{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 "" -1 256 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 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 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 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 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 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 "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 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 "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple P lot" -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 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 16 "Trapezoidal Rule" }}} {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 18 "1. Getting Started" }}{PARA 0 "" 0 "" {TEXT -1 38 "First we clear \+ all variables in Maple." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "r estart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 21 "2. Collection of Data" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "Next we enter the data points from the border of Virginia. We have identified 11 data points (and consequently a regular partition \+ of 10 subintervals)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "xbor der:=[24,69,114,159,204,249,294,339,384,429,474]:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 53 "yborder:=[98,122,160,152,160,217,243,298,282 ,211,98]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 25 "3. Visualization of Data" }}{PARA 0 "" 0 "" {TEXT -1 35 "We create a plot of the data points" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "borderdata:=PLOT(POINTS([xborder[t],yborder[t]] $t=1. .11)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "and also a plot a horiz ontal line for the southern border" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "sborder:=plot(98,x=xborder[1]..xborder[11],color=red, thickness=2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "And now display \+ the data points and southern border as a check. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 " Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display(borderdata,sborder,view=[0..480,0 ..300]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'P OINTSG6-7$\"#C\"#)*7$\"#p\"$A\"7$\"$9\"\"$g\"7$\"$f\"\"$_\"7$\"$/#F.7$ \"$\\#\"$<#7$\"$%H\"$V#7$\"$R$\"$)H7$\"$%Q\"$#G7$\"$H%\"$6#7$\"$u%F(-% 'CURVESG6%7S7$$F'\"\"!$F(FK7$$\"3c****\\P>(3Q$!#;FL7$$\"3)***\\ilLKMUF PFL7$$\"3]++Dc(=T>&FPFL7$$\"3K++v$Hw-;'FPFL7$$\"3\\**\\7`=%=7(FPFL7$$ \"39+]i:iL8!)FPFL7$$\"3:+]i!*pUO*)FPFL7$$\"3m**\\i!z)3\"*)*FPFL7$$\"3) **\\7y*)oU3\"!#:FL7$$\"35++vtE:#=\"FcoFL7$$\"3)***\\(ovo$o7FcoFL7$$\"3 =++DYwUl8FcoFL7$$\"3'****\\(o])GY\"FcoFL7$$\"3)****\\PN.ob\"FcoFL7$$\" 3*)*\\iS:!4U;FcoFL7$$\"3/++v3W]VFcoFL7$$\"3y****\\A-\"y,#FcoFL7$$\"3,+DcJV'[6#FcoFL7$$\"3.+vo %z#G2AFcoFL7$$\"36+]il+D\"G>$[2PFcoFL7$$\"3`***\\PCV+!QFcoFL7$$\"3i*\\i!R]%p*QFcoFL7$$ \"3G+++&)HF!*RFcoFL7$$\"3-+]P*G9d3%FcoFL7$$\"3E+Dc\"Hl.=%FcoFL7$$\"3?+ +]K(RtE%FcoFL7$$\"3/+](oDAqO%FcoFL7$$\"33+++&\\zhX%FcoFL7$$\"3Q+Dc1(R7 b%FcoFL7$$\"3&**\\P%eWAUYFcoFL7$$FDFKFL-%'COLOURG6&%$RGBG$\"*++++\"!\" )$FKFKF_w-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fhw-%%VIEWG6$;F_ w$\"$![FK;F_w$\"$+$FK" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Do the data points and southern border roughly outline the state o f Virginia?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 43 "4. Const ruction of Approximating Trapezoids" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 370 "For the trapezoidal rule, we construct an approximating linear function on each of the 10 subintervals by connecting the right and l eft hand endpoint of each subinterval. The following \"do loop\" find s a linear function on each of the subintervals from 1 to 10. The \"d o loop\" also creates a plot of each of the linear functions on each o f the subintervals from 1 to 10." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 to 10 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "line[i]:=x->b[i]*x+c[i]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "s[i]:=solve(\{line[i](xborder[i])=yborder[i],line[i](xborder[i+1])=yb order[i+1]\},\{b[i],c[i]\}):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ass ign(s[i]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "lplot[i]:=plot(line[i ](x),x=xborder[i]..xborder[i+1],color=red,thickness=2):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 260 44 "5. Visualization of Approximating Trapez oids" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 226 "To see the trapezoids, w e draw in vertical lines corresponding to our regular partition. The \+ following commands create a plot of these partition lines. Note there are 11 partition lines because there are 11 partition points." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 to 11 do" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "partition[i]:=PLOT(CURVES([[xborder [i],0],[xborder[i],yborder[i]]])):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "partitionlines:=d isplay(partition[k] $k=1..11):" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Now we display what we've cr eated so far." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "display(bo rderdata,sborder,partitionlines,lplot[t] $t=1..10,view=[0..480,0..310] );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6;-%'POINTSG 6-7$\"#C\"#)*7$\"#p\"$A\"7$\"$9\"\"$g\"7$\"$f\"\"$_\"7$\"$/#F.7$\"$\\# \"$<#7$\"$%H\"$V#7$\"$R$\"$)H7$\"$%Q\"$#G7$\"$H%\"$6#7$\"$u%F(-%'CURVE SG6%7S7$$F'\"\"!$F(FK7$$\"3c****\\P>(3Q$!#;FL7$$\"3)***\\ilLKMUFPFL7$$ \"3]++Dc(=T>&FPFL7$$\"3K++v$Hw-;'FPFL7$$\"3\\**\\7`=%=7(FPFL7$$\"39+]i 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$$\"3))**\\PCV+'>%Fco$\"3!RL3#)G/$eAFco7$$\"3z\\i!R]%p0UFco$\"3;^7GQ`, VAFco7$$\"3!)****\\)HF]@%Fco$\"3ynmOi-HGAFco7$$\"3<+v$*G9dCUFco$\"3\"p ;aauJK@#Fco7$$\"3.]i:Hl.MUFco$\"3m$ek1#zH)>#Fco7$$\"3>++DtRtUUFco$\"3S ++l'GvX=#Fco7$$\"3++voDAq_UFco$\"3_L3Z*fZ)o@Fco7$$\"3+++]\\zhhUFco$\"3 Snmco0ya@Fco7$$\"3K]ilqR7rUFco$\"3i\\7jz@yR@Fco7$$\"3;]P%eWA-G%Fco$\"3 P](ow!oUD@Fco7$$FAFK$FBFKFhvF`w-FF6%7SFabt7$$\"37+]P>(3)*H%Fco$\"3ul;z :#p`3#Fco7$$\"3#)\\ilLKM3VFco$\"3/$3_@5QR1#Fco7$$\"3')*\\iv=TzJ%Fco$\" 3fnTXtl$)R?Fco7$$\"3'**\\PHw-wK%Fco$\"3Gn\"zTGvb,#Fco7$$\"3**\\7`=%=sL %Fco$\"3;%3xXIH9*>Fco7$$\"3)*\\i:iL8YVFco$\"3OFco7$$\"3))\\i! *pUObVFco$\"3/](ob0je%>Fco7$$\"3')\\i!z)3\"\\O%Fco$\"3cFco7$$ \"3w\\7y*)oUuVFco$\"3E]P**)p%*z*=Fco7$$\"3;+]Pn_@%Q%Fco$\"3UL$e_*\\Tt= Fco7$$\"3'**\\(ovo$GR%Fco$\"3Um\"H)y^w^=Fco7$$\"3-+]ikFa-WFco$\"3*))* \\F8ERF=Fco7$$\"3r**\\(o])G7WFco$\"3Q,]ir*>H!=Fco7$$\"3!)**\\PN.o@WFco $\"3'***\\sYgLzzv\"Fco7$$\"37+](3W]. W%Fco$\"3cL$etL`Ct\"Fco7$$\"3/++]e:%*[WFco$\"3'HLLk3!)3r\"Fco7$$\"3)) \\ilq]$*eWFco$\"3M](=qE&y&o\"Fco7$$\"3)****\\A-\"ynWFco$\"3OLLo*4sNm\" Fco7$$\"3u\\i:Lk[xWFco$\"3p^(=Xt+#R;Fco7$$\"3u\\(o%z#Gn[%Fco$\"3$4Dc:V $*fh\"Fco7$$\"3%)*\\iluv/9Fco 7$$\"3#)*\\(ohm(4e%Fco$\"31+Dwr_Kz8Fco7$$\"37++D#)p2!f%Fco$\"3YLLobLZc 8Fco7$$\"37+](o^$z*f%Fco$\"3-+]idS2K8Fco7$$\"3>]iST\")f3YFco$\"3A;a8nY '*48Fco7$$\"3B+]i@RA=YFco$\"3++]ZjKz&G\"Fco7$$\"3@]ilI5GFYFco$\"3o$3_j &)\\IE\"Fco7$$\"3-]7G>$[nj%Fco$\"3B\\PpUkFR7Fco7$$\"3!***\\PCV+YYFco$ \"3;L$eaeLg@\"Fco7$$\"3!)\\i!R]%pbYFco$\"37^(ozY+<>\"Fco7$$\"3!)****\\ )HF]m%Fco$\"3Gnmw$yk#o6Fco7$$\"3=+v$*G9duYFco$\"39m\"z " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Does the red outline look like Virginia?" }}{PARA 0 "" 0 "" {TEXT -1 87 "Note that the trapezoids extend from the northern red bor der of Virginia to the x axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 263 43 "6. Area Calculation using Geometr ic Formula" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 410 "To find the area b etween the northern red border and the x axis, we use a geometric form ula to determine the area of each of the trapezoids on the 10 subinter vals and then sum these areas. Note that this sum includes the area o f the red region [state of Virginia] as well as the area of the rectan gular region between the southern boundary of Virginia and the x-axis. So, this approximation must be adjusted." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 44 "45*sum((yborder[j]+yborder[j+1])/2,j=1..10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"&Nu)" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "To calculate the area of the red outlined Virginia, we must subtract the area of the r ectangle with top edge defined by the southern border of Virginia." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "87435-98*(472-24);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"&JN%" }}}{PARA 0 "" 0 "" {TEXT -1 131 "Thus, the trapezoidal approximation(using 10 subintervals) of the area of Virginia(minus the Eastern Shore) is 43531 square pixels." }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 36 "7 . Area Calculation using Integrals" }}{PARA 0 "" 0 "" {TEXT -1 421 "T o find the area between the northern red border and the x axis, we det ermine the area between the approximating linear function and the x ax is on each subinterval. We use integrals for this calculation (Note: \+ Integration is not necessary when using trapezoids, but is necessary \+ when using parabolas in Simpson's rule). The following commands do th e integral calculation on each subinterval and then sum those results. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 to 10 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "area[i]:=int(b[i]*x+c[i],x=xborde r[i]..xborder[i+1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sum(area[j],j=1..10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"&Nu)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "To calculate the area of the red outlined Virginia, we must subtract the area of the rectangle with top edge defined by the southern border of Virginia." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "87435-98*(472-24);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&JN%" }}}{PARA 0 "" 0 "" {TEXT -1 131 "Thu s, the trapezoidal approximation(using 10 subintervals) of the area of Virginia(minus the Eastern Shore) is 43531 square pixels." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 11 "8. Solution" }}{PARA 0 "" 0 " " {TEXT 257 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "According to the map s cale, 80 miles = 87 pixels so " }{XPPEDIT 18 0 "80/87;" "6#*&\"#!)\" \"\"\"#()!\"\"" }{TEXT -1 23 " miles = 1 pixel so " }{XPPEDIT 18 0 "(80/87)^2;" "6#*$*&\"#!)\"\"\"\"#()!\"\"\"\"#" }{TEXT -1 31 " square \+ miles = 1 square pixel " }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "We convert 43531 square pixels to square \+ miles" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "43531.*(80/87)^2; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Q@y!o$!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "We conclude that the trapezoidal approximation (using 10 subintervals) of the area of Virginia (minus \+ the Eastern Shore) is 36808 square miles." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 23 "9. Trapezoidal Summary " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Is your trapezoidal approxima tion an under or over approximation? Explain." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Answer:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}}{MARK "2 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }