{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 "" -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 }{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 Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 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 28 "Left Endpoint Approximati ons" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 19 "1. Getting Started" }}{PARA 0 "" 0 "" {TEXT -1 39 "First, we \+ clear all variables in Maple." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 21 "2. Collection of Data" }}{PARA 0 "" 0 "" {TEXT -1 153 "Next, we enter the data points from the border of Virginia. We h ave identified 11 data points (and consequently a regular partition of 10 subintervals)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "xborde r:=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "yborder:=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 25 "3. Visu alization of Data" }}}{EXCHG {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 61 "and also a plot of a horizontal li ne 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 56 "And now display the data poin ts and the southern border." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name cha ngecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display(borderdata,sborder,view=[0..480,0..300]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Do the da ta points and southern border provide a rough outline of the state of \+ Virginia?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 49 "4. Construction of Approximating Left R ectangles" }}{PARA 0 "" 0 "" {TEXT -1 166 "We construct an approximati ng left endpoint rectangle on each of the 10 subintervals. The followi ng loop creates a plot of the upper edge of each of these rectangles. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 to 10 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "pplotred[i]:=plot(yborder[i],x=xb order[i]..xborder[i+1],color=red,thickness=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 256 "" 0 "" {TEXT -1 50 "5. Visualization of Approximati ng Left Rectangles" }}{PARA 0 "" 0 "" {TEXT -1 196 "To see the left re ctangles, we draw in vertical lines corresponding to our 10 subinterv als. The following commands create a plot of these partition lines (no te that there are 11 partition lines)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "partition[1]:=PLOT(CURVES([[xborder[1],0],[xborder[1] ,yborder[1]]])):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i f rom 2 to 11 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "partition[i]:=PLO T(CURVES([[xborder[i],0],[xborder[i],max(yborder[i-1],yborder[i])]])): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "partitionlines:=display(partition[k] $k=1..11):" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "N ow, we display what we've created so far." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 85 "display(sborder,borderdata,partitionlines,pplotred[ t] $t=1..10,view=[0..480,0..310]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Does the \+ region bounded above by the rectangles and below by the red horizontal line look like Virginia?" }}{PARA 0 "" 0 "" {TEXT -1 88 "Note that th e left rectangles extend from the northern border of Virginia to the x axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 41 "6. Area Calculation by Geometric Formula" }}{PARA 0 "" 0 "" {TEXT -1 400 "To find the area between the northern rectangular b order and the x axis, we use a geometric formula to determine the area of each of the left rectangles on each of the 10 subintervals and the n sum the areas. Note that this sum includes the area of Virginia as w ell as the area of the rectangular region between the southern border \+ of Virginia and the x axis. So, this approximation must be adjusted. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sum(45*yborder[j],j=1.. 10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "To calculate the area of the red outlined Virginia, we must subtract the area of the rectangle with top edge defined by the southern border of Virginia" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "?????-(474-24)*98;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "Thus, the left endpoint approximation (u sing 10 subintervals) of the area of Virginia (minus the Eastern Shore ) is _______ square pixels." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 13 "7. Solution " }}{PARA 0 "" 0 "" {TEXT -1 54 "According to the map scale, 80 miles = 87 pixels so " }{XPPEDIT 18 0 "80/87;" "6#*& \"#!)\"\"\"\"#()!\"\"" }{TEXT -1 23 " miles = 1 pixel so " } {XPPEDIT 18 0 "(80/87)^2;" "6#*$*&\"#!)\"\"\"\"#()!\"\"\"\"#" }{TEXT -1 31 " square miles = 1 square pixel " }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We convert 44190 square p ixels to square miles" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "?? ??????*(80/87)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "We conclude that the left endpoin t approximation(using 10 subintervals) of the area of Virginia (minus \+ the Eastern Shore) is ___________ square miles." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 25 "8. L eft Endpoint Summary" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Is your l eft endpoint approximation an under or over approximation? Explain. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Answer :" }}}}{MARK "38 0 0" 136 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }