{VERSION 2 3 "IBM INTEL NT" "2.3" } {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 "" 1 24 73 136 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 78 101 119 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 10 60 136 17 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 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 "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 " " 0 1 0 0 248 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 240 63 16 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 58 136 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 256 14 "Simpson's Rule" }}{PARA 256 " " 0 "" {TEXT -1 24 "(A better approximation)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 257 12 "by Greg Crow" }}{PARA 258 " " 0 "" {TEXT 258 22 "(please report errors)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 333 "In our previous lab we showed \+ that a quadratic can be fit through three points. The goal of this la boratory is to try and extend the idea of linear and trapezoidal appro ximation of definite integrals to a method that will find the area und er quadratic functions fit to each interval. We begin with the linear and trapezoidal rules." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 9 "Example 1" }{TEXT -1 40 ": Approximate the area und er the curve " }{XPPEDIT 18 0 "f(x)= cos(x)+x/6+ 2" "/-%\"fG6#%\"xG,(- %$cosG6#F&\"\"\"*&F&F+\"\"'!\"\"F+\"\"#F+" }{TEXT -1 76 " on the inter val [10,16] using the leftbox, rightbox and trapezoid commands." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Load the student package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7E%\"DG%%DiffG%*DoubleintG%$IntG%&LimitG%(Linein tG%(ProductG%$SumG%*TripleintG%*changevarG%(combineG%/completesquareG% )distanceG%'equateG%(extremaG%*integrandG%*interceptG%)intpartsG%(isol ateG%(leftboxG%(leftsumG%)makeprocG%)maximizeG%*middleboxG%*middlesumG 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33 "e valf(rightsum(f(x),x=10..16,2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+\">O\\j#!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 "When we turn to the trapezoid approximation algorithm we find that as predicted it is just the average of the left and right s ums. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "evalf(trapezoid(f(x),x=10..16,2));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+$QCF]#!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "evalf( (leftsum(f(x),x=10..16,2) + rightsum(f(x),x=10..16,2)) /2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$QCF]#!\")" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Find the linear ap proximation on each box for the trapezoid rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "p1:=x->evalf (((f(13)-f(10))/(13-10)),2)*(x-10)+evalf(f(10),2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "p2:=x->evalf(((f(16)-f(13))/(16-13)),2)*(x- 13)+evalf(f(13),2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "conv ert(If(x<13,p1(x),x<16,p2(x),undefined),piecewise);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%*PIECEWISEG6%7$,&%\"xG$\"#v!\"#$!$g%F+\"\"\"2F(\"#8 7$,&F($!#XF+$\"%&4\"F+F.2F(\"#;7$%*undefinedG%*otherwiseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 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7$$\"1+++/KV*H\"F1$\"1******H!