{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 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 "" 262 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 262 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 262 266 "" 0 1 0 0 0 0 0 0 1 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 "" 262 268 "" 0 1 0 0 0 0 0 0 1 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 "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 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 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 "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 }{PSTYLE "Normal " -1 258 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT 259 52 "Approximation of functions using partitions of unity" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, N anaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 26.3.200 7" }}{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 37 "Introduction - Bernstein po lynomials " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 52 "The Bernstein polynomial associated with a functio n " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(f(k/n)*B[k,n](x), k = 0 .. n);" "6#-%$SumG6$*&-%\"fG6#*&%\"kG\"\"\"%\"nG!\"\"F,-&%\"BG6$ F+F-6#%\"xGF,/F+;\"\"!F-" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "B[k,n](x) = matrix([[n], [k]])*(1-x)^(n-k)*x^k;" "6#/-&%\"BG6$%\"kG%\"nG6#%\"xG*( -%'matrixG6#7$7#F)7#F(\"\"\"),&F3F3F+!\"\",&F)F3F(F6F3)F+F(F3" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "matrix([[n], [k]])=n!/(k!*(n-k)!)" 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" form a " }{TEXT 260 18 "partition of un ity" }{TEXT -1 12 ", that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(B[k,n](x),k = 0 .. n) = 1;" "6#/-%$SumG6$-&%\"BG6$% \"kG%\"nG6#%\"xG/F+;\"\"!F,\"\"\"" }{TEXT -1 3 ", " }}{PARA 0 "" 0 " " {TEXT -1 5 " and " }{XPPEDIT 18 0 "B[k,n](x);" "6#-&%\"BG6$%\"kG%\"n G6#%\"xG" }{TEXT -1 27 " has its maximum value for " }{XPPEDIT 18 0 "0 <=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x=k/n" "6#/%\"xG*&%\"kG\"\"\"%\"nG!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 212 "In an effort to see why the Bernstein basis polynomials \+ can be used to approximate a continuous function, as in the following \+ example, we shall investigate similar families of functions in the fol lowing sections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 8 "Example:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = exp(-x^2); " "6#/-%\"fG6#%\"xG-%$expG6#,$*$F'\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 258 "" 0 "" {TEXT -1 38 "We construct the associated function " } {XPPEDIT 18 0 "g(n,x) = Sum(f(k/n)*B[k,n](x),k = 0 .. n);" "6#/-%\"gG6 $%\"nG%\"xG-%$SumG6$*&-%\"fG6#*&%\"kG\"\"\"F'!\"\"F2-&%\"BG6$F1F'6#F(F 2/F1;\"\"!F'" }{TEXT -1 7 ", when " }{XPPEDIT 18 0 "n=10" "6#/%\"nG\"# 5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "f := x -> exp(-x^2);\nn := 10;\ng := (n,x) - > Sum(f(k/n)*B(k,n,x),k=0..n);\nxvals := [seq(i/n,i=0..n)];\nyvals := \+ map(f,xvals);\npts := zip((x,y)->[x,y],xvals,yvals):\nplot([g(n,x),f(x ),pts],x=0..1,color=[red,magenta,blue], style=[line$2,point],sym bol=circle,linestyle=[1,2],thickness=[2,1],\n legend=[`g(n,x)`,` f(x)`,`f(x) points`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#% \"xG6\"6$%)operatorG%&arrowGF(-%$expG6#,$*$)9$\"\"#\"\"\"!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 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{TEXT -1 4 "Let " }{TEXT 271 1 "n" }{TEXT -1 45 " be a positive integer, and for each integer " }{TEXT 272 1 "k" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "0<=k" "6#1\"\"!%\"kG" }{XPPEDIT 18 0 "``<=n " "6#1%!G%\"nG" }{TEXT -1 39 " define a rectangular pulse centred at \+ " }{XPPEDIT 18 0 "x = k/n;" "6#/%\"xG*&%\"kG\"\"\"%\"nG!\"\"" }{TEXT -1 5 " by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[k,n] (x) = PIECEWISE([1, (2*k-1)/(2*n) < x and x < (2*k+1)/(2*n)],[0, other wise]);" "6#/-&%\"SG6$%\"kG%\"nG6#%\"xG-%*PIECEWISEG6$7$\"\"\"32*&,&*& \"\"#F0F(F0F0F0!\"\"F0*&F6F0F)F0F7F+2F+*&,&*&F6F0F(F0F0F0F0F0*&F6F0F)F 0F77$\"\"!%*otherwiseG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "S := (k,n,x)->piecewise(x>(2*k-1)/(2*n) and x<(2*k+1 )/(2*n),1,0):\n`S(k,n,x)`=S(k,n,x);\nplot(S(7,10,x),x=0..1,color=brown ,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%)S(k,n,x)G-%*PIECE WISEG6$7$\"\"\"32,&*&\"#5!\"\"%\"kGF)F)%\"xGF/#F)\"#?2,&F1F)*&F.F/F0F) F/F27$\"\"!%*otherwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 431 196 196 {PLOTDATA 2 "6'-%'CURVESG6#7co7$$\"\"!F)F(7$$\"3emmm;arz@!#>F(7$$\"3[L L$e9ui2%F-F(7$$\"3nmmm\"z_\"4iF-F(7$$\"3[mmmT&phN)F-F(7$$\"3CLLe*=)H\\ 5!#=F(7$$\"3gmm\"z/3uC\"F:F(7$$\"3%)***\\7LRDX\"F:F(7$$\"3]mm\"zR'ok;F :F(7$$\"3w***\\i5`h(=F:F(7$$\"3WLLL3En$4#F:F(7$$\"3qmm;/RE&G#F:F(7$$\" 3\")*****\\K]4]#F:F(7$$\"3$******\\PAvr#F:F(7$$\"3)******\\nHi#HF:F(7$ $\"3jmm\"z*ev:JF:F(7$$\"3?LLL347TLF:F(7$$\"3,LLLLY.KNF:F(7$$\"3w***\\7 o7Tv$F:F(7$$\"3'GLLLQ*o]RF:F(7$$\"3A++D\"=lj;%F:F(7$$\"31++vV&RY2aF:F(7$$\"39mm;zXu9cF:F(7$$\"3l******\\y 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mF*7$FeuF*7$FhuF*7$F[vF*7$$\"3y***\\(oTAq#*FgnF*7$F^vF*7$$\"31+DJ&**)4 A%*FgnF*7$$\"3sK$eRA5\\Z*FgnF*7$$\"31i!Rvi6:[*FgnF*7$$\"3U\"z>6.8\")[* FgnF*7$$\"3y?0qMWr%\\*FgnF*7$$\"3:]7GQeJ,&*FgnF(7$$\"3()3FWX'=X^*FgnF( 7$$\"3[mTg_9sF&*FgnF(7$$\"3!G3Fp1FTb*FgnF(F`v7$$\"3C+++]oi\"o*FgnF(Fcv Ffv-Fhv6$Fjv#F\\elFdal-F$6$7hnF^wF_wFawFQFWFZFhnF[oF^oFaoFdoFgoFjoF]pF `pFcpFfpFipF\\qF_qFbqFeqFhqF[rF^rFarFdrFgrFjrF]sF`sFcsFfsFisF\\tF_tFbt FetFhtF[uF^uFauFduFguFju7$F^dmF(F]v7$FbdmF(7$FedmF(7$FhdmF(7$F[emF(7$F ^emF(7$FaemF*7$FdemF*7$FgemF*7$FjemF*7$FavF*7$F]fmF*7$FdvF*7$F*F*-Fhv6 $Fjv#FdhlF\\^l-%+AXESLABELSG6$Q\"x6\"Q!Fjgm-%*THICKNESSG6#F\\\\m-%%VIE WG6$;F(F*%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Notice that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(S[k,n](x),k = 0 .. n) = 1;" "6#/-%$SumG6$ -&%\"SG6$%\"kG%\"nG6#%\"xG/F+;\"\"!F,\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 18 "for each value of " }{TEXT 273 1 "x" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "For each positive integer " }{TEXT 274 1 "n" }{TEXT -1 23 ", the s et of functions " }{XPPEDIT 18 0 "\{S[0,n], S[1,n], ` . . . `, S[n,n] \};" "6#<&&%\"SG6$\"\"!%\"nG&F%6$\"\"\"F(%(~.~.~.~G&F%6$F(F(" }{TEXT -1 8 " form a " }{TEXT 260 18 "partition of unity" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "n := 10;\nP := (n,x) -> Sum(S(k,n,x),k=0..n);\nplot(P(n,x),x=0..1, thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowG F)-%$SumG6$-%\"SG6%%\"kG9$9%/F3;\"\"!F4F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)$\"\"\"F) 7$$\"3emmm;arz@!#>F*7$$\"3[LL$e9ui2%F/F*7$$\"3nmmm\"z_\"4iF/F*7$$\"3[m mmT&phN)F/F*7$$\"3CLLe*=)H\\5!#=F*7$$\"3gmm\"z/3uC\"FY2aF " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Suppose that we are given a real valued function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 25 " defined on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\" \"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "Then the fun ction " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "g[n](x) = \+ Sum(f(k/n)*S[k,n](x),k = 0 .. n);" "6#/-&%\"gG6#%\"nG6#%\"xG-%$SumG6$* &-%\"fG6#*&%\"kG\"\"\"F(!\"\"F4-&%\"SG6$F3F(6#F*F4/F3;\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "provides a \"stepped\" or \"pi ecewise constant\" approximation for the function " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "g[n](x)" "6#-&%\"gG6#%\"nG6#%\"x G" }{TEXT -1 81 " is not continuous, in general, as there will usually be a jump discontinuity at " }{XPPEDIT 18 0 "x = (2*k+1)/(2*n);" "6#/ %\"xG*&,&*&\"\"#\"\"\"%\"kGF)F)F)F)F)*&F(F)%\"nGF)!\"\"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "k=0,` . . . `,n-1" "6%/%\"kG\"\"!%(~.~.~.~G,&% \"nG\"\"\"F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let \+ " }{XPPEDIT 18 0 "f(x)=exp(-x^2)" "6#/-%\"fG6#%\"xG-%$expG6#,$*$F'\"\" #!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "f := x -> exp(-x^2);\nn := 30;\ng \+ := (n,x) -> Sum(f(k/n)*S(k,n,x),k=0..n);\nplot([f(x),g(n,x)],x=0..1,y= 0..1,thickness=[1,2],\n color=[navy,red],nu mpoints=100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$% )operatorG%&arrowGF(-%$expG6#,$*$)9$\"\"#\"\"\"!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"gGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)-%$SumG6$*&-%\"fG6#*&% \"kG\"\"\"9$!\"\"F6-%\"SG6%F5F79%F6/F5;\"\"!F7F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7`q7$$\"\"!F)$\"\" \"F)7$$\"3C<<<<<$o0\"!#>$\"3w'y\"4pJ)))***!#=7$$\"3#QMMMMvj(>F/$\"3U)e nKq%4'***F27$$\"3aBBBB$)\\5IF/$\"3K\"e%f05%4***F27$$\"3!>9999w90%F/$\" 3;J1V0!*f$)**F27$$\"3ottttj](3&F/$\"3[Q(zdu]T(**F27$$\"37????!R![gF/$ \"3E7i&G/)[j**F27$$\"3CSRRR\\hUqF/$\"3]5D!>PC0&**F27$$\"3_yxxxn?r!)F/$ \"3z`%R1Nn]$**F27$$\"314444**\\'4*F/$\"3'eA$o%4&f<**F27$$\"30#>>>*R6:5 F2$\"3o/(3'fM[(*)*F27$$\"3YTTTTn+36F2$\"3cf#3%HE)z()*F27$$\"3$fddd(>e7 7F2$\"336hc)=SS&)*F27$$\"3=1111me<8F2$\"3(Qh_*yZ*y#)*F27$$\"3WCCCC!y(= 9F2$\"3D$ol?J>2!)*F27$$\"3HEEEE&p1^\"F2$\"3s7E(H5sVx*F27$$\"3G5555u$*> ;F2$\"3H<&o>c$*4u*F27$$\"3^SSSS;]7F2$\"3_M!e5&4uR'*F27$$\"3'eaaa%e0?? 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\"3N!*R_!38K$)*F2F^]q7$$\"3.*zz/w@h$)*F2Fcjl7$$\"3G:9R?\"R>%)*F2Fcjl7$ $\"3_JII![cx%)*F2Fcjl7$$\"3_'\\\\*>f-r)*F2Fcjl7$F^jlFcjl7$$\"3w!)zzzw9 Z**F2FcjlFbjl-Ffjl6&Fhjl$\"*++++\"F[[mF(F(-F_[m6#\"\"#-%+AXESLABELSG6$ Q\"x6\"Q\"yFhaq-%%VIEWG6$;F(F*F]bq" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "The approximating function is \+ assembled from multiples of the rectangle pulses determined by the val ues of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 22 " at the sample points " }{XPPEDIT 18 0 "x=k/n" "6#/%\"xG*&%\"kG \"\"\"%\"nG!\"\"" }{TEXT -1 6 ", for " }{XPPEDIT 18 0 "k=0,` . . . `,n " "6%/%\"kG\"\"!%(~.~.~.~G%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Approximation using narrow \"bell shape d\" pulses" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 96 "Let's try \"smoothing out\" the corners in the fun ctions which are pieced together to approximate " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 276 1 "n" }{TEXT -1 45 " be a positive integer and, for each integer " }{TEXT 275 1 "k" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "0<=k" "6#1\"\"!%\"kG" }{XPPEDIT 18 0 "``<=n " "6#1%!G%\"nG" }{TEXT -1 20 ", define a function " }{XPPEDIT 18 0 "U[ k,n];" "6#&%\"UG6$%\"kG%\"nG" }{TEXT -1 4 " by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "U[k,n](x) = PIECEWISE([(cos(Pi*(n*x-k)) +1)/2, (k-1)/n < x and x < (k+1)/n],[0, otherwise]);" "6#/-&%\"UG6$%\" kG%\"nG6#%\"xG-%*PIECEWISEG6$7$*&,&-%$cosG6#*&%#PiG\"\"\",&*&F)F7F+F7F 7F(!\"\"F7F7F7F7F7\"\"#F:32*&,&F(F7F7F:F7F)F:F+2F+*&,&F(F7F7F7F7F)F:7$ \"\"!%*otherwiseG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "n := 'n':\nU := (k,n,x) -> \+ piecewise(x>(k-1)/n and x<(k+1)/n,(cos(Pi*(n*x-k))+1)/2,0):\n`U(k,n,x) `=U(k,n,x);\nplot(U(3,5,x),x=0..1,color=brown,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%)U(k,n,x)G-%*PIECEWISEG6$7$,&*&#\"\"\"\"\" #F,-%$cosG6#*&%#PiGF,,&*&%\"nGF,%\"xGF,F,%\"kG!\"\"F,F,F,F+F,32,&*&,&F 7F,F,F8F,F5F8F,F6F8\"\"!2,&F6F,*&,&F7F,F,F,F,F5F8F8F>7$F>%*otherwiseG " }}{PARA 13 "" 1 "" {GLPLOT2D 411 200 200 {PLOTDATA 2 "6'-%'CURVESG6# 7`p7$$\"\"!F)F(7$$\"3emmm;arz@!#>F(7$$\"3[LL$e9ui2%F-F(7$$\"3nmmm\"z_ \"4iF-F(7$$\"3[mmmT&phN)F-F(7$$\"3CLLe*=)H\\5!#=F(7$$\"3gmm\"z/3uC\"F: F(7$$\"3%)***\\7LRDX\"F:F(7$$\"3]mm\"zR'ok;F:F(7$$\"3w***\\i5`h(=F:F(7 $$\"3WLLL3En$4#F:F(7$$\"3qmm;/RE&G#F:F(7$$\"3\")*****\\K]4]#F:F(7$$\"3 $******\\PAvr#F:F(7$$\"3)******\\nHi#HF:F(7$$\"3jmm\"z*ev:JF:F(7$$\"3? 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unity" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 7 " since " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(U[k,n](x),k = 0 . . n) = 1;" "6#/-%$SumG6$-&%\"UG6$%\"kG%\"nG6#%\"xG/F+;\"\"!F,\"\"\"" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 18 0 "0<= x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "n := 10;\nP := (n,x) -> Sum(U(k,n,x),k=0..n);\nplot(P (n,x),x=0..1,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\" #5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGf*6$%\"nG%\"xG6\"6$%)opera torG%&arrowGF)-%$SumG6$-%\"UG6%%\"kG9$9%/F3;\"\"!F4F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7iq7$$\"\"!F) $\"\"\"F)7$$\"3ILLL3x&)*3\"!#>F*7$$\"3emmm;arz@F/F*7$$\"3IL$e*)4bQl#F/ F*7$$\"3.++D\"y%*z7$F/F*7$$\"3/LeRAY1lLF/F*7$$\"3vm;ajW8-OF/F*7$$\"3E$ 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3?mm\"zpe*zqFepF*7$$\"31*\\(oa_VLrFepF*7$$\"3-L$e9\"=\"p=(FepF*7$$\"3c ]P%)*3]O@(FepF*7$$\"3)p;H#o$)QSsFepF*7$$\"3?v=U2vv`sFepF*7$$\"3S$e9mkE rE(FepF*7$$\"3'p$4@;7\"QF(FepF*7$$\"3i\"H2ey&\\!G(FepF*7$$\"3GYOSb.=(G (FepF*7$$\"3%)*****\\#\\'QH(FepFQ7$$\"3fy%\\Sea,I(FepFQ7$$\"3Ye*)4VUW1 tFepF*7$$\"3LQ%[@!Rt7tFepF*7$$\"33hN->tFepF*7$$\"3guoHzGgJtFepF*7$$ \"3BLeR(>#=WtFepF*7$$\"3O\\PfL3MptFepF*7$$\"3im;zp%*\\%R(FepF*7$$\"3*) )\\(=Un\"[W(FepF*7$$\"3GKLe9S8&\\(FepF*7$$\"3RmmT5hK+wFepF*7$$\"3R*** \\i?=bq(FepF*7$$\"3\"HLL$3s?6zFepF*7$$\"3Am;zpe()=!)FepF*7$$\"3a***\\7 `Wl7)FepF*7$$\"35mT5!R$Ry\")FepF*7$$\"3nK$e*[ACI#)FepF*7$$\"3'fT&Qym;c #)FepF*7$$\"3C*\\7y5\"4#G)FepF*7$$\"3WTg_AL0&H)FepF*7$$\"3k$eRs`:!3$)F epF*7$$\"3=ajfWm\\9$)FepFQ7$$\"3sCJ&>vx4K)FepF*7$$\"3F&*)4$f)euK)FepF* 7$$\"3#pmmm'*RRL)FepFQ7$$\"3gUg-NycS$)FepF*7$$\"3;Z$)Fep$\"2))** *************FS7$$\"3u\"zW^')FepF*7 $$\"3=LLe9tOc()FepF*7$$\"3u******\\Qk\\*)FepF*7$$\"3CLL$3dg6<*FepF*7$$ \"3ImmmmxGp$*FepF*7$$\"3A++D\"oK0e*FepF*7$$\"3A++v=5s#y*FepF*7$F*F*-%' COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fj`l-%*THICKNESS G6#\"\"#-%%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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Suppose that we are given a real valued " }{TEXT 260 10 " continuous" }{TEXT -1 34 " function defined on the interval " } {XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "Then the function " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "g[n](x) = Sum(f(k/n)*C[k,n](x),k = 0 .. n);" "6#/- &%\"gG6#%\"nG6#%\"xG-%$SumG6$*&-%\"fG6#*&%\"kG\"\"\"F(!\"\"F4-&%\"CG6$ F3F(6#F*F4/F3;\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 43 " provides an approximation for the function " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1] " "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let \+ " }{XPPEDIT 18 0 "f(x)=exp(-x^2)" "6#/-%\"fG6#%\"xG-%$expG6#,$*$F'\"\" #!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 16 "Example 1: n=10" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "In this case the graph \+ of " }{XPPEDIT 18 0 "y=g[n](x)" "6#/%\"yG-&%\"gG6#%\"nG6#%\"xG" } {TEXT -1 25 " has noticable \"wiggles\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "f := x -> exp(-x^2); \nn := 10;\ng := (n,x) -> Sum(f(k/n)*U(k,n,x),k=0..n);\nplot([f(x),g(n ,x)],x=0..1,thickness=[1,2],color=[black,red]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#,$*$) 9$\"\"#\"\"\"!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"nG%\"xG6\"6$%)operato rG%&arrowGF)-%$SumG6$*&-%\"fG6#*&%\"kG\"\"\"9$!\"\"F6-%\"UG6%F5F79%F6/ F5;\"\"!F7F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 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^v$\"38TxW)3tG6'F27$$\"3-L$e9\"=\"p=(F2$\"37O\\+CXzagF27$Fcv$\"3e;%zM&F27$Fbw$\"3EEDV?sT*G&F27$$\"3Am;zpe()=!)F2$\"3E/igY(*>s_F27$ Fgw$\"3NjH_$pz2C&F27$$\"3nK$e*[ACI#)F2$\"3!GPh:a>(p^F27$F\\x$\"3oA!4ip dh1&F27$Fax$\"3i#4L1ea8![F27$$\"3M+]PM&*>^')F2$\"3yM/d_xCsYF27$Ffx$\"3 -Rm[p;`jXF27$$\"3Ym;H#e0I&))F2$\"3AAT1IUv\"\\%F27$F[y$\"33'f1'yvs`WF27 $$\"3RL$3-.B]+*F2$\"3]E7]SF`[WF27$$\"31nmT5ASg!*F2$\"3mHLF8 " 0 "" {MPLTEXT 1 0 134 "f := x -> e xp(-x^2);\nn := 40;\ng := (n,x) -> Sum(f(k/n)*U(k,n,x),k=0..n);\nplot( [f(x),g(n,x)],x=0..1,thickness=[1,2],color=[black,red]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG 6#,$*$)9$\"\"#\"\"\"!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"nG\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"nG%\"xG6\"6$% )operatorG%&arrowGF)-%$SumG6$*&-%\"fG6#*&%\"kG\"\"\"9$!\"\"F6-%\"UG6%F 5F79%F6/F5;\"\"!F7F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "L et " }{TEXT 280 1 "n" }{TEXT -1 45 " be a positive integer and, for ea ch integer " }{TEXT 281 1 "k" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "-1 \+ <= k;" "6#1,$\"\"\"!\"\"%\"kG" }{XPPEDIT 18 0 "`` <= n+1;" "6#1%!G,&% \"nG\"\"\"F'F'" }{TEXT -1 20 ", define a function " }{XPPEDIT 18 0 "U[ k,n];" "6#&%\"UG6$%\"kG%\"nG" }{TEXT -1 4 " by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "U[k,n](x) = PIECEWISE([(cos(Pi/2*(n*x-k ))+1)/4, (k-2)/n < x and x < (k+2)/n],[0, otherwise]);" "6#/-&%\"UG6$% \"kG%\"nG6#%\"xG-%*PIECEWISEG6$7$*&,&-%$cosG6#*(%#PiG\"\"\"\"\"#!\"\", &*&F)F7F+F7F7F(F9F7F7F7F7F7\"\"%F932*&,&F(F7F8F9F7F)F9F+2F+*&,&F(F7F8F 7F7F)F97$\"\"!%*otherwiseG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "Now define " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "V[k,n] = PIECEWISE([U[-1,n](x)+U[0,n](x), k = 0],[U[k,n](x), 1 <= k and k <= n-1],[U[n,n](x)+U[n+1,n](x), k = n]);" "6#/&%\"VG6$%\"kG%\"n G-%*PIECEWISEG6%7$,&-&%\"UG6$,$\"\"\"!\"\"F(6#%\"xGF3-&F06$\"\"!F(6#F6 F3/F'F:7$-&F06$F'F(6#F631F3F'1F',&F(F3F3F47$,&-&F06$F(F(6#F6F3-&F06$,& F(F3F3F3F(6#F6F3/F'F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "n := 'n':\nU := (k,n,x) -> piecewise(x>(k-2)/n and x<(k+2)/n, (cos(Pi/2*(n*x-k))+1)/4,0):\n`U(k,n,x)`=U(k,n,x);\n\nV := proc( k,n,x)\n if k>0 and k%\" VGf*6%%\"kG%\"nG%\"xG6\"F*F*@)32\"\"!9$2F/9%-%\"UG6%F/F19&/F/F.,&-F36% !\"\"F1F5\"\"\"-F36%F.F1F5F;/F/F1,&-F36%F1F1F5F;-F36%,&F1F;F;F;F1F5F;% %FAILGF*F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The set of functions " }{XPPEDIT 18 0 "\{V[0,n], V[1,n], \+ ` . . . `, V[n,n]\};" "6#<&&%\"VG6$\"\"!%\"nG&F%6$\"\"\"F(%(~.~.~.~G&F %6$F(F(" }{TEXT -1 45 " can be plotted together in the same picture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "n := 8;\nplot([seq(V(k,n,x),k=0..n)],x=0..1,\n color=[seq(COLOR (HUE,k/(n+1)),k=0..n)],thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"nG\"\")" }}{PARA 13 "" 1 "" {GLPLOT2D 379 210 210 {PLOTDATA 2 "6 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\\F,7$Fax$\"3EeZNs*>***\\F,7$$\"3[;/^\")o_!y)F,$\"33;>YZ2;)*\\F,7$$\"3 \")*\\P%[ko/))F,$\"3c=l'**4*4%*\\F,7$$\"38$ek`,Y)G))F,$\"3!=>uKxQx)\\F ,7$F_`p$\"3is:OHc3z\\F,7$$\"34Le9;ZK,*)F,$\"3y:MF'*[$\\&\\F,7$Fdx$\"37 X)**R\"ft@\\F,7$Fhjp$\"3f[W%*RG@7[F,7$Fgx$\"3EMw&f,qzl%F,7$F`[q$\"3!yr S[yZX[%F,7$Fjx$\"3)3VtEp5/G%F,7$Fh[q$\"3&\\b0&z/\\KSF,7$F]y$\"3mZgNu>h dPF,7$F`\\q$\"3[+cx7*GPZ$F,7$F`y$\"3)pKfK8_T<$F,7$F]]q$\"3*RaC&G=CSGF, 7$FcyF]z-Ffy6$Fhy#\"\"(Fgel-F$6$7hnF[flF_emF`emFaemFbemFcemFeemFhcnFic nFjcnF[dnF\\dnFcrFirF\\sF_sFbsFesFhsF[tF^tFatFdtFgtFjtF]uF`uFcuFfuFiuF \\vF_vFbvFevFhvF[w7$Ff`oF(F^w7$F_gp$\"3)y^cBPdb\\%Fbr7$F]bo$\"3C0D%Q'f =%)>Fbq7$F\\hp$\"3#G'**)\\)H'=g%Fbq7$Fbw$\"3O\"R(**3e:\"H)Fbq7$Ff\\p$ \"3E%z>32%e`=F07$Few$\"3C>`4B\"=TE$F07$Fhw$\"3Eymq.YnctF07$F[x$\"3//qx H3l^7F,7$F^x$\"3gX_z6\"3i'=F,7$Fax$\"3x(Q=HK$3?DF,7$F_`p$\"3s'yvr08O%G F,7$Fdx$\"3ig`2:]!*)>$F,7$Fhjp$\"3\"3\"py(o.%QOF,7$Fgx$\"3FD:0[lB/TF,7 $F`[q$\"3@\\:Okq$e`%F,7$Fjx$\"3yE6$RV@Y(\\F,7$Fh[q$\"3UJ#4Zbq\\OrF,7$FcyF*-Ffy6$Fhy#\"\")Fgel-%+AXESLABELSG6$Q\"x6\"Q!6\" -%*THICKNESSG6#F[em-%%VIEWG6$;F(Fcy%(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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 26 "For each positive integer " }{TEXT 282 1 "n" }{TEXT -1 23 ", the set of functions " }{XPPEDIT 18 0 "\{V[0,n], V [1,n], ` . . . `, V[n,n]\};" "6#<&&%\"VG6$\"\"!%\"nG&F%6$\"\"\"F(%(~.~ .~.~G&F%6$F(F(" }{TEXT -1 8 " form a " }{TEXT 260 18 "partition of uni ty" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"! \"\"\"" }{TEXT -1 7 " since " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(V[k,n](x),k = 0 .. n) = 1;" "6#/-%$SumG6$-&%\"VG6$% \"kG%\"nG6#%\"xG/F+;\"\"!F,\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "n := 10;\nP := (n,x) - > Sum('V(k,n,x)',k=0..n);\nplot(P(n,x),x=0..1,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"PGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)-%$SumG6$.