(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1305497, 49538]*) (*NotebookOutlinePosition[ 1350824, 51108]*) (* CellTagsIndexPosition[ 1350682, 51100]*) (*WindowFrame->Normal*) Notebook[{ Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg"], "Graphics", ShowCellBracket->False, CellMargins->{{25, 24}, {5, 7}}, ImageSize->{81, 22}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "2D and 3D Measurements\n", StyleBox[ "Authors: Bill Davis, Horacio Porta and Jerry Uhl ", "Subtitle"], StyleBox["\[Copyright]1999", "Subtitle", FontSize->12], StyleBox["\nProducer: Bruce Carpenter\n", "Subtitle"], StyleBox["Publisher: ", "Subtitle", FontSize->12], StyleBox[ButtonBox["Math Everywhere, Inc.", ButtonData:>{ URL[ "http://www.matheverywhere.com"], None}, ButtonStyle->"MEIHyperlink"], FontSize->12], StyleBox[" Distributor: ", "Subtitle", FontSize->12], StyleBox[ButtonBox["Wolfram Research, Inc.", ButtonData:>{ URL[ "http://www.wolfram.com"], None}, ButtonStyle->"MEIHyperlink", ButtonNote->"Makers of Mathematica!"], FontSize->12] }], "PrefaceTitle", CellMargins->{{Inherited, Inherited}, {Inherited, 0}}], Cell[TextData[{ "VC.03 The Gradient\n", StyleBox["Samples", FontSize->16, FontSlant->"Italic"] }], "Title"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Initializations" }], "Special2"], Cell[BoxData[{ \(\(Off[General::spell];\)\), "\n", \(\(Off[General::spell1];\)\), "\n", \(\(Off[ParametricPlot3D::ppcom];\)\), "\n", \(\(Off[ParametricPlot::ppcom];\)\), "\n", \(\(Off[Plot::plnr];\)\), "\n", \(\(SetOptions[Limit, Analytic \[Rule] True];\)\)}], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[BoxData[ \(\(CMView = {2.7, 1.6, 1.2};\)\)], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Vector Drawer", "Subsubsection"], Cell[BoxData[ \(\( (*\ \( : Date : \ \ Copyright\ 1999\), \ Math\ Everywhere, \ \(Inc . \)\ \n*) \n (*\ \ \( : Drawbacks : \ \ Since\ the\ vectors\ are\ drawn\ out\ of\ \n\ \ \ \ \ context\), \ the\ shapes\ of\ the\ arrowheads\ are\ only\ \n\ \ \ \ \ correct\ with\ AspectRatio\ set\ to\ \(Automatic . \)\n*) \)\)], "Input"], Cell[BoxData[ \(\(BeginPackage["\", "\"];\)\)], "Input", InitializationCell->True], Cell[BoxData[{ \(\(Axes3D::"\" = "\";\)\), "\ \n", \(\(Perpend::"\" = "\";\)\), "\n", \(\(Arrow::"\" = "\";\)\), "\n", \(\(Vector::"\" = "\";\)\), "\n", \(\(ArrowHead::"\" = "\";\)\), "\n", \(\(Tail::"\" = "\";\)\), "\n", \(\(Norm::"\" = "\";\)\), "\n", \(\(Normalize::"\" = "\";\)\), "\n", \(\(Aperture::"\" = "\";\)\), "\n", \(\(TipSize::"\" = "\";\)\), "\n", \(\(TipRatio::"\" = "\";\)\), "\n", \(\(HeadRatio::"\" = "\";\)\), "\n\ ", \(\(HeadSize::"\" = "\";\)\), "\n", \(\(ScaleFactor::"\" = "\";\)\ \), "\n", \(\(ZeroVectorPointSize::"\" = "\";\)\), "\n", \(\(Options[ Vector] = \(Options[ Arrow] = \(Options[ArrowHead] = {HeadRatio \[Rule] 0.18, TipRatio \[Rule] 0.14, Aperture \[Rule] 0.3, ScaleFactor \[Rule] 1, HeadType \[Rule] Polygon, ShaftQ \[Rule] True, EdgesQ \[Rule] True, VectorColor \[Rule] RGBColor[0, 0, 1], ShaftWidth \[Rule] 0.005, ZeroVectorPointSize \[Rule] 0.01}\)\);\)\), "\n", \(\(Begin["\<`Private`\>"];\)\), "\n", \(\(SetOptions[ParametricPlot, AspectRatio \[Rule] Automatic];\)\), "\n", \(\(SetOptions[Plot, AspectRatio \[Rule] Automatic];\)\), "\n", \(\(SetOptions[Graphics, AspectRatio \[Rule] Automatic];\)\), "\n", \(\(Axes3D[u_, v_] := Graphics3D[{{Blue, Line[{{\(-\(u\/3\)\), 0, 0}, {u, 0, 0}}]}, Text["\", {u + v, 0, 0}], {Blue, Line[{{0, \(-\(u\/3\)\), 0}, {0, u, 0}}]}, Text["\", {0, u + v, 0}], {Blue, Line[{{0, 0, \(-\(u\/3\)\)}, {0, 0, u}}]}, Text["\", {0, 0, u + v}]}];\)\), "\n", \(\(Axes3D[u_] := Axes3D[u, u\/8];\)\), "\n", \(\(Norm[a_?VectorQ] := N[\@\(a . a\)];\)\), "\n", \(\(Normalize[a_?VectorQ] := If[a . a == 0, a, a\/Norm[a]];\)\), "\n", \(Perpend[a_?VectorQ] := Normalize[{a\[LeftDoubleBracket]2\[RightDoubleBracket], \(-a\ \[LeftDoubleBracket]1\[RightDoubleBracket]\)}] /; 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1.02742 m .33197 1.07227 L .30096 1.07526 L p .597 .603 .829 r F P 0 g s .30096 1.07526 m .28078 1.03447 L .31193 1.02742 L p .597 .603 .829 r F P 0 g s .58227 1.05038 m .60091 1.08739 L .56873 1.1208 L p .771 .71 .795 r F P 0 g s .56873 1.1208 m .54996 1.08789 L .58227 1.05038 L p .771 .71 .795 r F P 0 g s .49862 1.0807 m .51776 1.11759 L .48575 1.13947 L p .725 .694 .82 r F P 0 g s .48575 1.13947 m .46647 1.10667 L .49862 1.0807 L p .725 .694 .82 r F P 0 g s .76086 .80586 m .74269 .74884 L .77402 .66687 L p .81 .672 .711 r F P 0 g s .74269 .74884 m .76086 .80586 L .72926 .87701 L p .807 .676 .72 r F P 0 g s .6411 .03607 m .66043 .14677 L .62895 .24823 L p .766 .604 .678 r F P 0 g s .62895 .24823 m .60963 .14011 L .6411 .03607 L p .766 .604 .678 r F P 0 g s .7556 .59844 m .77402 .66687 L .74269 .74884 L p .801 .658 .704 r F P 0 g s .18942 .87561 m .20958 .93634 L .17927 .93631 L p .567 .539 .783 r F P 0 g s .17927 .93631 m .15898 .87967 L .18942 .87561 L p .567 .539 .783 r F P 0 g s 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s .67906 .89171 m .69738 .9409 L .66531 .99735 L p .798 .686 .742 r F P 0 g s .53095 1.0469 m .54996 1.08789 L .51776 1.11759 L p .747 .687 .794 r F P 0 g s .51776 1.11759 m .49862 1.0807 L .53095 1.0469 L p .747 .687 .794 r F P 0 g s .24047 .92878 m .2606 .98564 L .22984 .98911 L p .592 .563 .79 r F P 0 g s .22984 .98911 m .20958 .93634 L .24047 .92878 L p .592 .563 .79 r F P 0 g s .66531 .99735 m .6469 .95218 L .67906 .89171 L p .798 .686 .742 r F P 0 g s .72423 .68406 m .74269 .74884 L .71102 .82387 L p .798 .661 .712 r F P 0 g s .39528 1.04307 m .41499 1.08799 L .38327 1.1024 L p .671 .64 .81 r F P 0 g s .38327 1.1024 m .36343 1.06156 L .39528 1.04307 L p .671 .64 .81 r F P 0 g s .29187 .9745 m .31193 1.02742 L .28078 1.03447 L p .618 .588 .798 r F P 0 g s .28078 1.03447 m .2606 .98564 L .29187 .9745 L p .618 .588 .798 r F P 0 g s .67972 .25103 m .69893 .34856 L .66747 .44424 L p .775 .618 .685 r F P 0 g s .66747 .44424 m .64825 .3497 L .67972 .25103 L p .775 .618 .685 r F P 0 g s .34352 1.01263 m .36343 1.06156 L .33197 1.07227 L p .644 .614 .804 r F P 0 g s .33197 1.07227 m .31193 1.02742 L .34352 1.01263 L p .644 .614 .804 r F P 0 g s .71801 .43912 m .73691 .52247 L .7055 .6116 L p .787 .635 .693 r F P 0 g s .72423 .68406 m .7055 .6116 L .73691 .52247 L p .794 .646 .699 r F P 0 g s .69893 .34856 m .71801 .43912 L .68657 .53161 L p .781 .626 .689 r F P 0 g s .68657 .53161 m .66747 .44424 L .69893 .34856 L p .781 .626 .689 r F P 0 g s .7055 .6116 m .68657 .53161 L .71801 .43912 L p .787 .635 .693 r F P 0 g s .56338 1.0053 m .58227 1.05038 L .54996 1.08789 L p .762 .681 .773 r F P 0 g s .54996 1.08789 m .53095 1.0469 L .56338 1.0053 L p .762 .681 .773 r F P 0 g s .4793 1.0357 m .49862 1.0807 L .46647 1.10667 L p .723 .664 .791 r F P 0 g s .46647 1.10667 m .44702 1.06577 L .4793 1.0357 L p .723 .664 .791 r F P 0 g s .71102 .82387 m .69249 .76285 L .72423 .68406 L p .798 .661 .712 r F P 0 g s .69249 .76285 m .71102 .82387 L .67906 .89171 L p .793 .664 .72 r F P 0 g s .60963 .14011 m .62895 .24823 L .59705 .34372 L p .762 .603 .681 r F P 0 g s .59705 .34372 m .57773 .23839 L .60963 .14011 L p .762 .603 .681 r F P 0 g s .7055 .6116 m .72423 .68406 L .69249 .76285 L p .789 .647 .705 r F P 0 g s .22043 .86394 m .24047 .92878 L .20958 .93634 L p .609 .555 .769 r F P 0 g s .20958 .93634 m .18942 .87561 L .22043 .86394 L p .609 .555 .769 r F P 0 g s .59582 .95596 m .61461 1.00512 L .58227 1.05038 L p .773 .675 .756 r F P 0 g s .58227 1.05038 m .56338 1.0053 L .59582 .95596 L p .773 .675 .756 r F P 0 g s .67906 .89171 m .66046 .83458 L .69249 .76285 L p .793 .664 .72 r F P 0 g s .66046 .83458 m .67906 .89171 L .6469 .95218 L p .788 .667 .73 r F P 0 g s .42744 1.01674 m .44702 1.06577 L .41499 1.08799 L p .7 .641 .788 r F P 0 g s .41499 1.08799 m .39528 1.04307 L .42744 1.01674 L p .7 .641 .788 r F P 0 g s .62895 .24823 m .64825 .3497 L .61636 .44219 L p .766 .61 .685 r F P 0 g s .61636 .44219 m .59705 .34372 L .62895 .24823 L p .766 .61 .685 r F P 0 g 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.69405 m .69249 .76285 L .66046 .83458 L p .785 .649 .712 r F P 0 g s .64825 .3497 m .66747 .44424 L .63561 .53355 L p .771 .618 .689 r F P 0 g s .63561 .53355 m .61636 .44219 L .64825 .3497 L p .771 .618 .689 r F P 0 g s .66747 .44424 m .68657 .53161 L .65474 .61757 L p .776 .626 .694 r F P 0 g s .67372 .69405 m .65474 .61757 L .68657 .53161 L p .782 .636 .699 r F P 0 g s .65474 .61757 m .63561 .53355 L .66747 .44424 L p .776 .626 .694 r F P 0 g s .54428 .95212 m .56338 1.0053 L .53095 1.0469 L p .754 .658 .755 r F P 0 g s .53095 1.0469 m .51175 .9978 L .54428 .95212 L p .754 .658 .755 r F P 0 g s .45984 .98257 m .4793 1.0357 L .44702 1.06577 L p .721 .642 .769 r F P 0 g s .44702 1.06577 m .42744 1.01674 L .45984 .98257 L p .721 .642 .769 r F P 0 g s .66046 .83458 m .64162 .76954 L .67372 .69405 L p .785 .649 .712 r F P 0 g s .64162 .76954 m .66046 .83458 L .62821 .899 L p .779 .651 .721 r F P 0 g s .57773 .23839 m .59705 .34372 L .56481 .43291 L p .757 .602 .685 r F P 0 g s .56481 .43291 m .54547 .33057 L .57773 .23839 L p .757 .602 .685 r F P 0 g s .65474 .61757 m .67372 .69405 L .64162 .76954 L p .778 .637 .705 r F P 0 g s .57683 .89876 m .59582 .95596 L .56338 1.0053 L p .764 .655 .742 r F P 0 g s .56338 1.0053 m .54428 .95212 L .57683 .89876 L p .764 .655 .742 r F P 0 g s .25194 .84459 m .27186 .91353 L .24047 .92878 L p .64 .567 .756 r F P 0 g s .24047 .92878 m .22043 .86394 L .25194 .84459 L p .64 .567 .756 r F P 0 g s .62821 .899 m .6093 .83785 L .64162 .76954 L p .779 .651 .721 r F P 0 g s .6093 .83785 m .62821 .899 L .59582 .95596 L p .773 .653 .73 r F P 0 g s .40777 .9596 m .42744 1.01674 L .39528 1.04307 L p .7 .622 .766 r F P 0 g s .39528 1.04307 m .37549 .99002 L .40777 .9596 L p .7 .622 .766 r F P 0 g s .59582 .95596 m .57683 .89876 L .6093 .83785 L p .773 .653 .73 r F P 0 g s .59705 .34372 m .61636 .44219 L .58414 .5282 L p .761 .609 .689 r F P 0 g s .58414 .5282 m .56481 .43291 L .59705 .34372 L p .761 .609 .689 r F P 0 g s .63561 .53355 m .65474 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L .81 .711 .756 r F P 0 g s .66046 .83458 m .67906 .89171 L .6469 .95218 L p .788 .667 .73 r F P 0 g s .6469 .95218 m .62821 .899 L .66046 .83458 L p .788 .667 .73 r F P 0 g s .63366 .97389 m .65348 .87752 L .71432 .98086 L .485 .816 1 r F .6093 .83785 m .62821 .899 L .59582 .95596 L p .773 .653 .73 r F P 0 g s .59582 .95596 m .57683 .89876 L .6093 .83785 L p .773 .653 .73 r F P 0 g s .57075 .96536 m .65348 .87752 L .63366 .97389 L .485 .816 1 r F .63366 .97389 m .61461 1.00512 L p .61878 .9726 L .781 .671 .742 r F P 0 g s .47024 1.07208 m .18451 1.01733 L .36321 .96405 L .485 .816 1 r F .47024 1.07208 m .36321 .96405 L .36944 .96384 L F .35568 .9289 m .38807 .89439 L p .379 .96337 L .7 .608 .75 r F P 0 g s .37833 .9634 m .36908 .96345 L .37831 .96338 L .485 .816 1 r F .47024 1.07208 m .3693 .96369 L .37833 .9634 L F .3693 .96369 m .36908 .96345 L .37833 .9634 L F .38807 .89439 m .40777 .9596 L .40466 .96253 L p .7 .608 .75 r F P 0 g s .379 .96337 m .38807 .89439 L .40466 .96253 L .7 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r F P 0 g s .50034 .9641 m .49239 .9406 L p .51582 .96504 L .748 .64 .741 r F P 0 g s .51582 .96504 m .52502 .8909 L .53459 .96573 L .748 .64 .741 r F .53459 .96573 m .54428 .95212 L p .54229 .96629 L .754 .658 .755 r F P 0 g s .49239 .9406 m .50034 .9641 L p .48918 .9639 L .736 .641 .754 r F P 0 g s .54428 .95212 m .54949 .96661 L p .54229 .96629 L .754 .658 .755 r F P 0 g s .4793 1.0357 m .45984 .98257 L .47482 .96326 L p .736 .641 .754 r F P 0 g s .47482 .96326 m .49239 .9406 L p .48918 .9639 L .736 .641 .754 r F P 0 g s .4793 1.0357 m .47482 .96326 L .48918 .9639 L .736 .641 .754 r F .64162 .76954 m .66046 .83458 L .62821 .899 L p .779 .651 .721 r F P 0 g s .54229 .96629 m .54949 .96661 L .56533 .96798 L .485 .816 1 r F .54229 .96629 m .56533 .96798 L .56806 .96822 L F .54428 .95212 m .57683 .89876 L p .56806 .96822 L .764 .655 .742 r F P 0 g s .56338 1.0053 m .54949 .96661 L p .56806 .96822 L .764 .655 .742 r F P 0 g s .54949 .96661 m .54428 .95212 L p .56806 .96822 L .764 .655 .742 r F P 0 g s .17013 .97349 m .18286 .99624 L p .16203 .98767 L .151 .39 .839 r F P 0 g s .20958 .93634 m .22567 .97825 L p .19966 .98501 L .537 .543 .809 r F P 0 g s .18451 1.01733 m .17967 1.00158 L .21884 .98019 L .485 .816 1 r F .18451 1.01733 m .21884 .98019 L .22143 .97945 L F .20116 .98521 m .19966 .98501 L p .22567 .97825 L .537 .543 .809 r F P 0 g s .24047 .92878 m .25598 .9726 L p .23193 .97726 L .592 .563 .79 r F P 0 g s .20958 .93634 m .24047 .92878 L p .23193 .97726 L .592 .563 .79 r F P 0 g s .18451 1.01733 m .22163 .97925 L .23193 .97726 L .485 .816 1 r F .18451 1.01733 m .23193 .97726 L .24163 .97572 L F .22567 .97825 m .20958 .93634 L p .23193 .97726 L .592 .563 .79 r F P 0 g s .22163 .97925 m .22285 .978 L .24227 .97526 L .485 .816 1 r F .30431 .96709 m .32359 .9556 L p .32193 .96584 L .654 .598 .779 r F P 0 g s .32359 .9556 m .32699 .96532 L p .32193 .96584 L .654 .598 .779 r F P 0 g s .29811 .92995 m .31761 .89675 L .29185 .8346 L p .782 .495 .502 r F P 0 g s 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r F .28595 .96953 m .31053 .96701 L .31323 .96684 L F .31053 .96701 m .34729 .96324 L .31323 .96684 L F .36471 .96361 m .31323 .96684 L .37396 .96085 L F .34352 1.01263 m .32699 .96532 L p .35051 .9645 L .68 .604 .763 r F P 0 g s .28984 .96831 m .27186 .91353 L p .29277 .96812 L .66 .585 .76 r F P 0 g s .29277 .96812 m .30369 .89053 L .30431 .96709 L .66 .585 .76 r F .37396 .96085 m .28803 .96855 L .3965 .95413 L .485 .816 1 r F .29187 .9745 m .28984 .96831 L p .29277 .96812 L .66 .585 .76 r F P 0 g s .24047 .92878 m .27186 .91353 L p .26275 .97191 L .631 .576 .774 r F P 0 g s .18451 1.01733 m .24332 .97449 L .28803 .96855 L .485 .816 1 r F .25598 .9726 m .24047 .92878 L p .26275 .97191 L .631 .576 .774 r F P 0 g s .27186 .91353 m .28984 .96831 L p .26275 .97191 L .631 .576 .774 r F P 0 g s .33589 .85975 m .35568 .9289 L .32359 .9556 L p .682 .591 .747 r F P 0 g s .31761 .89675 m .33677 .85781 L .31119 .79369 L p .774 .497 .516 r F P 0 g s .31119 .79369 m .29185 .8346 L .31761 .89675 L 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Is this just an accident?\ \>", "Text"], Cell["Answer:", "Special1"], Cell[CellGroupData[{ Cell["This is no accident. ", "SmallText"], Cell["In Mathematics, there are no accidents.", "Accident"] }, Closed]], Cell[TextData[{ "Here's why:\nIf you parameterize the level curve \n ", Cell[BoxData[ \(f[x, y] = c\)]], " \nwith a parameterization ", Cell[BoxData[ \({x[t], y[t]}\)]], " and then you put\n ", Cell[BoxData[ \(g[t] = f[x[t], y[t]]\)]], ", \nthen you get ", Cell[BoxData[ \(g[t] = c\)]], " no matter what ", Cell[BoxData[ \(t\)]], " is.\nConsequently \n ", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["g", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "=", "0"}]]], " \nno matter what ", Cell[BoxData[ \(t\)]], " is. But the chain rule says \n ", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["g", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "=", RowBox[{\(gradf[x[t], y[t]]\), ".", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], ",", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "}"}]}]}]]], ". \nSo\n ", Cell[BoxData[ RowBox[{ RowBox[{\(gradf[x[t], y[t]]\), ".", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], ",", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "}"}]}], "=", "0"}]]], " \nno matter what ", Cell[BoxData[ \(t\)]], " is. This means that the gradient \n ", Cell[BoxData[ \(gradf[x[t], y[t]]\)]], " \nis perpendicular to the tangent vector \n ", Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], ",", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "}"}]]], " \nat ", Cell[BoxData[ \({x[t], y[t]}\)]], ". \nAs a result, no matter what ", Cell[BoxData[ \({x, y}\)]], " you go to on a level curve \n ", Cell[BoxData[ \(f[x, y] = c\)]], ", \nthe gradient ", Cell[BoxData[ \(gradf[x, y]\)]], " is perpendicular to the level curve at ", Cell[BoxData[ \({x, y}\)]], ".\nRead that again!\nCheck it out again, going with \n ", Cell[BoxData[ \(f[x, y] = x\^2 + \ 4 y\^2\)]], ":\nHere is a true-scale plot of the level curve \n ", Cell[BoxData[ \(f[x, y] = \(x\^2 + \ 4 y\^2 = 36\)\)]], " \nand some of the gradients ", Cell[BoxData[ \(gradf[x, y]\)]], " with tails at some points ", Cell[BoxData[ \({x, y}\)]], " on the 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0 g s .46858 .64315 m .48574 .66368 L .45803 .68029 L p .682 .697 .858 r F P 0 g s .45803 .68029 m .44073 .65675 L .46858 .64315 L p .682 .697 .858 r F P 0 g s .35017 .62309 m .36819 .65354 L .34121 .66373 L p .618 .617 .827 r F P 0 g s .34121 .66373 m .32319 .63029 L .35017 .62309 L p .618 .617 .827 r F P 0 g s .25984 .5631 m .27832 .60051 L .25211 .6046 L p .561 .561 .81 r F P 0 g s .25211 .6046 m .23373 .56427 L .25984 .5631 L p .561 .561 .81 r F P 0 g s .30491 .59636 m .32319 .63029 L .29656 .6374 L p .59 .588 .819 r F P 0 g s .29656 .6374 m .27832 .60051 L .30491 .59636 L p .59 .588 .819 r F P 0 g s .644 .5833 m .65919 .58833 L .63075 .61173 L p .773 .875 .922 r F P 0 g s .63075 .61173 m .61522 .60376 L .644 .5833 L p .773 .875 .922 r F P 0 g s .77514 .49294 m .78822 .4864 L .75939 .51337 L p .726 .99 .843 r F P 0 g s .75939 .51337 m .74582 .51706 L .77514 .49294 L p .726 .99 .843 r F P 0 g s .42313 .63279 m .44073 .65675 L .41319 .67021 L p .651 .665 .85 r F P 0 g s .41319 .67021 m .3955 .64323 L .42313 .63279 L p .651 .665 .85 r F P 0 g s .57052 .61287 m .5867 .62403 L .55841 .64413 L p .731 .794 .902 r F P 0 g s .55841 .64413 m .54198 .62997 L .57052 .61287 L p .731 .794 .902 r F P 0 g s .49673 .6294 m .51372 .6469 L .48574 .66368 L p .689 .723 .876 r F P 0 g s .48574 .66368 m .46858 .64315 L .49673 .6294 L p .689 .723 .876 r F P 0 g s .67303 .56267 m .68784 .56476 L .65919 .58833 L p .775 .918 .934 r F P 0 g s .65919 .58833 m .644 .5833 L .67303 .56267 L p .775 .918 .934 r F P 0 g s .80469 .46863 m .81725 .45925 L .78822 .4864 L p .63 .947 .742 r F P 0 g s .78822 .4864 m .77514 .49294 L .80469 .46863 L p .63 .947 .742 r F P 0 g s .37751 .61579 m .3955 .64323 L .36819 .65354 L p .62 .634 .842 r F P 0 g s .36819 .65354 m .35017 .62309 L .37751 .61579 L p .62 .634 .842 r F P 0 g s .2411 .52518 m .25984 .5631 L .23373 .56427 L p .529 .543 .814 r F P 0 g s .23373 .56427 m .21512 .52343 L .2411 .52518 L p .529 .543 .814 r F P 0 g s .33186 .59215 m .35017 .62309 L .32319 .63029 L p .59 .603 .833 r F P 0 g s .32319 .63029 m .30491 .59636 L .33186 .59215 L p .59 .603 .833 r F P 0 g s .28635 .56191 m .30491 .59636 L .27832 .60051 L p .559 .573 .824 r F P 0 g s .27832 .60051 m .25984 .5631 L .28635 .56191 L p .559 .573 .824 r F P 0 g s .59933 .59559 m .61522 .60376 L .5867 .62403 L p .738 .832 .923 r F P 0 g s .5867 .62403 m .57052 .61287 L .59933 .59559 L p .738 .832 .923 r F P 0 g s .70232 .54186 m .71672 .541 L .68784 .56476 L p .763 .958 .934 r F P 0 g s .68784 .56476 m .67303 .56267 L .70232 .54186 L p .763 .958 .934 r F P 0 g s .45109 .62223 m .46858 .64315 L .44073 .65675 L p .655 .688 .869 r F P 0 g s .44073 .65675 m .42313 .63279 L .45109 .62223 L p .655 .688 .869 r F P 0 g s .83447 .44413 m .84646 .43192 L .81725 .45925 L p .493 .849 .594 r F P 0 g s .81725 .45925 m .80469 .46863 L .83447 .44413 L p .493 .849 .594 r F P 0 g s .52519 .6155 m .54198 .62997 L .51372 .6469 L p .695 .753 .897 r F P 0 g s .51372 .6469 m .49673 .6294 L .52519 .6155 L p .695 .753 .897 r F P 0 g s .4052 .60839 m .42313 .63279 L .3955 .64323 L p .622 .653 .86 r F P 0 g s .3955 .64323 m .37751 .61579 L .4052 .60839 L p .622 .653 .86 r F P 0 g s .86448 .41944 m .87586 .40441 L .84646 .43192 L p 0 0 0 r F P 0 g s .84646 .43192 m .83447 .44413 L .86448 .41944 L p 0 0 0 r F P 0 g s .73186 .52086 m .74582 .51706 L .71672 .541 L p .728 .982 .909 r F P 0 g s .71672 .541 m .70232 .54186 L .73186 .52086 L p .728 .982 .909 r F P 0 g s .62844 .57815 m .644 .5833 L .61522 .60376 L p .741 .874 .942 r F P 0 g s .61522 .60376 m .59933 .59559 L .62844 .57815 L p .741 .874 .942 r F P 0 g s .2221 .48673 m .2411 .52518 L .21512 .52343 L p .492 .523 .817 r F P 0 g s .21512 .52343 m .19627 .48207 L .2221 .48673 L p .492 .523 .817 r F P 0 g s .35921 .58788 m .37751 .61579 L .35017 .62309 L p .589 .62 .85 r F P 0 g s .35017 .62309 m .33186 .59215 L .35921 .58788 L p .589 .62 .85 r F P 0 g s .4794 .61154 m .49673 .6294 L .46858 .64315 L p .659 .714 .889 r F P 0 g s .46858 .64315 m .45109 .62223 L .4794 .61154 L p .659 .714 .889 r F P 0 g s .55397 .60145 m .57052 .61287 L .54198 .62997 L p .7 .788 .919 r F P 0 g s .54198 .62997 m .52519 .6155 L .55397 .60145 L p .7 .788 .919 r F P 0 g s .26751 .52695 m .28635 .56191 L .25984 .5631 L p .525 .554 .828 r F P 0 g s .25984 .5631 m .2411 .52518 L .26751 .52695 L p .525 .554 .828 r F P 0 g s .31326 .56071 m .33186 .59215 L .30491 .59636 L p .557 .587 .839 r F P 0 g s .30491 .59636 m .28635 .56191 L .31326 .56071 L p .557 .587 .839 r F P 0 g s .89472 .39456 m .90546 .37672 L .87586 .40441 L p 0 0 0 r F P 0 g s .87586 .40441 m .86448 .41944 L .89472 .39456 L p 0 0 0 r F P 0 g s .76167 .49967 m .77514 .49294 L .74582 .51706 L p .655 .971 .84 r F P 0 g s .74582 .51706 m .73186 .52086 L .76167 .49967 L p .655 .971 .84 r F P 0 g s .65783 .56053 m .67303 .56267 L .644 .5833 L p .735 .917 .955 r F P 0 g s .644 .5833 m .62844 .57815 L .65783 .56053 L p .735 .917 .955 r F P 0 g s .43327 .6009 m .45109 .62223 L .42313 .63279 L p .624 .676 .879 r F P 0 g s .42313 .63279 m .4052 .60839 L .43327 .6009 L p .624 .676 .879 r F P 0 g s .58306 .58724 m .59933 .59559 L .57052 .61287 L p .703 .827 .942 r F P 0 g s .57052 .61287 m .55397 .60145 L .58306 .58724 L p .703 .827 .942 r F P 0 g s .50805 .60072 m .52519 .6155 L .49673 .6294 L p .662 .745 .911 r F P 0 g s .49673 .6294 m .4794 .61154 L .50805 .60072 L p .662 .745 .911 r F P 0 g s .20283 .44773 m .2221 .48673 L .19627 .48207 L p .45 .499 .818 r F P 0 g s .19627 .48207 m .17718 .44018 L .20283 .44773 L p .45 .499 .818 r F P 0 g s .79175 .4783 m .80469 .46863 L .77514 .49294 L p .538 .905 .718 r F P 0 g s .77514 .49294 m .76167 .49967 L .79175 .4783 L p .538 .905 .718 r F P 0 g s .38695 .58354 m .4052 .60839 L .37751 .61579 L p .588 .639 .868 r F P 0 g s .37751 .61579 m .35921 .58788 L .38695 .58354 L p .588 .639 .868 r F P 0 g s .68752 .54274 m .70232 .54186 L .67303 .56267 L p .713 .953 .951 r F P 0 g s .67303 .56267 m .65783 .56053 L .68752 .54274 L p .713 .953 .951 r F P 0 g s .24838 .49146 m .26751 .52695 L .2411 .52518 L p .485 .533 .831 r F P 0 g s .2411 .52518 m .2221 .48673 L .24838 .49146 L p .485 .533 .831 r F P 0 g s .34059 .55948 m .35921 .58788 L .33186 .59215 L p .553 .602 .857 r F P 0 g s .33186 .59215 m .31326 .56071 L .34059 .55948 L p .553 .602 .857 r F P 0 g s .29435 .52875 m .31326 .56071 L .28635 .56191 L p .519 .567 .844 r F P 0 g s .28635 .56191 m .26751 .52695 L .29435 .52875 L p .519 .567 .844 r F P 0 g s .4617 .5933 m .4794 .61154 L .45109 .62223 L p .624 .702 .901 r F P 0 g s .45109 .62223 m .43327 .6009 L .4617 .5933 L p .624 .702 .901 r F P 0 g s .8221 .45673 m .83447 .44413 L .80469 .46863 L p 0 0 0 r F P 0 g s .80469 .46863 m .79175 .4783 L .8221 .45673 L p 0 0 0 r F P 0 g s .61248 .57287 m .62844 .57815 L .59933 .59559 L p .699 .869 .962 r F P 0 g s .59933 .59559 m .58306 .58724 L .61248 .57287 L p .699 .869 .962 r F P 0 g s .53704 .58976 m .55397 .60145 L .52519 .6155 L p .663 .779 .936 r F P 0 g s .52519 .6155 m .50805 .60072 L .53704 .58976 L p .663 .779 .936 r F P 0 g s .71751 .52476 m .73186 .52086 L .70232 .54186 L p .66 .967 .915 r F P 0 g s .70232 .54186 m .68752 .54274 L .71751 .52476 L p .66 .967 .915 r F P 0 g s .18329 .40819 m .20283 .44773 L .17718 .44018 L p .402 .471 .816 r F P 0 g s .17718 .44018 m .15784 .39775 L .18329 .40819 L p .402 .471 .816 r F P 0 g s .41509 .57915 m .43327 .6009 L .4052 .60839 L p .586 .661 .889 r F P 0 g s .4052 .60839 m .38695 .58354 L .41509 .57915 L p .586 .661 .889 r F P 0 g s .85272 .43497 m .86448 .41944 L .83447 .44413 L p 0 0 0 r F P 0 g s .83447 .44413 m .8221 .45673 L .85272 .43497 L p 0 0 0 r F P 0 g s .22897 .45543 m .24838 .49146 L .2221 .48673 L p .438 .507 .832 r F P 0 g s .2221 .48673 m .20283 .44773 L .22897 .45543 L p .438 .507 .832 r F P 0 g s .36835 .55823 m .38695 .58354 L .35921 .58788 L p .549 .621 .876 r F P 0 g s .35921 .58788 m .34059 .55948 L .36835 .55823 L p .549 .621 .876 r F P 0 g s .64223 .55834 m .65783 .56053 L .62844 .57815 L p .684 .91 .973 r F P 0 g s .62844 .57815 m .61248 .57287 L .64223 .55834 L p .684 .91 .973 r F P 0 g s .27514 .49628 m .29435 .52875 L .26751 .52695 L p .475 .544 .847 r F P 0 g s .26751 .52695 m .24838 .49146 L .27514 .49628 L p .475 .544 .847 r F P 0 g s .32165 .53059 m .34059 .55948 L .31326 .56071 L p .512 .582 .862 r F P 0 g s .31326 .56071 m .29435 .52875 L .32165 .53059 L p .512 .582 .862 r F P 0 g s .49052 .58561 m .50805 .60072 L .4794 .61154 L p .622 .732 .925 r F P 0 g s .4794 .61154 m .4617 .5933 L .49052 .58561 L p .622 .732 .925 r F P 0 g s .74781 .50661 m .76167 .49967 L .73186 .52086 L p .565 .936 .826 r F P 0 g s .73186 .52086 m .71751 .52476 L .74781 .50661 L p .565 .936 .826 r F P 0 g s .5664 .57867 m .58306 .58724 L .55397 .60145 L p .659 .818 .96 r F P 0 g s .55397 .60145 m .53704 .58976 L .5664 .57867 L p .659 .818 .96 r F P 0 g s .88361 .41301 m .89472 .39456 L .86448 .41944 L p 0 0 0 r F P 0 g s .86448 .41944 m .85272 .43497 L .88361 .41301 L p 0 0 0 r F P 0 g s .16347 .36808 m .18329 .40819 L .15784 .39775 L p .347 .438 .812 r F P 0 g s .15784 .39775 m .13825 .35477 L .16347 .36808 L p .347 .438 .812 r F P 0 g s .44364 .57469 m .4617 .5933 L .43327 .6009 L p .582 .686 .913 r F P 0 g s .43327 .6009 m .41509 .57915 L .44364 .57469 L p .582 .686 .913 r F P 0 g s .67232 .54364 m .68752 .54274 L .65783 .56053 L p .646 .94 .963 r F P 0 g s .65783 .56053 m .64223 .55834 L .67232 .54364 L p .646 .94 .963 r F P 0 g s .77841 .48827 m .79175 .4783 L .76167 .49967 L p .428 .845 .679 r F P 0 g s .76167 .49967 m .74781 .50661 L .77841 .48827 L p .428 .845 .679 r F P 0 g s .20925 .41884 m .22897 .45543 L .20283 .44773 L p .385 .476 .83 r F P 0 g s .20283 .44773 m .18329 .40819 L .20925 .41884 L p .385 .476 .83 r F P 0 g s .51973 .57781 m .53704 .58976 L .50805 .60072 L p .617 .766 .951 r F P 0 g s .50805 .60072 m .49052 .58561 L .51973 .57781 L p .617 .766 .951 r F P 0 g s .39655 .55697 m .41509 .57915 L .38695 .58354 L p .541 .641 .898 r F P 0 g s .38695 .58354 m .36835 .55823 L .39655 .55697 L p .541 .641 .898 r F P 0 g s .59612 .56745 m .61248 .57287 L .58306 .58724 L p .646 .859 .98 r F P 0 g s .58306 .58724 m .5664 .57867 L .59612 .56745 L p .646 .859 .98 r F P 0 g s .2556 .46326 m .27514 .49628 L .24838 .49146 L p .424 .516 .848 r F P 0 g s .24838 .49146 m .22897 .45543 L .2556 .46326 L p .424 .516 .848 r F P 0 g s .34941 .53245 m .36835 .55823 L .34059 .55948 L p .502 .598 .882 r F P 0 g s .34059 .55948 m .32165 .53059 L .34941 .53245 L p .502 .598 .882 r F P 0 g s .30237 .50118 m .32165 .53059 L .29435 .52875 L p .463 .556 .865 r F P 0 g s .29435 .52875 m .27514 .49628 L .30237 .50118 L p .463 .556 .865 r F P 0 g s .14337 .3274 m .16347 .36808 L .13825 .35477 L p .284 .399 .803 r F P 0 g s .13825 .35477 m .1184 .31122 L .14337 .3274 L p .284 .399 .803 r F P 0 g s .80933 .46974 m .8221 .45673 L .79175 .4783 L p 0 0 0 r F P 0 g s .79175 .4783 m .77841 .48827 L .80933 .46974 L p 0 0 0 r F P 0 g s .70275 .52878 m .71751 .52476 L .68752 .54274 L p .573 .938 .91 r F P 0 g s .68752 .54274 m .67232 .54364 L .70275 .52878 L p .573 .938 .91 r F P 0 g s .4726 .57017 m .49052 .58561 L .4617 .5933 L p .574 .715 .938 r F P 0 g s .4617 .5933 m .44364 .57469 L .4726 .57017 L p .574 .715 .938 r F P 0 g s .62622 .55608 m .64223 .55834 L .61248 .57287 L p .618 .895 .988 r F P 0 g s .61248 .57287 m .59612 .56745 L .62622 .55608 L p .618 .895 .988 r F P 0 g s .54933 .5699 m .5664 .57867 L .53704 .58976 L p .604 .803 .976 r F P 0 g s .53704 .58976 m .51973 .57781 L .54933 .5699 L p .604 .803 .976 r F P 0 g s .18923 .38168 m .20925 .41884 L .18329 .40819 L p .324 .44 .825 r F P 0 g s .18329 .40819 m .16347 .36808 L .18923 .38168 L p .324 .44 .825 r F P 0 g s .42519 .55568 m .44364 .57469 L .41509 .57915 L p .531 .665 .922 r F P 0 g s .41509 .57915 m .39655 .55697 L .42519 .55568 L p .531 .665 .922 r F P 0 g s .84056 .45102 m .85272 .43497 L .8221 .45673 L p 0 0 0 r F P 0 g s .8221 .45673 m .80933 .46974 L .84056 .45102 L p 0 0 0 r F P 0 g s .73352 .51375 m .74781 .50661 L .71751 .52476 L p .455 .881 .796 r F P 0 g s .71751 .52476 m .70275 .52878 L .73352 .51375 L p .455 .881 .796 r F P 0 g s .23574 .4297 m .2556 .46326 L .22897 .45543 L p .365 .482 .845 r F P 0 g s .22897 .45543 m .20925 .41884 L .23574 .4297 L p .365 .482 .845 r F P 0 g s .37763 .53434 m .39655 .55697 L .36835 .55823 L p .488 .617 .904 r F P 0 g s .36835 .55823 m .34941 .53245 L .37763 .53434 L p .488 .617 .904 r F P 0 g s .12297 .28613 m .14337 .3274 L .1184 .31122 L p .213 .353 .79 r F P 0 g s .1184 .31122 m .0983 .2671 L .12297 .28613 L p .213 .353 .79 r F P 0 g s .28275 .47125 m .30237 .50118 L .27514 .49628 L p .405 .525 .866 r F P 0 g s .27514 .49628 m .2556 .46326 L .28275 .47125 L p .405 .525 .866 r F P 0 g s .3301 .50618 m .34941 .53245 L .32165 .53059 L p .447 .57 .885 r F P 0 g s .32165 .53059 m .30237 .50118 L .3301 .50618 L p .447 .57 .885 r F P 0 g s .502 .56558 m .51973 .57781 L .49052 .58561 L p .56 .747 .964 r F P 0 g s .49052 .58561 m .4726 .57017 L .502 .56558 L p .56 .747 .964 r F P 0 g s .65669 .54457 m .67232 .54364 L .64223 .55834 L p .561 .912 .965 r F P 0 g s .64223 .55834 m .62622 .55608 L .65669 .54457 L p .561 .912 .965 r F P 0 g s .87212 .4321 m .88361 .41301 L .85272 .43497 L p .005 0 0 r F P 0 g s .85272 .43497 m .84056 .45102 L .87212 .4321 L p .005 0 0 r F P 0 g s .57934 .56189 m .59612 .56745 L .5664 .57867 L p .579 .84 .994 r F P 0 g s .5664 .57867 m .54933 .5699 L .57934 .56189 L p .579 .84 .994 r F P 0 g s .76465 .49855 m .77841 .48827 L .74781 .50661 L p 0 0 0 r F P 0 g s .74781 .50661 m .73352 .51375 L .76465 .49855 L p 0 0 0 r F P 0 g s .16889 .34394 m .18923 .38168 L .16347 .36808 L p .254 .397 .814 r F P 0 g s .16347 .36808 m .14337 .3274 L .16889 .34394 L p .254 .397 .814 r F P 0 g s .45429 .55438 m .4726 .57017 L .44364 .57469 L p .515 .691 .948 r F P 0 g s .44364 .57469 m .42519 .55568 L .45429 .55438 L p .515 .691 .948 r F P 0 g s .10228 .24425 m .12297 .28613 L .0983 .2671 L p .133 .301 .769 r F P 0 g s .0983 .2671 m .07792 .2224 L .10228 .24425 L p .133 .301 .769 r F P 0 g s .21555 .39557 m .23574 .4297 L .20925 .41884 L p .296 .442 .838 r F P 0 g s .20925 .41884 m .18923 .38168 L .21555 .39557 L p .296 .442 .838 r F P 0 g s .68755 .53292 m .70275 .52878 L .67232 .54364 L p .463 .888 .889 r F P 0 g s .67232 .54364 m .65669 .54457 L .68755 .53292 L p .463 .888 .889 r F P 0 g s .40635 .53627 m .42519 .55568 L .39655 .55697 L p .47 .637 .928 r F P 0 g s .39655 .55697 m .37763 .53434 L .40635 .53627 L p .47 .637 .928 r F P 0 g s .53184 .56092 m .54933 .5699 L .51973 .57781 L p .536 .78 .988 r F P 0 g s .51973 .57781 m .502 .56558 L .53184 .56092 L p .536 .78 .988 r F P 0 g s .26278 .44079 m .28275 .47125 L .2556 .46326 L p .339 .488 .861 r F P 0 g s .2556 .46326 m .23574 .4297 L .26278 .44079 L p .339 .488 .861 r F P 0 g s .60977 .55377 m .62622 .55608 L .59612 .56745 L p .533 .867 .993 r F P 0 g s .59612 .56745 m .57934 .56189 L .60977 .55377 L p .533 .867 .993 r F P 0 g s .79614 .48317 m .80933 .46974 L .77841 .48827 L p 0 0 0 r F P 0 g s .77841 .48827 m .76465 .49855 L .79614 .48317 L p 0 0 0 r F P 0 g s .35834 .51126 m .37763 .53434 L .34941 .53245 L p .425 .585 .907 r F P 0 g s .34941 .53245 m .3301 .50618 L .35834 .51126 L p .425 .585 .907 r F P 0 g s .31043 .4794 m .3301 .50618 L .30237 .50118 L p .382 .535 .885 r F P 0 g s .30237 .50118 m .28275 .47125 L .31043 .4794 L p .382 .535 .885 r F P 0 g s .14824 .30561 m .16889 .34394 L .14337 .3274 L p .175 .347 .797 r F P 0 g s .14337 .3274 m .12297 .28613 L .14824 .30561 L p .175 .347 .797 r F P 0 g s .48386 .55305 m .502 .56558 L .4726 .57017 L p .49 .719 .973 r F P 0 g s .4726 .57017 m .45429 .55438 L .48386 .55305 L p .49 .719 .973 r F P 0 g s .7188 .52111 m .73352 .51375 L .70275 .52878 L p .324 .804 .747 r F P 0 g s .70275 .52878 m .68755 .53292 L .7188 .52111 L p .324 .804 .747 r F P 0 g s .82799 .46761 m .84056 .45102 L .80933 .46974 L p 0 0 0 r F P 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m .14824 .30561 L .12297 .28613 L p .087 .289 .771 r F P 0 g s .12297 .28613 m .10228 .24425 L .12725 .26667 L p .087 .289 .771 r F P 0 g s .29037 .45211 m .31043 .4794 L .28275 .47125 L p .306 .493 .878 r F P 0 g s .28275 .47125 m .26278 .44079 L .29037 .45211 L p .306 .493 .878 r F P 0 g s .33865 .4877 m .35834 .51126 L .3301 .50618 L p .35 .546 .904 r F P 0 g s .3301 .50618 m .31043 .4794 L .33865 .4877 L p .35 .546 .904 r F P 0 g s .75045 .50916 m .76465 .49855 L .73352 .51375 L p 0 0 0 r F P 0 g s .73352 .51375 m .7188 .52111 L .75045 .50916 L p 0 0 0 r F P 0 g s .86022 .45187 m .87212 .4321 L .84056 .45102 L p .098 0 0 r F P 0 g s .84056 .45102 m .82799 .46761 L .86022 .45187 L p .098 0 0 r F P 0 g s .51391 .5517 m .53184 .56092 L .502 .56558 L p .45 .745 .991 r F P 0 g s .502 .56558 m .48386 .55305 L .51391 .5517 L p .45 .745 .991 r F P 0 g s .67189 .53718 m .68755 .53292 L .65669 .54457 L p .33 .814 .847 r F P 0 g s .65669 .54457 m .64062 .54553 L .67189 .53718 L p .33 .814 .847 r F P 0 g s .59287 .55139 m .60977 .55377 L .57934 .56189 L p .425 .82 .984 r F P 0 g s .57934 .56189 m .56212 .55619 L .59287 .55139 L p .425 .82 .984 r F P 0 g s .17412 .32556 m .19501 .36086 L .16889 .34394 L p .13 .337 .802 r F P 0 g s .16889 .34394 m .14824 .30561 L .17412 .32556 L p .13 .337 .802 r F P 0 g s .46528 .54023 m .48386 .55305 L .45429 .55438 L p .403 .679 .973 r F P 0 g s .45429 .55438 m .43556 .53823 L .46528 .54023 L p .403 .679 .973 r F P 0 g s .78251 .49705 m .79614 .48317 L .76465 .49855 L p 0 0 0 r F P 0 g s .76465 .49855 m .75045 .50916 L .78251 .49705 L p 0 0 0 r F P 0 g s .22173 .37818 m .24244 .40977 L .21555 .39557 L p .173 .388 .832 r F P 0 g s .21555 .39557 m .19501 .36086 L .22173 .37818 L p .173 .388 .832 r F P 0 g s .41639 .52171 m .43556 .53823 L .40635 .53627 L p .356 .616 .949 r F P 0 g s .40635 .53627 m .3871 .51644 L .41639 .52171 L p .356 .616 .949 r F P 0 g s .54445 .55033 m .56212 .55619 L .53184 .56092 L p .388 .76 .992 r F P 0 g s .53184 .56092 m .51391 .5517 L .54445 .55033 L p .388 .76 .992 r F P 0 g s .26993 .42428 m .29037 .45211 L .26278 .44079 L p .218 .44 .862 r F P 0 g s .26278 .44079 m .24244 .40977 L .26993 .42428 L p .218 .44 .862 r F P 0 g s .70361 .52871 m .7188 .52111 L .68755 .53292 L p 0 0 0 r F P 0 g s .68755 .53292 m .67189 .53718 L .70361 .52871 L p 0 0 0 r F P 0 g s .36743 .49617 m .3871 .51644 L .35834 .51126 L p .309 .554 .921 r F P 0 g s .35834 .51126 m .33865 .4877 L .36743 .49617 L p .309 .554 .921 r F P 0 g s .31855 .46366 m .33865 .4877 L .31043 .4794 L p .263 .496 .892 r F P 0 g s .31043 .4794 m .29037 .45211 L .31855 .46366 L p .263 .496 .892 r F P 0 g s .62408 .54651 m .64062 .54553 L .60977 .55377 L p .32 .795 .917 r F P 0 g s .60977 .55377 m .59287 .55139 L .62408 .54651 L p .32 .795 .917 r F P 0 g s .81498 .48478 m .82799 .46761 L .79614 .48317 L p .096 0 0 r F P 0 g s .79614 .48317 m .78251 .49705 L .81498 .48478 L p .096 0 0 r F P 0 g s .15287 .28965 m .17412 .32556 L .14824 .30561 L p .032 .272 .77 r F P 0 g s .14824 .30561 m .12725 .26667 L .15287 .28965 L p .032 .272 .77 r F P 0 g s .49552 .54226 m .51391 .5517 L .48386 .55305 L p .345 .694 .982 r F P 0 g s .48386 .55305 m .46528 .54023 L .49552 .54226 L p .345 .694 .982 r F P 0 g s .20064 .34601 m .22173 .37818 L .19501 .36086 L p .075 .324 .803 r F P 0 g s .19501 .36086 m .17412 .32556 L .20064 .34601 L p .075 .324 .803 r F P 0 g s .73579 .52012 m .75045 .50916 L .7188 .52111 L p 0 0 0 r F P 0 g s .7188 .52111 m .70361 .52871 L .73579 .52012 L p 0 0 0 r F P 0 g s .57549 .54894 m .59287 .55139 L .56212 .55619 L p .295 .751 .954 r F P 0 g s .56212 .55619 m .54445 .55033 L .57549 .54894 L p .295 .751 .954 r F P 0 g s .44624 .52709 m .46528 .54023 L .43556 .53823 L p .3 .626 .961 r F P 0 g s .43556 .53823 m .41639 .52171 L .44624 .52709 L p .3 .626 .961 r F P 0 g s .65577 .54156 m .67189 .53718 L .64062 .54553 L p .18 .715 .781 r F P 0 g s .64062 .54553 m .62408 .54651 L .65577 .54156 L p .18 .715 .781 r F P 0 g s 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.195 0 0 r F P 0 g s .78251 .49705 m .76842 .5114 L .80151 .50256 L p .195 0 0 r F P 0 g s .27708 .41405 m .29803 .43912 L .26993 .42428 L p .05 .362 .833 r F P 0 g s .26993 .42428 m .24908 .39591 L .27708 .41405 L p .05 .362 .833 r F P 0 g s .37672 .48752 m .39679 .50481 L .36743 .49617 L p .135 .488 .902 r F P 0 g s .36743 .49617 m .34733 .47546 L .37672 .48752 L p .135 .488 .902 r F P 0 g s .55763 .54643 m .57549 .54894 L .54445 .55033 L p .145 .657 .897 r F P 0 g s .54445 .55033 m .5263 .54432 L .55763 .54643 L p .145 .657 .897 r F P 0 g s .32678 .45429 m .34733 .47546 L .31855 .46366 L p .092 .423 .869 r F P 0 g s .31855 .46366 m .29803 .43912 L .32678 .45429 L p .092 .423 .869 r F P 0 g s .72064 .53143 m .73579 .52012 L .70361 .52871 L p .105 0 0 r F P 0 g s .70361 .52871 m .68795 .53654 L .72064 .53143 L p .105 0 0 r F P 0 g s .63915 .54609 m .65577 .54156 L .62408 .54651 L p 0 0 0 r F P 0 g s .62408 .54651 m .60706 .54753 L .63915 .54609 L p 0 0 0 r F P 0 g s .50766 .53815 m 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.6175 .57057 L .64269 .5497 L p .799 .869 .9 r F P 0 g s .55328 .59924 m .56761 .6119 L .54292 .63236 L p .745 .762 .866 r F P 0 g s .54292 .63236 m .5284 .61707 L .55328 .59924 L p .745 .762 .866 r F P 0 g s 1 0 0 r .005 w .35627 .65214 m .35669 .65123 L s .35669 .65123 m .35677 .65107 L s .5 Mabswid .42441 .62239 m .43976 .64625 L .41555 .66092 L p .671 .655 .825 r F P 0 g s .41555 .66092 m .40014 .6344 L .42441 .62239 L p .671 .655 .825 r F P 0 g s .4888 .61655 m .50371 .63477 L .47921 .65233 L p .706 .704 .846 r F P 0 g s .47921 .65233 m .46417 .63147 L .4888 .61655 L p .706 .704 .846 r F P 0 g s .78279 .43512 m .79408 .42721 L .76885 .45381 L p .724 .982 .775 r F P 0 g s .76885 .45381 m .75715 .45924 L .78279 .43512 L p .724 .982 .775 r F P 0 g s .66805 .52869 m .6811 .5308 L .65603 .55438 L p .806 .911 .912 r F P 0 g s .65603 .55438 m .64269 .5497 L .66805 .52869 L p .806 .911 .912 r F P 0 g s .38452 .60752 m .40014 .6344 L .3761 .6463 L p .644 .628 .819 r F P 0 g s .3761 .6463 m .36047 .61677 L .38452 .60752 L p .644 .628 .819 r F P 0 g s 1 0 0 r .005 w .46734 .64857 m .46742 .6492 L s .5 Mabswid .57835 .58127 m .59247 .59131 L .56761 .6119 L p .756 .794 .884 r F P 0 g s .56761 .6119 m .55328 .59924 L .57835 .58127 L p .756 .794 .884 r F P 0 g s .34463 .58685 m .36047 .61677 L .33666 .62592 L p .617 .602 .812 r F P 0 g s .33666 .62592 m .32085 .59336 L .34463 .58685 L p .617 .602 .812 r F P 0 g s .80858 .41085 m .81944 .40048 L .79408 .42721 L p .616 .919 .657 r F P 0 g s .79408 .42721 m .78279 .43512 L .80858 .41085 L p .616 .919 .657 r F P 0 g s .30484 .56042 m .32085 .59336 L .29733 .5998 L p .591 .576 .805 r F P 0 g s .29733 .5998 m .28139 .56425 L .30484 .56042 L p .591 .576 .805 r F P 0 g s .4489 .61027 m .46417 .63147 L .43976 .64625 L p .677 .674 .84 r F P 0 g s .43976 .64625 m .42441 .62239 L .4489 .61027 L p .677 .674 .84 r F P 0 g s .51363 .60151 m .5284 .61707 L .50371 .63477 L p .714 .729 .862 r F P 0 g s .50371 .63477 m .4888 .61655 L .51363 .60151 L p .714 .729 .862 r F P 0 g s .69358 .50754 m .7063 .50709 L .6811 .5308 L p .805 .952 .915 r F P 0 g s .6811 .5308 m .66805 .52869 L .69358 .50754 L p .805 .952 .915 r F P 0 g s .83452 .38645 m .84492 .37363 L .81944 .40048 L p 0 0 0 r F P 0 g s .81944 .40048 m .80858 .41085 L .83452 .38645 L p 0 0 0 r F P 0 g s .6036 .56317 m .6175 .57057 L .59247 .59131 L p .766 .83 .902 r F P 0 g s .59247 .59131 m .57835 .58127 L .6036 .56317 L p .766 .83 .902 r F P 0 g s .40883 .59818 m .42441 .62239 L .40014 .6344 L p .648 .645 .833 r F P 0 g s .40014 .6344 m .38452 .60752 L .40883 .59818 L p .648 .645 .833 r F P 0 g s .71928 .48625 m .73165 .48323 L .7063 .50709 L p .787 .984 .9 r F P 0 g s .7063 .50709 m .69358 .50754 L .71928 .48625 L p .787 .984 .9 r F P 0 g s .53868 .58634 m .55328 .59924 L .5284 .61707 L p .723 .758 .88 r F P 0 g s .5284 .61707 m .51363 .60151 L .53868 .58634 L p .723 .758 .88 r F P 0 g s .86061 .3619 m .87051 .34664 L .84492 .37363 L p 0 0 0 r F P 0 g s .84492 .37363 m .83452 .38645 L .86061 .3619 L p 0 0 0 r F P 0 g s .47363 .59803 m .4888 .61655 L .46417 .63147 L p .683 .696 .856 r F P 0 g s .46417 .63147 m .4489 .61027 L .47363 .59803 L p .683 .696 .856 r F P 0 g s .36869 .58026 m .38452 .60752 L .36047 .61677 L p .62 .616 .826 r F P 0 g s .36047 .61677 m .34463 .58685 L .36869 .58026 L p .62 .616 .826 r F P 0 g s .28865 .52708 m .30484 .56042 L .28139 .56425 L p .563 .561 .809 r F P 0 g s .28139 .56425 m .26527 .5283 L .28865 .52708 L p .563 .561 .809 r F P 0 g s .32859 .55655 m .34463 .58685 L .32085 .59336 L p .591 .588 .818 r F P 0 g s .32085 .59336 m .30484 .56042 L .32859 .55655 L p .591 .588 .818 r F P 0 g s .62906 .54492 m .64269 .5497 L .6175 .57057 L p .774 .871 .919 r F P 0 g s .6175 .57057 m .6036 .56317 L .62906 .54492 L p .774 .871 .919 r F P 0 g s .74515 .46481 m .75715 .45924 L .73165 .48323 L p .74 .992 .855 r F P 0 g s .73165 .48323 m .71928 .48625 L .74515 .46481 L p .74 .992 .855 r F P 0 g s .43339 .58874 m .4489 .61027 L .42441 .62239 L p .652 .664 .849 r F P 0 g s .42441 .62239 m .40883 .59818 L .43339 .58874 L p .652 .664 .849 r F P 0 g s .56394 .57104 m .57835 .58127 L .55328 .59924 L p .731 .791 .9 r F P 0 g s .55328 .59924 m .53868 .58634 L .56394 .57104 L p .731 .791 .9 r F P 0 g s .4986 .58568 m .51363 .60151 L .4888 .61655 L p .689 .722 .874 r F P 0 g s .4888 .61655 m .47363 .59803 L .4986 .58568 L p .689 .722 .874 r F P 0 g s .65471 .52654 m .66805 .52869 L .64269 .5497 L p .776 .913 .932 r F P 0 g s .64269 .5497 m .62906 .54492 L .65471 .52654 L p .776 .913 .932 r F P 0 g s .7712 .44323 m .78279 .43512 L .75715 .45924 L p .654 .96 .764 r F P 0 g s .75715 .45924 m .74515 .46481 L .7712 .44323 L p .654 .96 .764 r F P 0 g s .39302 .5736 m .40883 .59818 L .38452 .60752 L p .622 .633 .841 r F P 0 g s .38452 .60752 m .36869 .58026 L .39302 .5736 L p .622 .633 .841 r F P 0 g s .27226 .49334 m .28865 .52708 L .26527 .5283 L p .532 .544 .813 r F P 0 g s .26527 .5283 m .24898 .49195 L .27226 .49334 L p .532 .544 .813 r F P 0 g s .35263 .55262 m .36869 .58026 L .34463 .58685 L p .592 .603 .832 r F P 0 g s .34463 .58685 m .32859 .55655 L .35263 .55262 L p .592 .603 .832 r F P 0 g s 1 0 0 r .005 w .59462 .5747 m .59488 .57519 L s .5 Mabswid .31234 .52585 m .32859 .55655 L .30484 .56042 L p .562 .573 .823 r F P 0 g s .30484 .56042 m .28865 .52708 L .31234 .52585 L p .562 .573 .823 r F P 0 g s .58943 .55561 m .6036 .56317 L .57835 .58127 L p .738 .828 .92 r F P 0 g s .57835 .58127 m .56394 .57104 L .58943 .55561 L p .738 .828 .92 r F P 0 g s .68055 .50801 m .69358 .50754 L .66805 .52869 L p .768 .953 .934 r F P 0 g s .66805 .52869 m .65471 .52654 L .68055 .50801 L p .768 .953 .934 r F P 0 g s .45821 .5792 m .47363 .59803 L .4489 .61027 L p .656 .687 .867 r F P 0 g s .4489 .61027 m .43339 .58874 L .45821 .5792 L p .656 .687 .867 r F P 0 g s .79743 .4215 m .80858 .41085 L .78279 .43512 L p .527 .875 .627 r F P 0 g s .78279 .43512 m .7712 .44323 L .79743 .4215 L p .527 .875 .627 r F P 0 g s .5238 .5732 m .53868 .58634 L .51363 .60151 L p .696 .751 .894 r F P 0 g s .51363 .60151 m .4986 .58568 L .5238 .5732 L p .696 .751 .894 r F P 0 g s .41763 .56686 m .43339 .58874 L .40883 .59818 L p .624 .653 .858 r F P 0 g s .40883 .59818 m .39302 .5736 L .41763 .56686 L p .624 .653 .858 r F P 0 g s .82384 .39963 m .83452 .38645 L .80858 .41085 L p 0 0 0 r F P 0 g s .80858 .41085 m .79743 .4215 L .82384 .39963 L p 0 0 0 r F P 0 g s .7066 .48934 m .71928 .48625 L .69358 .50754 L p .738 .98 .914 r F P 0 g s .69358 .50754 m .68055 .50801 L .7066 .48934 L p .738 .98 .914 r F P 0 g s .61513 .54004 m .62906 .54492 L .6036 .56317 L p .742 .87 .939 r F P 0 g s .6036 .56317 m .58943 .55561 L .61513 .54004 L p .742 .87 .939 r F P 0 g s .25566 .45918 m .27226 .49334 L .24898 .49195 L p .497 .524 .815 r F P 0 g s .24898 .49195 m .2325 .4552 L .25566 .45918 L p .497 .524 .815 r F P 0 g s .37697 .54865 m .39302 .5736 L .36869 .58026 L p .592 .619 .848 r F P 0 g s .36869 .58026 m .35263 .55262 L .37697 .54865 L p .592 .619 .848 r F P 0 g s .4833 .56956 m .4986 .58568 L .47363 .59803 L p .66 .713 .887 r F P 0 g s .47363 .59803 m .45821 .5792 L .4833 .56956 L p .66 .713 .887 r F P 0 g s .54925 .56061 m .56394 .57104 L .53868 .58634 L p .701 .785 .916 r F P 0 g s .53868 .58634 m .5238 .5732 L .54925 .56061 L p .701 .785 .916 r F P 0 g s .29587 .49474 m .31234 .52585 L .28865 .52708 L p .528 .555 .827 r F P 0 g s .28865 .52708 m .27226 .49334 L .29587 .49474 L p .528 .555 .827 r F P 0 g s .33634 .52459 m .35263 .55262 L .32859 .55655 L p .56 .587 .838 r F P 0 g s .32859 .55655 m .31234 .52585 L .33634 .52459 L p .56 .587 .838 r F P 0 g s .85042 .3776 m .86061 .3619 L .83452 .38645 L p 0 0 0 r F P 0 g s .83452 .38645 m .82384 .39963 L .85042 .3776 L p 0 0 0 r F P 0 g s .73285 .47053 m .74515 .46481 L .71928 .48625 L p .673 .977 .855 r F P 0 g s .71928 .48625 m .7066 .48934 L .73285 .47053 L p .673 .977 .855 r F P 0 g s .64106 .52434 m .65471 .52654 L .62906 .54492 L p .738 .913 .953 r F P 0 g s .62906 .54492 m .61513 .54004 L .64106 .52434 L p .738 .913 .953 r F P 0 g s .44253 .56004 m .45821 .5792 L .43339 .58874 L p .625 .675 .877 r F P 0 g s .43339 .58874 m .41763 .56686 L .44253 .56004 L p .625 .675 .877 r F P 0 g s .57495 .54789 m .58943 .55561 L .56394 .57104 L p .704 .823 .939 r F P 0 g s .56394 .57104 m .54925 .56061 L .57495 .54789 L p .704 .823 .939 r F P 0 g s .50865 .55981 m .5238 .5732 L .4986 .58568 L p .664 .742 .909 r F P 0 g s .4986 .58568 m .4833 .56956 L .50865 .55981 L p .664 .742 .909 r F P 0 g s .23887 .4246 m .25566 .45918 L .2325 .4552 L p .457 .501 .816 r F P 0 g s .2325 .4552 m .21583 .41803 L .23887 .4246 L p .457 .501 .816 r F P 0 g s .75931 .45157 m .7712 .44323 L .74515 .46481 L p .566 .923 .743 r F P 0 g s .74515 .46481 m .73285 .47053 L .75931 .45157 L p .566 .923 .743 r F P 0 g s .40162 .54463 m .41763 .56686 L .39302 .5736 L p .591 .638 .866 r F P 0 g s .39302 .5736 m .37697 .54865 L .40162 .54463 L p .591 .638 .866 r F P 0 g s .66722 .50849 m .68055 .50801 L .65471 .52654 L p .719 .95 .952 r F P 0 g s .65471 .52654 m .64106 .52434 L .66722 .50849 L p .719 .95 .952 r F P 0 g s .27919 .46322 m .29587 .49474 L .27226 .49334 L p .49 .534 .83 r F P 0 g s .27226 .49334 m .25566 .45918 L .27919 .46322 L p .49 .534 .83 r F P 0 g s .36068 .52333 m .37697 .54865 L .35263 .55262 L p .557 .603 .855 r F P 0 g s .35263 .55262 m .33634 .52459 L .36068 .52333 L p .557 .603 .855 r F P 0 g s .31983 .49617 m .33634 .52459 L .31234 .52585 L p .523 .568 .842 r F P 0 g s .31234 .52585 m .29587 .49474 L .31983 .49617 L p .523 .568 .842 r F P 0 g s .46772 .55314 m .4833 .56956 L .45821 .5792 L p .626 .7 .899 r F P 0 g s .45821 .5792 m .44253 .56004 L .46772 .55314 L p .626 .7 .899 r F P 0 g s .78597 .43245 m .79743 .4215 L .7712 .44323 L p .422 .814 .586 r F P 0 g s .7712 .44323 m .75931 .45157 L .78597 .43245 L p .422 .814 .586 r F P 0 g s .6009 .53505 m .61513 .54004 L .58943 .55561 L p .702 .865 .959 r F P 0 g s .58943 .55561 m .57495 .54789 L .6009 .53505 L p .702 .865 .959 r F P 0 g s .53427 .54996 m .54925 .56061 L .5238 .5732 L p .665 .777 .933 r F P 0 g s .5238 .5732 m .50865 .55981 L .53427 .54996 L p .665 .777 .933 r F P 0 g s .69361 .49251 m .7066 .48934 L .68055 .50801 L p .673 .968 .922 r F P 0 g s .68055 .50801 m .66722 .50849 L .69361 .49251 L p .673 .968 .922 r F P 0 g s .22186 .3896 m .23887 .4246 L .21583 .41803 L p .411 .475 .815 r F P 0 g s .21583 .41803 m .19898 .38044 L .22186 .3896 L p .411 .475 .815 r F P 0 g s .42658 .54056 m .44253 .56004 L .41763 .56686 L p .589 .66 .887 r F P 0 g s .41763 .56686 m .40162 .54463 L .42658 .54056 L p .589 .66 .887 r F P 0 g s .81284 .41319 m .82384 .39963 L .79743 .4215 L p 0 0 0 r F P 0 g s .79743 .4215 m .78597 .43245 L .81284 .41319 L p 0 0 0 r F P 0 g s .26228 .43129 m .27919 .46322 L .25566 .45918 L p .446 .509 .831 r F P 0 g s .25566 .45918 m .23887 .4246 L .26228 .43129 L p .446 .509 .831 r F P 0 g s .38534 .52204 m .40162 .54463 L .37697 .54865 L p .552 .62 .874 r F P 0 g s .37697 .54865 m .36068 .52333 L .38534 .52204 L p .552 .62 .874 r F P 0 g s .6271 .52208 m .64106 .52434 L .61513 .54004 L p .689 .907 .972 r F P 0 g s .61513 .54004 m .6009 .53505 L .6271 .52208 L p .689 .907 .972 r F P 0 g s .30307 .46733 m .31983 .49617 L .29587 .49474 L p .481 .545 .846 r F P 0 g s .29587 .49474 m .27919 .46322 L .30307 .46733 L p .481 .545 .846 r F P 0 g s .34414 .49761 m .36068 .52333 L .33634 .52459 L p .516 .582 .86 r F P 0 g s .33634 .52459 m .31983 .49617 L .34414 .49761 L p .516 .582 .86 r F P 0 g s .4932 .54617 m .50865 .55981 L .4833 .56956 L p .625 .73 .923 r F P 0 g s .4833 .56956 m .46772 .55314 L .4932 .54617 L p .625 .73 .923 r F P 0 g s .72023 .47639 m .73285 .47053 L .7066 .48934 L p .588 .946 .844 r F P 0 g s .7066 .48934 m .69361 .49251 L .72023 .47639 L p .588 .946 .844 r F P 0 g s .56017 .54001 m .57495 .54789 L .54925 .56061 L p .662 .815 .957 r F P 0 g s .54925 .56061 m .53427 .54996 L .56017 .54001 L p .662 .815 .957 r F P 0 g s .83993 .39378 m .85042 .3776 L .82384 .39963 L p 0 0 0 r F P 0 g s .82384 .39963 m .81284 .41319 L .83993 .39378 L p 0 0 0 r F P 0 g s .20465 .35416 m .22186 .3896 L .19898 .38044 L p .359 .443 .812 r F P 0 g s .19898 .38044 m .18194 .34243 L .20465 .35416 L p .359 .443 .812 r F P 0 g s .45185 .53643 m .46772 .55314 L .44253 .56004 L p .585 .685 .91 r F P 0 g s .44253 .56004 m .42658 .54056 L .45185 .53643 L p .585 .685 .91 r F P 0 g s .65356 .50899 m .66722 .50849 L .64106 .52434 L p .656 .938 .965 r F P 0 g s .64106 .52434 m .6271 .52208 L .65356 .50899 L p .656 .938 .965 r F P 0 g s .74709 .46012 m .75931 .45157 L .73285 .47053 L p .459 .867 .709 r F P 0 g s .73285 .47053 m .72023 .47639 L .74709 .46012 L p .459 .867 .709 r F P 0 g s .24514 .39892 m .26228 .43129 L .23887 .4246 L p .395 .48 .829 r F P 0 g s .23887 .4246 m .22186 .3896 L .24514 .39892 L p .395 .48 .829 r F P 0 g s .51898 .5391 m .53427 .54996 L .50865 .55981 L p .62 .764 .948 r F P 0 g s .50865 .55981 m .4932 .54617 L .51898 .5391 L p .62 .764 .948 r F P 0 g s .41035 .52073 m .42658 .54056 L .40162 .54463 L p .546 .641 .895 r F P 0 g s .40162 .54463 m .38534 .52204 L .41035 .52073 L p .546 .641 .895 r F P 0 g s .58635 .52995 m .6009 .53505 L .57495 .54789 L p .651 .855 .978 r F P 0 g s .57495 .54789 m .56017 .54001 L .58635 .52995 L p .651 .855 .978 r F P 0 g s 1 0 0 r .005 w .70842 .48647 m .70858 .48674 L s .5 Mabswid .28608 .43808 m .30307 .46733 L .27919 .46322 L p .432 .518 .847 r F P 0 g s .27919 .46322 m .26228 .43129 L .28608 .43808 L p .432 .518 .847 r F P 0 g s .3688 .49908 m .38534 .52204 L .36068 .52333 L p .507 .599 .88 r F P 0 g s .36068 .52333 m .34414 .49761 L .3688 .49908 L p .507 .599 .88 r F P 0 g s .32734 .4715 m .34414 .49761 L .31983 .49617 L p .469 .558 .864 r F P 0 g s .31983 .49617 m .30307 .46733 L .32734 .4715 L p .469 .558 .864 r F P 0 g s .18721 .31828 m .20465 .35416 L .18194 .34243 L p .3 .407 .804 r F P 0 g s .18194 .34243 m .1647 .30398 L .18721 .31828 L p .3 .407 .804 r F P 0 g s .77418 .44371 m .78597 .43245 L .75931 .45157 L p 0 0 0 r F P 0 g s .75931 .45157 m .74709 .46012 L .77418 .44371 L p 0 0 0 r F P 0 g s .68028 .49576 m .69361 .49251 L .66722 .50849 L p .59 .943 .921 r F P 0 g s .66722 .50849 m .65356 .50899 L .68028 .49576 L p .59 .943 .921 r F P 0 g s .47745 .53225 m .4932 .54617 L .46772 .55314 L p .578 .713 .935 r F P 0 g s .46772 .55314 m .45185 .53643 L .47745 .53225 L p .578 .713 .935 r F P 0 g s .61282 .51977 m .6271 .52208 L .6009 .53505 L p .626 .893 .987 r F P 0 g s .6009 .53505 m .58635 .52995 L .61282 .51977 L p .626 .893 .987 r F P 0 g s .54507 .53196 m .56017 .54001 L .53427 .54996 L p .609 .801 .973 r F P 0 g s .53427 .54996 m .51898 .5391 L .54507 .53196 L p .609 .801 .973 r F P 0 g s .22778 .36611 m .24514 .39892 L .22186 .3896 L p .337 .446 .824 r F P 0 g s .22186 .3896 m .20465 .35416 L .22778 .36611 L p .337 .446 .824 r F P 0 g s .4357 .51941 m .45185 .53643 L .42658 .54056 L p .536 .664 .919 r F P 0 g s .42658 .54056 m .41035 .52073 L .4357 .51941 L p .536 .664 .919 r F P 0 g s .80153 .42715 m .81284 .41319 L .78597 .43245 L p 0 0 0 r F P 0 g s .78597 .43245 m .77418 .44371 L .80153 .42715 L p 0 0 0 r F P 0 g s .70727 .48241 m .72023 .47639 L .69361 .49251 L p .481 .897 .818 r F P 0 g s .69361 .49251 m .68028 .49576 L .70727 .48241 L p .481 .897 .818 r F P 0 g s .26884 .4084 m .28608 .43808 L .26228 .43129 L p .375 .486 .844 r F P 0 g s .26228 .43129 m .24514 .39892 L .26884 .4084 L p .375 .486 .844 r F P 0 g s .39383 .50057 m .41035 .52073 L .38534 .52204 L p .494 .617 .902 r F P 0 g s .38534 .52204 m .3688 .49908 L .39383 .50057 L p .494 .617 .902 r F P 0 g s .16956 .28194 m .18721 .31828 L .1647 .30398 L p .233 .364 .792 r F P 0 g s .1647 .30398 m .14726 .26509 L .16956 .28194 L p .233 .364 .792 r F P 0 g s .31028 .44498 m .32734 .4715 L .30307 .46733 L p .414 .528 .864 r F P 0 g s .30307 .46733 m .28608 .43808 L .31028 .44498 L p .414 .528 .864 r F P 0 g s .35199 .47574 m .3688 .49908 L .34414 .49761 L p .454 .572 .883 r F P 0 g s .34414 .49761 m .32734 .4715 L .35199 .47574 L p .454 .572 .883 r F P 0 g s .86235 .72038 m .87225 .71421 L .89267 .77539 L closepath p .505 0 0 r F P 0 g s .85049 .72775 m .86235 .72038 L .89267 .77539 L closepath p .673 0 0 r F P 0 g s .50338 .52802 m .51898 .5391 L .4932 .54617 L p .565 .745 .961 r F P 0 g s .4932 .54617 m .47745 .53225 L .50338 .52802 L p .565 .745 .961 r F P 0 g s .93189 .71377 m .94455 .71051 L .96171 .78037 L closepath p .523 0 0 r F P 0 g s .91811 .72162 m .93189 .71377 L .96171 .78037 L closepath p .696 0 0 r F P 0 g s .91112 .72955 m .91811 .72162 L .96171 .78037 L closepath p .83 0 0 r F P 0 g s .8435 .73208 m .85049 .72775 L .89267 .77539 L closepath p .801 0 0 r F P 0 g s .63957 .50949 m .65356 .50899 L .6271 .52208 L p .575 .914 .97 r F P 0 g s .6271 .52208 m .61282 .51977 L .63957 .50949 L p .575 .914 .97 r F P 0 g s .82911 .41044 m .83993 .39378 L .81284 .41319 L p 0 0 0 r F P 0 g s .81284 .41319 m .80153 .42715 L .82911 .41044 L p 0 0 0 r F P 0 g s .57148 .52473 m .58635 .52995 L .56017 .54001 L p .587 .838 .992 r F P 0 g s .56017 .54001 m .54507 .53196 L .57148 .52473 L p .587 .838 .992 r F P 0 g s .73453 .46892 m .74709 .46012 L .72023 .47639 L p .335 .791 .658 r F P 0 g s .72023 .47639 m .70727 .48241 L .73453 .46892 L p .335 .791 .658 r F P 0 g s .21017 .33286 m .22778 .36611 L .20465 .35416 L p .271 .405 .815 r F P 0 g s .20465 .35416 m .18721 .31828 L .21017 .33286 L p .271 .405 .815 r F P 0 g s .4614 .51807 m .47745 .53225 L .45185 .53643 L p .521 .69 .945 r F P 0 g s .45185 .53643 m .4357 .51941 L .4614 .51807 L p .521 .69 .945 r F P 0 g s .94455 .71051 m .94872 .71378 L .96171 .78037 L closepath p 0 0 0 r F P 0 g s .91506 .73296 m .91112 .72955 L .96171 .78037 L closepath p 0 0 0 r F P 0 g s .92779 .72975 m .91506 .73296 L .96171 .78037 L closepath p .544 0 0 r F P 0 g s .9418 .72177 m .92779 .72975 L .96171 .78037 L closepath p .751 0 0 r F P 0 g s .94872 .71378 m .9418 .72177 L .96171 .78037 L closepath p .942 0 0 r F P 0 g s .87225 .71421 m .87442 .71283 L .89267 .77539 L closepath p 0 0 0 r F P 0 g s .15169 .24514 m .16956 .28194 L .14726 .26509 L p .158 .316 .774 r F P 0 g s .14726 .26509 m .12963 .22574 L .15169 .24514 L p .158 .316 .774 r F P 0 g s .25134 .37829 m .26884 .4084 L .24514 .39892 L p .311 .448 .838 r F P 0 g s .24514 .39892 m .22778 .36611 L .25134 .37829 L p .311 .448 .838 r F P 0 g s .8455 .73081 m .8435 .73208 L .89267 .77539 L closepath p 0 0 0 r F P 0 g s 1 0 0 r .005 w .70858 .48674 m .87577 .74889 L s .5 Mabswid .66662 .49909 m .68028 .49576 L .65356 .50899 L p .485 .898 .904 r F P 0 g s .65356 .50899 m .63957 .50949 L .66662 .49909 L p .485 .898 .904 r F P 0 g s .41924 .50208 m .4357 .51941 L .41035 .52073 L p .477 .638 .926 r F P 0 g s .41035 .52073 m .39383 .50057 L .41924 .50208 L p .477 .638 .926 r F P 0 g s .52965 .52374 m .54507 .53196 L .51898 .5391 L p .543 .778 .985 r F P 0 g s .51898 .5391 m .50338 .52802 L .52965 .52374 L p .543 .778 .985 r F P 0 g s 1 0 0 r .005 w .89668 .66058 m .93271 .72695 L s .5 Mabswid .29296 .41805 m .31028 .44498 L .28608 .43808 L p .351 .492 .861 r F P 0 g s .28608 .43808 m .26884 .4084 L .29296 .41805 L p .351 .492 .861 r F P 0 g s .59819 .51741 m .61282 .51977 L .58635 .52995 L p .545 .867 .995 r F P 0 g s .58635 .52995 m .57148 .52473 L .59819 .51741 L p .545 .867 .995 r F P 0 g s .76206 .45529 m .77418 .44371 L .74709 .46012 L p 0 0 0 r F P 0 g s .74709 .46012 m .73453 .46892 L .76206 .45529 L p 0 0 0 r F P 0 g s .37703 .48004 m .39383 .50057 L .3688 .49908 L p .434 .587 .905 r F P 0 g s .3688 .49908 m .35199 .47574 L .37703 .48004 L p .434 .587 .905 r F P 0 g s .33489 .45201 m .35199 .47574 L .32734 .4715 L p .392 .539 .883 r F P 0 g s .32734 .4715 m .31028 .44498 L .33489 .45201 L p .392 .539 .883 r F P 0 g s .19233 .29915 m .21017 .33286 L .18721 .31828 L p .197 .358 .8 r F P 0 g s .18721 .31828 m .16956 .28194 L .19233 .29915 L p .197 .358 .8 r F P 0 g s .48746 .51671 m .50338 .52802 L .47745 .53225 L p .498 .718 .97 r F P 0 g s .47745 .53225 m .4614 .51807 L .48746 .51671 L p .498 .718 .97 r F P 0 g s .87442 .71283 m .86748 .71713 L .89267 .77539 L closepath p .908 0 0 r F P 0 g s .69397 .48858 m .70727 .48241 L .68028 .49576 L p .353 .824 .773 r F P 0 g s .68028 .49576 m .66662 .49909 L .69397 .48858 L p .353 .824 .773 r F P 0 g s .85545 .72461 m .8455 .73081 L .89267 .77539 L closepath p .576 0 0 r F P 0 g s .86748 .71713 m .85545 .72461 L .89267 .77539 L closepath p .738 0 0 r F P 0 g s .78987 .44153 m .80153 .42715 L .77418 .44371 L p 0 0 0 r F P 0 g s .77418 .44371 m .76206 .45529 L .78987 .44153 L p 0 0 0 r F P 0 g s .23359 .34773 m .25134 .37829 L .22778 .36611 L p .237 .403 .826 r F P 0 g s .22778 .36611 m .21017 .33286 L .23359 .34773 L p .237 .403 .826 r F P 0 g s .55626 .51939 m .57148 .52473 L .54507 .53196 L p .506 .808 .999 r F P 0 g s .54507 .53196 m .52965 .52374 L .55626 .51939 L p .506 .808 .999 r F P 0 g s .62524 .51001 m .63957 .50949 L .61282 .51977 L p .47 .871 .961 r F P 0 g s .61282 .51977 m .59819 .51741 L .62524 .51001 L p .47 .871 .961 r F P 0 g s .44503 .50361 m .4614 .51807 L .4357 .51941 L p .452 .659 .95 r F P 0 g s .4357 .51941 m .41924 .50208 L .44503 .50361 L p .452 .659 .95 r F P 0 g s .27536 .3907 m .29296 .41805 L .26884 .4084 L p .278 .449 .851 r F P 0 g s .26884 .4084 m .25134 .37829 L .27536 .3907 L p .278 .449 .851 r F P 0 g s .40247 .48441 m .41924 .50208 L .39383 .50057 L p .407 .603 .927 r F P 0 g s .39383 .50057 m .37703 .48004 L .40247 .48441 L p .407 .603 .927 r F P 0 g s .31751 .42788 m .33489 .45201 L .31028 .44498 L p .32 .498 .877 r F P 0 g s .31028 .44498 m .29296 .41805 L .31751 .42788 L p .32 .498 .877 r F P 0 g s .17424 .26497 m .19233 .29915 L .16956 .28194 L p .114 .304 .778 r F P 0 g s .16956 .28194 m .15169 .24514 L .17424 .26497 L p .114 .304 .778 r F P 0 g s .35993 .45915 m .37703 .48004 L .35199 .47574 L p .363 .549 .902 r F P 0 g s .35199 .47574 m .33489 .45201 L .35993 .45915 L p .363 .549 .902 r F P 0 g s .72163 .47795 m .73453 .46892 L .70727 .48241 L p 0 0 0 r F P 0 g s .70727 .48241 m .69397 .48858 L .72163 .47795 L p 0 0 0 r F P 0 g s .81797 .42762 m .82911 .41044 L .80153 .42715 L p .073 0 0 r F P 0 g s .80153 .42715 m .78987 .44153 L .81797 .42762 L p .073 0 0 r F P 0 g s .51389 .51533 m .52965 .52374 L .50338 .52802 L p .461 .745 .99 r F P 0 g s .50338 .52802 m .48746 .51671 L .51389 .51533 L p .461 .745 .99 r F P 0 g s .65261 .50251 m .66662 .49909 L .63957 .50949 L p .356 .829 .867 r F P 0 g s .63957 .50949 m .62524 .51001 L .65261 .50251 L p .356 .829 .867 r F P 0 g s .91734 .76493 m .92 .77219 L .92183 .83799 L closepath p 0 0 0 r F P 0 g s .90403 .76378 m .91734 .76493 L .92183 .83799 L closepath p .475 0 0 r F P 0 g s .88784 .76933 m .90403 .76378 L .92183 .83799 L closepath p .685 0 0 r F P 0 g s .87807 .77838 m .88784 .76933 L .92183 .83799 L closepath p .836 0 0 r F P 0 g s .88045 .78572 m .87807 .77838 L .92183 .83799 L closepath p .825 0 0 r F P 0 g s .89377 .787 m .88045 .78572 L .92183 .83799 L closepath p .412 0 0 r F P 0 g s .91024 .78137 m .89377 .787 L .92183 .83799 L closepath p .715 0 0 r F P 0 g s .92 .77219 m .91024 .78137 L .92183 .83799 L closepath p .925 0 0 r F P 0 g s .58322 .51499 m .59819 .51741 L .57148 .52473 L p .442 .824 .989 r F P 0 g s .57148 .52473 m .55626 .51939 L .58322 .51499 L p .442 .824 .989 r F P 0 g s 1 0 0 r .005 w .33492 .46025 m .3351 .46002 L s .5 Mabswid .21557 .31672 m .23359 .34773 L .21017 .33286 L p .154 .35 .806 r F P 0 g s .21017 .33286 m .19233 .29915 L .21557 .31672 L p .154 .35 .806 r F P 0 g s .7097 .7814 m .71724 .7733 L .72717 .82521 L closepath p .358 0 0 r F P 0 g s .6976 .7871 m .7097 .7814 L .72717 .82521 L closepath p .606 0 0 r F P 0 g s .68793 .78711 m .6976 .7871 L .72717 .82521 L closepath p .761 0 0 r F P 0 g s .47121 .50517 m .48746 .51671 L .4614 .51807 L p .415 .681 .972 r F P 0 g s .4614 .51807 m .44503 .50361 L .47121 .50517 L p .415 .681 .972 r F P 0 g s .74959 .4672 m .76206 .45529 L .73453 .46892 L p 0 0 0 r F P 0 g s .73453 .46892 m .72163 .47795 L .74959 .4672 L p 0 0 0 r F P 0 g s .25748 .36291 m .27536 .3907 L .25134 .37829 L p .195 .398 .835 r F P 0 g s .25134 .37829 m .23359 .34773 L .25748 .36291 L p .195 .398 .835 r F P 0 g s .42833 .48886 m .44503 .50361 L .41924 .50208 L p .369 .618 .948 r F P 0 g s .41924 .50208 m .40247 .48441 L .42833 .48886 L p .369 .618 .948 r F P 0 g s .71724 .7733 m .71579 .76749 L .72717 .82521 L closepath p .74 0 0 r F P 0 g s .5407 .51394 m .55626 .51939 L .52965 .52374 L p .404 .763 .994 r F P 0 g s .52965 .52374 m .51389 .51533 L .5407 .51394 L p .404 .763 .994 r F P 0 g s .29984 .40334 m .31751 .42788 L .29296 .41805 L p .236 .449 .864 r F P 0 g s .29296 .41805 m .27536 .3907 L .29984 .40334 L p .236 .449 .864 r F P 0 g s .68634 .78136 m .68793 .78711 L .72717 .82521 L closepath p 0 0 0 r F P 0 g s .68031 .49492 m .69397 .48858 L .66662 .49909 L p .208 .728 .708 r F P 0 g s .66662 .49909 m .65261 .50251 L .68031 .49492 L p .208 .728 .708 r F P 0 g s .38539 .46642 m .40247 .48441 L .37703 .48004 L p .324 .559 .921 r F P 0 g s .37703 .48004 m .35993 .45915 L .38539 .46642 L p .324 .559 .921 r F P 0 g s .34252 .43789 m .35993 .45915 L .33489 .45201 L p .279 .502 .893 r F P 0 g s .33489 .45201 m .31751 .42788 L .34252 .43789 L p .279 .502 .893 r F P 0 g s .61054 .51054 m .62524 .51001 L .59819 .51741 L p .343 .806 .931 r F P 0 g s .59819 .51741 m .58322 .51499 L .61054 .51054 L p .343 .806 .931 r F P 0 g s .77787 .45633 m .78987 .44153 L .76206 .45529 L p .072 0 0 r F P 0 g s .76206 .45529 m .74959 .4672 L .77787 .45633 L p .072 0 0 r F P 0 g s .19729 .28525 m .21557 .31672 L .19233 .29915 L p .062 .29 .778 r F P 0 g s .19233 .29915 m .17424 .26497 L .19729 .28525 L p .062 .29 .778 r F P 0 g s .49779 .50675 m .51389 .51533 L .48746 .51671 L p .361 .697 .983 r F P 0 g s .48746 .51671 m .47121 .50517 L .49779 .50675 L p .361 .697 .983 r F P 0 g s .71579 .76749 m .70611 .76743 L .72717 .82521 L closepath p .853 0 0 r F P 0 g s .69386 .77319 m .68634 .78136 L .72717 .82521 L closepath p .521 0 0 r F P 0 g s .23932 .33467 m .25748 .36291 L .23359 .34773 L p .101 .338 .809 r F P 0 g s .23359 .34773 m .21557 .31672 L .23932 .33467 L p .101 .338 .809 r F P 0 g s .70836 .48724 m .72163 .47795 L .69397 .48858 L p 0 0 0 r F P 0 g s .69397 .48858 m .68031 .49492 L .70836 .48724 L p 0 0 0 r F P 0 g s .56789 .51252 m .58322 .51499 L .55626 .51939 L p .316 .759 .963 r F P 0 g s .55626 .51939 m .5407 .51394 L .56789 .51252 L p .316 .759 .963 r F P 0 g s .45461 .49338 m .47121 .50517 L .44503 .50361 L p .316 .63 .962 r F P 0 g s .44503 .50361 m .42833 .48886 L .45461 .49338 L p .316 .63 .962 r F P 0 g s .63822 .50602 m .65261 .50251 L .62524 .51001 L p .207 .734 .806 r F P 0 g s .62524 .51001 m .61054 .51054 L .63822 .50602 L p .207 .734 .806 r F P 0 g s .80647 .44534 m .81797 .42762 L .78987 .44153 L p .166 0 0 r F P 0 g s .78987 .44153 m .77787 .45633 L .80647 .44534 L p .166 0 0 r F P 0 g s 1 0 0 r .005 w .78445 .45381 m .89668 .66058 L s .78076 .44702 m .7808 .44709 L s .5 Mabswid .70611 .76743 m .69386 .77319 L .72717 .82521 L closepath p .69 0 0 r F P 0 g s 1 0 0 r .005 w .59488 .57519 m .69849 .77101 L s .5 Mabswid .28186 .37839 m .29984 .40334 L .27536 .3907 L p .142 .39 .841 r F P 0 g s .27536 .3907 m .25748 .36291 L .28186 .37839 L p .142 .39 .841 r F P 0 g s .4113 .47381 m .42833 .48886 L .40247 .48441 L p .271 .565 .935 r F P 0 g s .40247 .48441 m .38539 .46642 L .4113 .47381 L p .271 .565 .935 r F P 0 g s .32479 .41624 m .34252 .43789 L .31751 .42788 L p .184 .445 .873 r F P 0 g s .31751 .42788 m .29984 .40334 L .32479 .41624 L p .184 .445 .873 r F P 0 g s .36798 .44808 m .38539 .46642 L .35993 .45915 L p .227 .503 .904 r F P 0 g s .35993 .45915 m .34252 .43789 L .36798 .44808 L p .227 .503 .904 r F P 0 g s .52478 .50835 m .5407 .51394 L .51389 .51533 L p .281 .699 .97 r F P 0 g s .51389 .51533 m .49779 .50675 L .52478 .50835 L p .281 .699 .97 r F P 0 g s .73675 .47947 m .74959 .4672 L .72163 .47795 L p .075 0 0 r F P 0 g s .72163 .47795 m .70836 .48724 L .73675 .47947 L p .075 0 0 r F P 0 g s .59548 .51108 m .61054 .51054 L .58322 .51499 L p .194 .713 .875 r F P 0 g s .58322 .51499 m .56789 .51252 L .59548 .51108 L p .194 .713 .875 r F P 0 g s .66628 .50144 m .68031 .49492 L .65261 .50251 L p 0 0 0 r F P 0 g s .65261 .50251 m .63822 .50602 L .66628 .50144 L p 0 0 0 r F P 0 g s .22086 .30597 m .23932 .33467 L .21557 .31672 L p 0 .27 .773 r F P 0 g s .21557 .31672 m .19729 .28525 L .22086 .30597 L p 0 .27 .773 r F P 0 g s .48132 .49797 m .49779 .50675 L .47121 .50517 L p .241 .631 .959 r F P 0 g s .47121 .50517 m .45461 .49338 L .48132 .49797 L p .241 .631 .959 r F P 0 g s .26357 .353 m .28186 .37839 L .25748 .36291 L p .037 .321 .806 r F P 0 g s .25748 .36291 m .23932 .33467 L .26357 .353 L p .037 .321 .806 r F P 0 g s .43766 .48133 m .45461 .49338 L .42833 .48886 L p .199 .563 .936 r F P 0 g s .42833 .48886 m .4113 .47381 L .43766 .48133 L p .199 .563 .936 r F P 0 g s .7655 .4716 m .77787 .45633 L .74959 .4672 L p .176 0 0 r F P 0 g s .74959 .4672 m .73675 .47947 L .7655 .4716 L p .176 0 0 r F P 0 g s .30674 .39419 m .32479 .41624 L .29984 .40334 L p .076 .376 .84 r F P 0 g s .29984 .40334 m .28186 .37839 L .30674 .39419 L p .076 .376 .84 r F P 0 g s .39392 .45846 m .4113 .47381 L .38539 .46642 L p .157 .497 .907 r F P 0 g s .38539 .46642 m .36798 .44808 L .39392 .45846 L p .157 .497 .907 r F P 0 g s .55218 .50998 m .56789 .51252 L .5407 .51394 L p .17 .67 .912 r F P 0 g s .5407 .51394 m .52478 .50835 L .55218 .50998 L p .17 .67 .912 r F P 0 g s .35023 .42938 m .36798 .44808 L .34252 .43789 L p .116 .435 .874 r F P 0 g s .34252 .43789 m .32479 .41624 L .35023 .42938 L p .116 .435 .874 r F P 0 g s .69471 .4968 m .70836 .48724 L .68031 .49492 L p .081 0 0 r F P 0 g s .68031 .49492 m .66628 .50144 L .69471 .4968 L p .081 0 0 r F P 0 g s .62346 .50962 m .63822 .50602 L .61054 .51054 L p .049 .616 .721 r F P 0 g s .61054 .51054 m .59548 .51108 L .62346 .50962 L p .049 .616 .721 r F P 0 g s .78655 .8217 m .7855 .82959 L .77551 .89106 L closepath p 0 0 0 r F P 0 g s .77558 .81706 m .78655 .8217 L .77551 .89106 L closepath p .404 0 0 r F P 0 g s .75908 .81829 m .77558 .81706 L .77551 .89106 L closepath p .66 0 0 r F P 0 g s .74652 .82467 m .75908 .81829 L .77551 .89106 L closepath p .826 0 0 r F P 0 g s .7452 .83255 m .74652 .82467 L .77551 .89106 L closepath p .962 0 0 r F P 0 g s .75609 .83732 m .7452 .83255 L .77551 .89106 L closepath p .241 0 0 r F P 0 g s .77286 .8361 m .75609 .83732 L .77551 .89106 L closepath p .651 0 0 r F P 0 g s .7855 .82959 m .77286 .8361 L .77551 .89106 L closepath p .87 0 0 r F P 0 g s .50848 .50264 m .52478 .50835 L .49779 .50675 L p .139 .611 .923 r F P 0 g s .49779 .50675 m .48132 .49797 L .50848 .50264 L p .139 .611 .923 r F P 0 g s .48809 .84408 m .50228 .8403 L .49615 .87456 L closepath p .489 0 0 r F P 0 g s .47525 .83948 m .48809 .84408 L .49615 .87456 L closepath p .752 0 0 r F P 0 g s .79461 .46362 m .80647 .44534 L .77787 .45633 L p .25 0 0 r F P 0 g s .77787 .45633 m .7655 .4716 L .79461 .46362 L p .25 0 0 r F P 0 g s .50228 .8403 m .50959 .83032 L .49615 .87456 L closepath p 0 0 0 r F P 0 g s .24497 .32718 m .26357 .353 L .23932 .33467 L p 0 .244 .759 r F P 0 g s .23932 .33467 m .22086 .30597 L .24497 .32718 L p 0 .244 .759 r F P 0 g s .47123 .82914 m .47525 .83948 L .49615 .87456 L closepath p 0 0 0 r F P 0 g s .46449 .48899 m .48132 .49797 L .45461 .49338 L p .104 .546 .913 r F P 0 g s .45461 .49338 m .43766 .48133 L .46449 .48899 L p .104 .546 .913 r F P 0 g s .58002 .51164 m .59548 .51108 L .56789 .51252 L p .033 .597 .792 r F P 0 g s .56789 .51252 m .55218 .50998 L .58002 .51164 L p .033 .597 .792 r F P 0 g s .72353 .4921 m .73675 .47947 L .70836 .48724 L p .19 0 0 r F P 0 g s .70836 .48724 m .69471 .4968 L .72353 .4921 L p .19 0 0 r F P 0 g s .50959 .83032 m .5057 .81993 L .49615 .87456 L closepath p .997 0 0 r F P 0 g s .28836 .37174 m .30674 .39419 L .28186 .37839 L p 0 .297 .794 r F P 0 g s .28186 .37839 m .26357 .353 L .28836 .37174 L p 0 .297 .794 r F P 0 g s .65185 .50814 m .66628 .50144 L .63822 .50602 L p .093 0 0 r F P 0 g s .63822 .50602 m .62346 .50962 L .65185 .50814 L p .093 0 0 r F P 0 g s .42034 .46903 m .43766 .48133 L .4113 .47381 L p .067 .48 .891 r F P 0 g s .4113 .47381 m .39392 .45846 L .42034 .46903 L p .067 .48 .891 r F P 0 g s .33214 .41032 m .35023 .42938 L .32479 .41624 L p 0 .354 .828 r F P 0 g s .32479 .41624 m .30674 .39419 L .33214 .41032 L p 0 .354 .828 r F P 0 g s .37618 .44279 m .39392 .45846 L .36798 .44808 L p .031 .415 .861 r F P 0 g s .36798 .44808 m .35023 .42938 L .37618 .44279 L p .031 .415 .861 r F P 0 g s .47849 .81907 m .47123 .82914 L .49615 .87456 L closepath p .364 0 0 r F P 0 g s .53609 .50738 m .55218 .50998 L .52478 .50835 L p .011 .557 .834 r F P 0 g s .52478 .50835 m .50848 .50264 L .53609 .50738 L p .011 .557 .834 r F P 0 g s .5057 .81993 m .49281 .81524 L .49615 .87456 L closepath p .818 0 0 r F P 0 g s .49281 .81524 m .47849 .81907 L .49615 .87456 L closepath p .633 0 0 r F P 0 g s 1 0 0 r .005 w .46742 .6492 m .48873 .81633 L s .5 Mabswid .75274 .48733 m .7655 .4716 L .73675 .47947 L p .27 0 0 r F P 0 g s .73675 .47947 m .72353 .4921 L .75274 .48733 L p .27 0 0 r F P 0 g s .6083 .51332 m .62346 .50962 L .59548 .51108 L p .108 0 0 r F P 0 g s .59548 .51108 m .58002 .51164 L .6083 .51332 L p .108 0 0 r F P 0 g s .49179 .49678 m .50848 .50264 L .48132 .49797 L p 0 .503 .849 r F P 0 g s .48132 .49797 m .46449 .48899 L .49179 .49678 L p 0 .503 .849 r F P 0 g s .68067 .50664 m .69471 .4968 L .66628 .50144 L p .208 0 0 r F P 0 g s .66628 .50144 m .65185 .50814 L .68067 .50664 L p .208 0 0 r F P 0 g s .26963 .34887 m .28836 .37174 L .26357 .353 L p 0 .21 .733 r F P 0 g s .26357 .353 m .24497 .32718 L .26963 .34887 L p 0 .21 .733 r F P 0 g s .44727 .47981 m .46449 .48899 L .43766 .48133 L p 0 .443 .843 r F P 0 g s .43766 .48133 m .42034 .46903 L .44727 .47981 L p 0 .443 .843 r F P 0 g s .31369 .39089 m .33214 .41032 L .30674 .39419 L p 0 .263 .766 r F P 0 g s .30674 .39419 m .28836 .37174 L .31369 .39089 L p 0 .263 .766 r F P 0 g s .40265 .45646 m .42034 .46903 L .39392 .45846 L p 0 .381 .824 r F P 0 g s .39392 .45846 m .37618 .44279 L .40265 .45646 L p 0 .381 .824 r F P 0 g s .35808 .4268 m .37618 .44279 L .35023 .42938 L p 0 .321 .797 r F P 0 g s .35023 .42938 m .33214 .41032 L .35808 .4268 L p 0 .321 .797 r F P 0 g s .56417 .51221 m .58002 .51164 L .55218 .50998 L p .127 0 0 r F P 0 g s .55218 .50998 m .53609 .50738 L .56417 .51221 L p .127 0 0 r F P 0 g s .78236 .4825 m .79461 .46362 L .7655 .4716 L p .327 0 .007 r F P 0 g s 1 0 0 r .005 w .78853 .47299 m .90217 .78413 L s .5 Mabswid .7655 .4716 m .75274 .48733 L .78236 .4825 L p .327 0 .007 r F P 0 g s .63702 .51502 m .65185 .50814 L .62346 .50962 L p .229 0 0 r F P 0 g s .62346 .50962 m .6083 .51332 L .63702 .51502 L p .229 0 0 r F P 0 g s .70991 .50511 m .72353 .4921 L .69471 .4968 L p .293 0 0 r F P 0 g s .69471 .4968 m .68067 .50664 L .70991 .50511 L p .293 0 0 r F P 0 g s .27547 .83555 m .29151 .83481 L .25945 .86058 L closepath p .429 0 0 r F P 0 g s .2616 .82611 m .27547 .83555 L .25945 .86058 L closepath p .82 0 0 r F P 0 g s .51959 .50472 m .53609 .50738 L .50848 .50264 L p 0 .429 .732 r F P 0 g s .50848 .50264 m .49179 .49678 L .51959 .50472 L p 0 .429 .732 r F P 0 g s .55605 .80898 m .5706 .81158 L .55223 .87787 L closepath p .656 0 0 r F P 0 g s .54313 .81013 m .55605 .80898 L .55223 .87787 L closepath p .823 0 0 r F P 0 g s .25796 .81196 m .2616 .82611 L .25945 .86058 L closepath p .898 0 0 r F P 0 g s .29151 .83481 m .30041 .82428 L .25945 .86058 L closepath p .679 0 0 r F P 0 g s .5706 .81158 m .57839 .81643 L .55223 .87787 L closepath p .369 0 0 r F P 0 g s .57839 .81643 m .57479 .82074 L .55223 .87787 L closepath p .838 0 0 r F P 0 g s .53933 .81438 m .54313 .81013 L .55223 .87787 L closepath p .996 0 0 r F P 0 g s .54702 .81929 m .53933 .81438 L .55223 .87787 L closepath p .197 0 0 r F P 0 g s .56177 .82194 m .54702 .81929 L .55223 .87787 L closepath p .62 0 0 r F P 0 g s .57479 .82074 m .56177 .82194 L .55223 .87787 L closepath p .817 0 0 r F P 0 g s .47471 .49079 m .49179 .49678 L .46449 .48899 L p 0 .381 .752 r F P 0 g s .46449 .48899 m .44727 .47981 L .47471 .49079 L p 0 .381 .752 r F P 0 g s .59272 .51712 m .6083 .51332 L .58002 .51164 L p .252 0 0 r F P 0 g s .58002 .51164 m .56417 .51221 L .59272 .51712 L p .252 0 0 r F P 0 g s .26676 .80133 m .25796 .81196 L .25945 .86058 L closepath p .12 0 0 r F P 0 g s .08471 .6831 m .09845 .68969 L .04393 .71422 L closepath p .7 0 0 r F P 0 g s .07455 .66967 m .08471 .6831 L .04393 .71422 L closepath p .896 0 0 r F P 0 g s .29488 .37107 m .31369 .39089 L .28836 .37174 L p 0 .166 .688 r F P 0 g s .28836 .37174 m .26963 .34887 L .29488 .37107 L p 0 .166 .688 r F P 0 g s .30041 .82428 m .29688 .81006 L .25945 .86058 L closepath p .955 0 0 r F P 0 g s .16198 .67611 m .17424 .68304 L .1259 .72013 L closepath p .718 0 0 r F P 0 g s 1 0 0 r .005 w .17063 .59846 m .28068 .49791 L s .5 Mabswid .15196 .66588 m .16198 .67611 L .1259 .72013 L closepath p .874 0 0 r F P 0 g s .42965 .47042 m .44727 .47981 L .42034 .46903 L p 0 .328 .752 r F P 0 g s .42034 .46903 m .40265 .45646 L .42965 .47042 L p 0 .328 .752 r F P 0 g s .12558 .75917 m .1413 .76266 L .09158 .78349 L closepath p .569 0 0 r F P 0 g s .11341 .74609 m .12558 .75917 L .09158 .78349 L closepath p .886 0 0 r F P 0 g s .73959 .50356 m .75274 .48733 L .72353 .4921 L p .353 0 0 r F P 0 g s .72353 .4921 m .70991 .50511 L .73959 .50356 L p .353 0 0 r F P 0 g s .33959 .41047 m .35808 .4268 L .33214 .41032 L p 0 .218 .715 r F P 0 g s .33214 .41032 m .31369 .39089 L .33959 .41047 L p 0 .218 .715 r F P 0 g s .66621 .51676 m .68067 .50664 L .65185 .50814 L p .319 0 0 r F P 0 g s .65185 .50814 m .63702 .51502 L .66621 .51676 L p .319 0 0 r F P 0 g s .38456 .44362 m .40265 .45646 L .37618 .44279 L p 0 .272 .738 r F P 0 g s .37618 .44279 m .35808 .4268 L .38456 .44362 L p 0 .272 .738 r F P 0 g s .07386 .65715 m .07455 .66967 L .04393 .71422 L