{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 "" -1 263 "" 0 1 0 0 0 0 0 1 0 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 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 272 "" 1 18 0 0 0 0 0 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 }{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 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 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 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 42 "The Bernstein polynomials associ ated with " }{XPPEDIT 18 0 "1, x, x^2" "6%\"\"\"%\"xG*$F$\"\"#" } {TEXT 269 5 " and " }{XPPEDIT 18 0 "x^3" "6#*$%\"xG\"\"$" }{TEXT 270 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Ca nada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 26 "Bernstein polynomials for " }{XPPEDIT 18 0 "f(x) = 1;" "6#/-%\"fG6#%\"xG\"\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = 1;" "6#/-%\"fG6#%\"xG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 273 1 "n" }{TEXT -1 41 " th Bernstein \+ polynomial associated with " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B (n,x) = Sum(matrix([[n], [k]])*(1-x)^(n-k)*x^k*f(k/n),k = 0 .. n);" "6 #/-%\"BG6$%\"nG%\"xG-%$SumG6$**-%'matrixG6#7$7#F'7#%\"kG\"\"\"),&F4F4F (!\"\",&F'F4F3F7F4)F(F3F4-%\"fG6#*&F3F4F'F7F4/F3;\"\"!F'" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum(matri x([[n], [k]])*(1-x)^(n-k)*x^k,k = 0 .. n);" "6#/%!G-%$SumG6$*(-%'matri xG6#7$7#%\"nG7#%\"kG\"\"\"),&F1F1%\"xG!\"\",&F.F1F0F5F1)F4F0F1/F0;\"\" !F." }{XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Bernstein polynomials for " }{XPPEDIT 18 0 "f(x) = x;" "6#/-%\"fG6#%\"xGF'" } {TEXT -1 3 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = x;" "6#/-%\"f G6#%\"xGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 274 1 "n" }{TEXT -1 41 " th Bernstein polynomial associated with " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B(n,x) = Sum(f(k/n)*matrix( [[n], [k]])*(1-x)^(n-k)*x^k,k = 0 .. n);" "6#/-%\"BG6$%\"nG%\"xG-%$Sum G6$**-%\"fG6#*&%\"kG\"\"\"F'!\"\"F2-%'matrixG6#7$7#F'7#F1F2),&F2F2F(F3 ,&F'F2F1F3F2)F(F1F2/F1;\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "We have " }{XPPEDIT 18 0 "f(k/n) = k/n;" "6#/-%\"fG6#*&% \"kG\"\"\"%\"nG!\"\"*&F(F)F*F+" }{TEXT -1 6 " for " }{XPPEDIT 18 0 "k =0, ` . . . `, n" "6%/%\"kG\"\"!%(~.~.~.~G%\"nG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "k<>0" "6#0%\"kG\"\"! " }{TEXT -1 7 ", then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(k/n)*matrix([[n], [k]]) = ``(k/n)*`.`*``(n!/(k!*(n-k)!));" "6# /*&-%\"fG6#*&%\"kG\"\"\"%\"nG!\"\"F*-%'matrixG6#7$7#F+7#F)F**(-%!G6#*& F)F*F+F,F*%\".GF*-F56#*&-%*factorialG6#F+F**&-F=6#F)F*-F=6#,&F+F*F)F,F *F,F*" }{TEXT 271 1 " " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (n-1)!/((k-1 )!*(n-k)!);" "6#/%!G*&-%*factorialG6#,&%\"nG\"\"\"F+!\"\"F+*&-F'6#,&% \"kGF+F+F,F+-F'6#,&F*F+F1F,F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ matrix([[n-1], [k-1]]);" "6#/%!G-%'matrixG6#7$7#,&%\"nG\"\"\"F,!\"\"7# ,&%\"kGF,F,F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = matrix([[m], [j] ]);" "6#/%!G-%'matrixG6#7$7#%\"mG7#%\"jG" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "m=n-1" "6#/%\"mG,&%\"nG\"\" \"F'!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "j=k-1" "6#/%\"jG,&%\"kG \"\"\"F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "k=0" "6#/%\"kG\"\"!" }{TEXT -1 8 ", then " }{XPPEDIT 18 0 "``(k/n)*`.`*matrix([[n], [k]]) = 0;" "6#/*(-%!G6#*&%\"kG\"\"\"% \"nG!\"\"F*%\".GF*-%'matrixG6#7$7#F+7#F)F*\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``(k/n)*`.`*matri x([[n], [k]])*(1-x)^(n-k)*x^k,k = 0 .. n) = Sum(matrix([[m], [j]])*(1- x)^(m-j)*x^(j+1),j = 0 .. m);" "6#/-%$SumG6$*,-%!G6#*&%\"kG\"\"\"%\"nG !\"\"F-%\".GF--%'matrixG6#7$7#F.7#F,F-),&F-F-%\"xGF/,&F.F-F,F/F-)F9F,F -/F,;\"\"!F.-F%6$*(-F26#7$7#%\"mG7#%\"jGF-),&F-F-F9F/,&FFF-FHF/F-)F9,& FHF-F-F-F-/FH;F>FF" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = x*`.`*Sum(matrix([[m], [j]])*(1-x)^(m-j)*x^j,j \+ = 0 .. m);" "6#/%!G*(%\"xG\"\"\"%\".GF'-%$SumG6$*(-%'matrixG6#7$7#%\"m G7#%\"jGF'),&F'F'F&!\"\",&F2F'F4F7F')F&F4F'/F4;\"\"!F2F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = x*`.`*1;" "6#/%!G*(%\"xG\"\"\"%\".GF'F'F'" } {XPPEDIT 18 0 "`` = x;" "6#/%!G%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 275 1 "n" } {TEXT -1 41 " th Bernstein polynomial associated with " }{XPPEDIT 18 0 "f(x)=x" "6#/-%\"fG6#%\"xGF'" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "B(n,x)=x" "6#/-%\"BG6$%\"nG%\"xGF(" } {TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 272 6 "_____ _" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "alias(C=binomial):n := 'n':\nf := x->x;\nSum (f(k/n)*C(n,k)*(1-x)^(n-k)*x^k,k=0..n);\nassume(n_,nonnegint);\nsimpli fy(value(subs(n=n_,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6# %\"xG6\"6$%)operatorG%&arrowGF(9$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*,%\"kG\"\"\"%\"nG!\"\"-%\"CG6$F)F'F(),&F(F(%\"xGF*,&F) F(F'F*F()F0F'F(/F';\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Bernstein polynomials for " }{XPPEDIT 18 0 "f(x)=x-x^2" "6#/-%\"fG6#%\"xG,&F'\"\"\"*$F'\"\" #!\"\"" }{TEXT -1 3 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=x-x^2" "6# /-%\"fG6#%\"xG,&F'\"\"\"*$F'\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 276 1 "n" }{TEXT -1 41 " th Bernstein po lynomial associated with " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B (n,x) = Sum(f(k/n)*matrix([[n], [k]])*(1-x)^(n-k)*x^k,k = 0 .. n);" "6 #/-%\"BG6$%\"nG%\"xG-%$SumG6$**-%\"fG6#*&%\"kG\"\"\"F'!\"\"F2-%'matrix G6#7$7#F'7#F1F2),&F2F2F(F3,&F'F2F1F3F2)F(F1F2/F1;\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "k<>0" "6#0%\"k G\"\"!" }{TEXT -1 7 ", then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(k/n)*matrix([[n], [k]]) = ``(k/n)*(1-k/n)*``(n!/(k!*( n-k)!));" "6#/*&-%\"fG6#*&%\"kG\"\"\"%\"nG!\"\"F*-%'matrixG6#7$7#F+7#F )F**(-%!G6#*&F)F*F+F,F*,&F*F**&F)F*F+F,F,F*-F56#*&-%*factorialG6#F+F** &-F>6#F)F*-F>6#,&F+F*F)F,F*F,F*" }{TEXT 263 1 " " }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = k*(n-k)*n!/(n^2*k!*(n-k)!);" "6#/%!G**%\"kG\"\"\", &%\"nGF'F&!\"\"F'-%*factorialG6#F)F'*(F)\"\"#-F,6#F&F'-F,6#,&F)F'F&F*F 'F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (n-1)!/(n*(k-1)!*(n-k-1)!);" "6 #/%!G*&-%*factorialG6#,&%\"nG\"\"\"F+!\"\"F+*(F*F+-F'6#,&%\"kGF+F+F,F+ -F'6#,(F*F+F1F,F+F,F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``((n-1)/n) *`.`*``((n-2)!/((k-1)!*(n-k-1)!));" "6#/%!G*(-F$6#*&,&%\"nG\"\"\"F+!\" \"F+F*F,F+%\".GF+-F$6#*&-%*factorialG6#,&F*F+\"\"#F,F+*&-F26#,&%\"kGF+ F+F,F+-F26#,(F*F+F:F,F+F,F+F,F+" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ ``((n-1)/n)*`.`*matrix([[n-2], [k-1]]);" "6#/%!G*(-F$6#*&,&%\"nG\"\"\" F+!\"\"F+F*F,F+%\".GF+-%'matrixG6#7$7#,&F*F+\"\"#F,7#,&%\"kGF+F+F,F+" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``((n-1)/n)*`.`*matrix([[m], [j]]) ;" "6#/%!G*(-F$6#*&,&%\"nG\"\"\"F+!\"\"F+F*F,F+%\".GF+-%'matrixG6#7$7# %\"mG7#%\"jGF+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "m = n-2;" "6#/%\"m G,&%\"nG\"\"\"\"\"#!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "j=k-1" "6#/%\"jG,&%\"kG\"\"\"F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "k=0" "6#/%\"kG\"\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "k=n" "6#/%\"kG%\"nG " }{TEXT -1 8 " then " }{XPPEDIT 18 0 "f(k/n)*matrix([[n], [k]]) = 0 ;" "6#/*&-%\"fG6#*&%\"kG\"\"\"%\"nG!\"\"F*-%'matrixG6#7$7#F+7#F)F*\"\" !" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B(n,x) = ``((n-1)/n)*`.`*Sum(mat rix([[m], [j]])*(1-x)^(m-j+1)*x^(j+1),j = 0 .. m);" "6#/-%\"BG6$%\"nG% \"xG*(-%!G6#*&,&F'\"\"\"F/!\"\"F/F'F0F/%\".GF/-%$SumG6$*(-%'matrixG6#7 $7#%\"mG7#%\"jGF/),&F/F/F(F0,(F;F/F=F0F/F/F/)F(,&F=F/F/F/F//F=;\"\"!F; F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 " " {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Sum(matrix([[m], [j]])*(1-x)^(m- j)*x^(j),j = 0 .. m) =1" "6#/-%$SumG6$*(-%'matrixG6#7$7#%\"mG7#%\"jG\" \"\"),&F0F0%\"xG!\"\",&F-F0F/F4F0)F3F/F0/F/;\"\"!F-F0" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "B(n,x) \+ = ``((n-1)/n)*`.`*(1-x)*x;" "6#/-%\"BG6$%\"nG%\"xG**-%!G6#*&,&F'\"\"\" F/!\"\"F/F'F0F/%\".GF/,&F/F/F(F0F/F(F/" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 264 15 "_______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "This se quence of Bernstein polynomials " }{XPPEDIT 18 0 "B(n,x)" "6#-%\"BG6$% \"nG%\"xG" }{TEXT -1 1 " " }{TEXT 260 9 "converges" }{TEXT -1 4 " to \+ " }{XPPEDIT 18 0 "f(x)=x-x^2" "6#/-%\"fG6#%\"xG,&F'\"\"\"*$F'\"\"#!\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 21 ": The convergence is " }{TEXT 260 7 "uniform" }{TEXT -1 17 " on the i nterval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 49 ", tha t is, the maximum error ( which occurs when " }{XPPEDIT 18 0 "x=1/2" " 6#/%\"xG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 15 " ) tends to 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "a lias(C=binomial):n := 'n':\nf := x-> x-x^2;\nSum(f(k/n)*C(n,k)*(1-x)^( n-k)*x^k,k=0..n);\nassume(n_,nonnegint);\nsubs(n_=n,simplify(value(sub s(n=n_,%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$% )operatorG%&arrowGF(,&9$\"\"\"*$)F-\"\"#F.!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$**,&*&%\"kG\"\"\"%\"nG!\"\"F**&F)\"\"#F+! \"#F,F*-%\"CG6$F+F)F*),&F*F*%\"xGF,,&F+F*F)F,F*)F5F)F*/F);\"\"!F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"xG\"\"\",&F&!\"\"F%F&F&,&%\"nGF &F&F(F&F*F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Exampl e 1: n=10" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "alias(C=binomial):\nn := 10;\nf := x -> x-x^2;\nSum( f(k/n)*C(n,k)*(1-x)^(n-k)*x^k,k=0..n);\ng := unapply(simplify(value(%) ),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"*$ )F-\"\"#F.!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$**,& *&#\"\"\"\"#5F*%\"kGF*F**&#F*\"$+\"F**$)F,\"\"#F*F*!\"\"F*-%\"CG6$F+F, F*),&F*F*%\"xGF3,&F+F*F,F3F*)F9F,F*/F,;\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&#\"\"*\"#5 \"\"\"9$F1F1*&#F/F0F1*$)F2\"\"#F1F1!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "n := 10;\nx vals := [seq(i/n,i=0..n)];\nyvals := map(f,xvals);\npts := zip((x,y)-> [x,y],xvals,yvals):\nplot([g(x),f(x),pts],x=0..1,color=[red,magenta,bl ue],\n style=[line$2,point],symbol=circle,linestyle=[1,2],thickne ss=[2,1],\n legend=[`g(x)`,`f(x)`,`f(x) points`]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% &xvalsG7-\"\"!#\"\"\"\"#5#F(\"\"&#\"\"$F)#\"\"#F+#F(F/#F-F+#\"\"(F)#\" \"%F+#\"\"*F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7-\"\"!#\" \"*\"$+\"#\"\"%\"#D#\"#@F)#\"\"'F,#\"\"\"F+F/F-F*F'F&" }}{PARA 13 "" 1 "" {GLPLOT2D 458 354 354 {PLOTDATA 2 "6(-%'CURVESG6(7S7$$\"\"!F)F(7$ $\"3emmm;arz@!#>$\"3166KTM)*=>F-7$$\"3[LL$e9ui2%F-$\"3'o4UIj-\">NF-7$$ \"3nmmm\"z_\"4iF-$\"3ZUw)pIb7C&F-7$$\"3[mmmT&phN)F-$\"3)3I=GYA@*oF-7$$ \"3CLLe*=)H\\5!#=$\"3?CMl%ofFX)F-7$$\"3gmm\"z/3uC\"FB$\"3bdM/x#[i#)*F- 7$$\"3%)***\\7LRDX\"FB$\"3;i/L_qR<6FB7$$\"3]mm\"zR'ok;FB$\"3-6#=e[6)[7 FB7$$\"3w***\\i5`h(=FB$\"3kinf_Aur8FB7$$\"3WLLL3En$4#FB$\"3)GK?$)\\$z* [\"FB7$$\"3qmm;/RE&G#FB$\"3Oj\")p8(=ne\"FB7$$\"3\")*****\\K]4]#FB$\"3g :%p\\cFzo\"FB7$$\"3$******\\PAvr#FB$\"3A#pB-jE6y\"FB7$$\"3)******\\nHi #HFB$\"3yXM_(*G&H'=FB7$$\"3jmm\"z*ev:JFB$\"3_v(QZFB7$$\"3?LLL347T LFB$\"3^03L93L-?FB7$$\"3,LLLLY.KNFB$\"3-[SS\"*p0c?FB7$$\"3w***\\7o7Tv$ FB$\"3C9!35$))H5@FB7$$\"3'GLLLQ*o]RFB$\"3cbvn]_!4:#FB7$$\"3A++D\"=lj;% FB$\"3?*32!pZX(=#FB7$$\"31++vV&R*fZ9AFB7$$\"3WLL$e9Ege%F B$\"3lvfK3jdMAFB7$$\"3GLLeR\"3Gy%FB$\"3?(=>E\\adC#FB7$$\"3cmm;/T1&*\\F B$\"3O4RK2y**\\AFB7$$\"3&em;zRQb@&FB$\"3e;@\"z))=eC#FB7$$\"3\\***\\(=> Y2aFB$\"3Nr\"H1td]B#FB7$$\"39mm;zXu9cFB$\"3/oY@>!))f@#FB7$$\"3l****** \\y))GeFB$\"3'Q,\"*Q/l\")=#FB7$$\"3'*)***\\i_QQgFB$\"3u_g>W!eH:#FB7$$ \"3@***\\7y%3TiFB$\"3Zou\"4xt86#FB7$$\"35****\\P![hY'FB$\"3MT@*Q*o`c?F B7$$\"3kKLL$Qx$omFB$\"35k.h@l[**>FB7$$\"3!)*****\\P+V)oFB$\"3-Q(4()3Z/ $>FB7$$\"3?mm\"zpe*zqFB$\"3MskMj%R1'=FB7$$\"3%)*****\\#\\'QH(FB$\"3KGp _LlVwR=`F-7$$\"3A++D\"oK0e*FB$\"3*=h+/K[oh$F-7$$\"3A++v=5s#y*FB $\"3'4FE6p@I\">F-7$$\"\"\"F)F(-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%&S TYLEG6#%%LINEG-%*THICKNESSG6#\"\"#-%*LINESTYLEG6#Fcz-%'LEGENDG6#%%g(x) G-F$6(7SF'7$F+$\"3GB,pBQ?K@F-7$F1$\"3dSMrOS65RF-7$F6$\"3tYSw2qhBeF-7$F ;$\"3c*)edZQ\"zl(F-7$F@$\"3Mr/<0_&>R*F-7$FF$\"3uRf6k`!=4\"FB7$FK$\"3C! 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" }}{PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 278 1 "n" }{TEXT -1 29 " th Bernstein polynomial for " } {XPPEDIT 18 0 "h(x) = x;" "6#/-%\"hG6#%\"xGF'" }{TEXT -1 22 " is the s ame function " }{XPPEDIT 18 0 "B[2](n,x) = x;" "6#/-&%\"BG6#\"\"#6$%\" nG%\"xGF+" }{TEXT -1 9 " for all " }{TEXT 279 1 "n" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 31 "The process of associating the " }{TEXT 280 1 "n" }{TEXT -1 60 " th Bernstein polynomial to a function is easi ly seen to be " }{TEXT 260 6 "linear" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 30 "Thus the Bernstein polynomial " }{XPPEDIT 18 0 "B(n,x) " "6#-%\"BG6$%\"nG%\"xG" }{TEXT -1 17 " associated with " }{XPPEDIT 18 0 "f(x)=x^2" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 4 " is " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "B(n,x)=B[2](n,x)-B[1 ](n,x)" "6#/-%\"BG6$%\"nG%\"xG,&-&F%6#\"\"#6$F'F(\"\"\"-&F%6#F/6$F'F(! \"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B(n,x) = ``((n-1)/n)* x^2+x/n;" "6#/-%\"BG6$%\"nG%\"xG,&*&-%!G6#*&,&F'\"\"\"F0!\"\"F0F'F1F0* $F(\"\"#F0F0*&F(F0F'F1F0" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 265 14 "______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "This sequence of Ber nstein polynomials " }{XPPEDIT 18 0 "B(n,x)" "6#-%\"BG6$%\"nG%\"xG" } {TEXT -1 1 " " }{TEXT 260 9 "converges" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 1 " " }{TEXT 260 9 "uniformly" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, 1]" "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "alias(C=binomial):n := 'n ':\nf := x-> x^2;\nSum(f(k/n)*C(n,k)*(1-x)^(n-k)*x^k,k=0..n);\nassume( n_,nonnegint);\nsubs(n_=n,simplify(value(subs(n=n_,%))));\ncollect(%,x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG% &arrowGF(*$)9$\"\"#\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$ SumG6$*,%\"kG\"\"#%\"nG!\"#-%\"CG6$F)F'\"\"\"),&F.F.%\"xG!\"\",&F)F.F' F2F.)F1F'F./F';\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,(*&%\"xG \"\"\"%\"nGF'F'F'F'F&!\"\"F'F&F'F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,&*(,&%\"nG\"\"\"F'!\"\"F'F&F(%\"xG\"\"#F'*&F&F(F)F'F'" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Example 1: n=10" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "alias(C=b inomial):\nn := 10;\nf := x -> x^2;\nSum(f(k/n)*C(n,k)*(1-x)^(n-k)*x^k ,k=0..n);\ng := unapply(simplify(value(%)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6 #%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,$*&#\"\"\"\"$+\"F)**)%\"kG\"\"#F)-%\"CG6 $\"#5F-F)),&F)F)%\"xG!\"\",&F2F)F-F6F))F5F-F)F)F)/F-;\"\"!F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&* &#\"\"\"\"#5F/9$F/F/*&#\"\"*F0F/*$)F1\"\"#F/F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "n := 1 0;\nxvals := [seq(i/n,i=0..n)];\nyvals := map(f,xvals);\npts := zip((x ,y)->[x,y],xvals,yvals):\nplot([g(x),f(x),pts],x=0..1,color=[red,magen ta,blue],\n style=[line$2,point],symbol=circle,linestyle=[1,2],th ickness=[2,1],\n legend=[`g(x)`,`f(x)`,`f(x) points`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7-\"\"!#\"\"\"\"#5#F(\"\"&#\"\"$F)#\"\"#F+#F(F /#F-F+#\"\"(F)#\"\"%F+#\"\"*F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% &yvalsG7-\"\"!#\"\"\"\"$+\"#F(\"#D#\"\"*F)#\"\"%F+#F(F/#F-F+#\"#\\F)# \"#;F+#\"#\")F)F(" }}{PARA 13 "" 1 "" {GLPLOT2D 458 354 354 {PLOTDATA 2 "6(-%'CURVESG6(7S7$$\"\"!F)F(7$$\"3emmm;arz@!#>$\"3$pbbMv>tg#!#?7$$ \"3[LL$e9ui2%F-$\"3'yO7z7:*yA9N $F-7$$\"3]mm\"zR'ok;FC$\"3=cX)47\\(eTF-7$$\"3w***\\i5`h(=FC$\"3CrB`O&3 T/&F-7$$\"3WLLL3En$4#FC$\"3G2,8+6zQgF-7$$\"3qmm;/RE&G#FC$\"3oM]o/>X&)p F-7$$\"3\")*****\\K]4]#FC$\"3\"\\%eI+wAI\")F-7$$\"3$******\\PAvr#FC$\" 3;xIwZu&RO*F-7$$\"3)******\\nHi#HFC$\"3YalZxnFj5FC7$$\"3jmm\"z*ev:JFC$ 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";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Example 2: n=50" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "alias(C=binomial):\nn := 50;\nf := x -> x^2;\nSum(f( k/n)*C(n,k)*(1-x)^(n-k)*x^k,k=0..n);\ng := unapply(simplify(value(%)), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\" \"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,$*&#\"\"\"\"%+DF )**)%\"kG\"\"#F)-%\"CG6$\"#]F-F)),&F)F)%\"xG!\"\",&F2F)F-F6F))F5F-F)F) F)/F-;\"\"!F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$% )operatorG%&arrowGF(,&*&#\"#\\\"#]\"\"\"*$)9$\"\"#F1F1F1*&#F1F0F1F4F1F 1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "n := 50;\nxvals := [seq(i/n,i=0..n)]:\nyvals := map( f,xvals):\npts := zip((x,y)->[x,y],xvals,yvals):\nplot([g(x),f(x),pts] ,x=0..1,color=[red,magenta,blue],\n style=[line$2,point],symbol=c ircle,linestyle=[1,2],thickness=[2,1],\n legend=[`g(x)`,`f(x)`,` f(x) points`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#]" }} {PARA 13 "" 1 "" {GLPLOT2D 518 409 409 {PLOTDATA 2 "6(-%'CURVESG6(7S7$ $\"\"!F)F(7$$\"3emmm;arz@!#>$\"3@'eM]%pc:!*!#@7$$\"3[LL$e9ui2%F-$\"3\\ :;a)*QiVC!#?7$$\"3nmmm\"z_\"4iF-$\"3`0!z0C\"3?]F67$$\"3[mmmT&phN)F-$\" 3L[HU5(RT^)F67$$\"3CLLe*=)H\\5!#=$\"39qiw%zl))G\"F-7$$\"3gmm\"z/3uC\"F D$\"3uc/$3\"zQuDbDaZ)HmF-7$$\"3$******\\PAvr#FD$\"3gFJXwPt!y(F-7$$\"3) ******\\nHi#HFD$\"39$fLOk4o(*)F-7$$\"3jmm\"z*ev:JFD$\"3\"yxn)HFp85FD7$ $\"3?LLL347TLFD$\"33y'GF803;\"FD7$$\"3,LLLLY.KNFD$\"3QZ6Z?q@$H\"FD7$$ \"3w***\\7o7Tv$FD$\"3Oi,$>tJiX\"FD7$$\"3'GLLLQ*o]RFD$\"33sx\"[b#f3;FD7 $$\"3A++D\"=lj;%FD$\"3;q6L))*pWy\"FD7$$\"31++vV&RFD 7$$\"3WLL$e9Ege%FD$\"3yEGKK3#G:#FD7$$\"3GLLeR\"3Gy%FD$\"3-'*zRD5VPBFD7 $$\"3cmm;/T1&*\\FD$\"3s)HZ<\\m]a#FD7$$\"3&em;zRQb@&FD$\"3ehBTk64qFFD7$ $\"3\\***\\(=>Y2aFD$\"3IzE%)*QKP(HFD7$$\"39mm;zXu9cFD$\"3v;H8Z)ziW$FD7$$\"3'*)***\\i_QQgFD$\"3mj4$*pJ0 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+++++++9FD$\"3%*************f>F-7$$\"3-+++++++;FD$\"39++++++gDF-7$$\"3 #**************z\"FD$\"3$)************RKF-7$$\"35+++++++?FDF\\fl7$$\"3 -+++++++AFD$\"3')************R[F-7$$\"3!**************R#FD$\"3%)****** ******fdF-7$$\"33+++++++EFD$\"3O************fnF-7$$\"3E+++++++GFD$\"3v ************RyF-7$$\"3))**************HFD$\"3o***************)F-7$$\"3 1+++++++KFD$\"30++++++C5FD7$$\"3C+++++++MFD$\"3%************f:\"FD7$$ \"3')*************f$FD$\"3#************fH\"FD7$$\"3-+++++++QFD$\"3++++ +++W9FD7$$\"3A+++++++SFDFjgl7$$\"3%)*************>%FD$\"3-++++++kFD7$$\"3=+++++++YFD$\"35++++++;@FD7 $$\"3#)*************z%FD$\"3%************RI#FD7$$\"3++++++++]FD$\"3+++ +++++DFD7$$\"3;+++++++_FD$\"3u***********Rq#FD7$$\"3M+++++++aFD$\"3E++ ++++;HFD7$$\"3a+++++++cFD$\"3!************f8$FD7$$\"3g*************z&F D$\"3w***********RO$FD7$$\"3w**************fFDFjjl7$$\"3%************* *>'FD$\"3>++++++WQFD7$$\"39+++++++kFD$\"3?++++++'4%FD7$$\"3I+++++++mFD $\"3))***********fN%FD7$$\"3[+++++++oFD$\"3y***********Ri%FD7$$\"3a*** ***********pFD$\"3!***************[FD7$$\"3u*************>(FD$\"3s**** *******R=&FD7$$\"3!**************R(FD$\"3w***********fZ&FD7$$\"33+++++ ++wFD$\"3-++++++wdFD7$$\"3E+++++++yFD$\"3_++++++%3'FD7$$\"3U+++++++!)F DF\\_m7$$\"3]*************>)FD$\"3)************Rs'FD7$$\"3o*********** **R)FD$\"31++++++cqFD7$$\"3')*************f)FD$\"3M++++++'R(FD7$$\"3-+ ++++++))FD$\"3x***********Ru(FD7$$\"3A+++++++!*FD$\"3a+++++++\")FD7$$ \"3S+++++++#*FD$\"3U++++++k%)FD7$$\"3Y*************R*FD$\"3_++++++O))F D7$$\"3k*************f*FD$\"3v***********f@*FD7$$\"3#)*************z*F D$\"3J++++++/'*FDFcz-Fgz6&FizF(F(Fjz-F^[l6#%&POINTGFa[lFe[l-Fi[l6#%,f( x)~pointsG-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q\"x6\"Q!Fcem-%%VIEWG6$ ;F(Fdz%(DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "g(x)" "f(x)" "f(x) points" }}}}{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 26 "Bernste in polynomials for " }{XPPEDIT 18 0 "f(x) = x-x^3;" "6#/-%\"fG6#%\"xG, &F'\"\"\"*$F'\"\"$!\"\"" }{TEXT -1 3 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = x-x^3;" "6#/-%\"fG6#%\"xG,&F'\"\"\"*$F'\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 281 1 "n" }{TEXT -1 41 " th Bernstein polynomial associated with " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B(n,x) = Sum(f(k/n)*matrix([[n], [k]])*(1-x)^(n-k) *x^k,k = 0 .. n);" "6#/-%\"BG6$%\"nG%\"xG-%$SumG6$**-%\"fG6#*&%\"kG\" \"\"F'!\"\"F2-%'matrixG6#7$7#F'7#F1F2),&F2F2F(F3,&F'F2F1F3F2)F(F1F2/F1 ;\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "k<>0" "6#0%\"kG\"\"!" }{TEXT -1 7 ", then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(k/n)*matrix([[n], [k]]) = ``(k/n)*(1-k/n)*(1+k/n)*``(n!/(k!*(n-k)!));" "6#/*&-%\"fG6#*&%\"kG\" \"\"%\"nG!\"\"F*-%'matrixG6#7$7#F+7#F)F***-%!G6#*&F)F*F+F,F*,&F*F**&F) F*F+F,F,F*,&F*F**&F)F*F+F,F*F*-F56#*&-%*factorialG6#F+F**&-F@6#F)F*-F@ 6#,&F+F*F)F,F*F,F*" }{TEXT 266 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(1+k/n)*``(k* (n-k)*n!/(n^2*k!*(n-k)!));" "6#/%!G*&-F$6#,&\"\"\"F)*&%\"kGF)%\"nG!\" \"F)F)-F$6#**F+F),&F,F)F+F-F)-%*factorialG6#F,F)*(F,\"\"#-F36#F+F)-F36 #,&F,F)F+F-F)F-F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(1+k/n)*``((n- 1)/n)*``((n-2)!/((k-1)!*(n-k-1)!));" "6#/%!G*(-F$6#,&\"\"\"F)*&%\"kGF) %\"nG!\"\"F)F)-F$6#*&,&F,F)F)F-F)F,F-F)-F$6#*&-%*factorialG6#,&F,F)\" \"#F-F)*&-F66#,&F+F)F)F-F)-F66#,(F,F)F+F-F)F-F)F-F)" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = ``(1+k/n)*``((n-1)/n)*matrix([[n-2], [k-1]]);" "6# /%!G*(-F$6#,&\"\"\"F)*&%\"kGF)%\"nG!\"\"F)F)-F$6#*&,&F,F)F)F-F)F,F-F)- %'matrixG6#7$7#,&F,F)\"\"#F-7#,&F+F)F)F-F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``((n-2)*(n-1)/(n^2))*``((n+k)/(n-2))*matrix([[n-2], [k-1]]" "6# /%!G*(-F$6#*(,&%\"nG\"\"\"\"\"#!\"\"F+,&F*F+F+F-F+*$F*F,F-F+-F$6#*&,&F *F+%\"kGF+F+,&F*F+F,F-F-F+-%'matrixG6#7$7#,&F*F+F,F-7#,&F4F+F+F-F+" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``((n-2)*(n-1)/(n^2))*``((m+3+j)/m)* matrix([[m], [j]])" "6#/%!G*(-F$6#*(,&%\"nG\"\"\"\"\"#!\"\"F+,&F*F+F+F -F+*$F*F,F-F+-F$6#*&,(%\"mGF+\"\"$F+%\"jGF+F+F4F-F+-%'matrixG6#7$7#F47 #F6F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``((n-2)*(n-1)/(n^2))*``((m+3 )/m+j/m)*matrix([[m], [j]]);" "6#/%!G*(-F$6#*(,&%\"nG\"\"\"\"\"#!\"\"F +,&F*F+F+F-F+*$F*F,F-F+-F$6#,&*&,&%\"mGF+\"\"$F+F+F5F-F+*&%\"jGF+F5F-F +F+-%'matrixG6#7$7#F57#F8F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "m = n-2; " "6#/%\"mG,&%\"nG\"\"\"\"\"#!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "j=k-1" "6#/%\"jG,&%\"kG\"\"\"F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 " k=0" "6#/%\"kG\"\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "k=n" "6#/%\"kG %\"nG" }{TEXT -1 8 " then " }{XPPEDIT 18 0 "f(k/n)*matrix([[n], [k]] ) = 0;" "6#/*&-%\"fG6#*&%\"kG\"\"\"%\"nG!\"\"F*-%'matrixG6#7$7#F+7#F)F *\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B(n,x) = ``((n-2)*(n-1)/(n^ 2))*`.`*Sum(((m+3)/m+j/m)*matrix([[m], [j]])*(1-x)^(m-j+1)*x^(j+1),j = 0 .. m);" "6#/-%\"BG6$%\"nG%\"xG*(-%!G6#*(,&F'\"\"\"\"\"#!\"\"F/,&F'F /F/F1F/*$F'F0F1F/%\".GF/-%$SumG6$**,&*&,&%\"mGF/\"\"$F/F/F " 0 "" {MPLTEXT 1 0 171 "alias(C=binomial):n := 'n':\nf := x-> x- x^3;\nSum(f(k/n)*C(n,k)*(1-x)^(n-k)*x^k,k=0..n);\nassume(n_,nonnegint) ;\nsubs(n_=n,simplify(value(subs(n=n_,%))));\ncollect(%,x,factor);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&9$\"\"\"*$)F-\"\"$F.!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%$SumG6$**,&*&%\"kG\"\"\"%\"nG!\"\"F**&F)\"\"$F+!\"$F,F*-%\"CG6$F+F )F*),&F*F*%\"xGF,,&F+F*F)F,F*)F5F)F*/F);\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,0*$)%\"nG\"\"#\"\"\"!\"\"F*F**(\"\"$F*F(F*%\"xGF*F **&F-F*F.F*F+*&F'F*)F.F)F*F**(F-F*F(F*F1F*F+*&F)F*F1F*F*F*F.F*F(!\"#F+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(**,&%\"nG\"\"\"F'!\"\"F',&F&F'\" \"#F(F'F&!\"#%\"xG\"\"$F(**F-F'F%F'F&F+F,F*F(**F%F',&F&F'F'F'F'F&F+F,F 'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Example 1: \+ n=10" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "alias(C=binomial):\nn := 10;\nf := x ->x-x^3;\nSum(f (k/n)*C(n,k)*(1-x)^(n-k)*x^k,k=0..n);\nsimplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"*$)F-\"\"$F.!\"\" F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$**,&*&#\"\"\"\"#5F* %\"kGF*F**&#F*\"%+5F**$)F,\"\"$F*F*!\"\"F*-%\"CG6$F+F,F*),&F*F*%\"xGF3 ,&F+F*F,F3F*)F9F,F*/F,;\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*& #\"#**\"$+\"\"\"\"%\"xGF(F(*&#\"#=\"#DF(*$)F)\"\"$F(F(!\"\"*&#\"#FF'F( *$)F)\"\"#F(F(F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The expression derived above gives the same result." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "n := 'n':\n(n-1)*(n+1+(n-2)*x)*x*(1-x)/(n^2);\ng := unapply(subs(n =10,%),x);\nexpand(g(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,,&%\"nG \"\"\"F&!\"\"F&,(F%F&F&F&*&,&F%F&\"\"#F'F&%\"xGF&F&F&F,F&,&F&F&F,F'F&F %!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operato rG%&arrowGF(,$*&#\"\"*\"$+\"\"\"\"*(,&\"#6F1*&\"\")F19$F1F1F1F7F1,&F1F 1F7!\"\"F1F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"#**\"$+ \"\"\"\"%\"xGF(F(*&#\"#=\"#DF(*$)F)\"\"$F(F(!\"\"*&#\"#FF'F(*$)F)\"\"# F(F(F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "n := 10;\nxvals := [seq(i/n,i=0..n)];\nyvals := map( f,xvals);\npts := zip((x,y)->[x,y],xvals,yvals):\nplot([g(x),f(x),pts] ,x=0..1,color=[red,magenta,blue],\n style=[line$2,point],symbol=c ircle,linestyle=[1,2],thickness=[2,1],\n legend=[`g(x)`,`f(x)`,` f(x) points`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7-\"\"!#\"\"\"\"#5#F(\"\"&#\" \"$F)#\"\"#F+#F(F/#F-F+#\"\"(F)#\"\"%F+#\"\"*F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7-\"\"!#\"#**\"%+5#\"#C\"$D\"#\"$t#F)#\"#UF,# \"\"$\"\")#\"#[F,#\"$d$F)#\"#OF,#\"$r\"F)F&" }}{PARA 13 "" 1 "" {GLPLOT2D 475 374 374 {PLOTDATA 2 "6(-%'CURVESG6(7U7$$\"\"!F)F(7$$\"3e mmm;arz@!#>$\"3OFUy([WV9#F-7$$\"3[LL$e9ui2%F-$\"3?MJ)H^rd)RF-7$$\"3nmm m\"z_\"4iF-$\"3!p;cy(3tDgF-7$$\"3[mmmT&phN)F-$\"3'=/5&f'o?/)F-7$$\"3CL Le*=)H\\5!#=$\"3O&f%y)Gf2+\"FB7$$\"3gmm\"z/3uC\"FB$\"3iqw#*Rg%*y6FB7$$ \"3%)***\\7LRDX\"FB$\"37LT$4\"=)*e8FB7$$\"3]mm\"zR'ok;FB$\"33E.#R*H+S: FB7$$\"3w***\\i5`h(=FB$\"33U6\"pN/[r\"FB7$$\"3WLLL3En$4#FB$\"3+jGJNSI) )=FB7$$\"3qmm;/RE&G#FB$\"3(HRm_Ewa.#FB7$$\"3\")*****\\K]4]#FB$\"3G%HT3 ;MW>#FB7$$\"3$******\\PAvr#FB$\"3'*fL(3TfkM#FB7$$\"3)******\\nHi#HFB$ \"3+n2*HNJFB7$$\"31++vV&RY2aF B$\"3E\"*)e)oEWDMFB7$$\"39mm;zXu9cFB$\"3(3%ogeL'HV$FB7$$\"3l******\\y) )GeFB$\"3\"))fI$RqMFMFB7$$\"3'*)***\\i_QQgFB$\"3A**Rge]G3MFB7$$\"3@*** \\7y%3TiFB$\"3%*)[XZ;#pwLFB7$$\"35****\\P![hY'FB$\"3r;.KV.-ELFB7$$\"3k KLL$Qx$omFB$\"3;6.pH;5mKFB7$$\"3!)*****\\P+V)oFB$\"3-z%=p+um=$FB7$$\"3 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" 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Example 2: n=50" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "alias(C=binomial):\nn := 50;\nf := x ->x-x^3;\nSum(f (k/n)*C(n,k)*(1-x)^(n-k)*x^k,k=0..n);\nsimplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"*$)F-\"\"$F.!\"\" F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$**,&*&#\"\"\"\"#]F* %\"kGF*F**&#F*\"'+]7F**$)F,\"\"$F*F*!\"\"F*-%\"CG6$F+F,F*),&F*F*%\"xGF 3,&F+F*F,F3F*)F9F,F*/F,;\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(* &#\"%*\\#\"%+D\"\"\"%\"xGF(F(*&#\"$)e\"$D'F(*$)F)\"\"$F(F(!\"\"*&#\"$Z \"F'F(*$)F)\"\"#F(F(F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The expression derived above gives the same result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "n := 'n':\n(n-1)*(n+1+(n-2)*x)*x*(1-x)/(n^2);\ng := unapply(su bs(n=50,%),x);\nexpand(g(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,,&% \"nG\"\"\"F&!\"\"F&,(F%F&F&F&*&,&F%F&\"\"#F'F&%\"xGF&F&F&F,F&,&F&F&F,F 'F&F%!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF(,$*&#\"#\\\"%+D\"\"\"*(,&\"#^F1*&\"#[F19$F1F1F1F7F1,&F 1F1F7!\"\"F1F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"%*\\# \"%+D\"\"\"%\"xGF(F(*&#\"$)e\"$D'F(*$)F)\"\"$F(F(!\"\"*&#\"$Z\"F'F(*$) F)\"\"#F(F(F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "n := 50;\nxvals := [seq(i/n,i=0..n)]:\nyvals := map(f,xvals):\npts := zip((x,y)->[x,y],xvals,yvals):\nplot([g(x),f(x) ,pts],x=0..1,color=[red,magenta,blue],\n style=[line$2,point],sym bol=circle,linestyle=[1,2],thickness=[2,1],\n legend=[`g(x)`,`f( x)`,`f(x) points`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#]" }} {PARA 13 "" 1 "" {GLPLOT2D 479 379 379 {PLOTDATA 2 "6(-%'CURVESG6(7U7$ $\"\"!F)F(7$$\"3emmm;arz@!#>$\"3/hJ()Rb2v@F-7$$\"3[LL$e9ui2%F-$\"3;w3< ]7]eSF-7$$\"3nmmm\"z_\"4iF-$\"3K%4;,Gy9;'F-7$$\"3[mmmT&phN)F-$\"3?V\"R sPwoD)F-7$$\"3CLLe*=)H\\5!#=$\"3Jf)z4KN:.\"FB7$$\"3gmm\"z/3uC\"FB$\"3+ q?7:()\\>7FB7$$\"3%)***\\7LRDX\"FB$\"3'G)f_i)>2T\"FB7$$\"3]mm\"zR'ok;F B$\"3_x_N$\\DVg\"FB7$$\"3w***\\i5`h(=FB$\"3]\\*[QAvDz\"FB7$$\"3WLLL3En $4#FB$\"3A-vNj!=2)>FB7$$\"3qmm;/RE&G#FB$\"3TaLZt3OT@FB7$$\"3\")*****\\ K]4]#FB$\"3]:E%*)H/gJ#FB7$$\"3$******\\PAvr#FB$\"3%z6owf0U[#FB7$$\"3)* *****\\nHi#HFB$\"3@ur`Ic(*QEFB7$$\"3jmm\"z*ev:JFB$\"3Qr%>*Rz&Gx#FB7$$ \"3?LLL347TLFB$\"31aB3fJDBHFB7$$\"3,LLLLY.KNFB$\"3nM'oVx@F/$FB7$$\"3w* **\\7o7Tv$FB$\"3%eHVM&H)><$FB7$$\"3'GLLLQ*o]RFB$\"3)4w+V)o@xKFB7$$\"3A ++D\"=lj;%FB$\"3_F\\qNAA#Q$FB7$$\"31++vV&RY2aFB$\"3U@MI&*GzXPFB7$$\"39mm;zXu9cFB$\"3'4[p*o$\\=w$FB7$$ \"3l******\\y))GeFB$\"3?&e(Q]&*fjPFB7$$\"3'*)***\\i_QQgFB$\"3m\"=0)[G> 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\"FB$\"3*)*********z;u\"FB7$$\"35+++++++?FB$\"3/++++++?>FB7$$\"3-+++++ ++AFB$\"33+++++_$4#FB7$$\"3!**************R#FB$\"3))*********fB(GFB 7$$\"3C+++++++MFB$\"3>+++++'p+$FB7$$\"3')*************f$FB$\"36+++++WL JFB7$$\"3-+++++++QFB$\"3s*********z7D$FB7$$\"3A+++++++SFB$\"3A++++++gL FB7$$\"3%)*************>%FB$\"3)**********>\"fMFB7$$\"3-+++++++WFB$\"3 ?+++++;[NFB7$$\"3=+++++++YFB$\"3')*********Rmi$FB7$$\"3#)************* z%FB$\"37+++++3%p$FB7$$\"3++++++++]FB$\"3+++++++]PFB7$$\"3;+++++++_FB$ \"31+++++#Rz$FB7$$\"3M+++++++aFB$\"3()*********f`#QFB7$$\"3a+++++++cFB $\"3.+++++%Q%QFB7$$\"3g*************z&FB$\"33+++++))[QFB7$$\"3w******* *******fFB$\"33++++++SQFB7$$\"3%**************>'FB$\"36+++++s;QFB7$$\" 39+++++++kFB$\"3C+++++cyPFB7$$\"3I+++++++mFB$\"3-+++++/DPFB7$$\"3[++++ +++oFB$\"3/+++++obOFB7$$\"3a**************pFB$\"3%)************pNFB7$$ \"3u*************>(FB$\"3/+++++_nMFB7$$\"3!**************R(FB$\"3=++++ +wZLFB7$$\"33+++++++wFB$\"3w*********R-@$FB7$$\"3E+++++++yFB$\"3)***** *****zW0$FB7$$\"3U+++++++!)FB$\"3y************zGFB7$$\"3]************* >)FB$\"3#)*********>jo#FB7$$\"3o*************R)FB$\"3))*********fHZ#FB 7$$\"3')*************f)FB$\"3/+++++WRAFB7$$\"3-+++++++))FB$\"35+++++G& )>FB7$$\"3A+++++++!*FB$\"38++++++5JT\"FB7$$\"3Y*************R*FB$\"3)**********fT4\"FB7$$\"3k********** ***f*FB$\"3w**********REvF-7$$\"3#)*************z*FB$\"3A+++++!3)QF-F[ [l-F_[l6&Fa[lF(F(Fb[l-Ff[l6#%&POINTGFi[lF]\\l-Fa\\l6#%,f(x)~pointsG-%' SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q\"x6\"Q!Fifm-%%VIEWG6$;F(F\\[l%(DEF AULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "g(x)" "f(x)" "f(x) points" }}}}{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 26 "Bernstein polynomia ls for " }{XPPEDIT 18 0 "f(x) = x^3;" "6#/-%\"fG6#%\"xG*$F'\"\"$" } {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "f(x) = x^3;" "6#/-% \"fG6#%\"xG*$F'\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "I n the previous section we saw that the " }{TEXT 282 1 "n" }{TEXT -1 43 " th Bernstein polynomial for the function " }{XPPEDIT 18 0 "x-x^3 ;" "6#,&%\"xG\"\"\"*$F$\"\"$!\"\"" }{TEXT -1 5 " is " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " B[1](n,x) = ``((n-2)*(n-1)/(n^2))*`.`*(1-x)*x*`.`*((n+1)/(n-2)+x);" "6 #/-&%\"BG6#\"\"\"6$%\"nG%\"xG*.-%!G6#*(,&F*F(\"\"#!\"\"F(,&F*F(F(F3F(* $F*F2F3F(%\".GF(,&F(F(F+F3F(F+F(F6F(,&*&,&F*F(F(F(F(,&F*F(F2F3F3F(F+F( F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "or " }}{PARA 256 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "B[1](n,x) = (n-1)*(n+1+(n-2)*x) *x*(1-x)/(n^2);" "6#/-&%\"BG6#\"\"\"6$%\"nG%\"xG*,,&F*F(F(!\"\"F(,(F*F (F(F(*&,&F*F(\"\"#F.F(F+F(F(F(F+F(,&F(F(F+F.F(*$F*F2F." }{TEXT -1 2 " \+ ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The \+ " }{TEXT 283 1 "n" }{TEXT -1 29 " th Bernstein polynomial for " } {XPPEDIT 18 0 "h(x) = x;" "6#/-%\"hG6#%\"xGF'" }{TEXT -1 22 " is the s ame function " }{XPPEDIT 18 0 "B[2](n,x) = x;" "6#/-&%\"BG6#\"\"#6$%\" nG%\"xGF+" }{TEXT -1 9 " for all " }{TEXT 284 1 "n" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Thus the \+ Bernstein polynomial " }{XPPEDIT 18 0 "B(n,x)" "6#-%\"BG6$%\"nG%\"xG" }{TEXT -1 17 " associated with " }{XPPEDIT 18 0 "f(x) = x^3;" "6#/-%\" fG6#%\"xG*$F'\"\"$" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "B(n,x)=B[2](n,x)-B[1](n,x)" "6#/-%\"BG6$%\"nG%\"x G,&-&F%6#\"\"#6$F'F(\"\"\"-&F%6#F/6$F'F(!\"\"" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "B(n,x) = x-(n-1)*(n+1+(n-2)*x)*x*(1-x)/(n^2);" "6#/-%\"BG6$%\"nG %\"xG,&F(\"\"\"*,,&F'F*F*!\"\"F*,(F'F*F*F**&,&F'F*\"\"#F-F*F(F*F*F*F(F *,&F*F*F(F-F**$F'F1F-F-" }{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 5 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B(n,x) = \+ x*(x^2*(n-1)*(n-2)+3*(n-1)*x+1)/(n^2);" "6#/-%\"BG6$%\"nG%\"xG*(F(\"\" \",(*(F(\"\"#,&F'F*F*!\"\"F*,&F'F*F-F/F*F**(\"\"$F*,&F'F*F*F/F*F(F*F*F *F*F**$F'F-F/" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B(n,x) = ``((n-1) *(n-2)/(n^2))*`.`*x^3+``(3*(n-1)/(n^2))*`.`*x^2+``(1/(n^2))*`.`*x;" "6 #/-%\"BG6$%\"nG%\"xG,(*(-%!G6#*(,&F'\"\"\"F0!\"\"F0,&F'F0\"\"#F1F0*$F' F3F1F0%\".GF0F(\"\"$F0*(-F,6#*(F6F0,&F'F0F0F1F0*$F'F3F1F0F5F0F(F3F0*(- F,6#*&F0F0*$F'F3F1F0F5F0F(F0F0" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 268 35 "___________________________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "This sequence of Bernstein polynomials " }{XPPEDIT 18 0 " B(n,x)" "6#-%\"BG6$%\"nG%\"xG" }{TEXT -1 1 " " }{TEXT 260 9 "converges " }{TEXT -1 4 " to " }{XPPEDIT 18 0 "f(x) = x^3;" "6#/-%\"fG6#%\"xG*$F '\"\"$" }{TEXT -1 1 " " }{TEXT 260 9 "uniformly" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "alias(C=binomial):n := 'n':\nf := x-> x^3;\nSum(f(k/n)*C(n,k)*( 1-x)^(n-k)*x^k,k=0..n);\nassume(n_,nonnegint);\nsubs(n_=n,simplify(val ue(subs(n=n_,%))));\ncollect(%,x,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"$\"\" \"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*,%\"kG\"\"$%\"nG !\"$-%\"CG6$F)F'\"\"\"),&F.F.%\"xG!\"\",&F)F.F'F2F.)F1F'F./F';\"\"!F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,.\"\"\"F%*(\"\"$F%%\"nGF%%\"xGF %F%*&F'F%F)F%!\"\"*&)F(\"\"#F%)F)F.F%F%*(F'F%F(F%F/F%F+*&F.F%F/F%F%F%F )F%F(!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(**,&%\"nG\"\"\"F'!\"\"F ',&F&F'\"\"#F(F'F&!\"#%\"xG\"\"$F'**F-F'F%F'F&F+F,F*F'*&F&F+F,F'F'" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Example 1: n=10" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "alias(C=binomial):\nn := 10;\nf := x -> x^3;\nSum(f(k/n)*C(n,k)*( 1-x)^(n-k)*x^k,k=0..n);\ng := unapply(simplify(value(%)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"$\"\"\"F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,$*&#\"\"\"\"%+5F)**)%\"kG\" \"$F)-%\"CG6$\"#5F-F)),&F)F)%\"xG!\"\",&F2F)F-F6F))F5F-F)F)F)/F-;\"\"! F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG% &arrowGF(,(*&#\"\"\"\"$+\"F/9$F/F/*&#\"#=\"#DF/*$)F1\"\"$F/F/F/*&#\"#F F0F/*$)F1\"\"#F/F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The expression derived above gives the same res ult." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "n := 'n':\nx*(x^2*(n-1)*(n-2)+3*(n-1)*x+1)/(n^2);\ng \+ := unapply(subs(n=10,%),x);\nexpand(g(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"xG\"\"\",(*()F$\"\"#F%,&%\"nGF%F%!\"\"F%,&F+F%F)F, F%F%*(\"\"$F%F*F%F$F%F%F%F%F%F+!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\"$+\"F/*&9$F/,(* &\"#sF/)F2\"\"#F/F/*&\"#FF/F2F/F/F/F/F/F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"$+\"F&%\"xGF&F&*&#\"#=\"#DF&*$)F(\"\"$F&F& F&*&#\"#FF'F&*$)F(\"\"#F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "n := 10;\nxvals := [seq(i/n ,i=0..n)];\nyvals := map(f,xvals);\npts := zip((x,y)->[x,y],xvals,yval s):\nplot([g(x),f(x),pts],x=0..1,color=[red,magenta,blue],\n styl e=[line$2,point],symbol=circle,linestyle=[1,2],thickness=[2,1],\n \+ legend=[`g(x)`,`f(x)`,`f(x) points`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7-\"\"!# \"\"\"\"#5#F(\"\"&#\"\"$F)#\"\"#F+#F(F/#F-F+#\"\"(F)#\"\"%F+#\"\"*F)F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7-\"\"!#\"\"\"\"%+5#F(\"$ D\"#\"#FF)#\"\")F+#F(F/#F-F+#\"$V$F)#\"#kF+#\"$H(F)F(" }}{PARA 13 "" 1 "" {GLPLOT2D 458 354 354 {PLOTDATA 2 "6(-%'CURVESG6(7S7$$\"\"!F)F(7$ $\"3emmm;arz@!#>$\"3T?RC))G4PN!#@7$$\"3[LL$e9ui2%F-$\"3m2*>]Gj-0*F07$$ \"3nmmm\"z_\"4iF-$\"3j'*\\5Q\">U$=!#?7$$\"3[mmmT&phN)F-$\"3bWic@)359$F ;7$$\"3CLLe*=)H\\5!#=$\"3o%zt)z+*Q&[F;7$$\"3gmm\"z/3uC\"FD$\"3s(f**))z +i%oF;7$$\"3%)***\\7LRDX\"FD$\"3-kmeJ?vb$*F;7$$\"3]mm\"zR'ok;FD$\"3)GS j*RS$oC\"F-7$$\"3w***\\i5`h(=FD$\"37x&)Q$\\([8;F-7$$\"3WLLL3En$4#FD$\" 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unapply(simplify(value(%)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\" 6$%)operatorG%&arrowGF(*$)9$\"\"$\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,$*&#\"\"\"\"'+]7F)**)%\"kG\"\"$F)-%\"CG6$\"#] F-F)),&F)F)%\"xG!\"\",&F2F)F-F6F))F5F-F)F)F)/F-;\"\"!F2" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&#\" \"\"\"%+DF/9$F/F/*&#\"$)e\"$D'F/*$)F1\"\"$F/F/F/*&#\"$Z\"F0F/*$)F1\"\" #F/F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The expression derived above gives the same result." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "n := 'n':\nx*(x^2*(n-1)*(n-2)+3*(n-1)*x+1)/(n^2);\ng := unapply(su bs(n=50,%),x);\nexpand(g(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\" xG\"\"\",(*()F$\"\"#F%,&%\"nGF%F%!\"\"F%,&F+F%F)F,F%F%*(\"\"$F%F*F%F$F %F%F%F%F%F+!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6 $%)operatorG%&arrowGF(,$*&#\"\"\"\"%+DF/*&9$F/,(*&\"%_BF/)F2\"\"#F/F/* &\"$Z\"F/F2F/F/F/F/F/F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(* &#\"\"\"\"%+DF&%\"xGF&F&*&#\"$)e\"$D'F&*$)F(\"\"$F&F&F&*&#\"$Z\"F'F&*$ )F(\"\"#F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 271 "n := 50;\nxvals := [seq(i/n,i=0..n)]:\nyvals \+ := map(f,xvals):\npts := zip((x,y)->[x,y],xvals,yvals):\nplot([g(x),f( x),pts],x=0..1,color=[red,magenta,blue],\n style=[line$2,point],s ymbol=circle,linestyle=[1,2],thickness=[2,1],\n legend=[`g(x)`,` f(x)`,`f(x) points`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#]" }}{PARA 13 "" 1 "" {GLPLOT2D 458 354 354 {PLOTDATA 2 "6(-%'CURVESG6(7S 7$$\"\"!F)F(7$$\"3emmm;arz@!#>$\"3]%o0Nzw)RY!#A7$$\"3[LL$e9ui2%F-$\"3] 5cCm&*Gx=%FG7$$\"3]mm\"zR'o k;FD$\"3t:*QhX!4OgFG7$$\"3w***\\i5`h(=FD$\"31)*\\5S#)yd$)FG7$$\"3WLLL3 En$4#FD$\"3(*4$e(\\aaH6F-7$$\"3qmm;/RE&G#FD$\"3)Q7LpII!R9F-7$$\"3\")** 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$$\"3=+++++++YFD$\"3f+++++gL(*F-7$$\"3#)*************z%FD$\"3'******** **>f5\"FD7$$\"3++++++++]FD$\"3+++++++]7FD7$$\"3;+++++++_FD$\"36+++++31 9FD7$$\"3M+++++++aFD$\"3%**********RYd\"FD7$$\"3a+++++++cFD$\"3%****** ****fhv\"FD7$$\"3g*************z&FD$\"33+++++7^>FD7$$\"3w************* *fFD$\"3)*************f@FD7$$\"3%**************>'FD$\"37+++++G$Q#FD7$$ Fj[mFD$\"3))*********R9i#FD7$$\"3I+++++++mFD$\"3u*********f\\(GFD7$$\" 3[+++++++oFD$\"3*)*********>V9$FD7$$\"3a**************pFD$\"3F++++++IM FD7$$\"3u*************>(FD$\"3C+++++[KPFD7$$\"3!**************R(FD$\"3 u*********RA0%FD7$$\"33+++++++wFD$\"3w*********f(*Q%FD7$$\"3E+++++++yF D$\"3u*********>bu%FD7$$\"3U+++++++!)FD$\"35++++++?^FD7$$\"3]********* ****>)FD$\"3p*********zO^&FD7$$\"3o*************R)FD$\"34+++++/FfFD7$$ \"3')*************f)FD$\"3a*********f0O'FD7$$\"3-+++++++))FD$\"3n***** ****>Z\"oFD7$$\"3A+++++++!*FD$\"3#)*************G(FD7$$\"3S+++++++#*FD $\"3Y+++++)oy(FD7$$\"3Y*************R*FD$\"3*)*********ReI)FD7$$\"3k** ***********f*FD$\"3m*********ft%))FD7$$\"3#)*************z*FD$\"3H++++ +#>T*FDFdz-Fhz6&FjzF(F(F[[l-F_[l6#%&POINTGFb[lFf[l-Fj[l6#%,f(x)~points G-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q\"x6\"Q!F[fm-%%VIEWG6$;F(Fez%(D EFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "g(x) " "f(x)" "f(x) points" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Summary " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 30 "The following table gives the " }{TEXT 285 1 "n" }{TEXT -1 41 " th Bernstein polynomial associated with " }{XPPEDIT 18 0 "1, x, x^2" "6%\"\"\"%\"xG*$F$\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x^3" "6#* $%\"xG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[f(x), Sum(f(k/n)*m atrix([[n], [k]])*(1-x)^(n-k)*x^k,k = 0 .. n)], [_______, ____________ ______________________________], [1, 1], [x, x], [x^2, ``((n-1)/n)*x^2 +``(1/n)*x], [x^3, ``((n-1)*(n-2)/(n^2))*x^3+``(3*(n-1)/(n^2))*x^2+``( 1/(n^2))*x]]);" "6#-%'matrixG6#7(7$-%\"fG6#%\"xG-%$SumG6$**-F)6#*&%\"k G\"\"\"%\"nG!\"\"F4-F$6#7$7#F57#F3F4),&F4F4F+F6,&F5F4F3F6F4)F+F3F4/F3; \"\"!F57$%(_______G%K__________________________________________G7$F4F4 7$F+F+7$*$F+\"\"#,&*&-%!G6#*&,&F5F4F4F6F4F5F6F4*$F+FJF4F4*&-FN6#*&F4F4 F5F6F4F+F4F47$*$F+\"\"$,(*&-FN6#*(,&F5F4F4F6F4,&F5F4FJF6F4*$F5FJF6F4*$ F+FYF4F4*&-FN6#*(FYF4,&F5F4F4F6F4*$F5FJF6F4*$F+FJF4F4*&-FN6#*&F4F4*$F5 FJF6F4F+F4F4" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 67 "We can get Maple to construct this table, and also add another row." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 429 "alias(C=binomial):n := 'n':\nassum e(n_,nonnegint);\nrws := ['f(x)',Sum(``('f(k/n)')*matrix([[n],[k]])*(1 -x)^(n-k)*x^k,k=0..n)],\n [`_______`,`_______________________________ ____________`],[1,1]:\nM := 4:\nfor m from 1 to M do\n Sum((k/n)^m*C (n,k)*(1-x)^(n-k)*x^k,k=0..n);\n bx := subs(n_=n,simplify(value(subs (n=n_,%))));\n rws := rws,[x^m,add(``(factor(coeff(bx,x^(m+1-k))))*x ^(m+1-k),k=1..m)];\nend do:\nmatrix(M+3,2,[rws]);\n \n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7)7$*$)%\"xG\"\"$\"\"\"-%$SumG6$**- %!G6#*&%\"kGF+%\"nG!\"$F,-F$6#7$7#F67#F5F,),&F,F,F*!\"\",&F6F,F5F?F,)F *F5F,/F5;\"\"!F67$%(_______G%L________________________________________ ___G7$F,F,7$F**&-F26#F,F,F*F,7$*$)F*\"\"#F,,&*&-F26#*&,&F6F,F,F?F,F6F? F,FOF,F,*&-F26#*&F,F,F6F?F,F*F,F,7$F(,(*&-F26#*(FVF,,&F6F,FPF?F,F6!\"# F,F)F,F,*&-F26#,$*(F+F,FVF,F6F\\oF,F,FOF,F,*&-F26#*&F,F,*$)F6FPF,F?F,F *F,F,7$*$)F*\"\"%F,,**&-F26#**FVF,F[oF,,&F6F,F+F?F,F6F7F,FjoF,F,*&-F26 #,$**\"\"'F,FVF,F[oF,F6F7F,F,F)F,F,*&-F26#,$*(\"\"(F,FVF,F6F7F,F,FOF,F ,*&-F26#*&F,F,*$)F6F+F,F?F,F*F,F," }}}{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 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 37 "(a) Find the general formula for the " }{TEXT 286 1 "n" }{TEXT -1 25 " th Bernstein polynomial " } {XPPEDIT 18 0 "B(n,x)" "6#-%\"BG6$%\"nG%\"xG" }{TEXT -1 30 " associate d with the function " }{XPPEDIT 18 0 "f(x) = 2*x-x^2;" "6#/-%\"fG6#%\" xG,&*&\"\"#\"\"\"F'F+F+*$F'F*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "(b) Plot the graphs of " }{XPPEDIT 18 0 "B(n,x)" "6#-%\"B G6$%\"nG%\"xG" }{TEXT -1 28 " together with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "n=10 " "6#/%\"nG\"#5" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n=50" "6#/%\"nG\" #]" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "___________________ ___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 38 "______________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 37 "(a) Find the general formula for the " }{TEXT 287 1 "n" }{TEXT -1 25 " th Bernstein polyno mial " }{XPPEDIT 18 0 "B(n,x)" "6#-%\"BG6$%\"nG%\"xG" }{TEXT -1 30 " a ssociated with the function " }{XPPEDIT 18 0 "f(x) = x^2-x^3;" "6#/-% \"fG6#%\"xG,&*$F'\"\"#\"\"\"*$F'\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "(b) Plot the graphs of " }{XPPEDIT 18 0 "B(n,x)" "6#-%\"BG6$%\"nG%\"xG" }{TEXT -1 28 " together with the graph of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "n=10" "6#/%\"nG\"#5" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n=50" " 6#/%\"nG\"#]" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "_________ _____________________________" }}{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 38 "_______________________________ _______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 46 "Use Ma ple to find the general formula for the " }{TEXT 288 1 "n" }{TEXT -1 25 " th Bernstein polynomial " }{XPPEDIT 18 0 "B(n,x)" "6#-%\"BG6$%\"n G%\"xG" }{TEXT -1 30 " associated with the function " }{XPPEDIT 18 0 " f(x) = x^5;" "6#/-%\"fG6#%\"xG*$F'\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "______________________________________" }}{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 38 "__ ____________________________________" }}{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 }