(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 58144, 1479]*) (*NotebookOutlinePosition[ 59388, 1518]*) (* CellTagsIndexPosition[ 59344, 1514]*) (*WindowFrame->Normal*) Notebook[{ Cell["Math 225: Calculus III\tQuiz 5 2/25&2/27/96", "Subsection", CellMargins->{{18, 73}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{27, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[BoxData[{ \(\(<< "\"; \)\), \(i = {1, 0, 0}; j = {0, 1, 0}; k = {0, 0, 1}; \)}], "Input", CellMargins->{{18, 73}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{27, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ "1. Find the equation of the plane tangent to the surface ", StyleBox["x^3 y - z^3 x == 2", "Input"], StyleBox[" at the point ", "Text"], StyleBox["{1,3,1}", "Input"], StyleBox[".", "Text"] }], "Subsubsection", CellMargins->{{Inherited, 2}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{27, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell["Solution", "Subsubsection", CellMargins->{{Inherited, 73}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{27, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "The normal vector to the surface at ", StyleBox["{1,3,1}", "Input"], " is given by the gradient:." }], "Text", CellMargins->{{18, 73}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{27, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ "Grad[", StyleBox["x^3 y - z^3 x", "Input"], "]" }], "Input", AspectRatioFixed->True], Cell[BoxData[ \({3\ x\^2\ y - z\^3, x\^3, \(-3\)\ x\ z\^2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["n = % /. {x->1, y->3, z->1}", "Input"], Cell[BoxData[ \({8, 1, \(-3\)}\)], "Output"] }, Open ]], Cell["The equation of thetangent plane is then given by", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["n.{x,y,z} == n.{1,3,1}", "Input"], Cell[BoxData[ \(8\ x + y - 3\ z == 8\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "2. Show that the points ", StyleBox["{1,1}", "Input"], " and ", StyleBox["{-1,-1}", "Input"], " are critical points of the\nfunction ", StyleBox["f[x_,y_] = 2y^3 - 3 y + 2 x^3 - 3x^2 y", "Input"], " and determine whether\nthese points are local maxima, local minima, or \ saddle points.\n" }], "Subsubsection", CellMargins->{{Inherited, 19}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{27, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell["Solution", "Subsubsection", CellMargins->{{Inherited, 73}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{27, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[StyleBox[ "f[x_,y_] = 2y^3 - 3 y + 2 x^3 - 3x^2 y;", "Input"]], "Input"] }, Open ]], Cell[TextData[{ "We must show that ", StyleBox["{\[PlusMinus]1,\[PlusMinus]1}", "Input"], " are solutions of" }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "eqns = { D[f[x,y],x] == 0,\n D[f[x,y],y] == 0 }", "Input"]], "Input"], Cell[BoxData[ \({6\ x\^2 - 6\ x\ y == 0, \(-3\) - 3\ x\^2 + 6\ y\^2 == 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["eqns /. {x->1, y->1}", "Input"], Cell[BoxData[ \({True, True}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["eqns /. {x->-1, y->-1}", "Input"], Cell[BoxData[ \({True, True}\)], "Output"] }, Open ]], Cell["We can also see this by solving the equations...", "Text"], Cell[CellGroupData[{ Cell["Solve[eqns]", "Input"], Cell[BoxData[ \({{x \[Rule] \(-1\), y \[Rule] \(-1\)}, {x \[Rule] 1, y \[Rule] 1}, { y \[Rule] \(-\(1\/\@2\)\), x \[Rule] 0}, {y \[Rule] 1\/\@2, x \[Rule] 0}}\)], "Output"] }, Open ]], Cell[TextData[{ "or using the function ", StyleBox["CriticalPoints[]", "Input"], ":" }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ "cp = CriticalPoints[", StyleBox["f[x,y]", "Input"], "]" }], "Input"], Cell[BoxData[ \({{x \[Rule] \(-1\), y \[Rule] \(-1\)}, {x \[Rule] 1, y \[Rule] 1}, { y \[Rule] \(-\(1\/\@2\)\), x \[Rule] 0}, {y \[Rule] 1\/\@2, x \[Rule] 0}}\)], "Output"] }, Open ]], Cell["\<\ To detect which are max/min or saddle points we use the second \ derivative test.\ \>", "Text"], Cell[CellGroupData[{ Cell["d = D[f[x,y],x,x]D[f[x,y],y,y] - D[f[x,y],x,y]^2", "Input"], Cell[BoxData[ \(\(-36\)\ x\^2 + 12\ \((12\ x - 6\ y)\)\ y\)], "Output"] }, Open ]], Cell[TextData[{ "Evaluate ", StyleBox["d", "Input"], " at the critical points ", StyleBox["{\[PlusMinus]1,\[PlusMinus]1}", "Input"], ":" }], "Text"], Cell[CellGroupData[{ Cell["d /. {x->1, y->1}", "Input"], Cell[BoxData[ \(36\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["d /. {x->-1, y->-1}", "Input"], Cell[BoxData[ \(36\)], "Output"] }, Open ]], Cell[TextData[{ "Since ", StyleBox[" d > 0", "Input"], ", the graph is uniformly curved at these points. To see if it is curved up \ or down, check a second partial derivative:" }], "Text"], Cell[CellGroupData[{ Cell["D[f[x,y],x,x] /. {x->1, y->1}", "Input"], Cell[BoxData[ \(6\)], "Output"] }, Open ]], Cell[TextData[{ "Positive means curved up at ", StyleBox["{1,1}", "Input"], ", so ", StyleBox["f[x,y]", "Input"], " has a local minimum at ", StyleBox["{1,1}.", "Input"] }], "Text"], Cell[CellGroupData[{ Cell["D[f[x,y],x,x] /. {x->-1, y->-1}", "Input"], Cell[BoxData[ \(\(-6\)\)], "Output"] }, Open ]], Cell[TextData[{ "Negative means curved down at ", StyleBox["{-1,-1}", "Input"], ", so ", StyleBox["f[x,y]", "Input"], " has a local maximum at ", StyleBox["{-1,-1}.", "Input"] }], "Text"], Cell[CellGroupData[{ Cell["\<\ Plot3D[f[x,y], {x,-2,2}, {y,-2,2}, BoxRatios->{1,1,1},ViewPoint->{2.560, 2.094, 0.713}]\ \>", "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .90361 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% SurfaceGraphics /Courier findfont 10 scalefont setfont % Scaling calculations -0.0190787 1.05917 0 1.05917 [ [.46909 .91685 -6.02221 0 ] [.46909 .91685 5.97779 9 ] [.36931 .91045 -6.07576 0 ] [.36931 .91045 5.92424 9 ] [.25901 .90336 -3.06752 0 ] [.25901 .90336 2.93248 9 ] [.13643 .89548 -3.1005 0 ] [.13643 .89548 2.8995 9 ] [-0.00061 .88668 -3.13742 0 ] [-0.00061 .88668 2.86258 9 ] [.03206 .17346 -5.72515 -9 ] [.03206 .17346 6.27485 0 ] [.14722 .13342 -5.7905 -9 ] [.14722 .13342 6.2095 0 ] [.27382 .0894 -2.93147 -9 ] [.27382 .0894 3.06853 0 ] [.41368 .04076 -2.97183 -9 ] [.41368 .04076 3.02817 0 ] [.56899 -0.01324 -3.01711 -9 ] [.56899 -0.01324 2.98289 0 ] [.01848 .19246 -18 -6.04727 ] [.01848 .19246 0 2.95273 ] [.01097 .35197 -18 -5.74929 ] [.01097 .35197 0 3.25071 ] [.00327 .51644 -6 -5.44398 ] [.00327 .51644 0 3.55602 ] [-0.00461 .68607 -12 -5.13108 ] [-0.00461 .68607 0 3.86892 ] [-0.01269 .86112 -12 -4.8103 ] [-0.01269 .86112 0 4.1897 ] [ 0 0 0 0 ] [ 1 .90361 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .46914 .90361 m 0 .87345 L s .46914 .90361 m .46917 .897 L s [(-2)] .46909 .91685 .0037 -1 Mshowa .36948 .89721 m .36956 .89059 L s [(-1)] .36931 .91045 .01263 -1 Mshowa .25931 .89012 m .25946 .8835 L s [(0)] .25901 .90336 .02251 -1 Mshowa .13687 .88225 m .13709 .87563 L s [(1)] .13643 .89548 .0335 -1 Mshowa 0 .87345 m .0003 .86684 L s [(2)] -0.00061 .88668 .04581 -1 Mshowa .125 Mabswid .44998 .90238 m .45 .89841 L s .43044 .90113 m .43047 .89715 L s .41052 .89985 m .41055 .89587 L s .3902 .89854 m .39025 .89457 L s .34834 .89585 m .3484 .89188 L s .32676 .89446 m .32683 .89049 L s .30474 .89304 m .30482 .88907 L s .28226 .8916 m .28234 .88763 L s .23587 .88862 m .23596 .88465 L s .21192 .88708 m .21202 .88311 L s .18745 .8855 m .18756 .88153 L s .16244 .88389 m .16256 .87992 L s .11073 .88057 m .11087 .8766 L s .08399 .87885 m .08414 .87488 L s .05664 .87709 m .0568 .87312 L s .02865 .87529 m .02882 .87132 L s .25 Mabswid .03146 .18669 m .56907 0 L s .03146 .18669 m .03116 .1933 L s [(-2)] .03206 .17346 -0.04581 1 Mshowa .14675 .14665 m .14652 .15327 L s [(-1)] .14722 .13342 -0.03492 1 Mshowa .27352 .10263 m .27337 .10925 L s [(0)] .27382 .0894 -0.02284 1 Mshowa .41356 .054 m .4135 .06062 L s [(1)] .41368 .04076 -0.00939 1 Mshowa .56907 0 m .56911 .00662 L s [(2)] .56899 -0.01324 .0057 1 Mshowa .125 Mabswid .05368 .17898 m .0535 .18294 L s .0763 .17112 m .07614 .17509 L s .09935 .16311 m .09919 .16708 L s .12283 .15496 m .12268 .15893 L s .17114 .13819 m .17101 .14216 L s .19599 .12955 m .19587 .13353 L s .22133 .12076 m .22122 .12473 L s .24717 .11178 m .24707 .11575 L s .3004 .0933 m .30032 .09727 L s .32783 .08377 m .32776 .08774 L s .35582 .07405 m .35576 .07802 L s .38439 .06413 m .38434 .0681 L s .44334 .04366 m .44332 .04763 L s .47377 .03309 m .47375 .03707 L s .50485 .0223 m .50485 .02627 L s .53661 .01127 m .53662 .01524 L s .25 Mabswid .03146 .18669 m 0 .87345 L s .031 .19677 m .03726 .19892 L s [(-20)] .01848 .19246 1 .34384 Mshowa .02373 .35551 m .0301 .35729 L s [(-10)] .01097 .35197 1 .27762 Mshowa .01623 .51915 m .02271 .52051 L s [(0)] .00327 .51644 1 .20977 Mshowa .0085 .68791 m .01505 .68883 L s [(10)] -0.00461 .68607 1 .14024 Mshowa .00052 .86203 m .00713 .86249 L s [(20)] -0.01269 .86112 1 .06896 Mshowa .125 Mabswid .02956 .22813 m .03333 .22938 L s .02811 .25969 m .0319 .26089 L s .02666 .29144 m .03046 .2926 L s .0252 .32338 m .02901 .32449 L s .02224 .38784 m .02608 .38886 L s .02075 .42037 m .02461 .42134 L s .01926 .4531 m .02312 .45401 L s .01775 .48602 m .02162 .48689 L s .0147 .55249 m .0186 .55325 L s .01317 .58603 m .01707 .58674 L s .01162 .61978 m .01554 .62044 L s .01006 .65374 m .01399 .65434 L s .00692 .7223 m .01086 .72279 L s .00534 .7569 m .00929 .75734 L s .00374 .79172 m .0077 .79211 L s .00214 .82677 m .0061 .8271 L s .25 Mabswid .47123 .33975 m .46914 .90361 L s .46914 .90361 m 1 .87996 L s 1 .87996 m .971 .2194 L s .971 .2194 m .47123 .33975 L s .03146 .18669 m .56907 0 L s .56907 0 m .57384 .83567 L s .57384 .83567 m 0 .87345 L s 0 .87345 m .03146 .18669 L s .47123 .33975 m .46914 .90361 L s .46914 .90361 m 0 .87345 L s 0 .87345 m .03146 .18669 L s .03146 .18669 m .47123 .33975 L s .971 .2194 m .56907 0 L s .56907 0 m .57384 .83567 L s .57384 .83567 m 1 .87996 L s 1 .87996 m .971 .2194 L s 0 0 m 1 0 L 1 .90361 L 0 .90361 L closepath clip newpath .5 Mabswid .449 0 0 r .47031 .58716 .50174 .60482 .47511 .60554 .44366 .57511 Metetra .614 .08 0 r .44366 .57511 .47511 .60554 .44778 .59438 .41641 .55302 Metetra .743 .252 0 r .41641 .55302 .44778 .59438 .41977 .57491 .38855 .52451 Metetra .838 .381 0 r .38855 .52451 .41977 .57491 .3911 .55089 .36005 .49335 Metetra .892 .468 .135 r .36005 .49335 .3911 .55089 .36169 .52627 .33085 .46344 Metetra .905 .516 .269 r .33085 .46344 .36169 .52627 .33146 .50514 .30083 .43886 Metetra .885 .532 .372 r .30083 .43886 .33146 .50514 .30025 .49181 .2698 .42383 Metetra .846 .526 .446 r .2698 .42383 .30025 .49181 .26786 .49083 .23754 .42286 Metetra .799 .508 .497 r .23754 .42286 .26786 .49083 .234 .50711 .20372 .44084 Metetra .748 .483 .53 r .20372 .44084 .234 .50711 .1983 .54609 .16794 .48318 Metetra .698 .455 .552 r .16794 .48318 .1983 .54609 .16027 .61396 .12969 .55612 Metetra .651 .426 .565 r .12969 .55612 .16027 .61396 .11929 .718 .08829 .66707 Metetra .605 .396 .572 r .08829 .66707 .11929 .718 .07454 .86707 .04287 .8251 Metetra .682 .387 .468 r .04287 .8251 .07454 .86707 .03355 .87124 0 .87345 Metetra .26 .766 .863 r .50174 .60482 .53395 .59952 .50748 .61303 .47511 .60554 Metetra .024 .543 .864 r .47511 .60554 .50748 .61303 .48021 .61288 .44778 .59438 Metetra 0 .292 .727 r .44778 .59438 .48021 .61288 .45219 .60253 .41977 .57491 Metetra .471 0 0 r .41977 .57491 .45219 .60253 .42344 .58571 .3911 .55089 Metetra .676 .153 0 r .3911 .55089 .42344 .58571 .39393 .56635 .36169 .52627 Metetra .801 .308 0 r .36169 .52627 .39393 .56635 .3636 .54858 .33146 .50514 Metetra .836 .396 .182 r .33146 .50514 .3636 .54858 .33231 .53672 .30025 .49181 Metetra .812 .428 .327 r .30025 .49181 .33231 .53672 .29989 .53535 .26786 .49083 Metetra .762 .427 .419 r .26786 .49083 .29989 .53535 .26608 .54939 .234 .50711 Metetra .705 .411 .476 r .234 .50711 .26608 .54939 .23055 .58425 .1983 .54609 Metetra .649 .388 .511 r .1983 .54609 .23055 .58425 .19284 .64604 .16027 .61396 Metetra .597 .362 .531 r .16027 .61396 .19284 .64604 .15239 .74188 .11929 .718 Metetra .549 .335 .542 r .11929 .718 .15239 .74188 .10874 .87174 .07454 .86707 Metetra .266 0 0 r .07454 .86707 .10874 .87174 .06804 .86897 .03355 .87124 Metetra .637 .962 .809 r .53395 .59952 .56678 .57444 .54061 .60067 .50748 .61303 Metetra .544 .899 .973 r .50748 .61303 .54061 .60067 .51353 .61151 .48021 .61288 Metetra .369 .749 .989 r .48021 .61288 .51353 .61151 .48562 .6103 .45219 .60253 Metetra .159 .584 .934 r .45219 .60253 .48562 .6103 .45691 .60068 .42344 .58571 Metetra 0 .405 .816 r .42344 .58571 .45691 .60068 .4274 .58654 .39393 .56635 Metetra .38 0 0 r .39393 .56635 .4274 .58654 .39706 .57201 .3636 .54858 Metetra .613 .044 0 r .3636 .54858 .39706 .57201 .36577 .5614 .33231 .53672 Metetra .695 .2 .065 r .33231 .53672 .36577 .5614 .3334 .5593 .29989 .53535 Metetra .676 .269 .266 r .29989 .53535 .3334 .5593 .29973 .57061 .26608 .54939 Metetra .625 .288 .379 r .26608 .54939 .29973 .57061 .26444 .60069 .23055 .58425 Metetra .571 .285 .443 r .23055 .58425 .26444 .60069 .22713 .65554 .19284 .64604 Metetra .519 .271 .479 r .19284 .64604 .22713 .65554 .18729 .74208 .15239 .74188 Metetra .471 .246 .49 r .15239 .74188 .18729 .74208 .14422 .86857 .10874 .87174 Metetra .188 .712 .863 r .10874 .87174 .14422 .86857 .10353 .86663 .06804 .86897 Metetra .717 .944 .64 r .56678 .57444 .6001 .53311 .57436 .57191 .54061 .60067 Metetra .744 .99 .882 r .54061 .60067 .57436 .57191 .5476 .59367 .51353 .61151 Metetra .66 .917 .977 r .51353 .61151 .5476 .59367 .5199 .60155 .48562 .6103 Metetra .541 .821 .998 r .48562 .6103 .5199 .60155 .49134 .5991 .45691 .60068 Metetra .407 .737 .992 r .45691 .60068 .49134 .5991 .46193 .59013 .4274 .58654 Metetra .241 .659 .961 r .4274 .58654 .46193 .59013 .43165 .57871 .39706 .57201 Metetra 0 .542 .836 r .39706 .57201 .43165 .57871 .40045 .56916 .36577 .5614 Metetra .287 0 0 r .36577 .5614 .40045 .56916 .3682 .56604 .3334 .5593 Metetra .443 0 0 r .3334 .5593 .3682 .56604 .33471 .57421 .29973 .57061 Metetra .463 .061 .19 r .29973 .57061 .33471 .57421 .29973 .59896 .26444 .60069 Metetra .439 .119 .325 r .26444 .60069 .29973 .59896 .26289 .64615 .22713 .65554 Metetra .405 .14 .396 r .22713 .65554 .26289 .64615 .22371 .72251 .18729 .74208 Metetra .369 .144 .435 r .18729 .74208 .22371 .72251 .18158 .83593 .14422 .86857 Metetra .761 .909 .941 r .14422 .86857 .18158 .83593 .14005 .86423 .10353 .86663 Metetra .657 .844 .439 r .6001 .53311 .63381 .47925 .60863 .53047 .57436 .57191 Metetra .786 .988 .743 r .57436 .57191 .60863 .53047 .5823 .56305 .5476 .59367 Metetra .778 .98 .909 r .5476 .59367 .5823 .56305 .55494 .57997 .5199 .60155 Metetra .705 .911 .967 r .5199 .60155 .55494 .57997 .52662 .58464 .49134 .5991 Metetra .624 .849 .985 r .49134 .5991 .52662 .58464 .49738 .58077 .46193 .59013 Metetra .544 .813 .996 r .46193 .59013 .49738 .58077 .46726 .57238 .43165 .57871 Metetra .455 .804 .998 r .43165 .57871 .46726 .57238 .4362 .56373 .40045 .56916 Metetra .315 .785 .931 r .40045 .56916 .4362 .56373 .40412 .55934 .3682 .56604 Metetra .3682 .56604 .40412 .55934 .37087 .56403 .33471 .57421 Metetra .131 0 0 r .33471 .57421 .37087 .56403 .33622 .58301 .29973 .59896 Metetra .217 0 .118 r .29973 .59896 .33622 .58301 .29987 .62201 .26289 .64615 Metetra .238 0 .266 r .26289 .64615 .29987 .62201 .26139 .6875 .22371 .72251 Metetra .234 .002 .343 r .22371 .72251 .26139 .6875 .22022 .78703 .18158 .83593 Metetra .827 1 .802 r .18158 .83593 .22022 .78703 .17765 .86175 .14005 .86423 Metetra .63381 .47925 .66787 .41674 .64337 .48021 .60863 .53047 Metetra .713 .911 .554 r .60863 .53047 .64337 .48021 .61759 .52353 .5823 .56305 Metetra .803 1 .814 r .5823 .56305 .61759 .52353 .59065 .54946 .55494 .57997 Metetra .777 .962 .926 r .55494 .57997 .59065 .54946 .56266 .56122 .52662 .58464 Metetra .717 .902 .962 r .52662 .58464 .56266 .56122 .53368 .56242 .49738 .58077 Metetra .662 .863 .976 r .49738 .58077 .53368 .56242 .50377 .55699 .46726 .57238 Metetra .619 .858 .988 r .46726 .57238 .50377 .55699 .4729 .54912 .4362 .56373 Metetra .578 .886 .991 r .4362 .56373 .4729 .54912 .44103 .54329 .40412 .55934 Metetra .501 .905 .914 r .40412 .55934 .44103 .54329 .40805 .54426 .37087 .56403 Metetra .37087 .56403 .40805 .54426 .37376 .55715 .33622 .58301 Metetra .33622 .58301 .37376 .55715 .33791 .58753 .29987 .62201 Metetra .015 0 .074 r .29987 .62201 .33791 .58753 .30012 .64167 .26139 .6875 Metetra .068 0 .218 r .26139 .6875 .30012 .64167 .2599 .72676 .22022 .78703 Metetra .22022 .78703 .2599 .72676 .21659 .85138 .17765 .86175 Metetra 0 0 .008 r .66787 .41674 .70229 .3495 .67858 .42513 .64337 .48021 Metetra .64337 .48021 .67858 .42513 .65345 .47914 .61759 .52353 Metetra .726 .951 .653 r .61759 .52353 .65345 .47914 .62703 .51408 .59065 .54946 Metetra .785 .995 .869 r .59065 .54946 .62703 .51408 .59944 .53295 .56266 .56122 Metetra .752 .944 .944 r .56266 .56122 .59944 .53295 .57079 .53923 .53368 .56242 Metetra .706 .893 .965 r .53368 .56242 .57079 .53923 .54114 .53673 .50377 .55699 Metetra .673 .874 .974 r .50377 .55699 .54114 .53673 .5105 .52958 .4729 .54912 Metetra .655 .891 .982 r .4729 .54912 .5105 .52958 .47887 .52221 .44103 .54329 Metetra .642 .936 .966 r .44103 .54329 .47887 .52221 .44617 .51929 .40805 .54426 Metetra .584 .945 .84 r .40805 .54426 .44617 .51929 .41224 .52586 .37376 .55715 Metetra .37376 .55715 .41224 .52586 .37687 .54737 .33791 .58753 Metetra .33791 .58753 .37687 .54737 .33975 .58984 .30012 .64167 Metetra 0 0 .062 r .30012 .64167 .33975 .58984 .30044 .66014 .2599 .72676 Metetra 0 0 .189 r .2599 .72676 .30044 .66014 .25836 .76634 .21659 .85138 Metetra 0 0 .196 r .70229 .3495 .73712 .28147 .7143 .36924 .67858 .42513 Metetra .67858 .42513 .7143 .36924 .68992 .434 .65345 .47914 Metetra .65345 .47914 .68992 .434 .6641 .47801 .62703 .51408 Metetra .687 .965 .741 r .62703 .51408 .6641 .47801 .637 .50409 .59944 .53295 Metetra .729 .977 .92 r .59944 .53295 .637 .50409 .60873 .51551 .57079 .53923 Metetra .701 .921 .967 r .57079 .53923 .60873 .51551 .57938 .516 .54114 .53673 Metetra .671 .883 .977 r .54114 .53673 .57938 .516 .54901 .50958 .5105 .52958 Metetra .658 .882 .981 r .5105 .52958 .54901 .50958 .51763 .50062 .47887 .52221 Metetra .66 .917 .977 r .47887 .52221 .51763 .50062 .4852 .49373 .44617 .51929 Metetra .657 .963 .925 r .44617 .51929 .4852 .49373 .45161 .49385 .41224 .52586 Metetra .589 .929 .73 r .41224 .52586 .45161 .49385 .41669 .50632 .37687 .54737 Metetra .37687 .54737 .41669 .50632 .38018 .53695 .33975 .58984 Metetra .33975 .58984 .38018 .53695 .34171 .5923 .30044 .66014 Metetra 0 0 .073 r .30044 .66014 .34171 .5923 .30078 .67997 .25836 .76634 Metetra .477 .261 .505 r .73712 .28147 .77372 .26691 .75064 .31661 .7143 .36924 Metetra .018 0 .202 r .7143 .36924 .75064 .31661 .7271 .39229 .68992 .434 Metetra .68992 .434 .7271 .39229 .70197 .44557 .6641 .47801 Metetra .6641 .47801 .70197 .44557 .67542 .47902 .637 .50409 Metetra .57 .937 .816 r .637 .50409 .67542 .47902 .64758 .49576 .60873 .51551 Metetra .617 .932 .964 r .60873 .51551 .64758 .49576 .61858 .49936 .57938 .516 Metetra .607 .883 .991 r .57938 .516 .61858 .49936 .58848 .49375 .54901 .50958 Metetra .6 .863 .993 r .54901 .50958 .58848 .49375 .55735 .48322 .51763 .50062 Metetra .608 .883 .991 r .51763 .50062 .55735 .48322 .52518 .47234 .4852 .49373 Metetra .625 .932 .967 r .4852 .49373 .52518 .47234 .49189 .46598 .45161 .49385 Metetra .616 .958 .854 r .45161 .49385 .49189 .46598 .45737 .46933 .41669 .50632 Metetra .524 .859 .584 r .41669 .50632 .45737 .46933 .42141 .48806 .38018 .53695 Metetra .38018 .53695 .42141 .48806 .38369 .52844 .34171 .5923 Metetra .34171 .5923 .38369 .52844 .34378 .59764 .30078 .67997 Metetra .454 .17 .391 r .77372 .26691 .81131 .25786 .78777 .27135 .75064 .31661 Metetra .223 .003 .355 r .75064 .31661 .78777 .27135 .76515 .35828 .7271 .39229 Metetra .167 0 .223 r .7271 .39229 .76515 .35828 .7408 .42117 .70197 .44557 Metetra .062 0 0 r .70197 .44557 .7408 .42117 .71487 .46232 .67542 .47902 Metetra .67542 .47902 .71487 .46232 .68751 .48465 .64758 .49576 Metetra .317 .806 .837 r .64758 .49576 .68751 .48465 .65887 .49158 .61858 .49936 Metetra .403 .819 .973 r .61858 .49936 .65887 .49158 .62907 .48694 .58848 .49375 Metetra .436 .8 .995 r .58848 .49375 .62907 .48694 .59817 .47496 .55735 .48322 Metetra .462 .811 .998 r .55735 .48322 .59817 .47496 .56622 .46014 .52518 .47234 Metetra .494 .857 .989 r .52518 .47234 .56622 .46014 .53319 .44728 .49189 .46598 Metetra .521 .912 .925 r .49189 .46598 .53319 .44728 .499 .44152 .45737 .46933 Metetra .496 .888 .723 r .45737 .46933 .499 .44152 .46349 .44835 .42141 .48806 Metetra .42141 .48806 .46349 .44835 .4264 .47383 .38369 .52844 Metetra .38369 .52844 .4264 .47383 .38738 .52474 .34378 .59764 Metetra .105 0 0 r .81131 .25786 .8498 .24859 .8259 .23769 .78777 .27135 Metetra .368 .153 .45 r .78777 .27135 .8259 .23769 .80433 .33641 .76515 .35828 Metetra .37 .107 .379 r .76515 .35828 .80433 .33641 .78083 .40943 .7408 .42117 Metetra .368 .031 .255 r .7408 .42117 .78083 .40943 .75558 .45876 .71487 .46232 Metetra .347 0 .025 r .71487 .46232 .75558 .45876 .72875 .4871 .68751 .48465 Metetra .26 0 0 r .68751 .48465 .72875 .4871 .7005 .49771 .65887 .49158 Metetra 0 .492 .708 r .65887 .49158 .7005 .49771 .67098 .49431 .62907 .48694 Metetra .045 .569 .867 r .62907 .48694 .67098 .49431 .64031 .48105 .59817 .47496 Metetra .136 .613 .92 r .59817 .47496 .64031 .48105 .60854 .46239 .56622 .46014 Metetra .202 .671 .94 r .56622 .46014 .60854 .46239 .5757 .44311 .53319 .44728 Metetra .259 .747 .92 r .53319 .44728 .5757 .44311 .54175 .42826 .499 .44152 Metetra .296 .792 .791 r .499 .44152 .54175 .42826 .50658 .42324 .46349 .44835 Metetra .46349 .44835 .50658 .42324 .46999 .43391 .4264 .47383 Metetra .4264 .47383 .46999 .43391 .43169 .46675 .38738 .52474 Metetra .545 .818 .996 r .8498 .24859 .88922 .23909 .8656 .22717 .8259 .23769 Metetra .498 .284 .52 r .8259 .23769 .8656 .22717 .84493 .33133 .80433 .33641 Metetra .498 .25 .469 r .80433 .33641 .84493 .33133 .82238 .41524 .78083 .40943 Metetra .536 .234 .402 r .78083 .40943 .82238 .41524 .79789 .47346 .75558 .45876 Metetra .583 .203 .288 r .75558 .45876 .79789 .47346 .77163 .50842 .72875 .4871 Metetra .621 .142 .095 r .72875 .4871 .77163 .50842 .7438 .5232 .7005 .49771 Metetra .6 .036 0 r .7005 .49771 .7438 .5232 .71456 .52141 .67098 .49431 Metetra .495 0 0 r .67098 .49431 .71456 .52141 .68407 .50714 .64031 .48105 Metetra .368 0 0 r .64031 .48105 .68407 .50714 .65242 .48482 .60854 .46239 Metetra 0 .287 .687 r .60854 .46239 .65242 .48482 .61969 .45919 .5757 .44311 Metetra 0 .378 .722 r .5757 .44311 .61969 .45919 .58588 .43529 .54175 .42826 Metetra .123 0 0 r .54175 .42826 .58588 .43529 .55092 .41846 .50658 .42324 Metetra .05 0 0 r .50658 .42324 .55092 .41846 .51468 .41442 .46999 .43391 Metetra .46999 .43391 .51468 .41442 .47692 .42941 .43169 .46675 Metetra .551 .812 .994 r .88922 .23909 .92961 .22937 .90653 .22301 .8656 .22717 Metetra .522 .31 .533 r .8656 .22717 .90653 .22301 .88739 .34807 .84493 .33133 Metetra .582 .344 .521 r .84493 .33133 .88739 .34807 .86589 .44396 .82238 .41524 Metetra .632 .354 .479 r .82238 .41524 .86589 .44396 .84223 .51204 .79789 .47346 Metetra .693 .363 .413 r .79789 .47346 .84223 .51204 .8166 .55446 .77163 .50842 Metetra .756 .362 .308 r .77163 .50842 .8166 .55446 .7892 .5741 .7438 .5232 Metetra .796 .34 .156 r .7438 .5232 .7892 .5741 .76024 .57446 .71456 .52141 Metetra .786 .286 0 r .71456 .52141 .76024 .57446 .72989 .55954 .68407 .50714 Metetra .73 .212 0 r .68407 .50714 .72989 .55954 .69831 .53377 .65242 .48482 Metetra .66 .134 0 r .65242 .48482 .69831 .53377 .66559 .50191 .61969 .45919 Metetra .602 .059 0 r .61969 .45919 .66559 .50191 .63179 .46899 .58588 .43529 Metetra .555 0 0 r .58588 .43529 .63179 .46899 .59689 .44035 .55092 .41846 Metetra .49 0 0 r .55092 .41846 .59689 .44035 .56082 .42161 .51468 .41442 Metetra .371 0 0 r .51468 .41442 .56082 .42161 .5234 .41887 .47692 .42941 Metetra .49 .848 .993 r .92961 .22937 .971 .2194 .94993 .25189 .90653 .22301 Metetra .598 .392 .572 r .90653 .22301 .94993 .25189 .93222 .39226 .88739 .34807 Metetra .64 .409 .553 r .88739 .34807 .93222 .39226 .91192 .50161 .86589 .44396 Metetra .692 .43 .523 r .86589 .44396 .91192 .50161 .88919 .58088 .84223 .51204 Metetra .752 .453 .478 r .84223 .51204 .88919 .58088 .86425 .63191 .8166 .55446 Metetra .813 .472 .41 r .8166 .55446 .86425 .63191 .83731 .65735 .7892 .5741 Metetra .863 .48 .315 r .7892 .5741 .83731 .65735 .80862 .66056 .76024 .57446 Metetra .885 .467 .2 r .76024 .57446 .80862 .66056 .77837 .64549 .72989 .55954 Metetra .875 .435 .083 r .72989 .55954 .77837 .64549 .74676 .61656 .69831 .53377 Metetra .844 .391 0 r .69831 .53377 .74676 .61656 .71394 .57858 .66559 .50191 Metetra .812 .344 0 r .66559 .50191 .71394 .57858 .68001 .53664 .63179 .46899 Metetra .787 .298 0 r .63179 .46899 .68001 .53664 .64501 .49608 .59689 .44035 Metetra .761 .247 0 r .59689 .44035 .64501 .49608 .60891 .46257 .56082 .42161 Metetra .701 .179 0 r .56082 .42161 .60891 .46257 .57159 .44207 .5234 .41887 Metetra 0 g .25 Mabswid .03146 .18669 m .56907 0 L s .56907 0 m .57384 .83567 L s .57384 .83567 m 0 .87345 L s 0 .87345 m .03146 .18669 L s .971 .2194 m .56907 0 L s .56907 0 m .57384 .83567 L s .57384 .83567 m 1 .87996 L s 1 .87996 m .971 .2194 L s .03146 .18669 m 0 .87345 L s .031 .19677 m .03726 .19892 L s [(-20)] .01848 .19246 1 .34384 Mshowa .02373 .35551 m .0301 .35729 L s [(-10)] .01097 .35197 1 .27762 Mshowa .01623 .51915 m .02271 .52051 L s [(0)] .00327 .51644 1 .20977 Mshowa .0085 .68791 m .01505 .68883 L s [(10)] -0.00461 .68607 1 .14024 Mshowa .00052 .86203 m .00713 .86249 L s [(20)] -0.01269 .86112 1 .06896 Mshowa .125 Mabswid .02956 .22813 m .03333 .22938 L s .02811 .25969 m .0319 .26089 L s .02666 .29144 m .03046 .2926 L s .0252 .32338 m .02901 .32449 L s .02224 .38784 m .02608 .38886 L s .02075 .42037 m .02461 .42134 L s .01926 .4531 m .02312 .45401 L s .01775 .48602 m .02162 .48689 L s .0147 .55249 m .0186 .55325 L s .01317 .58603 m .01707 .58674 L s .01162 .61978 m .01554 .62044 L s .01006 .65374 m .01399 .65434 L s .00692 .7223 m .01086 .72279 L s .00534 .7569 m .00929 .75734 L s .00374 .79172 m .0077 .79211 L s .00214 .82677 m .0061 .8271 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{272.625, 246.312}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^CLg=gIfM[Fe]OCdm?clo5D005Gooj[ooc/k>gYjN[Vi^OShn03o oclo?ginO[fm_C/k>gYjN[Vi^OShn000EED0EJX00?l003lo?ginO[fm_Oclo0<30d92@X61PL30`3lo ?ginO[fm_Ol0E@00ZUD0ZZX0E@<30d92@X61PL30`000E@L71dI6AXF5Q@<30d92@X61PL30`?l0ZP00 oeD0ojX0ZPL71dI6AXF5QLC4a000ZP/;2dY:BXV9R@L71dI6AXF5QLC4a?l0o`1E05EE0:X0o`/;2dY: BXV9RLS8b000o`l?3di>CXf=S@/;2dY:BXV9RLS8b?mE001EEEEEEJYE00l?3di>CXf=SLcCXf=SLc0003o`000300 0_lk0000>`000ol0000=00?o0`000ol0001<0003o`0003800_li0000>`000ol0000:00CoE0000ol0 000d0003o`0003H0000j00;o2P02o`<000?o0000DP000ol0000e00;o=P0004<00omJ0003o`0003L0 0_ld0000@003o`8000?o0000F0000ol0000i00;oP03o`80 00?o0000GP000ol0000m0003o`0002d0000N00Co5@03ofH000?o0000?P02obd0000O0003o`000180 0ol20003o`0006@000?o0000@002ob/0000H00Go0`000ol0000?00;o1@000ol0001T0003o`000480 0_lY00008@000ol0000;00?o00<0o`00J`000ol0001400;o9`0001h000Co003o2P03og4000?o0000 AP000ol0000T00007`02o`P00ol00`3o001a0003o`0004L00_lT00009P03ogL000?o0000B@02ob80 000S00?o00<0o`00M`000ol0001;00;o800000h01?l300;o2@03ogd000?o0000C@02oah0000?0003 o`00008000Co003o1@03o`8000?o0000N`000ol0001?00;o700000P01Ol30005o`000?l00P000ol0 000200;oPP000ol0001A0003o`0001T0000A0004o`00o`8000?o00000P000ol0o`02ogl000?o0000 DP02oaT0000>0004o`00o`8000Co003o10000ol0000200?oO0000ol0001D00;o5`0000l00_l400;o 1@03o`D00omi0003o`0005H00_lE000070000ol0000800?oMP000ol0001H00;o4`0001`000?o0000 2`03og<000?o0000FP000ol0000@000070000ol0000>00?oL0000ol0001K00;o400001`000?o0000 4@03ofd000?o0000G@02o`h0000L0003o`0001@00omZ0003o`0005d00ol=000070000ol0000G00?o I`000ol0001:00Wo1@05o`034];o00d0000L00;o6`03of@000?o0000A006o`09c2g<;Lco7OlB00Go 00HBdQ;B4]82o`d0000L0003o`0001d00omQ0003o`0004400ol03T;o@Te2CD9=@Te2oh9Z0_l03=8B dQ;B4]8Bd_l0o`d0000L0003o`0002000_mO0003o`0003h00ol07g6LLIcoc2g<;L`]c2g`03o`0G@Re2;D8]ode2CD9=@Te2CD;o PVZ2JX800_l01Q;B4]8Bo`8000?o00002`0001/000?o00009P03oeT000?o0000>P007Ol5og6LLIaa W77o;L`]c2g<;L`]oh8MPQf27H8M00;o00CB4];o0P000ol0000;00006`000ol0000Y00?oEP000ol0 000i000Sog4Qobe2;D8]@RgoCD9=@Te2CD9=ofZ2JX9ZPVZ2JX;od_l00`000ol0000;00006`000ol0 000/00?oD`000ol0000h000So`DQoiaaW76LLIaaobg<;L`]c2g<;On27H8MPQf27H8MP_l010000ol0 000;00006`000ol0000_00?oD0000ol0000f00;o02=a8G7o@Re2;D8]@RgoCD9=@Te2CD9=ofZ2JX9Z PVZ2JX9Zo`040003o`0000/0000K00?oR3ood/PBb3o8OoK8=/P ol`Qc27<8Lcoo`07HR9R8V8RHP02oaD03Ol0120A8143o`LQ00ooPTb2C89a0Ol01?l03L1Z`6[0J/1Z`6[0J/000ol01B9RoomR00;o300?o`044OlAo`PQ01OoPTb2C89oa8C4Q l@02o`054Q`A?14l4C`A?14olC `A<2o`05/86`PK000_l00eooo`0@o`TA00goLR9b8W8RLR9b8W8R00?o01a=LTebCG9=LTebCG;oPW:2 LX9bP_n2LX9bPW;o3`000ol0000800006@000ol0000T000Zogm=OdeoCGm=Odgo:ZlZ[bZ_:Zoo[df_ CJm=[omb[g:_L_mb[g:_oeoo4P04o`09lOoo0a<34`?o00TC0_l00a?B4`03o`0;`A?14l4C`Oo14l40 2_l00eooo`0>o`TA02goLR5b8G8QLR5b8G8QLR7oL_n2LX9bPW:2LX9bP_nbP[:2/X:bobf2;H8]P_l0 3`000ol0000800006@000ol0000S000/oh:bP[:2/X:bP[:2odeoCGm=OdeoCOnMKie_WFnMobf_;Jl] ofnOKiooGol@00Oo00ka4o4ClOlCdQ?B4m8CdP;o00R^4ZhB[Q:^4P?o00?14l401Ol01L4B`A;100Go 00=RGol03Ol94@0^og8RLR9b8W8RLR9b8W8Rog:2L_m=LTebCG9=LTgoLX9bPW:2LX9bog:2LX9bo`l0 00?o0000200001T00_lS000^ogm=OdeoCGm=OdeoCGoo[bZ_:ZlZ[b[oCJm=[df_COn_LZmb[on_LZmb ok9Oo`l01?l01[7o0a<34`;o1a<2o`0<4m8CdQ?B4m8CdQ?B1?l03L7o`A?14l4C`A?14l400ol00f;o HP0Fo`0>8G8QLR5b8G8QLR5b8Ol5LP0Joh9bPW:2LX9boh:bP[:2/X:bP[;o;H8]P_l?0003o`0000P0 000I0003o`00028001Oo/X:bP[:2/X:bP[:2/_moCGm=OdeoC@06o`0BKigo;Jl][bf_oim_Wfoo[eoo 3@03o`0?l[7olA?a4ooB4m8CdQ?B00;o00^^4ZhB[Q:^4ZhB[P03o`04dQ;B4PCo00@B`A;11Ol02M8B d_mRof;oG`05o`TA0_l0<54RDOlRLR9b8W8RLR9b8_mbPW:2LX9bodebCG9=LTgoLX9bPW:2LX9bPW;o LX9bo`l000?o0000200001T000?o00008@005_m]odeoCGm=OdeoCGm=Oon_:ZlZ[bX2o`06COn_LZmb 1Ol03Zmb[g;oLZmb[omO/Uoo2`04o`05/Ol34`<03?l02A?B4m8CdQ?B4`02o`0;dQ?B4m8CdQ?o4m80 1Ol03/4B`A;14/4Bof;oH_mR1ol00a7o4@03o`068W8RLR9b0_l02G8QLR5b8G8Qo`09LP0Goh9bPW:2 L_n2/X:bP[:2/X:bP[;o;Ol040000ol0000800006@000ol0000Q000Dob[oP[:2/X:bP[:2/X:bP_m= Odd2o`0=Kde_COm_LVmbKg;o[`03o`0;[bgoKim_WomO[ol02P04o`03/Ooa00?o00`C0a<34`?lM:QdZ7B[oP[:2/X:bP[:2/X;oP_m=Kde_CFm=Kde_ og9_LVmbKomR[f:_H_mOofoo[en_oj;o1004o`0C0o43oa<34`_lZ7BXM:QdZ7OnbP[:2/X:bP_n2Kh9_P_m=Kde_CFm=KdgoKg9_LVmbKomR [f:_H_mOH_mO[eooGj85o`030o4300Ko00KB4m8CdQ<6o`PC0_l84`?o11<2o`8C0ol:00;o00YRoioo Gb;o8V8R2Ol8LP0Poh9bPW:2LX9bPW:2LX;o[On2LX9bPW:2LX9bPW:2L_lA0003o`0000P0000B00;o 1@000ol0000L000eogm]OfeoKGm]OomoCGm=Odeoojeb[G:]L_mbCG9=LTebCG9=ojmb[g:_L_mOLUmb Gg;oGf800_l01;9Ooel4o`03KP?o00;o00@C0ooo1a<2o`04dQ?B4`Co01?B4ooodQ?B4m8CdQ?BoihC oa>N00;o00olM:QdZ7BXM:QdZoh:bP[:2oh9_PVn2Kh9_ ofm=Kde_CFm=KdgoKg9_LVmbojmR[f:_of9Oof;o[onR00?o0143oolBoA;moa?B4m8CdQ?Bo`0:4`Co 1a<2o`>4hhCSQ>>4`0>4hhCSQ>>om8CdQ?B4`02o`08cQ?>olko4`82oa@000OoH_nOH_mO00;o00aR8V;oHVmRKf9_ H_l9LP0Hoh9bP_n]Kje_[Fn]KjgoPW:2LX9bPW;o4`000ol00008000060000ol0000J000PofeoKGm] OfeoKGm]OfeoKOmb[G:]LZeb[G:]LZeb[G83o`ab00Ko8V8RHR82o`@R1_l01A?14l4C00?o00TC`A?1 4l4C`A<01?l02=8CdQ?B4m8C0_l035hCGQ=N4olC]Q>f4`;o00@Coooo5P02o`0bHUooGj9Oof9OH_mR Kf9_HVmRog9_LVmbKg9_L_mboae_7FlMKae_7FlMoh:bP[:2/_lE0003o`0000L0000H00;o6P00>olM :QdZ7BXM:QdZ7BXM:QgoGOm_PVn2Kh9_PVn2Kh9_oadb7OlRLR9b8W8RLR9b8W;o8W8RLR9bob8A00Go 00DB0a834P02o`TC0ol011?14l44o`0:4hkooa?B4m8CdP;o00DCcQ?>4`04o`03S_l001L00_l04V:O ofmOKomR8V;oHVmRKf9_o`Ub0_l05:e_[Fn]Kje_[Fn]KjgoPW:2LX;o5@000ol00007000060000ol0 000I000Pogm]OfeoKGm]OfeoKGm]OombP_mb[G:]LZeb[G:]L_l5LP;o2W86o`038V;o00Co004ol0 0olM00;o00UOH_nRGj;oXVl00ol08V9_HVmRKom_LVoo[Fn]KjgoKae_7FlMKae_7FlMKonbP_lF0003 o`0000L0000H0003o`0001P002[o/X;o7BXM:QdZ7BXM:QgoGH9MPUf2GOm_PVn2KolMool080003OmR GomOXUooKj9_XVl02?l05Ze_[Fn]Kje_oae_7FlMKae_7FlMKad2oaL000?o00001`0001P000?o0000 5`00=On2/X:bobXM:QdZ7BXM:_n2GH9MPUf2GH9Mofoo7C8M4ol0o`0Q000/of;oGfmO Kom_WfnOok9_/VnbKom_/VnbKk9_/_m_[Fn]Kje_[Fn]Kje_[OlG0003o`0000L0000H0003o`0001L0 01_oCH9=PTgoOfeoKGm]OombPW:2LX9bPW:2LX800_l;LP09ob8Q8R4R8B8Q00;o00>4ol30003o`00020002[oWf;oGfmOoim_WfooKk9_/Vnbok9_ /VnbKk9_/_m_[Fn]Kje_[Fn]KolH0003o`0000L0000H0003o`0001H0023oPTf2CH9=P_m]OfeoKOn2 LX9bPW:2LX9bPW;o7G8MLP;o17802OlQ8R4R8B8Q8P04o`034Olk00Co4`03o`03`Ooo00Go00OB4m8C dQ?B00;o00@CMQ=f0_l01A?>4lko00@000?o00008@00:OmOoenRGom_XVnRKom_/VnbKk;o[Fn]Kje_ [Foo7FlMKae_7FlMKago01P000?o00001`0001P000?o00005@008on2/X:bP[:2/_lZ7BXM:_n2GH9M PUf2GH9MP_mQ@V52HD9Q00;o00XMoa`b738L>00Co1@000ol0000R000WoiooKem_ofnOKioo/VnbKk9_ofnbKk9_/VnbKon]Kje_[Fn] Kjgo01T000?o00001`0001P000?o00005@009Om=PTf2CH9=PTgoOfeoog:2LX9bPW:2LX9bP_mb7G8M LQeb7G802_l01R0A817o>`Go5`02o`03`Ooo01Co1`000ol0000S00;o02BRGj;oXVnRKj;o/VnbKk9_ ofn]Kje_[Fn]KolMKae_7FlMKolI0003o`0000L0000H0003o`0001D002KoP[:2/X:bP[:2/_lZoh9M PUf2GH9MPUf2GOm2HD9Q@V52HD9Qo`PQ00Ko4JPAob05oaX01Ol021;B4]8BdQ;B0ol00m8Co`02o`T0 00?o00009002o`0RGfmOoim_WfooKk9_/VnbKom_/VnbKk9_/_m_[Fn]Kje_oaX000?o00001`0001P0 00?o00005000`Go7@05o`03`A?o00;o0a<3o`/000?o00009@02o`0QXUnRoj9_XVooKk9_/Vnboje_[Fn]Kje_ [Om_7FlMKago01X000?o00001`0001P000?o000050009OnbP[:2/X:bP[:2/X;oGH9MPUf2GH9MPUf2 of52HD9Q@V52HOl022401_lAZ17o80Ko800;o`h000?o00009P008_m_GfooKim_WonbKk9_/Vnbok9_ /VnbKk9_/_m_[Fn]KolJ0003o`0000L0000H0003o`0001@000goPTf2CH9=PTf2CH9=00;o02:2LX9b PW:2LX9boaeb7G8MLQeb7G;o8G8QLR5b8OlA817o0S/6ob0000?o00005P000ol0000W00;o01jRGom_ XVnRok9_/VnbKom_[Fn]Kje_[Foo7FlMKolK0003o`0000L0000H00;o50009On2/X:bP[:2/X:bP[;o LX;oGH9MPUf2GH9Mod9Q@V52HD9Q@_l01b400olAZ002o`038?lP00Go8@000ol0000F0003o`0002T0 01koGonOKim_WonbKk9_/VooKk9_/VnbKk9_oje_[OlK0003o`0000L0000H0003o`0001<0037oCH9= PTf2CH9=PTf2oh;`07obD0 00?o00005`000ol0000d00;o00A_/Vnb1?lU0003o`0000H0000G0003o`00014002;o/X:bP[:2/X:b P[:2oh9bPW:2LX9bPW;oLW5bLG9aLW7o1R401_lP8B0Q80;o00DPoooo8002obD000?o00005`000ol0 000f00Co:@000ol0000600005`000ol0000@000^odf2CH9=PTf2CH9=P_n2c8;

Kool01?lW0003o`0001L0 00?o0000H`000ol0000600002P000ol000020004o`00o`@000?o00003`008_n2L_n2/X:bP[:2/X;o PW:2LX9bPW:2L_mbLG9aLW5bLOl68@05ob0Q82400_l00h/Po`05obL000?o00005`000ol0001S0003 o`0000H0000:0003o`00008000Co003o10000ol0000?000`ofZ2odf2CH9=PTf2COo

000Qog:2LX9b oh:bP[:2oh9bPW:2LX9bPW;oLW5bLG9aLW7o00HQ00Wo827o8OlPRb001_lY0003o`0001L000?o0000 H`000ol0000600005`000ol0000>000`oh9ZPVZ2J_n2CH9=olb2c8;

000P og:2LX9bP_nbP[:2oh9bPW:2LX9bP_maLW5bLG9aL_l68@03ob7o001_l00b3o000f0003 o`0001L000?o0000I0000ol0000500005P000ol00007000Tog:2LX9bPW:2LX;ok4;/@^a2k4;oLR5b 8G8QLR7oc27<8Lco0R49ocP000?o00005`000ol0001T0003o`0000D0000F0003o`0000H000goJX9Z PVZ2JX9ZPV[o00MM01;o7C4Moc4M2Olj0003o`0000l0 3_mR0003o`0000@0000E0003o`0000@002OoLX9bPW;oLX9bPW:2ona2k4;/@^a2og8QLR5b8G8Qol`Q c?lC0Q<02Olk0003o`0000oo3P0;oeL000?o0000100001D000?o00000`003ombWG:MLYebofZ2JX9Z o`07G@0 1_lC013o70000ol0000l00_o>0000ol00004000020000ol000020004o`00o`@002Co0000oh9bPW:2 LX9bPW:2oh9bod;/@^a2k4;oLR5b8G8QLR42o`060Q<24`8C1?l6013o;0000ol0001700[o;P000ol0 000400002@001Ol0003o008000?o00000P04o`0=WG:MLYebWG:MLYebW@02o`QM00WoML_l07om>0003 o`0006H02_l?0003o`0000@0000800;o1002o`D02Ol50003o`0000<000?o00002P000ol0001>0003 o`0007002ol40003o`0000@0000N013o0P000ol000030003o`0000@000?o0000A`000ol0001d00co 1P0002h04Ol40003o`0000<000?o000020000ol0000`0003o`0005l05OlB0000?`0@o`D000?o0000 0`000ol0000Z0003o`0004T05_lW0000CP0Bo`D000?o00000P000ol000020003o`0001D000?o0000 <`0Focd0001O017o10000ol000020003o`0000/000?o00007P0Eoe<0000B00CoF@0Ao`<000Go0000 o`0:01KoJ00001<000?o0000J0000ol0o`0Aogh0000D0003o`0001d00ooI00005@000ol0000M0003 o`000=P0000B0004o`00oal000?o00006`02ok/0000C00;o80000ol0000J0004o`00okX0000e0003 o`0001X000Co003o6P03oid0000d00;o70001?l00?lK0003o`0001@01?n40000DP001?l00?lD00Go 0P000ol0000E0003o`0008@0001C00;o70000ol0000>00Go0`000ol000230000L@000ol0000G0003 o`000880001`00;o5P001?l00?n40000R@02ohD0003o01400000\ \>"], ImageRangeCache->{{{0, 271.625}, {245.312, 0}} -> {-0.0641716, -0.047273, 0.00386314, 0.00386314}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] SurfaceGraphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 1024}, {0, 748}}, AutoGeneratedPackage->None, ScreenStyleEnvironment->"Working", WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Manual, WindowSize->{576, 651}, WindowMargins->{{Automatic, 102}, {Automatic, 26}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->False, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", MacintoshSystemPageSetup->"\<\ 00<0004/0B`000003]X;>omOoel?O0_B0fL5:0?l0080004/0B`000000]P2:001 0000D00000400`<300000BL?00400@6Hd40000000000060801T1T000000P0000 00000000004000000000000000000000\>" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1709, 49, 230, 4, 44, "Subsection"], Cell[1942, 55, 291, 6, 43, "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[2258, 65, 404, 10, 60, "Subsubsection"], Cell[2665, 77, 202, 4, 41, "Subsubsection"], Cell[2870, 83, 295, 8, 31, "Text"], Cell[CellGroupData[{ Cell[3190, 95, 109, 5, 27, "Input"], Cell[3302, 102, 76, 1, 31, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3415, 108, 44, 0, 27, "Input"], Cell[3462, 110, 48, 1, 26, "Output"] }, Open ]], Cell[3525, 114, 65, 0, 30, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[3627, 119, 39, 0, 27, "Input"], Cell[3669, 121, 54, 1, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3760, 127, 510, 13, 86, "Subsubsection"], Cell[4273, 142, 202, 4, 41, "Subsubsection"], Cell[4478, 148, 86, 1, 27, "Input"] }, Open ]], Cell[4579, 152, 127, 4, 31, "Text"], Cell[CellGroupData[{ Cell[4731, 160, 102, 1, 32, "Input"], Cell[4836, 163, 91, 1, 22, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4964, 169, 37, 0, 22, "Input"], Cell[5004, 171, 46, 1, 22, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5087, 177, 39, 0, 22, "Input"], Cell[5129, 179, 46, 1, 22, "Output"] }, Open ]], Cell[5190, 183, 64, 0, 25, "Text"], Cell[CellGroupData[{ Cell[5279, 187, 28, 0, 22, "Input"], Cell[5310, 189, 193, 3, 31, "Output"] }, Open ]], Cell[5518, 195, 102, 4, 25, "Text"], Cell[CellGroupData[{ Cell[5645, 203, 91, 4, 22, "Input"], Cell[5739, 209, 193, 3, 31, "Output"] }, Open ]], Cell[5947, 215, 105, 3, 25, "Text"], Cell[CellGroupData[{ Cell[6077, 222, 65, 0, 22, "Input"], Cell[6145, 224, 75, 1, 22, "Output"] }, Open ]], Cell[6235, 228, 158, 6, 25, "Text"], Cell[CellGroupData[{ Cell[6418, 238, 34, 0, 22, "Input"], Cell[6455, 240, 36, 1, 22, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6528, 246, 36, 0, 22, "Input"], Cell[6567, 248, 36, 1, 22, "Output"] }, Open ]], Cell[6618, 252, 195, 5, 25, "Text"], Cell[CellGroupData[{ Cell[6838, 261, 46, 0, 22, "Input"], Cell[6887, 263, 35, 1, 22, "Output"] }, Open ]], Cell[6937, 267, 193, 7, 25, "Text"], Cell[CellGroupData[{ Cell[7155, 278, 48, 0, 22, "Input"], Cell[7206, 280, 40, 1, 22, "Output"] }, Open ]], Cell[7261, 284, 199, 7, 25, "Text"], Cell[CellGroupData[{ Cell[7485, 295, 112, 3, 30, "Input"], Cell[7600, 300, 50388, 1171, 254, 20920, 803, "GraphicsData", "PostScript", "Graphics"], Cell[57991, 1473, 137, 3, 22, "Output"] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)