(*^ ::[ frontEndVersion = "Microsoft Windows Mathematica Notebook Front End Version 2.2"; microsoftWindowsStandardFontEncoding; fontset = title, "Arial", 24, L0, center, nohscroll, bold; fontset = subtitle, "Arial", 18, L0, center, nohscroll, bold; fontset = subsubtitle, "Arial", 14, L0, center, nohscroll, bold; fontset = section, "Arial", 14, L0, bold, grayBox; fontset = subsection, "Arial", 12, L0, bold, blackBox; fontset = subsubsection, "Arial", 10, L0, bold, whiteBox; fontset = text, "Arial", 12, L0; fontset = smalltext, "Arial", 10, L0; fontset = input, "Courier New", 12, L0, nowordwrap, bold; fontset = output, "Courier New", 12, L0, nowordwrap; fontset = message, "Courier New", 10, L0, nowordwrap, R65280; fontset = print, "Courier New", 10, L0, nowordwrap; fontset = info, "Courier New", 10, L0, nowordwrap; fontset = postscript, "Courier New", 8, L0, nowordwrap; fontset = name, "Arial", 10, L0, nohscroll, italic, B65280; fontset = header, "Times New Roman", 10, L0, right, nohscroll; fontset = footer, "Times New Roman", 10, L0, right, nohscroll; fontset = help, "Arial", 10, L0, nohscroll; fontset = clipboard, "Arial", 12, L0, nohscroll; fontset = completions, "Arial", 12, L0, nowordwrap, nohscroll; fontset = graphics, "Courier New", 10, L0, nowordwrap, nohscroll; fontset = special1, "Arial", 12, L0, nowordwrap, nohscroll; fontset = special2, "Arial", 12, L0, center, nowordwrap, nohscroll; fontset = special3, "Arial", 12, L0, right, nowordwrap, nohscroll; fontset = special4, "Arial", 12, L0, nowordwrap, nohscroll; fontset = special5, "Arial", 12, L0, nowordwrap, nohscroll; fontset = leftheader, "Arial", 12, L0, nowordwrap, nohscroll; fontset = leftfooter, "Arial", 12, L0, nowordwrap, nohscroll; fontset = reserved1, "Courier New", 10, L0, nowordwrap, nohscroll;] :[font = subsubtitle; inactive; nohscroll; center; ] Mathematica Project on Picard Iteration 1-30-94 :[font = input; nowordwrap; ] :[font = input; startGroup; nowordwrap; ] ?NestList :[font = output; inactive; formatted; output; endGroup; nowordwrap; ] No Input Form Generated ;[o] NestList[f, expr, n] gives a list of the results of applying f to expr 0 through n times. :[font = section; inactive; ] Problem 1 :[font = text; inactive; ] (b) :[font = input; startGroup; nowordwrap; ] NestList[E^(-#^2)&, 1.0, 100] :[font = output; inactive; formatted; output; endGroup; nowordwrap; ] {1., 0.3678794411714423, 0.873423018493117, 0.4663271888497615, 0.804558944245307, 0.5234493035249298, 0.7603327037814905, 0.5609599196759167, 0.7300253411903657, 0.5868787737589396, 0.7086264960853117, 0.6052271051630286, 0.6932948863834245, 0.6183764902539214, 0.6822292842673777, 0.6278607980048144, 0.6742130084847677, 0.6347251675342744, 0.668394949561601, 0.6397026572723588, 0.6641684381772411, 0.6433156878944631, 0.6610967542393303, 0.6459398322521614, 0.6588639157843027, 0.6478463923826179, 0.6572407113150254, 0.6492318704853642, 0.6560606620646033, 0.6502388043160431, 0.6552027805468837, 0.6509706751528184, 0.6545791166333425, 0.6515026467141909, 0.6541257297515256, 0.6518893302631424, 0.6537961332951166, 0.6521704114758961, 0.6535565303996033, 0.6523747329115924, 0.6533823503843726, 0.6525232581189454, 0.6532557304616823, 0.6526312246823513, 0.6531636846934246, 0.6527097085530565, 0.6530967726712911, 0.6527667608307456, 0.653048131569901, 0.6528082339392926, 0.6530127724175295, 0.6528383821072899, 0.6529870684773119, 0.6528602978299478, 0.65296838330411, 0.6528762291238001, 0.6529548003498653, 0.6528878101375667, 0.6529449263954666, 0.6528962287847343, 0.6529377486537187, 0.6529023485981904, 0.6529325308898934, 0.6529067973089033, 0.6529287379067096, 0.6529100312359901, 0.6529259806490482, 0.6529123820936693, 0.6529239762982203, 0.6529140910166582, 0.652922519262855, 0.6529153332943192, 0.6529214600910254, 0.6529162363507315, 0.6529206901406561, 0.6529168928150134, 0.6529201304358596, 0.652917370022593, 0.6529197235661904, 0.6529177169220124, 0.6529194277979099, 0.6529179690957289, 0.6529192127932505, 0.6529181524099127, 0.6529190564985893, 0.6529182856676155, 0.6529189428823495, 0.6529183825374547, 0.6529188602905974, 0.6529184529556411, 0.6529188002516672, 0.6529185041451654, 0.6529187566071987, 0.6529185413566805, 0.6529187248804571, 0.6529185684070757, 0.6529187018171418, 0.6529185880709855, 0.6529186850515842, 0.6529186023653923, 0.6529186728640945} ;[o] {1., 0.367879, 0.873423, 0.466327, 0.804559, 0.523449, 0.760333, 0.56096, 0.730025, 0.586879, 0.708626, 0.605227, 0.693295, 0.618376, 0.682229, 0.627861, 0.674213, 0.634725, 0.668395, 0.639703, 0.664168, 0.643316, 0.661097, 0.64594, 0.658864, 0.647846, 0.657241, 0.649232, 0.656061, 0.650239, 0.655203, 0.650971, 0.654579, 0.651503, 0.654126, 0.651889, 0.653796, 0.65217, 0.653557, 0.652375, 0.653382, 0.652523, 0.653256, 0.652631, 0.653164, 0.65271, 0.653097, 0.652767, 0.653048, 0.652808, 0.653013, 0.652838, 0.652987, 0.65286, 0.652968, 0.652876, 0.652955, 0.652888, 0.652945, 0.652896, 0.652938, 0.652902, 0.652933, 0.652907, 0.652929, 0.65291, 0.652926, 0.652912, 0.652924, 0.652914, 0.652923, 0.652915, 0.652921, 0.652916, 0.652921, 0.652917, 0.65292, 0.652917, 0.65292, 0.652918, 0.652919, 0.652918, 0.652919, 0.652918, 0.652919, 0.652918, 0.652919, 0.652918, 0.652919, 0.652918, 0.652919, 0.652919, 0.652919, 0.652919, 0.652919, 0.652919, 0.652919, 0.652919, 0.652919, 0.652919, 0.652919} :[font = input; nowordwrap; ] Picard[f_, phi_, n_] := Integrate[f[x, Picard[f, phi, n-1][x]], {x, 0, #}]& /; n > 0 :[font = input; nowordwrap; ] Picard[f_, phi_, 0] := phi :[font = input; nowordwrap; ] f1[x_, y_] := x^2 + y^2 :[font = input; nowordwrap; ] p1[x_] := x :[font = input; startGroup; nowordwrap; ] example = Table[Picard[f1, p1, i][t], {i, 0, 3}] :[font = message; inactive; formatted; output; nowordwrap; ] Can't Format Cell ;[o] Get::noopen: Can't open Integrate`DefiniteIntegrate`. :[font = output; inactive; formatted; output; endGroup; nowordwrap; ] {t, (2*t^3)/3, t^3/3 + (4*t^7)/63, t^3/3 + t^7/63 + (8*t^11)/2079 + (16*t^15)/59535} ;[o] 3 3 7 3 7 11 15 2 t t 4 t t t 8 t 16 t {t, ----, -- + ----, -- + -- + ----- + ------} 3 3 63 3 63 2079 59535 :[font = input; startGroup; nowordwrap; ] Plot[Evaluate[example], {t, 0, 1}, AspectRatio->Automatic] :[font = postscript; inactive; output; BITMAP; PostScript; pictureLeft = 100; pictureTop = 0; pictureWidth = 300; pictureHeight = 200; nowordwrap; ] %! %%Creator: Mathematica %%AspectRatio: .66733 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 0.0158889 0.952381 [ [(0.2)] .21429 .01589 0 2 Msboxa [(0.4)] .40476 .01589 0 2 Msboxa [(0.6)] .59524 .01589 0 2 Msboxa [(0.8)] .78571 .01589 0 2 Msboxa [(1)] .97619 .01589 0 2 Msboxa [(0.1)] .01131 .11113 1 0 Msboxa [(0.2)] .01131 .20637 1 0 Msboxa [(0.3)] .01131 .3016 1 0 Msboxa [(0.4)] .01131 .39684 1 0 Msboxa [(0.5)] .01131 .49208 1 0 Msboxa [(0.6)] .01131 .58732 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 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.01472 m .63333 .01847 L s P p .001 w .67143 .01472 m .67143 .01847 L s P p .001 w .70952 .01472 m .70952 .01847 L s P p .001 w .74762 .01472 m .74762 .01847 L s P p .001 w .82381 .01472 m .82381 .01847 L s P p .001 w .8619 .01472 m .8619 .01847 L s P p .001 w .9 .01472 m .9 .01847 L s P p .001 w .9381 .01472 m .9381 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .10035 m .03006 .10035 L s P [(0.5)] .01131 .10035 1 0 Mshowa p .002 w .02381 .18599 m .03006 .18599 L s P [(1)] .01131 .18599 1 0 Mshowa p .002 w .02381 .27163 m .03006 .27163 L s P [(1.5)] .01131 .27163 1 0 Mshowa p .002 w .02381 .35727 m .03006 .35727 L s P [(2)] .01131 .35727 1 0 Mshowa p .002 w .02381 .44291 m .03006 .44291 L s P [(2.5)] .01131 .44291 1 0 Mshowa p .002 w .02381 .52855 m .03006 .52855 L s P [(3)] .01131 .52855 1 0 Mshowa p .002 w .02381 .61418 m .03006 .61418 L s P [(3.5)] .01131 .61418 1 0 Mshowa p .001 w .02381 .03184 m .02756 .03184 L s P p .001 w .02381 .04897 m .02756 .04897 L s P p 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.54567 m .02756 .54567 L s P p .001 w .02381 .5628 m .02756 .5628 L s P p .001 w .02381 .57993 m .02756 .57993 L s P p .001 w .02381 .59706 m .02756 .59706 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .01472 m .06349 .02929 L .10317 .04448 L .14286 .06033 L .18254 .07684 L .22222 .09406 L .2619 .11201 L .30159 .13072 L .34127 .15023 L .38095 .17057 L .42063 .19178 L .46032 .21389 L .5 .23694 L .53968 .26097 L .57937 .28602 L .61905 .31214 L .65873 .33936 L .69841 .36775 L .7381 .39735 L .77778 .4282 L .81746 .46037 L .85714 .49391 L .89683 .52887 L .93651 .56532 L .97619 .60332 L s P P % End of Graphics MathPictureEnd :[font = output; inactive; formatted; output; endGroup; nowordwrap; ] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; startGroup; nowordwrap; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; fontColorRed = 0; fontColorGreen = 0; fontColorBlue = 0; bold; fontName = "Courier New"; fontSize = 12; ] Plot[problem2[[5]], {t, 0, 1}] :[font = postscript; inactive; output; BITMAP; PostScript; pictureLeft = 100; pictureTop = 0; pictureWidth = 300; pictureHeight = 185; nowordwrap; ] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 0.0147151 0.171855 [ [(0.2)] .21429 .01472 0 2 Msboxa [(0.4)] .40476 .01472 0 2 Msboxa [(0.6)] .59524 .01472 0 2 Msboxa [(0.8)] .78571 .01472 0 2 Msboxa [(1)] .97619 .01472 0 2 Msboxa [(0.5)] .01131 .10064 1 0 Msboxa [(1)] .01131 .18657 1 0 Msboxa [(1.5)] .01131 .2725 1 0 Msboxa [(2)] .01131 .35843 1 0 Msboxa [(2.5)] .01131 .44435 1 0 Msboxa [(3)] .01131 .53028 1 0 Msboxa [(3.5)] .01131 .61621 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .01472 m .21429 .02097 L s P [(0.2)] .21429 .01472 0 2 Mshowa p .002 w .40476 .01472 m .40476 .02097 L s P [(0.4)] .40476 .01472 0 2 Mshowa p .002 w .59524 .01472 m .59524 .02097 L s P [(0.6)] .59524 .01472 0 2 Mshowa p .002 w .78571 .01472 m .78571 .02097 L s P [(0.8)] .78571 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(1)] .97619 .01472 0 2 Mshowa p .001 w .0619 .01472 m .0619 .01847 L s P p .001 w .1 .01472 m .1 .01847 L s P p .001 w .1381 .01472 m .1381 .01847 L s P p .001 w .17619 .01472 m .17619 .01847 L s P p .001 w .25238 .01472 m .25238 .01847 L s P p .001 w .29048 .01472 m .29048 .01847 L s P p .001 w .32857 .01472 m .32857 .01847 L s P p .001 w .36667 .01472 m .36667 .01847 L s P p .001 w .44286 .01472 m .44286 .01847 L s P p .001 w .48095 .01472 m .48095 .01847 L s P p .001 w .51905 .01472 m .51905 .01847 L s P p .001 w .55714 .01472 m .55714 .01847 L s P p .001 w .63333 .01472 m .63333 .01847 L s P p .001 w .67143 .01472 m .67143 .01847 L s P p .001 w .70952 .01472 m .70952 .01847 L s P p .001 w .74762 .01472 m .74762 .01847 L s P p .001 w .82381 .01472 m .82381 .01847 L s P p .001 w .8619 .01472 m .8619 .01847 L s P p .001 w .9 .01472 m .9 .01847 L s P p .001 w .9381 .01472 m .9381 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .10064 m .03006 .10064 L s P [(0.5)] .01131 .10064 1 0 Mshowa p .002 w .02381 .18657 m .03006 .18657 L s P [(1)] .01131 .18657 1 0 Mshowa p .002 w .02381 .2725 m .03006 .2725 L s P [(1.5)] .01131 .2725 1 0 Mshowa p .002 w .02381 .35843 m .03006 .35843 L s P [(2)] .01131 .35843 1 0 Mshowa p .002 w .02381 .44435 m .03006 .44435 L s P [(2.5)] .01131 .44435 1 0 Mshowa p .002 w .02381 .53028 m .03006 .53028 L s P [(3)] .01131 .53028 1 0 Mshowa p .002 w .02381 .61621 m .03006 .61621 L s P [(3.5)] .01131 .61621 1 0 Mshowa p .001 w .02381 .0319 m .02756 .0319 L s P p .001 w .02381 .04909 m .02756 .04909 L s P p .001 w .02381 .06627 m .02756 .06627 L s P p .001 w .02381 .08346 m .02756 .08346 L s P p .001 w .02381 .11783 m .02756 .11783 L s P p .001 w .02381 .13501 m .02756 .13501 L s P p .001 w .02381 .1522 m .02756 .1522 L s P p .001 w .02381 .16938 m .02756 .16938 L s P p .001 w .02381 .20376 m .02756 .20376 L s P p .001 w .02381 .22094 m .02756 .22094 L s P p .001 w .02381 .23813 m .02756 .23813 L s P p .001 w .02381 .25531 m .02756 .25531 L s P p .001 w .02381 .28968 m .02756 .28968 L s P p .001 w .02381 .30687 m .02756 .30687 L s P p .001 w .02381 .32405 m .02756 .32405 L s P p .001 w .02381 .34124 m .02756 .34124 L s P p .001 w .02381 .37561 m .02756 .37561 L s P p .001 w .02381 .3928 m .02756 .3928 L s P p .001 w .02381 .40998 m .02756 .40998 L s P p .001 w .02381 .42717 m .02756 .42717 L s P p .001 w .02381 .46154 m .02756 .46154 L s P p .001 w .02381 .47872 m .02756 .47872 L s P p .001 w .02381 .49591 m .02756 .49591 L s P p .001 w .02381 .51309 m .02756 .51309 L s P p .001 w .02381 .54747 m .02756 .54747 L s P p .001 w .02381 .56465 m .02756 .56465 L s P p .001 w .02381 .58184 m .02756 .58184 L s P p .001 w .02381 .59902 m .02756 .59902 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .01472 m .06349 .02934 L .10317 .04458 L .14286 .06048 L .18254 .07705 L .22222 .09433 L .2619 .11234 L .30159 .13111 L .34127 .15068 L .38095 .17109 L .42063 .19236 L .46032 .21453 L .5 .23763 L .53968 .26172 L .57937 .28682 L .61905 .31297 L .65873 .34023 L .69841 .36862 L .7381 .3982 L .77778 .42902 L .81746 .46112 L .85714 .49454 L .89683 .52935 L .93651 .56559 L .97619 .60332 L s P P % End of Graphics MathPictureEnd :[font = output; inactive; formatted; output; endGroup; nowordwrap; ] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; nowordwrap; ] f3[x_, y_] := x^2 y^2 - x y + 1 :[font = input; nowordwrap; ] p3[x_] := x :[font = input; startGroup; nowordwrap; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; fontColorRed = 0; fontColorGreen = 0; fontColorBlue = 0; bold; fontName = "Courier New"; fontSize = 12; ] problem3 = Table[Picard[f3, p3, i][t], {i, 0, 4}] :[font = output; inactive; formatted; output; endGroup; nowordwrap; ] {t, t - t^3/3 + t^5/5, t - t^3/3 + (4*t^5)/15 - (13*t^7)/105 + (23*t^9)/405 - (2*t^11)/165 + t^13/325, t - t^3/3 + (4*t^5)/15 - (2*t^7)/15 + (242*t^9)/2835 - (1367*t^11)/31185 + (43558*t^13)/2027025 - (159581*t^15)/18243225 + (2547731*t^17)/723647925 - (5492*t^19)/4621617 + (6458857*t^21)/17239847625 - (5002*t^23)/53794125 + (7906*t^25)/398165625 - (4*t^27)/1447875 + t^29/3063125, t - t^3/3 + (4*t^5)/15 - (2*t^7)/15 + (7*t^9)/81 - (1502*t^11)/31185 + (58403*t^13)/2027025 - (1442296*t^15)/91216125 + (2631623*t^17)/310134825 - (176895227*t^19)/41247931725 + (82595033288*t^21)/38979295480125 - (3655731151*t^23)/3659280800175 + (16109841853591*t^25)/35220577701684375 - (571155885403904*t^27)/2852866793836434375 + (23565693740489038*t^29)/278841313664235196875 - (98157780141948643*t^31)/2881360241197097034375 + (333298374520863343*t^33)/25351796987227914328125 - (576188825962418206*t^35)/119775018823955573203125 + (14555999042769170089*t^37)/8723403739399459063078125 - (79071659843655233726*t^39)/145341852127777632843140625 + (5640685276396411263398*t^41)/ 33869620497981685123660078125 - (12670040861971991038*t^43)/267081181260446077446796875 + (889649562173370491719*t^45)/71042094962456825708673046875 - (2293621419223578286*t^47)/759007450000616527069734375 + (14321575672566625706*t^49)/21666702030159443414579296875 - (7394364367919752*t^51)/57675338145943505096484375 + (232922410449782*t^53)/10710011001250195400390625 - (458720252*t^55)/148040790673166015625 + (2034772*t^57)/5631036696720703125 - (8*t^59)/261666304453125 + t^61/572346820703125} ;[o] 3 5 3 5 7 9 11 13 t t t 4 t 13 t 23 t 2 t t {t, t - -- + --, t - -- + ---- - ----- + ----- - ----- + ---, 3 5 3 15 105 405 165 325 3 5 7 9 11 13 t 4 t 2 t 242 t 1367 t 43558 t t - -- + ---- - ---- + ------ - -------- + --------- - 3 15 15 2835 31185 2027025 15 17 19 21 159581 t 2547731 t 5492 t 6458857 t ---------- + ----------- - -------- + ----------- - 18243225 723647925 4621617 17239847625 23 25 27 29 5002 t 7906 t 4 t t -------- + --------- - ------- + -------, 53794125 398165625 1447875 3063125 3 5 7 9 11 13 t 4 t 2 t 7 t 1502 t 58403 t t - -- + ---- - ---- + ---- - -------- + --------- - 3 15 15 81 31185 2027025 15 17 19 21 1442296 t 2631623 t 176895227 t 82595033288 t ----------- + ----------- - ------------- + --------------- - 91216125 310134825 41247931725 38979295480125 23 25 27 3655731151 t 16109841853591 t 571155885403904 t -------------- + ------------------ - ------------------- + 3659280800175 35220577701684375 2852866793836434375 29 31 23565693740489038 t 98157780141948643 t --------------------- - ---------------------- + 278841313664235196875 2881360241197097034375 33 35 333298374520863343 t 576188825962418206 t ----------------------- - ------------------------ + 25351796987227914328125 119775018823955573203125 37 39 14555999042769170089 t 79071659843655233726 t ------------------------- - --------------------------- + 8723403739399459063078125 145341852127777632843140625 41 43 5640685276396411263398 t 12670040861971991038 t ----------------------------- - --------------------------- + 33869620497981685123660078125 267081181260446077446796875 45 47 889649562173370491719 t 2293621419223578286 t ----------------------------- - --------------------------- + 71042094962456825708673046875 759007450000616527069734375 49 51 14321575672566625706 t 7394364367919752 t ----------------------------- - -------------------------- + 21666702030159443414579296875 57675338145943505096484375 53 55 232922410449782 t 458720252 t -------------------------- - --------------------- + 10710011001250195400390625 148040790673166015625 57 59 61 2034772 t 8 t t ------------------- - --------------- + ---------------} 5631036696720703125 261666304453125 572346820703125 :[font = input; startGroup; nowordwrap; ] Plot[problem3[[2]], {t, 0, 1}] :[font = postscript; inactive; output; BITMAP; PostScript; pictureLeft = 100; pictureTop = 0; pictureWidth = 300; pictureHeight = 185; nowordwrap; ] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 0.0147151 0.679158 [ [(0.2)] .21429 .01472 0 2 Msboxa [(0.4)] .40476 .01472 0 2 Msboxa [(0.6)] .59524 .01472 0 2 Msboxa [(0.8)] .78571 .01472 0 2 Msboxa [(1)] .97619 .01472 0 2 Msboxa [(0.2)] .01131 .15055 1 0 Msboxa [(0.4)] .01131 .28638 1 0 Msboxa [(0.6)] .01131 .42221 1 0 Msboxa [(0.8)] .01131 .55804 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .01472 m .21429 .02097 L s P [(0.2)] .21429 .01472 0 2 Mshowa p .002 w .40476 .01472 m .40476 .02097 L s P [(0.4)] .40476 .01472 0 2 Mshowa p .002 w .59524 .01472 m .59524 .02097 L s P [(0.6)] .59524 .01472 0 2 Mshowa p .002 w .78571 .01472 m .78571 .02097 L s P [(0.8)] .78571 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(1)] .97619 .01472 0 2 Mshowa p .001 w .0619 .01472 m .0619 .01847 L s P p .001 w .1 .01472 m .1 .01847 L s P p .001 w .1381 .01472 m .1381 .01847 L s P p .001 w .17619 .01472 m .17619 .01847 L s P p .001 w .25238 .01472 m .25238 .01847 L s P p .001 w .29048 .01472 m .29048 .01847 L s P p .001 w .32857 .01472 m .32857 .01847 L s P p .001 w .36667 .01472 m .36667 .01847 L s P p .001 w .44286 .01472 m .44286 .01847 L s P p .001 w .48095 .01472 m .48095 .01847 L s P p .001 w .51905 .01472 m .51905 .01847 L s P p .001 w .55714 .01472 m .55714 .01847 L s P p .001 w .63333 .01472 m .63333 .01847 L s P p .001 w .67143 .01472 m .67143 .01847 L s P p .001 w .70952 .01472 m .70952 .01847 L s P p .001 w .74762 .01472 m .74762 .01847 L s P p .001 w .82381 .01472 m .82381 .01847 L s P p .001 w .8619 .01472 m .8619 .01847 L s P p .001 w .9 .01472 m .9 .01847 L s P p .001 w .9381 .01472 m .9381 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .15055 m .03006 .15055 L s P [(0.2)] .01131 .15055 1 0 Mshowa p .002 w .02381 .28638 m .03006 .28638 L s P [(0.4)] .01131 .28638 1 0 Mshowa p .002 w .02381 .42221 m .03006 .42221 L s P [(0.6)] .01131 .42221 1 0 Mshowa p .002 w .02381 .55804 m .03006 .55804 L s P [(0.8)] .01131 .55804 1 0 Mshowa p .001 w .02381 .04188 m .02756 .04188 L s P p .001 w .02381 .06905 m .02756 .06905 L s P p .001 w .02381 .09621 m .02756 .09621 L s P p .001 w .02381 .12338 m .02756 .12338 L s P p .001 w .02381 .17771 m .02756 .17771 L s P p .001 w .02381 .20488 m .02756 .20488 L s P p .001 w .02381 .23205 m .02756 .23205 L s P p .001 w .02381 .25921 m .02756 .25921 L s P p .001 w .02381 .31354 m .02756 .31354 L s P p .001 w .02381 .34071 m .02756 .34071 L s P p .001 w .02381 .36788 m .02756 .36788 L s P p .001 w .02381 .39504 m .02756 .39504 L s P p .001 w .02381 .44938 m .02756 .44938 L s P p .001 w .02381 .47654 m .02756 .47654 L s P p .001 w .02381 .50371 m .02756 .50371 L s P p .001 w .02381 .53088 m .02756 .53088 L s P p .001 w .02381 .58521 m .02756 .58521 L s P p .001 w .02381 .61237 m .02756 .61237 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .01472 m .06349 .043 L .10317 .07118 L .14286 .09917 L .18254 .12688 L .22222 .15421 L .2619 .1811 L .30159 .20747 L .34127 .23328 L .38095 .25847 L .42063 .28303 L .46032 .30695 L .5 .33024 L .53968 .35295 L .57937 .37513 L .61905 .39687 L .65873 .4183 L .69841 .43955 L .7381 .46081 L .77778 .4823 L .81746 .50426 L .85714 .52699 L .89683 .55082 L .93651 .57612 L .97619 .60332 L s P P % End of Graphics MathPictureEnd :[font = output; inactive; formatted; output; endGroup; nowordwrap; ] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; startGroup; nowordwrap; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; fontColorRed = 0; fontColorGreen = 0; fontColorBlue = 0; bold; fontName = "Courier New"; fontSize = 12; ] problem3[[2]] /. t->1.0 :[font = output; inactive; formatted; output; endGroup; nowordwrap; ] 0.866666666666667 ;[o] 0.866667 ^*)