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fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times";paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] (C) Copyright 1994 D.C.M. Burbulla and C.T.J. Dodson Notebook to support the book Self-Tutor for Computer Calculus ;[s] 3:0,0;75,1;104,2;143,-1; 3:1,0,-9 -*-times-bold-r-normal-*-*-120-75-75-*-*-*-*,times,1,12,0,0,0;1,0,-9 -*-times-bold-i-normal-*-*-170-75-75-*-*-*-*,times,3,17,0,0,0;1,0,-9 -*-times-bold-r-normal-*-*-240-75-75-*-*-*-*,times,1,24,0,0,0; :[font = subtitle; inactive; preserveAspect] Using Mathematica 2 ;[s] 2:0,0;6,1;20,-1; 2:1,0,-9 -*-times-bold-r-normal-*-*-230-75-75-*-*-*-*,times,1,23,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-230-75-75-*-*-*-*,times,2,23,0,0,0; :[font = subsubtitle; inactive; preserveAspect; fontName = "Times"] by D. C. M. Burbulla and C. T. J. Dodson, University of Toronto Published by: Prentice-Hall Canada (English) ISBN 0130152803 and Prentice-Hall Japan (Japanese) ISBN4-8101-8558-3 C3055 P2880E ;[s] 5:0,0;128,1;129,2;145,3;192,4;222,-1; 5:1,0,-9 -*-times-medium-i-normal-*-*-120-75-75-*-*-*-*,times,2,12,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-110-75-75-*-*-*-*,times,2,11,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-110-75-75-*-*-*-*,courier,2,11,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-120-75-75-*-*-*-*,times,2,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-110-75-75-*-*-*-*,courier,2,11,0,0,0; :[font = section; inactive; Cclosed; preserveAspect; startGroup] Antiderivatives :[font = subsection; inactive; Cclosed; preserveAspect; fontName = "Times"; startGroup] Chap6.m :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] anti/: anti[tableofvalues_][1] = 0 anti/: anti[tableofvalues_][n_] := anti[tableofvalues][n] = anti[tableofvalues][n - 1] + ( (First[Transpose[tableofvalues]][[n]] - First[Transpose[tableofvalues]][[n - 1]])* (Last[Transpose[tableofvalues]][[n]] + Last[Transpose[tableofvalues]][[n - 1]]))/2 antiderivative/: antiderivative[tableofvalues_, k_] := Block[{i}, Table[{First[Transpose[tableofvalues]][[i]], anti[tableofvalues][i] + k}, {i, Length[tableofvalues]}]] viewantiderivative/: viewantiderivative[f_, {a_, b_}, initialvalue_] := (one = Block[{x}, Plot[f[x], {x, a, b}, DisplayFunction -> Identity]]; two = Nest[First, InputForm[one], 5]; Block[{x}, ListPlot[antiderivative[two, initialvalue], PlotJoined -> True, PlotLabel -> {"an approximate antiderivative of", f[x]}]]) viewantiderivative::usage = "viewantiderivative[f, {a, b}, initialvalue] plots the graph of an approximate antiderivative F of f on the interval {a, b} with initial condition F[a] = initialvalue. The approximation is based on applying the Mean Value Theorem to Mathematica's InputForm of the Plot of f on {a, b}." antiderivativefamily/: antiderivativefamily[f_, {a_, b_, k_:1}] := (one = Block[{x}, Plot[f[x], {x, a, b}, AspectRatio -> Automatic, DisplayFunction -> Identity]]; two = Block[{x}, Integrate[f[x], x] /. x -> a]; three = Block[{x, i}, Plot[Release[Table[Integrate[ f[x], x] + i, {i, two - 5, two + 5, k}]], {x, a, b}, DisplayFunction -> Identity]]; Block[{x}, Show[one, three, PlotLabel -> {"antiderivatives of", f[x]}, DisplayFunction -> $DisplayFunction]]) antiderivativefamily::usage = "antiderivativefamily[f, {a, b, k}] plots F[x] + c, and f[x], on the interval {a, b}, where F[x] = Integrate[f[x], x], for a few values of c differing by k. The default value of k is 1." :[font = output; output; inactive; preserveAspect] 0 ;[o] 0 :[font = output; output; inactive; preserveAspect] "viewantiderivative[f, {a, b}, initialvalue] plots\ the graph of an approximate antiderivative F of f\ on the interval {a, b} with initial condition F[a]\ = initialvalue. The approximation is based on\ applying the Mean Value Theorem to Mathematica's\ InputForm of the Plot of f on {a, b}." ;[o] viewantiderivative[f, {a, b}, initialvalue] plots the\ graph of an approximate antiderivative F of f on\ the interval {a, b} with initial condition F[a] =\ initialvalue. The approximation is based on\ applying the Mean Value Theorem to Mathematica's\ InputForm of the Plot of f on {a, b}. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] "antiderivativefamily[f, {a, b, k}] plots F[x] + c,\ and f[x], on the interval {a, b}, where F[x] =\ Integrate[f[x], x], for a few values of c differing\ by k. The default value of k is 1." ;[o] antiderivativefamily[f, {a, b, k}] plots F[x] + c,\ and f[x], on the interval {a, b}, where F[x] =\ Integrate[f[x], x], for a few values of c differing\ by k. The default value of k is 1. :[font = subsection; inactive; preserveAspect; fontName = "Times"] Antidifferentiation :[font = subsection; inactive; Cclosed; preserveAspect; fontName = "Times"; startGroup] Constructing Antiderivatives :[font = text; inactive; preserveAspect] In this section we shall consider briefly the problem of plotting antiderivatives for given functions. This will be particularly useful in the cases where Mathematica cannot integrate the given function. We will not bother finding a specific equation for the antiderivative; we will simply try to plot its graph. This is what we do all the time with paper and pencil when we use information about f ' to sketch the graph of f . Of course, we will need to have an initial condition to specify one particular antiderivative. Example 1. Since any two antiderivatives of f will differ by at most a constant, the family of antiderivatives of f must consist of a family of curves of the same shape, but shifted vertically from each other. There is a defined function, antiderivativefamily, in the package Chap6.m that will allow you to see this. ;[s] 19:0,0;155,1;166,2;242,3;250,4;398,5;399,6;425,7;426,8;525,9;535,10;571,11;572,12;643,13;644,14;770,15;792,16;808,17;815,18;849,-1; 19:1,0,-9 -*-times-medium-r-normal-*-*-120-75-75-*-*-*-*,times,0,12,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-120-75-75-*-*-*-*,times,2,12,0,0,0;1,0,-9 -*-times-medium-r-normal-*-*-120-75-75-*-*-*-*,times,0,12,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-120-75-75-*-*-*-*,times,2,12,0,0,0;1,0,-9 -*-times-medium-r-normal-*-*-120-75-75-*-*-*-*,times,0,12,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-120-75-75-*-*-*-*,times,2,12,0,0,0;1,0,-9 -*-times-medium-r-normal-*-*-120-75-75-*-*-*-*,times,0,12,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-120-75-75-*-*-*-*,times,2,12,0,0,0;1,0,-9 -*-times-medium-r-normal-*-*-120-75-75-*-*-*-*,times,0,12,0,0,0;1,0,-9 -*-times-bold-r-normal-*-*-120-75-75-*-*-*-*,times,1,12,0,0,0;1,0,-9 -*-times-medium-r-normal-*-*-120-75-75-*-*-*-*,times,0,12,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-120-75-75-*-*-*-*,times,2,12,0,0,0;1,0,-9 -*-times-medium-r-normal-*-*-120-75-75-*-*-*-*,times,0,12,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-120-75-75-*-*-*-*,times,2,12,0,0,0;1,0,-9 -*-times-medium-r-normal-*-*-120-75-75-*-*-*-*,times,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-times-medium-r-normal-*-*-120-75-75-*-*-*-*,times,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-times-medium-r-normal-*-*-120-75-75-*-*-*-*,times,0,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?antiderivativefamily :[font = print; inactive; preserveAspect; endGroup] antiderivativefamily[f, {a, b, k}] plots F[x] + c, and f[x], on the interval {a, b}, where F[x] = Integrate[f[x], x], for a few values of c differing by k. 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.10317 .40411 L .14286 .38002 L .18254 .34863 L .22222 .31207 L .2619 .27284 L .30159 .23361 L .34127 .19705 L .38095 .16566 L .40079 .15258 L .42063 .14157 L .44048 .1328 L .4504 .12931 L .46032 .12643 L .47024 .12417 L .4752 .12329 L .48016 .12256 L .48512 .12199 L .4876 .12177 L .49008 .12159 L .49256 .12144 L .4938 .12139 L .49504 .12134 L .49628 .12131 L .49752 .12128 L .49876 .12127 L .5 .12126 L .50124 .12127 L .50248 .12128 L .50372 .12131 L .50496 .12134 L .5062 .12139 L .50744 .12144 L .50992 .12159 L .5124 .12177 L .51488 .12199 L .51984 .12256 L Mistroke .5248 .12329 L .52976 .12417 L .53968 .12643 L .55952 .1328 L .57937 .14157 L .61905 .16566 L .65873 .19705 L .69841 .23361 L .7381 .27284 L .77778 .31207 L .81746 .34863 L .85714 .38002 L .87698 .39309 L .89683 .40411 L .91667 .41288 L .93651 .41925 L .94643 .4215 L .95139 .42239 L .95635 .42312 L .96131 .42368 L .96379 .42391 L .96627 .42409 L .96875 .42423 L .96999 .42429 L .97123 .42433 L .97247 .42437 L .97371 .42439 L .97495 .42441 L .97619 .42441 L Mfstroke P P p p .004 w .02381 .5002 m .02505 .5002 L .02629 .50018 L .02753 .50016 L .02877 .50012 L .03125 .50002 L .03373 .49988 L .03621 .49969 L .03869 .49947 L .04365 .4989 L .04861 .49818 L .05357 .49729 L .06349 .49504 L .08333 .48866 L .10317 .47989 L .14286 .45581 L .18254 .42441 L .22222 .38786 L .2619 .34863 L .30159 .30939 L .34127 .27284 L .38095 .24144 L .40079 .22837 L .42063 .21736 L .44048 .20859 L .4504 .20509 L .46032 .20221 L .47024 .19996 L .4752 .19907 L .48016 .19835 L .48512 .19778 L .4876 .19756 L .49008 .19737 L .49256 .19723 L .4938 .19718 L .49504 .19713 L .49628 .19709 L .49752 .19707 L .49876 .19705 L .5 .19705 L .50124 .19705 L .50248 .19707 L .50372 .19709 L .50496 .19713 L .5062 .19718 L .50744 .19723 L .50992 .19737 L .5124 .19756 L .51488 .19778 L .51984 .19835 L Mistroke .5248 .19907 L .52976 .19996 L .53968 .20221 L .55952 .20859 L .57937 .21736 L .61905 .24144 L .65873 .27284 L .69841 .30939 L .7381 .34863 L .77778 .38786 L .81746 .42441 L .85714 .45581 L .87698 .46888 L .89683 .47989 L .91667 .48866 L .93651 .49504 L .94643 .49729 L .95139 .49818 L .95635 .4989 L .96131 .49947 L .96379 .49969 L .96627 .49988 L .96875 .50002 L .96999 .50007 L .97123 .50012 L .97247 .50016 L .97371 .50018 L .97495 .5002 L .97619 .5002 L Mfstroke P P p p .004 w .02381 .57599 m .02505 .57598 L .02629 .57597 L .02753 .57594 L .02877 .57591 L .03125 .57581 L .03373 .57566 L .03621 .57548 L .03869 .57526 L .04365 .57469 L .04861 .57396 L .05357 .57308 L .06349 .57082 L .08333 .56445 L .10317 .55568 L .14286 .53159 L .18254 .5002 L .22222 .46364 L .2619 .42441 L .30159 .38518 L .34127 .34863 L .38095 .31723 L .40079 .30416 L .42063 .29314 L .44048 .28438 L .4504 .28088 L .46032 .278 L .47024 .27575 L .4752 .27486 L .48016 .27413 L .48512 .27357 L .4876 .27334 L .49008 .27316 L .49256 .27302 L .4938 .27296 L .49504 .27292 L .49628 .27288 L .49752 .27286 L .49876 .27284 L .5 .27284 L .50124 .27284 L .50248 .27286 L .50372 .27288 L .50496 .27292 L .5062 .27296 L .50744 .27302 L .50992 .27316 L .5124 .27334 L .51488 .27357 L .51984 .27413 L Mistroke .5248 .27486 L .52976 .27575 L .53968 .278 L .55952 .28438 L .57937 .29314 L .61905 .31723 L .65873 .34863 L .69841 .38518 L .7381 .42441 L .77778 .46364 L .81746 .5002 L .85714 .53159 L .87698 .54467 L .89683 .55568 L .91667 .56445 L .93651 .57082 L .94643 .57308 L .95139 .57396 L .95635 .57469 L .96131 .57526 L .96379 .57548 L .96627 .57566 L .96875 .57581 L .96999 .57586 L .97123 .57591 L .97247 .57594 L .97371 .57597 L .97495 .57598 L .97619 .57599 L Mfstroke P P p p .004 w .02381 .65178 m .02505 .65177 L .02629 .65176 L .02753 .65173 L .02877 .6517 L .03125 .65159 L .03373 .65145 L .03621 .65127 L .03869 .65105 L .04365 .65048 L .04861 .64975 L .05357 .64886 L .06349 .64661 L .08333 .64024 L .10317 .63147 L .14286 .60738 L .18254 .57599 L .22222 .53943 L .2619 .5002 L .30159 .46097 L .34127 .42441 L .38095 .39302 L .40079 .37995 L .42063 .36893 L .44048 .36016 L .4504 .35667 L .46032 .35379 L .47024 .35154 L .4752 .35065 L .48016 .34992 L .48512 .34935 L .4876 .34913 L .49008 .34895 L .49256 .34881 L .4938 .34875 L .49504 .34871 L .49628 .34867 L .49752 .34865 L .49876 .34863 L .5 .34863 L .50124 .34863 L .50248 .34865 L .50372 .34867 L .50496 .34871 L .5062 .34875 L .50744 .34881 L .50992 .34895 L .5124 .34913 L .51488 .34935 L .51984 .34992 L Mistroke .5248 .35065 L .52976 .35154 L .53968 .35379 L .55952 .36016 L .57937 .36893 L .61905 .39302 L .65873 .42441 L .69841 .46097 L .7381 .5002 L .77778 .53943 L .81746 .57599 L .85714 .60738 L .87698 .62045 L .89683 .63147 L .91667 .64024 L .93651 .64661 L .94643 .64886 L .95139 .64975 L .95635 .65048 L .96131 .65105 L .96379 .65127 L .96627 .65145 L .96875 .65159 L .96999 .65165 L .97123 .6517 L .97247 .65173 L .97371 .65176 L .97495 .65177 L .97619 .65178 L Mfstroke P P p p .004 w .02381 .72757 m .02505 .72756 L .02629 .72755 L .02753 .72752 L .02877 .72748 L .03125 .72738 L .03373 .72724 L .03621 .72706 L .03869 .72684 L .04365 .72627 L .04861 .72554 L .05357 .72465 L .06349 .7224 L .08333 .71603 L .10317 .70726 L .14286 .68317 L .18254 .65178 L .22222 .61522 L .2619 .57599 L .30159 .53676 L .34127 .5002 L .38095 .46881 L .40079 .45574 L .42063 .44472 L .44048 .43595 L .4504 .43246 L .46032 .42958 L .47024 .42733 L .4752 .42644 L .48016 .42571 L .48512 .42514 L .4876 .42492 L .49008 .42474 L .49256 .4246 L .4938 .42454 L .49504 .42449 L .49628 .42446 L .49752 .42443 L .49876 .42442 L .5 .42441 L .50124 .42442 L .50248 .42443 L .50372 .42446 L .50496 .42449 L .5062 .42454 L .50744 .4246 L .50992 .42474 L .5124 .42492 L .51488 .42514 L .51984 .42571 L Mistroke .5248 .42644 L .52976 .42733 L .53968 .42958 L .55952 .43595 L .57937 .44472 L .61905 .46881 L .65873 .5002 L .69841 .53676 L .7381 .57599 L .77778 .61522 L .81746 .65178 L .85714 .68317 L .87698 .69624 L .89683 .70726 L .91667 .71603 L .93651 .7224 L .94643 .72465 L .95139 .72554 L .95635 .72627 L .96131 .72684 L .96379 .72706 L .96627 .72724 L .96875 .72738 L .96999 .72744 L .97123 .72748 L .97247 .72752 L .97371 .72755 L .97495 .72756 L .97619 .72757 L Mfstroke P P p p .004 w .02381 .80335 m .02505 .80335 L .02629 .80333 L .02753 .80331 L .02877 .80327 L .03125 .80317 L .03373 .80303 L .03621 .80285 L .03869 .80262 L .04365 .80206 L .04861 .80133 L .05357 .80044 L .06349 .79819 L .08333 .79182 L .10317 .78305 L .14286 .75896 L .18254 .72757 L .22222 .69101 L .2619 .65178 L .30159 .61255 L .34127 .57599 L .38095 .5446 L .40079 .53152 L .42063 .52051 L .44048 .51174 L .4504 .50825 L .46032 .50537 L .47024 .50311 L .4752 .50223 L .48016 .5015 L .48512 .50093 L .4876 .50071 L .49008 .50053 L .49256 .50038 L .4938 .50033 L .49504 .50028 L .49628 .50025 L .49752 .50022 L .49876 .50021 L .5 .5002 L .50124 .50021 L .50248 .50022 L .50372 .50025 L .50496 .50028 L .5062 .50033 L .50744 .50038 L .50992 .50053 L .5124 .50071 L .51488 .50093 L .51984 .5015 L Mistroke .5248 .50223 L .52976 .50311 L .53968 .50537 L .55952 .51174 L .57937 .52051 L .61905 .5446 L .65873 .57599 L .69841 .61255 L .7381 .65178 L .77778 .69101 L .81746 .72757 L .85714 .75896 L .87698 .77203 L .89683 .78305 L .91667 .79182 L .93651 .79819 L .94643 .80044 L .95139 .80133 L .95635 .80206 L .96131 .80262 L .96379 .80285 L .96627 .80303 L .96875 .80317 L .96999 .80323 L .97123 .80327 L .97247 .80331 L .97371 .80333 L .97495 .80335 L .97619 .80335 L Mfstroke P P p p .004 w .02381 .87914 m .02505 .87914 L .02629 .87912 L .02753 .8791 L .02877 .87906 L .03125 .87896 L .03373 .87882 L .03621 .87863 L .03869 .87841 L .04365 .87784 L .04861 .87712 L .05357 .87623 L .06349 .87398 L .08333 .8676 L .10317 .85883 L .14286 .83475 L .18254 .80335 L .22222 .7668 L .2619 .72757 L .30159 .68833 L .34127 .65178 L .38095 .62038 L .40079 .60731 L .42063 .5963 L .44048 .58753 L .4504 .58403 L .46032 .58115 L .47024 .5789 L .4752 .57801 L .48016 .57729 L .48512 .57672 L .4876 .5765 L .49008 .57631 L .49256 .57617 L .4938 .57612 L .49504 .57607 L .49628 .57603 L .49752 .57601 L .49876 .57599 L .5 .57599 L .50124 .57599 L .50248 .57601 L .50372 .57603 L .50496 .57607 L .5062 .57612 L .50744 .57617 L .50992 .57631 L .5124 .5765 L .51488 .57672 L .51984 .57729 L Mistroke .5248 .57801 L .52976 .5789 L .53968 .58115 L .55952 .58753 L .57937 .5963 L .61905 .62038 L .65873 .65178 L .69841 .68833 L .7381 .72757 L .77778 .7668 L .81746 .80335 L .85714 .83475 L .87698 .84782 L .89683 .85883 L .91667 .8676 L .93651 .87398 L .94643 .87623 L .95139 .87712 L .95635 .87784 L .96131 .87841 L .96379 .87863 L .96627 .87882 L .96875 .87896 L .96999 .87901 L .97123 .87906 L .97247 .8791 L .97371 .87912 L .97495 .87914 L .97619 .87914 L Mfstroke P P p p .004 w .02381 .95493 m .02505 .95492 L .02629 .95491 L .02753 .95488 L .02877 .95485 L .03125 .95475 L .03373 .95461 L .03621 .95442 L .03869 .9542 L .04365 .95363 L .04861 .95291 L .05357 .95202 L .06349 .94976 L .08333 .94339 L .10317 .93462 L .14286 .91053 L .18254 .87914 L .22222 .84258 L .2619 .80335 L .30159 .76412 L .34127 .72757 L .38095 .69617 L .40079 .6831 L .42063 .67208 L .44048 .66332 L .4504 .65982 L .46032 .65694 L .47024 .65469 L .4752 .6538 L .48016 .65307 L .48512 .65251 L .4876 .65228 L .49008 .6521 L .49256 .65196 L .4938 .6519 L .49504 .65186 L .49628 .65182 L .49752 .6518 L .49876 .65178 L .5 .65178 L .50124 .65178 L .50248 .6518 L .50372 .65182 L .50496 .65186 L .5062 .6519 L .50744 .65196 L .50992 .6521 L .5124 .65228 L .51488 .65251 L .51984 .65307 L Mistroke .5248 .6538 L .52976 .65469 L .53968 .65694 L .55952 .66332 L .57937 .67208 L .61905 .69617 L .65873 .72757 L .69841 .76412 L .7381 .80335 L .77778 .84258 L .81746 .87914 L .85714 .91053 L .87698 .92361 L .89683 .93462 L .91667 .94339 L .93651 .94976 L .94643 .95202 L .95139 .95291 L .95635 .95363 L .96131 .9542 L .96379 .95442 L .96627 .95461 L .96875 .95475 L .96999 .9548 L .97123 .95485 L .97247 .95488 L .97371 .95491 L .97495 .95492 L .97619 .95493 L Mfstroke P P p p .004 w .02381 1.03072 m .02505 1.03071 L .02629 1.0307 L .02753 1.03067 L .02877 1.03064 L .03125 1.03054 L .03373 1.03039 L .03621 1.03021 L .03869 1.02999 L .04365 1.02942 L .04861 1.02869 L .05357 1.02781 L .06349 1.02555 L .08333 1.01918 L .10317 1.01041 L .14286 .98632 L .18254 .95493 L .22222 .91837 L .2619 .87914 L .30159 .83991 L .34127 .80335 L .38095 .77196 L .40079 .75889 L .42063 .74787 L .44048 .7391 L .4504 .73561 L .46032 .73273 L .47024 .73048 L .4752 .72959 L .48016 .72886 L .48512 .7283 L .4876 .72807 L .49008 .72789 L .49256 .72775 L .4938 .72769 L .49504 .72765 L .49628 .72761 L .49752 .72759 L .49876 .72757 L .5 .72757 L .50124 .72757 L .50248 .72759 L .50372 .72761 L .50496 .72765 L .5062 .72769 L .50744 .72775 L .50992 .72789 L .5124 .72807 L .51488 .7283 L .51984 .72886 L Mistroke .5248 .72959 L .52976 .73048 L .53968 .73273 L .55952 .7391 L .57937 .74787 L .61905 .77196 L .65873 .80335 L .69841 .83991 L .7381 .87914 L .77778 .91837 L .81746 .95493 L .85714 .98632 L .87698 .9994 L .89683 1.01041 L .91667 1.01918 L .93651 1.02555 L .94643 1.02781 L .95139 1.02869 L .95635 1.02942 L .96131 1.02999 L .96379 1.03021 L .96627 1.03039 L .96875 1.03054 L .96999 1.03059 L .97123 1.03064 L .97247 1.03067 L .97371 1.0307 L .97495 1.03071 L .97619 1.03072 L Mfstroke P P p p .004 w .02381 1.10651 m .02505 1.1065 L .02629 1.10649 L .02753 1.10646 L .02877 1.10642 L .03125 1.10632 L .03373 1.10618 L .03621 1.106 L .03869 1.10578 L .04365 1.10521 L .04861 1.10448 L .05357 1.10359 L .06349 1.10134 L .08333 1.09497 L .10317 1.0862 L .14286 1.06211 L .18254 1.03072 L .22222 .99416 L .2619 .95493 L .30159 .9157 L .34127 .87914 L .38095 .84775 L .40079 .83468 L .42063 .82366 L .44048 .81489 L .4504 .8114 L .46032 .80852 L .47024 .80627 L .4752 .80538 L .48016 .80465 L .48512 .80408 L .4876 .80386 L .49008 .80368 L .49256 .80354 L .4938 .80348 L .49504 .80343 L .49628 .8034 L .49752 .80337 L .49876 .80336 L .5 .80335 L .50124 .80336 L .50248 .80337 L .50372 .8034 L .50496 .80343 L .5062 .80348 L .50744 .80354 L .50992 .80368 L .5124 .80386 L .51488 .80408 L .51984 .80465 L Mistroke .5248 .80538 L .52976 .80627 L .53968 .80852 L .55952 .81489 L .57937 .82366 L .61905 .84775 L .65873 .87914 L .69841 .9157 L .7381 .95493 L .77778 .99416 L .81746 1.03072 L .85714 1.06211 L .87698 1.07518 L .89683 1.0862 L .91667 1.09497 L .93651 1.10134 L .94643 1.10359 L .95139 1.10448 L .95635 1.10521 L .96131 1.10578 L .96379 1.106 L .96627 1.10618 L .96875 1.10632 L .96999 1.10638 L .97123 1.10642 L .97247 1.10646 L .97371 1.10649 L .97495 1.1065 L .97619 1.10651 L Mfstroke P P p p .004 w .02381 1.18229 m .02505 1.18229 L .02629 1.18227 L .02753 1.18225 L .02877 1.18221 L .03125 1.18211 L .03373 1.18197 L .03621 1.18179 L .03869 1.18156 L .04365 1.181 L .04861 1.18027 L .05357 1.17938 L .06349 1.17713 L .08333 1.17076 L .10317 1.16199 L .14286 1.1379 L .18254 1.10651 L .22222 1.06995 L .2619 1.03072 L .30159 .99149 L .34127 .95493 L .38095 .92354 L .40079 .91046 L .42063 .89945 L .44048 .89068 L .4504 .88719 L .46032 .88431 L .47024 .88205 L .4752 .88117 L .48016 .88044 L .48512 .87987 L .4876 .87965 L .49008 .87947 L .49256 .87932 L .4938 .87927 L .49504 .87922 L .49628 .87919 L .49752 .87916 L .49876 .87915 L .5 .87914 L .50124 .87915 L .50248 .87916 L .50372 .87919 L .50496 .87922 L .5062 .87927 L .50744 .87932 L .50992 .87947 L .5124 .87965 L .51488 .87987 L .51984 .88044 L Mistroke .5248 .88117 L .52976 .88205 L .53968 .88431 L .55952 .89068 L .57937 .89945 L .61905 .92354 L .65873 .95493 L .69841 .99149 L .7381 1.03072 L .77778 1.06995 L .81746 1.10651 L .85714 1.1379 L .87698 1.15097 L .89683 1.16199 L .91667 1.17076 L .93651 1.17713 L .94643 1.17938 L .95139 1.18027 L .95635 1.181 L .96131 1.18156 L .96379 1.18179 L .96627 1.18197 L .96875 1.18211 L .96999 1.18217 L .97123 1.18221 L .97247 1.18225 L .97371 1.18227 L .97495 1.18229 L .97619 1.18229 L Mfstroke P P p p .004 w .02381 1.25808 m .02505 1.25808 L .02629 1.25806 L .02753 1.25804 L .02877 1.258 L .03125 1.2579 L .03373 1.25776 L .03621 1.25757 L .03869 1.25735 L .04365 1.25679 L .04861 1.25606 L .05357 1.25517 L .06349 1.25292 L .08333 1.24654 L .10317 1.23777 L .14286 1.21369 L .18254 1.18229 L .22222 1.14574 L .2619 1.10651 L .30159 1.06728 L .34127 1.03072 L .38095 .99933 L .40079 .98625 L .42063 .97524 L .44048 .96647 L .4504 .96297 L .46032 .96009 L .47024 .95784 L .4752 .95695 L .48016 .95623 L .48512 .95566 L .4876 .95544 L .49008 .95525 L .49256 .95511 L .4938 .95506 L .49504 .95501 L .49628 .95498 L .49752 .95495 L .49876 .95493 L .5 .95493 L .50124 .95493 L .50248 .95495 L .50372 .95498 L .50496 .95501 L .5062 .95506 L .50744 .95511 L .50992 .95525 L .5124 .95544 L .51488 .95566 L .51984 .95623 L Mistroke .5248 .95695 L .52976 .95784 L .53968 .96009 L .55952 .96647 L .57937 .97524 L .61905 .99933 L .65873 1.03072 L .69841 1.06728 L .7381 1.10651 L .77778 1.14574 L .81746 1.18229 L .85714 1.21369 L .87698 1.22676 L .89683 1.23777 L .91667 1.24654 L .93651 1.25292 L .94643 1.25517 L .95139 1.25606 L .95635 1.25679 L .96131 1.25735 L .96379 1.25757 L .96627 1.25776 L .96875 1.2579 L .96999 1.25796 L .97123 1.258 L .97247 1.25804 L .97371 1.25806 L .97495 1.25808 L .97619 1.25808 L Mfstroke P P p p .004 w .02381 1.33387 m .02505 1.33386 L .02629 1.33385 L .02753 1.33382 L .02877 1.33379 L .03125 1.33369 L .03373 1.33355 L .03621 1.33336 L .03869 1.33314 L .04365 1.33257 L .04861 1.33185 L .05357 1.33096 L .06349 1.32871 L .08333 1.32233 L .10317 1.31356 L .14286 1.28947 L .18254 1.25808 L .22222 1.22152 L .2619 1.18229 L .30159 1.14306 L .34127 1.10651 L .38095 1.07511 L .40079 1.06204 L .42063 1.05103 L .44048 1.04226 L .4504 1.03876 L .46032 1.03588 L .47024 1.03363 L .4752 1.03274 L .48016 1.03201 L .48512 1.03145 L .4876 1.03122 L .49008 1.03104 L .49256 1.0309 L .4938 1.03084 L .49504 1.0308 L .49628 1.03076 L .49752 1.03074 L .49876 1.03072 L .5 1.03072 L .50124 1.03072 L .50248 1.03074 L .50372 1.03076 L .50496 1.0308 L .5062 1.03084 L .50744 1.0309 L .50992 1.03104 L .5124 1.03122 L .51488 1.03145 L .51984 1.03201 L Mistroke .5248 1.03274 L .52976 1.03363 L .53968 1.03588 L .55952 1.04226 L .57937 1.05103 L .61905 1.07511 L .65873 1.10651 L .69841 1.14306 L .7381 1.18229 L .77778 1.22152 L .81746 1.25808 L .85714 1.28947 L .87698 1.30255 L .89683 1.31356 L .91667 1.32233 L .93651 1.32871 L .94643 1.33096 L .95139 1.33185 L .95635 1.33257 L .96131 1.33314 L .96379 1.33336 L .96627 1.33355 L .96875 1.33369 L .96999 1.33374 L .97123 1.33379 L .97247 1.33382 L .97371 1.33385 L .97495 1.33386 L .97619 1.33387 L Mfstroke P P p p .004 w .02381 1.40966 m .02505 1.40965 L .02629 1.40964 L .02753 1.40961 L .02877 1.40958 L .03125 1.40948 L .03373 1.40933 L .03621 1.40915 L .03869 1.40893 L .04365 1.40836 L .04861 1.40763 L .05357 1.40675 L .06349 1.40449 L .08333 1.39812 L .10317 1.38935 L .14286 1.36526 L .18254 1.33387 L .22222 1.29731 L .2619 1.25808 L .30159 1.21885 L .34127 1.18229 L .38095 1.1509 L .40079 1.13783 L .42063 1.12681 L .44048 1.11804 L .4504 1.11455 L .46032 1.11167 L .47024 1.10942 L .4752 1.10853 L .48016 1.1078 L .48512 1.10724 L .4876 1.10701 L .49008 1.10683 L .49256 1.10669 L .4938 1.10663 L .49504 1.10659 L .49628 1.10655 L .49752 1.10653 L .49876 1.10651 L .5 1.10651 L .50124 1.10651 L .50248 1.10653 L .50372 1.10655 L .50496 1.10659 L .5062 1.10663 L .50744 1.10669 L .50992 1.10683 L .5124 1.10701 L .51488 1.10724 L .51984 1.1078 L Mistroke .5248 1.10853 L .52976 1.10942 L .53968 1.11167 L .55952 1.11804 L .57937 1.12681 L .61905 1.1509 L .65873 1.18229 L .69841 1.21885 L .7381 1.25808 L .77778 1.29731 L .81746 1.33387 L .85714 1.36526 L .87698 1.37834 L .89683 1.38935 L .91667 1.39812 L .93651 1.40449 L .94643 1.40675 L .95139 1.40763 L .95635 1.40836 L .96131 1.40893 L .96379 1.40915 L .96627 1.40933 L .96875 1.40948 L .96999 1.40953 L .97123 1.40958 L .97247 1.40961 L .97371 1.40964 L .97495 1.40965 L .97619 1.40966 L Mfstroke P P p p .004 w .02381 1.48545 m .02505 1.48544 L .02629 1.48543 L .02753 1.4854 L .02877 1.48536 L .03125 1.48526 L .03373 1.48512 L .03621 1.48494 L .03869 1.48472 L .04365 1.48415 L .04861 1.48342 L .05357 1.48253 L .06349 1.48028 L .08333 1.47391 L .10317 1.46514 L .14286 1.44105 L .18254 1.40966 L .22222 1.3731 L .2619 1.33387 L .30159 1.29464 L .34127 1.25808 L .38095 1.22669 L .40079 1.21362 L .42063 1.2026 L .44048 1.19383 L .4504 1.19034 L .46032 1.18746 L .47024 1.18521 L .4752 1.18432 L .48016 1.18359 L .48512 1.18302 L .4876 1.1828 L .49008 1.18262 L .49256 1.18248 L .4938 1.18242 L .49504 1.18238 L .49628 1.18234 L .49752 1.18231 L .49876 1.1823 L .5 1.18229 L .50124 1.1823 L .50248 1.18231 L .50372 1.18234 L .50496 1.18238 L .5062 1.18242 L .50744 1.18248 L .50992 1.18262 L .5124 1.1828 L .51488 1.18302 L .51984 1.18359 L Mistroke .5248 1.18432 L .52976 1.18521 L .53968 1.18746 L .55952 1.19383 L .57937 1.2026 L .61905 1.22669 L .65873 1.25808 L .69841 1.29464 L .7381 1.33387 L .77778 1.3731 L .81746 1.40966 L .85714 1.44105 L .87698 1.45412 L .89683 1.46514 L .91667 1.47391 L .93651 1.48028 L .94643 1.48253 L .95139 1.48342 L .95635 1.48415 L .96131 1.48472 L .96379 1.48494 L .96627 1.48512 L .96875 1.48526 L .96999 1.48532 L .97123 1.48536 L .97247 1.4854 L .97371 1.48543 L .97495 1.48544 L .97619 1.48545 L Mfstroke P P p p .004 w .02381 1.56123 m .02505 1.56123 L .02629 1.56121 L .02753 1.56119 L .02877 1.56115 L .03125 1.56105 L .03373 1.56091 L .03621 1.56073 L .03869 1.5605 L .04365 1.55994 L .04861 1.55921 L .05357 1.55832 L .06349 1.55607 L .08333 1.5497 L .10317 1.54093 L .14286 1.51684 L .18254 1.48545 L .22222 1.44889 L .2619 1.40966 L .30159 1.37043 L .34127 1.33387 L .38095 1.30248 L .40079 1.2894 L .42063 1.27839 L .44048 1.26962 L .4504 1.26613 L .46032 1.26325 L .47024 1.26099 L .4752 1.26011 L .48016 1.25938 L .48512 1.25881 L .4876 1.25859 L .49008 1.25841 L .49256 1.25826 L .4938 1.25821 L .49504 1.25816 L .49628 1.25813 L .49752 1.2581 L .49876 1.25809 L .5 1.25808 L .50124 1.25809 L .50248 1.2581 L .50372 1.25813 L .50496 1.25816 L .5062 1.25821 L .50744 1.25826 L .50992 1.25841 L .5124 1.25859 L .51488 1.25881 L .51984 1.25938 L Mistroke .5248 1.26011 L .52976 1.26099 L .53968 1.26325 L .55952 1.26962 L .57937 1.27839 L .61905 1.30248 L .65873 1.33387 L .69841 1.37043 L .7381 1.40966 L .77778 1.44889 L .81746 1.48545 L .85714 1.51684 L .87698 1.52991 L .89683 1.54093 L .91667 1.5497 L .93651 1.55607 L .94643 1.55832 L .95139 1.55921 L .95635 1.55994 L .96131 1.5605 L .96379 1.56073 L .96627 1.56091 L .96875 1.56105 L .96999 1.56111 L .97123 1.56115 L .97247 1.56119 L .97371 1.56121 L .97495 1.56123 L .97619 1.56123 L Mfstroke P P p p .004 w .02381 1.63702 m .02505 1.63702 L .02629 1.637 L .02753 1.63698 L .02877 1.63694 L .03125 1.63684 L .03373 1.6367 L .03621 1.63652 L .03869 1.63629 L .04365 1.63573 L .04861 1.635 L .05357 1.63411 L .06349 1.63186 L .08333 1.62548 L .10317 1.61671 L .14286 1.59263 L .18254 1.56123 L .22222 1.52468 L .2619 1.48545 L .30159 1.44622 L .34127 1.40966 L .38095 1.37827 L .40079 1.36519 L .42063 1.35418 L .44048 1.34541 L .4504 1.34191 L .46032 1.33903 L .47024 1.33678 L .4752 1.33589 L .48016 1.33517 L .48512 1.3346 L .4876 1.33438 L .49008 1.33419 L .49256 1.33405 L .4938 1.334 L .49504 1.33395 L .49628 1.33392 L .49752 1.33389 L .49876 1.33388 L .5 1.33387 L .50124 1.33388 L .50248 1.33389 L .50372 1.33392 L .50496 1.33395 L .5062 1.334 L .50744 1.33405 L .50992 1.33419 L .5124 1.33438 L .51488 1.3346 L .51984 1.33517 L Mistroke .5248 1.33589 L .52976 1.33678 L .53968 1.33903 L .55952 1.34541 L .57937 1.35418 L .61905 1.37827 L .65873 1.40966 L .69841 1.44622 L .7381 1.48545 L .77778 1.52468 L .81746 1.56123 L .85714 1.59263 L .87698 1.6057 L .89683 1.61671 L .91667 1.62548 L .93651 1.63186 L .94643 1.63411 L .95139 1.635 L .95635 1.63573 L .96131 1.63629 L .96379 1.63652 L .96627 1.6367 L .96875 1.63684 L .96999 1.6369 L .97123 1.63694 L .97247 1.63698 L .97371 1.637 L .97495 1.63702 L .97619 1.63702 L Mfstroke P P p p .004 w .02381 1.71281 m .02505 1.71281 L .02629 1.71279 L .02753 1.71276 L .02877 1.71273 L .03125 1.71263 L .03373 1.71249 L .03621 1.7123 L .03869 1.71208 L .04365 1.71151 L .04861 1.71079 L .05357 1.7099 L .06349 1.70765 L .08333 1.70127 L .10317 1.6925 L .14286 1.66841 L .18254 1.63702 L .22222 1.60046 L .2619 1.56123 L .30159 1.522 L .34127 1.48545 L .38095 1.45405 L .40079 1.44098 L .42063 1.42997 L .44048 1.4212 L .4504 1.4177 L .46032 1.41482 L .47024 1.41257 L .4752 1.41168 L .48016 1.41095 L .48512 1.41039 L .4876 1.41017 L .49008 1.40998 L .49256 1.40984 L .4938 1.40978 L .49504 1.40974 L .49628 1.4097 L .49752 1.40968 L .49876 1.40966 L .5 1.40966 L .50124 1.40966 L .50248 1.40968 L .50372 1.4097 L .50496 1.40974 L .5062 1.40978 L .50744 1.40984 L .50992 1.40998 L .5124 1.41017 L .51488 1.41039 L .51984 1.41095 L Mistroke .5248 1.41168 L .52976 1.41257 L .53968 1.41482 L .55952 1.4212 L .57937 1.42997 L .61905 1.45405 L .65873 1.48545 L .69841 1.522 L .7381 1.56123 L .77778 1.60046 L .81746 1.63702 L .85714 1.66841 L .87698 1.68149 L .89683 1.6925 L .91667 1.70127 L .93651 1.70765 L .94643 1.7099 L .95139 1.71079 L .95635 1.71151 L .96131 1.71208 L .96379 1.7123 L .96627 1.71249 L .96875 1.71263 L .96999 1.71268 L .97123 1.71273 L .97247 1.71276 L .97371 1.71279 L .97495 1.71281 L .97619 1.71281 L Mfstroke P P p p .004 w .02381 1.7886 m .02505 1.78859 L .02629 1.78858 L .02753 1.78855 L .02877 1.78852 L .03125 1.78842 L .03373 1.78827 L .03621 1.78809 L .03869 1.78787 L .04365 1.7873 L .04861 1.78657 L .05357 1.78569 L .06349 1.78343 L .08333 1.77706 L .10317 1.76829 L .14286 1.7442 L .18254 1.71281 L .22222 1.67625 L .2619 1.63702 L .30159 1.59779 L .34127 1.56123 L .38095 1.52984 L .40079 1.51677 L .42063 1.50575 L .44048 1.49698 L .4504 1.49349 L .46032 1.49061 L .47024 1.48836 L .4752 1.48747 L .48016 1.48674 L .48512 1.48618 L .4876 1.48595 L .49008 1.48577 L .49256 1.48563 L .4938 1.48557 L .49504 1.48553 L .49628 1.48549 L .49752 1.48547 L .49876 1.48545 L .5 1.48545 L .50124 1.48545 L .50248 1.48547 L .50372 1.48549 L .50496 1.48553 L .5062 1.48557 L .50744 1.48563 L .50992 1.48577 L .5124 1.48595 L .51488 1.48618 L .51984 1.48674 L Mistroke .5248 1.48747 L .52976 1.48836 L .53968 1.49061 L .55952 1.49698 L .57937 1.50575 L .61905 1.52984 L .65873 1.56123 L .69841 1.59779 L .7381 1.63702 L .77778 1.67625 L .81746 1.71281 L .85714 1.7442 L .87698 1.75728 L .89683 1.76829 L .91667 1.77706 L .93651 1.78343 L .94643 1.78569 L .95139 1.78657 L .95635 1.7873 L .96131 1.78787 L .96379 1.78809 L .96627 1.78827 L .96875 1.78842 L .96999 1.78847 L .97123 1.78852 L .97247 1.78855 L .97371 1.78858 L .97495 1.78859 L .97619 1.7886 L Mfstroke P P p p .004 w .02381 1.86439 m .02505 1.86438 L .02629 1.86437 L .02753 1.86434 L .02877 1.86431 L .03125 1.8642 L .03373 1.86406 L .03621 1.86388 L .03869 1.86366 L .04365 1.86309 L .04861 1.86236 L .05357 1.86147 L .06349 1.85922 L .08333 1.85285 L .10317 1.84408 L .14286 1.81999 L .18254 1.7886 L .22222 1.75204 L .2619 1.71281 L .30159 1.67358 L .34127 1.63702 L .38095 1.60563 L .40079 1.59256 L .42063 1.58154 L .44048 1.57277 L .4504 1.56928 L .46032 1.5664 L .47024 1.56415 L .4752 1.56326 L .48016 1.56253 L .48512 1.56196 L .4876 1.56174 L .49008 1.56156 L .49256 1.56142 L .4938 1.56136 L .49504 1.56132 L .49628 1.56128 L .49752 1.56125 L .49876 1.56124 L .5 1.56123 L .50124 1.56124 L .50248 1.56125 L .50372 1.56128 L .50496 1.56132 L .5062 1.56136 L .50744 1.56142 L .50992 1.56156 L .5124 1.56174 L .51488 1.56196 L .51984 1.56253 L Mistroke .5248 1.56326 L .52976 1.56415 L .53968 1.5664 L .55952 1.57277 L .57937 1.58154 L .61905 1.60563 L .65873 1.63702 L .69841 1.67358 L .7381 1.71281 L .77778 1.75204 L .81746 1.7886 L .85714 1.81999 L .87698 1.83306 L .89683 1.84408 L .91667 1.85285 L .93651 1.85922 L .94643 1.86147 L .95139 1.86236 L .95635 1.86309 L .96131 1.86366 L .96379 1.86388 L .96627 1.86406 L .96875 1.8642 L .96999 1.86426 L .97123 1.86431 L .97247 1.86434 L .97371 1.86437 L .97495 1.86438 L .97619 1.86439 L Mfstroke P P P P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] antiderivativefamily[ArcTan, {-6, 6}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 175; pictureHeight = 287; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.63854 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.079365 0.150572 0.079365 [ [(-6)] .02381 .15057 0 2 Msboxa [(-4)] .18254 .15057 0 2 Msboxa [(-2)] .34127 .15057 0 2 Msboxa [(2)] .65873 .15057 0 2 Msboxa [(4)] .81746 .15057 0 2 Msboxa [(6)] .97619 .15057 0 2 Msboxa [({antiderivatives of, ArcTan[x]})] .5 1.63854 0 -2 Msboxa [(5)] .4875 .5474 1 0 Msboxa [(10)] .4875 .94422 1 0 Msboxa [(15)] .4875 1.34105 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.63954 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .15057 m .02381 .15682 L s P [(-6)] .02381 .15057 0 2 Mshowa p .002 w .18254 .15057 m .18254 .15682 L s P [(-4)] .18254 .15057 0 2 Mshowa p .002 w .34127 .15057 m .34127 .15682 L s P [(-2)] .34127 .15057 0 2 Mshowa p .002 w .65873 .15057 m .65873 .15682 L s P [(2)] .65873 .15057 0 2 Mshowa p .002 w .81746 .15057 m .81746 .15682 L s P [(4)] .81746 .15057 0 2 Mshowa p .002 w .97619 .15057 m .97619 .15682 L s P [(6)] .97619 .15057 0 2 Mshowa p .001 w .05556 .15057 m .05556 .15432 L s P p .001 w .0873 .15057 m .0873 .15432 L s P p .001 w .11905 .15057 m .11905 .15432 L s P p .001 w .15079 .15057 m .15079 .15432 L s P p .001 w .21429 .15057 m .21429 .15432 L s P p .001 w .24603 .15057 m .24603 .15432 L s P p .001 w .27778 .15057 m .27778 .15432 L s P p .001 w .30952 .15057 m .30952 .15432 L s P p .001 w .37302 .15057 m .37302 .15432 L s P p .001 w .40476 .15057 m .40476 .15432 L s P p .001 w .43651 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If F is an antiderivative of f on the interval [a, b], then the Mean Value Theorem implies there is a number c in the interval [a, b] such that F(b) = F(a) + f (c) (b - a). Unfortunately, the Mean Value Theorem simply implies the existence of such a c ; it does not state its value. However, in some special cases we can find c explicitly. Consider a general quadratic function F(x) = s + t x + u x 2 . ;[s] 8:0,0;9,1;44,2;46,3;65,4;66,5;68,6;69,7;514,-1; 8:1,0,-9 -*-times-bold-r-normal-*-*-120-75-75-*-*-*-*,times,1,12,0,0,0;1,0,-9 -*-times-medium-r-normal-*-*-120-75-75-*-*-*-*,times,0,12,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-120-75-75-*-*-*-*,times,2,12,0,0,0;1,0,-9 -*-times-medium-r-normal-*-*-120-75-75-*-*-*-*,times,0,12,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-120-75-75-*-*-*-*,times,2,12,0,0,0;1,0,-9 -*-times-medium-r-normal-*-*-120-75-75-*-*-*-*,times,0,12,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-120-75-75-*-*-*-*,times,2,12,0,0,0;1,0,-9 -*-times-medium-r-normal-*-*-120-75-75-*-*-*-*,times,0,12,0,0,0; :[font = input; preserveAspect; fontName = "Courier"] F[x_] := s + t x + u x^2 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Solve[F'[c] == (F[b] - F[a])/(b - a), c] // Simplify :[font = output; output; inactive; preserveAspect; endGroup] {{c -> (a + b)/2}} ;[o] a + b {{c -> -----}} 2 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.79266 .01536 L .79514 .01583 L .79762 .01644 L .80258 .01805 L .80754 .02021 L .81746 .02618 L .82738 .03443 L .8373 .045 L .85714 .07332 L .87698 .1114 L .89683 .15935 L .93651 .28429 L .97619 .44502 L Mfstroke P P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?anti :[font = print; inactive; preserveAspect; endGroup] Global`anti anti[tableofvalues_][1] = 0 anti[tableofvalues_][n_] := anti[tableofvalues][n] = anti[tableofvalues][n - 1] + ((First[Transpose[tableofvalues]][[n]] - First[Transpose[tableofvalues]][[n - 1]])* (Last[Transpose[tableofvalues]][[n]] + Last[Transpose[tableofvalues]][[n - 1]]))/2 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?antiderivative :[font = print; inactive; preserveAspect; endGroup] Global`antiderivative antiderivative[tableofvalues_, k_] := Block[{i}, Table[{First[Transpose[tableofvalu\ es]][[i]], anti[tableofvalues][i] + k}, {i, Length[tableofvalues]}]] :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] antiderivative[N[Table[{i, f[i]}, {i, 0, 2, 0.5}]], 0] :[font = output; output; inactive; preserveAspect; endGroup] {{0, 0}, {0.5, 0.4121803176750321}, {1., 0.824360635350064}, {1.5, -0.296061632234452}, {2., -1.416483899818969}} ;[o] {{0, 0}, {0.5, 0.41218}, {1., 0.824361}, {1.5, -0.296062}, {2., -1.41648}} :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ListPlot[%, PlotJoined -> True]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.47619 0.386784 0.262671 [ [(0.5)] .2619 .38678 0 2 Msboxa [(1)] .5 .38678 0 2 Msboxa [(1.5)] .7381 .38678 0 2 Msboxa [(2)] .97619 .38678 0 2 Msboxa [(-1)] .01131 .12411 1 0 Msboxa [(-0.5)] .01131 .25545 1 0 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.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 .38678 m .02505 .38679 L .02629 .38679 L .02753 .3868 L .02877 .38682 L .03001 .38684 L .03125 .38686 L .03373 .38692 L .03621 .387 L .03869 .3871 L .04365 .38735 L .04861 .38767 L .05357 .38807 L .06349 .38909 L .07341 .39043 L .08333 .39209 L .10317 .39639 L .12302 .40201 L .14286 .40898 L .18254 .42679 L .22222 .44932 L .2619 .47558 L .30159 .50411 L .34127 .533 L .38095 .55994 L .42063 .58239 L .44048 .5911 L .4504 .59468 L .46032 .5977 L .47024 .60011 L .48016 .60187 L .48512 .6025 L .4876 .60275 L .49008 .60295 L .49256 .60311 L .4938 .60317 L .49504 .60323 L .49628 .60327 L .49752 .6033 L .49876 .60331 L .5 .60332 L .50124 .60331 L .50248 .6033 L .50372 .60327 L .50496 .60322 L .5062 .60317 L .50744 .60311 L .50992 .60294 L .5124 .60273 L .51488 .60246 L Mistroke .51984 .60179 L .5248 .60091 L .52976 .59983 L .53968 .59704 L .5496 .5934 L .55952 .58889 L .57937 .57721 L .59921 .56192 L .61905 .543 L .65873 .49458 L .69841 .43334 L .7381 .36194 L .77778 .28438 L .81746 .20586 L .85714 .13262 L .87698 .10014 L .89683 .0716 L .91667 .04793 L .93651 .02999 L .94643 .02345 L .95139 .02083 L .95635 .01866 L .96131 .01695 L .96627 .01572 L .96875 .01528 L .96999 .01511 L .97123 .01497 L .97247 .01486 L .97371 .01478 L .97495 .01473 L .97619 .01472 L Mfstroke P P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Show[%, %%%]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.47619 0.386784 0.201488 [ [(0.5)] .2619 .38678 0 2 Msboxa [(1)] .5 .38678 0 2 Msboxa [(1.5)] .7381 .38678 0 2 Msboxa [(2)] .97619 .38678 0 2 Msboxa [(-1.5)] .01131 .08455 1 0 Msboxa [(-1)] .01131 .1853 1 0 Msboxa [(-0.5)] .01131 .28604 1 0 Msboxa [(0.5)] .01131 .48753 1 0 Msboxa [(1)] .01131 .58827 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 .2619 .38678 m .2619 .39303 L s P [(0.5)] .2619 .38678 0 2 Mshowa p .002 w .5 .38678 m .5 .39303 L s P [(1)] .5 .38678 0 2 Mshowa p .002 w .7381 .38678 m .7381 .39303 L s P [(1.5)] .7381 .38678 0 2 Mshowa p .002 w .97619 .38678 m .97619 .39303 L s P [(2)] .97619 .38678 0 2 Mshowa p .001 w .07143 .38678 m .07143 .39053 L s P p .001 w .11905 .38678 m .11905 .39053 L s P p .001 w .16667 .38678 m .16667 .39053 L s P p .001 w .21429 .38678 m .21429 .39053 L s P p .001 w .30952 .38678 m .30952 .39053 L s P p .001 w .35714 .38678 m .35714 .39053 L s P p .001 w .40476 .38678 m .40476 .39053 L s P p .001 w .45238 .38678 m .45238 .39053 L s P p .001 w .54762 .38678 m .54762 .39053 L s P p .001 w .59524 .38678 m .59524 .39053 L s P p .001 w .64286 .38678 m .64286 .39053 L s P p .001 w .69048 .38678 m .69048 .39053 L s P p .001 w .78571 .38678 m .78571 .39053 L s P p .001 w .83333 .38678 m .83333 .39053 L s P p .001 w .88095 .38678 m .88095 .39053 L s P p .001 w .92857 .38678 m .92857 .39053 L s P p .002 w 0 .38678 m 1 .38678 L s P p .002 w .02381 .08455 m .03006 .08455 L s P [(-1.5)] .01131 .08455 1 0 Mshowa p .002 w .02381 .1853 m .03006 .1853 L s P [(-1)] .01131 .1853 1 0 Mshowa p .002 w .02381 .28604 m .03006 .28604 L s P [(-0.5)] .01131 .28604 1 0 Mshowa p .002 w .02381 .48753 m .03006 .48753 L s P [(0.5)] .01131 .48753 1 0 Mshowa p .002 w .02381 .58827 m .03006 .58827 L s P [(1)] .01131 .58827 1 0 Mshowa p .001 w .02381 .1047 m .02756 .1047 L s P p .001 w .02381 .12485 m .02756 .12485 L s P p .001 w .02381 .145 m .02756 .145 L s P p .001 w .02381 .16515 m .02756 .16515 L s P p .001 w .02381 .20544 m .02756 .20544 L s P p .001 w .02381 .22559 m .02756 .22559 L s P p .001 w .02381 .24574 m .02756 .24574 L s P p .001 w .02381 .26589 m .02756 .26589 L s P p .001 w .02381 .30619 m .02756 .30619 L s P p .001 w .02381 .32634 m .02756 .32634 L s P p .001 w .02381 .34649 m .02756 .34649 L s P p .001 w .02381 .36663 m .02756 .36663 L s P p .001 w .02381 .40693 m .02756 .40693 L s P p .001 w .02381 .42708 m .02756 .42708 L s P p .001 w .02381 .44723 m .02756 .44723 L s P p .001 w .02381 .46738 m .02756 .46738 L s P p .001 w .02381 .50768 m .02756 .50768 L s P p .001 w .02381 .52783 m .02756 .52783 L s P p .001 w .02381 .54797 m .02756 .54797 L s P p .001 w .02381 .56812 m .02756 .56812 L s P p .001 w .02381 .0644 m .02756 .0644 L s P p .001 w .02381 .04425 m .02756 .04425 L s P p .001 w .02381 .0241 m .02756 .0241 L s P p .001 w .02381 .00396 m .02756 .00396 L s P p .001 w .02381 .60842 m .02756 .60842 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 p .004 w .02381 .38678 m .02505 .38679 L .02629 .38679 L .02753 .3868 L .02877 .38682 L .03001 .38684 L .03125 .38686 L .03373 .38692 L .03621 .387 L .03869 .3871 L .04365 .38735 L .04861 .38767 L .05357 .38807 L .06349 .38909 L .07341 .39043 L .08333 .39209 L .10317 .39639 L .12302 .40201 L .14286 .40898 L .18254 .42679 L .22222 .44932 L .2619 .47558 L .30159 .50411 L .34127 .533 L .38095 .55994 L .42063 .58239 L .44048 .5911 L .4504 .59468 L .46032 .5977 L .47024 .60011 L .48016 .60187 L .48512 .6025 L .4876 .60275 L .49008 .60295 L .49256 .60311 L .4938 .60317 L .49504 .60323 L .49628 .60327 L .49752 .6033 L .49876 .60331 L .5 .60332 L .50124 .60331 L .50248 .6033 L .50372 .60327 L .50496 .60322 L .5062 .60317 L .50744 .60311 L .50992 .60294 L .5124 .60273 L .51488 .60246 L Mistroke .51984 .60179 L .5248 .60091 L .52976 .59983 L .53968 .59704 L .5496 .5934 L .55952 .58889 L .57937 .57721 L .59921 .56192 L .61905 .543 L .65873 .49458 L .69841 .43334 L .7381 .36194 L .77778 .28438 L .81746 .20586 L .85714 .13262 L .87698 .10014 L .89683 .0716 L .91667 .04793 L .93651 .02999 L .94643 .02345 L .95139 .02083 L .95635 .01866 L .96131 .01695 L .96627 .01572 L .96875 .01528 L .96999 .01511 L .97123 .01497 L .97247 .01486 L .97371 .01478 L .97495 .01473 L .97619 .01472 L Mfstroke P P .004 w .02381 .38678 m .03869 .3871 L .05357 .38807 L .06845 .38973 L .08333 .3921 L .09821 .3952 L .1131 .39905 L .12798 .40364 L .14286 .40899 L .15774 .41507 L .17262 .42187 L .1875 .42936 L .20238 .43751 L .21726 .44627 L .23214 .45557 L .24702 .46536 L .2619 .47556 L .27679 .48608 L .29167 .49683 L .30655 .5077 L .32143 .51859 L .33631 .52938 L .35119 .53993 L .36607 .55013 L .38095 .55984 L .39583 .56891 L .41071 .57722 L .4256 .58461 L .44048 .59095 L .45536 .5961 L .47024 .59993 L .48512 .60231 L .5 .60313 L .51488 .60226 L .52976 .59962 L .54464 .59511 L .55952 .58867 L .5744 .58025 L .58929 .5698 L .60417 .5573 L .61905 .54278 L .63393 .52624 L .64881 .50775 L .66369 .48738 L .67857 .46524 L .69345 .44144 L .70833 .41615 L .72321 .38954 L .7381 .36182 L .75298 .33322 L Mistroke .76786 .304 L .78274 .27444 L .79762 .24484 L .8125 .21552 L .82738 .18682 L .84226 .1591 L .85714 .13272 L .87202 .10805 L .8869 .08547 L .90179 .06538 L .91667 .04814 L .93155 .03414 L .94643 .02373 L .96131 .01725 L .97619 .01504 L Mfstroke P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect] Example 4. :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?viewantiderivative :[font = print; inactive; preserveAspect; endGroup] viewantiderivative[f, {a, b}, initialvalue] plots the graph of an approximate antiderivative F of f on the interval {a, b} with initial condition F[a] = initialvalue. The approximation is based on applying the Mean Value Theorem to Mathematica's InputForm of the Plot of f on {a, b}. :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] viewantiderivative[f, {0, 2}, 0]; (* f as in Example 3 *) :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.47619 0.386783 0.20202 [ [(0.5)] .2619 .38678 0 2 Msboxa [(1)] .5 .38678 0 2 Msboxa [(1.5)] .7381 .38678 0 2 Msboxa [(2)] .97619 .38678 0 2 Msboxa [( x)({an approximate antiderivative of, E Sin[Pi x]})] .5 .61803 0 -2 Msboxa [(-1.5)] .01131 .08375 1 0 Msboxa [(-1)] .01131 .18476 1 0 Msboxa [(-0.5)] .01131 .28577 1 0 Msboxa [(0.5)] .01131 .48779 1 0 Msboxa [(1)] .01131 .5888 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 .2619 .38678 m .2619 .39303 L s P [(0.5)] .2619 .38678 0 2 Mshowa p .002 w .5 .38678 m .5 .39303 L s P [(1)] .5 .38678 0 2 Mshowa p .002 w .7381 .38678 m .7381 .39303 L s P [(1.5)] .7381 .38678 0 2 Mshowa p .002 w .97619 .38678 m .97619 .39303 L s P [(2)] .97619 .38678 0 2 Mshowa p .001 w .07143 .38678 m .07143 .39053 L s P p .001 w .11905 .38678 m .11905 .39053 L s P p .001 w .16667 .38678 m .16667 .39053 L s P p .001 w .21429 .38678 m .21429 .39053 L s P p .001 w .30952 .38678 m .30952 .39053 L s P p .001 w .35714 .38678 m .35714 .39053 L s P p .001 w .40476 .38678 m .40476 .39053 L s P p .001 w .45238 .38678 m .45238 .39053 L s P p .001 w .54762 .38678 m .54762 .39053 L s P p .001 w .59524 .38678 m .59524 .39053 L s P p .001 w .64286 .38678 m .64286 .39053 L s P p .001 w .69048 .38678 m .69048 .39053 L s P p .001 w .78571 .38678 m .78571 .39053 L s P p .001 w .83333 .38678 m .83333 .39053 L s P p .001 w .88095 .38678 m .88095 .39053 L s P p .001 w .92857 .38678 m .92857 .39053 L s P p .002 w 0 .38678 m 1 .38678 L s P [( x)({an approximate antiderivative of, E Sin[Pi x]})] .5 .61803 0 -2 Mshowa p .002 w .02381 .08375 m .03006 .08375 L s P [(-1.5)] .01131 .08375 1 0 Mshowa p .002 w .02381 .18476 m .03006 .18476 L s P [(-1)] .01131 .18476 1 0 Mshowa p .002 w .02381 .28577 m .03006 .28577 L s P [(-0.5)] .01131 .28577 1 0 Mshowa p .002 w .02381 .48779 m .03006 .48779 L s P [(0.5)] .01131 .48779 1 0 Mshowa p .002 w .02381 .5888 m .03006 .5888 L s P [(1)] .01131 .5888 1 0 Mshowa p .001 w .02381 .10396 m .02756 .10396 L s P p .001 w .02381 .12416 m .02756 .12416 L s P p .001 w .02381 .14436 m .02756 .14436 L s P p .001 w .02381 .16456 m .02756 .16456 L s P p .001 w .02381 .20497 m .02756 .20497 L s P p .001 w .02381 .22517 m .02756 .22517 L s P p .001 w .02381 .24537 m .02756 .24537 L s P p .001 w .02381 .26557 m .02756 .26557 L s P p .001 w .02381 .30598 m .02756 .30598 L s P p .001 w .02381 .32618 m .02756 .32618 L s P p .001 w .02381 .34638 m .02756 .34638 L s P p .001 w .02381 .36658 m .02756 .36658 L s P p .001 w .02381 .40699 m .02756 .40699 L s P p .001 w .02381 .42719 m .02756 .42719 L s P p .001 w .02381 .44739 m .02756 .44739 L s P p .001 w .02381 .46759 m .02756 .46759 L s P p .001 w .02381 .508 m .02756 .508 L s P p .001 w .02381 .5282 m .02756 .5282 L s P p .001 w .02381 .5484 m .02756 .5484 L s P p .001 w .02381 .5686 m .02756 .5686 L s P p .001 w .02381 .06355 m .02756 .06355 L s P p .001 w .02381 .04335 m .02756 .04335 L s P p .001 w .02381 .02315 m .02756 .02315 L s P p .001 w .02381 .00295 m .02756 .00295 L s P p .001 w .02381 .609 m .02756 .609 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 .004 w .02381 .38678 m .06349 .38915 L .10317 .39649 L .14286 .40911 L .18254 .42692 L .22222 .44943 L .24206 .46219 L .25198 .46889 L .2619 .47575 L .27183 .48275 L .28175 .48987 L .28671 .49347 L .29167 .49708 L .29415 .49889 L .29663 .50071 L .29911 .50253 L .30159 .50435 L .30283 .50526 L .30407 .50617 L .30531 .50708 L .30655 .50799 L .30779 .50891 L .30903 .50982 L .31027 .51073 L .31151 .51164 L .31275 .51255 L .31399 .51346 L .31647 .51528 L .31895 .5171 L .32143 .51892 L .32639 .52255 L .33135 .52616 L .34127 .53331 L .35119 .54034 L .36111 .54721 L .38095 .56029 L .40079 .57224 L .42063 .58274 L .46032 .59787 L .5 .60332 L .53968 .59688 L .57937 .57693 L .61905 .54264 L .65873 .49421 L .69841 .43303 L .71825 .39833 L .72817 .38014 L .7381 .3615 L .74802 .34246 L .75794 .3231 L Mistroke .7629 .31332 L .76786 .3035 L .77282 .29364 L .7753 .28869 L .77654 .28622 L .77778 .28374 L .77902 .28127 L .78026 .27879 L .7815 .27631 L .78274 .27384 L .78398 .27136 L .78522 .26888 L .78646 .2664 L .7877 .26392 L .78894 .26145 L .79018 .25897 L .79266 .25402 L .79514 .24907 L .79762 .24413 L .80258 .23428 L .80754 .22447 L .81746 .20503 L .82738 .18592 L .8373 .16725 L .85714 .13169 L .87698 .09919 L .89683 .07066 L .93651 .02953 L .97619 .01472 L Mfstroke % End of Graphics MathPictureEnd :[font = input; preserveAspect; fontName = "Courier"] g[x_] := (1 + x^2)^(1/3) :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Integrate[g[x], x] :[font = output; output; inactive; preserveAspect; endGroup] (3*x*(1 + x^2)^(1/3))/5 + (2*Integrate[(1 + x^2)^(-2/3), x])/5 ;[o] 2 1/3 2 -(2/3) 3 x (1 + x ) 2 Integrate[(1 + x ) , x] --------------- + ------------------------------ 5 5 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] viewantiderivative[g, {-3, 3}, 0]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.15873 0.014715 0.065223 [ [(-3)] .02381 .01472 0 2 Msboxa [(-2)] .18254 .01472 0 2 Msboxa [(-1)] .34127 .01472 0 2 Msboxa [(1)] .65873 .01472 0 2 Msboxa [(2)] .81746 .01472 0 2 Msboxa [(3)] .97619 .01472 0 2 Msboxa [( 2 1/3)({an approximate antiderivative of, \(1 + x \) })] .5 .61803 0 -2 Msboxa [(2)] .4875 .14516 1 0 Msboxa [(4)] .4875 .27561 1 0 Msboxa [(6)] .4875 .40605 1 0 Msboxa [(8)] .4875 .5365 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 .02381 .01472 m .02381 .02097 L s P [(-3)] .02381 .01472 0 2 Mshowa p .002 w .18254 .01472 m .18254 .02097 L s P [(-2)] .18254 .01472 0 2 Mshowa p .002 w .34127 .01472 m .34127 .02097 L s P [(-1)] .34127 .01472 0 2 Mshowa p .002 w .65873 .01472 m .65873 .02097 L s P [(1)] .65873 .01472 0 2 Mshowa p .002 w .81746 .01472 m .81746 .02097 L s P [(2)] .81746 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(3)] .97619 .01472 0 2 Mshowa p .001 w .05556 .01472 m .05556 .01847 L s P p .001 w .0873 .01472 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