lab4_tutor_curv_vel

lab4_tutor_curv_vel - (************** Content-type:...

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Unformatted text preview: (************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data f for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do o 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 C CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematicacompatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may t 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[ (*NotebookOutlinePosition[ (* CellTagsIndexPosition[ ( (*WindowFrame->Normal*) 1846235, 1860672, 1860628, 68674]*) 69201]*) 69197]*) Notebook[{ Cell[TextData[StyleBox["Lab 4-TUTORIAL", "Title", FontColor->RGBColor[1, 0, 0]]], "StudentName"], Cell[CellGroupData[{ Cell["\<\ Arclength,Velocity, Acceleration,Unit Tangents, Principal Normals,Curvature, \ Osculating Circles \ \>", "Subtitle"], Cell[TextData[StyleBox["Copyright 2006-2007 G. Pelekanos, Southern Illinois \ University, Edwardsville", FontColor->RGBColor[0, 0, 1]]], "Copyright"], Cell[TextData[{ StyleBox["I. History", "Title", FontSize->24], "\n\n", StyleBox["Aristotle: (350 BC) distinguishes between straight lines, circles \ and \[CloseCurlyDoubleQuote]mixed behavior\[CloseCurlyDoubleQuote]\nOresme: \ (14\[CloseCurlyQuote]th century): measure of twist called \ \[CloseCurlyDoubleQuote]curvitas\[CloseCurlyDoubleQuote]\nKepler: (15\ \[CloseCurlyQuote]th century): circle of curvature.\nHuygens: (16\ \[CloseCurlyQuote]th century): evolutes and involutes in connection with \ optics.\nNewton: (17\[CloseCurlyQuote]th century): circle has constant \ curvature inversely proportional to radius. (using infinitesimals)\nSimpson: \ (17\[CloseCurlyQuote]th century): string construction of evolutes, \ description using fluxions.\nEuler: (17\[CloseCurlyQuote]th century): first \ formulas of curvature using second derivatives.\nGauss: \ (18\[CloseCurlyQuote]th century): modern description, higher dimensional \ versions.\n", FontFamily->"Garamond", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] } }], "Text"], C Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Initialization Cells--No need to open this Group (In case that \ you hear 6 sounds do the following: Go to Kernel---->Quit \ kernel---->Local---->Start Kernel---->Local. Then open this cell and move the \ statement ", FontFamily->"Comic Sans MS", FontSize->18], "Needs[\"Graphics`Arrow`\"]; ", StyleBox["one line up or down and Save. 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Then, we may \ndescribe the \ particle's motion by writing its position vector ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(r\&\[RightArrow]\), "TraditionalForm"], " ", \((t)\)}], TraditionalForm]], FontWeight->"Bold"], StyleBox[", as a \nfunction of time t:\n\n\t\t\t\t", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(r\&\[RightArrow]\), "TraditionalForm"], " ", \((t)\)}], TraditionalForm]], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["=x[t]i + y[t]j", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["\n\t\t\t\t\t\t\t\nAs t varies and the initial point of the vector \ ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(r\&\[RightArrow]\), "TraditionalForm"], " ", \((t)\)}], TraditionalForm]], FontWeight->"Bold"], StyleBox[" is fixed at the origin, \nthe terminal point of ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(r\&\[RightArrow]\), "TraditionalForm"], " ", \((t)\)}], TraditionalForm]], FontWeight->"Bold"], StyleBox[" traces out the path of the particle. ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(r\&\[RightArrow]\), "TraditionalForm"], " ", \((t)\)}], TraditionalForm]], FontWeight->"Bold"], StyleBox[" is \na vector-valued function, since its range consists of \ vectors.\n\nWe illustrate this idea using the function ", FontWeight->"Bold"], StyleBox["PositionVector[x,y,a,b,n]", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[", \nwhere:\n\n(i) x and y are functions of t giving the position \ of the \nparticle as a function of time t, with a\[LessEqual]t\[LessEqual]b. \ \n \n(ii) n is the number of subintervals of the time interval a<t<b to \ be considered. \n(Thus, each subinterval has width", FontWeight->"Bold"], StyleBox[" h= (b-a)/n", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[".) \n\n", FontWeight->"Bold"], StyleBox["PositionVector[x,y,a,b,n]", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" will generate n+1 graphics. 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", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["\t\n\nThe position vectors have been drawn in blue, as have the \ dots.\n\nLet's consider the ", FontWeight->"Bold"], StyleBox["kite-shaped", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[" object given by the parametric equations:\n\n \t \ ", FontWeight->"Bold"], StyleBox[" x[t]=Cos[t]+0.65 Cos[2t]-0.65\t\n \t y[t]=1.5 \ Sin[t]", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["\n \t\t\t\t\t\t\t\t\t\t\t\t\t\nwhere t goes from t=0 to t=2\ \[Pi]. \n\nThen, the position vector of the particle is given by:\n\n \ \t\t", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(r\&\[RightArrow]\), "TraditionalForm"], " ", \((t)\)}], TraditionalForm]], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" =(Cos[t]+0.65 Cos[2t]-0.65)i + 1.5 Sin[t]j", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["\n\nLet's have Mathematica generate these graphics with n=8. \n", 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s .5 .43333 m .50375 .43333 s .5 .46667 m .50375 .46667 s .5 .53333 m .50375 .53333 s .5 .56667 m .50375 .56667 s .5 .6 m .50375 .6 L s .5 .63333 m .50375 .63333 s .5 .7 m .50375 .7 L s .5 .73333 m .50375 .73333 s .5 .76667 m .50375 .76667 s .5 .8 m .50375 .8 L s .5 .86667 m .50375 .86667 s .5 .9 m .50375 .9 L L L L L L L L L L L L L s .5 .93333 m .50375 .93333 L s .5 .96667 m .50375 .96667 L s .25 Mabswid .5 0 m .5 1 L s 00m 10L 11L 01L closepath clip newpath % Start of user PostScript /mathtops { gsave MBeginOrig moveto MEndOrig currentpoint grestore } bind def /MAtocoords { mathtops 4 2 roll mathtops 4 copy pop pop 3 -1 roll sub /arry exch def exch sub /arrx exch def arrx dup mul arry dup mul add sqrt /arrl exch def translate } bind def /MAarrowhead1 { gsave MAtocoords arrl 0. eq { 0 0 Mdot } { [ arrx arrl div arry arrl div -1 arry mul arrl div arrx arrl div 00] concat -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto fill -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto -0.05 0.0125 lineto stroke } ifelse grestore } def % End of user PostScript .5 Mabswid .66667 .5 m .66665 .50194 L .6666 .5037 L .66651 .50574 L .66638 .50766 L .66608 .51108 L .66562 .5148 L .66496 .51888 L .66408 .5232 L .66207 .53094 L .65939 .5389 L .65635 .54631 L .64751 .56303 L .63588 .57979 L .62309 .59477 L .58764 .627 L .5508 .65291 L .50635 .67844 L .45854 .70122 L .41561 .71845 L .37295 .73278 L .33774 .74225 L .32232 .74554 L .30697 .74812 L .29968 .74903 L .29636 .74936 L .29332 .74961 L .29171 .74972 L .28999 .74982 L .28859 .74988 L .28707 .74994 L .28624 .74996 L .28547 .74998 L .28467 .74999 L .28381 .75 L .28307 .75 L .28229 .75 L .28093 .74997 L .27964 .74993 L .27893 .74991 L .27828 .74987 L .27712 .74981 L .27588 .74972 L .27381 .74952 L .27171 .74925 L .26955 .74889 L .26736 .74842 L .26559 .74796 L .26378 .74738 L .26067 .74608 L Mistroke .25814 .7446 L .25607 .74294 L .25449 .7412 L .25366 .74001 L .25301 .73886 L .25242 .73752 L .25217 .7368 L .25194 .73602 L .25176 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.25132 .27062 L .25129 .26983 L .25128 .2691 L .25129 .26869 L .2513 .26827 L .25134 .26751 L .25138 .26706 L .25143 .26663 L .25154 .26585 L .25168 .26505 L .25184 .26433 L .25232 .26276 L .25281 .26157 L .25345 .26034 L .25511 .25805 L .25609 .25704 L .25728 .25602 L .25981 .25437 L .26272 .25302 L .26581 .25198 L .2679 .25145 L .26999 .25103 L .27197 .25072 L .27423 .25044 L .27557 .25031 L .27684 .25021 L .27821 .25013 L .279 .25009 L .27975 .25006 L Mistroke .28112 .25002 L .28239 .25 L .28379 .25 L .28459 .25001 L .28534 .25002 L .28677 .25005 L .28815 .2501 L .28942 .25015 L .29082 .25023 L .29405 .25045 L .29716 .25071 L .30308 .25136 L .30977 .25229 L .32667 .25532 L .34359 .25915 L .38389 .27062 L .42485 .28502 L .51103 .32402 L .55123 .34736 L .59102 .37568 L .62245 .40454 L .63572 .42 L .64766 .43722 L .65637 .45373 L .65938 .46106 L .66211 .4692 L .6641 .47688 L .66545 .4841 L .66599 .48809 L .66634 .49175 L .66647 .49367 L .66658 .49577 L .66662 .49683 L .66665 .49796 L .66667 .5 L Mfstroke 001r .02 w .66667 .5 Mdot .50952 .67678 Mdot .28333 .75 Mdot .5 Mabswid .5 .5 m .28333 .75 L s % Start of user PostScript 0. 0. -1.3 1.5 MAarrowhead1 % End of user PostScript % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 288}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.166667 0.5 0.166667 [ [0 .4875 -6 -9 ] [0 .4875 6 0 ] [.16667 .4875 -6 -9 ] [.16667 .4875 6 0 ] [.33333 .4875 -6 -9 ] [.33333 .4875 6 0 ] [.66667 .4875 -3 -9 ] [.66667 .4875 3 0 ] [.83333 .4875 -3 -9 ] [.83333 .4875 3 0 ] [1 .4875 -3 -9 ] [1 .4875 3 0 ] [.4875 0 -12 -4.5 ] [.4875 0 0 4.5 ] [.4875 .16667 -12 -4.5 ] [.4875 .16667 0 4.5 ] [.4875 .33333 -12 -4.5 ] [.4875 .33333 0 4.5 ] [.4875 .66667 -6 -4.5 ] [.4875 .66667 0 4.5 ] [.4875 .83333 -6 -4.5 ] [.4875 .83333 0 4.5 ] [.4875 1 -6 -4.5 ] [.4875 1 0 4.5 ] [0000] [1100] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0g .25 Mabswid [ ] 0 setdash 0 .5 m 0 .50625 L s [(-3)] 0 .4875 0 1 Mshowa .16667 .5 m .16667 .50625 L s [(-2)] .16667 .4875 0 1 Mshowa .33333 .5 m .33333 .50625 L s [(-1)] .33333 .4875 0 1 Mshowa .66667 .5 m .66667 .50625 L s [(1)] .66667 .4875 0 1 Mshowa .83333 .5 m .83333 .50625 L s [(2)] .83333 .4875 0 1 Mshowa 1 .5 m 1 .50625 L s [(3)] 1 .4875 0 1 Mshowa .125 Mabswid .03333 .5 m .03333 .50375 L s .06667 .5 m .06667 .50375 s .1 .5 m .1 .50375 L s .13333 .5 m .13333 .50375 s .2 .5 m .2 .50375 L s .23333 .5 m .23333 .50375 s .26667 .5 m .26667 .50375 s .3 .5 m .3 .50375 L s .36667 .5 m .36667 .50375 s .4 .5 m .4 .50375 L s .43333 .5 m .43333 .50375 s .46667 .5 m .46667 .50375 s .53333 .5 m .53333 .50375 s .56667 .5 m .56667 .50375 s .6 .5 m .6 .50375 L s .63333 .5 m .63333 .50375 s .7 .5 m .7 .50375 L s .73333 .5 m .73333 .50375 s .76667 .5 m .76667 .50375 s .8 .5 m .8 .50375 L s .86667 .5 m .86667 .50375 s L L L L L L L L L L L L L .9 .5 m .9 .50375 L s .93333 .5 m .93333 .50375 L s .96667 .5 m .96667 .50375 L s .25 Mabswid 0 .5 m 1 .5 L s .5 0 m .50625 0 L s [(-3)] .4875 0 1 0 Mshowa .5 .16667 m .50625 .16667 L s [(-2)] .4875 .16667 1 0 Mshowa .5 .33333 m .50625 .33333 L s [(-1)] .4875 .33333 1 0 Mshowa .5 .66667 m .50625 .66667 L s [(1)] .4875 .66667 1 0 Mshowa .5 .83333 m .50625 .83333 L s [(2)] .4875 .83333 1 0 Mshowa .5 1 m .50625 1 L s [(3)] .4875 1 1 0 Mshowa .125 Mabswid .5 .03333 m .50375 .03333 L s .5 .06667 m .50375 .06667 L s .5 .1 m .50375 .1 L s .5 .13333 m .50375 .13333 L s .5 .2 m .50375 .2 L s .5 .23333 m .50375 .23333 L s .5 .26667 m .50375 .26667 L s .5 .3 m .50375 .3 L s .5 .36667 m .50375 .36667 s .5 .4 m .50375 .4 L s .5 .43333 m .50375 .43333 s .5 .46667 m .50375 .46667 s .5 .53333 m .50375 .53333 s .5 .56667 m .50375 .56667 s .5 .6 m .50375 .6 L s .5 .63333 m .50375 .63333 s .5 .7 m .50375 .7 L s .5 .73333 m .50375 .73333 s .5 .76667 m .50375 .76667 s .5 .8 m .50375 .8 L s .5 .86667 m .50375 .86667 s .5 .9 m .50375 .9 L s .5 .93333 m .50375 .93333 s .5 .96667 m .50375 .96667 s .25 Mabswid .5 0 m .5 1 L s 00m 10L 11L 01L L L L L L L L L L L L closepath clip newpath % Start of user PostScript /mathtops { gsave MBeginOrig moveto MEndOrig currentpoint grestore } bind def /MAtocoords { mathtops 4 2 roll mathtops 4 copy pop pop 3 -1 roll sub /arry exch def exch sub /arrx exch def arrx dup mul arry dup mul add sqrt /arrl exch def translate } bind def /MAarrowhead1 { gsave MAtocoords arrl 0. eq { 0 0 Mdot } { [ arrx arrl div arry arrl div -1 arry mul arrl div arrx arrl div 00] concat -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto fill -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto -0.05 0.0125 lineto stroke } ifelse grestore } def % End of user PostScript .5 Mabswid .66667 .5 m .66665 .50194 L .6666 .5037 L .66651 .50574 L .66638 .50766 L .66608 .51108 L .66562 .5148 L .66496 .51888 L .66408 .5232 L .66207 .53094 L .65939 .5389 L .65635 .54631 L .64751 .56303 L .63588 .57979 L .62309 .59477 L .58764 .627 L .5508 .65291 L .50635 .67844 L .45854 .70122 L .41561 .71845 L .37295 .73278 L .33774 .74225 L .32232 .74554 L .30697 .74812 L .29968 .74903 L .29636 .74936 L .29332 .74961 L .29171 .74972 L .28999 .74982 L .28859 .74988 L .28707 .74994 L .28624 .74996 L .28547 .74998 L .28467 .74999 L .28381 .75 L .28307 .75 L .28229 .75 L .28093 .74997 L .27964 .74993 L .27893 .74991 L .27828 .74987 L .27712 .74981 L .27588 .74972 L .27381 .74952 L .27171 .74925 L .26955 .74889 L .26736 .74842 L .26559 .74796 L .26378 .74738 L .26067 .74608 L Mistroke .25814 .7446 L .25607 .74294 L .25449 .7412 L .25366 .74001 L .25301 .73886 L .25242 .73752 L .25217 .7368 L .25194 .73602 L .25176 .73531 L .25162 .73465 L 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.26983 L .25128 .2691 L .25129 .26869 L .2513 .26827 L .25134 .26751 L .25138 .26706 L .25143 .26663 L .25154 .26585 L .25168 .26505 L .25184 .26433 L .25232 .26276 L .25281 .26157 L .25345 .26034 L .25511 .25805 L .25609 .25704 L .25728 .25602 L .25981 .25437 L .26272 .25302 L .26581 .25198 L .2679 .25145 L .26999 .25103 L .27197 .25072 L .27423 .25044 L .27557 .25031 L .27684 .25021 L .27821 .25013 L .279 .25009 L .27975 .25006 L Mistroke .28112 .25002 L .28239 .25 L .28379 .25 L .28459 .25001 L .28534 .25002 L .28677 .25005 L .28815 .2501 L .28942 .25015 L .29082 .25023 L .29405 .25045 L .29716 .25071 L .30308 .25136 L .30977 .25229 L .32667 .25532 L .34359 .25915 L .38389 .27062 L .42485 .28502 L .51103 .32402 L .55123 .34736 L .59102 .37568 L .62245 .40454 L .63572 .42 L .64766 .43722 L .65637 .45373 L .65938 .46106 L .66211 .4692 L .6641 .47688 L .66545 .4841 L .66599 .48809 L .66634 .49175 L .66647 .49367 L .66658 .49577 L .66662 .49683 L .66665 .49796 L .66667 .5 L Mfstroke 001r .02 w .66667 .5 Mdot .50952 .67678 Mdot .28333 .75 Mdot .27382 .67678 Mdot .5 Mabswid .5 .5 m .27382 .67678 L s % Start of user PostScript 0. 0. -1.3571068 1.0606602 MAarrowhead1 % End of user PostScript % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 288}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.166667 0.5 0.166667 [ [0 .4875 -6 -9 ] [0 .4875 6 0 ] [.16667 .4875 -6 -9 ] [.16667 .4875 6 0 ] [.33333 .4875 -6 -9 ] [.33333 .4875 6 0 ] [.66667 .4875 -3 -9 ] [.66667 .4875 3 0 ] [.83333 .4875 -3 -9 ] [.83333 .4875 3 0 ] [1 .4875 -3 -9 ] [1 .4875 3 0 ] [.4875 0 -12 -4.5 ] [.4875 0 0 4.5 ] [.4875 .16667 -12 -4.5 ] [.4875 .16667 0 4.5 ] [.4875 .33333 -12 -4.5 ] [.4875 .33333 0 4.5 ] [.4875 .66667 -6 -4.5 ] [.4875 .66667 0 4.5 ] [.4875 .83333 -6 -4.5 ] [.4875 .83333 0 4.5 ] [.4875 1 -6 -4.5 ] [.4875 1 0 4.5 ] [0000] [1100] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0g .25 Mabswid [ ] 0 setdash 0 .5 m 0 .50625 L s [(-3)] 0 .4875 0 1 Mshowa .16667 .5 m .16667 .50625 L s [(-2)] .16667 .4875 0 1 Mshowa .33333 .5 m .33333 .50625 L s [(-1)] .33333 .4875 0 1 Mshowa .66667 .5 m .66667 .50625 L s [(1)] .66667 .4875 0 1 Mshowa .83333 .5 m .83333 .50625 L s [(2)] .83333 .4875 0 1 Mshowa 1 .5 m 1 .50625 L s [(3)] 1 .4875 0 1 Mshowa .125 Mabswid .03333 .5 m .03333 .50375 L s .06667 .5 m .06667 .50375 L s .1 .5 m .1 .50375 L s .13333 .5 m .13333 .50375 L s .2 .5 m .2 .50375 L s .23333 .5 m .23333 .50375 L s .26667 .5 m .26667 .50375 s .3 .5 m .3 .50375 L s .36667 .5 m .36667 .50375 s .4 .5 m .4 .50375 L s .43333 .5 m .43333 .50375 s .46667 .5 m .46667 .50375 s .53333 .5 m .53333 .50375 s .56667 .5 m .56667 .50375 s .6 .5 m .6 .50375 L s .63333 .5 m .63333 .50375 s .7 .5 m .7 .50375 L s .73333 .5 m .73333 .50375 s .76667 .5 m .76667 .50375 s .8 .5 m .8 .50375 L s .86667 .5 m .86667 .50375 s .9 .5 m .9 .50375 L s .93333 .5 m .93333 .50375 s .96667 .5 m .96667 .50375 s .25 Mabswid 0 .5 m 1 .5 L s .5 0 m L L L L L L L L L L L L .50625 0 L s [(-3)] .4875 0 1 0 Mshowa .5 .16667 m .50625 .16667 L s [(-2)] .4875 .16667 1 0 Mshowa .5 .33333 m .50625 .33333 L s [(-1)] .4875 .33333 1 0 Mshowa .5 .66667 m .50625 .66667 L s [(1)] .4875 .66667 1 0 Mshowa .5 .83333 m .50625 .83333 L s [(2)] .4875 .83333 1 0 Mshowa .5 1 m .50625 1 L s [(3)] .4875 1 1 0 Mshowa .125 Mabswid .5 .03333 m .50375 .03333 L s .5 .06667 m .50375 .06667 L s .5 .1 m .50375 .1 L s .5 .13333 m .50375 .13333 L s .5 .2 m .50375 .2 L s .5 .23333 m .50375 .23333 L s .5 .26667 m .50375 .26667 L s .5 .3 m .50375 .3 L s .5 .36667 m .50375 .36667 L s .5 .4 m .50375 .4 L s .5 .43333 m .50375 .43333 L s .5 .46667 m .50375 .46667 L s .5 .53333 m .50375 .53333 L s .5 .56667 m .50375 .56667 L s .5 .6 m .50375 .6 L s .5 .63333 m .50375 .63333 L s .5 .7 m .50375 .7 L s .5 .73333 m .50375 .73333 L s .5 .76667 m .50375 .76667 L s .5 .8 m .50375 .8 L s .5 .86667 m .50375 .86667 L s .5 .9 m .50375 .9 L s .5 .93333 m .50375 .93333 L s .5 .96667 m .50375 .96667 L s .25 Mabswid .5 0 m .5 1 L s 00m 10L 11L 01L closepath clip newpath % Start of user PostScript /mathtops { gsave MBeginOrig moveto MEndOrig currentpoint grestore } bind def /MAtocoords { mathtops 4 2 roll mathtops 4 copy pop pop 3 -1 roll sub /arry exch def exch sub /arrx exch def arrx dup mul arry dup mul add sqrt /arrl exch def translate } bind def /MAarrowhead1 { gsave MAtocoords arrl 0. eq { 0 0 Mdot } { [ arrx arrl div arry arrl div -1 arry mul arrl div arrx arrl div 00] concat -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto fill -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto -0.05 0.0125 lineto 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.5 L Mfstroke 001r .02 w .66667 .5 Mdot .50952 .67678 Mdot .28333 .75 Mdot .27382 .67678 Mdot .33333 .5 Mdot .5 Mabswid .5 .5 m .33333 .5 L s % Start of user PostScript 0. 0. -1. 0. 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.73333 .50375 L s .76667 .5 m .76667 .50375 L s .8 .5 m .8 .50375 L s .86667 .5 m .86667 .50375 L s .9 .5 m .9 .50375 L s .93333 .5 m .93333 .50375 L s .96667 .5 m .96667 .50375 L s .25 Mabswid 0 .5 m 1 .5 L s .5 0 m .50625 0 L s [(-3)] .4875 0 1 0 Mshowa .5 .16667 m .50625 .16667 L s [(-2)] .4875 .16667 1 0 Mshowa .5 .33333 m .50625 .33333 L s [(-1)] .4875 .33333 1 0 Mshowa .5 .66667 m .50625 .66667 L s [(1)] .4875 .66667 1 0 Mshowa .5 .83333 m .50625 .83333 L s [(2)] .4875 .83333 1 0 Mshowa .5 1 m .50625 1 L s [(3)] .4875 1 1 0 Mshowa .125 Mabswid .5 .03333 m .50375 .03333 L s .5 .06667 m .50375 .06667 L s .5 .1 m .50375 .1 L s .5 .13333 m .50375 .13333 L s .5 .2 m .50375 .2 L s .5 .23333 m .50375 .23333 L s .5 .26667 m .50375 .26667 L s .5 .3 m .50375 .3 L s .5 .36667 m .50375 .36667 L s .5 .4 m .50375 .4 L s .5 .43333 m .50375 .43333 L s .5 .46667 m .50375 .46667 L s .5 .53333 m .50375 .53333 L s .5 .56667 m .50375 .56667 L s .5 .6 m .50375 .6 L s .5 .63333 m .50375 .63333 L s .5 .7 m .50375 .7 L s .5 .73333 m .50375 .73333 L s .5 .76667 m .50375 .76667 L s .5 .8 m .50375 .8 L s .5 .86667 m .50375 .86667 L s .5 .9 m .50375 .9 L s .5 .93333 m .50375 .93333 L s .5 .96667 m .50375 .96667 L s .25 Mabswid .5 0 m .5 1 L s 00m 10L 11L 01L closepath clip newpath % Start of user PostScript /mathtops { gsave MBeginOrig moveto MEndOrig currentpoint grestore } bind def /MAtocoords { mathtops 4 2 roll mathtops 4 copy pop pop 3 -1 roll sub /arry exch def exch sub /arrx exch def arrx dup mul arry dup mul add sqrt /arrl exch def translate } bind def /MAarrowhead1 { gsave MAtocoords arrl 0. eq { 0 0 Mdot } { [ arrx arrl div arry arrl div -1 arry mul arrl div arrx arrl div 00] concat -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto fill -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto -0.05 0.0125 lineto stroke } ifelse grestore } def % End of user PostScript .5 Mabswid .66667 .5 m .66665 .50194 L .6666 .5037 L .66651 .50574 L .66638 .50766 L .66608 .51108 L .66562 .5148 L 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[.4875 .66667 0 4.5 ] [.4875 .83333 -6 -4.5 ] [.4875 .83333 0 4.5 ] [.4875 1 -6 -4.5 ] [.4875 1 0 4.5 ] [0000] [1100] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0g .25 Mabswid [ ] 0 setdash 0 .5 m 0 .50625 L s [(-3)] 0 .4875 0 1 Mshowa .16667 .5 m .16667 .50625 L s [(-2)] .16667 .4875 0 1 Mshowa .33333 .5 m setfont .33333 .50625 L s [(-1)] .33333 .4875 0 1 Mshowa .66667 .5 m .66667 .50625 L s [(1)] .66667 .4875 0 1 Mshowa .83333 .5 m .83333 .50625 L s [(2)] .83333 .4875 0 1 Mshowa 1 .5 m 1 .50625 L s [(3)] 1 .4875 0 1 Mshowa .125 Mabswid .03333 .5 m .03333 .50375 L s .06667 .5 m .06667 .50375 L s .1 .5 m .1 .50375 L s .13333 .5 m .13333 .50375 L s .2 .5 m .2 .50375 L s .23333 .5 m .23333 .50375 L s .26667 .5 m .26667 .50375 L s .3 .5 m .3 .50375 L s .36667 .5 m .36667 .50375 L s .4 .5 m .4 .50375 L s .43333 .5 m .43333 .50375 L s .46667 .5 m .46667 .50375 L s .53333 .5 m .53333 .50375 L s .56667 .5 m .56667 .50375 L s .6 .5 m .6 .50375 L s .63333 .5 m .63333 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m .50375 .63333 s .5 .7 m .50375 .7 L s .5 .73333 m .50375 .73333 s .5 .76667 m .50375 .76667 s .5 .8 m .50375 .8 L s L L L L L L L L L L L L L .5 .86667 m .50375 .86667 L s .5 .9 m .50375 .9 L s .5 .93333 m .50375 .93333 L s .5 .96667 m .50375 .96667 L s .25 Mabswid .5 0 m .5 1 L s 00m 10L 11L 01L closepath clip newpath % Start of user PostScript /mathtops { gsave MBeginOrig moveto MEndOrig currentpoint grestore } bind def /MAtocoords { mathtops 4 2 roll mathtops 4 copy pop pop 3 -1 roll sub /arry exch def exch sub /arrx exch def arrx dup mul arry dup mul add sqrt /arrl exch def translate } bind def /MAarrowhead1 { gsave MAtocoords arrl 0. eq { 0 0 Mdot } { [ arrx arrl div arry arrl div -1 arry mul arrl div arrx arrl div 00] concat -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto fill -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto -0.05 0.0125 lineto stroke } ifelse grestore } def % End of user PostScript .5 Mabswid .66667 .5 m .66665 .50194 L .6666 .5037 L .66651 .50574 L 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.5 L .28474 .5 L .28588 .5 L Mistroke .2871 .5 L .28825 .5 L .2893 .5 L .29053 .5 L .29122 .5 L .29186 .5 L .29303 .5 L .29427 .5 L .29549 .5 L .29618 .5 L .29682 .5 L .29799 .5 L .29923 .5 L .30034 .5 L .30138 .5 L .30252 .5 L .30375 .5 L .30499 .5 L .30569 .5 L .30634 .5 L .30765 .5 L .30839 .5 L .30906 .5 L .31026 .5 L .31156 .5 L .31278 .5 L .31392 .5 L .31497 .5 L .31611 .5 L .31718 .5 L .31816 .5 L .3193 .5 L .32055 .5 L .32173 .5 L .32282 .5 L .32406 .5 L .32522 .5 L .32626 .5 L .32739 .5 L .32862 .5 L .32992 .5 L .33103 .5 L .33225 .5 L .33356 .5 L .33477 .5 L .33541 .5 L .3361 .5 L .33683 .5 L .33751 .5 L .33876 .5 L Mistroke .33991 .5 L .34123 .5 L .34249 .5 L .3436 .5 L .3448 .5 L .34592 .5 L .34694 .5 L .34814 .5 L .34945 .5 L .35067 .5 L .35181 .5 L .35302 .5 L .35431 .5 L .35554 .5 L .35665 .5 L .35796 .5 L .3587 .5 L .35939 .5 L .36064 .5 L .36128 .5 L .36198 .5 L .36266 .5 L .36337 .5 L .36465 .5 L .36536 .5 L .36602 .5 L .36675 .5 L .36753 .5 L .3682 .5 L .36893 .5 L .36962 .5 L .37026 .5 L .37146 .5 L .37274 .5 L .37394 .5 L .37526 .5 L .37651 .5 L .37766 .5 L .3788 .5 L .37984 .5 L .38103 .5 L .38216 .5 L .38339 .5 L .3847 .5 L .38581 .5 L .38703 .5 L .38834 .5 L .38908 .5 L .38975 .5 L .39106 .5 L Mistroke .39229 .5 L .39334 .5 L .3945 .5 L .39571 .5 L .39685 .5 L .39791 .5 L .39888 .5 L .39994 .5 L .40106 .5 L .40221 .5 L .40342 .5 L .40457 .5 L .40562 .5 L .40685 .5 L .40754 .5 L .40818 .5 L .40936 .5 L .4106 .5 L .41181 .5 L .41313 .5 L .41438 .5 L .41554 .5 L .41669 .5 L .41772 .5 L .41892 .5 L .42005 .5 L .42129 .5 L .42259 .5 L .42371 .5 L .42493 .5 L .42624 .5 L .42698 .5 L .42765 .5 L .42896 .5 L .43019 .5 L .43125 .5 L .43241 .5 L .43362 .5 L .43476 .5 L .43582 .5 L .43679 .5 L .43791 .5 L .43898 .5 L .44012 .5 L .44134 .5 L .44249 .5 L .44354 .5 L .44477 .5 L .44546 .5 L .4461 .5 L Mistroke .44727 .5 L .44851 .5 L .44973 .5 L .45042 .5 L .45106 .5 L .45223 .5 L .45347 .5 L .45458 .5 L .45562 .5 L .45676 .5 L .45799 .5 L .45922 .5 L .45992 .5 L .46057 .5 L .46188 .5 L .46262 .5 L .46329 .5 L .46449 .5 L .46579 .5 L .46702 .5 L .46815 .5 L .4692 .5 L .47034 .5 L .47141 .5 L .47239 .5 L .47353 .5 L .47478 .5 L .47596 .5 L .47705 .5 L .47829 .5 L .47945 .5 L .48049 .5 L .48162 .5 L .48285 .5 L .48415 .5 L .48526 .5 L .48648 .5 L .48779 .5 L .489 .5 L .48963 .5 L .49033 .5 L .49105 .5 L .49174 .5 L .49299 .5 L .49413 .5 L .49539 .5 L .49671 .5 L .49783 .5 L .49902 .5 L .50014 .5 L Mistroke .50116 .5 L .50236 .5 L .50366 .5 L .50489 .5 L .50603 .5 L .50723 .5 L .50853 .5 L .50975 .5 L .51087 .5 L .51218 .5 L .51292 .5 L .51361 .5 L .51486 .5 L .5155 .5 L .5162 .5 L .51687 .5 L .51759 .5 L .51886 .5 L .51958 .5 L .52024 .5 L .52096 .5 L .52174 .5 L .52241 .5 L .52314 .5 L .52383 .5 L .52447 .5 L .52568 .5 L .52695 .5 L .52816 .5 L .52947 .5 L .53072 .5 L .53187 .5 L .53301 .5 L .53405 .5 L .53525 .5 L .53637 .5 L .5376 .5 L .53891 .5 L .54002 .5 L .54124 .5 L .54255 .5 L .54329 .5 L .54396 .5 L .54526 .5 L .5465 .5 L .54755 .5 L .54871 .5 L .54992 .5 L .55105 .5 L .55211 .5 L Mistroke .55308 .5 L .55415 .5 L .55527 .5 L .55641 .5 L .55763 .5 L .55878 .5 L .55982 .5 L .56105 .5 L .56175 .5 L .56238 .5 L .56356 .5 L .5648 .5 L .56601 .5 L .56734 .5 L .56858 .5 L .56974 .5 L .57088 .5 L .57192 .5 L .57312 .5 L .57425 .5 L .57548 .5 L .57679 .5 L .5779 .5 L .57912 .5 L .58044 .5 L .58118 .5 L .58185 .5 L .58315 .5 L .58439 .5 L .58544 .5 L .5866 .5 L .58782 .5 L .58895 .5 L .59001 .5 L .59098 .5 L .5921 .5 L .59317 .5 L .59431 .5 L .59553 .5 L .59668 .5 L .59773 .5 L .59896 .5 L .59965 .5 L .60029 .5 L .60146 .5 L .60271 .5 L .60392 .5 L .60461 .5 L .60525 .5 L .60642 .5 L Mistroke .60766 .5 L .60876 .5 L .6098 .5 L .61094 .5 L .61217 .5 L .61341 .5 L .6141 .5 L .61475 .5 L .61607 .5 L .6168 .5 L .61747 .5 L .61867 .5 L .61997 .5 L .6212 .5 L .62233 .5 L .62339 .5 L .62452 .5 L .62559 .5 L .62657 .5 L .62771 .5 L .62896 .5 L .63014 .5 L .63123 .5 L .63247 .5 L .63363 .5 L .63467 .5 L .6358 .5 L .63703 .5 L .63833 .5 L .63944 .5 L .64066 .5 L .64197 .5 L .64318 .5 L .64381 .5 L .6445 .5 L .64523 .5 L .64591 .5 L .64716 .5 L .64831 .5 L .64956 .5 L .65088 .5 L .652 .5 L .65319 .5 L .65431 .5 L .65533 .5 L .65653 .5 L .65784 .5 L .65906 .5 L .6602 .5 L .6614 .5 L Mistroke .6627 .5 L .66392 .5 L .66504 .5 L .66635 .5 L .66709 .5 L .66777 .5 L .66903 .5 L .66966 .5 L .67036 .5 L .67104 .5 L .67175 .5 L .67303 .5 L .67374 .5 L .6744 .5 L .67513 .5 L .67591 .5 L .67658 .5 L .67731 .5 L .678 .5 L .67864 .5 L .67984 .5 L .68112 .5 L .68232 .5 L .68364 .5 L .68488 .5 L .68603 .5 L .68718 .5 L .68821 .5 L .68941 .5 L .69053 .5 L .69176 .5 L .69307 .5 L .69418 .5 L .6954 .5 L .69671 .5 L .69745 .5 L .69812 .5 L .69942 .5 L .70066 .5 L .70171 .5 L .70286 .5 L .70407 .5 L .70521 .5 L .70627 .5 L .70724 .5 L .7083 .5 L .70942 .5 L .71056 .5 L .71178 .5 L .71293 .5 L Mistroke .71398 .5 L .71521 .5 L .7159 .5 L .71654 .5 L .71771 .5 L .71896 .5 L .72017 .5 L .72149 .5 L .72273 .5 L .72389 .5 L .72503 .5 L .72607 .5 L .72727 .5 L .7284 .5 L .72963 .5 L .73094 .5 L .73205 .5 L .73327 .5 L .73458 .5 L .73532 .5 L .73599 .5 L .7373 .5 L .73853 .5 L .73959 .5 L .74075 .5 L .74196 .5 L .7431 .5 L .74416 .5 L .74513 .5 L .74619 .5 L .74731 .5 L .74846 .5 L .74967 .5 L .75083 .5 L .75187 .5 L .7531 .5 L .75379 .5 L .75443 .5 L .75561 .5 L .75685 .5 L .75806 .5 L .75875 .5 L .75939 .5 L .76056 .5 L .7618 .5 L .7629 .5 L .76394 .5 L .76508 .5 L .76631 .5 L .76754 .5 L Mistroke .76824 .5 L .76889 .5 L .7702 .5 L .77094 .5 L .77161 .5 L .7728 .5 L .7741 .5 L .77533 .5 L .77646 .5 L .77752 .5 L .77865 .5 L .77972 .5 L .7807 .5 L .78184 .5 L .78309 .5 L .78427 .5 L .78536 .5 L .7866 .5 L .78776 .5 L .7888 .5 L .78993 .5 L .79116 .5 L .79246 .5 L .79357 .5 L .79479 .5 L .7961 .5 L .7973 .5 L .79793 .5 L .79863 .5 L .79936 .5 L .80004 .5 L .80129 .5 L .80243 .5 L .80369 .5 L .80501 .5 L .80612 .5 L .80731 .5 L .80843 .5 L .80945 .5 L .81066 .5 L .81196 .5 L .81318 .5 L .81432 .5 L .81552 .5 L .81682 .5 L .81804 .5 L .81916 .5 L .82047 .5 L .8212 .5 L .82189 .5 L Mistroke .82315 .5 L .82385 .5 L .82448 .5 L .82516 .5 L .82587 .5 L .82715 .5 L .82786 .5 L .82852 .5 L .82925 .5 L .83002 .5 L .8307 .5 L .83143 .5 L .83212 .5 L .83275 .5 L .83396 .5 L .83524 .5 L .83644 .5 L .83775 .5 L .839 .5 L .84015 .5 L .84129 .5 L .84232 .5 L .84352 .5 L .84464 .5 L .84587 .5 L .84718 .5 L .84829 .5 L .8495 .5 L .85082 .5 L .85155 .5 L .85222 .5 L .85353 .5 L .85476 .5 L .85581 .5 L .85697 .5 L .85818 .5 L .85932 .5 L .86037 .5 L .86134 .5 L .86241 .5 L .86353 .5 L .86467 .5 L .86589 .5 L .86704 .5 L .86808 .5 L .86931 .5 L .87 .5 L .87064 .5 L .87182 .5 L .87306 .5 L Mistroke .87427 .5 L .87559 .5 L .87684 .5 L .87799 .5 L .87914 .5 L .88017 .5 L .88137 .5 L .88249 .5 L .88373 .5 L .88503 .5 L .88615 .5 L .88737 .5 L .88868 .5 L .88942 .5 L .89009 .5 L .89139 .5 L .89263 .5 L .89368 .5 L .89484 .5 L .89605 .5 L .89719 .5 L .89825 .5 L .89922 .5 L .90029 .5 L .90141 .5 L .90255 .5 L .90377 .5 L .90492 .5 L .90596 .5 L .90719 .5 L .90789 .5 L .90852 .5 L .9097 .5 L .91094 .5 L .91216 .5 L .91284 .5 L .91348 .5 L .91464 .5 L .91589 .5 L .91699 .5 L .91803 .5 L .91917 .5 L .9204 .5 L .92163 .5 L .92233 .5 L .92297 .5 L .92429 .5 L .92502 .5 L .92569 .5 L .92689 .5 L Mistroke .92819 .5 L .92941 .5 L .93055 .5 L .9316 .5 L .93273 .5 L .9338