{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "Helvetica" 1 14 128 0 0 1 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 24 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Helvetica " 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 14 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 1 14 0 0 255 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "" 18 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 25 "Simpson's Rule and Cubi cs" }}{PARA 0 "" 0 "" {TEXT 256 20 "Mth 351 July 14 1999" }}{PARA 0 " " 0 "" {TEXT 258 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 259 27 "Filename: simpson_cubic.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 260 14 "Degree 3 and 2" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 387 "It is well known that \+ Simpson's quadrature is more precise than one at first expects and mor eover is exact for cubics. The second fact accounts for the first one. We can use Maple to demonstrate symbolically that Simpson's rule is \+ exact for cubics. Morever we can demonstrate a similar phenomenon for \+ higher degree polynomials and other compound quadrature rules based on interpolation." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 317 "To show that Simpson's rule is exact for cubics it \+ suffices to show that for any three points on a cubic, with equispaced abscissas, the area under the graph of the cubic from the first to th e last point is the same as the area under the graph, on the same inte rval, of the unique quadratic throught the three points." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We begin by constr ucting a general cubic polynomial (function)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "p3:=unapply(a*x^3+b*x^2+c*x+d,x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#p3GR6#%\"xG6\"6$%)operatorG%&arrowGF(,**&%\"a G\"\"\")9$\"\"$\"\"\"F/*&%\"bGF/)F1\"\"#F3F/*&%\"cGF/F1F/F/%\"dGF/F(F( F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Now let's compute the integ ral from " }{XPPEDIT 18 0 "r-h;" "6#,&%\"rG\"\"\"%\"hG!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "r+h;" "6#,&%\"rG\"\"\"%\"hGF%" }{TEXT -1 22 " for any real numbers " }{XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A3:=collect(int(p3(x),x=r-h..r+h),h);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3G,&*&,&%\"bG#\"\"#\"\"$*&%\"aG\" \"\"%\"rGF.F*F.)%\"hGF+\"\"\"F.*&,**&F-F2)F/F+F2F*%\"dGF**&F(F.)F/F*F2 F**&%\"cGF.F/F2F*F.F1F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Now \+ we compute the quadratic interpolation polynomial " }{XPPEDIT 18 0 "p2 ;" "6#%#p2G" }{TEXT -1 36 " through the points on the graph of " } {XPPEDIT 18 0 "p3;" "6#%#p3G" }{TEXT -1 16 " with abscissas " } {XPPEDIT 18 0 "r-h;" "6#,&%\"rG\"\"\"%\"hG!\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "r+h; " "6#,&%\"rG\"\"\"%\"hGF%" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "X2:=[r-h,r,r+h];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#X2G7%,&%\"rG\"\"\"%\"hG!\"\"F',&F'F(F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Y2:=map(p3,X2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Y2G7%,**&%\"aG\"\"\"),&%\"rGF)%\"hG!\"\"\"\"$\"\"\"F)*&%\"bGF))F +\"\"#F0F)*&%\"cGF)F+F)F)%\"dGF),**&F(F0)F,F/F0F)*&F2F0)F,F4F0F)*&F6F0 F,F)F)F7F),**&F(F0),&F,F)F-F)F/F0F)*&F2F0)FAF4F0F)*&F6F0FAF)F)F7F)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "q2:=unapply(interp(X2,Y2,x) ,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q2GR6#%\"xG6\"6$%)operatorG %&arrowGF(,,*&,&*&%\"aG\"\"\"%\"rGF1\"\"$%\"bGF1F1)9$\"\"#\"\"\"F1*&,( *&F0F8)%\"hGF7F8F1*&F0F8)F2F7F8!\"$%\"cGF1F1F6F1F1*&F0F8)F2F3F8F1*(F0F 8F2F8F " 0 "" {MPLTEXT 1 0 37 "B2:=collect(int(q2(x),x =r-h..r+h),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B2G,&*&,&%\"bG#\" \"#\"\"$*&%\"aG\"\"\"%\"rGF.F*F.)%\"hGF+\"\"\"F.*&,**&,&F,F+F(F.F.)F/F *F2F**&,&*&F-F2F7F2!\"$%\"cGF.F.F/F2F**&F-F2)F/F+F2F*%\"dGF*F.F1F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simplify(A3-B2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "It is now easy to see that Simpson's compound quadrature rule, that is, Simpson's rule, is exact for cubics." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 14 "Degree 5 and 4 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 394 "Let' s verify that similar results hold for higher degree. We first show th at given any quintic polynomial (degree 5) and five points on its grap h with equispaced abscissas, the area under the graph of the quintic f rom the first to the last point is the same as the area under the grap h, over the same interval, of the unique quartic (degree 4) interpolat ion polynomial through the given points." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We begin by constructing \+ a general quintic polynomial." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "p5:=unapply(a*x^5+b*x^4+c*x^3+d*x^2+e*x+f,x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#p5GR6#%\"xG6\"6$%)operatorG%&arrowGF(,.*&%\"aG\"\" \")9$\"\"&\"\"\"F/*&%\"bGF/)F1\"\"%F3F/*&%\"cGF/)F1\"\"$F3F/*&%\"dGF/) F1\"\"#F3F/*&%\"eGF/F1F/F/%\"fGF/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Now let's compute the integral from " }{XPPEDIT 18 0 "r-2 *h;" "6#,&%\"rG\"\"\"*&\"\"#F%%\"hGF%!\"\"" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "r+2*h;" "6#,&%\"rG\"\"\"*&\"\"#F%%\"hGF%F%" }{TEXT -1 22 " for any real numbers " }{XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A5:=collect(int(p5(x),x=r-2*h..r+2*h),h); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A5G,(*&,&%\"bG#\"#k\"\"&*&%\"aG \"\"\"%\"rGF.F*F.)%\"hGF+\"\"\"F.*&,**&F-F2)F/\"\"$F2#\"$g\"F7%\"dG#\" #;F7*&F(F.)F/\"\"#F2\"#K*&%\"cGF.F/F2FF2FH*&%\"eGF.F/F2FH%\"fGFHF.F1F. F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Now we compute the quartic \+ interpolation polynomial " }{XPPEDIT 18 0 "p4;" "6#%#p4G" }{TEXT -1 36 " through the points on the graph of " }{XPPEDIT 18 0 "p5;" "6#%#p5 G" }{TEXT -1 16 " with abscissas " }{XPPEDIT 18 0 "r-2*h;" "6#,&%\"rG \"\"\"*&\"\"#F%%\"hGF%!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "r-h;" "6 #,&%\"rG\"\"\"%\"hG!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "r;" "6#%\"r G" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "r+h;" "6#,&%\"rG\"\"\"%\"hGF%" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "r+2*h;" "6#,&%\"rG\"\"\"*&\"\"#F%% \"hGF%F%" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "X4:=[r-2*h,r-h,r,r+h,r+2*h];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#X4 G7',&%\"rG\"\"\"%\"hG!\"#,&F'F(F)!\"\"F',&F'F(F)F(,&F'F(F)\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Y4:=map(p5,X4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#Y4G7',.*&%\"aG\"\"\"),&%\"rGF)%\"hG!\"#\"\"& \"\"\"F)*&%\"bGF))F+\"\"%F0F)*&%\"cGF))F+\"\"$F0F)*&%\"dGF))F+\"\"#F0F )*&%\"eGF)F+F)F)%\"fGF),.*&F(F0),&F,F)F-!\"\"F/F0F)*&F2F0)FCF4F0F)*&F6 F0)FCF8F0F)*&F:F0)FCFF0FCF)F)F?F),.*&F(F0)F,F/F0F)*&F2F0)F,F4 F0F)*&F6F0)F,F8F0F)*&F:F0)F,FF0F,F)F)F?F),.*&F(F0),&F,F)F-F)F /F0F)*&F2F0)FYF4F0F)*&F6F0)FYF8F0F)*&F:F0)FYFF0FYF)F)F?F),.*& F(F0),&F,F)F-FF0F^oF)F)F?F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "q4:=una pply(interp(X4,Y4,x),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#q4GR6#% \"xG6\"6$%)operatorG%&arrowGF(,2*&,&%\"bG\"\"\"*&%\"aGF0%\"rGF0\"\"&F0 )9$\"\"%\"\"\"F0*&,(%\"cGF0*&F2F8)F3\"\"#F8!#5*&F2F8)%\"hGF>F8F4F0)F6 \"\"$F8F0*&,(*(F2F8F3F8FAF8!#:*&F2F8)F3FDF8\"#5%\"dGF0F0)F6F>F8F0*&,** &F2F8)FBF7F8!\"%%\"eGF0*(F2F8F=F8FAF8\"#:*&F2F8)F3F7F8!\"&F0F6F0F0*(F2 F8FJF8FAF8FX%\"fGF0*&F2F8)F3F4F8F0*(F2F8F3F8FQF8F7F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Let's compute the integral from " } {XPPEDIT 18 0 "r-2*h;" "6#,&%\"rG\"\"\"*&\"\"#F%%\"hGF%!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "r+2*h;" "6#,&%\"rG\"\"\"*&\"\"#F%%\"hGF%F% " }{TEXT -1 22 " for any real numbers " }{XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "B4:=collect(int(q4(x),x=r-2* h..r+2*h),h);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#B4G,(*&,&%\"bG#\"# k\"\"&*&%\"aG\"\"\"%\"rGF.F*F.)%\"hGF+\"\"\"F.*&,**&,&%\"cGF.*&F-F2)F/ \"\"#F2!#5F.F/F2\"#;*&F-F2)F/\"\"$F2#\"$g\"F?%\"dG#FF2FL*&,&%\"eGF.*&F-F2FKF2!\"&F.F/F2FL%\"fGFLF.F1F.F." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simplify(A5-B4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "The algebra would be tedious (and error prone) to do by hand, but Maple has no problem with it, so let's try one more example." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 14 "Degree 7 and 6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 392 "Now let's verify that similar results hold for degree 7 and 6. We show that given any polynomial of degree 7 and seven points on its gr aph with equispaced abscissas, the area under the graph of the degree \+ 7 polynomial from the first to the last point is the same as the area \+ under the graph, over the same interval, of the unique interpolation p olynomial of degree 6 through the given points." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "We begin by construc ting the general polynomial of degree 7." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "p7:=una pply(a*x^7+b*x^6+c*x^5+d*x^4+e*x^3+f*x^2+g*x^1+h,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p7GR6#%\"xG6\"6$%)operatorG%&arrowGF(,2*&%\"aG\" \"\")9$\"\"(\"\"\"F/*&%\"bGF/)F1\"\"'F3F/*&%\"cGF/)F1\"\"&F3F/*&%\"dGF /)F1\"\"%F3F/*&%\"eGF/)F1\"\"$F3F/*&%\"fGF/)F1\"\"#F3F/*&%\"gGF/F1F/F/ %\"hGF/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Now let's comput e the integral from " }{XPPEDIT 18 0 "r-3*h;" "6#,&%\"rG\"\"\"*&\"\"$F %%\"hGF%!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "r+3*h;" "6#,&%\"rG\" \"\"*&\"\"$F%%\"hGF%F%" }{TEXT -1 22 " for any real numbers " } {XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h;" " 6#%\"hG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " A7:=collect(int(p7(x),x=r-3*h..r+3*h),h);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#A7G,,*&,&*&%\"aG\"\"\"%\"rGF*\"%uV%\"bG#F,\"\"(F*)% \"hGF/\"\"\"F**&,**&F)F2)F+\"\"$F2\"%-M*&F-F*)F+\"\"#F2\"%e9%\"dG#\"$' [\"\"&*&%\"cGF*F+F2F?F*)F1F@F2F**&,.*&%\"eGF*F+F2\"#a*&FBF2F6F2\"$!=*& F)F2)F+F@F2\"$y$*&F=F*F:F2\"$3\"%\"fG\"#=*&F-F2)F+\"\"%F2\"$q#F*)F1F7F 2F**$)F1F;F2\"\"'*&,0*&F)F2)F+F/F2FY*&FGF2F6F2FY*&FPF*F:F2FY*&F-F2)F+F YF2FY*&%\"gGF*F+F2FY*&F=F2FSF2FY*&FBF2FLF2FYF*F1F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Now we compute the interpolation polynomial " } {XPPEDIT 18 0 "p6;" "6#%#p6G" }{TEXT -1 48 " of degree 6 through the p oints on the graph of " }{XPPEDIT 18 0 "p7;" "6#%#p7G" }{TEXT -1 16 " \+ with abscissas " }{XPPEDIT 18 0 "r-3*h;" "6#,&%\"rG\"\"\"*&\"\"$F%%\"h GF%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "r-2*h;" "6#,&%\"rG\"\"\"*& \"\"#F%%\"hGF%!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "r-h;" "6#,&%\"rG \"\"\"%\"hG!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "r;" "6#%\"rG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "r+h;" "6#,&%\"rG\"\"\"%\"hGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "r+2*h;" "6#,&%\"rG\"\"\"*&\"\"#F%%\"hGF%F%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "r+3*h;" "6#,&%\"rG\"\"\"*&\"\"$F%% \"hGF%F%" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "X6:=[r-3*h,r-2*h,r-h,r,r+h,r+2*h,r+3*h];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#X6G7),&%\"rG\"\"\"%\"hG!\"$,&F'F(F)!\"#,&F'F(F)!\"\" F',&F'F(F)F(,&F'F(F)\"\"#,&F'F(F)\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Y6:=map(p7,X6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% #Y6G7),2*&%\"aG\"\"\"),&%\"rGF)%\"hG!\"$\"\"(\"\"\"F)*&%\"bGF))F+\"\"' F0F)*&%\"cGF))F+\"\"&F0F)*&%\"dGF))F+\"\"%F0F)*&%\"eGF))F+\"\"$F0F)*&% \"fGF))F+\"\"#F0F)*&%\"gGF)F+F)F)F-F),2*&F(F0),&F,F)F-!\"#F/F0F)*&F2F0 )FJF4F0F)*&F6F0)FJF8F0F)*&F:F0)FJFF0)FJF@F0F)*&FBF0)FJFDF0F)* &FFF0FJF)F)F-F),2*&F(F0),&F,F)F-!\"\"F/F0F)*&F2F0)FZF4F0F)*&F6F0)FZF8F 0F)*&F:F0)FZFF0)FZF@F0F)*&FBF0)FZFDF0F)*&FFF0FZF)F)F-F),2*&F( F0)F,F/F0F)*&F2F0)F,F4F0F)*&F6F0)F,F8F0F)*&F:F0)F,FF0)F,F@F0F )*&FBF0)F,FDF0F)*&FFF0F,F)F)F-F),2*&F(F0),&F,F)F-F)F/F0F)*&F2F0)FbpF4F 0F)*&F6F0)FbpF8F0F)*&F:F0)FbpFF0)FbpF@F0F)*&FBF0)FbpFDF0F)*&F FF0FbpF)F)F-F),2*&F(F0),&F,F)F-FDF/F0F)*&F2F0)FaqF4F0F)*&F6F0)FaqF8F0F )*&F:F0)FaqFF0)FaqF@F0F)*&FBF0)FaqFDF0F)*&FFF0FaqF)F)F-F),2*& F(F0),&F,F)F-F@F/F0F)*&F2F0)F`rF4F0F)*&F6F0)F`rF8F0F)*&F:F0)F`rFF0)F`rF@F0F)*&FBF0)F`rFDF0F)*&FFF0F`rF)F)F-F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "q6:=unapply(interp(X6,Y6,x),x);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%#q6GR6#%\"xG6\"6$%)operatorG%&arrowGF(,8*&,&*& %\"aG\"\"\"%\"rGF1\"\"(%\"bGF1F1)9$\"\"'\"\"\"F1*&,(*&F0F8)F2\"\"#F8!# @%\"cGF1*&F0F8)%\"hGF=F8\"#9F1)F6\"\"&F8F1*&,(%\"dGF1*(F0F8F2F8FAF8!#q *&F0F8)F2\"\"$F8\"#NF1)F6\"\"%F8F1*&,**&F0F8)F2FPF8!#N%\"eGF1*&F0F8)FB FPF8!#\\*(F0F8F " 0 "" {MPLTEXT 1 0 41 "B6:=collect(int(q6(x),x=r-3*h..r+3*h),h);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#B6G,,*&,&*&%\"aG\"\"\"%\"rGF*\"%uV%\"bG#F,\"\"(F*)% \"hGF/\"\"\"F**&,**&,&*&F)F2)F+\"\"#F2!#@%\"cGF*F*F+F2\"$'[*&F)F2)F+\" \"$F2\"%-M%\"dG#F<\"\"&*&,&F(F/F-F*F*F8F2\"%e9F*)F1FCF2F**&,.*&F)F2)F+ FCF2\"$y$*&F6F2F>F2\"$!=*&FEF2)F+\"\"%F2\"$q#*&,&FAF*F=\"#NF*F8F2\"$3 \"*&,&*&F)F2FPF2!#N%\"eGF*F*F+F2\"#a%\"fG\"#=F*)F1F?F2F**$)F1F9F2\"\"' *&,0*&FEF2)F+F\\oF2F\\o*&F)F2)F+F/F2F\\o*&,&FgnF*FJ\"#@F*F8F2F\\o*&F6F 2FKF2F\\o*&,&*&F)F2F`oF2!\"(%\"gGF*F*F+F2F\\o*&FTF2FPF2F\\o*&FXF2F>F2F \\oF*F1F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simplify(A7- B6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 391 "Are you beginning to suspect a general fact? Note the direct approach taken above is not suitable for hand calculation even for fairly low degree. Eventually Maple will give up too. But no fini te number of cases will suffice to prove the general fact anyway. For \+ a proof, what we really need is a suitable error estimate for the comp ound interpolation quadrature rules with equispaced nodes." }}}}{MARK "39 0 0" 274 }{VIEWOPTS 1 1 0 1 1 1803 }