(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 297352, 11334]*) (*NotebookOutlinePosition[ 298407, 11370]*) (* CellTagsIndexPosition[ 298363, 11366]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell[TextData[{ StyleBox[ " (C) Copyright 1994 D.C.M. Burbulla and C.T.J. Dodson\n \ ", Evaluatable->False, AspectRatioFixed->True, FontSize->12, PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox[" Notebook to support the book", Evaluatable->False, AspectRatioFixed->True, FontSize->17, FontSlant->"Italic", PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["\n Self-Tutor for Computer Calculus", Evaluatable->False, AspectRatioFixed->True, PrivateFontOptions->{"FontPostScriptName"->Automatic}] }], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Using ", Evaluatable->False, AspectRatioFixed->True, FontSize->23, PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["Mathematica 2 ", Evaluatable->False, AspectRatioFixed->True, FontSize->23, FontWeight->"Plain", FontSlant->"Italic", PrivateFontOptions->{"FontPostScriptName"->Automatic}] }], "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " by\n D. C. M. Burbulla and C. T. J. 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The \ approximation is based on applying the Mean Value Theorem \nto Mathematica's \ InputForm of the Plot of f on {a, b}.\"\n\n\nantiderivativefamily/: \ antiderivativefamily[f_, {a_, b_, k_:1}] :=\n (one = \ Block[{x}, Plot[f[x], {x, a, b}, AspectRatio ->\n\t\t Automatic, \ DisplayFunction -> Identity]];\n\t\t two = Block[{x}, Integrate[f[x], \ x] /. x -> a];\n\t\t three = Block[{x, i}, \ Plot[Release[Table[Integrate[\n\t\t f[x], x] + i, {i, two - 5, two + 5, \ k}]], {x, a, b},\n\t\t DisplayFunction -> Identity]];\n\t\t \ Block[{x}, Show[one, three, PlotLabel ->\n\t\t {\"antiderivatives of\", \ f[x]}, \n\t\t DisplayFunction -> $DisplayFunction]])\n\t\t \n\ antiderivativefamily::usage = \"antiderivativefamily[f, {a, b, k}] plots F[x] \ + c, and f[x], \non the interval {a, b}, where F[x] = Integrate[f[x], x], for \ a few values of c differing by k. \nThe default value of k is 1.\""], "Input",\ AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ 0\ \>", "\<\ 0\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData[ "\<\ \"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. 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.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 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If F is an antiderivative of ", Evaluatable->False, AspectRatioFixed->True, PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic", PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox[" on the interval [", Evaluatable->False, AspectRatioFixed->True, PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["a", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic", PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True, PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["b", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic", PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox[ "], then the Mean Value Theorem implies there is a number c in the \ interval [a, b] such that\n\n\t F(b) = F(a) \ + f (c) (b - a).\n\nUnfortunately, 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. 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