-%\"VG6%%\"k G9$9%/F4;\"\"!F5F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7aq7$$\"\"!F)$\"\"\"F)7$$\"3emmm;arz@!#>F*7 $$\"3[LL$e9ui2%F/F*7$$\"3nmmm\"z_\"4iF/F*7$$\"3[mmmT&phN)F/F*7$$\"3CLL e*=)H\\5!#=F*7$$\"3gmm\"z/3uC\"F%FT%F6`U%Fq4_WFY2aFvx4 K)FZ$)F^')F " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Supp ose that we are given a real valued " }{TEXT 260 10 "continuous" } {TEXT -1 34 " function defined on the interval " }{XPPEDIT 18 0 "[0,1] " "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "The n the function " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "g [n](x) = Sum(f(k/n)*V[k,n](x),k = 0 .. n);" "6#/-&%\"gG6#%\"nG6#%\"xG- %$SumG6$*&-%\"fG6#*&%\"kG\"\"\"F(!\"\"F4-&%\"VG6$F3F(6#F*F4/F3;\"\"!F( " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "provides an approxima tion for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 16 "Example 1: n=10" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The graph below compares \+ the graphs of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "y = g[n](x);" "6#/%\"yG-&%\"gG6#%\" nG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "f := x -> exp(-x^2);\nn := 10;\ng \+ := (n,x) -> Sum(f(k/n)*'V(k,n,x)',k=0..n);\nplot([f(x),g(n,x)],x=0..1, thickness=[1,2],color=[black,red]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#,$*$)9$\"\"#\"\"\"! \"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)-% $SumG6$*&-%\"fG6#*&%\"kG\"\"\"9$!\"\"F6.-%\"VG6%F5F79%F6/F5;\"\"!F7F)F )F)" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVES G6%7S7$$\"\"!F)$\"\"\"F)7$$\"3emmm;arz@!#>$\"3E^$*>p*\\_***!#=7$$\"3[L L$e9ui2%F/$\"3')3i.'y(R$)**F27$$\"3nmmm\"z_\"4iF/$\"3wzL6X1_h**F27$$\" 3[mmmT&phN)F/$\"3DIMoWwTI**F27$$\"3CLLe*=)H\\5F2$\"3cW'zBC,0*)*F27$$\" 3gmm\"z/3uC\"F2$\"3CHCVP%*)F27$$\"3,LLLLY.KNF2$ \"30EjjH=:F))F27$$\"3w***\\7o7Tv$F2$\"3_ZLFP'pao)F27$$\"3'GLLLQ*o]RF2$ \"3ip.v^?\"\\b)F27$$\"3A++D\"=lj;%F2$\"3X!zc[i[kS)F27$$\"31++vV&RY2aF2$\"3K`jN!y]YY(F27$$\"39mm;zXu9cF2 $\"3q#H6VZQgH(F27$$\"3l******\\y))GeF2$\"3YL[)4T;%>rF27$$\"3'*)***\\i_ QQgF2$\"3LqIM]#)fWpF27$$\"3@***\\7y%3TiF2$\"3;)4&>/x(Qx'F27$$\"35**** \\P![hY'F2$\"3gYgW)[$*Ge'F27$$\"3kKLL$Qx$omF2$\"3\"e96&f8M5kF27$$\"3!) *****\\P+V)oF2$\"3#zz$pi9ZDiF27$$\"3?mm\"zpe*zqF2$\"3%z*R=?.odgF27$$\" 3%)*****\\#\\'QH(F2$\"3W$p)H9?EueF27$$\"3GKLe9S8&\\(F2$\"3J2vqZs)>q&F2 7$$\"3R***\\i?=bq(F2$\"33\"y9P!R_AbF27$$\"3\"HLL$3s?6zF2$\"3!\\:nq')[z M&F27$$\"3a***\\7`Wl7)F2$\"3MN0zV)3k;&F27$$\"3#pmmm'*RRL)F2$\"3Ow+u?I, $*\\F27$$\"3Qmm;a<.Y&)F2$\"3bbaJa*>u\"[F27$$\"3=LLe9tOc()F2$\"3ceR^+5D XYF27$$\"3u******\\Qk\\*)F2$\"3#olk)RD(*)[%F27$$\"3CLL$3dg6<*F2$\"3dr* G\"**4N7VF27$$\"3ImmmmxGp$*F2$\"39])z5o;o:%F27$$\"3A++D\"oK0e*F2$\"3qR v(*HOq$*RF27$$\"3A++v=5s#y*F2$\"3%fyE[Fq.%QF27$F*$\"3MBWr6WzyOF2-%'COL OURG6&%$RGBGF)F)F)-%*THICKNESSG6#F+-F$6%7S7$F($\"3#3#HP%eC^(**F27$F-$ \"3qx)GH/$3h**F27$F4$\"3Ok]oNd%3%**F27$F9$\"3DuAe=l[6**F27$F>$\"3%Q!R* Qsy!y)*F27$FC$\"3fKx&G;pS%)*F27$FH$\"3uqp:$H))3!)*F27$FM$\"3fu7JZ[d`-'*F27$Ffn$\"3=h]ZV`1L&*F27$F[o$ \"3!)GqB*F27$Fjo$\" 3NhcV[*oY8*F27$F_p$\"37z80&=/t/*F27$Fdp$\"3vn\\uKl'f\"*)F27$Fip$\"3S') p)R')F27$Fcq$\"3&evrJ0/&>&)F27$Fhq$\"3%R#pt< l:#R)F27$F]r$\"3baT-gsJU#)F27$Fbr$\"3]<9$\\\"RGr!)F27$Fgr$\"36!GviaIs \"zF27$F\\s$\"3sQTH.%e:x(F27$Fas$\"3?k)y+<5nh(F27$Ffs$\"37\")\\(=O(pbu F27$F[t$\"3qv['e]m=F(F27$F`t$\"3[Nt#y=P64(F27$Fet$\"3E_i2'zT6%pF27$Fjt $\"3A53Xv#zJy'F27$F_u$\"3m&oLdPU+e'F27$Fdu$\"31vI@jEk#R'F27$Fiu$\"3`B4 pJXQ4iF27$F^v$\"3#zA.6]!*)ogF27$Fcv$\"3)pz$e:@%>*eF27$Fhv$\"3i\\6!HYyg q&F27$F]w$\"3@odrH[57bF27$Fbw$\"3Fh>()Q\">EM&F27$Fgw$\"3_mlWvI]*=&F27$ F\\x$\"3*[i>XdLl,&F27$Fax$\"3y\\6hv[7D[F27$Ffx$\"3yS%\\;<#QTYF27$F[y$ \"33\\;Ki@-&\\%F27$F`y$\"3!HemYsV%\\VF27$Fey$\"3+Fh&\\<4B@%F27$Fjy$\"3 ga@C=8\\sSF27$F_z$\"3OwOekFkeRF27$F*$\"3GqJMu4CrQF2-Fgz6&Fiz$\"*++++\" !\")F(F(-F[[l6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!F_el-%%VIEWG6$;F(F*%(DEFA ULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 16 "Example 2: n=50" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 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{MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = 1/2-abs(x-1/2)" "6#/-%\"fG6#%\"xG,&*&\"\"\"F*\"\"#!\"\"F*-%$ab sG6#,&F'F**&F*F*F+F,F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "(a) Use the1st \"bell-shaped\" fa mily of functions described above to provide an approximation for the \+ function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on t he interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 117 "(b) Use the 2nd wider \"bell-shaped \" family of functions described above to provide an approximation for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 2 ". " }}{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 30 "______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = abs(abs(abs(x -1/4)-1/2)-1/4);" "6#/-%\"fG6#%\"xG-%$absG6#,&-F)6#,&-F)6#,&F'\"\"\"*& F2F2\"\"%!\"\"F5F2*&F2F2\"\"#F5F5F2*&F2F2F4F5F5" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "(a) Use \+ the1st \"bell-shaped\" family of functions described above to provide \+ an approximation for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\" \"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 117 "(b) Use th e 2nd wider \"bell-shaped\" family of functions described above to pro vide an approximation for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\" fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1]" "6# 7$\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "_______ _______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 30 "______________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = x*sqrt(1-x^2);" "6#/-%\"fG6#%\"xG*&F'\"\"\"-%%sqrtG6#,&F)F)*$F '\"\"#!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 110 "(a) Use the1st \"bell-shaped\" family of functions described above to provide an approximation for the functio n " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the inte rval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 117 "(b) Use the 2nd wider \"bell-shaped\" fa mily of functions described above to provide an approximation for the \+ function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on t he interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 30 "______________________________" }} {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 30 "______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }