(*********************************************************************** 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[ 30268, 1088]*) (*NotebookOutlinePosition[ 31419, 1124]*) (* CellTagsIndexPosition[ 31375, 1120]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[ "Math 325: Differential\tEquations Quiz 2 9/20/96"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell["<True, AspectRatioFixed->True], Cell[TextData[{ "1. Solve the initial value problem and sketch the solution\n", StyleBox[ " y''''[t] + y''[t] = DiracDelta[t - Pi],\n y[0] == 0, y'[0] == 0, \ y''[0] == 0, y'''[0] == 0", "Input"] }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell["F[s_] := LaplaceTransform[y[t],t,s];", "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ eqn = LaplaceTransform[y''''[t] + y''[t] == DiracDelta[t-Pi],t,s] \ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{ \(s\^2\ LaplaceTransform[y[t], t, s]\), "+", \(s\^4\ LaplaceTransform[y[t], t, s]\), "-", \(s\ y[0]\), "-", \(s\^3\ y[0]\), "-", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "0", "]"}], "-", RowBox[{\(s\^2\), " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "0", "]"}]}], "-", RowBox[{"s", " ", RowBox[{ SuperscriptBox["y", "\[DoublePrime]", MultilineFunction->None], "[", "0", "]"}]}], "-", RowBox[{ SuperscriptBox["y", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "0", "]"}]}], "==", \(E\^\(\(-\[Pi]\)\ s\)\)}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ tran = Solve[eqn/.{y[0]->0,y'[0]->0,y''[0]->0,y'''[0]->0}, \ F[s]]\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{LaplaceTransform[y[t], t, s] \[Rule] E\^\(\(-\[Pi]\)\ s\)\/\(s\^2\ \((1 + s\^2)\)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Apart[1/(s^2(s^2+1))]", "Input"], Cell[BoxData[ \(1\/s\^2 - 1\/\(1 + s\^2\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["soln = InverseLaplaceTransform[F[s]/.First[tran],s,t]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\((\(-\[Pi]\) + t + Sin[t])\)\ UnitStep[\(-\[Pi]\) + t]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Plot[Evaluate[soln], {t,0,8Pi}]", "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 /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.037894 0.0147151 0.0267655 [ [.21328 .00222 -3 -9 ] [.21328 .00222 3 0 ] [.40275 .00222 -6 -9 ] [.40275 .00222 6 0 ] [.59222 .00222 -6 -9 ] [.59222 .00222 6 0 ] [.78169 .00222 -6 -9 ] [.78169 .00222 6 0 ] [.97116 .00222 -6 -9 ] [.97116 .00222 6 0 ] [.01131 .14854 -6 -4.5 ] [.01131 .14854 0 4.5 ] [.01131 .28237 -12 -4.5 ] [.01131 .28237 0 4.5 ] [.01131 .4162 -12 -4.5 ] [.01131 .4162 0 4.5 ] [.01131 .55002 -12 -4.5 ] [.01131 .55002 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21328 .01472 m .21328 .02097 L s [(5)] .21328 .00222 0 1 Mshowa .40275 .01472 m .40275 .02097 L s [(10)] .40275 .00222 0 1 Mshowa .59222 .01472 m .59222 .02097 L s [(15)] .59222 .00222 0 1 Mshowa .78169 .01472 m .78169 .02097 L s [(20)] .78169 .00222 0 1 Mshowa .97116 .01472 m .97116 .02097 L s [(25)] .97116 .00222 0 1 Mshowa .125 Mabswid .0617 .01472 m .0617 .01847 L s .0996 .01472 m .0996 .01847 L s .13749 .01472 m .13749 .01847 L s .17539 .01472 m .17539 .01847 L s .25117 .01472 m .25117 .01847 L s .28907 .01472 m .28907 .01847 L s .32696 .01472 m .32696 .01847 L s .36486 .01472 m .36486 .01847 L s .44064 .01472 m .44064 .01847 L s .47854 .01472 m .47854 .01847 L s .51643 .01472 m .51643 .01847 L s .55433 .01472 m .55433 .01847 L s .63011 .01472 m .63011 .01847 L s .66801 .01472 m .66801 .01847 L s .7059 .01472 m .7059 .01847 L s .7438 .01472 m .7438 .01847 L s .81958 .01472 m .81958 .01847 L s .85748 .01472 m .85748 .01847 L s .89537 .01472 m .89537 .01847 L s .93327 .01472 m .93327 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .02381 .14854 m .03006 .14854 L s [(5)] .01131 .14854 1 0 Mshowa .02381 .28237 m .03006 .28237 L s [(10)] .01131 .28237 1 0 Mshowa .02381 .4162 m .03006 .4162 L s [(15)] .01131 .4162 1 0 Mshowa .02381 .55002 m .03006 .55002 L s [(20)] .01131 .55002 1 0 Mshowa .125 Mabswid .02381 .04148 m .02756 .04148 L s .02381 .06825 m .02756 .06825 L s .02381 .09501 m .02756 .09501 L s .02381 .12178 m .02756 .12178 L s .02381 .17531 m .02756 .17531 L s .02381 .20207 m .02756 .20207 L s .02381 .22884 m .02756 .22884 L s .02381 .2556 m .02756 .2556 L s .02381 .30914 m .02756 .30914 L s .02381 .3359 m .02756 .3359 L s .02381 .36267 m .02756 .36267 L s .02381 .38943 m .02756 .38943 L s .02381 .44296 m .02756 .44296 L s .02381 .46973 m .02756 .46973 L s .02381 .49649 m .02756 .49649 L s .02381 .52326 m .02756 .52326 L s .02381 .57679 m .02756 .57679 L s .02381 .60356 m .02756 .60356 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .01472 m .06244 .01472 L .08255 .01472 L .10458 .01472 L .11448 .01472 L .12507 .01472 L .1302 .01472 L .13506 .01472 L .13942 .01472 L .14061 .01472 L .14188 .01472 L .14296 .01472 L .14415 .01472 L .14544 .01472 L .14664 .01472 L .14795 .01473 L .14868 .01473 L .14935 .01474 L .15057 .01475 L .15189 .01478 L .15314 .0148 L .15429 .01484 L .15644 .01492 L .1588 .01504 L .16126 .01522 L .16356 .01543 L .16627 .01575 L .16883 .01612 L .17364 .01703 L .17881 .01836 L .18444 .02026 L .19384 .02464 L .20421 .03132 L .21505 .04042 L .22522 .05084 L .26391 .10163 L .28356 .12857 L .29395 .14146 L .30505 .15358 L .3158 .16334 L .32558 .17038 L .33029 .17314 L .3354 .17568 L .34467 .17915 L .34956 .18044 L .35414 .18135 L .35916 .18205 L .36197 .18233 L .36455 .18253 L .36585 .18261 L Mistroke .36723 .18268 L .36971 .18277 L .37124 .18281 L .37267 .18284 L .37395 .18286 L .37468 .18287 L .37534 .18287 L .37657 .18288 L .37772 .18288 L .37898 .18289 L .37969 .18289 L .38033 .18289 L .38163 .18289 L .38236 .18289 L .38303 .18289 L .38428 .18289 L .38559 .1829 L .38673 .1829 L .38798 .18292 L .38929 .18293 L .3905 .18296 L .39267 .18302 L .39388 .18306 L .39499 .18311 L .39749 .18325 L .40019 .18346 L .40275 .18372 L .4051 .18402 L .40934 .18471 L .41399 .18573 L .41885 .18713 L .42339 .18877 L .43383 .19388 L .44372 .20056 L .46227 .21787 L .50057 .26778 L .51993 .29449 L .54132 .31989 L .55105 .32913 L .56155 .33717 L .57146 .3429 L .58055 .34662 L .58528 .34803 L .59034 .34918 L .59279 .34961 L .59511 .34996 L .59945 .35045 L .60196 .35066 L .60461 .35082 L .60688 .35091 L Mistroke .60817 .35096 L .60938 .35099 L .61077 .35101 L .61209 .35103 L .61327 .35104 L .61457 .35105 L .6153 .35106 L .61597 .35106 L .61674 .35106 L .61746 .35106 L .61875 .35106 L .61948 .35106 L .62014 .35106 L .62141 .35106 L .62261 .35106 L .62368 .35107 L .62485 .35108 L .62612 .35109 L .62676 .3511 L .62747 .35111 L .62872 .35113 L .62989 .35116 L .63217 .35124 L .63427 .35135 L .63656 .3515 L .63898 .3517 L .64146 .35197 L .6441 .35232 L .64886 .35317 L .65451 .35456 L .65965 .35624 L .66964 .36077 L .67885 .36654 L .68896 .37467 L .69965 .38526 L .73756 .43439 L .75866 .46349 L .77792 .48651 L .78746 .49582 L .79785 .50412 L .80767 .51012 L .81676 .51413 L .82186 .51579 L .82737 .51713 L .8299 .51762 L .83257 .51804 L .83487 .51834 L .83734 .5186 L .83975 .51881 L .84194 .51895 L Mistroke .84415 .51905 L .84623 .51913 L .84741 .51916 L .84866 .51918 L .84972 .5192 L .8509 .51921 L .85216 .51922 L .85287 .51923 L .85351 .51923 L .85477 .51923 L .85597 .51923 L .85721 .51923 L .85791 .51923 L .85855 .51923 L .85974 .51923 L .86099 .51924 L .86211 .51924 L .86316 .51925 L .86431 .51926 L .86555 .51928 L .86686 .51931 L .86829 .51935 L .86964 .51939 L .87089 .51944 L .87325 .51957 L .87576 .51976 L .87793 .51996 L .88027 .52023 L .8845 .52087 L .88953 .52192 L .89415 .52319 L .89888 .52484 L .90394 .52702 L .91306 .53209 L .9239 .54011 L .93376 .54926 L .97373 .59984 L .97619 .60332 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[ooonICXf=SLcII03=Voc=Vc3=V VC=VIS=V0000ol0003@0000 4P000ol0000i0003o`000P000ol0003>0000 4P000ol0000k0003o`0000003o`0007X0000B00;oT003ogT0000B0003o`0009800_mg00004P000ol0002D00;o M@00018000?o0000UP04og40000B0003o`0009X03omR00004P000ol0002Y00CoGP00018000?o0000 [@000ol0001K00004P02ojl000?o0000FP00018000?o0000[`02oeX0000B0003o`000;4000?o0000 E`00018000?o0000/P000ol0001F00004P000ol0002c0003o`0005D0000B0003o`000;@000?o0000 E000018000?o0000]@000ol0001C00004P02okH000?o0000D`00018000?o0000]P000ol0001B0000 4P000ol0002g0003o`000540000B0003o`000;P000?o0000D000018000?o0000^0000ol0001@0000 0`03o`<00ol60003o`000;T000?o0000C`0000@000?o00001@000ol000030003o`000;X000?o0000 CP0000@000?o00001@000ol0000300;o_0000ol0001=000010000ol0000200?o1P000ol0002l0003 o`0004`000040003o`00008000?o00001P000ol0002l0003o`0004`0000300;o1004o`D000?o0000 _@000ol0001;00004P000ol0002n0003o`0004X0000B0003o`000;l000?o0000B@00018000?o0000 _`000ol0001900004P000ol000300003o`0004P0000B00;o`P000ol0001700004P000ol000320003 o`0004H0000B0003o`000<8000?o0000AP00018000?o0000``000ol0001500004P000ol000340003 o`0004@0000B0003o`000000018000?o0000 d@000ol0000g00004P000ol0003B00?o=P00018000?o0000e@000ol0000c00004P000ol0003F00Oo ;P00018000?o0000g00>ob40000B00;oj`03oah0000B0003o`000>d00_lL00004P000ol0003_00;o 6P00018000?o0000l@000ol0000G00004P000ol0003b0003o`0001H0000300Co0`02o`H000?o0000 l`000ol0000E000010000ol000020004o`00o`D000?o0000m0000ol0000D00001@001Ol0003o0080 00?o00000`02ooH000?o00004`0000H000Co003o0P000ol000030003o`000?D000?o00004`0000<0 00Co003o0P001?l00?l50003o`000?H000?o00004P0000@00_l400;o1P000ol0003g0003o`000140 000B0003o`000?P000?o00004000018000?o0000n@000ol0000?00004P000ol0003i0003o`0000l0 000B00;on`000ol0000>00004P000ol0003k0003o`0000d0000B0003o`000?`000?o000030000180 00?o0000o0000ol0000<00004P000ol0003m0003o`0000/0000B0003o`000?h000?o00002P000180 00?o0000o`000ol0000900004P000ol0003o004000?o0000200001800_oo00d0000B0003o`000?l0 3000018000?o0000o`0<00004P000ol0003o00`00000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-1.75124, -1.69997, 0.0967683, 0.137002}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ " 2. Express the solution to the initial value problem\n", StyleBox[ " y''[t] + 8 y'[t] + 25y[t] == Tan[t],\n y[0] == 0, y'[0] == 0", "Input"], "\nin terms of a convolution integral." }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell["F[s_] := LaplaceTransform[y[t],t,s];", "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ "eqn = LaplaceTransform[", StyleBox["y''[t] + 8 y'[t] + 25y[t] == Tan[t]", "Input"], ",t,s]\n" }], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{ \(25\ LaplaceTransform[y[t], t, s]\), "+", \(s\^2\ LaplaceTransform[y[t], t, s]\), "+", \(8\ \((s\ LaplaceTransform[y[t], t, s] - y[0])\)\), "-", \(s\ y[0]\), "-", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "0", "]"}]}], "==", \(LaplaceTransform[Tan[t], t, s]\)}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["tran = Solve[eqn/.{y[0]->0,y'[0]->0},F[s]]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{LaplaceTransform[y[t], t, s] \[Rule] LaplaceTransform[Tan[t], t, s]\/\(25 + 8\ s + s\^2\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["soln = InverseLaplaceTransform[F[s]/.First[tran],s,t]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(Integrate::"gener" \( : \ \) "Unable to check convergence"\)], "Message"], Cell[BoxData[ \(\(1\/1275\(( 68 - 200\ Cos[2\ t] + \((200 - 50\ I)\)\ E\^\(I\ t\)\ Cos[3\ t]\ Hypergeometric2F1[1\/2 - 2\ I, 1, 3\/2 - 2\ I, \(-E\^\(2\ I\ t\)\)] + 50\ Sin[2\ t])\)\) - \(1\/2550\(( I\ E\^\(\(-4\)\ t\)\ \(( Cos[3\ t]\ \((264\ I + 425\ PolyGamma[0, 1\/4 - I] - 425\ PolyGamma[0, 3\/4 - I])\) + 2\ I\ Sin[3\ t])\))\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "We know the inverse Laplace tranform of ", Cell[BoxData[ StyleBox[\(LaplaceTransform[Tan[t], t, s]\/\(25 + 8\ s + s\^2\)\), "Input"]]], "is the convolution of ", StyleBox["Tan[t]", "Input"], " and" }], "Text"], Cell[CellGroupData[{ Cell["h[t_] = InverseLaplaceTransform[1/(s^2+8s+25),s,t]", "Input"], Cell[BoxData[ \(1\/3\ E\^\(\(-4\)\ t\)\ Sin[3\ t]\)], "Output"] }, Open ]], Cell[TextData[{ "(Note: ", StyleBox["s^2+8s+25 == (s+4)^2 + 3^2", "Input"], " .) Therefore," }], "Text"], Cell[CellGroupData[{ Cell["y[t_] = Integrate[Tan[t-u] (1/3)E^(-4u) Sin[3u], {u,0,t}]", "Input"], Cell[BoxData[ \(1\/3\ \(\[Integral]\_0\%t\( E\^\(\(-4\)\ u\)\ Sin[3\ u]\ Tan[t - u]\) \[DifferentialD]u\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Plot[Evaluate[y[t]], {t,0, Pi/4}]", "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 /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 1.21261 0.0147151 28.6545 [ [.26633 .00222 -9 -9 ] [.26633 .00222 9 0 ] [.50885 .00222 -9 -9 ] [.50885 .00222 9 0 ] [.75138 .00222 -9 -9 ] [.75138 .00222 9 0 ] [.9939 .00222 -9 -9 ] [.9939 .00222 9 0 ] [.01131 .15799 -30 -4.5 ] [.01131 .15799 0 4.5 ] [.01131 .30126 -24 -4.5 ] [.01131 .30126 0 4.5 ] [.01131 .44453 -30 -4.5 ] [.01131 .44453 0 4.5 ] [.01131 .58781 -24 -4.5 ] [.01131 .58781 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .26633 .01472 m .26633 .02097 L s [(0.2)] .26633 .00222 0 1 Mshowa .50885 .01472 m .50885 .02097 L s [(0.4)] .50885 .00222 0 1 Mshowa .75138 .01472 m .75138 .02097 L s [(0.6)] .75138 .00222 0 1 Mshowa .9939 .01472 m .9939 .02097 L s [(0.8)] .9939 .00222 0 1 Mshowa .125 Mabswid .08444 .01472 m .08444 .01847 L s .14507 .01472 m .14507 .01847 L s .2057 .01472 m .2057 .01847 L s .32696 .01472 m .32696 .01847 L s .38759 .01472 m .38759 .01847 L s .44822 .01472 m .44822 .01847 L s .56948 .01472 m .56948 .01847 L s .63011 .01472 m .63011 .01847 L s .69074 .01472 m .69074 .01847 L s .81201 .01472 m .81201 .01847 L s .87264 .01472 m .87264 .01847 L s .93327 .01472 m .93327 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .02381 .15799 m .03006 .15799 L s [(0.005)] .01131 .15799 1 0 Mshowa .02381 .30126 m .03006 .30126 L s [(0.01)] .01131 .30126 1 0 Mshowa .02381 .44453 m .03006 .44453 L s [(0.015)] .01131 .44453 1 0 Mshowa .02381 .58781 m .03006 .58781 L s [(0.02)] .01131 .58781 1 0 Mshowa .125 Mabswid .02381 .04337 m .02756 .04337 L s .02381 .07202 m .02756 .07202 L s .02381 .10068 m .02756 .10068 L s .02381 .12933 m .02756 .12933 L s .02381 .18664 m .02756 .18664 L s .02381 .2153 m .02756 .2153 L s .02381 .24395 m .02756 .24395 L s .02381 .27261 m .02756 .27261 L s .02381 .32992 m .02756 .32992 L s .02381 .35857 m .02756 .35857 L s .02381 .38722 m .02756 .38722 L s .02381 .41588 m .02756 .41588 L s .02381 .47319 m .02756 .47319 L s .02381 .50184 m .02756 .50184 L s .02381 .5305 m .02756 .5305 L s .02381 .55915 m .02756 .55915 L s .02381 .61646 m .02756 .61646 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .01472 m .02499 .01472 L .02605 .01472 L .02729 .01472 L .02846 .01472 L .02954 .01472 L .03053 .01472 L .03163 .01472 L .03279 .01472 L .03395 .01472 L .0352 .01472 L .03746 .01472 L .03884 .01472 L .04016 .01473 L .04262 .01473 L .045 .01474 L .04753 .01475 L .0521 .01477 L .05752 .01481 L .06244 .01486 L .06787 .01493 L .07287 .01501 L .08414 .01525 L .09408 .01554 L .10458 .01595 L .12422 .01701 L .14255 .0184 L .1629 .02045 L .18146 .02282 L .22131 .02965 L .25964 .03863 L .30042 .05088 L .33968 .06535 L .37743 .0817 L .41764 .10169 L .45632 .12335 L .49746 .14888 L .53709 .17579 L .5752 .20371 L .61576 .23552 L .6548 .26809 L .6963 .30471 L .73628 .34188 L .77475 .37935 L .81567 .42104 L .85507 .46297 L .89296 .50496 L .9333 .55155 L .97212 .59831 L .97619 .60332 L Mistroke Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[ooonICXf=SLcII03=Voc=Vc3=V VC=VIS=VP000ol0000k0003o`0000L0000Q0003o`0002P01Oo?00008@000ol0000]00GobP000240 00?o0000`00024000?o0000_`000ol0000j00008@000ol000300003o`0003T0000Q0003o`000<40 00?o0000>000024000?o0000`P000ol0000g00008@02ol@000?o0000=P00024000?o0000a0000ol0 000e00008@000ol000350003o`0003@0000Q0003o`0000003o`0002/0000Q0003o`0008000?o00005`00024000?o0000hP000ol0000G00008@02on@0 00?o00005P00024000?o0000i0000ol0000E00008@000ol0003U0003o`0001@0000Q0003o`000>H0 00?o00004`00024000?o0000i`000ol0000B00008@000ol0003W0003o`000180000800;o1002o`@0 0_l300Co10000ol0003X0003o`00014000070004o`00o`<00_l30004o`00o`<000?o000010000ol0 003Y0003o`00010000070004o`00o`P000Co003o10000ol0000300;oj`000ol0000?00001`001?l0 0?l80004o`00o`D000?o00000P000ol0003[0003o`0000h000070004o`00o`P000Co003o0P001?l0 0?l40003o`000>`000?o00003@0000P00_l:00;o1002o`D000?o0000k0000ol0000=00008@000ol0 003]0003o`0000`0000Q0003o`000?`0000Q0003o`000?`0000Q00;oo@000?l08@000?l08@000?l0 8@000?l08@000?l08@000?l08@000001\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.108334, -0.00254554, 0.00326741, 0.000138271}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 1024}, {0, 748}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Manual, WindowSize->{651, 639}, WindowMargins->{{4, Automatic}, {Automatic, 1}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@0000000000000006P801T1T00000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1709, 49, 135, 3, 44, "Subsection"], Cell[1847, 54, 99, 2, 27, "Input", InitializationCell->True], Cell[1949, 58, 276, 7, 71, "Subsubsection"], Cell[2228, 67, 91, 2, 41, "Subsubsection"], Cell[2322, 71, 79, 1, 27, "Input"], Cell[CellGroupData[{ Cell[2426, 76, 117, 4, 42, "Input"], Cell[2546, 82, 848, 22, 61, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3431, 109, 117, 4, 27, "Input"], Cell[3551, 115, 132, 2, 45, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3720, 122, 38, 0, 27, "Input"], Cell[3761, 124, 59, 1, 42, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3857, 130, 96, 1, 27, "Input"], Cell[3956, 133, 89, 1, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4082, 139, 48, 0, 27, "Input"], Cell[4133, 141, 12759, 496, 186, 5993, 408, "GraphicsData", "PostScript", "Graphics"], Cell[16895, 639, 130, 3, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17062, 647, 287, 8, 86, "Subsubsection"], Cell[17352, 657, 91, 2, 41, "Subsubsection"] }, Open ]], Cell[17458, 662, 79, 1, 27, "Input"], Cell[CellGroupData[{ Cell[17562, 667, 155, 5, 42, "Input"], Cell[17720, 674, 417, 10, 58, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[18174, 689, 85, 1, 27, "Input"], Cell[18262, 692, 138, 2, 42, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[18437, 699, 96, 1, 27, "Input"], Cell[18536, 702, 93, 1, 22, "Message"], Cell[18632, 705, 459, 11, 110, "Output"] }, Open ]], Cell[19106, 719, 250, 8, 34, "Text"], Cell[CellGroupData[{ Cell[19381, 731, 67, 0, 27, "Input"], Cell[19451, 733, 67, 1, 42, "Output"] }, Open ]], Cell[19533, 737, 110, 4, 31, "Text"], Cell[CellGroupData[{ Cell[19668, 745, 74, 0, 27, "Input"], Cell[19745, 747, 141, 3, 43, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[19923, 755, 50, 0, 27, "Input"], Cell[19976, 757, 10143, 323, 186, 3381, 235, "GraphicsData", "PostScript", "Graphics"], Cell[30122, 1082, 130, 3, 26, "Output"] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)