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Plot the vector field corresponding to this system and a \ representative sample of trajectories on a single graph. " }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["F[x_, y_] = y + x y;\nG[x_, y_] = x + y^2;"], "Input", AspectRatioFixed->True], Cell[TextData["Find the critical points:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["cp = Solve[{F[x,y] == 0, G[x,y] == 0}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {{x -> -1, y -> -1}, {x -> -1, y -> 1}, {x -> 0, y -> 0}}\ \>", "\<\ {{x -> -1, y -> -1}, {x -> -1, y -> 1}, {x -> 0, y -> 0}}\ \>"], "Output",\ Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "The critical points are thus ", StyleBox["{-1,-1}", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["{-1,1}", FontFamily->"Courier", FontWeight->"Bold"], ", and ", StyleBox["{0,0}", FontFamily->"Courier", FontWeight->"Bold"], ". To analyze them we need to examine the matrix of partial derivatives \ evaluated at each." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "A = {{D[F[x,y],x], D[F[x,y],y]},\n {D[G[x,y],x], D[G[x,y],y]}};\n\ A//MatrixForm"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ MatrixForm[{{y, 1 + x}, {1, 2*y}}]\ \>", "\<\ y 1 + x 1 2 y\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "At the critical point ", StyleBox["{-1,-1}", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["A1 = A /. cp[[1]];\nA1//MatrixForm"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ MatrixForm[{{-1, 0}, {1, -2}}]\ \>", "\<\ -1 0 1 -2\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Eigensystem[A1]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{-2, -1}, {{0, 1}, {1, 1}}}\ \>", "\<\ {{-2, -1}, {{0, 1}, {1, 1}}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "Since the eigenvalues have the same signs, the point ", StyleBox["{-1,-1}", FontFamily->"Courier", FontWeight->"Bold"], " is an improper node. It is asymptotically stable since the eigenvalues \ are negative.\nAt the point ", StyleBox["{-1,1}", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["A2 = A /. cp[[2]];\nA2//MatrixForm"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ MatrixForm[{{1, 0}, {1, 2}}]\ \>", "\<\ 1 0 1 2\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Eigensystem[A2]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{1, 2}, {{-1, 1}, {0, 1}}}\ \>", "\<\ {{1, 2}, {{-1, 1}, {0, 1}}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "The eigenvalues have the same signs, so the point ", StyleBox["{-1,0}", FontFamily->"Courier", FontWeight->"Bold"], " is again an improper node. 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StyleBox[ "\n x'[t] == y[t]\n y'[t] == -2.0 Sin[x[t]] - 0.75 y[t]\n\n", FontFamily->"Courier", FontWeight->"Bold"], "Here ", StyleBox["x[t]", FontFamily->"Courier", FontWeight->"Bold"], " is the angle of the pendulum from the down position and ", StyleBox["y[t] = x'[t]", FontFamily->"Courier", FontWeight->"Bold"], " is its angular velocity. Plot the vector field corresponding to this \ system and a representative sample of trajectories on a single graph. Then \ plot the trajectories for initial position down, ", StyleBox["theta[0] == 0", FontFamily->"Courier", FontWeight->"Bold"], ", and the initial velocity equal to ", StyleBox["b == 3.0", FontFamily->"Courier", FontWeight->"Bold"], ", to ", StyleBox["5.0", FontFamily->"Courier", FontWeight->"Bold"], " in steps of ", StyleBox["0.5", FontFamily->"Courier", FontWeight->"Bold"], ". Experiment to find a value for the initial velocity ", StyleBox["b", FontFamily->"Courier", FontWeight->"Bold"], " such that the trajectory approaches the unstable equilibrium point ", StyleBox["{Pi,0}", FontFamily->"Courier", FontWeight->"Bold"], " corresponding to the pendulum being stationary in an upright position." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "PlotVectorField[{y,-2.0Sin[x]-0.75y},{x,-2,8},{y,-2,5},\n \ ScaleFunction->(Log[50#+1]&), Frame->True]"], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .67201 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 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.08485 .23778 L .08685 .23762 L .08885 .23739 L .09084 .23708 L .09281 .2367 L .09672 .23571 L .10056 .23446 L .10794 .23119 L .11484 .22702 L .12114 .22212 L .12675 .21665 L .13162 .21075 L .1357 .20461 L .13896 .19835 L .14029 .19522 L .14141 .19212 L Mistroke .14233 .18906 L .14305 .18605 L .14357 .18311 L .14376 .18167 L .1439 .18025 L .144 .17885 L .14405 .17748 L .14406 .17613 L .14402 .17481 L .14394 .17351 L .14382 .17225 L .14366 .17102 L .14346 .16981 L .14294 .1675 L .14227 .16533 L .14147 .16329 L .14053 .1614 L .13948 .15967 L .13832 .15808 L .13706 .15665 L .13571 .15538 L .13279 .15332 L .13124 .15252 L .12964 .15188 L .128 .1514 L .12717 .15121 L .12634 .15106 L .1255 .15095 L .12466 .15088 L .12382 .15084 L .12297 .15083 L .12213 .15086 L .12129 .15093 L .12045 .15102 L .11961 .15115 L .11796 .1515 L .11633 .15198 L .11318 .15325 L .11023 .15492 L .10752 .15691 L .10508 .15916 L .10116 .16417 L .09971 .16681 L .09911 .16813 L .0986 .16945 L .09817 .17076 L .09783 .17204 L .0977 .17267 L .09758 .1733 L .09748 .17392 L Mistroke .0974 .17453 L .09735 .17513 L .09731 .17572 L .09729 .1763 L .09729 .17687 L .09731 .17743 L .09735 .17798 L .0974 .17852 L .09747 .17904 L .09756 .17955 L .09767 .18004 L .09793 .18099 L .09824 .18187 L .09862 .1827 L .09952 .18415 L .10003 .18478 L .10059 .18534 L .10118 .18584 L .1018 .18626 L .10245 .18662 L .10312 .18691 L .10346 .18703 L .1038 .18713 L .10415 .18722 L .1045 .1873 L .10485 .18735 L .10521 .18739 L .10556 .18742 L .10592 .18743 L .10627 .18742 L .10663 .18741 L .10698 .18737 L .10733 .18733 L .10803 .18719 L .10872 .18701 L .1094 .18677 L .11006 .1865 L .11131 .18582 L .11247 .18499 L .11352 .18406 L .11521 .18196 L .11584 .18085 L .11633 .17974 L .11667 .17864 L .11678 .17811 L .11683 .17784 L .11687 .17758 L .1169 .17733 L .11692 .17708 L .11693 .17683 L Mistroke .11693 .17658 L .11693 .17635 L .11692 .17611 L .1169 .17588 L .11687 .17566 L .11684 .17544 L .1168 .17523 L .1167 .17483 L .11657 .17445 L .11642 .17409 L .11624 .17376 L .11605 .17346 L .1156 .17295 L .11536 .17273 L .1151 .17254 L .11483 .17239 L .11455 .17226 L .11426 .17215 L .11397 .17208 L .11382 .17205 L .11367 .17203 L .11352 .17202 L .11337 .17201 L .11322 .17201 L .11307 .17201 L .11292 .17203 L .11278 .17204 L .11263 .17206 L .11248 .17209 L .11219 .17217 L .1119 .17226 L .11162 .17237 L .11109 .17265 L .11059 .17299 L .11015 .17338 L .10976 .17381 L .10942 .17426 L .10915 .17472 L .10894 .17519 L .10885 .17543 L .10878 .17566 L .10873 .17588 L .10869 .17611 L .10868 .17621 L .10867 .17632 L .10866 .17643 L .10866 .17653 L .10866 .17663 L .10866 .17673 L .10867 .17683 L Mistroke .10868 .17692 L .10869 .17702 L .10871 .17711 L .10875 .17728 L .1088 .17744 L .10886 .1776 L .10893 .17774 L .10901 .17787 L .10919 .17809 L .1093 .17818 L .1094 .17827 L .10952 .17833 L .10963 .17839 L .10975 .17844 L .10988 .17847 L .10994 .17848 L .11 .17849 L .11006 .1785 L .11013 .17851 L .11019 .17851 L .11025 .17851 L .11032 .1785 L .11038 .1785 L .11044 .17849 L .1105 .17848 L .11063 .17845 L .11075 .17841 L .11087 .17837 L .1111 .17825 L .11131 .17811 L .1115 .17795 L .11166 .17777 L .11181 .17758 L .11193 .17739 L .11202 .17719 L .11206 .17709 L .11209 .17699 L .11211 .1769 L .11213 .1768 L .11213 .17676 L .11214 .17671 L .11214 .17667 L .11214 .17662 L .11215 .17658 L .11214 .17654 L .11214 .1765 L .11214 .17646 L .11213 .17642 L .11213 .17638 L .11211 .1763 L Mistroke .11209 .17623 L .11207 .17617 L .11204 .17611 L .112 .17605 L .11193 .17596 L .11189 .17591 L .11184 .17588 L .11179 .17585 L .11175 .17582 L .1117 .1758 L .11164 .17579 L .11162 .17578 L .11159 .17578 L .11156 .17577 L .11154 .17577 L .11151 .17577 L .11149 .17577 L .11146 .17577 L .11143 .17577 L .11141 .17577 L .11138 .17578 L .11133 .17579 L .11128 .17581 L .11123 .17582 L .11113 .17587 L .11104 .17593 L .11096 .176 L .11082 .17615 L .11077 .17623 L .11073 .17632 L .1107 .1764 L .11069 .17644 L .11069 .17646 L .11068 .17648 L .11068 .1765 L .11068 .17652 L .11068 .17654 L .11068 .17655 L .11068 .17657 L .11068 .17659 L .11068 .17661 L .11068 .17663 L .11068 .17664 L .11068 .17666 L .11069 .17669 L .1107 .17672 L .11071 .17675 L .11072 .17677 L .11073 .1768 L .11076 .17684 L Mistroke .11078 .17686 L .1108 .17687 L .11082 .17689 L .11084 .1769 L .11086 .17691 L .11088 .17691 L .11089 .17692 L .1109 .17692 L .11091 .17692 L .11093 .17692 L .11094 .17692 L .11095 .17692 L .11096 .17692 L .11097 .17692 L .11098 .17692 L .11099 .17692 L .11101 .17692 L .11104 .17691 L .11106 .1769 L .1111 .17688 L .11114 .17686 L .11117 .17683 L Mfstroke P P p p .004 w .11111 .57389 m .3761 .30223 L .44642 .20821 L .45364 .20061 L .45661 .19801 L .45929 .19605 L .46054 .19528 L .46174 .19464 L .46291 .19413 L .46404 .19372 L .4646 .19356 L .46516 .19342 L .46571 .19331 L .46625 .19323 L .4668 .19317 L .46734 .19313 L .46788 .19311 L .46842 .19312 L .46897 .19315 L .46951 .1932 L .4706 .19336 L .47116 .19348 L .47171 .19361 L .47284 .19394 L .47517 .19483 L .47764 .19603 L .48315 .19938 L .48971 .20405 L .53447 .23955 L .57506 .26981 L .60179 .28739 L .63316 .3049 L .65051 .31289 L .66885 .31988 L .68802 .32543 L .69785 .32753 L .70281 .32839 L .7078 .32912 L .7128 .32971 L .71783 .33016 L .72287 .33045 L .72791 .33059 L .73296 .33058 L .73801 .33041 L .74304 .33007 L .74807 .32957 L .75307 .32891 L .75805 .32808 L .76791 .32592 L .77761 .3231 L Mistroke .7871 .31964 L .80531 .31093 L .82223 .30007 L .83761 .28748 L .85125 .27361 L .86301 .25889 L .87282 .24376 L .88063 .22859 L .88379 .22109 L .88646 .21369 L .88866 .20642 L .89038 .19931 L .89164 .19238 L .8921 .18899 L .89245 .18566 L .8927 .18238 L .89283 .17916 L .89287 .17599 L .8928 .17289 L .89263 .16986 L .89236 .16688 L .89199 .16398 L .89153 .16114 L .89034 .15568 L .8888 .15052 L .88693 .14565 L .88475 .1411 L .88228 .13686 L .87955 .13296 L .87657 .1294 L .87337 .12617 L .8664 .12078 L .86267 .11861 L .8588 .1168 L .85484 .11535 L .85282 .11475 L .85078 .11425 L .84873 .11383 L .84667 .1135 L .8446 .11325 L .84252 .11309 L .84044 .11302 L .83836 .11303 L .83628 .11312 L .8342 .11329 L .83213 .11354 L .83007 .11387 L .82598 .11476 L .82197 .11593 L .81804 .11739 L Mistroke .81051 .12105 L .80352 .12559 L .79717 .13085 L .78673 .14287 L .78272 .14929 L .77957 .15578 L .77727 .1622 L .77643 .16535 L .77609 .1669 L .77579 .16843 L .77555 .16994 L .77536 .17143 L .77521 .1729 L .77512 .17435 L .77507 .17577 L .77506 .17717 L .77511 .17853 L .77519 .17987 L .77532 .18118 L .77549 .18245 L .77596 .18491 L .77658 .18723 L .77735 .18941 L .77929 .19331 L .78045 .19502 L .78171 .19658 L .78307 .19797 L .78451 .1992 L .78603 .20027 L .78761 .20117 L .78925 .2019 L .79093 .20248 L .79178 .20271 L .79264 .2029 L .7935 .20305 L .79437 .20316 L .79525 .20324 L .79612 .20328 L .797 .20328 L .79787 .20325 L .79874 .20318 L .79961 .20308 L .80134 .20277 L .8022 .20257 L .80304 .20234 L .80471 .20179 L .80793 .20035 L .81093 .19853 L .81367 .19639 L .81823 .19142 L Mistroke .82 .18873 L .82141 .18599 L .82246 .18326 L .82286 .18191 L .82316 .18059 L .82338 .1793 L .82346 .17867 L .82352 .17805 L .82356 .17744 L .82358 .17683 L .82357 .17624 L .82355 .17566 L .82351 .1751 L .82346 .17454 L .82329 .17348 L .82318 .17297 L .82305 .17247 L .82275 .17152 L .82239 .17063 L .82197 .16981 L .82099 .16838 L .82043 .16776 L .81983 .16722 L .8192 .16674 L .81854 .16634 L .81786 .166 L .81716 .16574 L .8168 .16563 L .81644 .16554 L .81608 .16547 L .81571 .16541 L .81535 .16537 L .81498 .16534 L .81461 .16533 L .81424 .16534 L .81387 .16536 L .81351 .16539 L .81278 .16551 L .81241 .16558 L .81206 .16567 L .81135 .16589 L .80998 .16647 L .8087 .16722 L .80753 .1681 L .80649 .1691 L .80558 .17017 L .80481 .1713 L .80419 .17246 L .80394 .17304 L .80373 .17361 L Mistroke .80355 .17418 L .80341 .17474 L .80336 .17501 L .80331 .17529 L .80327 .17556 L .80324 .17582 L .80322 .17608 L .80321 .17634 L .80321 .17659 L .80321 .17684 L .80322 .17708 L .80324 .17732 L .80331 .17777 L .80335 .17799 L .8034 .1782 L .80352 .17861 L .80366 .17899 L .80383 .17935 L .80424 .17997 L .80447 .18023 L .80472 .18047 L .80498 .18068 L .80525 .18086 L .80554 .18101 L .80583 .18112 L .80598 .18117 L .80613 .18122 L .80629 .18125 L .80644 .18128 L .80659 .1813 L .80675 .18131 L .8069 .18132 L .80706 .18132 L .80722 .18132 L .80737 .1813 L .80752 .18129 L .80768 .18126 L .80798 .1812 L .80828 .18111 L .80886 .18088 L .80941 .18057 L .80991 .1802 L .81035 .17979 L .81074 .17934 L .81108 .17886 L .81134 .17838 L .81145 .17813 L .81155 .17789 L .81162 .17765 L .81169 .17741 L Mistroke .81171 .1773 L .81173 .17718 L .81175 .17707 L .81177 .17696 L .81178 .17684 L .81178 .17674 L .81179 .17663 L .81179 .17652 L .81178 .17642 L .81178 .17632 L .81177 .17622 L .81175 .17612 L .81174 .17603 L .81172 .17594 L .81167 .17577 L .81161 .1756 L .81154 .17545 L .81137 .17518 L .81128 .17507 L .81118 .17496 L .81107 .17487 L .81095 .1748 L .81083 .17473 L .81071 .17468 L .81065 .17466 L .81058 .17464 L .81052 .17462 L .81045 .17461 L .81039 .1746 L .81032 .17459 L .81026 .17459 L .81019 .17458 L .81013 .17458 L .81006 .17459 L .81 .17459 L .80993 .1746 L .80987 .17461 L .8098 .17463 L .80968 .17466 L .80955 .17471 L .80943 .17476 L .8092 .17488 L .80899 .17504 L .80863 .1754 L .80849 .17559 L .80837 .1758 L .80828 .176 L .80825 .17611 L .80822 .17621 L .8082 .1763 L Mistroke .80819 .17635 L .80818 .1764 L .80818 .17645 L .80817 .17649 L .80817 .17654 L .80817 .17658 L .80817 .17663 L .80817 .17667 L .80818 .17671 L .80818 .17675 L .80819 .17679 L .8082 .17683 L .80822 .17691 L .80824 .17698 L .80827 .17704 L .80834 .17716 L .80838 .17721 L .80842 .17725 L .80846 .17729 L .80851 .17733 L .80856 .17735 L .80861 .17738 L .80866 .1774 L .80869 .1774 L .80872 .17741 L .80875 .17742 L .80877 .17742 L .8088 .17742 L .80883 .17742 L .80886 .17742 L .80888 .17742 L .80891 .17742 L .80894 .17742 L .80897 .17741 L .80899 .17741 L .80905 .17739 L .8091 .17738 L .80915 .17736 L .80925 .1773 L .80934 .17724 L Mfstroke P P p p .004 w .11111 .61803 m .30037 .45352 L .41041 .33284 L .44686 .30109 L .46247 .29089 L .46983 .287 L .47696 .28384 L .48391 .28137 L .48733 .28038 L .49072 .27954 L .49408 .27885 L .49742 .2783 L .50075 .2779 L .50406 .27763 L .50737 .27749 L .51068 .27748 L .51399 .27758 L .5173 .27781 L .52396 .2786 L .52731 .27916 L .53068 .27982 L .58967 .30124 L .62429 .31608 L .64305 .32327 L .66271 .32968 L .68314 .33481 L .69357 .33673 L .69884 .33751 L .70412 .33815 L .70942 .33865 L .71474 .33901 L .72007 .33922 L .7254 .33927 L .73073 .33916 L .73605 .33889 L .74137 .33845 L .74666 .33784 L .75194 .33706 L .75718 .33611 L .76756 .33368 L .77776 .33058 L .78773 .3268 L .80683 .31738 L .82455 .30573 L .85483 .27756 L .86706 .262 L .87721 .24607 L .88527 .23014 L .88853 .22228 L .89127 .21454 L Mistroke .89351 .20695 L .89525 .19953 L .89652 .1923 L .89698 .18877 L .89732 .1853 L .89755 .18188 L .89767 .17853 L .89768 .17523 L .89758 .17201 L .89738 .16885 L .89707 .16576 L .89667 .16274 L .89617 .15979 L .89488 .15411 L .89323 .14873 L .89123 .14367 L .88892 .13893 L .88341 .13046 L .87689 .12338 L .86953 .11775 L .8656 .11548 L .86153 .11359 L .85735 .11206 L .85522 .11144 L .85308 .11091 L .85092 .11047 L .84875 .11012 L .84656 .10987 L .84437 .1097 L .84218 .10962 L .83999 .10963 L .83779 .10972 L .8356 .1099 L .83342 .11016 L .83125 .11051 L .82695 .11144 L .82272 .11267 L .81858 .1142 L .81064 .11805 L .80327 .12283 L .79658 .12836 L .79066 .13448 L .78556 .14101 L .78134 .14777 L .77801 .1546 L .77668 .158 L .77557 .16136 L .77469 .16467 L .77401 .16791 L .77375 .16951 L Mistroke .77355 .17108 L .77339 .17263 L .77329 .17415 L .77323 .17564 L .77323 .17711 L .77327 .17855 L .77336 .17996 L .77349 .18134 L .77367 .18268 L .77415 .18527 L .7748 .18772 L .77561 .19001 L .77765 .19412 L .77886 .19594 L .78018 .19758 L .78161 .19905 L .78312 .20035 L .78638 .20243 L .7881 .20322 L .78987 .20383 L .79167 .20428 L .79258 .20444 L .7935 .20457 L .79442 .20465 L .79534 .20469 L .79626 .2047 L .79718 .20467 L .7981 .2046 L .79902 .20449 L .79993 .20435 L .80084 .20418 L .80263 .20373 L .80439 .20315 L .80778 .20165 L .81095 .19974 L .81384 .19749 L .81642 .19498 L .81866 .19227 L .82053 .18944 L .82202 .18655 L .82263 .18511 L .82314 .18368 L .82356 .18226 L .82389 .18087 L .82402 .18019 L .82412 .17951 L .82421 .17885 L .82427 .17819 L .82431 .17754 L .82434 .17691 L Mistroke .82434 .17629 L .82432 .17567 L .82428 .17508 L .82422 .17449 L .82404 .17337 L .82393 .17283 L .8238 .1723 L .82348 .1713 L .82311 .17036 L .82267 .1695 L .82164 .16798 L .82105 .16733 L .82043 .16675 L .81977 .16625 L .81907 .16582 L .81836 .16546 L .81762 .16518 L .81724 .16506 L .81686 .16497 L .81648 .16489 L .8161 .16482 L .81571 .16478 L .81532 .16475 L .81494 .16474 L .81455 .16474 L .81416 .16476 L .81377 .1648 L .81339 .16485 L .813 .16491 L .81224 .16508 L .8115 .16531 L .81006 .16592 L .80871 .1667 L .80747 .16763 L .80637 .16867 L .8054 .1698 L .80459 .17099 L .80394 .17221 L .80367 .17282 L .80345 .17342 L .80326 .17402 L .80311 .17461 L .80305 .1749 L .803 .17519 L .80296 .17547 L .80293 .17575 L .8029 .17603 L .80289 .1763 L .80289 .17656 L .80289 .17682 L Mistroke .8029 .17708 L .80292 .17733 L .80299 .17781 L .80303 .17804 L .80308 .17827 L .80321 .1787 L .80336 .1791 L .80354 .17947 L .80374 .17982 L .80396 .18013 L .80446 .18067 L .80474 .18089 L .80503 .18108 L .80533 .18123 L .80564 .18136 L .80579 .18141 L .80595 .18146 L .80611 .18149 L .80627 .18152 L .80644 .18155 L .8066 .18156 L .80676 .18157 L .80693 .18157 L .80709 .18157 L .80725 .18156 L .80742 .18154 L .80758 .18151 L .8079 .18145 L .80822 .18136 L .80853 .18124 L .80883 .18111 L .8094 .18079 L .80993 .18041 L .8104 .17997 L .81081 .1795 L .81116 .179 L .81145 .17849 L .81166 .17797 L .81175 .17772 L .81181 .17747 L .81184 .17735 L .81186 .17723 L .81188 .17711 L .8119 .17699 L .81191 .17687 L .81192 .17676 L .81192 .17664 L .81192 .17653 L .81192 .17642 L .81191 .17632 L Mistroke .8119 .17621 L .81189 .17611 L .81185 .17592 L .8118 .17573 L .81174 .17556 L .81167 .1754 L .81158 .17525 L .81149 .17512 L .81128 .17488 L .81117 .17479 L .81105 .1747 L .81092 .17464 L .81079 .17458 L .81073 .17456 L .81066 .17453 L .81059 .17452 L .81053 .1745 L .81046 .17449 L .81039 .17448 L .81032 .17448 L .81025 .17448 L .81018 .17448 L .81011 .17448 L .81004 .17449 L .80998 .1745 L .80984 .17452 L .80971 .17456 L .80958 .1746 L .80945 .17466 L .8092 .17479 L .80898 .17495 L .80878 .17513 L .8086 .17533 L .80845 .17554 L .80833 .17575 L .80823 .17597 L .8082 .17608 L .80817 .17618 L .80814 .17629 L .80813 .17639 L .80812 .17644 L .80812 .17649 L .80811 .17653 L .80811 .17658 L .80811 .17663 L .80812 .17667 L .80812 .17672 L .80813 .17676 L .80813 .1768 L .80814 .17684 L Mistroke .80816 .17692 L .80818 .177 L .80821 .17706 L .80825 .17713 L .80829 .17719 L .80837 .17729 L .80842 .17733 L .80847 .17736 L .80852 .1774 L .80858 .17742 L .80863 .17744 L .80869 .17745 L .80872 .17746 L .80875 .17746 L .80878 .17747 L .8088 .17747 L .80883 .17747 L .80886 .17747 L .80889 .17746 L .80892 .17746 L .80895 .17746 L .80898 .17745 L .80903 .17744 L .80909 .17742 L .80914 .1774 L .80925 .17734 L .80934 .17728 L .80943 .1772 L .80951 .17712 L .80957 .17703 L .80962 .17694 L .80966 .17685 L .80969 .17676 L .8097 .17672 L .80971 .17669 L .80971 .17667 L .80971 .17665 L .80972 .17663 L .80972 .17661 L .80972 .17659 L .80972 .17657 L .80972 .17655 L .80972 .17653 L .80971 .17651 L .80971 .1765 L .80971 .17648 L .8097 .17645 L .80969 .17641 L .80968 .17638 L .80967 .17636 L Mistroke .80965 .17633 L .80961 .17629 L .80959 .17627 L .80957 .17625 L .80955 .17624 L .80953 .17623 L .80951 .17622 L .80948 .17621 L .80947 .17621 L .80946 .17621 L .80945 .17621 L .80943 .17621 L .80942 .17621 L .80941 .17621 L .8094 .17621 L .80938 .17621 L .80937 .17621 L .80936 .17621 L .80934 .17622 L .80931 .17623 L .80929 .17624 L .80925 .17626 L .8092 .17628 L .80917 .17632 L .80914 .17635 L .80908 .17643 L Mfstroke P P P % End of Graphics MathPictureEnd \ \>"], "Graphics", Evaluatable->False, AspectRatioFixed->True, ImageSize->{306, 189}, ImageMargins->{{34, Inherited}, {Inherited, Inherited}}, ImageCacheValid->False], Cell[OutputFormData["\<\ Graphics[\"<<>>\"]\ \>", "\<\ -Graphics-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "Here the critical velocity appears to be something less than ", StyleBox["4.5", FontFamily->"Courier", FontWeight->"Bold"], ". For initial velocities below this, the pendulum swings back and forth \ with ever decreasing angles ", StyleBox["x", FontFamily->"Courier", FontWeight->"Bold"], " and velocity ", StyleBox["y", FontFamily->"Courier", FontWeight->"Bold"], " coordinates. For initial velocities above ", StyleBox["4.5", FontFamily->"Courier", FontWeight->"Bold"], ", the pendulum swings around once, bringing all the angles to a range \ between ", StyleBox["Pi", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["3Pi", FontFamily->"Courier", FontWeight->"Bold"], ". The damping causes the pendulum to slow down and again the velocity \ decreases to ", StyleBox["0", FontFamily->"Courier", FontWeight->"Bold"], " while the the angles approach ", StyleBox["2Pi", FontFamily->"Courier", FontWeight->"Bold"], ". Let's plot some solutions near the critical velocity." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "ParametricPlot[Evaluate[{x[t],y[t]} /. 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.42475 L .22607 .4248 L .22727 .42484 L .22847 .42488 L .22968 .42489 L .23088 .4249 L .23208 .42489 L Mistroke .23328 .42487 L .23448 .42484 L .23568 .42479 L .23688 .42474 L .23928 .42458 L .24168 .42438 L .24645 .42383 L .2512 .42308 L Mfstroke P P p p .004 w .18549 .37194 m .19085 .37533 L .19648 .37831 L .20234 .38087 L .20839 .38295 L .21148 .38382 L .2146 .38455 L .21774 .38516 L .22091 .38563 L .2225 .38582 L .2241 .38598 L .2257 .3861 L .2265 .38615 L .2273 .38619 L .2281 .38622 L .2289 .38625 L .2297 .38626 L .2305 .38627 L .2313 .38627 L .2321 .38626 L .2329 .38624 L .2337 .38621 L .23451 .38618 L .23531 .38614 L .2369 .38603 L .2385 .38588 L .24009 .3857 L .24327 .38525 L .24642 .38466 L .25263 .38311 L .2587 .38106 L .27022 .37559 L .27561 .37224 L .28071 .36852 L .2899 .36012 L .29761 .35071 L .3037 .34054 L .30609 .33526 L .30805 .32987 L .30955 .3244 L .31013 .32164 L .3106 .31887 L .31095 .3161 L .31108 .31471 L .31119 .31331 L .31126 .31192 L .31129 .31122 L .31131 .31053 L .31132 .30983 L .31132 .30913 L Mistroke .31132 .30844 L .31131 .30774 L .31129 .30704 L .31127 .30635 L .3112 .30495 L .3111 .30356 L .31097 .30217 L .31063 .29939 L .31018 .29663 L .30892 .29112 L .30721 .28569 L .30505 .28034 L .30245 .27511 L .29598 .26509 L .28792 .25589 L .27841 .24777 L .26767 .24104 L .26191 .23828 L .25594 .23598 L .2498 .23415 L .24668 .23343 L .24353 .23283 L .24036 .23236 L .23877 .23218 L .23717 .23203 L .23557 .23191 L .23477 .23187 L .23397 .23183 L .23317 .2318 L .23237 .23178 L .23157 .23177 L .23077 .23176 L .22997 .23177 L .22917 .23178 L .22837 .2318 L .22757 .23183 L .22676 .23187 L .22596 .23191 L .22437 .23203 L .22277 .23218 L .22118 .23236 L .21801 .23283 L .21486 .23343 L .21174 .23415 L .2056 .23598 L .19963 .23828 L .19387 .24104 L .18313 .24777 L .17362 .25589 L .16555 .26509 L Mistroke .16211 .27002 L .15909 .27511 L .15649 .28034 L .15433 .28569 L .15262 .29112 L .15136 .29663 L .15091 .29939 L .15072 .30078 L .15056 .30217 L .15044 .30356 L .15034 .30495 L .1503 .30565 L .15027 .30635 L .15024 .30704 L .15023 .30774 L .15022 .30844 L .15022 .30913 L .15022 .30983 L .15023 .31053 L .15025 .31122 L .15028 .31192 L .15035 .31331 L .15046 .31471 L .15059 .3161 L .15094 .31887 L .15141 .32164 L .15199 .3244 L .15349 .32987 L .15545 .33526 L .15784 .34054 L .16393 .35071 L .17164 .36012 L .18083 .36852 L .18593 .37224 L .19131 .37559 L .19696 .37854 L .20284 .38106 L .20891 .38311 L .21512 .38466 L .21827 .38525 L .22145 .3857 L .22304 .38588 L .22463 .38603 L .22543 .38608 L .22623 .38614 L .22703 .38618 L .22783 .38621 L .22863 .38624 L .22944 .38626 L .23024 .38627 L Mistroke .23104 .38627 L .23184 .38626 L .23264 .38625 L .23344 .38622 L .23424 .38619 L .23584 .3861 L .23744 .38598 L .23904 .38582 L .24063 .38563 L .2438 .38516 L .24694 .38455 L .25315 .38295 L .2592 .38087 L .26506 .37831 L .27605 .37194 L Mfstroke P P p p .004 w .20295 .33585 m .20777 .34004 L .21042 .34183 L .21321 .3434 L .21612 .34474 L .21913 .34584 L .22066 .34629 L .22222 .34668 L .22379 .347 L .22537 .34726 L .22616 .34737 L .22696 .34745 L .22776 .34753 L .22816 .34755 L .22856 .34758 L .22896 .3476 L .22936 .34762 L .22976 .34763 L .23016 .34764 L .23056 .34764 L .23096 .34764 L .23136 .34764 L .23176 .34763 L .23216 .34762 L .23256 .3476 L .23296 .34758 L .23336 .34756 L .23496 .34741 L .23575 .34732 L .23655 .34721 L .23813 .34693 L .24124 .34619 L .24429 .34519 L .24724 .34394 L .25008 .34245 L .25534 .33882 L .25993 .33441 L .26373 .32934 L .2653 .3266 L .26664 .32376 L .26775 .32082 L .2682 .31932 L .2686 .31781 L .26893 .31629 L .2692 .31475 L .26931 .31398 L .26941 .3132 L .26949 .31243 L .26955 .31165 L .26958 .31126 L Mistroke .2696 .31087 L .26961 .31048 L .26963 .31009 L .26964 .3097 L .26964 .30931 L .26964 .30892 L .26964 .30853 L .26963 .30814 L .26962 .30776 L .26961 .30737 L .26959 .30698 L .26947 .30542 L .26939 .30465 L .26929 .30387 L .26904 .30233 L .26836 .29928 L .26743 .29629 L .26625 .29338 L .26319 .28787 L .26133 .28531 L .25926 .28289 L .25456 .27859 L .24921 .2751 L .24634 .27368 L .24335 .27251 L .24028 .27159 L .23872 .27122 L .23715 .27092 L .23556 .27069 L .23477 .2706 L .23397 .27052 L .23317 .27047 L .23277 .27044 L .23237 .27042 L .23197 .27041 L .23157 .2704 L .23117 .27039 L .23077 .27039 L .23037 .27039 L .22997 .2704 L .22957 .27041 L .22917 .27042 L .22877 .27044 L .22837 .27047 L .22757 .27052 L .22677 .2706 L .22597 .27069 L .22439 .27092 L .22281 .27122 L .22125 .27159 L Mistroke .21818 .27251 L .2152 .27368 L .21233 .2751 L .20698 .27859 L .20227 .28289 L .19834 .28787 L .1967 .29057 L .19529 .29338 L .19411 .29629 L .19318 .29928 L .1928 .3008 L .19249 .30233 L .19225 .30387 L .19207 .30542 L .192 .3062 L .19197 .30659 L .19195 .30698 L .19193 .30737 L .19192 .30776 L .19191 .30814 L .1919 .30853 L .1919 .30892 L .1919 .30931 L .1919 .3097 L .19191 .31009 L .19192 .31048 L .19194 .31087 L .19199 .31165 L .19205 .31243 L .19213 .3132 L .19233 .31475 L .1926 .31629 L .19294 .31781 L .19379 .32082 L .19489 .32376 L .19624 .3266 L .19781 .32934 L .20161 .33441 L .2062 .33882 L .20876 .34074 L .21146 .34245 L .2143 .34394 L .21725 .34519 L .22029 .34619 L .22185 .34659 L .22341 .34693 L .22499 .34721 L .22578 .34732 L .22658 .34741 L .22738 .34749 L Mistroke .22818 .34756 L .22858 .34758 L .22898 .3476 L .22938 .34762 L .22978 .34763 L .23018 .34764 L .23058 .34764 L .23098 .34764 L .23138 .34764 L .23178 .34763 L .23218 .34762 L .23258 .3476 L .23298 .34758 L .23458 .34745 L .23537 .34737 L .23617 .34726 L .23775 .347 L .23932 .34668 L .24241 .34584 L .24542 .34474 L .24833 .3434 L .25377 .34004 L .25858 .33585 L Mfstroke P P p p .004 w .23077 .30902 m .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L .23077 .30902 L s P P p p .004 w .25858 .28218 m .25377 .278 L .25112 .27621 L .24833 .27463 L .24542 .27329 L .24241 .2722 L .24087 .27175 L .23932 .27136 L .23775 .27103 L .23617 .27077 L .23537 .27067 L .23458 .27058 L .23378 .27051 L .23338 .27048 L .23298 .27045 L .23258 .27043 L .23218 .27042 L .23178 .2704 L .23138 .27039 L .23098 .27039 L .23058 .27039 L .23018 .27039 L .22978 .2704 L .22938 .27042 L .22898 .27043 L .22858 .27045 L .22818 .27048 L .22658 .27062 L .22578 .27072 L .22499 .27083 L .22341 .2711 L .22029 .27185 L .21725 .27285 L .2143 .2741 L .21146 .27558 L .2062 .27921 L .20161 .28362 L .19781 .28869 L .19624 .29143 L .19489 .29428 L .19379 .29721 L .19333 .29871 L .19294 .30022 L .1926 .30175 L .19233 .30329 L .19222 .30406 L .19213 .30483 L .19205 .30561 L .19199 .30639 L .19196 .30677 L Mistroke .19194 .30716 L .19192 .30755 L .19191 .30794 L .1919 .30833 L .1919 .30872 L .1919 .30911 L .1919 .3095 L .19191 .30989 L .19192 .31028 L .19193 .31067 L .19195 .31106 L .19207 .31261 L .19215 .31339 L .19225 .31416 L .19249 .3157 L .19318 .31875 L .19411 .32174 L .19529 .32465 L .19834 .33016 L .20021 .33273 L .20227 .33514 L .20698 .33944 L .21233 .34294 L .2152 .34435 L .21818 .34552 L .22125 .34644 L .22281 .34681 L .22439 .34711 L .22597 .34734 L .22677 .34743 L .22757 .34751 L .22837 .34757 L .22877 .34759 L .22917 .34761 L .22957 .34763 L .22997 .34764 L .23037 .34764 L .23077 .34764 L .23117 .34764 L .23157 .34764 L .23197 .34763 L .23237 .34761 L .23277 .34759 L .23317 .34757 L .23397 .34751 L .23477 .34743 L .23556 .34734 L .23715 .34711 L .23872 .34681 L .24028 .34644 L Mistroke .24335 .34552 L .24634 .34435 L .24921 .34294 L .25456 .33944 L .25926 .33514 L .26319 .33016 L .26484 .32747 L .26625 .32465 L .26743 .32174 L .26836 .31875 L .26873 .31723 L .26904 .3157 L .26929 .31416 L .26947 .31261 L .26954 .31183 L .26956 .31145 L .26959 .31106 L .26961 .31067 L .26962 .31028 L .26963 .30989 L .26964 .3095 L .26964 .30911 L .26964 .30872 L .26964 .30833 L .26963 .30794 L .26961 .30755 L .2696 .30716 L .26955 .30639 L .26949 .30561 L .26941 .30483 L .2692 .30329 L .26893 .30175 L .2686 .30022 L .26775 .29721 L .26664 .29428 L .2653 .29143 L .26373 .28869 L .25993 .28362 L .25534 .27921 L .25278 .27729 L .25008 .27558 L .24724 .2741 L .24429 .27285 L .24124 .27185 L .23969 .27144 L .23813 .2711 L .23655 .27083 L .23575 .27072 L .23496 .27062 L .23416 .27054 L Mistroke .23336 .27048 L .23296 .27045 L .23256 .27043 L .23216 .27042 L .23176 .2704 L .23136 .27039 L .23096 .27039 L .23056 .27039 L .23016 .27039 L .22976 .2704 L .22936 .27042 L .22896 .27043 L .22856 .27045 L .22696 .27058 L .22616 .27067 L .22537 .27077 L .22379 .27103 L .22222 .27136 L .21913 .2722 L .21612 .27329 L .21321 .27463 L .20777 .278 L .20295 .28218 L Mfstroke P P p p .004 w .27605 .2461 m .27069 .24271 L .26506 .23972 L .2592 .23717 L .25315 .23508 L .25006 .23422 L .24694 .23348 L .2438 .23288 L .24063 .2324 L .23904 .23221 L .23744 .23205 L .23584 .23193 L .23504 .23188 L .23424 .23184 L .23344 .23181 L .23264 .23179 L .23184 .23177 L .23104 .23176 L .23024 .23176 L .22944 .23177 L .22863 .23179 L .22783 .23182 L .22703 .23185 L .22623 .2319 L .22463 .23201 L .22304 .23215 L .22145 .23233 L .21827 .23279 L .21512 .23337 L .20891 .23493 L .20284 .23697 L .19131 .24244 L .18593 .2458 L .18083 .24952 L .17164 .25791 L .16393 .26732 L .15784 .27749 L .15545 .28278 L .15349 .28817 L .15199 .29364 L .15141 .29639 L .15094 .29916 L .15059 .30194 L .15046 .30333 L .15035 .30472 L .15028 .30611 L .15025 .30681 L .15023 .30751 L .15022 .3082 L .15022 .3089 L Mistroke .15022 .3096 L .15023 .31029 L .15024 .31099 L .15027 .31169 L .15034 .31308 L .15044 .31447 L .15056 .31586 L .15091 .31864 L .15136 .32141 L .15262 .32691 L .15433 .33235 L .15649 .33769 L .15909 .34293 L .16555 .35294 L .17362 .36215 L .18313 .37026 L .19387 .377 L .19963 .37975 L .2056 .38206 L .21174 .38388 L .21486 .38461 L .21801 .3852 L .22118 .38567 L .22277 .38585 L .22437 .386 L .22596 .38612 L .22676 .38617 L .22757 .3862 L .22837 .38623 L .22917 .38625 L .22997 .38627 L .23077 .38627 L .23157 .38627 L .23237 .38625 L .23317 .38623 L .23397 .3862 L .23477 .38617 L .23557 .38612 L .23717 .386 L .23877 .38585 L .24036 .38567 L .24353 .3852 L .24668 .38461 L .2498 .38388 L .25594 .38206 L .26191 .37975 L .26767 .377 L .27841 .37026 L .28792 .36215 L .29598 .35294 L Mistroke .29943 .34802 L .30245 .34293 L .30505 .33769 L .30721 .33235 L .30892 .32691 L .31018 .32141 L .31063 .31864 L .31082 .31725 L .31097 .31586 L .3111 .31447 L .3112 .31308 L .31124 .31238 L .31127 .31169 L .31129 .31099 L .31131 .31029 L .31132 .3096 L .31132 .3089 L .31132 .3082 L .31131 .30751 L .31129 .30681 L .31126 .30611 L .31119 .30472 L .31108 .30333 L .31095 .30194 L .3106 .29916 L .31013 .29639 L .30955 .29364 L .30805 .28817 L .30609 .28278 L .3037 .27749 L .29761 .26732 L .2899 .25791 L .28071 .24952 L .27561 .2458 L .27022 .24244 L .26457 .23949 L .2587 .23697 L .25263 .23493 L .24642 .23337 L .24327 .23279 L .24009 .23233 L .2385 .23215 L .2369 .23201 L .2361 .23195 L .23531 .2319 L .23451 .23185 L .2337 .23182 L .2329 .23179 L .2321 .23177 L .2313 .23176 L Mistroke .2305 .23176 L .2297 .23177 L .2289 .23179 L .2281 .23181 L .2273 .23184 L .2257 .23193 L .2241 .23205 L .2225 .23221 L .22091 .2324 L .21774 .23288 L .2146 .23348 L .20839 .23508 L .20234 .23717 L .19648 .23972 L .18549 .2461 L Mfstroke P P p p .004 w .2512 .19496 m .24645 .19421 L .24407 .19391 L .24168 .19365 L .23928 .19345 L .23688 .1933 L .23568 .19324 L .23448 .1932 L .23328 .19316 L .23208 .19314 L .23088 .19314 L .22968 .19314 L .22847 .19316 L .22727 .19319 L .22607 .19323 L .22487 .19329 L .22247 .19344 L .22008 .19363 L .21768 .19388 L .21292 .19452 L .20819 .19536 L .20349 .19639 L .19425 .19898 L .18525 .20225 L .17655 .20614 L .16022 .21547 L .14559 .22639 L .13286 .23834 L .12216 .25087 L .11357 .26368 L .11008 .27011 L .10712 .27655 L .10469 .28298 L .10279 .2894 L .10205 .2926 L .10143 .2958 L .10095 .299 L .10076 .3006 L .1006 .30219 L .10048 .30379 L .10043 .30459 L .10038 .30539 L .10035 .30619 L .10032 .30698 L .10031 .30778 L .1003 .30858 L .1003 .30938 L .10031 .31018 L .10032 .31098 L .10035 .31178 L Mistroke .10042 .31337 L .10047 .31417 L .10053 .31497 L .10084 .31817 L .10105 .31976 L .10129 .32136 L .10258 .32777 L .1044 .33418 L .10675 .34061 L .11306 .35348 L .12151 .36629 L .13206 .37886 L .14466 .39086 L .15917 .40186 L .17539 .41132 L .18405 .41529 L .19301 .41865 L .20222 .42134 L .2069 .42241 L .21163 .4233 L .21638 .424 L .21877 .42427 L .22116 .4245 L .22356 .42467 L .22476 .42474 L .22596 .4248 L .22716 .42484 L .22837 .42487 L .22957 .42489 L .23077 .4249 L .23197 .42489 L .23317 .42487 L .23437 .42484 L .23558 .4248 L .23678 .42474 L .23798 .42467 L .24037 .4245 L .24277 .42427 L .24515 .424 L .24991 .4233 L .25464 .42241 L .25932 .42134 L .26853 .41865 L .27749 .41529 L .28615 .41132 L .30237 .40186 L .31688 .39086 L .32948 .37886 L .34003 .36629 L .34847 .35348 L Mistroke .35478 .34061 L .35714 .33418 L .35896 .32777 L .35967 .32456 L .36025 .32136 L .36049 .31976 L .3607 .31817 L .36087 .31657 L .36101 .31497 L .36107 .31417 L .36112 .31337 L .36116 .31257 L .36119 .31178 L .36122 .31098 L .36123 .31018 L .36124 .30938 L .36124 .30858 L .36123 .30778 L .36121 .30698 L .36119 .30619 L .36115 .30539 L .36106 .30379 L .36094 .30219 L .36078 .3006 L .36059 .299 L .36011 .2958 L .35874 .2894 L .35685 .28298 L .35442 .27655 L .34797 .26368 L .33938 .25087 L .32868 .23834 L .31595 .22639 L .30131 .21547 L .28499 .20614 L .27629 .20225 L .26729 .19898 L .25805 .19639 L .25335 .19536 L .24862 .19452 L .24385 .19388 L .24146 .19363 L .23907 .19344 L .23667 .19329 L .23547 .19323 L .23427 .19319 L .23306 .19316 L .23186 .19314 L .23066 .19314 L .22946 .19314 L Mistroke .22826 .19316 L .22706 .1932 L .22585 .19324 L .22465 .1933 L .22225 .19345 L .21986 .19365 L .21508 .19421 L .21034 .19496 L Mfstroke P P p p .004 w s s 0 .31995 m 8e-05 .32003 L s 8e-05 .32003 m .00441 .32436 L .01045 .33039 L .01885 .3387 L .0305 .35009 L .04655 .36537 L .06841 .38518 L .09757 .40915 L .13505 .43457 L .15686 .44604 L .16844 .45102 L .18041 .45532 L .1927 .45882 L .19895 .46023 L .20525 .4614 L .21159 .46233 L .21477 .46269 L .21796 .46299 L .22116 .46322 L .22276 .46332 L .22436 .46339 L .22596 .46345 L .22756 .46349 L .22917 .46352 L .23077 .46353 L .23237 .46352 L .23397 .46349 L .23558 .46345 L .23718 .46339 L .23878 .46332 L .24038 .46322 L .24357 .46299 L .24677 .46269 L .24995 .46233 L .25629 .4614 L .26884 .45882 L .28113 .45532 L .30467 .44604 L .32648 .43457 L .36396 .40915 L .39313 .38518 L .41499 .36537 L .43104 .35009 L .44269 .3387 L .45109 .33039 L .45712 .32436 L .46146 .32003 L .46457 .31691 L .4668 .31468 L s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s 0 .42542 m .00023 .42543 L s .00023 .42543 m .00265 .42569 L .00507 .426 L .00994 .42676 L .01979 .42887 L .02985 .43175 L .04018 .43537 L .06188 .4446 L .11074 .46867 L .13829 .48125 L .15282 .48701 L .16781 .49211 L .18318 .49634 L .19099 .49807 L .19886 .49951 L .20679 .50066 L .21077 .50111 L .21476 .50148 L .21876 .50178 L .22076 .50189 L .22276 .50199 L .22476 .50206 L .22676 .50211 L .22877 .50214 L .23077 .50215 L .23277 .50214 L .23478 .50211 L .23678 .50206 L .23878 .50199 L .24078 .50189 L .24278 .50178 L .24678 .50148 L .25077 .50111 L .25475 .50066 L .26267 .49951 L .27836 .49634 L .29373 .49211 L .32325 .48125 L .3508 .46867 L .39966 .4446 L .42136 .43537 L .43169 .43175 L .44175 .42887 L .44669 .42772 L .4516 .42676 L .45647 .426 L .46131 .42543 L .46372 .42523 L .46493 .42514 L .46613 .42507 L Mistroke .46733 .42501 L .46854 .42496 L .46974 .42493 L .47094 .42491 L .47214 .4249 L .47334 .4249 L .47455 .42492 L .47575 .42495 L .47695 .42499 L .47815 .42504 L .48056 .42519 L .48297 .42538 L .48539 .42563 L .49024 .42626 L .49512 .4271 L .50004 .42814 L .52027 .43417 L .54174 .44301 L .56487 .45413 L .58998 .46668 L .61721 .47937 L .63159 .48528 L .64644 .49061 L .66171 .49513 L .66947 .49703 L .6773 .49865 L .6852 .49999 L .68917 .50054 L .69314 .50101 L .69713 .5014 L .70113 .50171 L .70312 .50184 L .70513 .50194 L .70713 .50203 L .70913 .50209 L .71113 .50213 L .71313 .50215 L .71514 .50215 L .71714 .50213 L .71914 .50209 L .72115 .50202 L .72315 .50194 L .72515 .50183 L .73314 .50121 L .73712 .50077 L .74109 .50026 L .75685 .49744 L .77233 .49352 L .80212 .48309 L .85576 .45792 L Mistroke Mfstroke P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s 0 .48211 m .00313 .48233 L s .00313 .48233 m .00673 .48266 L .01034 .48307 L .01758 .48409 L .02487 .48539 L .03964 .48881 L .05473 .49321 L .08619 .50434 L .11965 .51706 L .13717 .5233 L .1552 .52902 L .17366 .5339 L .18303 .53593 L .19248 .53763 L .202 .53899 L .20677 .53953 L .21156 .53998 L .21636 .54033 L .21876 .54047 L .22116 .54058 L .22356 .54067 L .22596 .54073 L .22837 .54077 L .23077 .54078 L .23317 .54077 L .23558 .54073 L .23798 .54067 L .24038 .54058 L .24278 .54047 L .24518 .54033 L .24998 .53998 L .25954 .53899 L .26906 .53763 L .28788 .5339 L .30634 .52902 L .37535 .50434 L .40681 .49321 L .4219 .48881 L .42932 .48697 L .43667 .48539 L .44396 .48409 L .4512 .48307 L .45481 .48266 L .45841 .48233 L .462 .48208 L .46379 .48198 L .46559 .4819 L .46738 .48184 L .46917 .48179 L .47096 .48177 L Mistroke .47276 .48176 L .47455 .48178 L .47634 .48181 L .47813 .48186 L .47992 .48193 L .48351 .48212 L .48711 .48239 L .49431 .48315 L .50156 .48419 L .50885 .48552 L .52364 .48899 L .55426 .49871 L .6213 .52357 L .63935 .52926 L .65783 .5341 L .67667 .53777 L .68619 .5391 L .69096 .53962 L .69575 .54005 L .69815 .54023 L .70055 .54038 L .70295 .54051 L .70535 .54061 L .70775 .54069 L .71016 .54075 L .71256 .54077 L .71496 .54078 L .71737 .54076 L .71977 .54071 L .72217 .54064 L .72458 .54054 L .72938 .54027 L .73417 .53991 L .73895 .53944 L .74373 .53888 L .75324 .53749 L .77205 .5337 L .79049 .52878 L .80849 .52302 L .84297 .51034 L .87535 .4982 L .89082 .49299 L .9059 .48863 L .91331 .48682 L .92066 .48526 L .92794 .48398 L .93518 .48299 L .93878 .4826 L .94238 .48228 L .94418 .48215 L Mistroke .94597 .48204 L .94777 .48195 L .94956 .48187 L .95135 .48182 L .95314 .48178 L .95494 .48177 L .95673 .48177 L .95852 .48179 L .96031 .48182 L .9621 .48188 L .9639 .48196 L .96749 .48216 L .97108 .48245 L .97468 .4828 L .97829 .48323 L .98554 .4843 L .99284 .48566 L Mfstroke 1 .48736 m .99284 .48566 L s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s 1 .13079 m .99953 .13091 L s .99953 .13091 m .99217 .13251 L .98488 .13384 L .97763 .13489 L .97403 .1353 L .97043 .13565 L .96683 .13591 L .96325 .13611 L .96145 .13618 L .95966 .13623 L .95787 .13626 L .95608 .13627 L .95428 .13627 L .95249 .13624 L .9507 .1362 L .94891 .13614 L .94532 .13595 L .94173 .1357 L .93813 .13537 L .93452 .13497 L .92728 .13395 L .91999 .13264 L .90522 .12923 L .89013 .12482 L .85867 .1137 L .82521 .10098 L .80769 .09474 L .78967 .08901 L .7712 .08413 L .76183 .08211 L .75238 .0804 L .74286 .07904 L .73809 .0785 L .7333 .07805 L .7285 .07771 L .7261 .07757 L .7237 .07746 L .7213 .07737 L .7189 .0773 L .71649 .07727 L .71409 .07725 L .71169 .07727 L .70928 .0773 L .70688 .07737 L .70448 .07746 L .70208 .07757 L .69968 .07771 L .69488 .07805 L .68532 .07904 L .67581 .0804 L Mistroke .65698 .08413 L .63852 .08901 L .56951 .1137 L .53805 .12482 L .52296 .12923 L .51554 .13107 L .50819 .13264 L .5009 .13395 L .49366 .13497 L .49005 .13537 L .48646 .1357 L .48286 .13596 L .48107 .13605 L .47927 .13614 L .47748 .1362 L .47569 .13624 L .4739 .13627 L .47211 .13627 L .47031 .13626 L .46852 .13623 L .46673 .13618 L .46494 .13611 L .46135 .13591 L .45775 .13565 L .45055 .13489 L .4433 .13384 L .43601 .13251 L .42123 .12905 L .3906 .11933 L .32356 .09446 L .30552 .08877 L .28703 .08394 L .2682 .08026 L .25868 .07894 L .2539 .07841 L .24911 .07798 L .24671 .07781 L .24431 .07765 L .24191 .07752 L .23951 .07742 L .23711 .07734 L .23471 .07729 L .2323 .07726 L .2299 .07726 L .2275 .07728 L .22509 .07732 L .22269 .0774 L .22029 .07749 L .21549 .07776 L .21069 .07813 L Mistroke .20591 .07859 L .20113 .07915 L .19163 .08054 L .17282 .08433 L .15437 .08926 L .13637 .09501 L .1019 .10769 L .06952 .11984 L .05404 .12504 L .03896 .1294 L .03155 .13122 L .02421 .13277 L .01692 .13405 L .00968 .13505 L .00608 .13544 L .00248 .13575 L .00069 .13588 L Mfstroke 0 .13592 m .00069 .13588 L s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s 1 .18505 m .99335 .18725 L s .99335 .18725 m .98336 .1899 L .97844 .19093 L .97356 .19177 L .97113 .19211 L .96871 .19241 L .9663 .19265 L .96388 .19285 L .96268 .19293 L .96148 .19299 L .96027 .19305 L .95907 .19309 L .95787 .19312 L .95667 .19313 L .95546 .19314 L .95426 .19313 L .95306 .1931 L .95186 .19307 L .95065 .19302 L .94945 .19296 L .94463 .1926 L .94221 .19234 L .93979 .19204 L .93492 .19127 L .92507 .18916 L .91501 .18628 L .90468 .18267 L .88298 .17343 L .83412 .14937 L .80657 .13678 L .79204 .13103 L .77705 .12593 L .76168 .1217 L .75387 .11997 L .746 .11852 L .73807 .11738 L .73409 .11692 L .7301 .11655 L .7261 .11626 L .7241 .11614 L .7221 .11605 L .7201 .11598 L .7181 .11592 L .71609 .11589 L .71409 .11588 L .71209 .11589 L .71009 .11592 L .70808 .11598 L .70608 .11605 L .70408 .11614 L Mistroke .70208 .11626 L .69808 .11655 L .6941 .11692 L .69011 .11738 L .68219 .11852 L .6665 .1217 L .65113 .12593 L .62161 .13678 L .59406 .14937 L .5452 .17343 L .5235 .18267 L .51317 .18628 L .50311 .18916 L .49817 .19032 L .49326 .19127 L .48839 .19204 L .48355 .1926 L .48114 .19281 L .47994 .19289 L .47873 .19296 L .47753 .19302 L .47633 .19307 L .47512 .19311 L .47392 .19313 L .47272 .19314 L .47152 .19313 L .47031 .19312 L .46911 .19309 L .46791 .19305 L .46671 .19299 L .4643 .19285 L .46189 .19265 L .45947 .19241 L .45462 .19177 L .44974 .19093 L .44482 .1899 L .42459 .18387 L .40313 .17503 L .37999 .1639 L .35488 .15136 L .32766 .13866 L .31327 .13275 L .29842 .12743 L .28316 .1229 L .2754 .12101 L .26756 .11938 L .25967 .11805 L .2557 .1175 L .25172 .11702 L .24773 .11663 L Mistroke .24374 .11632 L .24174 .1162 L .23974 .11609 L .23774 .11601 L .23574 .11595 L .23373 .1159 L .23173 .11588 L .22973 .11588 L .22772 .11591 L .22572 .11595 L .22372 .11601 L .22172 .1161 L .21972 .1162 L .21173 .11683 L .20775 .11726 L .20377 .11777 L .18802 .12059 L .17254 .12452 L .14274 .13494 L .08911 .16012 L Mfstroke P P p p .004 w .95011 .30337 m .94788 .30113 L .94478 .29801 L .94044 .29367 L .9344 .28765 L .92601 .27933 L .91436 .26795 L .89831 .25266 L .87645 .23285 L .84729 .20889 L .80981 .18346 L .788 .172 L .77642 .16702 L .76445 .16271 L .75216 .15921 L .74591 .1578 L .73961 .15663 L .73327 .15571 L .73009 .15534 L .7269 .15504 L .7237 .15481 L .7221 .15472 L .7205 .15464 L .7189 .15458 L .7173 .15454 L .71569 .15452 L .71409 .15451 L .71249 .15452 L .71089 .15454 L .70928 .15458 L .70768 .15464 L .70608 .15472 L .70448 .15481 L .70129 .15504 L .69809 .15534 L .69491 .15571 L .68857 .15663 L .67602 .15921 L .66373 .16271 L .64019 .172 L .61838 .18346 L .5809 .20889 L .55173 .23285 L .52987 .25266 L .51382 .26795 L .50218 .27933 L .49378 .28765 L .48774 .29367 L .48341 .29801 L .4803 .30113 L Mistroke .47807 .30337 L Mfstroke P P p p .004 w .69366 .42308 m .6984 .42383 L .70079 .42413 L .70318 .42438 L .70558 .42458 L .70797 .42473 L .70917 .42479 L .71038 .42484 L .71158 .42487 L .71278 .42489 L .71398 .4249 L .71518 .42489 L .71638 .42487 L .71759 .42484 L .71879 .4248 L .71999 .42475 L .72239 .4246 L .72478 .4244 L .72717 .42415 L .73194 .42351 L .73667 .42267 L .74137 .42165 L .75061 .41906 L .75961 .41578 L .76831 .4119 L .78463 .40256 L .79927 .39165 L .812 .3797 L .8227 .36716 L .83129 .35436 L .83478 .34792 L .83774 .34148 L .84017 .33506 L .84207 .32864 L .84281 .32544 L .84343 .32224 L .84391 .31904 L .8441 .31744 L .84426 .31584 L .84438 .31424 L .84443 .31345 L .84448 .31265 L .84451 .31185 L .84453 .31105 L .84455 .31025 L .84456 .30945 L .84456 .30865 L .84455 .30786 L .84454 .30706 L .84451 .30626 L Mistroke .84444 .30466 L .84439 .30386 L .84433 .30306 L .84402 .29987 L .84381 .29827 L .84357 .29667 L .84228 .29027 L .84046 .28385 L .83811 .27743 L .8318 .26456 L .82335 .25174 L .8128 .23918 L .8002 .22717 L .78569 .21617 L .76947 .20671 L .76081 .20275 L .75185 .19939 L .74264 .1967 L .73796 .19562 L .73323 .19473 L .72848 .19404 L .72609 .19376 L .7237 .19354 L .7213 .19336 L .7201 .19329 L .7189 .19324 L .7177 .19319 L .71649 .19316 L .71529 .19314 L .71409 .19314 L .71289 .19314 L .71169 .19316 L .71049 .19319 L .70928 .19324 L .70808 .19329 L .70688 .19336 L .70449 .19354 L .70209 .19376 L .69971 .19404 L .69495 .19473 L .69023 .19562 L .68554 .1967 L .67633 .19939 L .66737 .20275 L .65871 .20671 L .64249 .21617 L .62798 .22717 L .61538 .23918 L .60483 .25174 L .59639 .26456 L Mistroke .59008 .27743 L .58772 .28385 L .5859 .29027 L .58519 .29347 L .58461 .29667 L .58437 .29827 L .58416 .29987 L .58399 .30147 L .58385 .30306 L .58379 .30386 L .58374 .30466 L .5837 .30546 L .58367 .30626 L .58365 .30706 L .58363 .30786 L .58362 .30865 L .58362 .30945 L .58363 .31025 L .58365 .31105 L .58367 .31185 L .58371 .31265 L .5838 .31424 L .58392 .31584 L .58408 .31744 L .58427 .31904 L .58475 .32224 L .58612 .32864 L .58801 .33506 L .59044 .34148 L .5969 .35436 L .60548 .36716 L .61618 .3797 L .62891 .39165 L .64355 .40256 L .65987 .4119 L .66857 .41578 L .67757 .41906 L .68682 .42165 L .69151 .42267 L .69624 .42351 L .70101 .42415 L .7034 .4244 L .7058 .4246 L .70819 .42475 L .7094 .4248 L .7106 .42484 L .7118 .42487 L .713 .42489 L .7142 .4249 L .7154 .42489 L Mistroke .7166 .42487 L .71781 .42484 L .71901 .42479 L .72021 .42473 L .72261 .42458 L .725 .42438 L .72978 .42383 L .73453 .42308 L Mfstroke P P p p .004 w .66881 .37194 m .67417 .37532 L .6798 .37831 L .68566 .38087 L .69171 .38295 L .6948 .38381 L .69792 .38455 L .70106 .38516 L .70423 .38564 L .70582 .38582 L .70742 .38598 L .70902 .3861 L .70982 .38615 L .71062 .38619 L .71142 .38622 L .71222 .38625 L .71302 .38626 L .71382 .38627 L .71462 .38627 L .71542 .38626 L .71622 .38624 L .71703 .38622 L .71783 .38618 L .71863 .38614 L .72022 .38603 L .72182 .38588 L .72341 .3857 L .72659 .38525 L .72974 .38466 L .73595 .38311 L .74202 .38106 L .75355 .37559 L .75893 .37224 L .76403 .36852 L .77322 .36012 L .78093 .35071 L .78702 .34054 L .78941 .33526 L .79137 .32987 L .79287 .3244 L .79345 .32164 L .79392 .31887 L .79427 .3161 L .7944 .31471 L .79451 .31331 L .79458 .31192 L .79461 .31122 L .79463 .31053 L .79464 .30983 L .79464 .30913 L Mistroke .79464 .30844 L .79463 .30774 L .79462 .30704 L .79459 .30635 L .79452 .30495 L .79442 .30356 L .7943 .30217 L .79395 .29939 L .7935 .29663 L .79224 .29112 L .79053 .28569 L .78837 .28034 L .78577 .27511 L .77931 .26509 L .77124 .25589 L .76173 .24777 L .75099 .24104 L .74523 .23828 L .73926 .23598 L .73312 .23415 L .73 .23343 L .72685 .23283 L .72368 .23236 L .72209 .23218 L .72049 .23203 L .7189 .23191 L .7181 .23187 L .7173 .23183 L .71649 .2318 L .71569 .23178 L .71489 .23177 L .71409 .23176 L .71329 .23177 L .71249 .23178 L .71169 .2318 L .71089 .23183 L .71009 .23187 L .70929 .23191 L .70769 .23203 L .70609 .23218 L .7045 .23236 L .70133 .23283 L .69818 .23343 L .69506 .23415 L .68892 .23598 L .68295 .23828 L .67719 .24104 L .66645 .24777 L .65694 .25589 L .64888 .26509 L Mistroke .64543 .27002 L .64241 .27511 L .63981 .28034 L .63765 .28569 L .63594 .29112 L .63469 .29663 L .63423 .29939 L .63404 .30078 L .63389 .30217 L .63376 .30356 L .63366 .30495 L .63362 .30565 L .63359 .30635 L .63357 .30704 L .63355 .30774 L .63354 .30844 L .63354 .30913 L .63354 .30983 L .63355 .31053 L .63357 .31122 L .6336 .31192 L .63368 .31331 L .63378 .31471 L .63391 .3161 L .63426 .31887 L .63473 .32164 L .63531 .3244 L .63681 .32987 L .63877 .33526 L .64117 .34054 L .64725 .35071 L .65496 .36012 L .66416 .36852 L .66925 .37224 L .67464 .37559 L .68029 .37854 L .68616 .38106 L .69223 .38311 L .69845 .38466 L .7016 .38525 L .70477 .3857 L .70636 .38588 L .70796 .38603 L .70876 .38609 L .70956 .38614 L .71036 .38618 L .71116 .38622 L .71196 .38624 L .71276 .38626 L .71356 .38627 L Mistroke .71436 .38627 L .71516 .38626 L .71596 .38625 L .71676 .38622 L .71757 .38619 L .71917 .3861 L .72076 .38598 L .72236 .38582 L .72395 .38564 L .72712 .38516 L .73027 .38455 L .73647 .38295 L .74252 .38087 L .74838 .37831 L .75937 .37194 L Mfstroke P P p p .004 w .68628 .33585 m .69109 .34004 L .69374 .34183 L .69653 .3434 L .69944 .34474 L .70245 .34584 L .70399 .34629 L .70554 .34668 L .70711 .347 L .70869 .34726 L .70949 .34737 L .71028 .34745 L .71108 .34753 L .71148 .34755 L .71188 .34758 L .71228 .3476 L .71268 .34762 L .71308 .34763 L .71348 .34764 L .71388 .34764 L .71428 .34764 L .71468 .34764 L .71508 .34763 L .71548 .34762 L .71588 .3476 L .71628 .34758 L .71668 .34756 L .71828 .34741 L .71908 .34732 L .71987 .34721 L .72145 .34693 L .72457 .34619 L .72761 .34519 L .73056 .34394 L .7334 .34245 L .73866 .33882 L .74325 .33441 L .74705 .32934 L .74862 .3266 L .74997 .32376 L .75107 .32082 L .75153 .31932 L .75192 .31781 L .75226 .31629 L .75253 .31475 L .75264 .31398 L .75273 .3132 L .75281 .31243 L .75287 .31165 L .7529 .31126 L Mistroke .75292 .31087 L .75294 .31048 L .75295 .31009 L .75296 .3097 L .75296 .30931 L .75296 .30892 L .75296 .30853 L .75295 .30814 L .75294 .30776 L .75293 .30737 L .75291 .30698 L .75279 .30542 L .75271 .30465 L .75261 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.17091 .30429 L .16304 .307 L .15542 .3103 L .14113 .31855 L Mfstroke P P p p .004 w .22656 .25236 m .22039 .25139 L .21729 .251 L .21418 .25067 L .21107 .25041 L .20795 .25021 L .20639 .25014 L .20483 .25008 L .20327 .25004 L .2017 .25001 L .20014 .25 L .19858 .25001 L .19702 .25003 L .19545 .25007 L .19389 .25012 L .19233 .2502 L .18921 .25039 L .1861 .25064 L .18299 .25097 L .1768 .2518 L .17064 .25288 L .16454 .25421 L .15252 .25756 L .14083 .2618 L .12951 .26683 L .10829 .27891 L .08926 .29304 L .07271 .30851 L .05881 .32474 L .04764 .34131 L .0431 .34964 L .03925 .35797 L .0361 .3663 L .03363 .3746 L .03266 .37875 L .03186 .38289 L .03123 .38703 L .03099 .3891 L .03078 .39117 L .03062 .39324 L .03055 .39427 L .0305 .3953 L .03046 .39634 L .03042 .39737 L .0304 .3984 L .03039 .39944 L .03039 .40047 L .0304 .4015 L .03042 .40254 L .03045 .40357 L Mistroke .03055 .40564 L .03061 .40667 L .03069 .40771 L .03109 .41184 L .03136 .41391 L .03167 .41598 L .03335 .42427 L .03572 .43257 L .03878 .44089 L .04698 .45755 L .05796 .47414 L .07168 .4904 L .08805 .50594 L .10692 .52018 L .128 .53243 L .13926 .53756 L .15091 .54191 L .16289 .54539 L .16897 .54678 L .17512 .54793 L .1813 .54883 L .1844 .54919 L .18751 .54948 L .19063 .54971 L .19219 .5498 L .19375 .54987 L .19531 .54993 L .19688 .54997 L .19844 .54999 L .2 .55 L .20156 .54999 L .20312 .54997 L .20469 .54993 L .20625 .54987 L .20781 .5498 L .20937 .54971 L .21249 .54948 L .2156 .54919 L .2187 .54883 L .22488 .54793 L .23103 .54678 L .23711 .54539 L .24909 .54191 L .26074 .53756 L .272 .53243 L .29308 .52018 L .31195 .50594 L .32832 .4904 L .34204 .47414 L .35302 .45755 L Mistroke .36122 .44089 L .36428 .43257 L .36665 .42427 L .36757 .42012 L .36833 .41598 L .36864 .41391 L .36891 .41184 L .36913 .40977 L .36931 .40771 L .36939 .40667 L .36945 .40564 L .36951 .4046 L .36955 .40357 L .36958 .40254 L .3696 .4015 L .36961 .40047 L .36961 .39944 L .3696 .3984 L .36958 .39737 L .36954 .39634 L .3695 .3953 L .36938 .39324 L .36922 .39117 L .36901 .3891 L .36877 .38703 L .36814 .38289 L .36637 .3746 L .3639 .3663 L .36075 .35797 L .35236 .34131 L .34119 .32474 L .32729 .30851 L .31074 .29304 L .29171 .27891 L .27049 .26683 L .25917 .2618 L .24748 .25756 L .23546 .25421 L .22936 .25288 L .2232 .2518 L .21701 .25097 L .2139 .25064 L .21079 .25039 L .20767 .2502 L .20611 .25012 L .20455 .25007 L .20298 .25003 L .20142 .25001 L .19986 .25 L .1983 .25001 L Mistroke .19673 .25004 L .19517 .25008 L .19361 .25014 L .19205 .25021 L .18893 .25041 L .18582 .25067 L .17961 .25139 L .17344 .25236 L Mfstroke P P p p .004 w s s s s s s s s s 0 .50764 m .02685 .52961 L s .02685 .52961 m .07557 .56253 L .10392 .57736 L .11898 .58381 L .13453 .58938 L .15051 .59391 L .15863 .59574 L .16682 .59725 L .17506 .59845 L .1792 .59892 L .18335 .59931 L .18751 .59961 L .18959 .59973 L .19167 .59983 L .19375 .5999 L .19583 .59996 L .19792 .59999 L .2 .6 L .20208 .59999 L .20417 .59996 L .20625 .5999 L .20833 .59983 L .21041 .59973 L .21249 .59961 L .21665 .59931 L .2208 .59892 L .22494 .59845 L .23318 .59725 L .24949 .59391 L .26547 .58938 L .29608 .57736 L .32443 .56253 L .37315 .52961 L .41107 .49859 L .43949 .47295 L .46035 .45316 L .47549 .43843 L .48641 .42766 L .49426 .41987 L .4999 .41425 L .50394 .41022 L .50683 .40732 L s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s 0 .5851 m .04396 .60666 L s .04396 .60666 m .07977 .62295 L .09867 .63039 L .11815 .63699 L .13814 .64247 L .14829 .64471 L .15852 .64659 L .16883 .64807 L .17401 .64865 L .17919 .64914 L .18439 .64951 L .18699 .64966 L .18959 .64978 L .19219 .64988 L .19479 .64995 L .1974 .64999 L .2 .65 L .2026 .64999 L .20521 .64995 L .20781 .64988 L .21041 .64978 L .21301 .64966 L .21561 .64951 L .22081 .64914 L .22599 .64865 L .23117 .64807 L .24148 .64659 L .26186 .64247 L .28185 .63699 L .32023 .62295 L .35604 .60666 L .41956 .5755 L .44777 .56355 L .46119 .55887 L .47427 .55515 L .4807 .55365 L .48708 .55241 L .49341 .55142 L .4997 .55069 L .50284 .55043 L .5044 .55032 L .50597 .55022 L .50753 .55015 L .5091 .55009 L .51066 .55004 L .51222 .55001 L .51379 .55 L .51535 .55 L .51691 .55003 L .51847 .55006 L Mistroke .52004 .55012 L .5216 .55018 L .52473 .55037 L .52787 .55062 L .53101 .55094 L .53731 .55177 L .54366 .55285 L .55006 .55419 L .57635 .562 L .60426 .57345 L .63433 .58784 L .66698 .60408 L .70237 .62051 L .72107 .62816 L .74038 .63506 L .76022 .64091 L .77031 .64337 L .78049 .64547 L .79076 .6472 L .79592 .64791 L .80109 .64852 L .80627 .64903 L .81146 .64943 L .81406 .64959 L .81666 .64973 L .81926 .64984 L .82187 .64992 L .82447 .64997 L .82708 .65 L .82968 .65 L .83228 .64997 L .83489 .64991 L .83749 .64983 L .84009 .64972 L .84269 .64959 L .85308 .64878 L .85825 .64822 L .86342 .64755 L .88391 .6439 L .90403 .63882 L .94276 .62533 L Mfstroke 1 .59858 m .94276 .62533 L s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s 0 .64859 m .01204 .65283 L s .01204 .65283 m .05554 .66929 L .07832 .67737 L .10175 .68478 L .12576 .6911 L .13794 .69372 L .15023 .69593 L .1626 .69769 L .16881 .69839 L .17503 .69896 L .18126 .69942 L .18438 .69959 L .1875 .69974 L .19063 .69985 L .19375 .69993 L .19688 .69998 L .2 .7 L .20312 .69998 L .20625 .69993 L .20937 .69985 L .2125 .69974 L .21562 .69959 L .21874 .69942 L .22497 .69896 L .2374 .69769 L .24977 .69593 L .27424 .6911 L .29825 .68478 L .38796 .65283 L .42885 .63843 L .44847 .63273 L .45812 .63035 L .46767 .6283 L .47715 .62661 L .48656 .62529 L .49125 .62477 L .49593 .62435 L .5006 .62402 L .50293 .62389 L .50526 .62378 L .50759 .6237 L .50992 .62365 L .51225 .62362 L .51458 .62361 L .51691 .62362 L .51924 .62366 L .52157 .62373 L .5239 .62382 L .52857 .62407 L .53324 .62442 L Mistroke .54261 .6254 L .55203 .62675 L .56151 .62847 L .58073 .63296 L .62054 .64554 L .70769 .67772 L .73115 .68509 L .75518 .69135 L .77967 .69611 L .79204 .69782 L .79825 .6985 L .80448 .69906 L .80759 .69929 L .81071 .69948 L .81383 .69965 L .81696 .69979 L .82008 .69989 L .8232 .69996 L .82633 .69999 L .82945 .7 L .83258 .69997 L .8357 .69991 L .83883 .69982 L .84195 .69969 L .84819 .69934 L .85442 .69887 L .86064 .69827 L .86685 .69755 L .87921 .69575 L .90366 .69084 L .92764 .68446 L .95104 .67701 L .99586 .6606 L Mfstroke 1 .65905 m .99586 .6606 L s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s 1 .14101 m .97277 .13071 L s .97277 .13071 m .94999 .12263 L .92656 .11522 L .90256 .10891 L .89038 .10628 L .87809 .10407 L .86572 .10231 L .85951 .10161 L .85329 .10104 L .84706 .10058 L .84394 .10041 L .84081 .10026 L .83769 .10015 L .83457 .10007 L .83144 .10002 L .82832 .1 L .82519 .10002 L .82207 .10007 L .81895 .10015 L .81582 .10026 L .8127 .10041 L .80958 .10058 L .80335 .10104 L .79092 .10231 L .77855 .10407 L .75408 .10891 L .73007 .11522 L .64036 .14717 L .59947 .16157 L .57985 .16727 L .5702 .16965 L .56065 .1717 L .55117 .17339 L .54176 .17471 L .53707 .17523 L .53239 .17565 L .52772 .17598 L .52539 .17611 L .52306 .17622 L .52073 .1763 L .5184 .17635 L .51607 .17639 L .51374 .17639 L .51141 .17638 L .50908 .17634 L .50675 .17627 L .50442 .17618 L .49975 .17593 L .49508 .17558 L .48571 .1746 L Mistroke .47629 .17325 L .46681 .17153 L .44759 .16704 L .40778 .15446 L .32063 .12228 L .29717 .11491 L .27314 .10865 L .24866 .10389 L .23628 .10218 L .23007 .1015 L .22384 .10095 L .22073 .10072 L .21761 .10052 L .21449 .10035 L .21137 .10022 L .20824 .10011 L .20512 .10004 L .20199 .10001 L .19887 .1 L .19574 .10003 L .19262 .10009 L .1895 .10018 L .18637 .10031 L .18013 .10066 L .1739 .10113 L .16768 .10173 L .16148 .10245 L .14911 .10425 L .12466 .10916 L .10068 .11554 L .07728 .12299 L .03247 .1394 L Mfstroke 0 .15152 m .03247 .1394 L s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s 1 .20101 m .98436 .19334 L s .98436 .19334 m .94854 .17705 L .92965 .16961 L .91017 .16301 L .89018 .15753 L .88003 .15529 L .86979 .15342 L .85949 .15194 L .85431 .15135 L .84913 .15086 L .84393 .15049 L .84133 .15034 L .83873 .15022 L .83613 .15012 L .83353 .15005 L .83092 .15001 L .82832 .15 L .82571 .15001 L .82311 .15005 L .82051 .15012 L .8179 .15022 L .8153 .15034 L .8127 .15049 L .80751 .15086 L .80232 .15135 L .79715 .15194 L .78684 .15342 L .76646 .15753 L .74647 .16301 L .70809 .17705 L .67228 .19334 L .60876 .2245 L .58055 .23645 L .56713 .24113 L .55405 .24486 L .54762 .24635 L .54124 .24759 L .53491 .24858 L .52862 .24931 L .52548 .24958 L .52392 .24968 L .52235 .24978 L .52079 .24986 L .51922 .24992 L .51766 .24996 L .5161 .24999 L .51453 .25 L .51297 .25 L .51141 .24998 L .50985 .24994 L Mistroke .50828 .24989 L .50672 .24982 L .50359 .24963 L .50045 .24938 L .49731 .24906 L .49101 .24823 L .48466 .24715 L .47827 .24581 L .45197 .238 L .42406 .22656 L .39399 .21216 L .36135 .19593 L .32595 .17949 L .30725 .17184 L .28795 .16494 L .26811 .15909 L .25802 .15663 L .24783 .15453 L .23757 .15281 L .23241 .15209 L .22724 .15148 L .22206 .15097 L .21686 .15057 L .21426 .15041 L .21166 .15027 L .20906 .15016 L .20646 .15008 L .20385 .15003 L .20125 .15 L .19865 .15 L .19604 .15003 L .19344 .15009 L .19084 .15017 L .18823 .15028 L .18563 .15041 L .17525 .15122 L .17007 .15178 L .16491 .15245 L .14442 .1561 L .1243 .16118 L .08557 .17467 L .01584 .20726 L Mfstroke P P p p .004 w s s s s s s s s s s 1 .2694 m .95275 .23748 L s .95275 .23748 m .92439 .22264 L .90934 .21619 L .89378 .21062 L .87781 .20609 L .86969 .20426 L .8615 .20275 L .85325 .20155 L .84911 .20108 L .84497 .20069 L .84081 .20039 L .83873 .20027 L .83665 .20017 L .83457 .2001 L .83248 .20004 L .8304 .20001 L .82832 .2 L .82624 .20001 L .82415 .20004 L .82207 .2001 L .81999 .20017 L .81791 .20027 L .81583 .20039 L .81167 .20069 L .80752 .20108 L .80338 .20155 L .79514 .20275 L .77883 .20609 L .76285 .21062 L .73224 .22264 L .70389 .23748 L .65517 .27039 L .61725 .30141 L .58883 .32705 L .56797 .34684 L .55283 .36157 L .54191 .37234 L .53406 .38014 L .52843 .38575 L .52439 .38979 L .5215 .39269 L s P P p p .004 w .80175 .54764 m .80793 .54861 L .81103 .549 L .81413 .54933 L .81725 .54959 L .82037 .54979 L .82193 .54986 L .82349 .54992 L .82505 .54996 L .82661 .54999 L .82818 .55 L .82974 .54999 L .8313 .54997 L .83286 .54993 L .83442 .54987 L .83598 .5498 L .8391 .54961 L .84222 .54935 L .84533 .54903 L .85152 .5482 L .85767 .54712 L .86378 .54579 L .87579 .54244 L .88749 .5382 L .8988 .53317 L .92003 .52109 L .93905 .50696 L .9556 .49149 L .96951 .47526 L .98067 .45869 L .98522 .45036 L .98907 .44203 L .99222 .43371 L .99468 .4254 L .99566 .42125 L .99646 .41711 L .99708 .41297 L .99733 .4109 L .99754 .40883 L .9977 .40677 L .99776 .40573 L .99782 .4047 L .99786 .40366 L .99789 .40263 L .99792 .4016 L .99793 .40056 L .99793 .39953 L .99792 .3985 L .9979 .39746 L .99787 .39643 L Mistroke .99777 .39436 L .99771 .39333 L .99763 .3923 L .99722 .38816 L .99696 .38609 L .99664 .38402 L .99497 .37573 L .9926 .36743 L .98954 .35911 L .98133 .34245 L .97036 .32586 L .95664 .3096 L .94026 .29406 L .9214 .27982 L .90031 .26757 L .88906 .26244 L .87741 .25809 L .86543 .25461 L .85934 .25322 L .8532 .25207 L .84702 .25117 L .84392 .25081 L .8408 .25052 L .83769 .25029 L .83613 .2502 L .83457 .25013 L .83301 .25007 L .83144 .25003 L .82988 .25001 L .82832 .25 L .82676 .25001 L .82519 .25003 L .82363 .25007 L .82207 .25013 L .82051 .2502 L .81895 .25029 L .81583 .25052 L .81272 .25081 L .80962 .25117 L .80343 .25207 L .79729 .25322 L .79121 .25461 L .77923 .25809 L .76758 .26244 L .75632 .26757 L .73524 .27982 L .71637 .29406 L .7 .3096 L .68628 .32586 L .6753 .34245 L Mistroke .6671 .35911 L .66404 .36743 L .66167 .37573 L .66075 .37988 L .65999 .38402 L .65968 .38609 L .65941 .38816 L .65919 .39023 L .65901 .3923 L .65893 .39333 L .65887 .39436 L .65881 .3954 L .65877 .39643 L .65874 .39746 L .65872 .3985 L .65871 .39953 L .65871 .40056 L .65872 .4016 L .65874 .40263 L .65878 .40366 L .65882 .4047 L .65894 .40677 L .6591 .40883 L .65931 .4109 L .65955 .41297 L .66018 .41711 L .66195 .4254 L .66442 .43371 L .66757 .44203 L .67596 .45869 L .68713 .47526 L .70103 .49149 L .71758 .50696 L .73661 .52109 L .75783 .53317 L .76915 .5382 L .78085 .54244 L .79286 .54579 L .79896 .54712 L .80512 .5482 L .81131 .54903 L .81442 .54935 L .81753 .54961 L .82065 .5498 L .82221 .54987 L .82378 .54993 L .82534 .54997 L .8269 .54999 L .82846 .55 L .83002 .54999 L Mistroke .83159 .54996 L .83315 .54992 L .83471 .54986 L .83627 .54979 L .83939 .54959 L .8425 .54933 L .84871 .54861 L .85488 .54764 L Mfstroke P P p p .004 w .76945 .48145 m .77642 .48583 L .78374 .4897 L .79136 .493 L .79923 .49571 L .80324 .49682 L .80729 .49777 L .81138 .49856 L .8155 .49918 L .81757 .49942 L .81965 .49962 L .82172 .49978 L .82276 .49985 L .8238 .4999 L .82484 .49994 L .82588 .49997 L .82693 .49999 L .82797 .5 L .82901 .5 L .83005 .49999 L .83109 .49996 L .83213 .49993 L .83317 .49988 L .83421 .49983 L .83629 .49968 L .83837 .49949 L .84044 .49926 L .84456 .49868 L .84866 .49792 L .85674 .4959 L .86462 .49325 L .87961 .48618 L .88661 .48183 L .89323 .47702 L .90519 .46615 L .91521 .45397 L .92312 .44081 L .92624 .43396 L .92878 .42699 L .93073 .41991 L .93149 .41634 L .9321 .41276 L .93255 .40916 L .93272 .40736 L .93286 .40556 L .93296 .40376 L .93299 .40286 L .93302 .40196 L .93303 .40105 L .93304 .40015 L Mistroke .93303 .39925 L .93302 .39835 L .933 .39745 L .93297 .39654 L .93288 .39474 L .93275 .39294 L .93258 .39114 L .93214 .38754 L .93155 .38396 L .92991 .37684 L .92769 .3698 L .92488 .36288 L .92151 .35611 L .9131 .34314 L .90261 .33123 L .89025 .32072 L .87629 .31201 L .8688 .30844 L .86104 .30546 L .85306 .30309 L .849 .30215 L .84491 .30138 L .84079 .30078 L .83872 .30054 L .83664 .30035 L .83456 .3002 L .83352 .30014 L .83248 .30009 L .83144 .30005 L .8304 .30002 L .82936 .30001 L .82832 .3 L .82728 .30001 L .82624 .30002 L .82519 .30005 L .82415 .30009 L .82311 .30014 L .82207 .3002 L .81999 .30035 L .81792 .30054 L .81585 .30078 L .81173 .30138 L .80764 .30215 L .80358 .30309 L .7956 .30546 L .78784 .30844 L .78035 .31201 L .76638 .32072 L .75402 .33123 L .74354 .34314 L Mistroke .73906 .34952 L .73513 .35611 L .73175 .36288 L .72895 .3698 L .72673 .37684 L .72509 .38396 L .7245 .38754 L .72426 .38934 L .72405 .39114 L .72389 .39294 L .72376 .39474 L .72371 .39564 L .72367 .39654 L .72364 .39745 L .72362 .39835 L .7236 .39925 L .7236 .40015 L .72361 .40105 L .72362 .40196 L .72365 .40286 L .72368 .40376 L .72378 .40556 L .72391 .40736 L .72408 .40916 L .72454 .41276 L .72515 .41634 L .7259 .41991 L .72786 .42699 L .7304 .43396 L .73351 .44081 L .74142 .45397 L .75145 .46615 L .7634 .47702 L .77002 .48183 L .77703 .48618 L .78437 .49 L .79201 .49325 L .7999 .4959 L .80798 .49792 L .81208 .49868 L .8162 .49926 L .81827 .49949 L .82034 .49968 L .82138 .49976 L .82242 .49983 L .82346 .49988 L .8245 .49993 L .82554 .49996 L .82659 .49999 L .82763 .5 L Mistroke .82867 .5 L .82971 .49999 L .83075 .49997 L .83179 .49994 L .83283 .4999 L .83491 .49978 L .83699 .49962 L .83907 .49942 L .84113 .49918 L .84525 .49856 L .84935 .49777 L .85741 .49571 L .86528 .493 L .87289 .4897 L .88719 .48145 L Mfstroke P P p p .004 w .79216 .43474 m .79841 .44015 L .80186 .44247 L .80549 .44451 L .80927 .44624 L .81318 .44766 L .81518 .44825 L .8172 .44875 L .81924 .44917 L .8213 .4495 L .82233 .44964 L .82337 .44975 L .82441 .44985 L .82492 .44988 L .82544 .44992 L .82596 .44994 L .82648 .44997 L .82701 .44998 L .82753 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p .004 w s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s P P P % End of Graphics MathPictureEnd \ \>"], "Graphics", Evaluatable->False, AspectRatioFixed->True, ImageSize->{282, 174}, ImageMargins->{{34, Inherited}, {Inherited, Inherited}}, ImageCacheValid->False], Cell[OutputFormData["\<\ Graphics[\"<<>>\"]\ \>", "\<\ -Graphics-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Show[p,v,PlotRange->{{-2,8},{-3,3}}]]"], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.2 0.1 0.309017 0.103006 [ [(-2)] 0 .30902 0 2 Msboxa [(2)] .4 .30902 0 2 Msboxa [(4)] .6 .30902 0 2 Msboxa [(6)] .8 .30902 0 2 Msboxa [(8)] 1 .30902 0 2 Msboxa [(-3)] .1875 0 1 0 Msboxa [(-2)] .1875 .10301 1 0 Msboxa [(-1)] .1875 .20601 1 0 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For the ", StyleBox["x", FontFamily->"Courier", FontWeight->"Bold"], " population that size is ", StyleBox["3", FontFamily->"Courier", FontWeight->"Bold"], " and for the ", StyleBox["y", FontFamily->"Courier", FontWeight->"Bold"], " population it is ", StyleBox["3/4", FontFamily->"Courier", FontWeight->"Bold"], ". Both of these points are unstable. 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StyleBox["-1", FontFamily->"Courier", FontWeight->"Bold"], ", so the critical point ", StyleBox["{0,0}", FontFamily->"Courier", FontWeight->"Bold"], " is an unstable saddle point." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["A2 = A/.cp[[2]];\nA2//MatrixForm"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ MatrixForm[{{0, -3/2}, {4/3, 0}}]\ \>", "\<\ 3 -(-) 0 2 4 - 3 0\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Eigenvalues[A2]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {-I*2^(1/2), I*2^(1/2)}\ \>", "\<\ {-I Sqrt[2], I Sqrt[2]}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "The eigenvalues are imaginary, so the critical point ", StyleBox["{3/2,2}", FontFamily->"Courier", FontWeight->"Bold"], " is a center or a spiral point and has indeterminate stability." }], "Text", Evaluatable->False, 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.11348 .56012 m .1083 .51948 L .10446 .48123 L .10297 .46301 L .10175 .44541 L .10076 .42842 L .1 .41202 L .09945 .39622 L .09911 .38101 L .09895 .36637 L .09897 .35228 L .09917 .33874 L .09953 .32574 L .10004 .31325 L .10072 .30126 L .10154 .28977 L .10251 .27874 L .10488 .25804 L .10781 .23904 L .11128 .22162 L .1153 .20567 L .11985 .19109 L .13059 .16559 L .13678 .15449 L .14353 .14438 L .15872 .12682 L .17619 .11234 L .19594 .10049 L .21789 .09086 L .24192 .08315 L .26777 .07709 L .29514 .07247 L .30926 .07065 L .3236 .06913 L .33808 .06789 L .35263 .06693 L .35992 .06655 L .3672 .06624 L .37446 .06599 L .3817 .0658 L .3889 .06567 L .39606 .06561 L .40316 .06561 L .4102 .06566 L .41716 .06578 L .42405 .06596 L .43085 .06619 L .43755 .06649 L .45062 .06726 L .4632 .06826 L Mistroke .47524 .06951 L .48667 .07099 L .50756 .07468 L .51694 .0769 L .52556 .07936 L .5334 .08209 L .54045 .08507 L .5521 .09182 L .55669 .09558 L .56047 .09962 L .56343 .10391 L .5646 .10615 L .56558 .10846 L .56636 .11083 L .56694 .11326 L .56733 .11575 L .56753 .1183 L .56754 .12091 L .56736 .12358 L .567 .1263 L .56646 .12907 L .56485 .13476 L .56257 .14063 L .55611 .15282 L .54737 .1654 L .52443 .19066 L .49673 .214 L .46737 .23309 L .45297 .24051 L .43916 .24634 L .43255 .24865 L .42618 .25056 L .42006 .25207 L .41421 .25319 L .41139 .25362 L .40865 .25395 L .40598 .25419 L .40339 .25434 L .40087 .25441 L .39843 .2544 L .39608 .25431 L .3938 .25414 L .3916 .2539 L .38949 .25359 L .3855 .25276 L .38183 .25169 L .37848 .25039 L .37546 .24889 L .37274 .24722 L .37033 .24539 L Mistroke .36823 .24343 L .36488 .23922 L .36362 .237 L .36262 .23473 L .36188 .23243 L .3616 .23128 L .36138 .23012 L .36121 .22896 L .3611 .22781 L .36105 .22666 L .36105 .22551 L .3611 .22437 L .3612 .22324 L .36135 .22212 L .36155 .221 L .36208 .21882 L .36278 .21668 L .36463 .21262 L .36702 .20887 L .36985 .20546 L .37301 .20242 L .37641 .19977 L .37998 .19751 L .38362 .19565 L .38725 .19417 L .3908 .19306 L .39253 .19265 L .39422 .19231 L .39504 .19218 L .39585 .19207 L .39665 .19197 L .39743 .1919 L .3982 .19184 L .39895 .19181 L .39969 .19179 L .40041 .19179 L .40111 .19181 L .40179 .19185 L .40245 .1919 L .40309 .19197 L .40432 .19216 L .40547 .1924 L .40654 .1927 L .40751 .19306 L .40841 .19346 L .40921 .1939 L .41056 .19491 L .41111 .19546 L .41157 .19603 L .41194 .19664 L Mistroke .41224 .19726 L .41236 .19757 L .41246 .19789 L .41254 .19821 L .4126 .19854 L .41264 .19887 L .41266 .19919 L .41267 .19952 L .41266 .19985 L .41263 .20018 L .41259 .20051 L .41253 .20084 L .41246 .20117 L .41202 .20245 L .41172 .20308 L .41137 .20369 L .4096 .20594 L .40855 .20692 L .40742 .20778 L .40626 .20852 L .40508 .20914 L .40391 .20963 L .40278 .20999 L .40224 .21013 L .40171 .21024 L .4012 .21032 L .40096 .21035 L .40072 .21037 L .40048 .21039 L .40025 .21039 L .40002 .2104 L .3998 .2104 L .39959 .21039 L .39938 .21037 L .39918 .21035 L .39899 .21033 L .39862 .21026 L .39827 .21018 L .39796 .21007 L .39767 .20995 L .39741 .20982 L .39717 .20967 L .39696 .20951 L .39678 .20934 L .39663 .20916 L .3965 .20897 L .39639 .20878 L .39632 .20858 L Mfstroke P P p p .004 w s s s s s s s s s s s s s 1 .49571 m .84984 .57174 L s .84984 .57174 m 0 0 rlineto .75929 .61803 m .84984 .57174 L s s s s s s s s s s s s s s s s s .20779 .61803 m .19529 .59271 L s .19529 .59271 m .17268 .52942 L .16464 .49844 L .15833 .46841 L .15352 .43955 L .15 .41202 L .14867 .39879 L .14761 .38592 L .14681 .37343 L .14624 .3613 L .14589 .34955 L .14577 .33817 L .14585 .32716 L .14612 .31652 L .14659 .30623 L .14724 .29631 L .14908 .27749 L .15025 .26859 L .15158 .26002 L .15473 .24383 L .1585 .22885 L .16287 .21502 L .17333 .19054 L .1794 .17976 L .18601 .16987 L .20083 .15252 L .21767 .13809 L .23639 .12618 L .25682 .11648 L .27871 .10873 L .30174 .10268 L .32555 .09816 L .33762 .09643 L .34972 .09502 L .3618 .09393 L .36782 .0935 L .3738 .09315 L .37976 .09287 L .38566 .09266 L .39152 .09252 L .39733 .09246 L .40307 .09246 L .40874 .09253 L .41433 .09267 L .41983 .09287 L .42525 .09315 L .43057 .09349 L .44089 .09436 L .45075 .0955 L .46011 .09689 L Mistroke .46892 .09853 L .47716 .10042 L .49181 .10495 L .49817 .10758 L .50386 .11045 L .50888 .11355 L .51322 .11689 L .51688 .12044 L .51985 .12422 L .52109 .12618 L .52216 .12819 L .52306 .13026 L .52379 .13237 L .52437 .13452 L .52478 .13672 L .52503 .13897 L .52513 .14125 L .52508 .14357 L .52487 .14592 L .52452 .14831 L .52402 .15073 L .52338 .15318 L .5226 .15565 L .52065 .16066 L .51526 .17086 L .50813 .18111 L .49955 .19118 L .47929 .2099 L .457 .22536 L .44583 .23148 L .43496 .23639 L .42461 .24008 L .41968 .24147 L .41494 .24256 L .4104 .24336 L .40821 .24365 L .40608 .24388 L .404 .24404 L .40198 .24414 L .40002 .24417 L .39812 .24414 L .39628 .24405 L .3945 .2439 L .39279 .2437 L .39113 .24345 L .38801 .24278 L .38513 .24194 L .3825 .24092 L .38012 .23975 L .37798 .23844 L Mistroke .37609 .23702 L .37443 .23549 L .373 .23389 L .37179 .23221 L .3708 .23048 L .37002 .22871 L .36944 .22691 L .36922 .22601 L .36905 .2251 L .36893 .22419 L .36885 .22329 L .36881 .22238 L .36882 .22148 L .36887 .22059 L .36896 .21969 L .36908 .21881 L .36925 .21793 L .36968 .21621 L .37026 .21452 L .37096 .21288 L .37269 .20978 L .37481 .20693 L .37723 .20436 L .3827 .20013 L .38559 .19848 L .38851 .19715 L .39138 .19612 L .39279 .19571 L .39416 .19538 L .3955 .19512 L .39616 .19501 L .3968 .19492 L .39744 .19485 L .39806 .19479 L .39867 .19475 L .39926 .19472 L .39984 .19471 L .40041 .19472 L .40096 .19473 L .4015 .19477 L .40203 .19481 L .40254 .19487 L .40351 .19502 L .40441 .19522 L .40525 .19547 L .40672 .19608 L .40735 .19644 L .40791 .19683 L .4084 .19725 L .40882 .19769 L Mistroke .40918 .19816 L .40947 .19864 L .4097 .19914 L .40979 .19939 L .40986 .19964 L .40992 .1999 L .40996 .20016 L .40999 .20042 L .41001 .20068 L .41001 .20094 L .41 .20121 L .40998 .20147 L .40994 .20173 L .40989 .20199 L .40983 .20225 L .40967 .20276 L .40947 .20327 L .40923 .20377 L .40864 .20472 L .40793 .2056 L .40627 .20714 L .40535 .20776 L .40442 .20829 L .4035 .20872 L .40259 .20905 L .40215 .20918 L .40172 .20929 L .4013 .20937 L .4009 .20943 L .40071 .20945 L .40052 .20947 L .40033 .20948 L .40015 .20949 L .39997 .20949 L .3998 .20949 L .39963 .20948 L .39947 .20947 L .39931 .20945 L .39916 .20943 L .39887 .20937 L .3986 .2093 L .39835 .20922 L .39812 .20912 L .39792 .20902 L .39757 .20877 L .39743 .20863 L .39731 .20849 L .39721 .20834 L .39713 .20819 L .3971 .20811 L Mistroke .39707 .20803 L .39705 .20795 L .39703 .20787 L .39701 .20779 L .397 .20771 L .397 .20763 L .397 .20755 L Mfstroke P P p p .004 w s s s s s s s s s s s s s s s 1 .35204 m .68894 .50576 L s .68894 .50576 m .55033 .56621 L .52087 .57628 L .49334 .58436 L .46766 .59051 L .44373 .5948 L .42149 .59732 L .40083 .59817 L .38167 .59748 L .36394 .59536 L .34753 .59193 L .33238 .58731 L .31839 .58163 L .3055 .57499 L .28271 .5593 L .26346 .54108 L .24729 .52105 L .23378 .49983 L .22257 .47794 L .21337 .45579 L .20592 .43373 L .2 .41202 L .19544 .39087 L .19363 .38055 L .1921 .37042 L .19084 .3605 L .18984 .35079 L .18908 .3413 L .18856 .33205 L .18826 .32302 L .18818 .31423 L .1883 .30569 L .18862 .29738 L .18913 .28932 L .18983 .28149 L .1907 .27391 L .19175 .26656 L .19434 .25257 L .19757 .2395 L .20581 .21598 L .21077 .20546 L .21625 .19571 L .2287 .17841 L .24297 .16378 L .25887 .15154 L .27618 .14142 L .29463 .13323 L .31393 .12674 L .33376 .12181 L .34375 .11988 L Mistroke .35375 .11829 L .36369 .11703 L .36862 .11652 L .37353 .11608 L .3784 .11572 L .38323 .11543 L .38801 .11521 L .39274 .11507 L .39741 .11499 L .40202 .11499 L .40656 .11505 L .41102 .11518 L .4154 .11538 L .4197 .11565 L .42803 .11637 L .43597 .11734 L .44348 .11857 L .45055 .12003 L .45713 .12172 L .46879 .12579 L .47383 .12815 L .47833 .13072 L .48228 .13348 L .48568 .13644 L .48853 .13957 L .49084 .14287 L .49178 .14458 L .4926 .14632 L .49328 .1481 L .49383 .14992 L .49425 .15177 L .49454 .15364 L .49471 .15555 L .49475 .15748 L .49468 .15943 L .49448 .16141 L .49417 .1634 L .49375 .16542 L .49321 .16744 L .49257 .16948 L .49098 .17358 L .48666 .18181 L .481 .18993 L .47426 .19778 L .45853 .21199 L .45002 .21807 L .44139 .22332 L .43284 .22768 L .42456 .2311 L .41669 .23359 L Mistroke .41295 .23449 L .40935 .23518 L .40761 .23544 L .40592 .23564 L .40426 .2358 L .40265 .2359 L .40108 .23596 L .39955 .23597 L .39808 .23594 L .39664 .23585 L .39526 .23573 L .39392 .23556 L .39263 .23536 L .39138 .23511 L .38904 .23451 L .38689 .23378 L .38493 .23292 L .38316 .23195 L .38158 .23089 L .38019 .22974 L .37898 .22852 L .37795 .22724 L .37709 .22592 L .3764 .22455 L .37612 .22386 L .37587 .22316 L .37567 .22246 L .3755 .22175 L .37537 .22104 L .37528 .22033 L .37522 .21962 L .3752 .21892 L .37521 .21821 L .37525 .21751 L .37533 .21681 L .37543 .21611 L .37573 .21474 L .37614 .2134 L .37666 .21209 L .37727 .21083 L .37875 .20842 L .38052 .20622 L .38251 .20424 L .38693 .20101 L .38924 .19977 L .39155 .19877 L .39382 .19801 L .39492 .19772 L .39599 .19749 L .39704 .19731 L Mistroke .39754 .19724 L .39804 .19719 L .39853 .19714 L .39901 .19711 L .39948 .19709 L .39994 .19708 L .40039 .19709 L .40082 .1971 L .40125 .19713 L .40166 .19717 L .40206 .19722 L .40245 .19727 L .40318 .19742 L .40386 .1976 L .40449 .19781 L .40559 .19834 L .40605 .19864 L .40646 .19896 L .40682 .19931 L .40712 .19967 L .40737 .20004 L .40757 .20043 L .40765 .20063 L .40771 .20083 L .40777 .20104 L .40781 .20124 L .40785 .20145 L .40787 .20165 L .40788 .20186 L .40788 .20207 L .40787 .20227 L .40785 .20248 L .40777 .20289 L .40772 .2031 L .40766 .2033 L .40752 .2037 L .40714 .20448 L .40664 .20522 L .40606 .2059 L .40472 .20706 L .404 .20753 L .40326 .20793 L .40253 .20824 L .40183 .20848 L .40149 .20857 L .40115 .20864 L .40083 .20869 L .40068 .20871 L .40053 .20873 L .40038 .20874 L Mistroke .40023 .20875 L .40009 .20875 L .39995 .20875 L .39981 .20875 L .39968 .20874 L .39955 .20873 L .39943 .20872 L .39931 .2087 L .39919 .20868 L .39897 .20863 L .39877 .20857 L .39859 .2085 L .39842 .20842 L .39827 .20833 L .39801 .20812 L .39791 .20801 L .39783 .2079 L .39776 .20778 L .3977 .20766 L .39768 .20759 L .39766 .20753 L .39765 .20747 L .39764 .2074 L .39763 .20734 L .39763 .20728 L .39763 .20721 L .39763 .20715 L Mfstroke P P p p .004 w s s s s s s s s s s s s s s s s s 1 .25671 m .74948 .37573 L s .74948 .37573 m .62817 .4363 L .57553 .4598 L .52794 .47828 L .48518 .49169 L .46554 .49653 L .447 .50017 L .42955 .50267 L .41312 .50406 L .3977 .50441 L .38323 .50378 L .36967 .50223 L .35699 .49983 L .34514 .49665 L .33408 .49276 L .31418 .48309 L .29699 .47133 L .28222 .45795 L .26961 .44337 L .25894 .42795 L .25 .41202 L .24261 .39584 L .23661 .37964 L .23186 .36359 L .22825 .34784 L .22684 .34012 L .22567 .33251 L .22474 .32503 L .22403 .31768 L .22354 .31047 L .22326 .30341 L .22318 .29651 L .22329 .28976 L .22359 .28317 L .22406 .27675 L .22471 .2705 L .22553 .2644 L .22764 .25273 L .23036 .24172 L .23365 .23137 L .23747 .22166 L .24662 .20414 L .25756 .18899 L .27008 .17604 L .28395 .16511 L .29891 .15603 L .3147 .14863 L .33105 .14277 L .34766 .13833 L .35597 .13661 L Mistroke .36422 .1352 L .37239 .13409 L .37643 .13364 L .38043 .13327 L .38439 .13296 L .38831 .13272 L .39218 .13256 L .39599 .13245 L .39975 .13242 L .40344 .13245 L .40707 .13254 L .41063 .13269 L .41752 .13318 L .42085 .13351 L .42409 .1339 L .43031 .13485 L .43617 .13603 L .44669 .139 L .45133 .14079 L .45554 .14277 L .45931 .14492 L .46263 .14724 L .46551 .14972 L .46794 .15234 L .46993 .1551 L .47076 .15653 L .47148 .15799 L .4721 .15947 L .4726 .16098 L .47301 .16252 L .47331 .16407 L .47351 .16565 L .47361 .16724 L .47361 .16885 L .47352 .17048 L .47333 .17212 L .47306 .17376 L .47224 .17708 L .47109 .18042 L .46962 .18376 L .46584 .19037 L .46108 .19677 L .44943 .20847 L .43621 .218 L .42946 .22175 L .42283 .22479 L .41644 .2271 L .41338 .22798 L .41041 .22869 L .40756 .22923 L Mistroke .40618 .22943 L .40483 .2296 L .40351 .22972 L .40223 .22981 L .40098 .22985 L .39976 .22986 L .39858 .22984 L .39743 .22978 L .39632 .22969 L .39525 .22956 L .39322 .22921 L .39226 .229 L .39133 .22875 L .38802 .22752 L .38659 .22677 L .38531 .22594 L .38418 .22504 L .3832 .22409 L .38235 .22309 L .38165 .22204 L .38108 .22097 L .38064 .21987 L .38047 .21931 L .38033 .21875 L .38022 .21819 L .38014 .21763 L .38009 .21707 L .38006 .2165 L .38007 .21594 L .3801 .21538 L .38015 .21483 L .38023 .21427 L .38046 .21318 L .38079 .21211 L .38119 .21106 L .38223 .20905 L .38354 .2072 L .38505 .20551 L .38849 .20269 L .39033 .20157 L .3922 .20066 L .39404 .19994 L .39494 .19965 L .39583 .19941 L .39669 .19922 L .39753 .19907 L .39794 .19901 L .39834 .19896 L .39874 .19892 L .39912 .1989 L Mistroke .3995 .19888 L .39987 .19887 L .40023 .19887 L .40058 .19888 L .40092 .1989 L .40125 .19893 L .40157 .19897 L .40188 .19901 L .40248 .19912 L .40303 .19927 L .40353 .19944 L .404 .19963 L .40441 .19985 L .40479 .20009 L .40512 .20035 L .40541 .20063 L .40565 .20092 L .40586 .20122 L .40602 .20153 L .40609 .20169 L .40614 .20185 L .40619 .20201 L .40623 .20217 L .40625 .20234 L .40627 .2025 L .40628 .20267 L .40628 .20283 L .40628 .203 L .40626 .20317 L .40624 .20333 L .40621 .2035 L .40612 .20382 L .40601 .20415 L .40587 .20446 L .40552 .20507 L .4051 .20564 L .40407 .20662 L .4035 .20703 L .40292 .20738 L .40234 .20766 L .40176 .20789 L .40148 .20797 L .40121 .20805 L .40094 .2081 L .40069 .20815 L .40056 .20817 L .40044 .20818 L .40032 .20819 L .4002 .2082 L .40009 .2082 L Mistroke .39998 .2082 L .39987 .2082 L .39976 .20819 L .39966 .20819 L .39956 .20818 L .39937 .20815 L .39928 .20813 L .3992 .20811 L .39903 .20806 L .39888 .208 L .39875 .20794 L .39851 .20779 L .39842 .20771 L .39834 .20762 L .39827 .20753 L .39821 .20743 L .39817 .20734 L .39813 .20724 L .39812 .20718 L .39811 .20713 L .39811 .20708 L .3981 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