\\d9&F-7$$\"1+++Vq1+8F1$\"1++]1$)p*4&F-7 $$\"1+++#)3q+8F1$\"1*****4.Yo4&F-7$$\"1++++?H48F1$\"1********f=e]F-7$F cr$\"1****\\tf_>]F-7$$\"1+++T,(yL\"F1$\"1****\\lVeH\\F-7$$\"1+++X?cb8F 1$\"1****\\(zq*\\[F-7$$\"1+++mG(\\P\"F1$\"1*****H5AEw%F-7$$\"1+++flX$R \"F1$\"1++]%[X%zYF-7$$\"1+++`Bu79F1$\"1++]6%fEf%F-7$$\"1+++LFXI9F1$\"1 ****\\,F'H^%F-7$$\"1+++pdb\\9F1$\"1****\\R!**pU%F-7$$\"1+++c%)Rp9F1$\" 1*****z%pqPVF-7$$\"1+++t:n'[\"F1$\"1****\\@z(*fUF-7$$\"1+++7qK0:F1$\"1 *****fWGg<%F-7$$\"1+++2**fC:F1$\"1****\\=/I*3%F-7$$\"1+++uYXV:F1$\"1** ***p'RX/SF-7$$\"1+++Iwph:F1$\"1*****\\mgB#RF-7$$\"1+++BL&>e\"F1$\"1++] Y+@JQF-7$%%FAILGFj[m-Fa^l6&Fc^lF*Fd^lF*-%*THICKNESSG6#\"\"#-%+AXESLABE LSG6$%\"xG%\"yG-%%VIEWG6$;F(F\\^l;F*$\"\"(F*" 2 384 384 384 2 0 1 2 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 }}}{PARA 0 "" 0 "" {TEXT 261 9 "Example 2 " }{TEXT -1 40 ": Approximate the area under the curve " }{XPPEDIT 18 0 "f(x)= cos(x)+x/6+ 2" "/-%\"fG6#%\"xG,(-%$cosG6#F&\"\"\"*&F&F+\" \"'!\"\"F+\"\"#F+" }{TEXT -1 46 " on the interval [10,16] using Simpso n's rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "This section will create the 1st approximating quadratic." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "x:=array([10,11.5,13]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y:=array([f(x[1]),f(x[2]), f(x[3])]):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "p1:=x->a*x^2+b*x+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G:6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&%\"aG\"\"\"9$\"\"#F/ *&%\"bGF/F0F/F/%\"cGF/F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "solve(\{p1(x[1])=y[1], p1(x[2])=y[2], p1(x[3])=y[3]\}, \{a,b,c\}); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"aG$!+Zh2'*>!#5/%\"bG$\"+wX\" )R`!\"*/%\"cG$!+:*y41$!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(\",p1(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p1 : = unapply(\",x):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "The next section creates the 2nd approximating quadratic. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "x:=array([13,14.5,16]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y:=array([f(x[1]),f(x[2]), f(x[3])]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p2:=x->a*x^2+b*x+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G:6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&%\"aG\"\"\" 9$\"\"#F/*&%\"bGF/F0F/F/%\"cGF/F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "solve(\{p2(x[1])=y[1], p2(x[2])=y[2], p2(x[3])=y[3]\} , \{a,b,c\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"aG$\"+5u&eY\"!# 5/%\"cG$\"+/3%z9%!\")/%\"bG$!+4>-1Z!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(\",p2(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p2 := unapply(\",x):" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "We need to put the two quadrat ic functions together to graph them. The commands that do this are " } {MPLTEXT 1 0 7 "convert" }{TEXT -1 5 " and " }{MPLTEXT 1 0 9 "piecewis e" }{TEXT -1 1 "." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 55 "convert(If(x<13,p1(x),x<16,p2(x),undefined), piecewise):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "g := unapply (\",x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(g(x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6%7$,(*$%\"xG\"\"#$!+ Zh2'*>!#5F)$\"+\"e9)R`!\"*$!+:*y41$!\")\"\"\"2F)$\"#8\"\"!7$,(F($\"+5u &eY\"F-F)$!+4>-1ZF0$\"+/3%z9%F3F42F)$\"#;F87$%*undefinedG1FBF)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([f(x),g(x)],x=10..19,y= 0..7,thickness=2);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6$7hn7$$ \"#5\"\"!$\"19-fP^fFG!#:7$$\"1++vQuh>5!#9$\"1d1\")HXU#)HF-7$$\"1+DJnko O5F1$\"1JG9y\"4(RJF-7$$\"1+]7vB)e0\"F1$\"1>Xs%p1oL$F-7$$\"1+](e_0_2\"F 1$\"1p()*e*)*)3b$F-7$$\"1+D1PoV%4\"F1$\"1G!4E&y(Gx$F-7$$\"1+DJCnE76F1$ \"1O=Ge*G0)RF-7$$\"1+D\")R&G28\"F1$\"1*zSMIN7>%F-7$$\"1+D\"ex@)\\6F1$ \"1Go#3N8\")R%F-7$$\"1+DczP&)o6F1$\"1;#4,h4pe%F-7$$\"1++vM0V)=\"F1$\"1 Vl*Q***)pv%F-7$$\"1+]P^Pn07F1$\"1wb78)zB)[F-7$$\"1++DHb3D7F1$\"1)p#*G \"eW#*\\F-7$$\"1++]r7$[B\"F1$\"1k#*p^:PM]F-7$$\"1++v8qdW7F1$\"1&*[P&fJ q1&F-7$$\"1++DU)oRD\"F1$\"1HS&=7#f*3&F-7$$\"1++vq1Oj7F1$\"1j`?B;M.^F-7 $$\"1]7y]$*)=F\"F1$\"1#HF(*\\1#3^F-7$$\"1+D\"3.=/G\"F1$\"1soOPj)e5&F-7 $$\"1]7Gc%f0H\"F1$\"1A;#>hXR4&F-7$$\"1++v\")3q+8F1$\"1Ol*fW9B2&F-7$$\" 1+]P**>H48F1$\"1.v[m'*pY]F-7$$\"1+++f<\\F-7$$\"1++]W?cb8F1$\"1LLPTpe3[F-7$$\"1+DJmG(\\P \"F1$\"1)yJUDR%pYF-7$$\"1+v$*elX$R\"F1$\"1\"H%)=vXO_%F-7$$\"1+]7`Bu79F 1$\"1HHGWQJkVF-7$$\"1+DcKFXI9F1$\"1Bmw*o2v@%F-7$$\"1+]Ppdb\\9F1$\"1)oQ Sfe^1%F-7$$\"1+D\"eX)Rp9F1$\"1lp(R?50#RF-7$$\"1+vos:n'[\"F1$\"1I-o?ED6 QF-7$$\"1+]77qK0:F1$\"1\\XDSNk:PF-7$$\"1++]1**fC:F1$\"1)z(yy5#ek$F-7$$ \"1+D1!HFS`\"F1$\"1\"pEJ4_Ni$F-7$$\"1+]itYXV:F1$\"1#)[?z/d4OF-7$$\"1]( o>:wDb\"F1$\"1t'z&*yzTg$F-7$$\"1+DJIwph:F1$\"1!ysG%e'pg$F-7$$\"1]P%oZD =d\"F1$\"1bm4&3i(>OF-7$$\"1+]PBL&>e\"F1$\"1a(*QZj!Gk$F-7$$\"1+v$Rk`5f \"F1$\"1B6o\\R?sOF-7$$\"1++]kR:+;F1$\"1O!Rw>3(4PF-7$$\"1++vLqe>;F1$\"1 _.C5e*f\"QF-7$$\"1+D\"GG'>P;F1$\"1MJg\\n7TRF-7$$\"1++DVyWc;F1$\"11=fZ> n0TF-7$$\"1+DJh?cu;F1$\"11>7(e(p#G%F-7$$\"1+DcQm\\$p\"F1$\"1X2iZOV&[%F -7$$\"1++v['3?r\"F1$\"1N.!fiR`p%F-7$$\"1+D\"y+*QJ'[4dF-7$$\"1+]P^ WSD=F1$\"19\\I%\\--(eF-7$$\"1+++**eBV=F1$\"1WBzj\")G')fF-7$$\"1+DJTzCi =F1$\"1fvT#Hv!ygF-7$$\"1++];kMr=F1$\"1&p/]\\k'4hF-7$$\"1+vo\"*[W!)=F1$ \"1:H-rw0LhF-7$$\"1]P%eWA-*=F1$\"1\"3wi2%)*[hF-7$$\"#>F*$\"1OL&[Gr`:'F --%'COLOURG6&%$RGBG$F)!\"\"F*F*-F$6$7I7$F($\"15++S^fFGF-7$$\"+?(3)45! \")$\"*6jy&HF__l7$$\"1+++Ruh>5F1$\"10%[&)=!H%3$F-7$$\"1+++`>:G5F1$\"1) p\")3%)o6>$F-7$$\"1+++nkoO5F1$\"1!*G:#pR^H$F-7$$\"1+++@WGY5F1$\"1v2OI> f3MF-7$$\"1+++vB)e0\"F1$\"1gD4plO=NF-7$$\"1+++^Ral5F1$\"1)*QgCY:DOF-7$ $\"1+++Eb?v5F1$\"1=C8`h@GPF-7$$\"1+++PoV%4\"F1$\"16'G0RoA#RF-7$$\"1+++ CnE76F1$\"1O^\\V'*)*)3%F-7$$\"1+++S&G28\"F1$\"1g.aEfC[UF-7$$\"1+++w<#) \\6F1$\"1zcH%=O')R%F-7$$\"1+++!y`)o6F1$\"1,!>whgS`%F-7$$\"1+++N0V)=\"F 1$\"1IWTjUFeYF-7$$\"1+++^Pn07F1$\"1C\")RD(3]v%F-7$$\"1+++Hb3D7F1$\"1u< Nqfq\\[F-7$$\"1+++9qdW7F1$\"1)\\f'fhlH\\F-7$$\"1+++r1Oj7F1$\"1)pSrC_B* \\F-7$$\"1+++J!=/G\"F1$\"1_4wEK3P]F-7$$\"1+++d%f0H\"F1$\"1P9q#4s\"e]F- 7$$\"1+++#)3q+8F1$\"135C7#\\y1&F-7$$\"1++++?H48F1$\"1k'G%f\\B#*\\F-7$F cr$\"1:7%[`%y=\\F-7$$\"1+++T,(yL\"F1$\"1]1+vnFcZF-7$$\"1+++X?cb8F1$\"1 \"4nQW,Ai%F-7$$\"1+++mG(\\P\"F1$\"1J#G5fcc[%F-7$$\"1+++flX$R\"F1$\"1\" H7j!4!fO%F-7$$\"1+++`Bu79F1$\"1O?&[$fi^UF-7$$\"1+++LFXI9F1$\"1!G*fq.Hc TF-7$$\"1+++pdb\\9F1$\"1[p)eqmP1%F-7$$\"1+++c%)Rp9F1$\"1of%=e))*yRF-7$ $\"1+++t:n'[\"F1$\"16j^rme9RF-7$$\"1+++7qK0:F1$\"1^ixXf&[&QF-7$$\"1+++ 2**fC:F1$\"12\\X6N'Q!QF-7$$\"1+++uYXV:F1$\"1&RG/C:Xw$F-7$$\"1+++Iwph:F 1$\"12]`!Gkjt$F-7$$\"1+++BL&>e\"F1$\"1^rmS%Qlr$F-7$%%FAILGFejl-Fa^l6&F c^lF*Fd^lF*-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$%\"xG%\"yG-%%VIEWG6$;F( F\\^l;F*$\"\"(F*" 2 384 384 384 2 0 1 2 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 -12478 -798 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf(simpson(f(x ),x=10..16,2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6cI$o#!\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 8 "Example \+ " }{TEXT -1 67 "3: Compare the errors on the leftsum and trapezoid ap proximations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:=x->cos(x) + x/6+2;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%)operatorG%&arrowGF(,(-%$cos G6#9$\"\"\"F0#F1\"\"'\"\"#F1F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "L:=[n, \n10^n,\nevalf( leftsum(f(x),x=10..16,(10^n)),15),\nevalf( ( leftsum(f(x),x=10..16,1 0^n)-evalf(int(f(x),x=10..16),15) ), 15)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7&\"\"\"\"#5$\"0l#e9kR)\\#!#8$!.yR[O:s#F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "L:=[n, \n10^n,\nevalf(leftsum(f(x),x=10..16,(10^n)),15),\nevalf( \+ ( leftsum(f(x),x=10..16,10^n)-evalf(int(f(x),x=10..16),15) ), 15)] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7&\"\"#\"$+\"$\"0I\"Gf)fH_# !#8$!-8T,#>l#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=3;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "L:=[n, \n10^n,\nevalf(leftsum(f(x),x=10..16,(10^n )),15),\nevalf( ( leftsum(f(x),x=10..16,10^n)-evalf(int(f(x),x=10..1 6),15) ), 15)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7&\"\"$\"%+ 5$\"0Us*ysMDD!#8$!,,]/]k#F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "Doing ten times as many subintervals buys us o ne more decimal place of accuracy. This can be seen by comparing" }} {PARA 0 "" 0 "" {TEXT -1 33 "the ratios of consecutive errors:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "-.265192014113e-1/(-.2721536483978);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+f??W(*!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "-.26 450045001e-2/(-.265192014113e-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+P@#R(**!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "What this means is that if we work ten times as long, we \+ can get 10 times the accuracy." }}{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 -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Question 1: Draw a Simpson's rule picture for " } {XPPEDIT 18 0 "f(x)=sin(x)+3" "/-%\"fG6#%\"xG,&-%$sinG6#F&\"\"\"\"\"$F +" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,2Pi]" "7$\"\"!* &\"\"#\"\"\"%#PiGF&" }{TEXT -1 19 " with 2 parabolas. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Question 2: Compute the actual value of " }{XPPEDIT 18 0 " Int(sin(x)+3,x=0..2Pi" "-%$Int G6$,&-%$sinG6#%\"xG\"\"\"\"\"$F*/F);\"\"!*&\"\"#F*%#PiGF*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "Question 3: How much accuracy do we gain by using a tenfold increase \+ in the number of subdivisions with the trapezoid rule?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Question 4: Repeat q uestion 3 for Simpson's rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 117 "Question 5: For the stout hearted: Veri fy the formula for Simpson's rule in problem 20 of section 7.6. (Use \+ Maple)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "85 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }