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18 1 {CSTYLE " " -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 360 24 "Calculus Single Variable " }{TEXT 363 3 " \n" }{TEXT 361 35 "Brian E. Blank and Steven G. Kran tz" }{TEXT 362 49 "\n\nSection 5.7\nNumerical Techniques of Integratio n" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "0. Load" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "This worksheet makes use \+ of some Maple packages that are not loaded at start-up. We do so now. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "w ith(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changeco ords has been redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "If the restart command is executed in this worksh eet, then this package will have to be reloaded." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "The following home-made f unction will also be used in this worksheet; it must be executed. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1594 "trapezoids := proc()\noption `Copyright (c) 2000-2006 by Brian E . Blank. \nAll rights to the code are reserved. \nHowever, all use of \+ the code including distribution is permitted.`:\nlocal a, b, ep, Delta , i, plot_struct, area, u, List, N, fnctn:\nglobal trapezoidalArea:\ni f nargs <> 4 then \nerror \"trapezoids requires four arguments.\";\nel if not type(args[1], \{name,procedure\}) then \nerror \"trapezoids exp ects its first argument to be a function, the name of a function, or a procedure.\";\nelif not type(args[2],realcons) then \nerror \"trapezo ids expects its second argument to be real.\";\nelif not type(args[3], realcons) then \nerror \"trapezoids\002 expects its third argument to \+ be real.\";\nelif evalf(args[2]) > evalf(args[3])\nthen \nerror \"trap ezoids expects its third argument to be greater than its second argume nt.\";\nelif not type( args[4] , posint) \nthen \nerror \"trapezoids i ts fourth argument, the number of subintervals, to be a positive integ er.\";\nelse\nfnctn := args[1]:\na := args[2]:\nb := args[3]:\nN := ar gs[4]:\nplot_struct[0] := plot(fnctn(u),u=a..b,color=RED,thickness=2): \nList := [plot_struct[0]]:\nDelta := (b-a)/N;\nep[0] := a:\ntrapezoid alArea := 0:\nfor i from 1 to N do\nep[i] := ep[i-1]+Delta:\ntrapezoid alArea := trapezoidalArea + (fnctn(ep[i])+fnctn(ep[i-1]))*Delta/2:\npl ot_struct[i] := plottools[polygon]([[ep[i-1],0],[ep[i],0],[ep[i],(args [1])(ep[i])],[ep[i-1],(args[1])(ep[i-1])],[ep[i-1],0]],color=COLOR(RGB ,.845,.920,.99)):\nList := [op(List),plot_struct[i]]:\nod:\nprintf(\"E nter <> for the area of the trapezoidal approximation .\\n\"):\nreturn plots[display](List);\nfi;\nend:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "1. Approximations by Rectangles" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 260 1 " " } {XPPEDIT 261 1 "f" "6#%\"fG" }{TEXT 262 1 " " }{TEXT -1 39 " be a func tion defined on an interval " }{TEXT 263 5 "[a,b]" }{TEXT -1 64 ". Pa rtition the interval into N equal subintervals by the points" }}{PARA 0 "" 0 "" {TEXT -1 13 " " }{TEXT 256 7 " " }}{PARA 0 "" 0 "" {TEXT 259 25 " " }{XPPEDIT 257 1 "a= x[0],x[1]..x[N-1],x[N]=b" "6%/%\"aG&%\"xG6#\"\"!;&F&6#\"\"\"&F&6#,&%\" NGF,F,!\"\"/&F&6#F0%\"bG" }{TEXT 258 2 ".\n" }{TEXT -1 3 "Let" }} {PARA 0 "" 0 "" {TEXT 268 25 " " }{XPPEDIT 265 1 "Delta*x=(b-a)/N" "6#/*&%&DeltaG\"\"\"%\"xGF&*&,&%\"bGF&%\"aG!\" \"F&%\"NGF," }{TEXT 264 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 51 "denote t he common length of the subintervals. Let " }{TEXT 266 1 " " } {XPPEDIT 267 1 "xi[j]" "6#&%#xiG6#%\"jG" }{TEXT -1 81 " denote a poin t in the j'th subinterval. \nThen the Riemann sum\n\n \+ " }{TEXT 269 13 " " }{XPPEDIT 270 1 "Sum(f(xi[j])*Delta*x, j=1..N" "6#-%$SumG6$*(-%\"fG6#&%#xiG6#%\"jG\"\"\"%&DeltaGF.%\"xGF./F-; F.%\"NG" }{TEXT -1 40 "\n\nis an approximation to the integral " } {XPPEDIT 19 1 "Int(f(x),x=a..b)" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\" bG" }{TEXT -1 12 " . Although " }{XPPEDIT 271 1 "xi[j]" "6#&%#xiG6#%\" jG" }{TEXT -1 85 " could be any point in the j'th subinterval, three \+ special choices stand out: \n\nthe " }{TEXT 277 13 "left endpoint" } {TEXT -1 2 " " }{TEXT 275 36 " \n " }{XPPEDIT 276 1 "xi[j] = x[j-1]" "6#/&%#xiG6#%\"jG&%\"xG6#,&F'\"\"\"F, !\"\"" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "the " }{TEXT 278 8 "midpoint" }{TEXT -1 8 " " }{TEXT 273 2 " " }}{PARA 0 " " 0 "" {TEXT 280 27 " " }{XPPEDIT 274 1 "xi[ j] = (x[j-1]+x[j])/2" "6#/&%#xiG6#%\"jG*&,&&%\"xG6#,&F'\"\"\"F.!\"\"F. &F+6#F'F.F.\"\"#F/" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 8 "an d the " }{TEXT 279 14 "right endpoint" }{TEXT -1 1 " " }{TEXT 272 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 35 " \+ " }{TEXT 281 4 " " }{XPPEDIT 282 1 "xi[j]=x[j]" "6#/&%#xiG6#%\"jG&% \"xG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "There are built-in commands in Maple for displaying these three approximations (" } {HYPERLNK 17 "leftbox" 2 "leftbox" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "middlebox" 2 "middlebox" "" }{TEXT -1 7 ", and " }{HYPERLNK 17 "righ tbox" 2 "rightbox" "" }{TEXT -1 28 ") and for calculating them (" } {HYPERLNK 17 "leftsum" 2 "leftsum" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "middlesum" 2 "middlesum" "" }{TEXT -1 6 ", and " }{HYPERLNK 17 "right sum" 2 "rightsum" "" }{TEXT -1 4 "). \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 8 "Example:" }{TEXT -1 124 " Use the \+ three rectangular approximations (left endpoint, midpoint, right endpo int) , with six subintervals to estimate " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 " " }}{PARA 0 " " 0 "" {TEXT -1 29 " " }{XPPEDIT 18 0 "int (1+sin(x)*cos(x^(3/2)),x = 2 .. 4);" "6#-%$intG6$,&\"\"\"F'*&-%$sinG6# %\"xGF'-%$cosG6#)F,*&\"\"$F'\"\"#!\"\"F'F'/F,;F3\"\"%" }{TEXT -1 1 " \+ " }{TEXT 375 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{TEXT 284 6 " " }}{PARA 0 "" 0 "" {TEXT -1 44 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*olx&*GFep$\"3I)z-Fa+?0\"Fep7$$\"3emm;H_?T-6u1\"Fep7$$\"3KLL$3#G,**HFep$\"3-fJ$egod1\"Fep7 $$\"3YL$3-Dg5-$Fep$\"3kV2oIJnh5Fep7$$\"3w)\\y$e\"pFap7$$\"3Cmm\"H!o-*\\$Fep$ \"3RbwHIjm?mFap7$$\"3))***\\7k.6a$Fep$\"3)**H$4%3q$)Q'Fap7$$\"3+L$e9as ;c$Fep$\"3&oDN\"RUf2jFap7$$\"3emmmT9C#e$Fep$\"3?Q3a>lY]iFap7$$\"3CL$eR 0$zjsY;,\"Fep7$$\"3-+](=5s#y RFep$\"3w/!H/s8+1\"Fep7$Fgo$\"3eYPl)y9,6\"Fep-%'COLOURG6&F8$\"*++++\"! \")F+F+-%&STYLEG6#%%LINEG-%*THICKNESSG6#F)-%+AXESLABELSG6$Q\"x6\"Q!Fcd l-%%VIEWG6$;F(Fgo%(DEFAULTG" 1 2 0 1 10 0 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " leftsum(expr, x = a..b, 6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\" \"\"\"\"$F&-%$SumG6$,&F&F&*&-%$sinG6#,&\"\"#F&*&F'!\"\"%\"iGF&F&F&-%$c osG6#*$)F0#F'F1F&F&F&/F4;\"\"!\"\"&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "leftapprox := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+leftapproxG$\"+*4:WF\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 287 33 "The Right Endpoint Approximation:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rightbox(expr, x = a..b, 6); " }}{PARA 13 "" 1 "" {GLPLOT2D 441 193 193 {PLOTDATA 2 "6+-%'CURVESG6& 7jn7$$\"\"#\"\"!$\"3%fN$=c&z#\\8!#=7$$\"3Rmm;arz@?!#<$\"35'e.Cn<$=8F-7 $$\"3ALLL3VfV?F1$\"3MU*\\lZp>J\"F-7$$\"3?+]i&*)fD1#F1$\"3@#*HA)o%\\E8F -7$$\"3smm\"H[D:3#F1$\"3q!*pM2Gif8F-7$$\"3J+]PpU&G5#F1$\"3%=AUy^^*=9F- 7$$\"3XLL$e0$=C@F1$\"33_VW8CQ,:F-7$$\"3QLL$3RBr;#F1$\"3Lgz_AavNLLezs$HL#F1$\"3&*RhBwW=+MF-7$$\"3 1++D@1BvBF1$\"3u&Q.E&yFyRF-7$$\"3\"pmm;_M(=CF1$\"3d52/?[H;YF-7$$\"37LL $3y_qX#F1$\"3U\")f\"4HrS?&F-7$$\"3'******\\1!>+DF1$\"3)p%RLB&*)>)eF-7$ $\"3w*****\\Z/Na#F1$\"3aYw]mY'fc'F-7$$\"35+++NfC&e#F1$\"3'3:FG(QO9sF-7 $$\"36LLez6:BEF1$\"3WkwS#HQGy(F-7$$\"3_mmm\"=C#oEF1$\"3h\")=-)fx*>%)F- 7$$\"3QmmmEpS1FF1$\"30Xv!f`eg\"*)F-7$$\"3%)***\\i`A3v#F1$\"3R9EpJ1&3V* F-7$$\"3Ymmmwy8!z#F1$\"34[mlmem@)*F-7$$\"3/++DOIFLGF1$\"3d=6`)e(=<5F17 $$\"3!****\\(3zMuGF1$\"3iY4di!oA/\"F17$$\"3YL$e*olx&*GF1$\"3I)z-Fa+?0 \"F17$$\"3emm;H_?T-6u1\"F17 $$\"3KLL$3#G,**HF1$\"3-fJ$egod1\"F17$$\"3YL$3-Dg5-$F1$\"3kV2oIJnh5F17$ $\"3w)\\y$e\"pF-7$$\"3Cmm\" H!o-*\\$F1$\"3RbwHIjm?mF-7$$\"3))***\\7k.6a$F1$\"3)**H$4%3q$)Q'F-7$$\" 3+L$e9as;c$F1$\"3&oDN\"RUf2jF-7$$\"3emmmT9C#e$F1$\"3?Q3a>lY]iF-7$$\"3C L$eR0$zjsY;,\"F17$$\"3-+](=5s#yRF1$\"3w/!H/s8+1\"F17$$ \"\"%F*$\"3eYPl)y9,6\"F1-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fa_l-%&ST YLEG6#%%LINEG-%*THICKNESSG6#F)-%)POLYGONSG6$7&7$F(Fa_l7$F($\"+<'o`S$!# 57$$\"+LLLLB!\"*F_`l7$Fc`lFa_l-%&COLORG6&F]_l$\"\"(!\"\"$\"\"*F\\alFj` l-Fj_l6$7&Ff`l7$Fc`l$\"+6+#))R)Fa`l7$$\"+nmmmEFe`lFcal7$FfalFa_lFg`l-F j_l6$7&Fhal7$Ffal$\"+**ojl5Fe`l7$$\"\"$F*F]bl7$F`blFa_lFg`l-Fj_l6$7&Fb bl7$F`bl$\"+O7KJ\")Fa`l7$$\"+LLLLLFe`lFgbl7$FjblFa_lFg`l-Fj_l6$7&F\\cl 7$Fjbl$\"+ZXH\"H'Fa`l7$$\"+nmmmOFe`lFacl7$FdclFa_lFg`l-Fj_l6$7&Ffcl7$F dcl$\"+*y9,6\"Fe`l7$Ff^lF[dl7$Ff^lFa_lFg`l-%+AXESLABELSG6$Q\"x6\"Q!Fcd l-%%VIEWG6$;F(Ff^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " rightsum(expr, x = a..b, 6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&# \"\"\"\"\"$F&-%$SumG6$,&F&F&*&-%$sinG6#,&\"\"#F&*&F'!\"\"%\"iGF&F&F&-% $cosG6#*$)F0#F'F1F&F&F&/F4;F&\"\"'F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "rightapprox := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,rightapproxG$\"+wtZ*f\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 288 27 "The Midpoint Approximation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "middlebox(expr, x = a..b, 6);" }}{PARA 13 "" 1 "" {GLPLOT2D 486 208 208 {PLOTDATA 2 "6+-%'CURVESG6&7jn7$$\"\"#\"\"!$\"3%fN$=c&z#\\8!#= 7$$\"3Rmm;arz@?!#<$\"35'e.Cn<$=8F-7$$\"3ALLL3VfV?F1$\"3MU*\\lZp>J\"F-7 $$\"3?+]i&*)fD1#F1$\"3@#*HA)o%\\E8F-7$$\"3smm\"H[D:3#F1$\"3q!*pM2Gif8F -7$$\"3J+]PpU&G5#F1$\"3%=AUy^^*=9F-7$$\"3XLL$e0$=C@F1$\"33_VW8CQ,:F-7$ $\"3QLL$3RBr;#F1$\"3Lgz_AavNLLezs$HL#F1$\"3&*RhBwW=+MF-7$$\"31++D@1BvBF1$\"3u&Q.E&yFyRF-7$$\"3\"p mm;_M(=CF1$\"3d52/?[H;YF-7$$\"37LL$3y_qX#F1$\"3U\")f\"4HrS?&F-7$$\"3'* *****\\1!>+DF1$\"3)p%RLB&*)>)eF-7$$\"3w*****\\Z/Na#F1$\"3aYw]mY'fc'F-7 $$\"35+++NfC&e#F1$\"3'3:FG(QO9sF-7$$\"36LLez6:BEF1$\"3WkwS#HQGy(F-7$$ \"3_mmm\"=C#oEF1$\"3h\")=-)fx*>%)F-7$$\"3QmmmEpS1FF1$\"30Xv!f`eg\"*)F- 7$$\"3%)***\\i`A3v#F1$\"3R9EpJ1&3V*F-7$$\"3Ymmmwy8!z#F1$\"34[mlmem@)*F -7$$\"3/++DOIFLGF1$\"3d=6`)e(=<5F17$$\"3!****\\(3zMuGF1$\"3iY4di!oA/\" F17$$\"3YL$e*olx&*GF1$\"3I)z-Fa+?0\"F17$$\"3emm;H_?T-6u1\"F17$$\"3KLL$3#G,**HF1$\"3-fJ$egod1\"F17 $$\"3YL$3-Dg5-$F1$\"3kV2oIJnh5F17$$\"3w)\\y$e\"pF-7$$\"3Cmm\"H!o-*\\$F1$\"3RbwHIjm?mF-7$$\"3))** *\\7k.6a$F1$\"3)**H$4%3q$)Q'F-7$$\"3+L$e9as;c$F1$\"3&oDN\"RUf2jF-7$$\" 3emmmT9C#e$F1$\"3?Q3a>lY]iF-7$$\"3CL$eR0$zjsY;,\"F17$$ \"3-+](=5s#yRF1$\"3w/!H/s8+1\"F17$$\"\"%F*$\"3eYPl)y9,6\"F1-%'COLOURG6 &%$RGBG$\"*++++\"!\")$F*F*Fa_l-%&STYLEG6#%%LINEG-%*THICKNESSG6#F)-%)PO LYGONSG6$7&7$F(Fa_l7$F($\"+;DzK \+ " 0 "" {MPLTEXT 1 0 29 "middlesum(expr, x = a..b, 6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&-%$SumG6$,&F&F&*&-%$sinG6#,&#\"# 8\"\"'F&*&F'!\"\"%\"iGF&F&F&-%$cosG6#*$)F0#F'\"\"#F&F&F&/F6;\"\"!\"\"& F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "midapprox := evalf( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*midapproxG$\"+)[G1S\"!\"*" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 109 "Three different geometric approximations lead, naturally enough, to three different numerical approximations. " }}{PARA 0 "" 0 "" {TEXT -1 14 "Which is best?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 "Let us not answer this q uestion just yet. Instead, let us first answer the less specific quest ion: Which geometric approximation is " }{TEXT 289 6 "likely" }{TEXT -1 518 " to be best? If you look back to the illustrations of the endp oint approximations, you will see that on each subinterval the area of the approximating rectangle is clearly either too large or too small. Because functions are generically increasing or decreasing over small subintervals, endpoint approximations tend to select either the least or the greatest value of the function in the subinterval. By contrast , the Midpoint Method tends to pick an intermediate value, as can be s een in the third illustration above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 173 "Before returning to the question, \"Wh ich approximation is best?\", let us next note that an approximation i s necessary. Maple cannot compute the exact value of this integral:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(J);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$intG6$,&\"\"\"F'*&-%$sinG6#%\"xGF'-%$cosG6#*$)F,# \"\"$\"\"#F'F'F'/F,;F4\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "We can use Maple's built-in numerical approxima tion to get many digits of accuracy:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "approx := evalf(J);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'approxG$\"+BZ(GT\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 111 "\nTo exp licitly obtain the error for each of our three approximations by recta ngles, we must subtract each from " }{MPLTEXT 1 0 6 "approx" }{TEXT -1 97 " and take the absolute value of each difference. In other word s, me must evaluate the function " }{MPLTEXT 1 0 18 "z -> abs(approx- z)" }{TEXT -1 119 " at each of our three approximations. Maple provid es us with a concise way of doing this repetitive operation via the " }{HYPERLNK 17 "map" 2 "map" "" }{TEXT -1 10 " command:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "map(z - > abs(approx-z) , [leftapprox, rightapprox, midapprox]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%$\"*C'f%Q\"!\"*$\"*`Eg'=F&$\")NiC7F&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "As expect ed, the Midpoint Approximation is closest to the exact answer." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 87 "Among th e rectangular approximations, the Midpoint Rule is generally the most \+ accurate." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "2. Approximation by Trape zoids" }}{PARA 0 "" 0 "" {TEXT -1 554 "The idea behind the rectangle a pproximations is this: we use a (simpler) geometric object, one whose \+ area is already known, to approximate the graph of a function over an \+ interval, and thereby to approximate its Riemann integral over that in terval.\n\nSince graphs of functions are not typically flat over subin tervals, rectangles have an obvious defect. The next idea is to approx imate the graph of a function by line segmants that are secants. 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::::::::::::::::::::::::::::::::::::::::::::::::tywoYxiyUqoatmqEiimpSu nXTIyQyyI>yymw?YJsiIuAY\\yxyytqI>Y\\yyuIwawuyryLqyyyyyI::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::cyyLIkYxyIxmqyYIiaxMysy\\p;>Z:>:1:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "The area of the trapezoidal approximati on is the product of the base by the average height:" }}{PARA 0 "" 0 " " {TEXT -1 22 " " }}{PARA 0 "" 0 "" {TEXT -1 37 " " }{XPPEDIT 18 0 "(f(x[j-1])+f(x[ j]))/2" "6#*&,&-%\"fG6#&%\"xG6#,&%\"jG\"\"\"F-!\"\"F--F&6#&F)6#F,F-F- \"\"#F." }{TEXT -1 2 " " }{XPPEDIT 18 0 "Delta*x;" "6#*&%&DeltaG\"\" \"%\"xGF%" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 291 3 " " }} {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 42 "The tra pezoidal approximation is therefore" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 24 " " }{XPPEDIT 18 0 "(f(x[0])+f(x[1]))/2;" "6#*&,&-%\"fG6#&%\"xG6#\"\"!\"\"\"-F&6#&F)6#F ,F,F,\"\"#!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Delta*x;" "6#*&%&Del taG\"\"\"%\"xGF%" }{TEXT -1 9 " + " }{XPPEDIT 18 0 "(f(x[1])+f(x [2]))/2;" "6#*&,&-%\"fG6#&%\"xG6#\"\"\"F+-F&6#&F)6#\"\"#F+F+F0!\"\"" } {TEXT -1 2 " " }{XPPEDIT 18 0 "Delta*x;" "6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT -1 7 " + " }{TEXT 366 1 " " }{TEXT 367 3 "..." }{TEXT -1 6 " + " }{XPPEDIT 18 0 "(f(x[N-1])+f(x[N]))/2;" "6#*&,&-%\"fG6#&%\"xG 6#,&%\"NG\"\"\"F-!\"\"F--F&6#&F)6#F,F-F-\"\"#F." }{TEXT -1 2 " " } {XPPEDIT 18 0 "Delta*x;" "6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Notice t hat there is a common factor " }{XPPEDIT 18 0 "Delta*x/2;" "6#*(%&De ltaG\"\"\"%\"xGF%\"\"#!\"\"" }{TEXT -1 154 " and that each functiona l evaluation appears twice - once as a right endpoint, once as the lef t endpoint of the next subinterval - with the exception of" }{TEXT 292 3 " " }{XPPEDIT 18 0 "f(x[0]);" "6#-%\"fG6#&%\"xG6#\"\"!" } {TEXT -1 78 ", which appear only once, as the left endpoint of the fir st subinterval, and " }{XPPEDIT 18 0 "f(x[N]);" "6#-%\"fG6#&%\"xG6#% \"NG" }{TEXT -1 93 ", which appears only once, as the right endpoint of the last subinterval. Thus, we have the" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 30 "Trapezoidal Rule Approximation " }}{PARA 0 "" 0 "" {TEXT 368 1 " " }}{PARA 0 "" 0 "" {TEXT 369 29 " \+ " }{XPPEDIT 370 0 "Delta*x/2;" "6#*(%&Delta G\"\"\"%\"xGF%\"\"#!\"\"" }{TEXT 371 3 " " }{TEXT 364 1 "(" }{TEXT 372 2 " " }{XPPEDIT 373 0 "f(x[0])+2*f(x[1])+2*f(x[2])+` ... `+2*f(x[ N-1])+f(x[N]);" "6#,.-%\"fG6#&%\"xG6#\"\"!\"\"\"*&\"\"#F+-F%6#&F(6#F+F +F+*&F-F+-F%6#&F(6#F-F+F+%&~...~GF+*&F-F+-F%6#&F(6#,&%\"NGF+F+!\"\"F+F +-F%6#&F(6#F>F+" }{TEXT 374 2 " " }{TEXT 365 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }}{PARA 0 "" 0 "" {TEXT -1 22 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Ma ple has a command, " }{HYPERLNK 17 "trapezoid" 2 "student[trapezoid] " "" }{TEXT -1 139 ", that calculates this formula.\nMaple does not h ave a built-in procedure for illustrating the Trapezoidal Rule. We wil l use the function " }{TEXT 393 10 "trapezoids" }{TEXT -1 88 " pres ented in Section 0. Go back and execute it now if you have not alread y done so.\n" }}{PARA 0 "" 0 "" {TEXT -1 92 "Let us apply the Trapezoi dal Rule approximation to the integral we considered in Section 1.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a := 2:\nb := 4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "J := Int(1+sin(x)*cos(x^(3/2 )), x = a..b):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "The function has been written to take a function or func tion name, not an expression, as its first argument, so\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "trapezoids( x -> 1+sin(x)*cos(x^(3/ 2)), 2, 4, 6);" }}{PARA 6 "" 1 "" {TEXT -1 72 "Enter <> for the area of the trapezoidal approximation." }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6+-%'CURVESG6%7jn7$$\"\"#\"\"!$\"3% fN$=c&z#\\8!#=7$$\"3Rmm;arz@?!#<$\"35'e.Cn<$=8F-7$$\"3ALLL3VfV?F1$\"3M U*\\lZp>J\"F-7$$\"3?+]i&*)fD1#F1$\"3@#*HA)o%\\E8F-7$$\"3smm\"H[D:3#F1$ \"3q!*pM2Gif8F-7$$\"3J+]PpU&G5#F1$\"3%=AUy^^*=9F-7$$\"3XLL$e0$=C@F1$\" 33_VW8CQ,:F-7$$\"3QLL$3RBr;#F1$\"3Lgz_AavNLLezs$HL#F1$\"3&*RhBwW=+MF-7$$\"31++D@1BvBF1$\"3u&Q.E&y FyRF-7$$\"3\"pmm;_M(=CF1$\"3d52/?[H;YF-7$$\"37LL$3y_qX#F1$\"3U\")f\"4H rS?&F-7$$\"3'******\\1!>+DF1$\"3)p%RLB&*)>)eF-7$$\"3w*****\\Z/Na#F1$\" 3aYw]mY'fc'F-7$$\"35+++NfC&e#F1$\"3'3:FG(QO9sF-7$$\"36LLez6:BEF1$\"3Wk wS#HQGy(F-7$$\"3_mmm\"=C#oEF1$\"3h\")=-)fx*>%)F-7$$\"3QmmmEpS1FF1$\"30 Xv!f`eg\"*)F-7$$\"3%)***\\i`A3v#F1$\"3R9EpJ1&3V*F-7$$\"3Ymmmwy8!z#F1$ \"34[mlmem@)*F-7$$\"3/++DOIFLGF1$\"3d=6`)e(=<5F17$$\"3!****\\(3zMuGF1$ \"3iY4di!oA/\"F17$$\"3YL$e*olx&*GF1$\"3I)z-Fa+?0\"F17$$\"3emm;H_?T-6u1\"F17$$\"3KLL$3#G,**HF1$\" 3-fJ$egod1\"F17$$\"3YL$3-Dg5-$F1$\"3kV2oIJnh5F17$$\"3w)\\y$e\"pF-7$$\"3Cmm\"H!o-*\\$F1$\"3RbwHIjm ?mF-7$$\"3))***\\7k.6a$F1$\"3)**H$4%3q$)Q'F-7$$\"3+L$e9as;c$F1$\"3&oDN \"RUf2jF-7$$\"3emmmT9C#e$F1$\"3?Q3a>lY]iF-7$$\"3CL$eR0 $zjsY;,\"F17$$\"3-+](=5s#yRF1$\"3w/!H/s8+1\"F17$$\"\"%F*$\"3eYPl)y9,6 \"F1-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fa_l-%*THICKNESSG6#F)-%)POLYG ONSG6$7'7$F(Fa_l7$$\"+LLLLB!\"*Fa_l7$F[`l$\"+;'o`S$!#57$F($\"+e&z#\\8F a`lFi_l-%&COLORG6&F]_l$\"$X)!\"$$\"$?*Fj`l$\"#**!\"#-Ff_l6$7'Fj_l7$$\" +nmmmEF]`lFa_l7$Fdal$\"+>+#))R)Fa`lF^`lFj_lFe`l-Ff_l6$7'Fcal7$$\"\"$F* Fa_l7$F]bl$\"+**ojl5F]`lFfalFcalFe`l-Ff_l6$7'F\\bl7$$\"+LLLLLF]`lFa_l7 $Ffbl$\"+R7KJ\")Fa`lF_blF\\blFe`l-Ff_l6$7'Febl7$$\"+nmmmOF]`lFa_l7$F_c l$\"+VXH\"H'Fa`lFhblFeblFe`l-Ff_l6$7'F^cl7$Ff^lFa_l7$Ff^l$\"+*y9,6\"F] `lFaclF^clFe`l-%+AXESLABELSG6$Q\"u6\"Q!F_dl-%%VIEWG6$;F(Ff^l%(DEFAULTG " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "trapezoidalArea;" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#,2\"\"#\"\"\"*&#F%\"\"$F%*&-%$sinG6##\"\"(F(F%-%$cosG 6#,$**F.F%\"\"*!\"\"F.#F%F$F(F6F%F%F%F%*&#F%\"\"'F%*&-F+6#F$F%-F06#,$* &F$F%F$F6F%F%F%F%*&F'F%*&-F+6##\"\")F(F%-F06#,$**FFF%F4F5FFF6F(F6F%F%F %F%*&F'F%*&-F+6#F(F%-F06#,$*&F(F%F(F6F%F%F%F%*&F'F%*&-F+6##\"#5F(F%-F0 6#,$**FXF%F4F5FXF6F(F6F%F%F%F%*&F'F%*&-F+6##\"#6F(F%-F06#,$**F\\oF%F4F 5F\\oF6F(F6F%F%F%F%*&F8F%*&-F+6#\"\"%F%-F06#,$*&FeoF%FeoF6F%F%F%F%" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf( trapezoidalArea);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Qi%pV\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 16 "\nRemember that " }{MPLTEXT 1 0 6 "approx" }{TEXT -1 136 " is the highly accurate approximation that Maple obtains automat ically.\nLet us see how accurate our Trapezoidal Rule approximation is :\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "abs(approx-%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$ absG6#,&%'approxG\"\"\"$\"+Qi%pV\"!\"*!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 284 "\n\nCompare this error with those produced by the rectangular \+ approximations of Section 1. Notice that the Trapezoidal Rule error is significantly smaller than those produced by the two End Point Rules, but about twice as big as the error produced by the Midpoint Rule. Th at is typical!" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 294 233 "The \+ Trapezoidal Rule Approximation is generally significantly more accurat e than either End Point Rule Approximation. However, the Trapezoidal R ule Approximation is usually slightly less accurate than the\nMidpoint Rule Approximation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 42 "3. An Example of Trapezoidal Approximatio n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 245 "Alt hough the Midpoint Rule is generically a bit more accurate than the Tr apezoidal Rule, the Trapezoidal Rule may be useful when midpoint value s are not available. That often happens with discrete functions, as th e following example illustrates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 376 35 "The Lorenz Curve and the Gini Index" }} {PARA 0 "" 0 "" {TEXT -1 10 "For each " }{XPPEDIT 18 0 "x;" "6#%\"xG " }{TEXT -1 29 " between 0 and 100, let " }{XPPEDIT 18 0 "L(x);" "6#-%\"LG6#%\"xG" }{TEXT -1 96 " denote the percentage of a nation's \+ total annual income that is earned by the lowest-earning " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 90 "% of its inhabitants during a spec ified year. Economists call the increasing function " }{XPPEDIT 18 0 "L;" "6#%\"LG" }{TEXT -1 5 " a " }{TEXT 377 16 " Lorenz function" }{TEXT -1 30 ". The graph of the equation " }{XPPEDIT 18 0 "y = L(x) ;" "6#/%\"yG-%\"LG6#%\"xG" }{TEXT -1 15 " is called a " }{TEXT 378 12 "Lorenz curve" }{TEXT -1 52 ". Notice that \n \n \+ " }{XPPEDIT 18 0 "L(0) = 0;" "6#/-%\"LG6#\"\"!F'" } {TEXT -1 38 " and " }{XPPEDIT 18 0 "L (100) = 100;" "6#/-%\"LG6#\"$+\"F'" }{TEXT -1 19 ".\n\nIn other words, " }{XPPEDIT 18 0 "L(x) = x;" "6#/-%\"LG6#%\"xGF'" }{TEXT -1 8 " when " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "x = 100;" "6#/%\"xG\"$+\"" }{TEXT -1 53 ", the smalles t and largest values in the domain of " }{XPPEDIT 18 0 "L;" "6#%\"LG " }{TEXT -1 105 ". In a (hypothetical!) economy in which all personal income was equally distributed, then the equation " }{XPPEDIT 18 0 " L(x) = x;" "6#/-%\"LG6#%\"xGF'" }{TEXT -1 23 " would hold for all \+ " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 65 " in [0,100]. That is, \+ the Lorenz curve would be the graph of " }{XPPEDIT 18 0 "y = x;" "6# /%\"yG%\"xG" }{TEXT -1 56 ". In actuality, the Lorenz curve lies belo w the line " }{XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" }{TEXT -1 12 " \+ for \n" }{XPPEDIT 18 0 "0 < x;" "6#2\"\"!%\"xG" }{XPPEDIT 18 0 " ` ` < 100;" "6#2%\"~G\"$+\"" }{TEXT -1 17 ". \n\n\nThe area " } {XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 136 " of the region below the Lorenz curve is used to measure the disparity of income distribution. The number\n\n " }{XPPEDIT 18 0 "G = 1-A/50 00;" "6#/%\"GG,&\"\"\"F&*&%\"AGF&\"%+]!\"\"F*" }{TEXT -1 18 " \n\nis \+ called the " }{TEXT 379 25 "coefficient of inequality" }{TEXT -1 8 " o r the " }{TEXT 380 17 "Gini coefficient " }{TEXT -1 9 " or the " } {TEXT 381 10 "Gini index" }{TEXT -1 17 ". Notice that " }{XPPEDIT 18 0 "G;" "6#%\"GG" }{TEXT -1 73 " is a number that lies between 0 \+ and 1. When the Lorenz curve is " }{XPPEDIT 18 0 "y = x;" "6#/%\" yG%\"xG" }{TEXT -1 51 " (perfectly equal income distribution), the n " }{XPPEDIT 18 0 "A = 5000;" "6#/%\"AG\"%+]" }{TEXT -1 9 " and \+ " }{XPPEDIT 18 0 "G = 0;" "6#/%\"GG\"\"!" }{TEXT -1 90 ". When income distribution becomes increasingly unequal, the Lorenz curve approach es the " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 8 "-axis, " } {XPPEDIT 18 0 "y = 0;" "6#/%\"yG\"\"!" }{TEXT -1 101 ", and the verti cal line segment between the points (100,0) and (100,100). In this c ase the area " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 51 " approa ches 0, and the coefficient of inequality " }{XPPEDIT 18 0 "G;" "6#%\" GG" }{TEXT -1 16 " approaches 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 382 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 90 "U se the tabulated data to estimate the coefficient of inequality for th e United States in " }{TEXT 392 4 "1990" }{TEXT -1 15 ": \n \+ " }}{PARA 0 "" 0 "" {TEXT -1 16 " " }{OLE 1 130564 1 " [xm]Br=WfoRrB:::wk;nyyI;G:;:J;j:J:ZJJ;j:JyyyyyI<:^:vyyyyy=:::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::uyyyyy<:wAwy yyqxyIG:I:K:M:O:Q:S:UJ:n;v;;JBB:]:_:a:c:e:g:i:k:m:o:q:s:u:w:y:;[:F>N>V >^>f>n>v>>?F?N?V?^?f?n?v?>`:B:];_;a;c;e;g;i;k;m;o;q;s;u;wK:vA>BGFGNGVG^GfGnGvG>HnUykeQgTqr;V:>Z;Z:j:VBYmp>HY LkNG>::::::::N:F:YLpfF]KRnC=MtFGgml>:;::::::JGNZ ;F:V::Ceu=:;j::F@FCcmlfGW:;::::::wJ;@jyyyyyI:;Z:: ::::J;j:wAwAjy;::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::j:nyyYZDjysyOj;B:;JZC:bKIZ:>d::::::::::j:b:B::::::::::::::::::::::::::::::: ::::::::::::::::::::::=^:NJVJ^JfJnJvJ>KFKNK_>a>c>e>g> 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Data" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "for j from 1 to 5 do\nLorenz2005[j] := Lorenz2005[j-1 ] + quintiles2005[j];\nend: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "A := (20/2)*(Lorenz2005[0]+Lorenz2005[5]+2*sum(Lorenz2005['j'],' j'=2..4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG$\"&Sx#!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Gini := 1 - A/5500.0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%GiniG$\"+POOc\\!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "quintiles := [0,20,40,60,80,100]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "plot([x,interp(quintiles , [seq(Lorenz1990[j],j=0..5)], x), interp(quintiles, [seq(Lorenz2005[j ],j=0..5)], x)], x = 0 .. 100, scaling = constrained,color=[BLACK,RED, MAROON]);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6(-% 'CURVESG6$7S7$$\"\"!F)F(7$$\"3ymmm;arz@!#Y2aF: Fbp7$$\"3%pmm\"zXu9cF:Fep7$$\"3F+++]y))GeF:Fhp7$$\"3I++]i_QQgF:F[q7$$ \"3>++D\"y%3TiF:F^q7$$\"3m****\\P![hY'F:Faq7$$\"3MLLL$Qx$omF:Fdq7$$\"3 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2>!R+7#*F:Fct-Fgt6&Fit$\")viobFc^l$\")!\\DP\"Fc^l$\")%yg>%Fc^l-%(SCALI NGG6#%,CONSTRAINEDG-%+AXESLABELSG6$Q\"x6\"Q!Fdhl-%%VIEWG6$;F(Fdt%(DEFA ULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "Here is a plot of t he Lorenz curves for 1990 and 2005. The Lorenz curve for 1990 is plott ed in red. The Lorenz curve for 2005 is plotted in maroon." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 35 "4. Approximations By Parabolic Arcs" }} {PARA 0 "" 0 "" {TEXT -1 924 "\nAlthough the Midpoint Rule Approximati on and the Trapezoidal Rule Approximation are reasonably accurate, the y are nevertheless compromised by one obvious defect: each uses a poly gonal figure to approximate a geometric figure that has a curved bound ary. Polygonal figures have the advantage that their areas are easy to calculate. However, it stands to reason that a figure with a curved b oundary can be a better approximation than a polygon. The trick is to \+ get a better approximation without greatly increasing the computationa l cost. The natural choice is to use the simplest curve other than a s traight line: in other words, a parabola.\n\nThe first complication is that a parabola, unlike the secants of the Trapezoidal Rule Approxima tion, is determined by three, not two, points. 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395 12 "Figure 4.2\n\n" }}{PARA 0 "" 0 "" {TEXT 384 48 "E xample (Close-fitting interpolating quadratic)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "F := x -> e xp(x)*sin(1+x)/(1+x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#% \"xG6\"6$%)operatorG%&arrowGF(*(-%$expG6#9$\"\"\"-%$sinG6#,&F1F1F0F1F1 ,&F1F1*$)F0\"\"#F1F1!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "evalf(interpQuadratic(F, [-1 /4,0,1/4], x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&$\")eC " 0 "" {MPLTEXT 1 0 83 "plot([F(x), interpQuadratic( F, [-1/4,0,1/4], x)], x = -1/4..1/4, color=[NAVY,RED]);" }}{PARA 13 " " 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!3+++++++ +D!#=$\"3YK0\">%pL'*\\F*7$$!3mmm;HU,\"R#F*$\"3(Q.g'*eVc8&F*7$$!3=L$3FH '='H#F*$\"3UXo3=I3e_F*7$$!3ommTgBa*=#F*$\"3'H>\"*4+$3(R&F*7$$!3_mm\"H_ \">#3#F*$\"3Ad;4;(G$QbF*7$$!3YL$3_!4Nv>F*$\"3S=E!)zn9!o&F*7$$!3qm;/wfH 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83 "plot([F(x), interpQuadratic(F, [-1/4,0,1/4], \+ x)], x = -1/4..1/4, color=[NAVY,RED]);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6&-%'CURVESG6$7Y7$$!3++++++++D!#=$!3='*****\\ dI\\*)F*7$$!3mmm;HU,\"R#F*$!3c+d #)GF*7$$!3ommTgBa*=#F*$!3/fcQMd#\\H\"F*7$$!3_mm\"H_\">#3#F*$!36Z/.7k)> \"Q!#>7$$!3YL$3_!4Nv>F*$\"31>32&pOxZ(!#?7$$!3qm;/wfHw=F*$\"3qw4ae4[>CF A7$$!3%***\\PM.ttc\"F*$\"3g\"*Q@\"eJ)p6FA7$$!3ELL$epjJX\"F*$\"377i8['zGI$F G7$$!3mmm\"z/otN\"F*$!3Y])R[rU_;#FG7$$!35++]P[_\\7F*$!3m4GJ-zQLcFG7$$! 3/++]7)Q79\"F*$!3y/9'fO;nX'FG7$$!3,++]i^)o.\"F*$!3_uO6!fQ1N&FG7$$!3'om ;/^?7U*FA$!31V\\z))4A&Q$FG7$$!3/MLLeaR%H)FA$!3[J^apKdin!#@7$$!3(\\LLL$ o#)RtFA$\"3E=))eH-9k8FG7$$!3D,+v$fO%HiFA$\"3;v5`Sx?dHFG7$$!3qNLL$3`lC& FA$\"3[l'pXG&*yX$FG7$$!3#*)**\\P4u\"oTFA$\"3zsL.omN=IFG7$$!3n***\\7G-8 9$FA$\"34'euZC;N#=FG7$$!3sKL$3Fp)p?FA$\"3_;Cwf)=L<\"Fip7$$!3eLL3-$ff3 \"FA$!311F([Kjq[\"FG7$$!3)Grmm\"z%zY#Fip$!3()Ha,px 5FA$!3Ac$z#)f*faMFG7$$\"3V(**\\Pf4t.#FA$!3&\\Bmh9<_7$FG7$$\"3qIL$e*Gst 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1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "\nIf the distance " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 31 " between consecutive nodes " } {XPPEDIT 18 0 "c-h,c,c+h;" "6%,&%\"cG\"\"\"%\"hG!\"\"F$,&F$F%F&F%" } {TEXT -1 103 " is small enough, then the approximation of the inter polating quadratic over the entire interval " }{XPPEDIT 18 0 "[c-h, c+h];" "6#7$,&%\"cG\"\"\"%\"hG!\"\",&F%F&F'F&" }{TEXT -1 113 " will be good. Let us return to the preceding example. We will examine a n eighborhood centered at the value " }{XPPEDIT 18 0 "x;" "6#%\"xG" } {TEXT -1 5 " = 0" }{TEXT 387 1 "." }{TEXT -1 81 "14 wwhere the appr oximation is currently poor and see that by using a smaller " } {XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 41 " we can obtain a good appr oximation. \n\n\n" }{TEXT 386 7 "Example" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "F := x -> 10^5*(x+0.2)*(x+0.14)*(x +0.08)*(x+0.02)*(x-0.04)*(x-0.10)*(x-0.16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*2\"'++5\"\" \",&9$F/$\"\"#!\"\"F/F/,&F1F/$\"#9!\"#F/F/,&F1F/$\"\")F8F/F/,&F1F/$F3F 8F/F/,&F1F/$\"\"%F8F4F/,&F1F/$\"#5F8F4F/,&F1F/$\"#;F8F4F/F/F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "c := 0.14; h := 0.02;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"#9!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"#!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot([F(x), interpQuadratic(F, [c-h,c,c+h], x)], x = c-h .. c+ h, color=[NAVY,RED]);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"3%**************>\"!#=$!3W********R% 4\\\"!#>7$$\"3kmmmh)=(37F*$!3goy\"e65Wc\"F-7$$\"3ILLe'40j@\"F*$!3Tzm2@ f.G;F-7$$\"3cmm;6m$[A\"F*$!3-DSV8f5*p\"F-7$$\"3imm;yYUL7F*$!3'Q+P1-e*p (>C\"F*$!3&4$pQd7gR=F-7$$\"3emm\">K'*)\\7F*$!3hy5^(=BK! 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The next example shows the potential of th e parabolic approximation.\n\n\n" }}{PARA 0 "" 0 "" {TEXT 397 8 "Examp le:" }{TEXT -1 55 " Approximate \n\n \+ " }{TEXT 396 4 " " }{XPPEDIT 257 1 "Int(3-sin(2*x)*exp(x),x=- 1..1)" "6#-%$IntG6$,&\"\"$\"\"\"*&-%$sinG6#*&\"\"#F(%\"xGF(F(-%$expG6# F/F(!\"\"/F/;,$F(F3F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "by using a region bounded above by a parabola to appro ximate the graph of the integrand." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT 398 9 "Soultion:" }{TEXT -1 159 "\nHere is wha t the graph of the integrand looks like (in red) together with the gra ph of the interpolating polynomial:\n \+ " }{METAFILE 319 332 332 1 "iXmGh`B:VyFNjG>w=Z:>jOG:MZ=veLO:fb LO:^:VZ;>;E:Sb:qgykS:E:Qb:B:E<=t<;K`gb;>a;Z:VqxAA;Z:>?>a;j\\@J:=;J ::::jyyyyyI::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 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::::::::::::::::::::::::::::::::::::j;J:3:" }{TEXT -1 45 "\n\n\n \+ " }{TEXT 401 12 "Figure 4.3\n\n" } }{PARA 0 "" 0 "" {TEXT -1 74 "\nTo find the approximation we may integ rate the interpolating polynomial,\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "parabola_approx := int(int erpQuadratic(x -> 3-sin(2*x)*exp(x),[-1,0,1],x), x=-1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0parabola_approxG,(*&#\"\"\"\"\"$F(*&-%$si nG6#\"\"#F(-%$expG6#F(F(F(!\"\"*&#F(F)F(*&F+F(-F06#F2F(F(F(\"\"'F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(parabola_approx);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+_]f(G&!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The exact value is\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " exact := int(3-sin(2*x)*exp(x),x=-1..1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&exactG,,\"\"'\"\"\"*&#\"\"#\"\"&F'*&-%$ex pG6#F'F'-%$cosG6#F*F'F'F'*&#F'F+F'*&-%$sinGF2F'F-F'F'!\"\"*&#F*F+F'*&- F.6#F8F'F0F'F'F8*&#F'F+F'*&F6F'F " 0 "" {MPLTEXT 1 0 27 "near_exact := evalf(exact);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+near_exactG$\"+$H2v/&!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 9 "\nfor a " }{TEXT 400 14 "relative error" }{TEXT -1 26 " (or \+ per cent error) of\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "abs( evalf(parabola_approx - near_exact))/near_exact*100*percent;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$\"+B4ccZ!\"*\"\"\"%(percentGF(F(" }}} {PARA 0 "" 0 "" {TEXT -1 38 "\n\nNow an approximation that produces a " }{TEXT 399 6 " 5% " }{TEXT -1 208 " error is no great shakes, but do not forget that we used only two subintervals to obtain the approx imation. Greater accuracy will come from increasing the number of subi ntervals. Let us see how to do that.\n" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 83 "We first observe that the value of th e integral of the interpolating quadratic of " }{XPPEDIT 18 0 "f;" "6 #%\"fG" }{TEXT -1 21 " over the interval " }{XPPEDIT 18 0 "[c-h, c+h ];" "6#7$,&%\"cG\"\"\"%\"hG!\"\",&F%F&F'F&" }{TEXT -1 34 " can be ex pressed in terms of " }{XPPEDIT 18 0 "f(c-h),f(c);" "6$-%\"fG6#,&%\" cG\"\"\"%\"hG!\"\"-F$6#F'" }{TEXT -1 9 ", and " }{XPPEDIT 18 0 "f(c +h);" "6#-%\"fG6#,&%\"cG\"\"\"%\"hGF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "c := 'c': h := 'h': #Removes the values assigned to c and h" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "factor(int(interpQuadratic(f,[c-h,c ,c+h],x),x=c-h .. c+h));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\" \"\"\"$F&*&%\"hGF&,(-%\"fG6#,&%\"cGF&F)F&F&*&\"\"%F&-F,6#F/F&F&-F,6#,& F/F&F)!\"\"F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 12 "If we let " }{XPPEDIT 18 0 "c = x[j];" "6#/%\"cG&%\"x G6#%\"jG" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "h = Delta*x;" "6#/%\"hG* &%&DeltaG\"\"\"%\"xGF'" }{TEXT -1 15 ", and let " }{XPPEDIT 18 0 "p;" "6#%\"pG" }{TEXT -1 43 " denote the interpolating quadratic of \+ " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 32 " passing through th e points " }{XPPEDIT 18 0 "[x[j-1], f(x[j-1])];" "6#7$&%\"xG6#,&%\"jG \"\"\"F)!\"\"-%\"fG6#&F%6#,&F(F)F)F*" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "[x[j], f(x[j])];" "6#7$&%\"xG6#%\"jG-%\"fG6#&F%6#F'" }{TEXT -1 8 ", and " }{XPPEDIT 18 0 "[x[j+1], f(x[j+1])];" "6#7$&%\"xG6#,&%\"jG\" \"\"F)F)-%\"fG6#&F%6#,&F(F)F)F)" }{TEXT -1 39 ". then we have\n\n \+ " }{TEXT 304 4 " " }{XPPEDIT 256 1 "Int(p(t),t = \+ x[j-1] .. x[j+1])" "6#-%$IntG6$-%\"pG6#%\"tG/F);&%\"xG6#,&%\"jG\"\"\"F 1!\"\"&F-6#,&F0F1F1F1" }{TEXT 404 9 " = " }{TEXT 405 1 " " } {XPPEDIT 406 0 "Delta*x/3;" "6#*(%&DeltaG\"\"\"%\"xGF%\"\"$!\"\"" } {TEXT 403 4 " ( " }{XPPEDIT 407 0 "f(x[j-1])+4*f(x[j])+f(x[j+1]);" "6 #,(-%\"fG6#&%\"xG6#,&%\"jG\"\"\"F,!\"\"F,*&\"\"%F,-F%6#&F(6#F+F,F,-F%6 #&F(6#,&F+F,F,F,F," }{TEXT 402 3 " ) " }{TEXT 408 14 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " \n" }}{PARA 0 "" 0 "" {TEXT -1 19 "Noti \n\n\n " }{TEXT -1 235 " \n\n \n When we add \+ this up for two consecutive pairs of intervals, the righmost node of t he pair on the left is also the leftmost node of the pair on the right . This double weighting shows up as coefficients of 2 in the" } {TEXT -1 67 " resulting approximation rule, which is known as Simpson 's Rule.\n\n" }{TEXT 311 28 "Simpson's Rule Approximation" }}{PARA 0 " " 0 "" {TEXT -1 3 "\n " }{XPPEDIT 18 0 "int(f(x),x = x[0] .. x[N]) = \+ ` `;" "6#/-%$intG6$-%\"fG6#%\"xG/F*;&F*6#\"\"!&F*6#%\"NG%\"~G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Delta*x/3;" "6#*(%&DeltaG\"\"\"%\"xGF%\"\"$! \"\"" }{TEXT -1 1 " " }{TEXT 409 1 "(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x[0])+4*f(x[1])+2*f(x[2])+4*f(x[3]);" "6#,*-%\"fG6#&%\"xG6#\"\"!\" \"\"*&\"\"%F+-F%6#&F(6#F+F+F+*&\"\"#F+-F%6#&F(6#F3F+F+*&F-F+-F%6#&F(6# \"\"$F+F+" }{TEXT -1 4 " + " }{TEXT 390 4 "... " }{TEXT -1 2 "+ " } {XPPEDIT 18 0 "4*f(x[N-3])+2*f(x[N-2])+4*f(x[N-1])+f(x[N]);" "6#,**&\" \"%\"\"\"-%\"fG6#&%\"xG6#,&%\"NGF&\"\"$!\"\"F&F&*&\"\"#F&-F(6#&F+6#,&F .F&F2F0F&F&*&F%F&-F(6#&F+6#,&F.F&F&F0F&F&-F(6#&F+6#F.F&" }{TEXT -1 1 " " }{TEXT 410 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Remember that Simp son's Rule " }{TEXT 312 8 "requires" }{TEXT -1 4 " an " }{TEXT 313 4 " even" }{TEXT -1 191 " number of subintervals. Since an even number of \+ intervals are situated between an odd number of points, when Simpson's Rule is used with discrete data, the integrand must be specified at a n " }{TEXT 314 3 "odd" }{TEXT -1 171 " number of equally spaced points . Look at the formula another time to notice that the interpolating po lynomials do not appear in the final formula - caught a break there! " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 23 "Maple has a command, " }{HYPERLNK 17 "s impson" 2 "student[simpson]" "" }{TEXT -1 10 ", in the " }{HYPERLNK 17 "student" 2 "student" "" }{TEXT -1 59 " package for calculating th e Simpson Rule Approximation\n\n\n" }{TEXT 317 8 "Example:" }{TEXT -1 35 " Use Simpson's Rule to approximate" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 " " } {TEXT 315 4 " " }{XPPEDIT 316 1 "Int(3-sin(2*x)*exp(x),x=-1..1)" "6 #-%$IntG6$,&\"\"$\"\"\"*&-%$sinG6#*&\"\"#F(%\"xGF(F(-%$expG6#F/F(!\"\" /F/;,$F(F3F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "with a relative error less than " }{TEXT 318 6 "0.001%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 319 9 "Soluti on:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "J := Int(3-sin(2*x)*exp(x),x = -1 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$,&\"\"$\"\"\"*&-%$sinG6#,$*&\"\" #F*%\"xGF*F*F*-%$expG6#F2F*!\"\"/F2;F6F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "relative_error := N-> abs(evalf(simpson(integrand(J), x=-1..1,N)) - near_exact)/near_exact*100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/relative_errorGf*6#%\"NG6\"6$%)operatorG%&arrowGF(,$ *(\"$+\"\"\"\"-%$absG6#,&-%&evalfG6#-%(simpsonG6%-%*integrandG6#%\"JG/ %\"xG;!\"\"F/9$F/%+near_exactGFAF/FCFAF/F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "for N from 4 to infinity by 2 \nwhile relative _error(N) > 0.001\ndo\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "relative_error(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+=nXXy!#8 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 218 "As an illustration of \+ how much more efficient Simpson's Rule is than our previously most eff icient rule, the Midpoint Rule, let us see what value of N is required to achieve comparable accuracy with the Midpoint Rule:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "N := 'N':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "Relative_error := N-> abs(evalf(middlesum(integran d(J),x=-1..1,N)) - near_exact)/near_exact*100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/Relative_errorGf*6#%\"NG6\"6$%)operatorG%&arrowGF(,$ *(\"$+\"\"\"\"-%$absG6#,&-%&evalfG6#-%*middlesumG6%-%*integrandG6#%\"J G/%\"xG;!\"\"F/9$F/%+near_exactGFAF/FCFAF/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for N from 4 to infinity\nwhile Relative_erro r(N) > 0.001\ndo\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Re lative_error(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+2Z-Q'*!#8" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#a" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 57 "Finally, let us put the end point rules forever to rest :\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "N := 'N':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "slug := N-> abs(evalf(rightsum(inte grand(J),x=-1..1,N)) - near_exact)/near_exact*100;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%slugGf*6#%\"NG6\"6$%)operatorG%&arrowGF(,$*(\"$+\" \"\"\"-%$absG6#,&-%&evalfG6#-%)rightsumG6%-%*integrandG6#%\"JG/%\"xG;! \"\"F/9$F/%+near_exactGFAF/FCFAF/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "slug(1000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Re @gb!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "5. Error Estimates" }}{PARA 0 "" 0 "" {TEXT -1 161 "Approximati on techniques are most useful when a priori error estimates let you kn ow the greatest possible inaccuracy that may be incurred by using the \+ technique." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{TEXT 320 10 "M(k,f,a,b)" }{TEXT -1 31 " denote a numbe r such that " }{XPPEDIT 19 1 "abs((D@@k)(f)(x))<=M(k,f,a,b)" "6#1-% $absG6#---%#@@G6$%\"DG%\"kG6#%\"fG6#%\"xG-%\"MG6&F-F/%\"aG%\"bG" } {TEXT -1 13 " for every " }{TEXT 321 1 "x" }{TEXT -1 17 " in the int erval " }{TEXT 322 5 "[a,b]" }{TEXT -1 159 ". (Once you have such a n umber then any greater number will also satisfy the same inequality, b ut, as you will see, the idea is to find the least such value.)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 323 40 "Bounds o n the Approximation Rule Errors " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 15 "Midpoint Rule: " }{TEXT 325 3 " " } {TEXT 324 10 " " }{XPPEDIT 326 1 "M(2,f,a,b)*(b-a)^3/24/N^2" "6#**-%\"MG6&\"\"#%\"fG%\"aG%\"bG\"\"\"*$,&F*F+F)!\"\"\"\"$F+\"#CF.*$% \"NGF'F." }}{PARA 0 "" 0 "" {TEXT -1 18 "\nTrapezoidal Rule:" }{TEXT 328 1 " " }{TEXT 327 10 " " }{XPPEDIT 329 1 "M(2,f,a,b)*(b-a) ^3/12/N^2" "6#**-%\"MG6&\"\"#%\"fG%\"aG%\"bG\"\"\"*$,&F*F+F)!\"\"\"\"$ F+\"#7F.*$%\"NGF'F." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Simpson's Rule: " } {TEXT 330 10 " " }{XPPEDIT 331 1 "M(4,f,a,b)*(b-a)^5/180/N^4 " "6#**-%\"MG6&\"\"%%\"fG%\"aG%\"bG\"\"\"*$,&F*F+F)!\"\"\"\"&F+\"$!=F. *$%\"NGF'F." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Each estimate represents the " }{TEXT 332 8 "greatest" }{TEXT -1 93 " possible absolute error that ca n occur when the stated approximation technique is used with " }{TEXT 335 1 "N" }{TEXT -1 45 " subintervals to approximate the integral of \+ " }{TEXT 333 4 "f(x)" }{TEXT -1 19 " over the interval " }{TEXT 334 5 "[a,b]" }{TEXT -1 59 ". Often the error is less - sometimes significa ntly less. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "These error estimates can be used to determine the smallest " } {TEXT 336 1 "N" }{TEXT -1 113 " that is guaranteed to yield a required accuracy of approximation. (This is not to say that no smaller value \+ of " }{TEXT 337 1 "N" }{TEXT -1 61 " will do. However, any method of finding a smaller value of " }{TEXT 338 1 " " }{TEXT 340 1 "N" } {TEXT -1 49 " will have to depend on the particular function " } {TEXT 339 1 "f" }{TEXT -1 23 " under consideration.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 341 8 "Example:" }{TEXT -1 84 " Earlier in this worksheet we used Simpson's Rule to calculate \+ \n\n " }{TEXT 342 5 " " }{XPPEDIT 343 1 "Int(3-sin( 2*x)*exp(x),x=-1..1)" "6#-%$IntG6$,&\"\"$\"\"\"*&-%$sinG6#*&\"\"#F(%\" xGF(F(-%$expG6#F/F(!\"\"/F/;,$F(F3F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "to within " }{TEXT 344 1 " " }{XPPEDIT 345 1 "5*10^(-5)" "6#*&\"\"&\"\"\")\"#5,$F$!\"\"F%" }{TEXT -1 221 " . \+ Our method was by trial and error: we started with 2 subintervals and increased this number until we obtained the required accuracy. That p rocess involved unnecessary computations. Fortunately it ended quickly enough." }}{PARA 0 "" 0 "" {TEXT -1 68 "Use the error estimate for Si mpson's Rule to find a suitable number " }{TEXT 346 1 "N" }{TEXT -1 17 " of subintervals." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 347 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "First we have to determine an acceptable value for " }{TEXT 348 10 "M(4,f,a,b)" }{TEXT -1 15 ". Plotting the " }{TEXT 349 14 "abso lute value" }{TEXT -1 97 " of the fourth derivative over the interval \+ of integration is usually the easiest way to proceed:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart: \+ with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f := x -> 3-sin(2*x)*exp(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG 6\"6$%)operatorG%&arrowGF(,&\"\"$\"\"\"*&-%$sinG6#,$9$\"\"#F.-%$expG6# F4F.!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot( ab s( (D@@4)(f)(x)), x = -1 .. 1 );" }}{PARA 13 "" 1 "" {GLPLOT2D 371 197 197 {PLOTDATA 2 "6%-%'CURVESG6$7bo7$$!\"\"\"\"!$\"1oi&3e(y:g!#:7$$ !1nmm;p0k&*!#;$\"1-/)3w3si&F-7$$!1LL$3SF-7$$!1ML$3i.9!zF1$ \"1y'z2V:'zKF-7$$!1nm;/R=0vF1$\"19<]+^,2DF-7$$!1++]P8#\\4(F1$\"1K(GZS@ $=;F-7$$!1ML$3FuF)oF1$\"1NN!pdpL7\"F-7$$!1nm;/siqmF1$\"1\\SvM\\$Q/'F17 $$!1+++]Q*[c'F1$\"13'3oCHzO$F17$$!1ML$e\\g\"fkF1$\"1iV%>h>$Hj!#<7$$!1o 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\"%F(!\"\"$\"+66rwd!\"*#F(\"&++#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(inequality,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$,$%)infinityG!\"\"-%%OpenG6#$!+MlkV=!\")-F$6$-F*6#$ \"+MlkV=F.F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Since we are looking for a positive number we use the ope n interval with left endpoint " }{TEXT 353 11 "18.43646534" }{TEXT -1 49 ". Since N is an integer we must round. Although " }{TEXT 355 2 "1 8" }{TEXT -1 14 " is closer to " }{TEXT 354 11 "18.43646534" }{TEXT -1 10 " than to " }{TEXT 356 2 "19" }{TEXT -1 216 " we must neverthel ess round up. (That is because is outside the interval RealRange(Open (18.43646534),infinity). Finally, we recall that Simpson's Rule requir es an even number of subintervals. So we will have to take" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "N : = 20:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "absolute_error := \+ abs(evalf(simpson(f(x),x=-1..1,N))-evalf(Int(3-sin(2*x)*exp(x),x = -1 \+ .. 1))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "is( absolute_er ror < 5*10^(-5) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "As we have seen, even " }{TEXT 357 6 "N = 16" }{TEXT -1 9 " will do." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "N := 16: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "absolute_error := abs(e valf(simpson(f(x),x=-1..1,N))-evalf(Int(3-sin(2*x)*exp(x),x = -1 .. 1) )):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "is( absolute_error < 5*10^(-5) );" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%true G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "We \+ know from our trial and error process that " }{TEXT 358 6 "N = 16" } {TEXT -1 148 " is the smallest number of subintervals that will give \+ us the required accuracy. Although the error estimate does not yield t he smallest value of " }{TEXT 359 1 "N" }{TEXT -1 121 " it does give us a value that is in the right ballpark without requiring us to reso rt to computationally costly efforts." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "6. Code for the Figures" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Code for \+ Figures 4.1 and 4.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ff := x -> x^3+2*x+1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "pts := [[-2,ff(-2)], [0,ff(0)], [2,ff(2)]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "ptPlot := plot(pts, style =POINT, symbol = CIRCLE, color = MAROON):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "curve := plot(x^3+2*x+1,x=-2.5..2.5, axes=NONE):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display(ptPlot, curve);" } }{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6'-%'CURVESG6&7%7 $$!\"#\"\"!$!#6F*7$$F*F*$\"\"\"F*7$$\"\"#F*$\"#8F*-%'COLOURG6&%$RGBG$ \")viob!\")$\")!\\DP\"F<$\")%yg>%F<-%'SYMBOLG6#%'CIRCLEG-%&STYLEG6#%&P OINTG-F$6$7S7$$!3++++++++D!#<$!3++++++]i>!#;7$$!3smm;HU,\"R#FO$!3cLV,( \\L^u\"FR7$$!3=L$3FH'='H#FO$!3iyO8N\\*)p:FR7$$!3gmmTgBa*=#FO$!3_\">5T/ 'f(Q\"FR7$$!3wmm\"H_\">#3#FO$!3]L+*R#p<>7FR7$$!3ML$3_!4Nv>FO$!3sM;R;V& e1\"FR7$$!3km;/wfHw=FO$!3?Q^1un1e$*FO7$$!3;+]PM.tttpFO7$$!3%)**\\(oWB>c\"FO$!3mdJ;;HKMfFO7$$!3; LL$epjJX\"FO$!3/Cg#Q(4&\\(\\FO7$$!3amm\"z/otN\"FO$!3\\l@ukhh:UFO7$$!3) )****\\P[_\\7FO$!3)HsYt>[*\\MFO7$$!3#*****\\7)Q79\"FO$!3'Q2,wvc)oFFO7$ $!3'*****\\i^)o.\"FO$!3IcE'4ad&)=#FO7$$!3vlmT50A@%*!#=$!3E]TGVfY?Ffr7$$ !3q'**\\P4u\"oTFfr$\"3uM]@0y'[R*!#>7$$!3*z**\\7G-89$Ffr$\"3DHVbQ'=uS$F fr7$$!3%)GL$3Fp)p?Ffr$\"3)eD9@$3erdFfr7$$!3YKL3-$ff3\"Ffr$\"3Nw\")=&fu _\"yFfr7$$!3)Grmm\"z%zY#!#?$\"3?5(\\8*3k]**Ffr7$$\"3!fL$e*)>px5Ffr$\"3 *y^t40!z;7FO7$$\"3w++v$f4t.#Ffr$\"3B&)[%y.=fT\"FO7$$\"3OPL$e*GstIFfr$ \"30c7j8WyV;FO7$$\"3Y+++]#RW9%Ffr$\"3&feb(*Hu+!>FO7$$\"3Y,+]7j#>>&Ffr$ \"3SF3KT$R$y@FO7$$\"3t-+D1RU0iFfr$\"35LsY8)Q+[#FO7$$\"3+++](=S2L(Ffr$ \"3-R(G\\>+,'GFO7$$\"3:jmm;p)=M)Ffr$\"3!=K&G8\\')[KFO7$$\"3O-++v=]@%*F fr$\"3S+(*oRrf?PFO7$$\"35L$e*[$z*R5FO$\"3xGsu!pbZ?%FO7$$\"3e++]iC$p9\" FO$\"3Qkm'[!>g-[FO7$$\"3[m;H2qcZ7FO$\"3*zs6[awoV&FO7$$\"3O+]7.\"fFN\"F O$\"3)puA[X45='FO7$$\"3Ymm;/Ogb9FO$\"3%oQ@#4PJ&*pFO7$$\"3w**\\ilAFj:FO $\"3SpRIN3!p%zFO7$$\"3yLLL$)*ppm\"FO$\"3)f4=_p(4m*)FO7$$\"3)RL$3xe,t,\"FR7$$\"3Cn;HdO=y=FO$\"3ku90())z\"Q6FR7$$\"3a+++D>#[(>F O$\"3qm*[,YH^E\"FR7$$\"3SnmT&G!e&3#FO$\"3le$4'p%pUU\"FR7$$\"3#RLLL)Qk% =#FO$\"3g$=kz'pez:FR7$$\"37+]iSjE!H#FO$\"3/.B#>Br$fFR7$$\"3++++++++DFO$\"3++++++]i@FR-F76&F9$\"#5!\"\"F. 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Author Information" }}{PARA 0 "" 0 "" {TEXT -1 571 "Copyright \+ and Author Information\nM132L7R5.mws A MapleV Release 5 worksheet. \n\nAuthor: Brian E. Blank (18 February 2000)\n\nThis document may n ot be distributed by any medium,\nincluding print, disk, and electroni c transfer, without\nprior written permission of the author.\n\nFor mo re information, please contact the author:\n \n Department of \+ Mathematics, \n Washington University in St. Louis\n St. Louis , MO 63130\n \n Telephone: (314) 935-6763\n e-m ail: brian@math.wustl.edu\n\nCopyright: \251 2000 Brian E. Blank, \+ All Rights Reserved.\n" }}}}{MARK "6" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }