(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, cellOutline, groupLikeTitle, center, M18, O486, R65535, e8, 24, "CalcMath"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, bold, R21845, G21845, B21845, e6, 12, "CalcMath"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, R21845, G21845, B21845, e6, 12, "CalcMath"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "CalcMath"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "CalcMath"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "CalcMath"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "CalcMath"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, B65535, 12, "CalcMath"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M36, N23, O486, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M36, N23, O486, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M18, O486, l34, w351, h314, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = leftheader, inactive, M18, O486, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M18, O486, 12, "Times"; fontset = leftfooter, inactive, M18, O486, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, whiteBox, M18, O486, bold, R21845, G21845, B21845, 12, "CalcMath"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, R21845, G21845, B21845, 12, "CalcMath"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, 10, "Courier"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 10, "Courier"; paletteColors = 256; automaticGrouping; currentKernel; ] :[font = input; preserveAspect] :[font = input; preserveAspect] :[font = title; inactive; preserveAspect; startGroup] Mini Project -2 Ellipses :[font = text; inactive; preserveAspect] by Shirley Treadway :[font = section; inactive; preserveAspect; startGroup] Introduction :[font = text; inactive; preserveAspect; endGroup] One of the more important conic sections is called an ellipse. If it is located on the coordinate axes so that the center is at the origin it has a very nice equation, (xÛ/aÛ) + (yÛ/bÛ) = 1 or (xÛ/bÛ) + (yÛ/aÛ)= 1. There is a special relationship between a and b and another variable usually referred to as c, that is that aÛ= bÛ + cÛ. There are two points on the x-axis that have coordinates (c,0) and (-c,0) or in the second case on the y-axis with coordinates (0,c) and (0,-c). We would like to investigate the relationship between the distances from any point on the ellipse, which we will refer to as P, and the two points, (c,0) and (-c,0) or (0,c) and (0,-c). As we construct our first ellipse, you'll will notice the what points have "a" and have "b" as a part of their respective coordinates. :[font = section; inactive; Cclosed; preserveAspect; startGroup] Investigate Ellipse (major axis--- x-axis) :[font = text; inactive; preserveAspect] ** Activate the code below until you get to the first questions. :[font = input; initialization; preserveAspect] *) Needs["Graphics`Colors`"] (* :[font = input; preserveAspect] dist[{x1_,y1_},{x2_,y2_}] = Sqrt[(x2 - x1)^2 + (y2 - y1)^2]; midpoint[{x1_,y1_},{x2_,y2_}] = {(x2+x1)/2,(y2+y1)/2}; :[font = input; preserveAspect; startGroup] Clear[x,y,t] a= 4; b = 2; x[t_]= a Cos[t]; y[t_] = b Sin[t]; ellipse=ParametricPlot[{x[t],y[t]},{t,0,2 Pi}, PlotStyle->{{Thickness[0.005]}}, AspectRatio->Automatic, PlotRange->{{-a-2,a+2},{-b-2,b+2}}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 234; endGroup] %! %%Creator: Mathematica %%AspectRatio: .66667 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0833333 0.333333 0.0833333 [ [(-6)] 0 .33333 0 2 Msboxa [(-4)] .16667 .33333 0 2 Msboxa [(-2)] .33333 .33333 0 2 Msboxa [(2)] .66667 .33333 0 2 Msboxa [(4)] .83333 .33333 0 2 Msboxa [(6)] 1 .33333 0 2 Msboxa [(-4)] .4875 0 1 0 Msboxa [(-3)] .4875 .08333 1 0 Msboxa [(-2)] .4875 .16667 1 0 Msboxa [(-1)] .4875 .25 1 0 Msboxa [(1)] .4875 .41667 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(3)] .4875 .58333 1 0 Msboxa [(4)] .4875 .66667 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .66767 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .33333 m 0 .33958 L s P [(-6)] 0 .33333 0 2 Mshowa p .002 w .16667 .33333 m .16667 .33958 L s P [(-4)] .16667 .33333 0 2 Mshowa p .002 w .33333 .33333 m .33333 .33958 L s P [(-2)] .33333 .33333 0 2 Mshowa p .002 w .66667 .33333 m .66667 .33958 L s P [(2)] .66667 .33333 0 2 Mshowa p .002 w .83333 .33333 m .83333 .33958 L s P [(4)] .83333 .33333 0 2 Mshowa p .002 w 1 .33333 m 1 .33958 L s P [(6)] 1 .33333 0 2 Mshowa p .001 w .03333 .33333 m .03333 .33708 L s P p .001 w .06667 .33333 m .06667 .33708 L s P p .001 w .1 .33333 m .1 .33708 L s P p .001 w .13333 .33333 m .13333 .33708 L s P p .001 w .2 .33333 m .2 .33708 L s P p .001 w .23333 .33333 m .23333 .33708 L s P p .001 w .26667 .33333 m .26667 .33708 L s P p .001 w .3 .33333 m .3 .33708 L s P p .001 w .36667 .33333 m .36667 .33708 L s P p .001 w .4 .33333 m .4 .33708 L s P p .001 w .43333 .33333 m .43333 .33708 L s P p .001 w .46667 .33333 m .46667 .33708 L s P p .001 w .53333 .33333 m .53333 .33708 L s P p .001 w .56667 .33333 m .56667 .33708 L s P p .001 w .6 .33333 m .6 .33708 L s P p .001 w .63333 .33333 m .63333 .33708 L s P p .001 w .7 .33333 m .7 .33708 L s P p .001 w .73333 .33333 m .73333 .33708 L s P p .001 w .76667 .33333 m .76667 .33708 L s P p .001 w .8 .33333 m .8 .33708 L s P p .001 w .86667 .33333 m .86667 .33708 L s P p .001 w .9 .33333 m .9 .33708 L s P p .001 w .93333 .33333 m .93333 .33708 L s P p .001 w .96667 .33333 m .96667 .33708 L s P p .002 w 0 .33333 m 1 .33333 L s P p .002 w .5 0 m .50625 0 L s P [(-4)] .4875 0 1 0 Mshowa p .002 w .5 .08333 m .50625 .08333 L s P [(-3)] .4875 .08333 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-2)] .4875 .16667 1 0 Mshowa p .002 w .5 .25 m .50625 .25 L s P [(-1)] .4875 .25 1 0 Mshowa p .002 w .5 .41667 m .50625 .41667 L s P [(1)] .4875 .41667 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .58333 m .50625 .58333 L s P [(3)] .4875 .58333 1 0 Mshowa p .002 w .5 .66667 m .50625 .66667 L s P [(4)] .4875 .66667 1 0 Mshowa p .001 w .5 .01667 m .50375 .01667 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .05 m .50375 .05 L s P p .001 w .5 .06667 m .50375 .06667 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .11667 m .50375 .11667 L s P p .001 w .5 .13333 m .50375 .13333 L s P p .001 w .5 .15 m .50375 .15 L s P p .001 w .5 .18333 m .50375 .18333 L s P p .001 w .5 .2 m .50375 .2 L s P p .001 w .5 .21667 m .50375 .21667 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .26667 m .50375 .26667 L s P p .001 w .5 .28333 m .50375 .28333 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .31667 m .50375 .31667 L s P p .001 w .5 .35 m .50375 .35 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .38333 m .50375 .38333 L s P p .001 w .5 .4 m .50375 .4 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45 m .50375 .45 L s P p .001 w .5 .46667 m .50375 .46667 L s P p .001 w .5 .48333 m .50375 .48333 L s P p .001 w .5 .51667 m .50375 .51667 L s P p .001 w .5 .53333 m .50375 .53333 L s P p .001 w .5 .55 m .50375 .55 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .6 m .50375 .6 L s P p .001 w .5 .61667 m .50375 .61667 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65 m .50375 .65 L s P p .002 w .5 0 m .5 .66667 L s P P 0 0 m 1 0 L 1 .66667 L 0 .66667 L closepath clip newpath p p .005 w .83333 .33333 m .83332 .3347 L .83329 .33606 L .83323 .33742 L .83315 .33879 L .83293 .34151 L .83262 .34423 L .83222 .34695 L .83173 .34967 L .83048 .35509 L .82888 .36048 L .82693 .36585 L .82198 .37647 L .81564 .38691 L .80796 .39711 L .78868 .41667 L .76445 .43479 L .7357 .45118 L .70292 .46556 L .66667 .47767 L .64743 .48281 L .62756 .48731 L .60715 .49116 L .58627 .49432 L .56503 .4968 L .5543 .49777 L .54351 .49857 L .53267 .4992 L .52724 .49944 L .5218 .49964 L .51636 .4998 L .51363 .49986 L .51091 .49991 L .50818 .49995 L .50545 .49998 L .50273 .49999 L .5 .5 L .49727 .49999 L .49455 .49998 L .49182 .49995 L .48909 .49991 L .48637 .49986 L .48364 .4998 L .4782 .49964 L .47276 .49944 L .46733 .4992 L .45649 .49857 L .4457 .49777 L .43497 .4968 L .41373 .49432 L Mistroke .39285 .49116 L .37244 .48731 L .33333 .47767 L .29708 .46556 L .2643 .45118 L .23555 .43479 L .21132 .41667 L .20104 .40705 L .19204 .39711 L .18436 .38691 L .17802 .37647 L .17307 .36585 L .17112 .36048 L .16952 .35509 L .16827 .34967 L .16778 .34695 L .16738 .34423 L .16707 .34151 L .16695 .34015 L .16685 .33879 L .16677 .33742 L .16671 .33606 L .16668 .3347 L .16667 .33333 L .16668 .33197 L .16671 .33061 L .16677 .32924 L .16685 .32788 L .16707 .32516 L .16738 .32243 L .16778 .31971 L .16827 .317 L .16952 .31158 L .17112 .30618 L .17307 .30082 L .17802 .2902 L .18436 .27976 L .19204 .26955 L .21132 .25 L .23555 .23187 L .2643 .21548 L .29708 .20111 L .33333 .189 L .35257 .18385 L .37244 .17935 L .39285 .17551 L .41373 .17235 L .43497 .16987 L .4457 .16889 L .45649 .16809 L Mistroke .46733 .16747 L .47276 .16722 L .4782 .16702 L .48364 .16687 L .48637 .16681 L .48909 .16676 L .49182 .16672 L .49455 .16669 L .49727 .16667 L .5 .16667 L .50273 .16667 L .50545 .16669 L .50818 .16672 L .51091 .16676 L .51636 .16687 L .5218 .16702 L .52724 .16722 L .53267 .16747 L .54351 .16809 L .5543 .16889 L .56503 .16987 L .58627 .17235 L .60715 .17551 L .62756 .17935 L .66667 .189 L .70292 .20111 L .7357 .21548 L .76445 .23187 L .78868 .25 L .79896 .25962 L .80796 .26955 L .81564 .27976 L .82198 .2902 L .82693 .30082 L .82888 .30618 L .83048 .31158 L .83173 .317 L .83222 .31971 L .83262 .32243 L .83293 .32516 L .83305 .32652 L .83315 .32788 L .83323 .32924 L .83329 .33061 L .83332 .33197 L .83333 .33333 L Mfstroke P P % End of Graphics MathPictureEnd :[font = input; preserveAspect; startGroup] {c1,c2} = c/.Solve[a^2 == b^2 + c^2,c] :[font = output; output; inactive; preserveAspect; endGroup] {-2*3^(1/2), 2*3^(1/2)} ;[o] {-2 Sqrt[3], 2 Sqrt[3]} :[font = input; preserveAspect] Clear[c1plot,c2plot,c1label,c2label,c1val,c2val] c1plot[t_] := Graphics[{RGBColor[1,0,0],Line[{{c1,0},{x[t],y[t]}}]}] c2plot[t_] := Graphics[{Blue,Line[{{c2,0},{x[t],y[t]}}]}] c1label[t_] := Graphics[Text["c1", {c1,-.2}]] c2label[t_] := Graphics[Text["c2", {c2,-.2}]] c1val[t_] := Graphics[ Text["= d(f1,p)" N[dist[{c1,0},{x[t],y[t]}]], {- 3a/4,b + 3/4}]]; c2val[t_] := Graphics[ Text["= d(f2,p)" N[dist[{c2,0},{x[t],y[t]}]], {3a/4,b+3/4}]]; ctotal[t_] := Graphics[ Text["= d(f1,p) + d(f2,p)" N[dist[{c2,0},{x[t],y[t]}] + dist[{c1,0},{x[t],y[t]}]], {a/2,-b-1/2}]] ppoint[t_] := Graphics[ {Maroon,PointSize[.02],Point[{x[t],y[t]}]} ] ppointlabel[t_] := Graphics[ Text["p",{x[t]+.2,y[t]+.2}] ] plotall[t_] := Show[ellipse,c1plot[t],c2plot[t], c1label[t],c2label[t],c1val[t],c2val[t], ctotal[t],ppoint[t],ppointlabel[t], PlotRange->{{-a-5,a+5},{-b-5,b+5}}]; :[font = input; preserveAspect; startGroup] plotall[0]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; endGroup] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .72222 .38889 m .72221 .3898 L .72219 .39071 L .72216 .39162 L .7221 .39252 L .72195 .39434 L .72175 .39616 L .72148 .39797 L .72115 .39978 L .72032 .40339 L .71925 .40699 L .71795 .41057 L .71465 .41765 L .71043 .4246 L .70531 .43141 L .69245 .44444 L .6763 .45653 L .65713 .46746 L .63528 .47704 L .61111 .48511 L .59829 .48854 L .58504 .49154 L .57143 .4941 L .55752 .49621 L .54335 .49787 L .5362 .49852 L .52901 .49905 L .52178 .49946 L .51816 .49963 L .51453 .49976 L .5109 .49987 L .50909 .49991 L .50727 .49994 L .50545 .49997 L .50364 .49999 L .50182 .5 L .5 .5 L .49818 .5 L .49636 .49999 L .49455 .49997 L .49273 .49994 L .49091 .49991 L .4891 .49987 L .48547 .49976 L .48184 .49963 L .47822 .49946 L .47099 .49905 L .4638 .49852 L .45665 .49787 L .44248 .49621 L Mistroke .42857 .4941 L .41496 .49154 L .38889 .48511 L .36472 .47704 L .34287 .46746 L .3237 .45653 L .30755 .44444 L .30069 .43803 L .29469 .43141 L .28957 .4246 L .28535 .41765 L .28205 .41057 L .28075 .40699 L .27968 .40339 L .27885 .39978 L .27852 .39797 L .27825 .39616 L .27805 .39434 L .27796 .39343 L .2779 .39252 L .27784 .39162 L .27781 .39071 L .27779 .3898 L .27778 .38889 L .27779 .38798 L .27781 .38707 L .27784 .38616 L .2779 .38525 L .27805 .38344 L .27825 .38162 L .27852 .37981 L .27885 .378 L .27968 .37439 L .28075 .37079 L .28205 .36721 L .28535 .36013 L .28957 .35317 L .29469 .34637 L .30755 .33333 L .3237 .32125 L .34287 .31032 L .36472 .30074 L .38889 .29266 L .40171 .28924 L .41496 .28624 L .42857 .28367 L .44248 .28156 L .45665 .27991 L .4638 .27926 L .47099 .27873 L Mistroke .47822 .27831 L .48184 .27815 L .48547 .27802 L .4891 .27791 L .49091 .27787 L .49273 .27784 L .49455 .27781 L .49636 .27779 L .49818 .27778 L .5 .27778 L .50182 .27778 L .50364 .27779 L .50545 .27781 L .50727 .27784 L .5109 .27791 L .51453 .27802 L .51816 .27815 L .52178 .27831 L .52901 .27873 L .5362 .27926 L .54335 .27991 L .55752 .28156 L .57143 .28367 L .58504 .28624 L .61111 .29266 L .63528 .30074 L .65713 .31032 L .6763 .32125 L .69245 .33333 L .69931 .33975 L .70531 .34637 L .71043 .35317 L .71465 .36013 L .71795 .36721 L .71925 .37079 L .72032 .37439 L .72115 .378 L .72148 .37981 L .72175 .38162 L .72195 .38344 L .72204 .38435 L .7221 .38525 L .72216 .38616 L .72219 .38707 L .72221 .38798 L .72222 .38889 L Mfstroke P P p 1 0 0 r .004 w .30755 .38889 m .72222 .38889 L s P p 0 0 1 r .004 w .69245 .38889 m .72222 .38889 L s P [(c1)] .30755 .37778 0 0 Mshowa [(c2)] .69245 .37778 0 0 Mshowa [(7.4641 = d\(f1,p\))] .33333 .54167 0 0 Mshowa [(0.535898 = d\(f2,p\))] .66667 .54167 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .61111 .25 0 0 Mshowa p .69 .188 .377 r .02 w .72222 .38889 Mdot P [(p)] .73333 .4 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = input; preserveAspect; startGroup] plotall[Pi/4]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; endGroup] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .72222 .38889 m .72221 .3898 L .72219 .39071 L .72216 .39162 L .7221 .39252 L .72195 .39434 L .72175 .39616 L .72148 .39797 L .72115 .39978 L .72032 .40339 L .71925 .40699 L .71795 .41057 L .71465 .41765 L .71043 .4246 L .70531 .43141 L .69245 .44444 L .6763 .45653 L .65713 .46746 L .63528 .47704 L .61111 .48511 L .59829 .48854 L .58504 .49154 L .57143 .4941 L .55752 .49621 L .54335 .49787 L .5362 .49852 L .52901 .49905 L .52178 .49946 L .51816 .49963 L .51453 .49976 L .5109 .49987 L .50909 .49991 L .50727 .49994 L .50545 .49997 L .50364 .49999 L .50182 .5 L .5 .5 L .49818 .5 L .49636 .49999 L .49455 .49997 L .49273 .49994 L .49091 .49991 L .4891 .49987 L .48547 .49976 L .48184 .49963 L .47822 .49946 L .47099 .49905 L .4638 .49852 L .45665 .49787 L .44248 .49621 L Mistroke .42857 .4941 L .41496 .49154 L .38889 .48511 L .36472 .47704 L .34287 .46746 L .3237 .45653 L .30755 .44444 L .30069 .43803 L .29469 .43141 L .28957 .4246 L .28535 .41765 L .28205 .41057 L .28075 .40699 L .27968 .40339 L .27885 .39978 L .27852 .39797 L .27825 .39616 L .27805 .39434 L .27796 .39343 L .2779 .39252 L .27784 .39162 L .27781 .39071 L .27779 .3898 L .27778 .38889 L .27779 .38798 L .27781 .38707 L .27784 .38616 L .2779 .38525 L .27805 .38344 L .27825 .38162 L .27852 .37981 L .27885 .378 L .27968 .37439 L .28075 .37079 L .28205 .36721 L .28535 .36013 L .28957 .35317 L .29469 .34637 L .30755 .33333 L .3237 .32125 L .34287 .31032 L .36472 .30074 L .38889 .29266 L .40171 .28924 L .41496 .28624 L .42857 .28367 L .44248 .28156 L .45665 .27991 L .4638 .27926 L .47099 .27873 L Mistroke .47822 .27831 L .48184 .27815 L .48547 .27802 L .4891 .27791 L .49091 .27787 L .49273 .27784 L .49455 .27781 L .49636 .27779 L .49818 .27778 L .5 .27778 L .50182 .27778 L .50364 .27779 L .50545 .27781 L .50727 .27784 L .5109 .27791 L .51453 .27802 L .51816 .27815 L .52178 .27831 L .52901 .27873 L .5362 .27926 L .54335 .27991 L .55752 .28156 L .57143 .28367 L .58504 .28624 L .61111 .29266 L .63528 .30074 L .65713 .31032 L .6763 .32125 L .69245 .33333 L .69931 .33975 L .70531 .34637 L .71043 .35317 L .71465 .36013 L .71795 .36721 L .71925 .37079 L .72032 .37439 L .72115 .378 L .72148 .37981 L .72175 .38162 L .72195 .38344 L .72204 .38435 L .7221 .38525 L .72216 .38616 L .72219 .38707 L .72221 .38798 L .72222 .38889 L Mfstroke P P p 1 0 0 r .004 w .30755 .38889 m .65713 .46746 L s P p 0 0 1 r .004 w .69245 .38889 m .65713 .46746 L s P [(c1)] .30755 .37778 0 0 Mshowa [(c2)] .69245 .37778 0 0 Mshowa [(6.44949 = d\(f1,p\))] .33333 .54167 0 0 Mshowa [(1.55051 = d\(f2,p\))] .66667 .54167 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .61111 .25 0 0 Mshowa p .69 .188 .377 r .02 w .65713 .46746 Mdot P [(p)] .66825 .47857 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect] l. We have prepared a graph of an ellipse with values for a and for b that we have chosen. For the first program we let a= 4 and b = 2. What points on the graph have values related to these two? 2. We have used the formula for the distance between two points to determine the distance between P and (c1,o) and P and (c2,0). The program we have written with Mathematica will give these distances and their sum in each case as the point moves around the ellipse. Please activate the "Table" command(below this text--it will move point P around the ellipse) and scroll through each graph to investigate the values of the distance from P to (c1,0) and the distance from P to (c2,0) as P changes location from `/2 to 2 ` in increments of `/4. In each case, please give the value of the sum of the two distances on your Investigation Sheet. 3. Does this number have any similarity to any other given value? ;[s] 3:0,0;361,1;372,0;914,-1; 2:2,24,16,CalcMath,0,12,0,0,0;1,24,16,CalcMath,2,12,0,0,0; :[font = input; preserveAspect; startGroup] Table[plotall[theta],{theta,0,2Pi,Pi/4}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; startGroup; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .72222 .38889 m .72221 .3898 L .72219 .39071 L .72216 .39162 L .7221 .39252 L .72195 .39434 L .72175 .39616 L .72148 .39797 L .72115 .39978 L .72032 .40339 L .71925 .40699 L .71795 .41057 L .71465 .41765 L .71043 .4246 L .70531 .43141 L .69245 .44444 L .6763 .45653 L .65713 .46746 L .63528 .47704 L .61111 .48511 L .59829 .48854 L .58504 .49154 L .57143 .4941 L .55752 .49621 L .54335 .49787 L .5362 .49852 L .52901 .49905 L .52178 .49946 L .51816 .49963 L .51453 .49976 L .5109 .49987 L .50909 .49991 L .50727 .49994 L .50545 .49997 L .50364 .49999 L .50182 .5 L .5 .5 L .49818 .5 L .49636 .49999 L .49455 .49997 L .49273 .49994 L .49091 .49991 L .4891 .49987 L .48547 .49976 L .48184 .49963 L .47822 .49946 L .47099 .49905 L .4638 .49852 L .45665 .49787 L .44248 .49621 L Mistroke .42857 .4941 L .41496 .49154 L .38889 .48511 L .36472 .47704 L .34287 .46746 L .3237 .45653 L .30755 .44444 L .30069 .43803 L .29469 .43141 L .28957 .4246 L .28535 .41765 L .28205 .41057 L .28075 .40699 L .27968 .40339 L .27885 .39978 L .27852 .39797 L .27825 .39616 L .27805 .39434 L .27796 .39343 L .2779 .39252 L .27784 .39162 L .27781 .39071 L .27779 .3898 L .27778 .38889 L .27779 .38798 L .27781 .38707 L .27784 .38616 L .2779 .38525 L .27805 .38344 L .27825 .38162 L .27852 .37981 L .27885 .378 L .27968 .37439 L .28075 .37079 L .28205 .36721 L .28535 .36013 L .28957 .35317 L .29469 .34637 L .30755 .33333 L .3237 .32125 L .34287 .31032 L .36472 .30074 L .38889 .29266 L .40171 .28924 L .41496 .28624 L .42857 .28367 L .44248 .28156 L .45665 .27991 L .4638 .27926 L .47099 .27873 L Mistroke .47822 .27831 L .48184 .27815 L .48547 .27802 L .4891 .27791 L .49091 .27787 L .49273 .27784 L .49455 .27781 L .49636 .27779 L .49818 .27778 L .5 .27778 L .50182 .27778 L .50364 .27779 L .50545 .27781 L .50727 .27784 L .5109 .27791 L .51453 .27802 L .51816 .27815 L .52178 .27831 L .52901 .27873 L .5362 .27926 L .54335 .27991 L .55752 .28156 L .57143 .28367 L .58504 .28624 L .61111 .29266 L .63528 .30074 L .65713 .31032 L .6763 .32125 L .69245 .33333 L .69931 .33975 L .70531 .34637 L .71043 .35317 L .71465 .36013 L .71795 .36721 L .71925 .37079 L .72032 .37439 L .72115 .378 L .72148 .37981 L .72175 .38162 L .72195 .38344 L .72204 .38435 L .7221 .38525 L .72216 .38616 L .72219 .38707 L .72221 .38798 L .72222 .38889 L Mfstroke P P p 1 0 0 r .004 w .30755 .38889 m .72222 .38889 L s P p 0 0 1 r .004 w .69245 .38889 m .72222 .38889 L s P [(c1)] .30755 .37778 0 0 Mshowa [(c2)] .69245 .37778 0 0 Mshowa [(7.4641 = d\(f1,p\))] .33333 .54167 0 0 Mshowa [(0.535898 = d\(f2,p\))] .66667 .54167 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .61111 .25 0 0 Mshowa p .69 .188 .377 r .02 w .72222 .38889 Mdot P [(p)] .73333 .4 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .72222 .38889 m .72221 .3898 L .72219 .39071 L .72216 .39162 L .7221 .39252 L .72195 .39434 L .72175 .39616 L .72148 .39797 L .72115 .39978 L .72032 .40339 L .71925 .40699 L .71795 .41057 L .71465 .41765 L .71043 .4246 L .70531 .43141 L .69245 .44444 L .6763 .45653 L .65713 .46746 L .63528 .47704 L .61111 .48511 L .59829 .48854 L .58504 .49154 L .57143 .4941 L .55752 .49621 L .54335 .49787 L .5362 .49852 L .52901 .49905 L .52178 .49946 L .51816 .49963 L .51453 .49976 L .5109 .49987 L .50909 .49991 L .50727 .49994 L .50545 .49997 L .50364 .49999 L .50182 .5 L .5 .5 L .49818 .5 L .49636 .49999 L .49455 .49997 L .49273 .49994 L .49091 .49991 L .4891 .49987 L .48547 .49976 L .48184 .49963 L .47822 .49946 L .47099 .49905 L .4638 .49852 L .45665 .49787 L .44248 .49621 L Mistroke .42857 .4941 L .41496 .49154 L .38889 .48511 L .36472 .47704 L .34287 .46746 L .3237 .45653 L .30755 .44444 L .30069 .43803 L .29469 .43141 L .28957 .4246 L .28535 .41765 L .28205 .41057 L .28075 .40699 L .27968 .40339 L .27885 .39978 L .27852 .39797 L .27825 .39616 L .27805 .39434 L .27796 .39343 L .2779 .39252 L .27784 .39162 L .27781 .39071 L .27779 .3898 L .27778 .38889 L .27779 .38798 L .27781 .38707 L .27784 .38616 L .2779 .38525 L .27805 .38344 L .27825 .38162 L .27852 .37981 L .27885 .378 L .27968 .37439 L .28075 .37079 L .28205 .36721 L .28535 .36013 L .28957 .35317 L .29469 .34637 L .30755 .33333 L .3237 .32125 L .34287 .31032 L .36472 .30074 L .38889 .29266 L .40171 .28924 L .41496 .28624 L .42857 .28367 L .44248 .28156 L .45665 .27991 L .4638 .27926 L .47099 .27873 L Mistroke .47822 .27831 L .48184 .27815 L .48547 .27802 L .4891 .27791 L .49091 .27787 L .49273 .27784 L .49455 .27781 L .49636 .27779 L .49818 .27778 L .5 .27778 L .50182 .27778 L .50364 .27779 L .50545 .27781 L .50727 .27784 L .5109 .27791 L .51453 .27802 L .51816 .27815 L .52178 .27831 L .52901 .27873 L .5362 .27926 L .54335 .27991 L .55752 .28156 L .57143 .28367 L .58504 .28624 L .61111 .29266 L .63528 .30074 L .65713 .31032 L .6763 .32125 L .69245 .33333 L .69931 .33975 L .70531 .34637 L .71043 .35317 L .71465 .36013 L .71795 .36721 L .71925 .37079 L .72032 .37439 L .72115 .378 L .72148 .37981 L .72175 .38162 L .72195 .38344 L .72204 .38435 L .7221 .38525 L .72216 .38616 L .72219 .38707 L .72221 .38798 L .72222 .38889 L Mfstroke P P p 1 0 0 r .004 w .30755 .38889 m .65713 .46746 L s P p 0 0 1 r .004 w .69245 .38889 m .65713 .46746 L s P [(c1)] .30755 .37778 0 0 Mshowa [(c2)] .69245 .37778 0 0 Mshowa [(6.44949 = d\(f1,p\))] .33333 .54167 0 0 Mshowa [(1.55051 = d\(f2,p\))] .66667 .54167 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .61111 .25 0 0 Mshowa p .69 .188 .377 r .02 w .65713 .46746 Mdot P [(p)] .66825 .47857 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .72222 .38889 m .72221 .3898 L .72219 .39071 L .72216 .39162 L .7221 .39252 L .72195 .39434 L .72175 .39616 L .72148 .39797 L .72115 .39978 L .72032 .40339 L .71925 .40699 L .71795 .41057 L .71465 .41765 L .71043 .4246 L .70531 .43141 L .69245 .44444 L .6763 .45653 L .65713 .46746 L .63528 .47704 L .61111 .48511 L .59829 .48854 L .58504 .49154 L .57143 .4941 L .55752 .49621 L .54335 .49787 L .5362 .49852 L .52901 .49905 L .52178 .49946 L .51816 .49963 L .51453 .49976 L .5109 .49987 L .50909 .49991 L .50727 .49994 L .50545 .49997 L .50364 .49999 L .50182 .5 L .5 .5 L .49818 .5 L .49636 .49999 L .49455 .49997 L .49273 .49994 L .49091 .49991 L .4891 .49987 L .48547 .49976 L .48184 .49963 L .47822 .49946 L .47099 .49905 L .4638 .49852 L .45665 .49787 L .44248 .49621 L Mistroke .42857 .4941 L .41496 .49154 L .38889 .48511 L .36472 .47704 L .34287 .46746 L .3237 .45653 L .30755 .44444 L .30069 .43803 L .29469 .43141 L .28957 .4246 L .28535 .41765 L .28205 .41057 L .28075 .40699 L .27968 .40339 L .27885 .39978 L .27852 .39797 L .27825 .39616 L .27805 .39434 L .27796 .39343 L .2779 .39252 L .27784 .39162 L .27781 .39071 L .27779 .3898 L .27778 .38889 L .27779 .38798 L .27781 .38707 L .27784 .38616 L .2779 .38525 L .27805 .38344 L .27825 .38162 L .27852 .37981 L .27885 .378 L .27968 .37439 L .28075 .37079 L .28205 .36721 L .28535 .36013 L .28957 .35317 L .29469 .34637 L .30755 .33333 L .3237 .32125 L .34287 .31032 L .36472 .30074 L .38889 .29266 L .40171 .28924 L .41496 .28624 L .42857 .28367 L .44248 .28156 L .45665 .27991 L .4638 .27926 L .47099 .27873 L Mistroke .47822 .27831 L .48184 .27815 L .48547 .27802 L .4891 .27791 L .49091 .27787 L .49273 .27784 L .49455 .27781 L .49636 .27779 L .49818 .27778 L .5 .27778 L .50182 .27778 L .50364 .27779 L .50545 .27781 L .50727 .27784 L .5109 .27791 L .51453 .27802 L .51816 .27815 L .52178 .27831 L .52901 .27873 L .5362 .27926 L .54335 .27991 L .55752 .28156 L .57143 .28367 L .58504 .28624 L .61111 .29266 L .63528 .30074 L .65713 .31032 L .6763 .32125 L .69245 .33333 L .69931 .33975 L .70531 .34637 L .71043 .35317 L .71465 .36013 L .71795 .36721 L .71925 .37079 L .72032 .37439 L .72115 .378 L .72148 .37981 L .72175 .38162 L .72195 .38344 L .72204 .38435 L .7221 .38525 L .72216 .38616 L .72219 .38707 L .72221 .38798 L .72222 .38889 L Mfstroke P P p 1 0 0 r .004 w .30755 .38889 m .5 .5 L s P p 0 0 1 r .004 w .69245 .38889 m .5 .5 L s P [(c1)] .30755 .37778 0 0 Mshowa [(c2)] .69245 .37778 0 0 Mshowa [(4. = d\(f1,p\))] .33333 .54167 0 0 Mshowa [(4. = d\(f2,p\))] .66667 .54167 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .61111 .25 0 0 Mshowa p .69 .188 .377 r .02 w .5 .5 Mdot P [(p)] .51111 .51111 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .72222 .38889 m .72221 .3898 L .72219 .39071 L .72216 .39162 L .7221 .39252 L .72195 .39434 L .72175 .39616 L .72148 .39797 L .72115 .39978 L .72032 .40339 L .71925 .40699 L .71795 .41057 L .71465 .41765 L .71043 .4246 L .70531 .43141 L .69245 .44444 L .6763 .45653 L .65713 .46746 L .63528 .47704 L .61111 .48511 L .59829 .48854 L .58504 .49154 L .57143 .4941 L .55752 .49621 L .54335 .49787 L .5362 .49852 L .52901 .49905 L .52178 .49946 L .51816 .49963 L .51453 .49976 L .5109 .49987 L .50909 .49991 L .50727 .49994 L .50545 .49997 L .50364 .49999 L .50182 .5 L .5 .5 L .49818 .5 L .49636 .49999 L .49455 .49997 L .49273 .49994 L .49091 .49991 L .4891 .49987 L .48547 .49976 L .48184 .49963 L .47822 .49946 L .47099 .49905 L .4638 .49852 L .45665 .49787 L .44248 .49621 L Mistroke .42857 .4941 L .41496 .49154 L .38889 .48511 L .36472 .47704 L .34287 .46746 L .3237 .45653 L .30755 .44444 L .30069 .43803 L .29469 .43141 L .28957 .4246 L .28535 .41765 L .28205 .41057 L .28075 .40699 L .27968 .40339 L .27885 .39978 L .27852 .39797 L .27825 .39616 L .27805 .39434 L .27796 .39343 L .2779 .39252 L .27784 .39162 L .27781 .39071 L .27779 .3898 L .27778 .38889 L .27779 .38798 L .27781 .38707 L .27784 .38616 L .2779 .38525 L .27805 .38344 L .27825 .38162 L .27852 .37981 L .27885 .378 L .27968 .37439 L .28075 .37079 L .28205 .36721 L .28535 .36013 L .28957 .35317 L .29469 .34637 L .30755 .33333 L .3237 .32125 L .34287 .31032 L .36472 .30074 L .38889 .29266 L .40171 .28924 L .41496 .28624 L .42857 .28367 L .44248 .28156 L .45665 .27991 L .4638 .27926 L .47099 .27873 L Mistroke .47822 .27831 L .48184 .27815 L .48547 .27802 L .4891 .27791 L .49091 .27787 L .49273 .27784 L .49455 .27781 L .49636 .27779 L .49818 .27778 L .5 .27778 L .50182 .27778 L .50364 .27779 L .50545 .27781 L .50727 .27784 L .5109 .27791 L .51453 .27802 L .51816 .27815 L .52178 .27831 L .52901 .27873 L .5362 .27926 L .54335 .27991 L .55752 .28156 L .57143 .28367 L .58504 .28624 L .61111 .29266 L .63528 .30074 L .65713 .31032 L .6763 .32125 L .69245 .33333 L .69931 .33975 L .70531 .34637 L .71043 .35317 L .71465 .36013 L .71795 .36721 L .71925 .37079 L .72032 .37439 L .72115 .378 L .72148 .37981 L .72175 .38162 L .72195 .38344 L .72204 .38435 L .7221 .38525 L .72216 .38616 L .72219 .38707 L .72221 .38798 L .72222 .38889 L Mfstroke P P p 1 0 0 r .004 w .30755 .38889 m .34287 .46746 L s P p 0 0 1 r .004 w .69245 .38889 m .34287 .46746 L s P [(c1)] .30755 .37778 0 0 Mshowa [(c2)] .69245 .37778 0 0 Mshowa [(1.55051 = d\(f1,p\))] .33333 .54167 0 0 Mshowa [(6.44949 = d\(f2,p\))] .66667 .54167 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .61111 .25 0 0 Mshowa p .69 .188 .377 r .02 w .34287 .46746 Mdot P [(p)] .35398 .47857 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .72222 .38889 m .72221 .3898 L .72219 .39071 L .72216 .39162 L .7221 .39252 L .72195 .39434 L .72175 .39616 L .72148 .39797 L .72115 .39978 L .72032 .40339 L .71925 .40699 L .71795 .41057 L .71465 .41765 L .71043 .4246 L .70531 .43141 L .69245 .44444 L .6763 .45653 L .65713 .46746 L .63528 .47704 L .61111 .48511 L .59829 .48854 L .58504 .49154 L .57143 .4941 L .55752 .49621 L .54335 .49787 L .5362 .49852 L .52901 .49905 L .52178 .49946 L .51816 .49963 L .51453 .49976 L .5109 .49987 L .50909 .49991 L .50727 .49994 L .50545 .49997 L .50364 .49999 L .50182 .5 L .5 .5 L .49818 .5 L .49636 .49999 L .49455 .49997 L .49273 .49994 L .49091 .49991 L .4891 .49987 L .48547 .49976 L .48184 .49963 L .47822 .49946 L .47099 .49905 L .4638 .49852 L .45665 .49787 L .44248 .49621 L Mistroke .42857 .4941 L .41496 .49154 L .38889 .48511 L .36472 .47704 L .34287 .46746 L .3237 .45653 L .30755 .44444 L .30069 .43803 L .29469 .43141 L .28957 .4246 L .28535 .41765 L .28205 .41057 L .28075 .40699 L .27968 .40339 L .27885 .39978 L .27852 .39797 L .27825 .39616 L .27805 .39434 L .27796 .39343 L .2779 .39252 L .27784 .39162 L .27781 .39071 L .27779 .3898 L .27778 .38889 L .27779 .38798 L .27781 .38707 L .27784 .38616 L .2779 .38525 L .27805 .38344 L .27825 .38162 L .27852 .37981 L .27885 .378 L .27968 .37439 L .28075 .37079 L .28205 .36721 L .28535 .36013 L .28957 .35317 L .29469 .34637 L .30755 .33333 L .3237 .32125 L .34287 .31032 L .36472 .30074 L .38889 .29266 L .40171 .28924 L .41496 .28624 L .42857 .28367 L .44248 .28156 L .45665 .27991 L .4638 .27926 L .47099 .27873 L Mistroke .47822 .27831 L .48184 .27815 L .48547 .27802 L .4891 .27791 L .49091 .27787 L .49273 .27784 L .49455 .27781 L .49636 .27779 L .49818 .27778 L .5 .27778 L .50182 .27778 L .50364 .27779 L .50545 .27781 L .50727 .27784 L .5109 .27791 L .51453 .27802 L .51816 .27815 L .52178 .27831 L .52901 .27873 L .5362 .27926 L .54335 .27991 L .55752 .28156 L .57143 .28367 L .58504 .28624 L .61111 .29266 L .63528 .30074 L .65713 .31032 L .6763 .32125 L .69245 .33333 L .69931 .33975 L .70531 .34637 L .71043 .35317 L .71465 .36013 L .71795 .36721 L .71925 .37079 L .72032 .37439 L .72115 .378 L .72148 .37981 L .72175 .38162 L .72195 .38344 L .72204 .38435 L .7221 .38525 L .72216 .38616 L .72219 .38707 L .72221 .38798 L .72222 .38889 L Mfstroke P P p 1 0 0 r .004 w .30755 .38889 m .27778 .38889 L s P p 0 0 1 r .004 w .69245 .38889 m .27778 .38889 L s P [(c1)] .30755 .37778 0 0 Mshowa [(c2)] .69245 .37778 0 0 Mshowa [(0.535898 = d\(f1,p\))] .33333 .54167 0 0 Mshowa [(7.4641 = d\(f2,p\))] .66667 .54167 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .61111 .25 0 0 Mshowa p .69 .188 .377 r .02 w .27778 .38889 Mdot P [(p)] .28889 .4 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .72222 .38889 m .72221 .3898 L .72219 .39071 L .72216 .39162 L .7221 .39252 L .72195 .39434 L .72175 .39616 L .72148 .39797 L .72115 .39978 L .72032 .40339 L .71925 .40699 L .71795 .41057 L .71465 .41765 L .71043 .4246 L .70531 .43141 L .69245 .44444 L .6763 .45653 L .65713 .46746 L .63528 .47704 L .61111 .48511 L .59829 .48854 L .58504 .49154 L .57143 .4941 L .55752 .49621 L .54335 .49787 L .5362 .49852 L .52901 .49905 L .52178 .49946 L .51816 .49963 L .51453 .49976 L .5109 .49987 L .50909 .49991 L .50727 .49994 L .50545 .49997 L .50364 .49999 L .50182 .5 L .5 .5 L .49818 .5 L .49636 .49999 L .49455 .49997 L .49273 .49994 L .49091 .49991 L .4891 .49987 L .48547 .49976 L .48184 .49963 L .47822 .49946 L .47099 .49905 L .4638 .49852 L .45665 .49787 L .44248 .49621 L Mistroke .42857 .4941 L .41496 .49154 L .38889 .48511 L .36472 .47704 L .34287 .46746 L .3237 .45653 L .30755 .44444 L .30069 .43803 L .29469 .43141 L .28957 .4246 L .28535 .41765 L .28205 .41057 L .28075 .40699 L .27968 .40339 L .27885 .39978 L .27852 .39797 L .27825 .39616 L .27805 .39434 L .27796 .39343 L .2779 .39252 L .27784 .39162 L .27781 .39071 L .27779 .3898 L .27778 .38889 L .27779 .38798 L .27781 .38707 L .27784 .38616 L .2779 .38525 L .27805 .38344 L .27825 .38162 L .27852 .37981 L .27885 .378 L .27968 .37439 L .28075 .37079 L .28205 .36721 L .28535 .36013 L .28957 .35317 L .29469 .34637 L .30755 .33333 L .3237 .32125 L .34287 .31032 L .36472 .30074 L .38889 .29266 L .40171 .28924 L .41496 .28624 L .42857 .28367 L .44248 .28156 L .45665 .27991 L .4638 .27926 L .47099 .27873 L Mistroke .47822 .27831 L .48184 .27815 L .48547 .27802 L .4891 .27791 L .49091 .27787 L .49273 .27784 L .49455 .27781 L .49636 .27779 L .49818 .27778 L .5 .27778 L .50182 .27778 L .50364 .27779 L .50545 .27781 L .50727 .27784 L .5109 .27791 L .51453 .27802 L .51816 .27815 L .52178 .27831 L .52901 .27873 L .5362 .27926 L .54335 .27991 L .55752 .28156 L .57143 .28367 L .58504 .28624 L .61111 .29266 L .63528 .30074 L .65713 .31032 L .6763 .32125 L .69245 .33333 L .69931 .33975 L .70531 .34637 L .71043 .35317 L .71465 .36013 L .71795 .36721 L .71925 .37079 L .72032 .37439 L .72115 .378 L .72148 .37981 L .72175 .38162 L .72195 .38344 L .72204 .38435 L .7221 .38525 L .72216 .38616 L .72219 .38707 L .72221 .38798 L .72222 .38889 L Mfstroke P P p 1 0 0 r .004 w .30755 .38889 m .34287 .31032 L s P p 0 0 1 r .004 w .69245 .38889 m .34287 .31032 L s P [(c1)] .30755 .37778 0 0 Mshowa [(c2)] .69245 .37778 0 0 Mshowa [(1.55051 = d\(f1,p\))] .33333 .54167 0 0 Mshowa [(6.44949 = d\(f2,p\))] .66667 .54167 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .61111 .25 0 0 Mshowa p .69 .188 .377 r .02 w .34287 .31032 Mdot P [(p)] .35398 .32143 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .72222 .38889 m .72221 .3898 L .72219 .39071 L .72216 .39162 L .7221 .39252 L .72195 .39434 L .72175 .39616 L .72148 .39797 L .72115 .39978 L .72032 .40339 L .71925 .40699 L .71795 .41057 L .71465 .41765 L .71043 .4246 L .70531 .43141 L .69245 .44444 L .6763 .45653 L .65713 .46746 L .63528 .47704 L .61111 .48511 L .59829 .48854 L .58504 .49154 L .57143 .4941 L .55752 .49621 L .54335 .49787 L .5362 .49852 L .52901 .49905 L .52178 .49946 L .51816 .49963 L .51453 .49976 L .5109 .49987 L .50909 .49991 L .50727 .49994 L .50545 .49997 L .50364 .49999 L .50182 .5 L .5 .5 L .49818 .5 L .49636 .49999 L .49455 .49997 L .49273 .49994 L .49091 .49991 L .4891 .49987 L .48547 .49976 L .48184 .49963 L .47822 .49946 L .47099 .49905 L .4638 .49852 L .45665 .49787 L .44248 .49621 L Mistroke .42857 .4941 L .41496 .49154 L .38889 .48511 L .36472 .47704 L .34287 .46746 L .3237 .45653 L .30755 .44444 L .30069 .43803 L .29469 .43141 L .28957 .4246 L .28535 .41765 L .28205 .41057 L .28075 .40699 L .27968 .40339 L .27885 .39978 L .27852 .39797 L .27825 .39616 L .27805 .39434 L .27796 .39343 L .2779 .39252 L .27784 .39162 L .27781 .39071 L .27779 .3898 L .27778 .38889 L .27779 .38798 L .27781 .38707 L .27784 .38616 L .2779 .38525 L .27805 .38344 L .27825 .38162 L .27852 .37981 L .27885 .378 L .27968 .37439 L .28075 .37079 L .28205 .36721 L .28535 .36013 L .28957 .35317 L .29469 .34637 L .30755 .33333 L .3237 .32125 L .34287 .31032 L .36472 .30074 L .38889 .29266 L .40171 .28924 L .41496 .28624 L .42857 .28367 L .44248 .28156 L .45665 .27991 L .4638 .27926 L .47099 .27873 L Mistroke .47822 .27831 L .48184 .27815 L .48547 .27802 L .4891 .27791 L .49091 .27787 L .49273 .27784 L .49455 .27781 L .49636 .27779 L .49818 .27778 L .5 .27778 L .50182 .27778 L .50364 .27779 L .50545 .27781 L .50727 .27784 L .5109 .27791 L .51453 .27802 L .51816 .27815 L .52178 .27831 L .52901 .27873 L .5362 .27926 L .54335 .27991 L .55752 .28156 L .57143 .28367 L .58504 .28624 L .61111 .29266 L .63528 .30074 L .65713 .31032 L .6763 .32125 L .69245 .33333 L .69931 .33975 L .70531 .34637 L .71043 .35317 L .71465 .36013 L .71795 .36721 L .71925 .37079 L .72032 .37439 L .72115 .378 L .72148 .37981 L .72175 .38162 L .72195 .38344 L .72204 .38435 L .7221 .38525 L .72216 .38616 L .72219 .38707 L .72221 .38798 L .72222 .38889 L Mfstroke P P p 1 0 0 r .004 w .30755 .38889 m .5 .27778 L s P p 0 0 1 r .004 w .69245 .38889 m .5 .27778 L s P [(c1)] .30755 .37778 0 0 Mshowa [(c2)] .69245 .37778 0 0 Mshowa [(4. = d\(f1,p\))] .33333 .54167 0 0 Mshowa [(4. = d\(f2,p\))] .66667 .54167 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .61111 .25 0 0 Mshowa p .69 .188 .377 r .02 w .5 .27778 Mdot P [(p)] .51111 .28889 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .72222 .38889 m .72221 .3898 L .72219 .39071 L .72216 .39162 L .7221 .39252 L .72195 .39434 L .72175 .39616 L .72148 .39797 L .72115 .39978 L .72032 .40339 L .71925 .40699 L .71795 .41057 L .71465 .41765 L .71043 .4246 L .70531 .43141 L .69245 .44444 L .6763 .45653 L .65713 .46746 L .63528 .47704 L .61111 .48511 L .59829 .48854 L .58504 .49154 L .57143 .4941 L .55752 .49621 L .54335 .49787 L .5362 .49852 L .52901 .49905 L .52178 .49946 L .51816 .49963 L .51453 .49976 L .5109 .49987 L .50909 .49991 L .50727 .49994 L .50545 .49997 L .50364 .49999 L .50182 .5 L .5 .5 L .49818 .5 L .49636 .49999 L .49455 .49997 L .49273 .49994 L .49091 .49991 L .4891 .49987 L .48547 .49976 L .48184 .49963 L .47822 .49946 L .47099 .49905 L .4638 .49852 L .45665 .49787 L .44248 .49621 L Mistroke .42857 .4941 L .41496 .49154 L .38889 .48511 L .36472 .47704 L .34287 .46746 L .3237 .45653 L .30755 .44444 L .30069 .43803 L .29469 .43141 L .28957 .4246 L .28535 .41765 L .28205 .41057 L .28075 .40699 L .27968 .40339 L .27885 .39978 L .27852 .39797 L .27825 .39616 L .27805 .39434 L .27796 .39343 L .2779 .39252 L .27784 .39162 L .27781 .39071 L .27779 .3898 L .27778 .38889 L .27779 .38798 L .27781 .38707 L .27784 .38616 L .2779 .38525 L .27805 .38344 L .27825 .38162 L .27852 .37981 L .27885 .378 L .27968 .37439 L .28075 .37079 L .28205 .36721 L .28535 .36013 L .28957 .35317 L .29469 .34637 L .30755 .33333 L .3237 .32125 L .34287 .31032 L .36472 .30074 L .38889 .29266 L .40171 .28924 L .41496 .28624 L .42857 .28367 L .44248 .28156 L .45665 .27991 L .4638 .27926 L .47099 .27873 L Mistroke .47822 .27831 L .48184 .27815 L .48547 .27802 L .4891 .27791 L .49091 .27787 L .49273 .27784 L .49455 .27781 L .49636 .27779 L .49818 .27778 L .5 .27778 L .50182 .27778 L .50364 .27779 L .50545 .27781 L .50727 .27784 L .5109 .27791 L .51453 .27802 L .51816 .27815 L .52178 .27831 L .52901 .27873 L .5362 .27926 L .54335 .27991 L .55752 .28156 L .57143 .28367 L .58504 .28624 L .61111 .29266 L .63528 .30074 L .65713 .31032 L .6763 .32125 L .69245 .33333 L .69931 .33975 L .70531 .34637 L .71043 .35317 L .71465 .36013 L .71795 .36721 L .71925 .37079 L .72032 .37439 L .72115 .378 L .72148 .37981 L .72175 .38162 L .72195 .38344 L .72204 .38435 L .7221 .38525 L .72216 .38616 L .72219 .38707 L .72221 .38798 L .72222 .38889 L Mfstroke P P p 1 0 0 r .004 w .30755 .38889 m .65713 .31032 L s P p 0 0 1 r .004 w .69245 .38889 m .65713 .31032 L s P [(c1)] .30755 .37778 0 0 Mshowa [(c2)] .69245 .37778 0 0 Mshowa [(6.44949 = d\(f1,p\))] .33333 .54167 0 0 Mshowa [(1.55051 = d\(f2,p\))] .66667 .54167 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .61111 .25 0 0 Mshowa p .69 .188 .377 r .02 w .65713 .31032 Mdot P [(p)] .66825 .32143 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; endGroup; endGroup; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .72222 .38889 m .72221 .3898 L .72219 .39071 L .72216 .39162 L .7221 .39252 L .72195 .39434 L .72175 .39616 L .72148 .39797 L .72115 .39978 L .72032 .40339 L .71925 .40699 L .71795 .41057 L .71465 .41765 L .71043 .4246 L .70531 .43141 L .69245 .44444 L .6763 .45653 L .65713 .46746 L .63528 .47704 L .61111 .48511 L .59829 .48854 L .58504 .49154 L .57143 .4941 L .55752 .49621 L .54335 .49787 L .5362 .49852 L .52901 .49905 L .52178 .49946 L .51816 .49963 L .51453 .49976 L .5109 .49987 L .50909 .49991 L .50727 .49994 L .50545 .49997 L .50364 .49999 L .50182 .5 L .5 .5 L .49818 .5 L .49636 .49999 L .49455 .49997 L .49273 .49994 L .49091 .49991 L .4891 .49987 L .48547 .49976 L .48184 .49963 L .47822 .49946 L .47099 .49905 L .4638 .49852 L .45665 .49787 L .44248 .49621 L Mistroke .42857 .4941 L .41496 .49154 L .38889 .48511 L .36472 .47704 L .34287 .46746 L .3237 .45653 L .30755 .44444 L .30069 .43803 L .29469 .43141 L .28957 .4246 L .28535 .41765 L .28205 .41057 L .28075 .40699 L .27968 .40339 L .27885 .39978 L .27852 .39797 L .27825 .39616 L .27805 .39434 L .27796 .39343 L .2779 .39252 L .27784 .39162 L .27781 .39071 L .27779 .3898 L .27778 .38889 L .27779 .38798 L .27781 .38707 L .27784 .38616 L .2779 .38525 L .27805 .38344 L .27825 .38162 L .27852 .37981 L .27885 .378 L .27968 .37439 L .28075 .37079 L .28205 .36721 L .28535 .36013 L .28957 .35317 L .29469 .34637 L .30755 .33333 L .3237 .32125 L .34287 .31032 L .36472 .30074 L .38889 .29266 L .40171 .28924 L .41496 .28624 L .42857 .28367 L .44248 .28156 L .45665 .27991 L .4638 .27926 L .47099 .27873 L Mistroke .47822 .27831 L .48184 .27815 L .48547 .27802 L .4891 .27791 L .49091 .27787 L .49273 .27784 L .49455 .27781 L .49636 .27779 L .49818 .27778 L .5 .27778 L .50182 .27778 L .50364 .27779 L .50545 .27781 L .50727 .27784 L .5109 .27791 L .51453 .27802 L .51816 .27815 L .52178 .27831 L .52901 .27873 L .5362 .27926 L .54335 .27991 L .55752 .28156 L .57143 .28367 L .58504 .28624 L .61111 .29266 L .63528 .30074 L .65713 .31032 L .6763 .32125 L .69245 .33333 L .69931 .33975 L .70531 .34637 L .71043 .35317 L .71465 .36013 L .71795 .36721 L .71925 .37079 L .72032 .37439 L .72115 .378 L .72148 .37981 L .72175 .38162 L .72195 .38344 L .72204 .38435 L .7221 .38525 L .72216 .38616 L .72219 .38707 L .72221 .38798 L .72222 .38889 L Mfstroke P P p 1 0 0 r .004 w .30755 .38889 m .72222 .38889 L s P p 0 0 1 r .004 w .69245 .38889 m .72222 .38889 L s P [(c1)] .30755 .37778 0 0 Mshowa [(c2)] .69245 .37778 0 0 Mshowa [(7.4641 = d\(f1,p\))] .33333 .54167 0 0 Mshowa [(0.535898 = d\(f2,p\))] .66667 .54167 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .61111 .25 0 0 Mshowa p .69 .188 .377 r .02 w .72222 .38889 Mdot P [(p)] .73333 .4 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = input; preserveAspect] :[font = text; inactive; preserveAspect] 4. Now let's look to see if this is just a special case or if this observation is true for all ellipses. So that our graphs are manageable let me suggest that you go back to the beginning and let a =5 and b = 3Ž3. ;[s] 2:0,0;212,1;215,-1; 2:1,24,16,CalcMath,0,12,0,0,0;1,16,12,Monaco,0,12,0,0,0; :[font = text; inactive; preserveAspect; endGroup] 5. Repeat the activations from above for these values of a and record these values on the Investigation Sheet. Be sure to also use the "Table" command too. 6. What similarity do you see this time? Is it the same as last? 7. If you wish try again with new values of a and b go ahead-- but be sure to make a larger than b. :[font = section; inactive; Cclosed; preserveAspect; startGroup] Investigate Ellipse (major axis---y-axis) :[font = text; inactive; preserveAspect] l. Now let's look at another ellipse. Notice that this time we have changed our code to say that x(t)= b Cos[t] and y(t)= a Sin[t]. Activate all of the commands below down through the "ellipse" command. How is this ellipse different than the ones above? 2. What points have special relationships with the values of a and b? :[font = input; preserveAspect] :[font = input; preserveAspect] dist[{x1_,y1_},{x2_,y2_}] = Sqrt[(x2 - x1)^2 + (y2 - y1)^2]; midpoint[{x1_,y1_},{x2_,y2_}] = {(x2+x1)/2,(y2+y1)/2}; :[font = input; preserveAspect; startGroup] Clear[x,y,t] a= 4; b = 2; x[t_]= b Cos[t]; y[t_] = a Sin[t]; ellipse=ParametricPlot[{x[t],y[t]},{t,0,2 Pi}, PlotStyle->{{Thickness[0.005]}}, AspectRatio->Automatic, PlotRange->{{-a-2,a+2},{-b-2,b+2}}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 234; endGroup] %! %%Creator: Mathematica %%AspectRatio: .66667 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0833333 0.333333 0.0833333 [ [(-6)] 0 .33333 0 2 Msboxa [(-4)] .16667 .33333 0 2 Msboxa [(-2)] .33333 .33333 0 2 Msboxa [(2)] .66667 .33333 0 2 Msboxa [(4)] .83333 .33333 0 2 Msboxa [(6)] 1 .33333 0 2 Msboxa [(-4)] .4875 0 1 0 Msboxa [(-3)] .4875 .08333 1 0 Msboxa [(-2)] .4875 .16667 1 0 Msboxa [(-1)] .4875 .25 1 0 Msboxa [(1)] .4875 .41667 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(3)] .4875 .58333 1 0 Msboxa [(4)] .4875 .66667 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .66767 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .33333 m 0 .33958 L s P [(-6)] 0 .33333 0 2 Mshowa p .002 w .16667 .33333 m .16667 .33958 L s P [(-4)] .16667 .33333 0 2 Mshowa p .002 w .33333 .33333 m .33333 .33958 L s P [(-2)] .33333 .33333 0 2 Mshowa p .002 w .66667 .33333 m .66667 .33958 L s P [(2)] .66667 .33333 0 2 Mshowa p .002 w .83333 .33333 m .83333 .33958 L s P [(4)] .83333 .33333 0 2 Mshowa p .002 w 1 .33333 m 1 .33958 L s P [(6)] 1 .33333 0 2 Mshowa p .001 w .03333 .33333 m .03333 .33708 L s P p .001 w .06667 .33333 m .06667 .33708 L s P p .001 w .1 .33333 m .1 .33708 L s P p .001 w .13333 .33333 m .13333 .33708 L s P p .001 w .2 .33333 m .2 .33708 L s P p .001 w .23333 .33333 m .23333 .33708 L s P p .001 w .26667 .33333 m .26667 .33708 L s P p .001 w .3 .33333 m .3 .33708 L s P p .001 w .36667 .33333 m .36667 .33708 L s P p .001 w .4 .33333 m .4 .33708 L s P p .001 w .43333 .33333 m .43333 .33708 L s P p .001 w .46667 .33333 m .46667 .33708 L s P p .001 w .53333 .33333 m .53333 .33708 L s P p .001 w .56667 .33333 m .56667 .33708 L s P p .001 w .6 .33333 m .6 .33708 L s P p .001 w .63333 .33333 m .63333 .33708 L s P p .001 w .7 .33333 m .7 .33708 L s P p .001 w .73333 .33333 m .73333 .33708 L s P p .001 w .76667 .33333 m .76667 .33708 L s P p .001 w .8 .33333 m .8 .33708 L s P p .001 w .86667 .33333 m .86667 .33708 L s P p .001 w .9 .33333 m .9 .33708 L s P p .001 w .93333 .33333 m .93333 .33708 L s P p .001 w .96667 .33333 m .96667 .33708 L s P p .002 w 0 .33333 m 1 .33333 L s P p .002 w .5 0 m .50625 0 L s P [(-4)] .4875 0 1 0 Mshowa p .002 w .5 .08333 m .50625 .08333 L s P [(-3)] .4875 .08333 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-2)] .4875 .16667 1 0 Mshowa p .002 w .5 .25 m .50625 .25 L s P [(-1)] .4875 .25 1 0 Mshowa p .002 w .5 .41667 m .50625 .41667 L s P [(1)] .4875 .41667 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .58333 m .50625 .58333 L s P [(3)] .4875 .58333 1 0 Mshowa p .002 w .5 .66667 m .50625 .66667 L s P [(4)] .4875 .66667 1 0 Mshowa p .001 w .5 .01667 m .50375 .01667 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .05 m .50375 .05 L s P p .001 w .5 .06667 m .50375 .06667 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .11667 m .50375 .11667 L s P p .001 w .5 .13333 m .50375 .13333 L s P p .001 w .5 .15 m .50375 .15 L s P p .001 w .5 .18333 m .50375 .18333 L s P p .001 w .5 .2 m .50375 .2 L s P p .001 w .5 .21667 m .50375 .21667 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .26667 m .50375 .26667 L s P p .001 w .5 .28333 m .50375 .28333 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .31667 m .50375 .31667 L s P p .001 w .5 .35 m .50375 .35 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .38333 m .50375 .38333 L s P p .001 w .5 .4 m .50375 .4 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45 m .50375 .45 L s P p .001 w .5 .46667 m .50375 .46667 L s P p .001 w .5 .48333 m .50375 .48333 L s P p .001 w .5 .51667 m .50375 .51667 L s P p .001 w .5 .53333 m .50375 .53333 L s P p .001 w .5 .55 m .50375 .55 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .6 m .50375 .6 L s P p .001 w .5 .61667 m .50375 .61667 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65 m .50375 .65 L s P p .002 w .5 0 m .5 .66667 L s P P 0 0 m 1 0 L 1 .66667 L 0 .66667 L closepath clip newpath p p .005 w .66667 .33333 m .66666 .33606 L .66664 .33879 L .66662 .34151 L .66658 .34424 L .66647 .34969 L .66631 .35513 L .66611 .36057 L .66586 .36601 L .66524 .37684 L .66444 .38763 L .66346 .39836 L .66099 .41961 L .65782 .44048 L .65398 .46089 L .64434 .5 L .63223 .53625 L .61785 .56904 L .60146 .59778 L .58333 .62201 L .57371 .63229 L .56378 .64129 L .55357 .64898 L .54314 .65531 L .53252 .66026 L .52715 .66221 L .52175 .66381 L .51634 .66506 L .51362 .66555 L .5109 .66595 L .50818 .66627 L .50682 .66639 L .50545 .66649 L .50409 .66657 L .50273 .66662 L .50136 .66666 L .5 .66667 L .49864 .66666 L .49727 .66662 L .49591 .66657 L .49455 .66649 L .49318 .66639 L .49182 .66627 L .4891 .66595 L .48638 .66555 L .48366 .66506 L .47825 .66381 L .47285 .66221 L .46748 .66026 L .45686 .65531 L Mistroke .44643 .64898 L .43622 .64129 L .41667 .62201 L .39854 .59778 L .38215 .56904 L .36777 .53625 L .35566 .5 L .35052 .48076 L .34602 .46089 L .34218 .44048 L .33901 .41961 L .33654 .39836 L .33556 .38763 L .33476 .37684 L .33414 .36601 L .33389 .36057 L .33369 .35513 L .33353 .34969 L .33347 .34696 L .33342 .34424 L .33338 .34151 L .33336 .33879 L .33334 .33606 L .33333 .33333 L .33334 .33061 L .33336 .32788 L .33338 .32515 L .33342 .32243 L .33353 .31698 L .33369 .31153 L .33389 .30609 L .33414 .30066 L .33476 .28982 L .33556 .27903 L .33654 .2683 L .33901 .24706 L .34218 .22619 L .34602 .20577 L .35566 .16667 L .36777 .13041 L .38215 .09763 L .39854 .06888 L .41667 .04466 L .42629 .03438 L .43622 .02537 L .44643 .01769 L .45686 .01136 L .46748 .0064 L .47285 .00445 L .47825 .00285 L Mistroke .48366 .00161 L .48638 .00111 L .4891 .00071 L .49182 .0004 L .49318 .00028 L .49455 .00018 L .49591 .0001 L .49727 4e-05 L .49864 1e-05 L .5 0 L .50136 1e-05 L .50273 4e-05 L .50409 .0001 L .50545 .00018 L .50818 .0004 L .5109 .00071 L .51362 .00111 L .51634 .00161 L .52175 .00285 L .52715 .00445 L .53252 .0064 L .54314 .01136 L .55357 .01769 L .56378 .02537 L .58333 .04466 L .60146 .06888 L .61785 .09763 L .63223 .13041 L .64434 .16667 L .64948 .1859 L .65398 .20577 L .65782 .22619 L .66099 .24706 L .66346 .2683 L .66444 .27903 L .66524 .28982 L .66586 .30066 L .66611 .30609 L .66631 .31153 L .66647 .31698 L .66653 .3197 L .66658 .32243 L .66662 .32515 L .66664 .32788 L .66666 .33061 L .66667 .33333 L Mfstroke P P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect] 3. Activate the Plotall command for 0 and also for `/4. Be sure to take note of the differences between this and those in the first set of exercises. :[font = text; inactive; preserveAspect] 4. Please activate the "Table" command(below this text) and scroll through each graph to investigate the values of the distance from P to (0,c1) and the distance from P to (0,c2). This will give the values of points on around the circle to 2 `. In each case, please give the value of the sum of the two distances on your Investigation Sheet. 5. Does this number have any similarity to any other given value? :[font = input; preserveAspect; startGroup] {c1,c2} = c/.Solve[a^2 == b^2 + c^2,c] :[font = output; output; inactive; preserveAspect; endGroup] {-2*3^(1/2), 2*3^(1/2)} ;[o] {-2 Sqrt[3], 2 Sqrt[3]} :[font = input; preserveAspect] Clear[c1plot,c2plot,c1label,c2label,c1val,c2val] c1plot[t_] := Graphics[{RGBColor[1,0,0],Line[{{0,c1},{x[t],y[t]}}]}] c2plot[t_] := Graphics[{Blue,Line[{{0,c2},{x[t],y[t]}}]}] c1label[t_] := Graphics[Text["c1", {-.2,c1}]] c2label[t_] := Graphics[Text["c2", {0,c2}]] c1val[t_] := Graphics[ Text["= d(f1,p)" N[dist[{0,c1},{x[t],y[t]}]], {3a/4,a+3/2}]]; c2val[t_] := Graphics[ Text["= d(f2,p)" N[dist[{0,c2},{x[t],y[t]}]], {-3a/4,a+3/ 2}]]; ctotal[t_] := Graphics[ Text["= d(f1,p) + d(f2,p)" N[dist[{0,c2},{x[t],y[t]}] + dist[{0,c1},{x[t],y[t]}]], {a/3,-a-3/2}]] ppoint[t_] := Graphics[ {Maroon,PointSize[.02],Point[{x[t],y[t]}]} ] ppointlabel[t_] := Graphics[ Text["p",{x[t]+.2,y[t]+.2}] ] plotall[t_] := Show[ellipse,c1plot[t],c2plot[t], c1label[t],c2label[t],c1val[t],c2val[t], ctotal[t],ppoint[t],ppointlabel[t], PlotRange->{{-a-5,a+5},{-b-5,b+5}}]; :[font = input; preserveAspect; startGroup] plotall[0]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; endGroup] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .61111 .38889 m .61111 .39071 L .6111 .39252 L .61108 .39434 L .61105 .39616 L .61098 .39979 L .61087 .40342 L .61074 .40705 L .61058 .41067 L .61016 .41789 L .60963 .42509 L .60898 .43224 L .60733 .4464 L .60521 .46032 L .60265 .47393 L .59623 .5 L .58815 .52417 L .57857 .54602 L .56764 .56519 L .55556 .58134 L .54914 .58819 L .54252 .5942 L .53572 .59932 L .52876 .60354 L .52168 .60684 L .5181 .60814 L .5145 .60921 L .51089 .61004 L .50908 .61037 L .50727 .61064 L .50545 .61084 L .50454 .61093 L .50364 .61099 L .50273 .61104 L .50182 .61108 L .50091 .6111 L .5 .61111 L .49909 .6111 L .49818 .61108 L .49727 .61104 L .49636 .61099 L .49546 .61093 L .49455 .61084 L .49273 .61064 L .49092 .61037 L .48911 .61004 L .4855 .60921 L .4819 .60814 L .47832 .60684 L .47124 .60354 L Mistroke .46428 .59932 L .45748 .5942 L .44444 .58134 L .43236 .56519 L .42143 .54602 L .41185 .52417 L .40377 .5 L .40035 .48718 L .39735 .47393 L .39479 .46032 L .39267 .4464 L .39102 .43224 L .39037 .42509 L .38984 .41789 L .38942 .41067 L .38926 .40705 L .38913 .40342 L .38902 .39979 L .38898 .39798 L .38895 .39616 L .38892 .39434 L .3889 .39252 L .38889 .39071 L .38889 .38889 L .38889 .38707 L .3889 .38525 L .38892 .38344 L .38895 .38162 L .38902 .37798 L .38913 .37435 L .38926 .37073 L .38942 .36711 L .38984 .35988 L .39037 .35269 L .39102 .34554 L .39267 .33137 L .39479 .31746 L .39735 .30385 L .40377 .27778 L .41185 .25361 L .42143 .23175 L .43236 .21259 L .44444 .19644 L .45086 .18958 L .45748 .18358 L .46428 .17846 L .47124 .17424 L .47832 .17094 L .4819 .16963 L .4855 .16857 L Mistroke .48911 .16774 L .49092 .16741 L .49273 .16714 L .49455 .16693 L .49546 .16685 L .49636 .16679 L .49727 .16673 L .49818 .1667 L .49909 .16667 L .5 .16667 L .50091 .16667 L .50182 .1667 L .50273 .16673 L .50364 .16679 L .50545 .16693 L .50727 .16714 L .50908 .16741 L .51089 .16774 L .5145 .16857 L .5181 .16963 L .52168 .17094 L .52876 .17424 L .53572 .17846 L .54252 .18358 L .55556 .19644 L .56764 .21259 L .57857 .23175 L .58815 .25361 L .59623 .27778 L .59965 .2906 L .60265 .30385 L .60521 .31746 L .60733 .33137 L .60898 .34554 L .60963 .35269 L .61016 .35988 L .61058 .36711 L .61074 .37073 L .61087 .37435 L .61098 .37798 L .61102 .3798 L .61105 .38162 L .61108 .38344 L .6111 .38525 L .61111 .38707 L .61111 .38889 L Mfstroke P P p 1 0 0 r .004 w .5 .19644 m .61111 .38889 L s P p 0 0 1 r .004 w .5 .58134 m .61111 .38889 L s P [(c1)] .48889 .19644 0 0 Mshowa [(c2)] .5 .58134 0 0 Mshowa [(4. = d\(f1,p\))] .66667 .69444 0 0 Mshowa [(4. = d\(f2,p\))] .33333 .69444 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .57407 .08333 0 0 Mshowa p .69 .188 .377 r .02 w .61111 .38889 Mdot P [(p)] .62222 .4 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = input; preserveAspect; startGroup; animationSpeed = 30] plotall[Pi/4]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; endGroup] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .61111 .38889 m .61111 .39071 L .6111 .39252 L .61108 .39434 L .61105 .39616 L .61098 .39979 L .61087 .40342 L .61074 .40705 L .61058 .41067 L .61016 .41789 L .60963 .42509 L .60898 .43224 L .60733 .4464 L .60521 .46032 L .60265 .47393 L .59623 .5 L .58815 .52417 L .57857 .54602 L .56764 .56519 L .55556 .58134 L .54914 .58819 L .54252 .5942 L .53572 .59932 L .52876 .60354 L .52168 .60684 L .5181 .60814 L .5145 .60921 L .51089 .61004 L .50908 .61037 L .50727 .61064 L .50545 .61084 L .50454 .61093 L .50364 .61099 L .50273 .61104 L .50182 .61108 L .50091 .6111 L .5 .61111 L .49909 .6111 L .49818 .61108 L .49727 .61104 L .49636 .61099 L .49546 .61093 L .49455 .61084 L .49273 .61064 L .49092 .61037 L .48911 .61004 L .4855 .60921 L .4819 .60814 L .47832 .60684 L .47124 .60354 L Mistroke .46428 .59932 L .45748 .5942 L .44444 .58134 L .43236 .56519 L .42143 .54602 L .41185 .52417 L .40377 .5 L .40035 .48718 L .39735 .47393 L .39479 .46032 L .39267 .4464 L .39102 .43224 L .39037 .42509 L .38984 .41789 L .38942 .41067 L .38926 .40705 L .38913 .40342 L .38902 .39979 L .38898 .39798 L .38895 .39616 L .38892 .39434 L .3889 .39252 L .38889 .39071 L .38889 .38889 L .38889 .38707 L .3889 .38525 L .38892 .38344 L .38895 .38162 L .38902 .37798 L .38913 .37435 L .38926 .37073 L .38942 .36711 L .38984 .35988 L .39037 .35269 L .39102 .34554 L .39267 .33137 L .39479 .31746 L .39735 .30385 L .40377 .27778 L .41185 .25361 L .42143 .23175 L .43236 .21259 L .44444 .19644 L .45086 .18958 L .45748 .18358 L .46428 .17846 L .47124 .17424 L .47832 .17094 L .4819 .16963 L .4855 .16857 L Mistroke .48911 .16774 L .49092 .16741 L .49273 .16714 L .49455 .16693 L .49546 .16685 L .49636 .16679 L .49727 .16673 L .49818 .1667 L .49909 .16667 L .5 .16667 L .50091 .16667 L .50182 .1667 L .50273 .16673 L .50364 .16679 L .50545 .16693 L .50727 .16714 L .50908 .16741 L .51089 .16774 L .5145 .16857 L .5181 .16963 L .52168 .17094 L .52876 .17424 L .53572 .17846 L .54252 .18358 L .55556 .19644 L .56764 .21259 L .57857 .23175 L .58815 .25361 L .59623 .27778 L .59965 .2906 L .60265 .30385 L .60521 .31746 L .60733 .33137 L .60898 .34554 L .60963 .35269 L .61016 .35988 L .61058 .36711 L .61074 .37073 L .61087 .37435 L .61098 .37798 L .61102 .3798 L .61105 .38162 L .61108 .38344 L .6111 .38525 L .61111 .38707 L .61111 .38889 L Mfstroke P P p 1 0 0 r .004 w .5 .19644 m .57857 .54602 L s P p 0 0 1 r .004 w .5 .58134 m .57857 .54602 L s P [(c1)] .48889 .19644 0 0 Mshowa [(c2)] .5 .58134 0 0 Mshowa [(6.44949 = d\(f1,p\))] .66667 .69444 0 0 Mshowa [(1.55051 = d\(f2,p\))] .33333 .69444 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .57407 .08333 0 0 Mshowa p .69 .188 .377 r .02 w .57857 .54602 Mdot P [(p)] .58968 .55713 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = input; preserveAspect; startGroup; animationSpeed = 30] Table[plotall[theta],{theta,0,7Pi/4,Pi/4}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; startGroup; animationSpeed = 12] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .61111 .38889 m .61111 .39071 L .6111 .39252 L .61108 .39434 L .61105 .39616 L .61098 .39979 L .61087 .40342 L .61074 .40705 L .61058 .41067 L .61016 .41789 L .60963 .42509 L .60898 .43224 L .60733 .4464 L .60521 .46032 L .60265 .47393 L .59623 .5 L .58815 .52417 L .57857 .54602 L .56764 .56519 L .55556 .58134 L .54914 .58819 L .54252 .5942 L .53572 .59932 L .52876 .60354 L .52168 .60684 L .5181 .60814 L .5145 .60921 L .51089 .61004 L .50908 .61037 L .50727 .61064 L .50545 .61084 L .50454 .61093 L .50364 .61099 L .50273 .61104 L .50182 .61108 L .50091 .6111 L .5 .61111 L .49909 .6111 L .49818 .61108 L .49727 .61104 L .49636 .61099 L .49546 .61093 L .49455 .61084 L .49273 .61064 L .49092 .61037 L .48911 .61004 L .4855 .60921 L .4819 .60814 L .47832 .60684 L .47124 .60354 L Mistroke .46428 .59932 L .45748 .5942 L .44444 .58134 L .43236 .56519 L .42143 .54602 L .41185 .52417 L .40377 .5 L .40035 .48718 L .39735 .47393 L .39479 .46032 L .39267 .4464 L .39102 .43224 L .39037 .42509 L .38984 .41789 L .38942 .41067 L .38926 .40705 L .38913 .40342 L .38902 .39979 L .38898 .39798 L .38895 .39616 L .38892 .39434 L .3889 .39252 L .38889 .39071 L .38889 .38889 L .38889 .38707 L .3889 .38525 L .38892 .38344 L .38895 .38162 L .38902 .37798 L .38913 .37435 L .38926 .37073 L .38942 .36711 L .38984 .35988 L .39037 .35269 L .39102 .34554 L .39267 .33137 L .39479 .31746 L .39735 .30385 L .40377 .27778 L .41185 .25361 L .42143 .23175 L .43236 .21259 L .44444 .19644 L .45086 .18958 L .45748 .18358 L .46428 .17846 L .47124 .17424 L .47832 .17094 L .4819 .16963 L .4855 .16857 L Mistroke .48911 .16774 L .49092 .16741 L .49273 .16714 L .49455 .16693 L .49546 .16685 L .49636 .16679 L .49727 .16673 L .49818 .1667 L .49909 .16667 L .5 .16667 L .50091 .16667 L .50182 .1667 L .50273 .16673 L .50364 .16679 L .50545 .16693 L .50727 .16714 L .50908 .16741 L .51089 .16774 L .5145 .16857 L .5181 .16963 L .52168 .17094 L .52876 .17424 L .53572 .17846 L .54252 .18358 L .55556 .19644 L .56764 .21259 L .57857 .23175 L .58815 .25361 L .59623 .27778 L .59965 .2906 L .60265 .30385 L .60521 .31746 L .60733 .33137 L .60898 .34554 L .60963 .35269 L .61016 .35988 L .61058 .36711 L .61074 .37073 L .61087 .37435 L .61098 .37798 L .61102 .3798 L .61105 .38162 L .61108 .38344 L .6111 .38525 L .61111 .38707 L .61111 .38889 L Mfstroke P P p 1 0 0 r .004 w .5 .19644 m .61111 .38889 L s P p 0 0 1 r .004 w .5 .58134 m .61111 .38889 L s P [(c1)] .48889 .19644 0 0 Mshowa [(c2)] .5 .58134 0 0 Mshowa [(4. = d\(f1,p\))] .66667 .69444 0 0 Mshowa [(4. = d\(f2,p\))] .33333 .69444 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .57407 .08333 0 0 Mshowa p .69 .188 .377 r .02 w .61111 .38889 Mdot P [(p)] .62222 .4 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; animationSpeed = 12] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .61111 .38889 m .61111 .39071 L .6111 .39252 L .61108 .39434 L .61105 .39616 L .61098 .39979 L .61087 .40342 L .61074 .40705 L .61058 .41067 L .61016 .41789 L .60963 .42509 L .60898 .43224 L .60733 .4464 L .60521 .46032 L .60265 .47393 L .59623 .5 L .58815 .52417 L .57857 .54602 L .56764 .56519 L .55556 .58134 L .54914 .58819 L .54252 .5942 L .53572 .59932 L .52876 .60354 L .52168 .60684 L .5181 .60814 L .5145 .60921 L .51089 .61004 L .50908 .61037 L .50727 .61064 L .50545 .61084 L .50454 .61093 L .50364 .61099 L .50273 .61104 L .50182 .61108 L .50091 .6111 L .5 .61111 L .49909 .6111 L .49818 .61108 L .49727 .61104 L .49636 .61099 L .49546 .61093 L .49455 .61084 L .49273 .61064 L .49092 .61037 L .48911 .61004 L .4855 .60921 L .4819 .60814 L .47832 .60684 L .47124 .60354 L Mistroke .46428 .59932 L .45748 .5942 L .44444 .58134 L .43236 .56519 L .42143 .54602 L .41185 .52417 L .40377 .5 L .40035 .48718 L .39735 .47393 L .39479 .46032 L .39267 .4464 L .39102 .43224 L .39037 .42509 L .38984 .41789 L .38942 .41067 L .38926 .40705 L .38913 .40342 L .38902 .39979 L .38898 .39798 L .38895 .39616 L .38892 .39434 L .3889 .39252 L .38889 .39071 L .38889 .38889 L .38889 .38707 L .3889 .38525 L .38892 .38344 L .38895 .38162 L .38902 .37798 L .38913 .37435 L .38926 .37073 L .38942 .36711 L .38984 .35988 L .39037 .35269 L .39102 .34554 L .39267 .33137 L .39479 .31746 L .39735 .30385 L .40377 .27778 L .41185 .25361 L .42143 .23175 L .43236 .21259 L .44444 .19644 L .45086 .18958 L .45748 .18358 L .46428 .17846 L .47124 .17424 L .47832 .17094 L .4819 .16963 L .4855 .16857 L Mistroke .48911 .16774 L .49092 .16741 L .49273 .16714 L .49455 .16693 L .49546 .16685 L .49636 .16679 L .49727 .16673 L .49818 .1667 L .49909 .16667 L .5 .16667 L .50091 .16667 L .50182 .1667 L .50273 .16673 L .50364 .16679 L .50545 .16693 L .50727 .16714 L .50908 .16741 L .51089 .16774 L .5145 .16857 L .5181 .16963 L .52168 .17094 L .52876 .17424 L .53572 .17846 L .54252 .18358 L .55556 .19644 L .56764 .21259 L .57857 .23175 L .58815 .25361 L .59623 .27778 L .59965 .2906 L .60265 .30385 L .60521 .31746 L .60733 .33137 L .60898 .34554 L .60963 .35269 L .61016 .35988 L .61058 .36711 L .61074 .37073 L .61087 .37435 L .61098 .37798 L .61102 .3798 L .61105 .38162 L .61108 .38344 L .6111 .38525 L .61111 .38707 L .61111 .38889 L Mfstroke P P p 1 0 0 r .004 w .5 .19644 m .57857 .54602 L s P p 0 0 1 r .004 w .5 .58134 m .57857 .54602 L s P [(c1)] .48889 .19644 0 0 Mshowa [(c2)] .5 .58134 0 0 Mshowa [(6.44949 = d\(f1,p\))] .66667 .69444 0 0 Mshowa [(1.55051 = d\(f2,p\))] .33333 .69444 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .57407 .08333 0 0 Mshowa p .69 .188 .377 r .02 w .57857 .54602 Mdot P [(p)] .58968 .55713 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; animationSpeed = 12] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .61111 .38889 m .61111 .39071 L .6111 .39252 L .61108 .39434 L .61105 .39616 L .61098 .39979 L .61087 .40342 L .61074 .40705 L .61058 .41067 L .61016 .41789 L .60963 .42509 L .60898 .43224 L .60733 .4464 L .60521 .46032 L .60265 .47393 L .59623 .5 L .58815 .52417 L .57857 .54602 L .56764 .56519 L .55556 .58134 L .54914 .58819 L .54252 .5942 L .53572 .59932 L .52876 .60354 L .52168 .60684 L .5181 .60814 L .5145 .60921 L .51089 .61004 L .50908 .61037 L .50727 .61064 L .50545 .61084 L .50454 .61093 L .50364 .61099 L .50273 .61104 L .50182 .61108 L .50091 .6111 L .5 .61111 L .49909 .6111 L .49818 .61108 L .49727 .61104 L .49636 .61099 L .49546 .61093 L .49455 .61084 L .49273 .61064 L .49092 .61037 L .48911 .61004 L .4855 .60921 L .4819 .60814 L .47832 .60684 L .47124 .60354 L Mistroke .46428 .59932 L .45748 .5942 L .44444 .58134 L .43236 .56519 L .42143 .54602 L .41185 .52417 L .40377 .5 L .40035 .48718 L .39735 .47393 L .39479 .46032 L .39267 .4464 L .39102 .43224 L .39037 .42509 L .38984 .41789 L .38942 .41067 L .38926 .40705 L .38913 .40342 L .38902 .39979 L .38898 .39798 L .38895 .39616 L .38892 .39434 L .3889 .39252 L .38889 .39071 L .38889 .38889 L .38889 .38707 L .3889 .38525 L .38892 .38344 L .38895 .38162 L .38902 .37798 L .38913 .37435 L .38926 .37073 L .38942 .36711 L .38984 .35988 L .39037 .35269 L .39102 .34554 L .39267 .33137 L .39479 .31746 L .39735 .30385 L .40377 .27778 L .41185 .25361 L .42143 .23175 L .43236 .21259 L .44444 .19644 L .45086 .18958 L .45748 .18358 L .46428 .17846 L .47124 .17424 L .47832 .17094 L .4819 .16963 L .4855 .16857 L Mistroke .48911 .16774 L .49092 .16741 L .49273 .16714 L .49455 .16693 L .49546 .16685 L .49636 .16679 L .49727 .16673 L .49818 .1667 L .49909 .16667 L .5 .16667 L .50091 .16667 L .50182 .1667 L .50273 .16673 L .50364 .16679 L .50545 .16693 L .50727 .16714 L .50908 .16741 L .51089 .16774 L .5145 .16857 L .5181 .16963 L .52168 .17094 L .52876 .17424 L .53572 .17846 L .54252 .18358 L .55556 .19644 L .56764 .21259 L .57857 .23175 L .58815 .25361 L .59623 .27778 L .59965 .2906 L .60265 .30385 L .60521 .31746 L .60733 .33137 L .60898 .34554 L .60963 .35269 L .61016 .35988 L .61058 .36711 L .61074 .37073 L .61087 .37435 L .61098 .37798 L .61102 .3798 L .61105 .38162 L .61108 .38344 L .6111 .38525 L .61111 .38707 L .61111 .38889 L Mfstroke P P p 1 0 0 r .004 w .5 .19644 m .5 .61111 L s P p 0 0 1 r .004 w .5 .58134 m .5 .61111 L s P [(c1)] .48889 .19644 0 0 Mshowa [(c2)] .5 .58134 0 0 Mshowa [(7.4641 = d\(f1,p\))] .66667 .69444 0 0 Mshowa [(0.535898 = d\(f2,p\))] .33333 .69444 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .57407 .08333 0 0 Mshowa p .69 .188 .377 r .02 w .5 .61111 Mdot P [(p)] .51111 .62222 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; animationSpeed = 12] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .61111 .38889 m .61111 .39071 L .6111 .39252 L .61108 .39434 L .61105 .39616 L .61098 .39979 L .61087 .40342 L .61074 .40705 L .61058 .41067 L .61016 .41789 L .60963 .42509 L .60898 .43224 L .60733 .4464 L .60521 .46032 L .60265 .47393 L .59623 .5 L .58815 .52417 L .57857 .54602 L .56764 .56519 L .55556 .58134 L .54914 .58819 L .54252 .5942 L .53572 .59932 L .52876 .60354 L .52168 .60684 L .5181 .60814 L .5145 .60921 L .51089 .61004 L .50908 .61037 L .50727 .61064 L .50545 .61084 L .50454 .61093 L .50364 .61099 L .50273 .61104 L .50182 .61108 L .50091 .6111 L .5 .61111 L .49909 .6111 L .49818 .61108 L .49727 .61104 L .49636 .61099 L .49546 .61093 L .49455 .61084 L .49273 .61064 L .49092 .61037 L .48911 .61004 L .4855 .60921 L .4819 .60814 L .47832 .60684 L .47124 .60354 L Mistroke .46428 .59932 L .45748 .5942 L .44444 .58134 L .43236 .56519 L .42143 .54602 L .41185 .52417 L .40377 .5 L .40035 .48718 L .39735 .47393 L .39479 .46032 L .39267 .4464 L .39102 .43224 L .39037 .42509 L .38984 .41789 L .38942 .41067 L .38926 .40705 L .38913 .40342 L .38902 .39979 L .38898 .39798 L .38895 .39616 L .38892 .39434 L .3889 .39252 L .38889 .39071 L .38889 .38889 L .38889 .38707 L .3889 .38525 L .38892 .38344 L .38895 .38162 L .38902 .37798 L .38913 .37435 L .38926 .37073 L .38942 .36711 L .38984 .35988 L .39037 .35269 L .39102 .34554 L .39267 .33137 L .39479 .31746 L .39735 .30385 L .40377 .27778 L .41185 .25361 L .42143 .23175 L .43236 .21259 L .44444 .19644 L .45086 .18958 L .45748 .18358 L .46428 .17846 L .47124 .17424 L .47832 .17094 L .4819 .16963 L .4855 .16857 L Mistroke .48911 .16774 L .49092 .16741 L .49273 .16714 L .49455 .16693 L .49546 .16685 L .49636 .16679 L .49727 .16673 L .49818 .1667 L .49909 .16667 L .5 .16667 L .50091 .16667 L .50182 .1667 L .50273 .16673 L .50364 .16679 L .50545 .16693 L .50727 .16714 L .50908 .16741 L .51089 .16774 L .5145 .16857 L .5181 .16963 L .52168 .17094 L .52876 .17424 L .53572 .17846 L .54252 .18358 L .55556 .19644 L .56764 .21259 L .57857 .23175 L .58815 .25361 L .59623 .27778 L .59965 .2906 L .60265 .30385 L .60521 .31746 L .60733 .33137 L .60898 .34554 L .60963 .35269 L .61016 .35988 L .61058 .36711 L .61074 .37073 L .61087 .37435 L .61098 .37798 L .61102 .3798 L .61105 .38162 L .61108 .38344 L .6111 .38525 L .61111 .38707 L .61111 .38889 L Mfstroke P P p 1 0 0 r .004 w .5 .19644 m .42143 .54602 L s P p 0 0 1 r .004 w .5 .58134 m .42143 .54602 L s P [(c1)] .48889 .19644 0 0 Mshowa [(c2)] .5 .58134 0 0 Mshowa [(6.44949 = d\(f1,p\))] .66667 .69444 0 0 Mshowa [(1.55051 = d\(f2,p\))] .33333 .69444 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .57407 .08333 0 0 Mshowa p .69 .188 .377 r .02 w .42143 .54602 Mdot P [(p)] .43254 .55713 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; animationSpeed = 12] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .61111 .38889 m .61111 .39071 L .6111 .39252 L .61108 .39434 L .61105 .39616 L .61098 .39979 L .61087 .40342 L .61074 .40705 L .61058 .41067 L .61016 .41789 L .60963 .42509 L .60898 .43224 L .60733 .4464 L .60521 .46032 L .60265 .47393 L .59623 .5 L .58815 .52417 L .57857 .54602 L .56764 .56519 L .55556 .58134 L .54914 .58819 L .54252 .5942 L .53572 .59932 L .52876 .60354 L .52168 .60684 L .5181 .60814 L .5145 .60921 L .51089 .61004 L .50908 .61037 L .50727 .61064 L .50545 .61084 L .50454 .61093 L .50364 .61099 L .50273 .61104 L .50182 .61108 L .50091 .6111 L .5 .61111 L .49909 .6111 L .49818 .61108 L .49727 .61104 L .49636 .61099 L .49546 .61093 L .49455 .61084 L .49273 .61064 L .49092 .61037 L .48911 .61004 L .4855 .60921 L .4819 .60814 L .47832 .60684 L .47124 .60354 L Mistroke .46428 .59932 L .45748 .5942 L .44444 .58134 L .43236 .56519 L .42143 .54602 L .41185 .52417 L .40377 .5 L .40035 .48718 L .39735 .47393 L .39479 .46032 L .39267 .4464 L .39102 .43224 L .39037 .42509 L .38984 .41789 L .38942 .41067 L .38926 .40705 L .38913 .40342 L .38902 .39979 L .38898 .39798 L .38895 .39616 L .38892 .39434 L .3889 .39252 L .38889 .39071 L .38889 .38889 L .38889 .38707 L .3889 .38525 L .38892 .38344 L .38895 .38162 L .38902 .37798 L .38913 .37435 L .38926 .37073 L .38942 .36711 L .38984 .35988 L .39037 .35269 L .39102 .34554 L .39267 .33137 L .39479 .31746 L .39735 .30385 L .40377 .27778 L .41185 .25361 L .42143 .23175 L .43236 .21259 L .44444 .19644 L .45086 .18958 L .45748 .18358 L .46428 .17846 L .47124 .17424 L .47832 .17094 L .4819 .16963 L .4855 .16857 L Mistroke .48911 .16774 L .49092 .16741 L .49273 .16714 L .49455 .16693 L .49546 .16685 L .49636 .16679 L .49727 .16673 L .49818 .1667 L .49909 .16667 L .5 .16667 L .50091 .16667 L .50182 .1667 L .50273 .16673 L .50364 .16679 L .50545 .16693 L .50727 .16714 L .50908 .16741 L .51089 .16774 L .5145 .16857 L .5181 .16963 L .52168 .17094 L .52876 .17424 L .53572 .17846 L .54252 .18358 L .55556 .19644 L .56764 .21259 L .57857 .23175 L .58815 .25361 L .59623 .27778 L .59965 .2906 L .60265 .30385 L .60521 .31746 L .60733 .33137 L .60898 .34554 L .60963 .35269 L .61016 .35988 L .61058 .36711 L .61074 .37073 L .61087 .37435 L .61098 .37798 L .61102 .3798 L .61105 .38162 L .61108 .38344 L .6111 .38525 L .61111 .38707 L .61111 .38889 L Mfstroke P P p 1 0 0 r .004 w .5 .19644 m .38889 .38889 L s P p 0 0 1 r .004 w .5 .58134 m .38889 .38889 L s P [(c1)] .48889 .19644 0 0 Mshowa [(c2)] .5 .58134 0 0 Mshowa [(4. = d\(f1,p\))] .66667 .69444 0 0 Mshowa [(4. = d\(f2,p\))] .33333 .69444 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .57407 .08333 0 0 Mshowa p .69 .188 .377 r .02 w .38889 .38889 Mdot P [(p)] .4 .4 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; animationSpeed = 12] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .61111 .38889 m .61111 .39071 L .6111 .39252 L .61108 .39434 L .61105 .39616 L .61098 .39979 L .61087 .40342 L .61074 .40705 L .61058 .41067 L .61016 .41789 L .60963 .42509 L .60898 .43224 L .60733 .4464 L .60521 .46032 L .60265 .47393 L .59623 .5 L .58815 .52417 L .57857 .54602 L .56764 .56519 L .55556 .58134 L .54914 .58819 L .54252 .5942 L .53572 .59932 L .52876 .60354 L .52168 .60684 L .5181 .60814 L .5145 .60921 L .51089 .61004 L .50908 .61037 L .50727 .61064 L .50545 .61084 L .50454 .61093 L .50364 .61099 L .50273 .61104 L .50182 .61108 L .50091 .6111 L .5 .61111 L .49909 .6111 L .49818 .61108 L .49727 .61104 L .49636 .61099 L .49546 .61093 L .49455 .61084 L .49273 .61064 L .49092 .61037 L .48911 .61004 L .4855 .60921 L .4819 .60814 L .47832 .60684 L .47124 .60354 L Mistroke .46428 .59932 L .45748 .5942 L .44444 .58134 L .43236 .56519 L .42143 .54602 L .41185 .52417 L .40377 .5 L .40035 .48718 L .39735 .47393 L .39479 .46032 L .39267 .4464 L .39102 .43224 L .39037 .42509 L .38984 .41789 L .38942 .41067 L .38926 .40705 L .38913 .40342 L .38902 .39979 L .38898 .39798 L .38895 .39616 L .38892 .39434 L .3889 .39252 L .38889 .39071 L .38889 .38889 L .38889 .38707 L .3889 .38525 L .38892 .38344 L .38895 .38162 L .38902 .37798 L .38913 .37435 L .38926 .37073 L .38942 .36711 L .38984 .35988 L .39037 .35269 L .39102 .34554 L .39267 .33137 L .39479 .31746 L .39735 .30385 L .40377 .27778 L .41185 .25361 L .42143 .23175 L .43236 .21259 L .44444 .19644 L .45086 .18958 L .45748 .18358 L .46428 .17846 L .47124 .17424 L .47832 .17094 L .4819 .16963 L .4855 .16857 L Mistroke .48911 .16774 L .49092 .16741 L .49273 .16714 L .49455 .16693 L .49546 .16685 L .49636 .16679 L .49727 .16673 L .49818 .1667 L .49909 .16667 L .5 .16667 L .50091 .16667 L .50182 .1667 L .50273 .16673 L .50364 .16679 L .50545 .16693 L .50727 .16714 L .50908 .16741 L .51089 .16774 L .5145 .16857 L .5181 .16963 L .52168 .17094 L .52876 .17424 L .53572 .17846 L .54252 .18358 L .55556 .19644 L .56764 .21259 L .57857 .23175 L .58815 .25361 L .59623 .27778 L .59965 .2906 L .60265 .30385 L .60521 .31746 L .60733 .33137 L .60898 .34554 L .60963 .35269 L .61016 .35988 L .61058 .36711 L .61074 .37073 L .61087 .37435 L .61098 .37798 L .61102 .3798 L .61105 .38162 L .61108 .38344 L .6111 .38525 L .61111 .38707 L .61111 .38889 L Mfstroke P P p 1 0 0 r .004 w .5 .19644 m .42143 .23175 L s P p 0 0 1 r .004 w .5 .58134 m .42143 .23175 L s P [(c1)] .48889 .19644 0 0 Mshowa [(c2)] .5 .58134 0 0 Mshowa [(1.55051 = d\(f1,p\))] .66667 .69444 0 0 Mshowa [(6.44949 = d\(f2,p\))] .33333 .69444 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .57407 .08333 0 0 Mshowa p .69 .188 .377 r .02 w .42143 .23175 Mdot P [(p)] .43254 .24287 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; animationSpeed = 12] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .61111 .38889 m .61111 .39071 L .6111 .39252 L .61108 .39434 L .61105 .39616 L .61098 .39979 L .61087 .40342 L .61074 .40705 L .61058 .41067 L .61016 .41789 L .60963 .42509 L .60898 .43224 L .60733 .4464 L .60521 .46032 L .60265 .47393 L .59623 .5 L .58815 .52417 L .57857 .54602 L .56764 .56519 L .55556 .58134 L .54914 .58819 L .54252 .5942 L .53572 .59932 L .52876 .60354 L .52168 .60684 L .5181 .60814 L .5145 .60921 L .51089 .61004 L .50908 .61037 L .50727 .61064 L .50545 .61084 L .50454 .61093 L .50364 .61099 L .50273 .61104 L .50182 .61108 L .50091 .6111 L .5 .61111 L .49909 .6111 L .49818 .61108 L .49727 .61104 L .49636 .61099 L .49546 .61093 L .49455 .61084 L .49273 .61064 L .49092 .61037 L .48911 .61004 L .4855 .60921 L .4819 .60814 L .47832 .60684 L .47124 .60354 L Mistroke .46428 .59932 L .45748 .5942 L .44444 .58134 L .43236 .56519 L .42143 .54602 L .41185 .52417 L .40377 .5 L .40035 .48718 L .39735 .47393 L .39479 .46032 L .39267 .4464 L .39102 .43224 L .39037 .42509 L .38984 .41789 L .38942 .41067 L .38926 .40705 L .38913 .40342 L .38902 .39979 L .38898 .39798 L .38895 .39616 L .38892 .39434 L .3889 .39252 L .38889 .39071 L .38889 .38889 L .38889 .38707 L .3889 .38525 L .38892 .38344 L .38895 .38162 L .38902 .37798 L .38913 .37435 L .38926 .37073 L .38942 .36711 L .38984 .35988 L .39037 .35269 L .39102 .34554 L .39267 .33137 L .39479 .31746 L .39735 .30385 L .40377 .27778 L .41185 .25361 L .42143 .23175 L .43236 .21259 L .44444 .19644 L .45086 .18958 L .45748 .18358 L .46428 .17846 L .47124 .17424 L .47832 .17094 L .4819 .16963 L .4855 .16857 L Mistroke .48911 .16774 L .49092 .16741 L .49273 .16714 L .49455 .16693 L .49546 .16685 L .49636 .16679 L .49727 .16673 L .49818 .1667 L .49909 .16667 L .5 .16667 L .50091 .16667 L .50182 .1667 L .50273 .16673 L .50364 .16679 L .50545 .16693 L .50727 .16714 L .50908 .16741 L .51089 .16774 L .5145 .16857 L .5181 .16963 L .52168 .17094 L .52876 .17424 L .53572 .17846 L .54252 .18358 L .55556 .19644 L .56764 .21259 L .57857 .23175 L .58815 .25361 L .59623 .27778 L .59965 .2906 L .60265 .30385 L .60521 .31746 L .60733 .33137 L .60898 .34554 L .60963 .35269 L .61016 .35988 L .61058 .36711 L .61074 .37073 L .61087 .37435 L .61098 .37798 L .61102 .3798 L .61105 .38162 L .61108 .38344 L .6111 .38525 L .61111 .38707 L .61111 .38889 L Mfstroke P P p 1 0 0 r .004 w .5 .19644 m .5 .16667 L s P p 0 0 1 r .004 w .5 .58134 m .5 .16667 L s P [(c1)] .48889 .19644 0 0 Mshowa [(c2)] .5 .58134 0 0 Mshowa [(0.535898 = d\(f1,p\))] .66667 .69444 0 0 Mshowa [(7.4641 = d\(f2,p\))] .33333 .69444 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .57407 .08333 0 0 Mshowa p .69 .188 .377 r .02 w .5 .16667 Mdot P [(p)] .51111 .17778 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 273; endGroup; endGroup; animationSpeed = 12] %! %%Creator: Mathematica %%AspectRatio: .77778 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0555556 0.388889 0.0555556 [ [(-7.5)] .08333 .38889 0 2 Msboxa [(-5)] .22222 .38889 0 2 Msboxa [(-2.5)] .36111 .38889 0 2 Msboxa [(2.5)] .63889 .38889 0 2 Msboxa [(5)] .77778 .38889 0 2 Msboxa [(7.5)] .91667 .38889 0 2 Msboxa [(-6)] .4875 .05556 1 0 Msboxa [(-4)] .4875 .16667 1 0 Msboxa [(-2)] .4875 .27778 1 0 Msboxa [(2)] .4875 .5 1 0 Msboxa [(4)] .4875 .61111 1 0 Msboxa [(6)] .4875 .72222 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .77878 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .08333 .38889 m .08333 .39514 L s P [(-7.5)] .08333 .38889 0 2 Mshowa p .002 w .22222 .38889 m .22222 .39514 L s P [(-5)] .22222 .38889 0 2 Mshowa p .002 w .36111 .38889 m .36111 .39514 L s P [(-2.5)] .36111 .38889 0 2 Mshowa p .002 w .63889 .38889 m .63889 .39514 L s P [(2.5)] .63889 .38889 0 2 Mshowa p .002 w .77778 .38889 m .77778 .39514 L s P [(5)] .77778 .38889 0 2 Mshowa p .002 w .91667 .38889 m .91667 .39514 L s P [(7.5)] .91667 .38889 0 2 Mshowa p .001 w .11111 .38889 m .11111 .39264 L s P p .001 w .13889 .38889 m .13889 .39264 L s P p .001 w .16667 .38889 m .16667 .39264 L s P p .001 w .19444 .38889 m .19444 .39264 L s P p .001 w .25 .38889 m .25 .39264 L s P p .001 w .27778 .38889 m .27778 .39264 L s P p .001 w .30556 .38889 m .30556 .39264 L s P p .001 w .33333 .38889 m .33333 .39264 L s P p .001 w .38889 .38889 m .38889 .39264 L s P p .001 w .41667 .38889 m .41667 .39264 L s P p .001 w .44444 .38889 m .44444 .39264 L s P p .001 w .47222 .38889 m .47222 .39264 L s P p .001 w .52778 .38889 m .52778 .39264 L s P p .001 w .55556 .38889 m .55556 .39264 L s P p .001 w .58333 .38889 m .58333 .39264 L s P p .001 w .61111 .38889 m .61111 .39264 L s P p .001 w .66667 .38889 m .66667 .39264 L s P p .001 w .69444 .38889 m .69444 .39264 L s P p .001 w .72222 .38889 m .72222 .39264 L s P p .001 w .75 .38889 m .75 .39264 L s P p .001 w .80556 .38889 m .80556 .39264 L s P p .001 w .83333 .38889 m .83333 .39264 L s P p .001 w .86111 .38889 m .86111 .39264 L s P p .001 w .88889 .38889 m .88889 .39264 L s P p .001 w .05556 .38889 m .05556 .39264 L s P p .001 w .02778 .38889 m .02778 .39264 L s P p .001 w .94444 .38889 m .94444 .39264 L s P p .001 w .97222 .38889 m .97222 .39264 L s P p .002 w 0 .38889 m 1 .38889 L s P p .002 w .5 .05556 m .50625 .05556 L s P [(-6)] .4875 .05556 1 0 Mshowa p .002 w .5 .16667 m .50625 .16667 L s P [(-4)] .4875 .16667 1 0 Mshowa p .002 w .5 .27778 m .50625 .27778 L s P [(-2)] .4875 .27778 1 0 Mshowa p .002 w .5 .5 m .50625 .5 L s P [(2)] .4875 .5 1 0 Mshowa p .002 w .5 .61111 m .50625 .61111 L s P [(4)] .4875 .61111 1 0 Mshowa p .002 w .5 .72222 m .50625 .72222 L s P [(6)] .4875 .72222 1 0 Mshowa p .001 w .5 .07778 m .50375 .07778 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .12222 m .50375 .12222 L s P p .001 w .5 .14444 m .50375 .14444 L s P p .001 w .5 .18889 m .50375 .18889 L s P p .001 w .5 .21111 m .50375 .21111 L s P p .001 w .5 .23333 m .50375 .23333 L s P p .001 w .5 .25556 m .50375 .25556 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .32222 m .50375 .32222 L s P p .001 w .5 .34444 m .50375 .34444 L s P p .001 w .5 .36667 m .50375 .36667 L s P p .001 w .5 .41111 m .50375 .41111 L s P p .001 w .5 .43333 m .50375 .43333 L s P p .001 w .5 .45556 m .50375 .45556 L s P p .001 w .5 .47778 m .50375 .47778 L s P p .001 w .5 .52222 m .50375 .52222 L s P p .001 w .5 .54444 m .50375 .54444 L s P p .001 w .5 .56667 m .50375 .56667 L s P p .001 w .5 .58889 m .50375 .58889 L s P p .001 w .5 .63333 m .50375 .63333 L s P p .001 w .5 .65556 m .50375 .65556 L s P p .001 w .5 .67778 m .50375 .67778 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .03333 m .50375 .03333 L s P p .001 w .5 .01111 m .50375 .01111 L s P p .001 w .5 .74444 m .50375 .74444 L s P p .001 w .5 .76667 m .50375 .76667 L s P p .002 w .5 0 m .5 .77778 L s P P 0 0 m 1 0 L 1 .77778 L 0 .77778 L closepath clip newpath p p p .005 w .61111 .38889 m .61111 .39071 L .6111 .39252 L .61108 .39434 L .61105 .39616 L .61098 .39979 L .61087 .40342 L .61074 .40705 L .61058 .41067 L .61016 .41789 L .60963 .42509 L .60898 .43224 L .60733 .4464 L .60521 .46032 L .60265 .47393 L .59623 .5 L .58815 .52417 L .57857 .54602 L .56764 .56519 L .55556 .58134 L .54914 .58819 L .54252 .5942 L .53572 .59932 L .52876 .60354 L .52168 .60684 L .5181 .60814 L .5145 .60921 L .51089 .61004 L .50908 .61037 L .50727 .61064 L .50545 .61084 L .50454 .61093 L .50364 .61099 L .50273 .61104 L .50182 .61108 L .50091 .6111 L .5 .61111 L .49909 .6111 L .49818 .61108 L .49727 .61104 L .49636 .61099 L .49546 .61093 L .49455 .61084 L .49273 .61064 L .49092 .61037 L .48911 .61004 L .4855 .60921 L .4819 .60814 L .47832 .60684 L .47124 .60354 L Mistroke .46428 .59932 L .45748 .5942 L .44444 .58134 L .43236 .56519 L .42143 .54602 L .41185 .52417 L .40377 .5 L .40035 .48718 L .39735 .47393 L .39479 .46032 L .39267 .4464 L .39102 .43224 L .39037 .42509 L .38984 .41789 L .38942 .41067 L .38926 .40705 L .38913 .40342 L .38902 .39979 L .38898 .39798 L .38895 .39616 L .38892 .39434 L .3889 .39252 L .38889 .39071 L .38889 .38889 L .38889 .38707 L .3889 .38525 L .38892 .38344 L .38895 .38162 L .38902 .37798 L .38913 .37435 L .38926 .37073 L .38942 .36711 L .38984 .35988 L .39037 .35269 L .39102 .34554 L .39267 .33137 L .39479 .31746 L .39735 .30385 L .40377 .27778 L .41185 .25361 L .42143 .23175 L .43236 .21259 L .44444 .19644 L .45086 .18958 L .45748 .18358 L .46428 .17846 L .47124 .17424 L .47832 .17094 L .4819 .16963 L .4855 .16857 L Mistroke .48911 .16774 L .49092 .16741 L .49273 .16714 L .49455 .16693 L .49546 .16685 L .49636 .16679 L .49727 .16673 L .49818 .1667 L .49909 .16667 L .5 .16667 L .50091 .16667 L .50182 .1667 L .50273 .16673 L .50364 .16679 L .50545 .16693 L .50727 .16714 L .50908 .16741 L .51089 .16774 L .5145 .16857 L .5181 .16963 L .52168 .17094 L .52876 .17424 L .53572 .17846 L .54252 .18358 L .55556 .19644 L .56764 .21259 L .57857 .23175 L .58815 .25361 L .59623 .27778 L .59965 .2906 L .60265 .30385 L .60521 .31746 L .60733 .33137 L .60898 .34554 L .60963 .35269 L .61016 .35988 L .61058 .36711 L .61074 .37073 L .61087 .37435 L .61098 .37798 L .61102 .3798 L .61105 .38162 L .61108 .38344 L .6111 .38525 L .61111 .38707 L .61111 .38889 L Mfstroke P P p 1 0 0 r .004 w .5 .19644 m .57857 .23175 L s P p 0 0 1 r .004 w .5 .58134 m .57857 .23175 L s P [(c1)] .48889 .19644 0 0 Mshowa [(c2)] .5 .58134 0 0 Mshowa [(1.55051 = d\(f1,p\))] .66667 .69444 0 0 Mshowa [(6.44949 = d\(f2,p\))] .33333 .69444 0 0 Mshowa [(8. = d\(f1,p\) + d\(f2,p\))] .57407 .08333 0 0 Mshowa p .69 .188 .377 r .02 w .57857 .23175 Mdot P [(p)] .58968 .24287 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = input; preserveAspect] :[font = input; preserveAspect; animationSpeed = 30] :[font = text; inactive; preserveAspect; animationSpeed = 30; infiniteLoop; loopDistance = -11] 6. Now let's look to see if this is just a special case or if this observation is true for all ellipses. So that our graphs are manageable let me again suggest that you go back to the beginning and let a =5 and b = 3Ž3. ;[s] 2:0,0;219,1;224,-1; 2:1,24,16,CalcMath,0,12,0,0,0;1,16,12,Monaco,0,12,0,0,0; :[font = text; inactive; preserveAspect] 7. Repeat the activations from above for these values of a and b and record these values on the Investigation Sheet. Be sure to also use the "Table" command too. 8. What similarity do you see this time? Is it the same as last? 9. If you wish try again with new values of a and b go ahead-- but be sure to make a larger than b. :[font = text; inactive; preserveAspect; endGroup; endGroup] 10. In summary write a short paragraph that contrasts the two different cases of ellipses, noting the vertices(intersections with the axes) and the focii(c1,c2). 11. Please construct your very own definition of an ellipse based on our computer investigation. ^*)