$$ \"4nmmm;9@BM)Fbp$\"5%o8>f2_OmP#F/7$$\"5++++]P$[/a)F/$\"5$y`;Uk8?&QDF/7 $$\"4LLLLL`v&Q()Fbp$\"5%3sJx&*[U%*p#F/7$$\"4++++DOl5;*Fbp$\"5A;'HLM:Ov .$F/7$$\"4++++v.Uac*Fbp$\"5bfZ6Gi([BN$F/7$$\"\"\"F*$\"5g@BWr6WzyOF/-%' COLOURG6&%$RGBG$\"*++++\"!\")F+F+-%*THICKNESSG6#Fhy-F$6%7X7$F($\"0^KTO hu%GF/7$F-$\"5b`k_y&p2['=!#E7$F1$!5_=ClxU%4'zfF^[l7$F4$\"5tgNhlV:w\"f$ F^[l7$F7$!5,$H0%*)pqe2CF^[l7$F:$\"5TnMSe)o)GuIF^[l7$F=$!564th6nTaj\\F^ [l7$F@$\"5&R'R8#y8(fx[F^[l7$Fhn$\"5nfaSHESsSYF^[l7$F[o$!5g\\#RU)*e#>)3( F^[l7$F^o$!5YKt6ep>FoAF^[l7$Fao$\"5V&=PQC1(fB')F^[l7$Fdo$!5w\\:f!**HGyGan`Fjs7$Fft$\"5%odqTS(GXR')Fjs7$F[u$\"5j9n)>fA,K!=F_u7$Fau$ \"5E**HZ45N^'G$F_u7$Ffu$\"5o'Rs)RFj%[B&F_u7$F[v$\"5BOh!)=L>3]xF_u7$F`v $\"5MR6E\"*fB'[0\"F/7$Fev$\"5EocZ+0$R%p8F/7$Fjv$\"5/X#3UE_^.q\"F/7$F_w $\"5LRh^H;iJc=F/7$Fdw$\"5YATY\")HYs8?F/7$Fiw$\"5T(fm98\"*)3&>#F/7$F^x$ \"5****3+@l;kwBF/7$Fcx$\"53(G5q-%*G&QDF/7$Fhx$\"5`k7?dIvV*p#F/7$F]y$\" 5wIX\"))e!faPIF/7$Fby$\"5=e>q)\\\"fM_LF/7$Fgy$\"5Rc&\\#>_)[(yOF/-F\\z6 &F^zF+F_zF+-Fcz6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fdel-%%VIEWG6$;F(Fgy%(D EFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "The error curve shows that the maximum absolut error occu rs when " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "evalf(plot(g(x)-S(x),x=-1..1,color=blue),20);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7c`l7$$!\"\"\"\"!$!0^ KTOhu%G!#?7$$!5MLLekynP')**F-$!5G'z\\8)*[8%HV!#E7$$!5nmm;HdNvs**F-$\"5 Jt(=y$\\h\"*)[(F37$$!5,++v$fLI\"f**F-$\"5K%)3pnJC&z8\"!#D7$$!5MLLLe9r] X**F-$\"50R;:![NqX0\"F>7$$!5nmm\"HK*Q)=$**F-$\"5,r!=]\"3A)fD(F37$$!5,+ +](=ng#=**F-$\"5N:qck>DzdIF37$$!5MLL3_]uj/**F-$!58C&RC_(3%*[5F37$$!5nm mm;HU,\"*)*F-$!5-QaS#[&y#>Z%F37$$!5,++D\"y+\"Rx)*F-$!5ytoLE1C)[\"pF37$ $!5MLL$ekynP')*F-$!5463D2RgR*G)F37$$!5nmmT5lX9])*F-$!51%3)*>4&QGY')F37 $$!5,+++vV8_O)*F-$!5\"G;%GvZi$H7)F37$$!5nmm;/,\\F4)*F-$!5qPd#3B\\j5>&F 37$$!5MLLLLe%G?y*F-$!5zkkRbSh9o5F37$$!5nmmm\"HdNvs*F-$\"5Rw7\"*psP>$y& F37$$!5,+++](oUIn*F-$\"5kvJN@W#o!\\qF37$$!5MLLL3-)\\&='*F-$\"5L0XlJ?78 \"R$F37$$!5nmmmm;p0k&*F-$!5b`k_y&p2['=F37$$!5,++vV[Hk;&*F-$!50IQaoj$\\ aH&F37$$!5MLL$3-)*G#p%*F-$!5zKTBY_7!o]'F37$$!5nmm\"z>,:=U*F-$!5lI'pMp( ='HV&F37$$!5++++vV5Su$*F-$!5\\VPj()4``KFF37$$!5mmm;H2Jdz#*F-$\"5[&ffT9 L\">yNF37$$!5LLLL$3A )F-$!5AJ:y%z!***[l\"F37$$!5++++Dc[3:\")F-$!5-%pz:MK_=o%F37$$!5mm;H#=qu $)3)F-$!5/h8=KS;'42&F37$$!5LLLeRZXmh!)F-$!5AIgj&*e,Z%G&F37$$!5****\\(o HRa\\.)F-$!5Y')*)e7F!y(=`F37$$!5mmm;aQUC3!)F-$!57P[g+ASIw^F37$$!5***** \\(oHR#[&zF-$!5PM2T]%[!e)R%F37$$!5LLLL$3i.9!zF-$!5TnMSe)o)GuIF37$$!5** ******\\iPH.xF-$\"5Y8mfkdel'H$F37$$!5mmmm;/R=0vF-$\"564th6nTaj\\F37$$! 5LLLL$3i_+I(F-$!5G.QhK@l8v5!#F7$$!5++++]P8#\\4(F-$!5&R'R8#y#zS$Q7[%F37$$!5LLLL$3FuF)oF-$!5B+F:1x)>v W$F37$$!5********\\P2qwnF-$!5HLU#*p0<-OgF_[l7$$!5mmmm;/siqmF-$\"5vG'yP L)*3dU#F37$$!5LLLL3F0wll1e\")\\F37$$!5mmmm\"H_F37$$!5******\\PMQf&['F-$\"5q#p1jJ%o2_`F37$$!5LLLL$ e\\g\"fkF-$\"5NZIzGx&4uP&F37$$!5mmmmmTrU`jF-$\"5&z\")>l'HjmTWF37$$!5++ ++](y$pZiF-$\"5@W(fHW?*Qe@F37$$!5mmmmT&Gu,.'F-$!5UcP&[ool!*f$F37$$!5LL LLL$yaE\"eF-$!5F))))\\%*Q2ax`F37$$!5********\\([j5i&F-$!5F)y\\eoMWu-#F 37$$!5mmmmm\">s%HaF-$\"58KFL9loidIF37$$!5LLL$eke]bP&F-$\"5cL2N&*GF37$$!5+++++]_&\\c%F-$!5NYS)Hk]=^/'F37$$!5+++++]zCcVF-$!5J [$H&*4(z*p'QF37$$!5+++++]1aZTF-$\"55jlSw?$yNl\"F37$$!5mmm;aQvx_SF-$\"5 ()4****)*p=eYSF37$$!5LLLL3FW,eRF-$\"5ot%p8--B=v&F37$$!5****\\(=#\\OKMR F-$\"58:4PZqJSJgF37$$!5mmmTNrGj5RF-$\"5&3bM]nrs[C'F37$$!5LL$e*[$4Up)QF -$\"5g95,$Q7h(*Q'F37$$!5******\\i:8DjQF-$\"53y()>l-ANkkF37$$!5mm;/wP0c RQF-$\"5v`:@TE4qnkF37$$!5LLLe*)f(pe\"QF-$\"5))))Hq@f=k*R'F37$$!5****\\ 7.#)*y@z$F-$\"5B-,EUi\"*ygiF37$$!5mmmm;/#)[oPF-$\"5r[`&pEBGD0'F37$$!5* ******\\P>BJa$F-$\"5;[*=Q*36;&G\"F37$$!5LLLLL$=exJ$F-$!532mf&>8(fx[F37 $$!5KLLL3_(H+F$F-$!5UFo(*Qa[L*z&F37$$!5KLLL$3K,BA$F-$!52ir8gVU.\"['F37 $$!5KLL$3_5P%)>$F-$!5B=)=>q-w=s'F37$$!5KLLLe*)GduJF-$!5*p6q3xb$4#*oF37 $$!5KLL$eRn32:$F-$!5zyt(=[NT'*)pF37$$!5KLLLLeW%o7$F-$!5a@^B,!QOK,(F37$ $!5KLLL$ef(QJIF-$!5O:3J*)[AlpjF37$$!5LLLLLL2$f$HF-$!5nfaSHESsSYF37$$!5 mmmmT&o_Qr#F-$\"5`))y?X&G2C#=F37$$!5********\\PYx\"\\#F-$\"5g\\#RU)*e# >)3(F37$$!5LL$eR(\\D?nCF-$\"5OqEX*\\1+lO(F37$$!5mmm\"z>YIEW#F-$\"5(zMc !y3)fjc(F37$$!5****\\(=UPe!=CF-$\"5)>#f![ko(>&o(F37$$!5LLL$ekG'[$R#F-$ \"5<5>()e2'*H@xF37$$!5mm;zp)>9*oBF-$\"5tx%4$efC$Qn(F37$$!5*****\\P46UV M#F-$\"56y`jK$zbGa(F37$$!5LL$3xJ-q(>BF-$\"5?g9\")\\ITKHtF37$$!5mmmmTNz >&H#F-$\"5(3>&f4e)z].(F37$$!5******\\P%e4p>#F-$\"5)[sI?<\"Hs<^F37$$!5L LLLLL7i)4#F-$\"5YKt6ep>FoAF37$$!5mmmmTNa%H)=F-$!5:Gm9?=ANd[F37$$!5**** ****\\P'psm\"F-$!5V&=PQC1(fB')F37$$!5*********\\F&*=Y\"F-$!5'yYwri!fBN hF37$$!5********\\74_c7F-$\"5w\\:f!**HE(R2)F37$$!5GLL$e*)fEm))*! #@$\"5]u#[O[.S&G\"*F37$$!5&******\\P454N*F\\am$\"5-kad'e86qu*F37$$!5IL Le9T=0$3*F\\am$\"5^OIbA![C%z)*F37$$!5jmm;a)e$>:))F\\am$\"5!y[fWD.Y'*)) *F37$$!5(****\\PfLNta)F\\am$\"5GJg?we#zix*F37$$!4LLLLL3x%z#)F-$\"50\"o TM7*)Q%R&*F37$$!5ILLLe9rc&H(F\\am$\"5)e%4gQFgt_wF37$$!5ILLL$e9d;J'F\\a m$\"5?Env#fO.bR%F37$$!5ILLL3xruF`F\\am$\"5@g0X-//2#f#F_[l7$$!4LLLL$3s$ QM%F-$!5=Y,8DoOY'4%F37$$!5jmmmT&QdDG$F\\am$!5;7[[rP.11#)F37$$!5&****** *\\ivF@AF\\am$!5Bmz(zg&))*))3\"F>7$$!5HLL3x1w&f&>F\\am$!5j7$$!5imm;/^wj!p\"F\\am$!5e<`7$$!5&****\\7`p<`U\"F\\am$!5&4 \"zfVP*e*e6F>7$$!5GLLLeRx**f6F\\am$!5<.*49g%oEa6F>7$$!5S*****\\7GyNH'! #A$!5SHDHr$3)y,6F>7$$!2mmmmm\"zr)*F-$!5R<#zxQj$GI**F37$$\"5ILLL3_Nl.5F \\am$!5daSWBG%G82'F37$$\"5?LLL$3-Dg5#F\\am$!5m*HJ.yZn6#oF_[l7$$\"55LLL e*['R3KF\\am$\"59f'=#=i&)\\k^F37$$\"3LLLLezw5V!#>$\"5D:!Q(RvM`A5F>7$$ \"5v******\\ibQq_F\\am$\"5-e;![z+(338F>7$$\"5]mmmmmJ+IiF\\am$\"5k_Ohn- kT*Q\"F>7$$\"5DLLL$3x?'*=(F\\am$\"5TI4&*>1RcS7F>7$$\"3++++v$Q#\\\")Fgg m$\"5nwnS0&\\OWs)F37$$\"5DLLL3x;l&=*F\\am$\"57$$ \"4LLLL$e\"*[H7Fggm$!59PcnLS5K::F>7$$\"5kmm;/w\\-$G\"F-$!5\"e2F!)3Gx[l \"F>7$$\"5)******\\PzglL\"F-$!5*oB\\u7$$\"5KLL$e9h'4!R\"F-$!5 il3M'p(=Ym;F>7$$\"5lmmm;HCjV9F-$!5R.bP=7dWI:F>7$$\"5)*****\\(oCor\\\"F -$!5p1h_%[^%4.8F>7$$\"5KLLLekSq]:F-$!5$Q(oceD#[^$**F37$$\"5mmm;H#))RUg \"F-$!5r#*Rz0>;%*fhF37$$\"4+++++qvxl\"Fggm$!5a]#)f([_1\\*=F37$$\"5+++] 7`+:57$$\"5++++]7JFn=F-$\"5G\"=jBb?6v[\"F>7$$ \"5+++]iluk>>F-$\"5rN%GKSuC_y\"F>7$$\"5++++v==-s>F-$\"5\\Y'H'z;xc*)>F> 7$$\"5+++DJ&**3#)*>F-$\"56P&yIOq5:0#F>7$$\"5+++](=<'RC?F-$\"5_vBfa:p`% 3#F>7$$\"5+++vV[Le]?F-$\"5ePql+;Zc(3#F>7$$\"4++++]_qn2#Fggm$\"5w<6g#Gy U*f?F>7$$\"5+++vo/aWF@F-$\"5Lam/H%yG\">>F>7$$\"5+++]P%G?\"y@F-$\"5'*\\ EQY/k&om\"F>7$$\"5+++D1k^zGAF-$\"5Lbx!fv7ONJ\"F>7$$\"5++++vV+ZzAF-$\"5 Bb9x\\#H&3h()F37$$\"5+++vVB\\9IBF-$\"5aZ#ffE>7Ox$F37$$\"5+++]7.)>3Q#F- $!5)Hrr)R)Hm>b\"F37$$\"5+++D\"Go%\\JCF-$!5*p3jf\\\\'f3pF37$$\"4++++Dcp @[#Fggm$!5*>LB\"pW4d(>\"F>7$$\"5+++Dcw`VQDF-$!5x3[+BgxD)o\"F>7$$\"5+++ ]i!>,Zf#F-$!5+y\"*H:^a%f1#F>7$$\"5+++vo/q'4l#F-$!5&fUF*pjCa*H#F>7$$\"5 ++++v=GB2FF-$!5^Xu7^aMUoBF>7$$\"5+++D\"Gj)\\jFF-$!5W#)R$y%yAYkAF>7$$\" 5+++](oWk(>GF-$!5vu%y=2.GI*>F>7$$\"5+++v$4EIg(GF-$!5Ge*Gnwq>Gd\"F>7$$ \"4++++]2'HKHFggm$!5v5\\**y&yIX.\"F>7$$\"5MLL$e9T`G)HF-$!35VLDmv!>$[!# C7$$\"5ommm\"zu5M.$F-$\"2][1R#QCI(*Faen7$$\"5-++]P%3oR3$F-$\"3nv#>^*[! Gr'Faen7$$\"5NLLL$3UDX8$F-$\"4/wkqk_#\\.7Faen7$$\"5qmm;HdF3&=$F-$\"4.3 bd!yXTh;Faen7$$\"5.+++v$4ScB$F-$\"4J7XDq@bt,#Faen7$$\"5PLL$3-V(>'G$F-$ \"4+Cr,398-D#Faen7$$\"4nmmmmwanL$Fggm$\"3u[Y_%prnM#!#B7$$\"5PLLLe9bt!R $F-$\"32pD*eC8WH#Fegn7$$\"5.+++]iirWMF-$\"3,\"RD:^qq3#Fegn7$$\"5qmmmT5 qp)\\$F-$\"3v:^'[.&4TF]fm7$$\"5lmmm\"H2fU'RF-$!2 1.=YY#Q$4#F]fm7$$\"5(*****\\PM#)3iSF-$!2,#yx\"y*3$*>F]fm7$$\"4LLLLeR<* fTFggm$!2V'[))Gh%F-$\"2,Zns4`B\"=F]fm7$$\"5,+voH/yYDYF- $\"2-:E5k!)G#=F]fm7$$\"5MLLeRAr/QYF-$\"2++5bG&[E=F]fm7$$\"5nm\"z%\\Ski ]YF-$\"2=!Ggqz=B=F]fm7$$\"5,+]Pfed?jYF-$\"1o![pRNI\"=F\\am7$$\"5ML3Fpw ]yvYF-$\"1F#Hju*4'z\"F\\am7$$\"5omm;z%Rk$)o%F-$\"1B<^4$yCx\"F\\am7$$\" 5-++v=n;oQZF-$\"1$44<$4$Rh\"F\\am7$$\"5NLLLeR*)**)y%F-$\"1C4)3_KBO\"F \\am7$$\"5qmm\"z>@;$R[F-$\"1[/QYL^M5F\\am7$$\"5.++]P%[L'*)[F-$\"0`;i=9 d^'F\\am7$$\"5PLL3xc2&*R\\F-$\"0*[N@IjvBF\\am7$$\"4nmmm\"H!o-*\\Fggm$! 0M1FIL'>=F\\am7$$\"5PLLL3xS'G/&F-$!0H;*[\\N()fF\\am7$$\"5.++++D,Y&4&F- $!0b-M2(Rk'*F\\am7$$\"5qmmm\"HS7Yh7F\\am7$$\"5NLLL$3A_1?& F-$!1Bp\")4PLl9F\\am7$$\"5pmm;zW-&pA&F-$!1bSAJ?-H:F\\am7$$\"5-+++vo#[K D&F-$!1=O?Qz!fc\"F\\am7$$\"5NLL$3FHY&z_F-$!1%y7<3$ev:F\\am7$$\"5ommmm; V%eI&F-$!1u\"=hU-\"e:F\\am7$$\"5MLLLek.We`F-$!1l?&>W(=W9F\\am7$$\"4+++ +DTO5T&Fggm$!1[jUzReL7F\\am7$$\"5MLL3-j'eCY&F-$!0,S()[%H%\\*F\\am7$$\" 5omm;a84)Q^&F-$!0q*fCklygF\\am7$$\"5-++D1kJIlbF-$!0P%[g\\IBBF\\am7$$\" 5NLLLe9as;cF-$\"0'>a_yw;:F\\am7$$\"5.++]i:*p&>dF-$\"0+v/K\">H%)F\\am7$ $\"4nmmmmT9C#eFggm$\"1D#\\6))GWG\"F\\am7$$\"5.+]7.#eJ$\\eF-$\"1k\"=yR_ DM\"F\\am7$$\"5PLLeRZ([i(eF-$\"1D[M0\"3VP\"F\\am7$$\"5qm;/w7f;.fF-$\"1 CsOFHAz8F\\am7$$\"5.++]7yI3IfF-$\"1VzG\")pLd8F\\am7$$\"5qmmT&)3u\"R)fF -$\"1Ehs[K)eB\"F\\am7$$\"5NLLLeR)*>BO*F\\am7$$\"5lmmm\"z\\%[gkF-$!15:v'G%)HA\"F\\am7$$\" 5(*****\\i:A=klF-$!1LPhN\"zx7\"F\\am7$$\"4LLLLL$*zym'Fggm$!/CjJsHCoF-7 $$\"5ILLL3Fe#Rx'F-$!.NbU`_Q\"F-7$$\"5ILLL$3sr*zoF-$\"/='=%=!\\W'F-7$$ \"5ILL$3xm%*H$pF-$\"/$e9XB,'*)F-7$$\"5ILLLe9w,')pF-$\"0uv.q6M1\"F-7$$ \"5ILL3-)3HD,(F-$\"0b#ejlJ56F-7$$\"5ILL$e9cS!RqF-$\"0*\\qDD'48\"F-7$$ \"5ILLe*[._b1(F-$\"05**=W%zC6F-7$$\"4LLLL$3N1#4(Fggm$\"08&fK/)=4\"F-7$ $\"5lmm;a)GJs>(F-$\"/0;@jG0sF-7$$\"5++++vo!*R-tF-$\".jqC!pw%)F-7$$\"5N LL$e*[oc2uF-$!/wBG`MbdF-7$$\"4nmmm\"HYt7vFggm$!/wJ-)\\>***F-7$$\"5.+]P 4'>%*o`(F-$!098\"yoPT5F-7$$\"5PLL3-jP0hvF-$!0en;>$zf5F-7$$\"5qm;z%*HL@ &e(F-$!0*fQ0\"F-7$$\"5.++](o*GP4wF-$!0j[%3)RN-\"F-7$$\"5qmm\"H2.#p dwF-$!/Q%*y'\\R#*)F-7$$\"5NLLLek6,1xF-$!/)>#*)z7snF-7$$\"5omm;HK%\\E!y F-$!.@4SGo$yF-7$$\"4+++++q(G**yFggm$\"/AGU*45]&F-7$$\"5MLL$3-)omazF-$ \"/U=)H!eZ#)F-7$$\"5ommmTgg/5!)F-$\"/Q%H2^c&)*F-7$$\"5NLL3_]ctP!)F-$\" 0@s$z*GX,\"F-7$$\"5-++]iS_Ul!)F-$\"08K?iTn+\"F-7$$\"5pmm\"H2$[6$4)F-$ \"/X$>wr%>'*F-7$$\"5NLLL$3U/37)F-$\"/FueZT7))F-7$$\"5qmm;/,O=w\")F-$\" /^?gtyKiF-7$$\"5.+++D\"yi:B)F-$\"/F)='GozEF-7$$\"5PLL$e9'>%pG)F-$!/%)f R:oA8F-7$$\"4nmmm;9@BM)Fggm$!/:j<3XW^F-7$$\"5.++]iSH&=R)F-$!/q/3Y'H)yF -7$$\"5NLLLeRZQT%)F-$!/GbMz5i&*F-7$$\"5-++D1R1:m%)F-$!/TU'p#G6**F-7$$ \"5omm;aQl\"4\\)F-$!/km/D#R!**F-7$$\"5MLL3-QCo:&)F-$!/w)*fK!G`*F-7$$\" 5++++]P$[/a)F-$!/D\\Pz#Q!))F-7$$\"5KLL$ek8!)**e)F-$!/lS1_\"RO'F-7$$\"5 lmmmTN>^R')F-$!/9el(G(**GF-7$$\"5(*****\\PMP/*o)F-$\"/g@NB?&3\"F-7$$\" 4LLLLL`v&Q()Fggm$\"/Jc/`+f\\F-7$$\"5jmm\"z>w'Q\"z)F-$\"/I%*e\"*)R@)F-7 $$\"5(*****\\i!*z>W))F-$\"/m@$>`\"p)*F-7$$\"5ILeky(z+u&))F-$\"/6))R*)) =(**F-7$$\"5jm;z%\\g.1())F-$\"/iy?jlS**F-7$$\"5(**\\P4@T1Q)))F-$\"/I0d Gat(*F-7$$\"5ILL3F>#4q*))F-$\"/$\\(fNVq%*F-7$$\"5(***\\PfL[TB*)F-$\"/2 Ssr*F-$!04)Gvq( G-\"F-7$$\"5+++](oW7;@*F-$!0_A%[-'R+\"F-7$$\"5+++D1*)f)oB*F-$!/P(HKHo7 *F-7$$\"5++++DJ&f@E*F-$!/I4io-HvF-7$$\"5+++]i:mq7$*F-$!/wm:M:_EF-7$$\" 5++++++PDj$*F-$\"/LO)Ga'GLF-7$$\"5+++]P%y+QT*F-$\"/p;7k96&)F-7$$\"5+++ +voyMk%*F-$\"01B6Q%=&3\"F-7$$\"5+++]7`\\*[^*F-$\"/4g#f*F-$!/#o;j!\\u9F-7$$\"5+++D\"G yh(>'*F-$!/6,*)H(G\"eF-7$$\"5++](oal@pk*F-$!/+*\\x;*\\%*F-7$$\"5+++]7G :3u'*F-$!0<*oQ0Wj6F-7$$\"5++DJXk9m(o*F-$!0gA1\"z*p>\"F-7$$\"5++]7y+9C, (*F-$!04y`&\\[r6F-7$$\"5++v$4rL@[r*F-$!0SwfYOJ3\"F-7$$\"5+++vVt7SG(*F- $!/V]x!45J*F-7$$\"5++]P4Y6cb(*F-$!/$)Q7h%z]%F-7$$\"5++++v=5s#y*F-$\"/) ph.d8'>F-7$$\"5++]iS\"*3))4)*F-$\"/'\\l)eY0')F-7$$\"5+++D1k2/P)*F-$\"0 LX,y$\\B8F-7$$\"5++D1R+2i])*F-$\"0?y^Je.S\"F-7$$\"5++](=nj+U')*F-$\"0: m(zYhM8F-7$$\"5++vo/t0yx)*F-$\"0m%p:*e_5\"F-7$$\"5+++]P40O\"*)*F-$\"/- rq'3V0(F-7$$\"5++DJqX/%\\!**F-$\"/[h'4'R2:F-7$$\"5++]7.#Q?&=**F-$!/\"3 WQ^$*4&F-7$$\"5++v$f$=.5K**F-$!0aDB`T1=\"F-7$$\"5+++voa-oX**F-$!0CX!)4 X4q\"F-7$$\"5++Dc,\">g#f**F-$!0^HZ$**>C=F-7$$\"5++]PMF,%G(**F-$!05VA4y 1>\"F-7$$\"5++v=nj+U')**F-$\"/\"H#>t)Q0(F-7$$\"\"\"F*$\"0@lF>_fb%F--%' COLOURG6&%$RGBG$F*F*Fbgq$\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!Fjgq-%% VIEWG6$;F(Fjfq%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "xx := 1;\nevalf(abs(g(xx)-S(xx))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&gb%!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "This is great approximation in comparison to the Taylor polynomial, which failed awfully in making an approximation fo r " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Bernstein polynomials " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 126 "In the introductio n, we gave the usual definition for the Bernstein polynomials. These p olynomials are defined for a function " }{XPPEDIT 18 0 "g(x)" "6#-%\"g G6#%\"xG" }{TEXT -1 18 " over an interval " }{XPPEDIT 18 0 "[0, 1]" "6 #7$\"\"!\"\"\"" }{TEXT -1 183 ". For the reader unfamiliar with these \+ polynomials, it is perhaps instructive to look at graphs of the indivi dual terms of the sums and to visualize how they might add to approxim ate " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{TEXT 280 1 "x" }{TEXT -1 4 " in " } {XPPEDIT 18 0 "[0, 1]" "6#7$\"\"!\"\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B(n,x) = Sum(g(k/n)*matrix([[n] , [k]])*x^k*(1-x)^(n-k),k = 0 .. n);" "6#/-%\"BG6$%\"nG%\"xG-%$SumG6$* *-%\"gG6#*&%\"kG\"\"\"F'!\"\"F2-%'matrixG6#7$7#F'7#F1F2)F(F1F2),&F2F2F (F3,&F'F2F1F3F2/F1;\"\"!F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "matrix([[n], [k]])= n!/ (k!*(n-k!))" "6# /-%'matrixG6#7$7#%\"nG7#%\"kG*&-%*factorialG6#F)\"\"\"*&-F.6#F+F0,&F)F 0-F.6#F+!\"\"F0F7" }{TEXT -1 30 " is the binomial coefficient. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "For the p urpose of illustration, we define a function " }{XPPEDIT 18 0 "g(x)" " 6#-%\"gG6#%\"xG" }{TEXT -1 68 " and draw the terms of the sum, as well as the sum and the graph of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG " }{TEXT -1 36 ". The reader might choose to modify " }{XPPEDIT 18 0 " g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 105 ", or to change the number of te rms in the sum to increase intuition. In the graph which follows, we p lot " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 4 " in " } {TEXT 282 4 "navy" }{TEXT -1 26 " and the approximation in " }{TEXT 281 5 "brown" }{TEXT -1 97 ". The terms of the sum that makes the appr oximation are coloured with a hue which varies from 0 (" }{TEXT 261 3 "red" }{TEXT -1 10 ") through " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\" $!\"\"" }{TEXT -1 2 " (" }{TEXT 257 5 "green" }{TEXT -1 6 ") and " } {XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 2 " (" } {TEXT 256 4 "blue" }{TEXT -1 8 ") to 1 (" }{TEXT 261 3 "red" }{TEXT -1 23 ", again). [35, 35, 142]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "g := x -> 1+x^2;\nn := 7;\n alias(C=binomial):\nB := (k,x)->g(k/n)*C(n,k)*x^k*(1-x)^(n-k);\nSum(B( k,x),k=0..n);\ng := unapply(simplify(value(%)),x);\nplot([seq(B(k,x),k =0..n),Sum(B(k,x),k=0..n),g(x)],x=0..1,\n color=[seq(COLOR(HUE,k/ (n+1)),k=0..n),brown,navy],\n thickness=2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&*$)9$\"\"#\"\"\"F1F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"nG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BGf*6$%\"kG%\"xG6\"6 $%)operatorG%&arrowGF)**-%\"gG6#*&9$\"\"\"%\"nG!\"\"F3-%\"CG6$F4F2F3)9 %F2F3),&F3F3F:F5,&F4F3F2F5F3F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$SumG6$**,&*&\"#\\!\"\"%\"kG\"\"#\"\"\"F-F-F--%\"CG6$\"\"(F+F-)%\"xG F+F-),&F-F-F3F*,&F1F-F+F*F-/F+;\"\"!F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(\"\"\"F-*&#F-\"\"(F-9 $F-F-*&#\"\"'F0F-*$)F1\"\"#F-F-F-F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 435 325 325 {PLOTDATA 2 "6/-%'CURVESG6$7W7$$\"\"!F)$\"\"\"F)7$$\"3ILLL 3x&)*3\"!#>$\"3Y'=*f8]fh#*!#=7$$\"3emmm;arz@F/$\"3cK,ytoUq&)F27$$\"3.+ +D\"y%*z7$F/$\"3_n0i&)>\\0!)F27$$\"3[LL$e9ui2%F/$\"3/pjDQ8ysuF27$$\"33 ++voMrU^F/$\"3D2#ox#yE5pF27$$\"3nmmm\"z_\"4iF/$\"3C4+Y4fW%Q'F27$$\"3em mmm6m#G(F/$\"3CU[nnZ:!*eF27$$\"3[mmmT&phN)F/$\"3qCbw')[.HaF27$$\"3CLLe *=)H\\5F2$\"3kQ,>SF27$$\"3 qmm;/RE&G#F2$\"3-.$p6[vki\"F27$$\"3\")*****\\K]4]#F2$\"3NC>7b`lL8F27$$ \"3$******\\PAvr#F2$\"3w:udQ3J'3\"F27$$\"3)******\\nHi#HF2$\"3/>N=&))3 D'))F/7$$\"3jmm\"z*ev:JF2$\"3s\\3W!>T\"GtF/7$$\"3?LLL347TLF2$\"3[m6-Mf 20eF/7$$\"3,LLLLY.KNF2$\"3+HEm*y'eNZF/7$$\"3w***\\7o7Tv$F2$\"3-p8A#pk \"3PF/7$$\"3'GLLLQ*o]RF2$\"3#35(zn*HW'HF/7$$\"3A++D\"=lj;%F2$\"3w=%)[/ !*=*H#F/7$$\"31++vV&RY2aF2$\"3Rx R%zCH*3VFdt7$$\"39mm;zXu9cF2$\"3Y)\\;t\"Hj=JFdt7$$\"3l******\\y))GeF2$ \"3=8'Q&\\6m'>#Fdt7$$\"3'*)***\\i_QQgF2$\"3?vMMV+YJ:Fdt7$$\"3@***\\7y% 3TiF2$\"3#>'32b=Kg5Fdt7$$\"35****\\P![hY'F2$\"3c-Q-KrP#)o!#@7$$\"3kKLL $Qx$omF2$\"3)e#3=VB2cXFhv7$$\"3!)*****\\P+V)oF2$\"3)*H/Uk,H]GFhv7$$\"3 ?mm\"zpe*zqF2$\"399A)H*\\=5=Fhv7$$\"3%)*****\\#\\'QH(F2$\"31Iu,Wryi5Fh v7$$\"3GKLe9S8&\\(F2$\"3+E!>Rrhr='!#A7$$\"3R***\\i?=bq(F2$\"3+C-Z0H0[L Fbx7$$\"3\"HLL$3s?6zF2$\"3(f+n3Ls[t\"Fbx7$$\"3a***\\7`Wl7)F2$\"3g*)3(4 (pM+\")!#B7$$\"3#pmmm'*RRL)F2$\"3kNCeOM:jNFby7$$\"3Qmm;a<.Y&)F2$\"3%=l =)[knt8Fby7$$\"3=LLe9tOc()F2$\"3x_h**[A#4g%!#C7$$\"3u******\\Qk\\*)F2$ \"3a/fk\\ZW59Fbz7$$\"3CLL$3dg6<*F2$\"3wg*[8P+&G#!#F7$$\"3A++v=5s#y*F 2$\"3a?%*pd:D'G#!#H7$F*F(-%'COLOURG6$%$HUEGF)-F$6$7`o7$F(F(7$$\"3WmmmT &)G\\aFdt$\"3\"\\D]by!QJQF/7$F-$\"3zp+w!zuUT(F/7$$\"3$*****\\ilyM;F/$ \"3G7ZUG4)e2\"F27$F4$\"31cw&=k\"[(Q\"F27$$\"3IL$e*)4bQl#F/$\"3uu0]UZvS ;F27$F9$\"3S3.<=41y=F27$$\"3vm;ajW8-OF/$\"3q/sq2a***4#F27$F>$\"3Pq\\W: y82BF27$$\"3xm;H2Q\\4YF/$\"3i$\\=h(47BDF27$FC$\"3;q5E9\"*)=s#F27$$\"3Q L$3-8Lfn&F/$\"3V.M?w%)>/HF27$FH$\"3YZU`BAyqIF27$FM$\"3yT\"H;J>8O$F27$F R$\"3'RWiL5/lf$F27$$\"36++v=ddC%*F/$\"3m\"[V'R-v!y$F27$FW$\"3v-XB?S3?R F27$$\"3')***\\(=JN[6F2$\"3*[(>mvL/8SF27$Ffn$\"3'>\\8-@xX2%F27$$\"3#)* **\\(o3p)H\"F2$\"3!e.`JBE^4%F27$$\"33LLe*ot*\\8F2$\"3(*4JnXZ]3TF27$$\" 3KmmT5lD,9F2$\"3GUUQ026:TF27$F[o$\"3=Lb@&GG`6%F27$$\"3emm\"z4wb]\"F2$ \"3H4+dCxA4TF27$$\"3/LLekGhe:F2$\"3'zp$HJg3(4%F27$$\"3x***\\7j\\;h\"F2 $\"3y_tcAQGzSF27$F`o$\"3cL\\Y:u=cSF27$$\"39LL3_(>/x\"F2$\"3ohJ5d7t&*RF 27$Feo$\"3EVWA%*QD=RF27$Fjo$\"3$Qy_FF]ar$F27$F_p$\"3?G'yfL#R+NF27$Fdp$ \"3T&*)=t+U9B$F27$Fip$\"3%G)*4XD8^%HF27$F^q$\"3mRB)y<%fjEF27$Fcq$\"3_Y 06T_l4CF27$Fhq$\"38(3*Rl8=;@F27$F]r$\"3+![\\R=>)y=F27$Fbr$\"3]'**)eMaH >;F27$Fgr$\"3ATFyqCd19F27$F\\s$\"3!zBTr9\"F27$Fas$\"3S\"zddC4'45F27 $Ffs$\"3cqqvk3\\!R)F/7$F[t$\"3t%>aM=8u+(F/7$F`t$\"3)eE=OY'3/dF/7$Fft$ \"3%*\\il%Q%=XXF/7$F[u$\"31<'*>ZM2'o$F/7$F`u$\"3;.$)fFr.,HF/7$Feu$\"3) 4Mhx)fBIAF/7$Fju$\"3;J[pe4$fp\"F/7$F_v$\"3;x*z6*[0z7F/7$Fdv$\"3'\\Ul)z !Q$\\\"*Fdt7$Fjv$\"3pGzt$4^`i'Fdt7$F_w$\"3u(HvYD\"fvXFdt7$Fdw$\"3#=@Ni JO()=$Fdt7$Fiw$\"3u2-))4V<\"3#Fdt7$F^x$\"3Uv\"G>xb]M\"Fdt7$Fdx$\"3A[FI Ty*)o\")Fhv7$Fix$\"3?(353w]Qx%Fhv7$F^y$\"39`%4+O>Gb#Fhv7$Fdy$\"3?mQ^zb $\\H\"Fhv7$Fiy$\"39*eIgezg'eFbx7$F^z$\"3!H%HKi\"*e`BFbx7$Fdz$\"3GzO2`^ IJ()Fby7$Fiz$\"3G0;!G=L-;#Fby7$F_[l$\"3C!Qs9Ns\\G%Fbz7$Fe[l$\"3ObAv5fo \"z$F][l7$F[\\l$\"3i%p)4dFfyuFi[lF`\\l-Fb\\l6$Fd\\l#F+\"\")-F$6$7aoFh \\l7$$\"3ALL$3FWYs#Fdt$\"30w;46bA3F/7$F>$\"3Mz11&))* zZJF/7$FC$\"3$Gmo*ppFF2$ \"3cJVYCt(e`$F27$$\"3n*****\\-w=#GF2$\"3s$)o@%y;)RNF27$$\"3c******\\G0 uGF2$\"3n,(RUZ-/a$F27$F^q$\"3pM>p`gqPNF27$$\"3-L$eky#*4-$F2$\"3\"GvQiK kY_$F27$Fcq$\"3z_@NyO`,NF27$$\"3>+]7.%Q%GKF2$\"3W?Bj#[>;Y$F27$Fhq$\"3< )yroX.\"4MF27$F]r$\"3)Qtgs%H6%H$F27$Fbr$\"3%=PEsQ&*[7$F27$Fgr$\"3%epr( HMM\\HF27$F\\s$\"3$z5)y>IlNFF27$Fas$\"3o]/Q)y[y^#F27$Ffs$\"3sB/S*pQ>G# F27$F[t$\"3Q3X&*>#HD1#F27$F`t$\"3'Q9%QLSyF=F27$Fft$\"3M5d\\:Nz!f\"F27$ F[u$\"3_r.az0[$R\"F27$F`u$\"3gmgge?d#>\"F27$Feu$\"33&)f5ppk+5F27$Fju$ \"3[h_S*3?&*H)F/7$F_v$\"3]1I6=xS=oF/7$Fdv$\"3[/Z@W&e]P&F/7$Fjv$\"3!3)y R)\\XwD%F/7$F_w$\"3!RGwjx$*fC$F/7$Fdw$\"3q@<-jJJ#[#F/7$Fiw$\"3')egQtn* 4!=F/7$F^x$\"3'*GuYW]?#H\"F/7$Fdx$\"3!R`5+U#*z!))Fdt7$Fix$\"3z&R$\"3C*oJAxKfW#Fdt7$FC$\"3i'py`b&)op%Fdt7$FH$\"38]#*z=x7 ,zFdt7$FR$\"39miljaVb!>\"RCBfH:F27$F_p$\"3gSkx\"p)>.=F27$Fdp$\"3jT&\\VBQ+6#F27$Fip$\"3 \"fcaU\\5wS#F27$F^q$\"3g9fjsx&fn#F27$Fcq$\"3Gxr&Qy!y(*GF27$Fhq$\"3qu!= -sWx7$F27$F]r$\"3crXpl=C*G$F27$Fbr$\"3I%zs(yFQMMF27$$\"3Im;HK5S_QF2$\" 3I8t;@o5$[$F27$Fgr$\"3:A[fPZ-ANF27$$\"3Em;H#GF&eSF2$\"3!)zY%yP,Lb$F27$ F\\s$\"3)>oq5'pbsNF27$$\"3')****\\iB0pUF2$\"3A&y[n;J(zNF27$Fas$\"3#H@n G9,hd$F27$$\"3wm;zWG))yWF2$\"3%z4R8dg4c$F27$Ffs$\"3w5a*)G3ZMNF27$$\"3m L$3F9jkttX$F27$F`t$\"3v_1O/-bNLF27$ Fft$\"3E$3II)4)3<$F27$F[u$\"32Wp7%4T,+$F27$F`u$\"3k\"3M8w$)z\"F27$Fjv$\"3-3OVq%Q#e:F27$F_w$\"3cR\"o^p]9J\"F27$Fdw$\"34'*p '\\0@05\"F27$Fiw$\"3cIt(\\e!3w))F/7$F^x$\"3w[\"\\*\\O6qqF/7$Fdx$\"3$pv vib;(3aF/7$Fix$\"3W(eiea8.-%F/7$F^y$\"3^>H=c.&*>GF/7$Fdy$\"3dpRW,QB->F /7$Fiy$\"3kBduc5x*=\"F/7$F^z$\"3U[CLK4+]oFdt7$Fdz$\"3/!ei?xj9s$Fdt7$Fi z$\"3s!=,*R8w_:Fdt7$F_[l$\"3yt;\\NLI^bFhv7$Fe[l$\"3q*4MT[77;\"Fhv7$F[ \\l$\"3?U\"e\"zj?+*)FbyF`\\l-Fb\\l6$Fd\\l#\"\"$F`[m-F$6$7ZFh\\l7$F4$\" 3a+A*Rxxo+\"Fbx7$F>$\"3wl*[WET7;\"Fhv7$FH$\"3gDSbgP)Q%eFhv7$FR$\"3I7\\ w;^D)y\"Fdt7$FW$\"3&*\\is#oaC9%Fdt7$Ffn$\"3AlM6^0LOxFdt7$F[o$\"3.H'Q]U )oC8F/7$F`o$\"3\"*Gw$RMq#>@F/7$Feo$\"3-RfgW%\\b;$F/7$Fjo$\"3<#*z,w,LDX F/7$F_p$\"3m\")*['RnfnfF/7$Fdp$\"3%H!zu61%>'yF/7$Fip$\"3Q&pFN**QP+\"F2 7$F^q$\"3ahl%\\(otO7F27$Fcq$\"3sWcEo_Dl9F27$Fhq$\"3/2k.krK`9HiI#F27$Fgr$\"3%pf->+/)pDF27$F\\s$\"3EVZ(3 L51&GF27$Fas$\"3)G)f*p/[L5$F27$Ffs$\"3p0a&y?<\\M$F27$F[t$\"3k\\d$QAe5a $F27$F`t$\"3;/N**p$4#>PF27$Fft$\"31<5yh=xhQF27$$\"3ALLLe,]6`F2$\"3W/tJ :3'*3RF27$F[u$\"3#z!\\\"G#))fYRF27$$\"3PL$e*[K56bF2$\"3-2ArC*fh(RF27$F `u$\"3w&*o=OX)Q*RF27$$\"3*GL$e9i\"=s&F2$\"3y$=t$)3\"\\**RF27$Feu$\"3qn pV$y()>*RF27$$\"3I***\\ibOO$fF2$\"3[]W%e$G&=(RF27$Fju$\"3@xc'HmV!RRF27 $$\"34**\\(=-N(RhF2$\"3QCHTCyH&*QF27$F_v$\"3O\\?U%[*))RQF27$Fdv$\"3A\\ 0)*4MOwOF27$Fjv$\"3G>Q1Et[%[$F27$F_w$\"3D-A>HKSPKF27$Fdw$\"3q)R!e<%G6) HF27$Fiw$\"3M>F/7$F_[l$\"3O$*eU.\"3K@*Fdt7$$\"3sK$eRA5\\Z*F2$\"3I\\X)Q6>-c&Fdt7 $Fe[l$\"3KiPHLI3jHFdt7$$\"3C+++]oi\"o*F2$\"3kX-s&e=6N\"Fdt7$F[\\l$\"3U rwrgR&pZ%FhvF`\\l-Fb\\l6$Fd\\l#F+\"\"#-F$6$7`oFh\\l7$F4$\"3*4^DTW/?_\" Fbz7$F>$\"3/g#43!zbZLFby7$FH$\"3*e\")[\\#3ZCEFbx7$FR$\"3?b&4>\")=h5\"F hv7$FW$\"3Ygq$e+OVH$Fhv7$Ffn$\"3sso>J[azuFhv7$F[o$\"3j=8(**z=r_\"Fdt7$ F`o$\"33l8U/7@rGFdt7$Feo$\"3u=z$GlS$f\\Fdt7$Fjo$\"3\"ev&e'e)GH\")Fdt7$ F_p$\"3#pk*\\Dj<*>\"F/7$Fdp$\"3!37\"Qphnyn?)=( 4\"Q\"F27$Fas$\"3\"zLAI4I_j\"F27$Ffs$\"34J'p\"394A>F27$F[t$\"3%['*yJ6` @?#F27$F`t$\"3uA(3UGL!=DF27$Fft$\"3Ci'fJAbd&GF27$F[u$\"3'Rn!y(>KB:$F27 $F`u$\"35Up*=$p&*oMF27$Feu$\"3#R@0JG]Vy$F27$Fju$\"3!Qz!)>wOH2%F27$F_v$ \"38<qip:UD&[F27$$\"3WK$ek`H@)pF2$\"3;L O&*eiR&)[F27$Fdw$\"3-5xD-`K.\\F27$$\"31*\\(oa_VLrF2$\"3E<5Yy*\\l!\\F27 $$\"3-L$e9\"=\"p=(F2$\"3'Re'Hl9)\\!\\F27$$\"3)p;H#o$)QSsF2$\"33E^)o&Q^ )*[F27$Fiw$\"3!4p$Gp\"[q)[F27$$\"3im;zp%*\\%R(F2$\"3e/k_NRg^[F27$F^x$ \"3czA!=w6wz%F27$Fdx$\"3[H@9Md9BYF27$Fix$\"3c)G7;)p#3P%F27$F^y$\"3mzD% z0$Q@SF27$$\"3nK$e*[ACI#)F2$\"3/Zw'y(3SBQF27$Fdy$\"3!Qdn[y#Q2OF27$$\"3 lmmTge)*R%)F2$\"3Sdjb]U=pLF27$Fiy$\"3FjgKx&R_6$F27$$\"3M+]PM&*>^')F2$ \"3Uzk9\")G%*\\GF27$F^z$\"3A:.VE8ttDF27$$\"3Ym;H#e0I&))F2$\"3Q!*H'Rr&y 7BF27$Fdz$\"3Q-')o'*\\nZ?F27$$\"31nmT5ASg!*F2$\"37tadu2`U$\"3u8$H* [;S(R&F][l7$FH$\"3c.V`:)QAf'Fbz7$FR$\"3ML\")*Q(eqEQFby7$FW$\"3]9/rU$3` Y\"Fbx7$Ffn$\"3TYlbAQ^WSFbx7$F[o$\"3?$)Qk$>Bl%)*Fbx7$F`o$\"3uvX<5&zc<# Fhv7$Feo$\"3)*[M)*R.gXVFhv7$Fjo$\"3EQ401N\"y;)Fhv7$F_p$\"33t%HOrvxM\"F dt7$Fdp$\"3%z9%3alo]AFdt7$Fip$\"332dJ))GY(f$Fdt7$F^q$\"33y`wh8GZaFdt7$ Fcq$\"39fh+3uHDxFdt7$Fhq$\"3X](=`:Uh8\"F/7$F]r$\"3)o>dcCm-a\"F/7$Fbr$ \"3sev=a@XW@F/7$Fgr$\"3:?MU)=?6#GF/7$F\\s$\"3/Yn!fX)\\UPF/7$Fas$\"3D&= nKrO#>[F/7$Ffs$\"3QZ*)[=N^xhF/7$F[t$\"35KpqB6sfwF/7$F`t$\"3'e60nJd]`*F /7$Fft$\"3C2k)*=N:\"=\"F27$F[u$\"3![!338'*G39F27$F`u$\"3KY&QzA2_o\"F27 $Feu$\"3=#)>0ZW_1?F27$Fju$\"3!fiC+'*eaN#F27$F_v$\"3\"e&Rx:;fCFF27$Fdv$ \"3!Q*fijD6oJF27$Fjv$\"3[ZPM-\"G'F27$F^y$\"38woy2^Y=mF27$Fdio$\"3Oa\"Gqs#fc(oF27$F\\[p$\"3K0Hl]v/tnF27$Fdz$\" 37\"z'*)z/()>mF27$Fd[p$\"3)))[0$pxOvjF27$Fiz$\"3of*[#GJ:\\gF27$F\\\\p$ \"3qS#>&)>(z!o&F27$F_[l$\"3HZ(o)y/)HB&F27$Fj^o$\"3]?!pBc\\(fYF27$Fe[l$ \"3?IFr4^XyRF27$$\"3B+]il(z5j*F2$\"3\\n_&z`w7h$F27$Fb_o$\"3UzD9fs\"f@$ F27$$\"3B+]PMR\"z#F27$F[\\l$\"3Vv,==$**eL#F27$F`]p$ \"3auTOrt56=F27$Fe]p$\"35[1J!pkzC\"F27$F_^p$\"3cog3JzE[kF/F`\\l-Fb\\l6 $Fd\\l#FednFbim-F$6$7WFh\\l7$F4$\"3O\"y4?W0bn%F_\\l7$F>$\"3\\q#pq%z,SP Fi[l7$FH$\"3'Qd/%[cL;rFc[l7$FR$\"3w_F/&R'e*o&F][l7$FW$\"3)R`b&H*Fhv7$F]r$\"3!3LPA(z_r8Fdt7$Fbr$\"3 J,iOF.v,@Fdt7$Fgr$\"3&o_Q6_yU+$Fdt7$F\\s$\"3a(oAucU%eVFdt7$Fas$\"3Hb([ mxNR5'Fdt7$Ffs$\"3Bdh\"zN%oK&)Fdt7$F[t$\"3x,y2!y8]9\"F/7$F`t$\"37\")=$ =lM'HQR&*4#F/7$F[u$\"3.dIW70'Qq#F/7$F`u$\"3vzKn'*zN= NF/7$Feu$\"3IK_#))=\\Ad%F/7$Fju$\"3in))3J,JaeF/7$F_v$\"3A^<'[\"Q]wtF/7 $Fdv$\"3Eo:()37c_%*F/7$Fjv$\"3)3V$)y\\dE<\"F27$F_w$\"3_R&3@57dY\"F27$F dw$\"3IG&Rn\"[Q$y\"F27$Fiw$\"3K\">(QuT^'>#F27$F^x$\"3)>?3gtwvl#F27$Fdx $\"3ji%>+'4%eA$F27$Fix$\"3E6W,\\^3zQF27$F^y$\"3yRbX\\MN\"o%F27$Fdy$\"3 #y/g(4yZ%e&F27$Fiy$\"3I/!3pHy&emF27$F^z$\"3sJr-8A,%*yF27$Fdz$\"3'[F>;& *4v>*F27$Fiz$\"33`&[$G0W\"4\"!#<7$F\\\\p$\"37nfWz))ow6Fedq7$F_[l$\"3'4 wdj$>en7Fedq7$Fj^o$\"3!R)yrlw0r8Fedq7$Fe[l$\"3O)=&*y#3p\"[\"Fedq7$Fb_o $\"3A^'\\-,hYf\"Fedq7$F[\\l$\"3X%)**3Z]$\\r\"Fedq7$Fe]p$\"3-)*>r#*Hx_= Fedq7$F*$F\\`oF)-Fb\\l6$Fd\\l#\"\"(F`[m-F$6$7ao7$F($F+%*undefinedG7$$ \"3/LL3x1h6oFhv$\"3*e>&3d5=+5Fedq7$$\"3immTN@Ki8Fdt$\"3)4^Pfzi.+\"Fedq 7$$\"3#***\\7.K[V?Fdt$\"3%[%pb;_a+5Fedq7$Fe[m$\"3z'\\V*=$G2+\"Fedq7$$ \"3%)***\\iSmp3%Fdt$\"3ucz,pl4,5Fedq7$Fj\\l$\"3&**)3;YvY,5Fedq7$F]\\m$ \"3<#=_;o815Fedq7$F9$\"3%)=z'>h$\"3kO`-t`.75Fedq7$FH$\"3 9/'3.x0$>5Fedq7$FR$\"3*[RI7Z*HF5Fedq7$FW$\"3$>h$3*pFf.\"Fedq7$Ffn$\"3N KB9Ll_W5Fedq7$F[o$\"3cs/AZy.a5Fedq7$F`o$\"3?(*4.X[_k5Fedq7$Feo$\"3kX$H 7NOc2\"Fedq7$Fjo$\"3)H3_]Y^x3\"Fedq7$F_p$\"3]#fb@L)**)4\"Fedq7$Fdp$\"3 Ouj>F]I76Fedq7$Fip$\"3(>%oBXTNE6Fedq7$F^q$\"3hlq%*z^aS6Fedq7$Fcq$\"3?( y-V^')R:\"Fedq7$Fhq$\"3KoK:9llq6Fedq7$F]r$\"3%>%*zi2i`=\"Fedq7$Fbr$\"3 !f\"yzv>9.7Fedq7$Fgr$\"3F2sBO[[>7Fedq7$F\\s$\"3'R_z=yo!Q7Fedq7$Fas$\"3 3uT,//Sc7Fedq7$Ffs$\"31Gp1A#)=w7Fedq7$F[t$\"3ql;g;R&\\H\"Fedq7$F`t$\"3 cyG-?J$eJ\"Fedq7$Fft$\"3*>Rwco@#Q8Fedq7$F[u$\"3aT(y&*\\#He8Fedq7$F`u$ \"3;'GcnZx0Q\"Fedq7$Feu$\"3_Y88nIE/9Fedq7$Fju$\"3SrF<\\o3G9Fedq7$F_v$ \"3SD6gN9v^9Fedq7$Fdv$\"35&H$)4CM(y9Fedq7$Fjv$\"3[?T\")[Vh.:Fedq7$F_w$ \"3V2V0?D%3`\"Fedq7$Fdw$\"3#=G3S$p5c:Fedq7$Fiw$\"3i7LFl7P%e\"Fedq7$F^x $\"3\"RF#*Gsz:h\"Fedq7$Fdx$\"3.Xh,&oc1k\"Fedq7$Fix$\"3W4/k$\\8(p;Fedq7 $F^y$\"3+Sofs$*z+Fedq7$Fb_o$\"3wb?0,n^X> Fedq7$F[\\l$\"3_pRF)fbE'>Fedq7$F`]p$\"3MGtIkm#>(>Fedq7$Fe]p$\"3')>hg'3 T7)>Fedq7$Fj]p$\"3)G0bjb9f)>Fedq7$F_^p$\"3oV.Fedq7$$\"3ACc,\"> g#f**F2$\"3ItyA%3XH*>Fedq7$Fd^p$\"3s#*>08SH&*>Fedq7$$\"357y]&4I'z**F2$ \"3[Bli$\\pk*>Fedq7$$\"3uu=nj+U')**F2$\"3q,Fk^ck(*>Fedq7$$\"3OPf$=.5K* **F2$\"37F05([A))*>Fedq7$F*Fefq-Fb\\l6&%$RGBG$F[_p!\"\"$\"$l\"!\"$Ffdr -F$6$7SF'7$F4$\"3(*z@R@6_.5Fedq7$F>$\"3u.lGyuC25Fedq7$FH$\"3cgsq5[<75F edq7$FR$\"3A;g'QVAz,\"Fedq7$FW$\"3\"z:rdMFW-\"Fedq7$Ffn$\"3VVu'y[d6.\" Fedq7$F[o$\"3R%QOi;N)Q5Fedq7$F`o$\"3k%)3d7U`Z5Fedq7$Feo$\"3=q+L\">tp0 \"Fedq7$Fjo$\"3Q2_vl@[n5Fedq7$F_p$\"3-8aWA.Tx5Fedq7$Fdp$\"3UOCJE+M*3\" Fedq7$Fip$\"3F['f)H67-6Fedq7$F^q$\"3ew:-#)))>:6Fedq7$Fcq$\"3htzQo;sF6F edq7$Fhq$\"3@B*R!=RTV6Fedq7$F]r$\"3'H/r<^)Qd6Fedq7$Fbr$\"3q3A!>\"Fedq7$F\\s$\"3Dx_]#\\2$37Fedq7$Fas$\"3VNS&*y9FE7 Fedq7$Ffs$\"3u#>9L#eyX7Fedq7$F[t$\"3'o))3PV*Rk7Fedq7$F`t$\"3sYs&\\!4A& G\"Fedq7$Fft$\"3IzrlEjm28Fedq7$F[u$\"3$ypi3b$)yK\"Fedq7$F`u$\"3)>A'*)z zU]8Fedq7$Feu$\"3oOK:9@\\u8Fedq7$Fju$\"3%HB$*H!\\z)R\"Fedq7$F_v$\"3tXr %=EDIU\"Fedq7$Fdv$\"3mV))=ZTv]9Fedq7$Fjv$\"3Y<4c]/Tw9Fedq7$F_w$\"3DeL \"Q$zd/:Fedq7$Fdw$\"3w;xP6@zI:Fedq7$Fiw$\"3qIbtg??g:Fedq7$F^x$\"3c'fkk x!f)e\"Fedq7$Fdx$\"3j,![d\\2!>;Fedq7$Fix$\"3m@\\xb*y%\\;Fedq7$F^y$\"3= za&=Fedq7$F_[l$\"3 q?LcCuF')=Fedq7$Fe[l$\"30%*\\k$42O#>Fedq7$F[\\l$\"3](ztw%G0g>FedqF[fq- Fb\\l6&Fcdr$\"$P\"FhdrF[^s$\"$d&Fhdr-%*THICKNESSG6#F\\`o-%+AXESLABELSG 6$Q\"x6\"Q!Ff^s-%%VIEWG6$;F(F*%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We turn again to the function \+ " }}{PARA 256 "" 0 "" {TEXT -1 2 "\n " }{XPPEDIT 18 0 "g(x) = PIECEWIS E([0, x <= 0],[exp(-1/(x^2)), 0 < x]);" "6#/-%\"gG6#%\"xG-%*PIECEWISEG 6$7$\"\"!1F'F,7$-%$expG6#,$*&\"\"\"F4*$F'\"\"#!\"\"F72F,F'" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "which was approximated poorly with Taylor's polynomials, and well wit h Legendre's polynomials. " }}{PARA 0 "" 0 "" {TEXT -1 52 "Bernstein p olynomials can be defined for a function " }{XPPEDIT 18 0 "g(x)" "6#-% \"gG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 4 " by " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "B(n,x) = sum(matrix([[n], [k]])*g(2*k/n-1)*((1 +x)/2)^k*((1-x)/2)^(n-k),k = 0 .. n);" "6#/-%\"BG6$%\"nG%\"xG-%$sumG6$ **-%'matrixG6#7$7#F'7#%\"kG\"\"\"-%\"gG6#,&*(\"\"#F4F3F4F'!\"\"F4F4F;F 4)*&,&F4F4F(F4F4F:F;F3F4)*&,&F4F4F(F;F4F:F;,&F'F4F3F;F4/F3;\"\"!F'" } {TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 " matrix([[n], [k]])= n!/ (k!*(n-k!))" "6#/-%'matrixG6#7$7#%\"nG7#%\"kG* &-%*factorialG6#F)\"\"\"*&-F.6#F+F0,&F)F0-F.6#F+!\"\"F0F7" }{TEXT -1 30 " is the binomial coefficient. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 11 "Example 1: " }{XPPEDIT 18 0 "n=9" "6#/%\"nG\"\"*" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 269 "g := x -> piecewise(x<=0,0,exp(-1/(x^2))):\n'g( x)'=g(x);\nn := 9;\nalias(C=binomial):\nB := (k,x) -> C(n,k)*'g(2*k/n- 1)'*((1+x)/2)^k*((1-x)/2)^(n-k):\nSum(B(k,x),k=0..n);\nsimplify(evalf( value(%)));\nh := unapply(%,x):\nplot([h(x),g(x)],x=-1..1,color=[navy, brown],thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG -%*PIECEWISEG6$7$\"\"!1F'F,7$-%$expG6#,$*&\"\"\"F4*$)F'\"\"#F4!\"\"F8% *otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$**-%\"CG6$\"\"&%\"kG\"\"\"-%\"gG6#, &*(\"\"#F,F+F,%\"nG!\"\"F,F,F4F,),&*&F2F4%\"xGF,F,#F,F2F,F+F,),&F9F,*& F2F4F8F,F4,&F*F,F+F4F,/F+;\"\"!F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#, .*&$\"+ih@`&*!#6\"\"\")%\"xG\"\"$F(F(*&$\"+2T=sk)!#C7$$!3mmmmT%p\" e()F1$\"3epzfmD[tY!#B7$$!3:mmm\"4m(G$)F1$\"3N>]:LE'*Q:!#A7$$!3\"QLL3i. 9!zF1$\"3OM)*)=:07%QFF7$$!3\"ommT!R=0vF1$\"3Cx\"=(yVH*p(FF7$$!3u****\\ P8#\\4(F1$\"3Y.*or-l2U\"!#@7$$!3+nm;/siqmF1$\"3'4#ztu#p-Y#FV7$$!3[++]( y$pZiF1$\"3U\\t,-VM%)RFV7$$!33LLL$yaE\"eF1$\"3PAp@+:$G?'FV7$$!3hmmm\"> s%HaF1$\"3#*='*o$*[>M))FV7$$!3Q+++]$*4)*\\F1$\"3GzKm8'**>F\"!#?7$$!39+ ++]_&\\c%F1$\"3E2B7$=Q*zWcP)*f'Fjo7$$!3EM LLL7i)4#F1$\"3RPl<%y(*=7)Fjo7$$!3c****\\P'psm\"F1$\"3/eCxJ!4$35!#>7$$! 3')****\\74_c7F1$\"3NlmM\"*feE7Fcr7$$!3)3LLL3x%z#)Fcr$\"3k**Rdciu!\\\" Fcr7$$!3KMLL3s$QM%Fcr$\"3illk&)eUpZmI?DFcr7$$\"3s*)***\\PQ#\\\")Fcr$\"3f6\"F17$$\"3y)***\\7k.6aF1$\"3=l(4 kx93D\"F17$$\"3#emmmT9C#eF1$\"3yR4b%=1WR\"F17$$\"33****\\i!*3`iF1$\"3w >xB())*)yb\"F17$$\"3%QLLL$*zym'F1$\"3.xdTe%=)GF17$$\"3Nmm;HYt7vF1$\"3OB@ZcS0@@F17$$\"3Y*******p(G**yF1$ \"3@^<%\\#Go@BF17$$\"3]mmmT6KU$)F1$\"3'yK>y\"3:pDF17$$\"3fKLLLbdQ()F1$ \"37=fy$efr!GF17$$\"3[++]i`1h\"*F1$\"3iST%zyOF1-%'COLOURG6&%$RGBG$\"$P\"! \"$Fd[l$\"$d&Ff[l-F$6$7X7$F($F*F*7$F/F]\\l7$F6F]\\l7$F!$^#7$F^t$\"3N8S_Kf=>S!#$)7$Fct$\"3#4W$4K=Hi=!#Y7$Fht$\"3W%*oq)f v[d\"!#L7$F]u$\"3DC?m'4m9_)F-7$Fbu$\"3S3% *))F@7$F\\v$\"3-7>jR^2d7FV7$Fav$\"3w%[F\"zJu^()FV7$$\"3-mm\"H2fU'RF1$ \"3df0vqA$Qs\"Fjo7$Ffv$\"398blV`r#4$Fjo7$$\"3/mm\"HiBQP%F1$\"3G(QwJe'* zO&Fjo7$F[w$\"3A+tT\\MAT')Fjo7$F`w$\"3$\\$oeHI=.=Fcr7$Few$\"3Y%y'p^,R' G$Fcr7$Fjw$\"3?MoGqZ(\\B&Fcr7$F_x$\"3sZcJ:n/]xFcr7$Fdx$\"3kz`hNb&[0\"F 17$Fix$\"3A#34QA]%p8F17$F^y$\"31!G#pI:M+#F17$Fhy $\"3!>>f2_OmP#F17$$\"3a****\\P$[/a)F1$\"3,l@WO,_QDF17$F]z$\"3C " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 2: " }{XPPEDIT 18 0 "n=80" "6#/%\"nG\"#!)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "g := x -> piecewise(x<=0,0,exp(-1/(x^2))):\n'g(x)'=g(x);\nn := 80;\nDigits : = 20:\nalias(C=binomial):\nB := (k,x) -> C(n,k)*'g(2*k/n-1)'*((1+x)/2) ^k*((1-x)/2)^(n-k);\nSum(B(k,x),k=0..n);\nsimplify(evalf(value(%))):\n h := unapply(%,x):\nplot(['h(x)','g(x)'],x=-1..1,color=[navy,brown],th ickness=2);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6 #%\"xG-%*PIECEWISEG6$7$\"\"!1F'F,7$-%$expG6#,$*&\"\"\"F4*$)F'\"\"#F4! \"\"F8%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#!)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BGf*6$%\"kG%\"xG6\"6$%)operatorG%& arrowGF)**-%\"CG6$%\"nG9$\"\"\".-%\"gG6#,&*(\"\"#F3F2F3F1!\"\"F3F3F;F3 ),&*&#F3F:F39%F3F3F?F3F2F3),&F?F3*&#F3F:F3F@F3F;,&F1F3F2F;F3F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$**-%\"CG6$\"#!)%\"kG\"\"\"-% \"gG6#,&*(\"\"#F,F+F,%\"nG!\"\"F,F,F4F,),&*&F2F4%\"xGF,F,#F,F2F,F+F,), &F9F,*&F2F4F8F,F4,&F*F,F+F4F,/F+;\"\"!F*" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$!\"\"\"\"!$!+9gr ?5!#C7$$!5nmmmm;p0k&*!#?$!(YG^#F-7$$!5LLLL$3s%HaF1$\"&!f[F-7$$!5+++++]$*4)*\\F1$\"(*f\"e#F-7$$!5++ +++]_&\\c%F1$\")r`O#*F-7$$!5+++++]1aZTF1$\"+N4T>@F-7$$!5mmmm;/#)[oPF1$ \",h#Ht')GF-7$$!5LLLLL$=exJ$F1$\"-LqK`$*\\F-7$$!5LLLLLL2$f$HF1$\".)fMF Q0YF-7$$!5********\\PYx\"\\#F1$\"/%zOJ_1+&F-7$$!5LLLLLL7i)4#F1$\"1o0aD 026N!#D7$$!5********\\P'psm\"F1$\"2a`d]9m0a#Fhq7$$!5********\\74_c7F1$ \"3]=Xf<\\&4X\"Fhq7$$!4LLLLL3x%z#)F1$\"33l%Hk@A-!yFhq7$$!4LLLL$3s$QM%F 1$\"4ok:-eEsBE$Fhq7$$!2mmmmm\"zr)*F1$\"5lmr)**>$\"5bw3ls)f*)GN&Fhq7$$\"3++++v$Q#\\\")Fes$\"5:X'*3&pm'f2;F-7$$\"4L LLL$e\"*[H7Fes$\"5TikGEK0D=[F-7$$\"4+++++qvxl\"Fes$\"5u_Lwk[Kdk8!#B7$$ \"4++++]_qn2#Fes$\"5Nu'=l2)yokMFgt7$$\"4++++Dcp@[#Fes$\"5)>1\\Ss?*3**y Fgt7$$\"4++++]2'HKHFes$\"5`:_:rJXI7=!#A7$$\"4nmmmmwanL$Fes$\"5*GNN3&Qp \\cNFgu7$$\"4+++++v+'oPFes$\"5\"R@n3A98S!oFgu7$$\"4LLLLeR<*fTFes$\"5ml ]yG0,l`6!#@7$$\"4+++++&)Hxe%Fes$\"5'Ga>qe(32L>Fgv7$$\"4nmmm\"H!o-*\\Fe s$\"5B/$y!pP`evHFgv7$$\"4++++DTO5T&Fes$\"5%f'>L2\\R\"QV%Fgv7$$\"4nmmmm T9C#eFes$\"56aq)H#4fxYiFgv7$$\"4++++D1*3`iFes$\"5R7qhhQv(*[&)Fgv7$$\"4 LLLLL$*zym'Fes$\"5&z0#=%4w6A6\"F17$$\"4LLLL$3N1#4(Fes$\"5(fs8Cw%o`09F1 7$$\"4nmmm\"HYt7vFes$\"5@)[PY1-X$>!$` #7$Fis$\"5%oB_DD$f=>S!#&)7$F^t$\"55=bO4K=Hi=!#[7$Fct$\"5dq[pq)fv[d\"!# N7$Fit$\"5#Q0Uim4m9_)!#I7$F^u$\"5'3<\\8<_)3N*)!#F7$Fcu$\"5%[<#))=!*>3% *))Fhq7$Fiu$\"5!3/#>jR^2d7Fgt7$F^v$\"5Oy(\\F\"zJu^()Fgt7$$\"5lmmm\"H2f U'RF1$\"5Bgi0vqA$Qs\"Fgu7$Fcv$\"5%3L_bOM:F4$Fgu7$$\"5lmmm\"HiBQP%F1$\" 5)4nRwJe'*zO&Fgu7$Fiv$\"5#RII<%\\MAT')Fgu7$F^w$\"5H3SoeHI=.=Fgv7$Fcw$ \"5yN(y'p^,R'G$Fgv7$Fhw$\"5$*))QoGqZ(\\B&Fgv7$F]x$\"57\"el:`rY+v(Fgv7$ Fbx$\"55wz`hNb&[0\"F17$Fgx$\"5Rj#34QA]%p8F17$F\\y$\"5G8!G#pI:M+#F17$Ffy$\"5%o8>f2_OmP#F17$$\"5++++]P$[/a)F1$\"5$y`; Uk8?&QDF17$F[z$\"5%3sJx&*[U%*p#F17$F`z$\"5A;'HLM:Ov.$F17$Fez$\"5bfZ6Gi ([BN$F17$Fjz$\"5g@BWr6WzyOF1-F_[l6&Fa[l$\"\"&F)$\"$l\"Fd[lF_dl-%*THICK NESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fidl-%%VIEWG6$;F(Fjz%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "The convergence appears to be slow, as there is still a significa nt error even with a degree 80 Bernstein polynomial." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot('h(x) -g(x)',x=-1..1,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7]p7$$!\"\"\"\"!$!3=gzbx:5.x!#H7$$!3ommm;p0 k&*!#=$!3@*[X%RZ_;:!#I7$$!3wKL$3!#J7$$!3:mmm\"4m(G$)F1$\"3o4xT3s#\\F#!#L7$$!3\"QLL3i .9!zF1$!3Eyq\"pC1u)=!#K7$$!3\"ommT!R=0vF1$!39E`YA!>F?\"FK7$$!3u****\\P 8#\\4(F1$\"3:Bw,vb\"*4s% HaF1$\"3;,(G!\\sNl[!#N7$$!3Q+++]$*4)*\\F1$\"3K<]'*4'f7H(Feo7$$!39+++]_ &\\c%F1$\"37VgG&>h$\\!*Fen7$$!31+++]1aZTF1$\"3uo+#f3[$=@FK7$$!3umm;/#) [oPF1$\"3Q\"\\#\\nH\"o)GF?7$$!3hLLL$=exJ$F1$\"3S0\\!oNNN*\\F47$$!3*RLL LtIf$HF1$\"3<,)Qv%GQ0YF-7$$!3]++]PYx\"\\#F1$\"3;\\,@=Bl+]!#G7$$!3EMLLL 7i)4#F1$\"3SW#ec_q5^$!#F7$$!3c****\\P'psm\"F1$\"3!z+f]9m0a#!#E7$$!3')* ***\\74_c7F1$\"33L\\f<\\&4X\"!#D7$$!3)3LLL3x%z#)!#>$\"3Y0%Hk@A-!yF[s7$ $!3KMLL3s$QM%F_s$\"3mf@!eEsBE$!#C7$$!3]^omm;zr)*!#@$\"3!4()**>iPG-#)Fit7$$\"3I*******pvxl\"F1$\"3Mcgk[Kdk8F[t7$$\"3g)***\\7JFn=F1$ \"3K=eYYz9(>#F[t7$$\"3#z****\\_qn2#F1$\"3yh&='GqokMF[t7$$\"3=)**\\P/q% zAF1$\"3-*39K#G8!G&F[t7$$\"3U)***\\i&p@[#F1$\"3E%H)Q)p&>)*yF[t7$$\"3L) **\\(=GB2FF1$\"3So.b#yF$37!#?7$$\"3B)****\\2'HKHF1$\"3*yN@(\\/T.=Fbw7$ $\"3uJL$3UDX8$F1$\"3NKT;-.!>_#Fbw7$$\"3ElmmmZvOLF1$\"3\"zDW&*RglFbw7$$\"3-mm\"H2fU'RF1$\"3O)H yAEor>(Fbw7$$\"3A**\\PM#)3iSF1$\"3)*R)oB#odGyFbw7$$\"3UKL$eR<*fTF1$\"3 A\")HA4dyV%)Fbw7$$\"3C**\\P40(oE%F1$\"3Ov.^nm>&3*Fbw7$$\"3/mm\"HiBQP%F 1$\"3o/YiP/6\"o*Fbw7$$\"3%GLektw2[%F1$\"3L^hfC(Q>-\"F_s7$$\"3m******\\ )Hxe%F1$\"3?l%G4`[*o5F_s7$$\"37KLeR*)**)y%F1$\"30y27Bk0O6F_s7$$\"3ckm; H!o-*\\F1$\"3]TR53BSs6F_s7$$\"31JL3xS'G/&F1$\"31p![A/Wm<\"F_s7$$\"3c(* ***\\7ga4&F1$\"3c'=kE+8(y6F_s7$$\"31km\"HFbw7$$\"3\"HL$ek6, 1xF1$\"3afF%)H/Vw7Fbw7$$\"3Y*******p(G**yF1$\"3/VEQ`G$)fvF[t7$$\"3]mmm T6KU$)F1$!3mHX4!)pxZnFit7$$\"3a****\\P$[/a)F1$!3c(*)=Wx-L&GF[t7$$\"3fK LLLbdQ()F1$!36Tn:s%QeA%F[t7$$\"36+]i!*z>W))F1$!3)*\\H_e'f1m%F[t7$$\"3a mm\"zW?)\\*)F1$!3V=4u\"fBv!\\F[t7$$\"3'HL3_!HWb!*F1$!3O%QJ9`n0)\\F[t7$ $\"3[++]i`1h\"*F1$!3Y2#ph$Hj$*[F[t7$$\"3Y++++PDj$*F1$!3+8=YxncHVF[t7$$ \"3W++]P?Wl&*F1$!3^E:xv'3wJ$F[t7$$\"3A++v=5s#y*F1$!3nxv@okQC=F[t7$$\" \"\"F*$!3K'yd5z#fY=F--%'COLOURG6&%$RGBG$F*F*F]flFeel-%+AXESLABELSG6$Q \"x6\"Q!Fbfl-%%VIEWG6$;F(Feel%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Summary " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 654 "Taylor's Polynomials are better known a s polynomial approximations for a function on an interval. These polyn omials have the same derivatives as the function being approximated. L egendre Polynomials are polynomials that can be used to approximate a \+ function over an interval. They are chosen to minimize the area betwee n the polynomial approximation and the function being approximated. Be rnstein Polynomials will converge uniformly to the function being appr oximated, if the function is continuous. While the convergence of thes e Bernstein Polynomials may be uniform, it may also be slow.\nAt least two other polynomial fits could be constructed. Choose " }{TEXT 285 1 "n" }{TEXT -1 76 " points on the graph of the function and determine the polynomial of degree " }{XPPEDIT 18 0 "n-1" "6#,&%\"nG\"\"\"F%!\" \"" }{TEXT -1 34 " which exactly goes through those " }{TEXT 286 1 "n " }{TEXT -1 20 " points. Or, choose " }{TEXT 283 1 "n" }{TEXT -1 99 " \+ points on the graph of the function and find a regression polynomial o f degree anything less than " }{TEXT 284 1 "n" }{TEXT -1 151 " determi ned by those points. The reader might like to construct and compare th ese polynomials with the polynomials considered in the previous sectio ns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 29 "(a) Approximate the function " }{XPPEDIT 18 0 "f(x) = PIE CEWISE([0, x <= 0],[exp(-1/(x^2)), 0 < x]);" "6#/-%\"fG6#%\"xG-%*PIECE WISEG6$7$\"\"!1F'F,7$-%$expG6#,$*&\"\"\"F4*$F'\"\"#!\"\"F72F,F'" } {TEXT -1 87 " using an interpolating polynomial based on equally space d grid points in the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"! \"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 55 "(b) What happe ns as the number of points is increased? " }}{PARA 0 "" 0 "" {TEXT -1 31 "_______________________________" }}{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 31 "_______________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 53 "Find some polyno mial approximations for the function " }{XPPEDIT 18 0 "f(x) = PIECEWIS E([0, x <= 0],[x*exp(-1/(x^2)), 0 < x])" "6#/-%\"fG6#%\"xG-%*PIECEWISE G6$7$\"\"!1F'F,7$*&F'\"\"\"-%$expG6#,$*&F0F0*$F'\"\"#!\"\"F8F02F,F'" } {TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"! \"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "_______________ ________________" }}{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 31 "_______________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "9 0 \+ 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }