(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing 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). 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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[ 204342, 6143]*) (*NotebookOutlinePosition[ 205186, 6175]*) (* CellTagsIndexPosition[ 205111, 6169]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "How do I check results in ", StyleBox["Mathematica", FontSlant->"Italic"], "?" }], "Title"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is one of the most reliable pieces of software ever written and every \ version has passed a great number of tests before it is shipped. But no piece \ of software is And, on the user side, there may always be typos or errors in \ inputs. Of course, the first line of defense against incorrect results is \ common sense. For example, one should always ask: Is the order of magnitude \ plausible? Is the result plausible? \nAdditionally, there are some simple \ tests you can do in ", StyleBox["Mathematica", FontSlant->"Italic"], " to check your results without a lot of effort. In more complex cases, \ more complex tests might be necessary." }], "Text"], Cell[CellGroupData[{ Cell["Substitiute the result back into the problem", "Section"], Cell["\<\ In many cases, you can simply put the result back into the original problem \ and see if it really solves it.\ \>", "Text"], Cell["For example, take the result to this well known problem:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[a\ x\^2 + b\ x\ + \ c \[Equal] d, \ x]\)], "Input"], Cell[BoxData[ \({{x \[Rule] \(\(-b\) - \@\(b\^2 - 4\ a\ c + 4\ a\ d\)\)\/\(2\ a\)}, {x \ \[Rule] \(\(-b\) + \@\(b\^2 - 4\ a\ c + 4\ a\ d\)\)\/\(2\ a\)}}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["/.", FontFamily->"Courier"], " is shortcut notation for ", StyleBox[ButtonBox["ReplaceAll", ButtonData:>{ URL[ "http://documents.wolfram.com/v4/RefGuide/RefGuide/ReplaceAll.html"], None}, ButtonStyle->"Hyperlink"], FontFamily->"Courier"], ", % refers to the last result ", StyleBox["Mathematica", FontSlant->"Italic"], " returned. Using ", StyleBox[ButtonBox["FullSimplify", ButtonData:>{ URL[ "http://documents.wolfram.com/v4/RefGuide/RefGuide/FullSimplify.html"], None}, ButtonStyle->"Hyperlink"], FontFamily->"Courier"], ", gives ", StyleBox["{True, True}", FontFamily->"Courier"], ", showing that both results returned satisfy the equation." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(a\ x\^2 + b\ x\ + \ c \[Equal] d /. % // FullSimplify\)], "Input"], Cell[BoxData[ \({True, True}\)], "Output"] }, Open ]], Cell[TextData[{ "The same methode works for the differential equation solver ", StyleBox["DSolve", FontFamily->"Courier"], " and other commands that return rules." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"DSolve", "[", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "2"], "+", \(2\ y[x]\)}], "==", "0"}], ",", "y", ",", "x"}], "]"}]], "Input"], Cell[BoxData[ \({{y \[Rule] \((1\/2\ \((\(-#1\^2\) + 2\ #1\ C[1] - C[1]\^2)\) &)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "2"], "+", \(2\ y[x]\)}], "==", "0"}], "/.", "\[InvisibleSpace]", "%"}], "]"}]], "Input"], Cell[BoxData[ \({True}\)], "Output"] }, Open ]], Cell["You can easily substitue multiple results, too.", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"eqn", "=", RowBox[{"{", RowBox[{ RowBox[{\(y[x]\), "==", RowBox[{"-", RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], ",", RowBox[{\(z[x]\), "==", RowBox[{"-", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}]}], "}"}]}], ";"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(sol = DSolve[eqn, {y, z}, x]\)], "Input"], Cell[BoxData[ \({{y \[Rule] \((1\/2\ \[ExponentialE]\^\(-#1\)\ \((C[ 1] + \[ExponentialE]\^\(2\ #1\)\ C[1] + C[2] - \[ExponentialE]\^\(2\ #1\)\ C[2])\) &)\), z \[Rule] \((\(-\(1\/2\)\)\ \[ExponentialE]\^\(-#1\)\ \((\(-C[ 1]\) + \[ExponentialE]\^\(2\ #1\)\ C[1] - C[2] - \[ExponentialE]\^\(2\ #1\)\ C[ 2])\) &)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[eqn /. \[InvisibleSpace]sol]\)], "Input"], Cell[BoxData[ \({{True, True}}\)], "Output"] }, Open ]], Cell["\<\ For many other operations, you can simply use the inverse command. For \ example,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(res = Integrate[Sin[x^2] + Cos[x^3], x]\)], "Input"], Cell[BoxData[ \(\@\(\[Pi]\/2\)\ FresnelS[\@\(2\/\[Pi]\)\ x] + 1\/2\ \((\(-\(\(x\ Gamma[ 1\/3, \(-\[ImaginaryI]\)\ x\^3]\)\/\(3\ \((\(-\ \[ImaginaryI]\)\ x\^3)\)\^\(1/3\)\)\)\) - \(x\ Gamma[1\/3, \[ImaginaryI]\ \ x\^3]\)\/\(3\ \((\[ImaginaryI]\ x\^3)\)\^\(1/3\)\))\) + 1\/4\ \((\(x\ Gamma[1\/3, \(-\[ImaginaryI]\)\ x\^3]\)\/\(3\ \((\(-\ \[ImaginaryI]\)\ x\^3)\)\^\(1/3\)\) - \(x\ Gamma[1\/3, \[ImaginaryI]\ x\^3]\)\ \/\(3\ \((\[ImaginaryI]\ x\^3)\)\^\(1/3\)\))\) + 1\/4\ \((\(-\(\(x\ Gamma[ 1\/3, \(-\[ImaginaryI]\)\ x\^3]\)\/\(3\ \((\(-\ \[ImaginaryI]\)\ x\^3)\)\^\(1/3\)\)\)\) + \(x\ Gamma[1\/3, \[ImaginaryI]\ \ x\^3]\)\/\(3\ \((\[ImaginaryI]\ x\^3)\)\^\(1/3\)\))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[res, x] == Sin[x^2] + Cos[x^3] // FullSimplify\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Numerical tests", "Section"], Cell[CellGroupData[{ Cell["Substitute the result back into the original problem", "Subsection"], Cell["\<\ Let's compare the numerical solution to a differential equation with machine \ precisions with a numerical solution when using a higher precision \ setting.\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"lowsol", " ", "=", " ", RowBox[{ StyleBox["NDSolve", FontColor->RGBColor[0, 0, 1]], "[", \({\(y'\)[x] == \ 1\/\(2 y[x]\), y[ .01] == .1}, y, {x, .01, 1}\), "]"}]}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"highsol", " ", "=", " ", RowBox[{ StyleBox["NDSolve", FontColor->RGBColor[0, 0, 1]], "[", \({\(y'\)[x] == \ 1\/\(2 y[x]\), y[ .01] == .1}, y, {x, .01, 1}, WorkingPrecision \[Rule] 22\), "]"}]}], ";"}]], "Input", CellTags->"NDSolve"], Cell[TextData[{ "The following plot shows the logarithm of the difference between the \ numerical solution and the differential equation with the default setting of \ ", StyleBox["WorkingPrecision", FontFamily->"Courier"], " (black) and with ", StyleBox["WorkingPrecision", FontFamily->"Courier"], " set to ", StyleBox["22", FontFamily->"Courier"], " (", StyleBox["red", FontColor->RGBColor[1, 0, 0]], ")." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Plot[ Evaluate[\(-Log[ Abs[{\((\(y'\)[x] - \ 1\/\(2 y[x]\)\ )\) /. \ lowsol, \((\(y'\)[x] - \ 1\/\(2 y[x]\)\ )\) /. \ highsol}], 10]\)], {x, .01, 1}, PlotStyle \[Rule] {GrayLevel[0], RGBColor[1, 0, 0]}, \ AxesOrigin \[Rule] {0, 0}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 -0.118382 1.91968 [ [.21429 -0.13088 -9 -9 ] [.21429 -0.13088 9 0 ] [.40476 -0.13088 -9 -9 ] [.40476 -0.13088 9 0 ] 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.55144 .24954 L .55421 .23299 L .55557 .22336 L .55682 .21266 L .55796 .19828 L .55917 .14396 L .56038 .19859 L .56107 .21035 L .56171 .21836 L .563 .22983 L .56373 .23451 L .5644 .23794 L .56564 .24222 L .56633 .24355 L .56695 .24409 L .56814 .24331 L .56942 .2393 L .57063 .23114 L .57174 .21626 L .57304 .16944 L .57424 .22405 L .57494 .23652 L .57557 .24522 L .57698 .2599 L .57835 .27028 L .57983 .2789 L .58122 .28513 L .58251 .28953 L .58375 .29267 L .58492 .29479 L .58597 .29602 L .58712 .29669 L .58836 .29666 L .58906 .29633 L .58969 .29584 L .59092 .29438 L .59208 .29244 L .59432 .28726 L .59638 .28102 L .59863 .27276 L .60104 .26197 L .60232 .25495 L .60348 .24711 L .60466 .2366 L .60577 .22118 L .60706 .1546 L Mistroke .60826 .21921 L .60888 .23132 L .60957 .24111 L .61025 .24867 L .61096 .25511 L .61161 .25993 L .61232 .26429 L .61357 .26999 L .61427 .27223 L .61502 .27391 L .61573 .27481 L .61637 .27508 L .61763 .274 L .61833 .27242 L .61899 .27025 L .62018 .2642 L .62145 .25375 L .62265 .2367 L .62395 .17088 L .62519 .23329 L .62633 .25385 L .62746 .26661 L .62849 .275 L .62967 .28228 L .63079 .28739 L .63201 .29143 L .6333 .29421 L .6344 .29553 L .63561 .29595 L .63691 .29531 L .63764 .29448 L .6383 .29345 L .6396 .29073 L .64082 .28733 L .64497 .27074 L .64944 .24657 L .65069 .23855 L .652 .22845 L .65312 .21659 L .65434 .19195 L .6555 .18938 L .65673 .21846 L .65789 .23262 L .65895 .24154 L .66018 .24908 L .66131 .25394 L .66256 .25732 L .66318 .2583 L .66387 .25882 L .66503 .25838 L Mistroke .66627 .25593 L .66745 .25139 L .66852 .24476 L .66982 .23156 L .67102 .20602 L .67164 .14564 L .67233 .20715 L .67306 .22612 L .67373 .23733 L .67498 .25128 L .67611 .2602 L .67736 .26742 L .67868 .27294 L .67982 .27635 L .68086 .27851 L .68206 .28003 L .68319 .28058 L .68424 .28039 L .6852 .27964 L .68626 .27824 L .68739 .27612 L .68852 .27338 L .68973 .2698 L .69193 .26185 L .69693 .23832 L .69822 .23094 L .69944 .22282 L .70011 .21739 L .70084 .21012 L .70211 .1884 L .70333 .17973 L .70462 .20982 L .70572 .22267 L .70693 .23231 L .70766 .23673 L .70836 .24013 L .70966 .24476 L .71038 .24644 L .71106 .24745 L .71179 .24796 L .71258 .24781 L .71387 .24591 L .71509 .24197 L .71578 .23855 L .71642 .23446 L .71712 .22853 L .71785 .21983 L .7191 .18891 L .72023 .20047 L .72154 .22704 L Mistroke .72277 .24045 L .72392 .24905 L .72497 .25502 L .72613 .26 L .72735 .26385 L .72858 .2665 L .72928 .26751 L .72991 .26816 L .73113 .26871 L .7323 .26843 L .73295 .26795 L .73365 .2672 L .7349 .26524 L .73639 .26197 L .73774 .2582 L .74258 .23957 L .74502 .22783 L .74642 .22008 L .74769 .21153 L .74887 .20062 L .74998 .18212 L .75115 .16988 L .75242 .20062 L .75354 .21339 L .75472 .22257 L .75572 .22828 L .75684 .23288 L .75814 .23635 L .7588 .23737 L .75951 .23795 L .76078 .23759 L .76196 .23552 L .76268 .23333 L .76336 .23049 L .76463 .2225 L .76534 .21572 L .766 .20682 L .76672 .19046 L .76748 .15732 L .76867 .20734 L .76994 .22483 L .77114 .23519 L .77224 .24204 L .77352 .24801 L .7742 .25046 L .77492 .25263 L .77555 .25418 L .77624 .25556 L .77746 .25721 L .77866 .25795 L Mistroke .77978 .25792 L .78101 .25715 L .7817 .25639 L .78232 .25552 L .78358 .25322 L .78495 .24998 L .78744 .24234 L .78975 .23356 L .7922 .22285 L .79429 .2122 L .79548 .20469 L .79659 .19528 L .79787 .1758 L .79859 .13144 L .79926 .17242 L .80048 .19589 L .80179 .20914 L .80307 .21812 L .80428 .22445 L .80537 .22884 L .80654 .23243 L .80783 .23513 L .80857 .23611 L .80925 .23665 L .80997 .23683 L .81063 .23663 L .81139 .23594 L .81211 .23479 L .81339 .23134 L .81457 .22606 L .81586 .21653 L .81651 .20901 L .81723 .19594 L .81844 .17711 L .81956 .21051 L .82079 .22742 L .82148 .23431 L .82211 .23965 L .82337 .24867 L .82475 .25655 L .82724 .26751 L .82955 .27495 L .83082 .27812 L .83201 .28059 L .8341 .28389 L .83521 .28512 L .8364 .28608 L .83707 .28645 L .83778 .28673 L .83905 .28693 L Mistroke .83984 .28686 L .84057 .28667 L .84123 .2864 L .84194 .28601 L .84328 .28499 L .84468 .28353 L .8472 .28003 L .8519 .27074 L .85706 .2568 L .85956 .24806 L .86087 .24245 L .86227 .23517 L .86349 .22688 L .86418 .22075 L .86483 .21339 L .86604 .18885 L .8672 .18702 L .86831 .21311 L .86952 .228 L .87062 .23746 L .87164 .24432 L .87284 .25071 L .87396 .2555 L .87525 .25979 L .87643 .26268 L .87763 .26472 L .87831 .26548 L .87895 .26593 L .88023 .26608 L .88095 .26572 L .88161 .26508 L .88278 .26321 L .88406 .25996 L .88527 .25551 L .88638 .24982 L .88766 .24061 L .88886 .22727 L .88955 .21519 L .89018 .19581 L .89087 .17781 L .8916 .21155 L .89227 .22549 L .89298 .23572 L .89425 .24872 L .89503 .25474 L .89577 .25955 L .89715 .26688 L .89829 .27168 L .89952 .27586 L .90058 .27879 L Mistroke .90173 .28134 L .90243 .2826 L .90308 .28359 L .90436 .28504 L .90553 .28584 L .90677 .28618 L .90746 .28615 L .90809 .286 L .90879 .28568 L .90952 .2852 L .91076 .28403 L .91212 .28227 L .91462 .27779 L .91696 .27229 L .92578 .24305 L .92832 .23242 L .92957 .22637 L .93072 .21978 L .93177 .21228 L .93288 .20108 L .93409 .17649 L .93522 .17909 L .93635 .20265 L .93759 .21607 L .93879 .22471 L .94008 .23109 L .94128 .23507 L .94241 .23736 L .94342 .23831 L .94452 .23823 L .94573 .23681 L .94641 .23538 L .94705 .23361 L .94835 .22866 L .94907 .22496 L .94973 .22088 L .95094 .2111 L .95222 .19474 L .95332 .16115 L .95453 .18438 L .95564 .19918 L .95683 .20703 L .95795 .21085 L .95898 .21231 L .96011 .21211 L .96134 .20994 L .96263 .20549 L .96383 .19903 L .96497 .18999 L .96606 .17686 L Mistroke .96703 .15284 L .96808 .16016 L .96923 .1758 L .97045 .18163 L .97149 .18269 L .97264 .18069 L .97353 .17678 L .97435 .17085 L .97529 .15885 L .97619 .11016 L Mfstroke 1 0 0 r .03333 .29356 m .0345 .08155 L .03556 .0905 L .03678 .09173 L .03793 .09244 L .039 .09969 L .03999 .07312 L .04107 .07668 L .04222 .09135 L .04337 .09001 L .04461 .0726 L .04578 .0818 L .04685 .07468 L .0481 .07515 L .04881 .05722 L .04947 .05795 L .05067 .06808 L .05195 .08756 L .0526 .09638 L .05328 .08347 L .0545 .0745 L .05518 .08235 L .05581 .0865 L .05651 .07341 L .05725 .07962 L .0579 .06896 L .0586 .05896 L .05987 .09183 L .0611 .08751 L .06225 .09041 L .06336 .08152 L .0644 .07692 L .06564 .05906 L .06677 .06819 L .06788 .07373 L .06909 .08306 L .0703 .07632 L .07158 .07786 L .07229 .08483 L .07293 .08035 L .07364 .07611 L .0744 .07534 L .07566 .07158 L .07638 .06241 L .07706 .07648 L .07831 .07449 L .07945 .07249 L .0807 .07963 L .0814 .09157 L .08203 .06671 L Mistroke .08276 .07664 L .08345 .07311 L .08408 .08415 L .08477 .08018 L .08553 .07897 L .08633 .0781 L .08701 .07428 L .08776 .08138 L .08847 .06581 L .08911 .07813 L .08986 .05727 L .09057 .06528 L .09183 .0729 L .09254 .07258 L .09319 .06347 L .09434 .08175 L .09559 .0853 L .09676 .06936 L .09783 .08245 L .09902 .07117 L .10032 .07529 L .10098 .06915 L .10168 .07591 L .10295 .06901 L .10364 .07555 L .10429 .06898 L .10499 .07147 L .10574 .07858 L .10699 .0715 L .10771 .07693 L .10837 .06112 L .1096 .07458 L .11028 .06048 L .11092 .06939 L .11206 .08128 L .1133 .08001 L .11442 .08683 L .1156 .07271 L .1169 .08305 L .11809 .07504 L .12026 .05937 L .12146 .08059 L .12259 .04476 L .12322 .07612 L .12391 .07928 L .12512 .05962 L .12635 .07726 L .12749 .05377 L .12872 .07533 L .12943 .06207 L Mistroke .13009 .06346 L .13073 .06134 L .13141 .07093 L .13216 .05221 L .13285 .07006 L .13357 .07569 L .13426 .07838 L .13553 .07433 L .13676 .08507 L .13745 .07631 L .13808 .06263 L .13919 .07718 L .14041 .07551 L .14168 .07535 L .14287 .08214 L .14354 .07013 L .14426 .05445 L .14495 .07589 L .14559 .07497 L .14679 .06448 L .14807 .07414 L .14872 .07689 L .14943 .06853 L .15072 .07029 L .15146 .06557 L .15224 .07093 L .15291 .05891 L .15364 .06733 L .15495 .07358 L .15615 .04047 L .15678 .06819 L .15748 .06493 L .15821 .06785 L .15888 .05623 L .16011 .06936 L .16079 .07384 L .16143 .07848 L .16257 .06724 L .16382 .08146 L .16504 .08404 L .16615 .06633 L .16742 .07769 L .16862 .08178 L .16973 .0674 L .17076 .07239 L .17188 .07248 L .17307 .06989 L .17436 .06731 L .17554 .06805 L .1762 .07709 L Mistroke .1769 .06606 L .17819 .06879 L .1794 .07267 L .18067 .06342 L .18176 .06427 L .18297 .06604 L .18369 .0727 L .18438 .06185 L .18566 .06648 L .18634 .06544 L .18705 .0724 L .18784 .06379 L .18857 .05338 L .18925 .06283 L .18998 .06532 L .19068 .07025 L .19132 .06608 L .19253 .04622 L .19381 .06648 L .19502 .07749 L .1957 .07412 L .19633 .06981 L .19749 .07129 L .19872 .07147 L .19981 .08191 L .20084 .06748 L .20197 .06604 L .20319 .07169 L .20441 .07924 L .2051 .07046 L .20574 .06392 L .20705 .07422 L .20777 .07155 L .20844 .07707 L .20962 .06907 L .21091 .05576 L .21212 .07226 L .21325 .07623 L .21429 .06604 L .21541 .05016 L .21647 .07168 L .21744 .0704 L .21857 .06858 L .21981 .04974 L .22098 .06954 L .22206 .0695 L .22328 .06491 L .22443 .04479 L .22546 .06743 L .22658 .06669 L Mistroke .2278 .06414 L .22909 .03996 L .23019 .06647 L .23139 .06868 L .2326 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.29731 .06414 L .29837 .03587 L .29948 .07012 L .30061 .07143 L .30181 .07546 L .30295 .07423 L .30399 .06553 L .3052 .05169 L .30589 .06323 L .30652 .07051 L .30769 .06991 L .30892 .07529 L .31012 .07104 L .31142 .06014 L .31266 .05697 L .3138 .06956 L .31494 .05806 L .31596 .07309 L .31715 .06824 L .31826 .05975 L .31949 .0525 L .32078 .06843 L .32188 .05838 L .32309 .07201 L .32439 .06406 L .32511 .05871 L .32579 .05373 L .32642 .04991 L .3271 .06192 L .3283 .06649 L .32942 .06595 L .33048 .0693 L .33163 .06005 L Mistroke .33287 .05047 L .33417 .06149 L .33541 .06389 L .33652 .06601 L .33773 .066 L .33887 .05673 L .33991 .04764 L .34111 .05977 L .34224 .06329 L .34329 .06192 L .34425 .06664 L .34531 .05917 L .34643 .05199 L .34764 .0533 L .34832 .06054 L .34895 .06394 L .3501 .05089 L .35133 .07179 L .35243 .07258 L .35346 .06886 L .35459 .06486 L .3558 .05186 L .35702 .07185 L .35771 .07524 L .35836 .07411 L .35966 .07393 L .36038 .0782 L .36105 .07663 L .36223 .06733 L .36352 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.43001 .07178 L .43122 .07026 L .43236 .06679 L .43339 .06392 L .43404 .06067 L .43472 .04817 L .43594 .06669 L .43662 .07135 L Mistroke .43725 .07373 L .43794 .07346 L .43869 .06612 L .43933 .0667 L .44003 .07509 L .4413 .07568 L .44254 .06837 L .44368 .06195 L .44487 .05504 L .44616 .06056 L .44727 .06868 L .44848 .07011 L .44918 .06338 L .44983 .06414 L .45051 .07235 L .45124 .07434 L .45193 .07308 L .45257 .07006 L .45378 .06343 L .45508 .05795 L .45573 .04886 L .45645 .05721 L .45773 .06801 L .45892 .07025 L .45964 .06527 L .46033 .06099 L .46161 .07318 L .46233 .07243 L .46299 .06939 L .4645 .06026 L .46578 .05429 L .46642 .03785 L .46713 .05638 L .46838 .0657 L .46954 .06716 L .47019 .06225 L .47089 .05995 L .47217 .07071 L .47294 .06983 L .47365 .06664 L .47499 .05931 L .47608 .05496 L .47726 .04844 L .47856 .0627 L .47977 .06651 L .48107 .05212 L .48172 .0636 L .48243 .06872 L .48368 .06773 L .48485 .06123 L Mistroke .48601 .05598 L .48726 .04708 L .48844 .05607 L .48953 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The \ trade-off for using higher precision is, of course, increased calculation \ times.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Random testing", "Subsection"], Cell["\<\ Often, you can get a pretty good indication of whether a result is plausible \ by plugging in random numbers.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(res = Integrate[Sin[Tan[x]], x]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(\(1\/\(4\ \[ExponentialE]\)\)\((\[ExponentialE]\^2\ \ \(Ci(\[ImaginaryI] - tan(x))\) - Ci(\[ImaginaryI] - tan(x)) + \[ExponentialE]\^2\ \(Ci(\[ImaginaryI] + tan(x))\) - Ci(\[ImaginaryI] + tan(x)) + \[ExponentialE]\^2\ \[ImaginaryI]\ \(Si(\[ImaginaryI] - tan(x))\) + \[ImaginaryI]\ \(Si(\[ImaginaryI] - tan(x))\) + \[ExponentialE]\^2\ \[ImaginaryI]\ \(Si(\ \[ImaginaryI] + tan(x))\) + \[ImaginaryI]\ \(Si(\[ImaginaryI] + tan(x))\))\)\)\)], "Output"] }, Open ]], Cell["\<\ The following command calculated the difference between the differentiated \ result and the original integrand at 20 random points.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ Chop[\((D[res, x] - Sin[Tan[x]])\) /. x \[Rule] Random[Complex]], {20}]\)], "Input"], Cell[BoxData[ \({0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"4.1 for X", ScreenRectangle->{{0, 800}, {0, 600}}, WindowSize->{791, 547}, WindowMargins->{{Automatic, 3}, {Automatic, -6}}, CellLabelAutoDelete->True, Magnification->1.25, StyleDefinitions -> "ArticleClassic.nb" ] (*********************************************************************** Cached data follows. 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