closepath p .988 0 0 r F P 0 g s .11185 .73098 m .11341 .74609 L .09158 .78349 L closepath p .995 0 0 r F P 0 g s .33512 .73976 m .34767 .74501 L .3126 .798 L closepath p .684 0 0 r F P 0 g s .32368 .73492 m .33512 .73976 L .3126 .798 L closepath p .842 0 0 r F P 0 g s .33512 .73976 m .34767 .74501 L .3126 .798 L closepath p .684 0 0 r F P 0 g s .32368 .73492 m .33512 .73976 L .3126 .798 L closepath p .842 0 0 r F P 0 g s .17424 .68304 m .18167 .68266 L .1259 .72013 L closepath p .192 0 0 r F P 0 g s .09845 .68969 m .10783 .68559 L .04393 .71422 L closepath p 0 0 0 r F P 0 g s .28292 .80052 m .26676 .80133 L .25945 .86058 L closepath p .61 0 0 r F P 0 g s .15 .65826 m .15196 .66588 L .1259 .72013 L closepath p .984 0 0 r F P 0 g s .34767 .74501 m .35409 .74764 L .3126 .798 L closepath p .352 0 0 r F P 0 g s .34767 .74501 m .35409 .74764 L .3126 .798 L closepath p .352 0 0 r F P 0 g s .1413 .76266 m .15146 .75447 L .09158 .78349 L closepath p .629 0 0 r F P 0 g s .29688 .81006 m .28292 .80052 L .25945 .86058 L closepath p .815 0 0 r F P 0 g s 1 0 0 r .005 w .28628 .80282 m .35627 .65214 L s .5 Mabswid .54789 .51279 m .56417 .51221 L .53609 .50738 L p .274 0 0 r F P 0 g s .53609 .50738 m .51959 .50472 L .54789 .51279 L p .274 0 0 r F P 0 g s .31998 .7333 m .32368 .73492 L .3126 .798 L closepath p .996 0 0 r F P 0 g s .31998 .7333 m .32368 .73492 L .3126 .798 L closepath p .996 0 0 r F P 0 g s .1219 .72264 m .11185 .73098 L .09158 .78349 L closepath p .015 0 0 r F P 0 g s .08315 .65287 m .07386 .65715 L .04393 .71422 L closepath p .149 0 0 r F P 0 g s .10783 .68559 m .10729 .67307 L .04393 .71422 L closepath p .883 0 0 r F P 0 g s .18167 .68266 m .17987 .6751 L .1259 .72013 L closepath p .877 0 0 r F P 0 g s 1 0 0 r .005 w .13483 .73945 m .28938 .58208 L s .5 Mabswid .62177 .52211 m .63702 .51502 L .6083 .51332 L p .347 0 0 r F P 0 g s .6083 .51332 m .59272 .51712 L .62177 .52211 L p .347 0 0 r F P 0 g s .50267 .50198 m .51959 .50472 L .49179 .49678 L p .295 0 0 r F P 0 g s .49179 .49678 m .47471 .49079 L .50267 .50198 L p .295 0 0 r F P 0 g s .15146 .75447 m .15003 .73929 L .09158 .78349 L closepath p .912 0 0 r F P 0 g s 1 0 0 r .005 w .12752 .7469 m .13483 .73945 L s .5 Mabswid .76971 .50199 m .78236 .4825 L .75274 .48733 L p .395 0 .048 r F P 0 g s .75274 .48733 m .73959 .50356 L .76971 .50199 L p .395 0 .048 r F P 0 g s .15736 .65775 m .15 .65826 L .1259 .72013 L closepath p .269 0 0 r F P 0 g s 1 0 0 r .005 w .1371 .6291 m .17063 .59846 L s .5 Mabswid .35409 .74764 m .35056 .74608 L .3126 .798 L closepath p .888 0 0 r F P 0 g s .35409 .74764 m .35056 .74608 L .3126 .798 L closepath p .888 0 0 r F P 0 g s .69587 .51852 m .70991 .50511 L .68067 .50664 L p .382 0 0 r F P 0 g s .68067 .50664 m .66621 .51676 L .69587 .51852 L p .382 0 0 r F P 0 g s .13775 .72606 m .1219 .72264 L .09158 .78349 L closepath p .642 0 0 r F P 0 g s .4572 .48465 m .47471 .49079 L .44727 .47981 L p 0 .255 .64 r F P 0 g s .44727 .47981 m .42965 .47042 L .4572 .48465 L p 0 .255 .64 r F P 0 g s .32631 .73589 m .31998 .7333 L .3126 .798 L closepath p .261 0 0 r F P 0 g s .32631 .73589 m .31998 .7333 L .3126 .798 L closepath p .261 0 0 r F P 0 g s 1 0 0 r .005 w .32485 .7692 m .44658 .48283 L s .5 Mabswid .32072 .39379 m .33959 .41047 L .31369 .39089 L p 0 .114 .622 r F P 0 g s .31369 .39089 m .29488 .37107 L .32072 .39379 L p 0 .114 .622 r F P 0 g s .09705 .65946 m .08315 .65287 L .04393 .71422 L closepath p .676 0 0 r F P 0 g s .41162 .4608 m .42965 .47042 L .40265 .45646 L p 0 .209 .646 r F P 0 g s .40265 .45646 m .38456 .44362 L .41162 .4608 L p 0 .209 .646 r F P 0 g s .15003 .73929 m .13775 .72606 L .09158 .78349 L closepath p .822 0 0 r F P 0 g s .36608 .43049 m .38456 .44362 L .35808 .4268 L p 0 .161 .638 r F P 0 g s .35808 .4268 m .33959 .41047 L .36608 .43049 L p 0 .161 .638 r F P 0 g s .10729 .67307 m .09705 .65946 L .04393 .71422 L closepath p .817 0 0 r F P 0 g s 1 0 0 r .005 w .09982 .66315 m .1371 .6291 L s .5 Mabswid .57672 .52102 m .59272 .51712 L .56417 .51221 L p .375 0 0 r F P 0 g s .56417 .51221 m .54789 .51279 L .57672 .52102 L p .375 0 0 r F P 0 g s .33903 .7412 m .32631 .73589 L .3126 .798 L closepath p .638 0 0 r F P 0 g s .35056 .74608 m .33903 .7412 L .3126 .798 L closepath p .796 0 0 r F P 0 g s .33903 .7412 m .32631 .73589 L .3126 .798 L closepath p .638 0 0 r F P 0 g s .35056 .74608 m .33903 .7412 L .3126 .798 L closepath p .796 0 0 r F P 0 g s .17987 .6751 m .16978 .66474 L .1259 .72013 L closepath p .803 0 0 r F P 0 g s 1 0 0 r .005 w .17015 .66512 m .33492 .46025 L s .5 Mabswid .16978 .66474 m .15736 .65775 L .1259 .72013 L closepath p .67 0 0 r F P 0 g s .65132 .52719 m .66621 .51676 L .63702 .51502 L p .414 0 0 r F P 0 g s .63702 .51502 m .62177 .52211 L .65132 .52719 L p .414 0 0 r F P 0 g s .72601 .52032 m .73959 .50356 L .70991 .50511 L p .425 0 0 r F P 0 g s .70991 .50511 m .69587 .51852 L .72601 .52032 L p .425 0 0 r F P 0 g s 1 0 0 r .005 w .71947 .51782 m .71951 .51804 L s .5 Mabswid .53119 .5134 m .54789 .51279 L .51959 .50472 L p .4 0 0 r F P 0 g s .51959 .50472 m .50267 .50198 L .53119 .5134 L p .4 0 0 r F P 0 g s .48532 .49918 m .50267 .50198 L .47471 .49079 L p .421 0 0 r F P 0 g s .47471 .49079 m .4572 .48465 L .48532 .49918 L p .421 0 0 r F P 0 g s .60607 .5294 m .62177 .52211 L .59272 .51712 L p .446 0 0 r F P 0 g s .59272 .51712 m .57672 .52102 L .60607 .5294 L p .446 0 0 r F P 0 g s 1 0 0 r .005 w .59019 .52152 m .59021 .52125 L s .5 Mabswid .43927 .47836 m .4572 .48465 L .42965 .47042 L p .439 0 0 r F P 0 g s .42965 .47042 m .41162 .4608 L .43927 .47836 L p .439 0 0 r F P 0 g s .34717 .41706 m .36608 .43049 L .33959 .41047 L p 0 .054 .533 r F P 0 g s .33959 .41047 m .32072 .39379 L .34717 .41706 L p 0 .054 .533 r F P 0 g s .39317 .45096 m .41162 .4608 L .38456 .44362 L p .453 0 0 r F P 0 g s .38456 .44362 m .36608 .43049 L .39317 .45096 L p .453 0 0 r F P 0 g s .68139 .53235 m .69587 .51852 L .66621 .51676 L p .458 0 0 r F P 0 g s .66621 .51676 m .65132 .52719 L .68139 .53235 L p .458 0 0 r F P 0 g s .75664 .52214 m .76971 .50199 L .73959 .50356 L p .455 .002 .085 r F P 0 g s .73959 .50356 m .72601 .52032 L .75664 .52214 L p .455 .002 .085 r F P 0 g s .56026 .52503 m .57672 .52102 L .54789 .51279 L p .477 0 0 r F P 0 g s .54789 .51279 m .53119 .5134 L .56026 .52503 L p .477 0 0 r F P 0 g s 1 0 0 r .005 w .58969 .52618 m .58984 .52476 L s .55826 .82131 m .58969 .52618 L s .5 Mabswid .51403 .51401 m .53119 .5134 L .50267 .50198 L p .504 0 0 r F P 0 g s .50267 .50198 m .48532 .49918 L .51403 .51401 L p .504 0 0 r F P 0 g s .63598 .53793 m .65132 .52719 L .62177 .52211 L p .493 0 0 r F P 0 g s .62177 .52211 m .60607 .5294 L .63598 .53793 L p .493 0 0 r F P 0 g s .71198 .53761 m .72601 .52032 L .69587 .51852 L p .487 .008 .043 r F P 0 g s 1 0 0 r .005 w .72081 .52672 m .76732 .8365 L s .5 Mabswid .69587 .51852 m .68139 .53235 L .71198 .53761 L p .487 .008 .043 r F P 0 g s .46752 .49631 m .48532 .49918 L .4572 .48465 L p .525 0 0 r F P 0 g s .4572 .48465 m .43927 .47836 L .46752 .49631 L p .525 0 0 r F P 0 g s .42088 .47191 m .43927 .47836 L .41162 .4608 L p .542 0 0 r F P 0 g s .41162 .4608 m .39317 .45096 L .42088 .47191 L p .542 0 0 r F P 0 g s .37427 .44089 m .39317 .45096 L .36608 .43049 L p .554 .009 0 r F P 0 g s .36608 .43049 m .34717 .41706 L .37427 .44089 L p .554 .009 0 r F P 0 g s .58992 .5369 m .60607 .5294 L .57672 .52102 L p .527 0 0 r F P 0 g s .57672 .52102 m .56026 .52503 L .58992 .5369 L p .527 0 0 r F P 0 g s .66645 .54663 m .68139 .53235 L .65132 .52719 L p .522 .018 0 r F P 0 g s .65132 .52719 m .63598 .53793 L .66645 .54663 L p .522 .018 0 r F P 0 g s .54334 .52916 m .56026 .52503 L .53119 .5134 L p .559 0 0 r F P 0 g s .53119 .5134 m .51403 .51401 L .54334 .52916 L p .559 0 0 r F P 0 g s 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%%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (z) show 69.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 1.000 setlinewidth grestore % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 283.875}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell["\<\ The gradients are all perpendicular to the surface\[Dash]just as \ normal as Beaver, Wally, June, and Ward.\ \>", "SmallText"] }, Closed]] }, Closed]], Cell["Tutorial Problem", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "T.3) The gradient and maximization and minimization:\nThe ", StyleBox["FindMinimum", "Input", FontColor->GrayLevel[0]], " instruction" }], "Subsection", CellTags->"VC.03.T3"], Cell["\<\ When you plot a surface, your eyes look for crest and dips. Tops of \ crests are the tops of the hills. Bottoms of dips are the deepest points in \ depressions that could collect rain water. Fancy folks call the tops of crests by the name \"local maxima.\" The same \ folks call the bottoms of the dips by the name \"local minima.\" Most folks, fancy and down-to-earth, are interested in the maximum which is \ located at the top of the very highest crest. The same folks are also \ interested in the minimum which is located at the bottom of the deepest \ dip.\ \>", "Text"], Cell["T.3.a.i) Tops of the crests", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "How do you know that if ", Cell[BoxData[ \({x\_0, y\_0, f[x\_0, y\_0]}\)]], " sits at the top of a crest on the surface ", Cell[BoxData[ \(z = f[x, y]\)]], ", then \n ", Cell[BoxData[ \(gradf[x\_0, y\_0] = {0, 0}\)]], "?" }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "If you are at ", Cell[BoxData[ \({x\_0, y\_0}\)]], " and you leave ", Cell[BoxData[ \({x\_0, y\_0}\)]], " in the direction of ", Cell[BoxData[ \(gradf[x\_0, y\_0\ ]\)]], ", then ", Cell[BoxData[ \(f[x, y]\)]], " initially goes UP unless \n ", Cell[BoxData[ \(gradf[x\_0, y\_0] = {0, 0}\)]], ".\nThe upshot:\nIf ", Cell[BoxData[ \(gradf[x\_0, y\_0]\)]], " is not ", Cell[BoxData[ \({0, 0}\)]], ", then ", Cell[BoxData[ \({x\_0, y\_0, f[x\_0, y\_0]}\)]], " cannot sit at the top of a crest on the surface ", Cell[BoxData[ \(z = f[x, y]\)]], ". In other words, if ", Cell[BoxData[ \({x\_0, y\_0, f[x\_0, y\_0]}\)]], " sits at the top of a crest on the surface ", Cell[BoxData[ \(z = f[x, y]\)]], ", then \n ", Cell[BoxData[ \(gradf[x\_0, y\_0] = {0, 0}\)]], "." }], "SmallText"] }, Closed]], Cell["T.3.a.ii) Bottoms of the dips", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "How do you know that if ", Cell[BoxData[ \({x\_0, y\_0, f[x\_0, y\_0]}\)]], " sits at the bottom of a dip on the surface ", Cell[BoxData[ \(z = f[x, y]\)]], ", then \n ", Cell[BoxData[ \(gradf[x\_0, y\_0] = {0, 0}\)]], "?" }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "If you are at ", Cell[BoxData[ \({x\_0, y\_0}\)]], " and you leave ", Cell[BoxData[ \({x\_0, y\_0}\)]], " in the direction of", Cell[BoxData[ \(\(-gradf[x\_0, y\_0]\)\)]], ", then ", Cell[BoxData[ \(f[x, y]\)]], " initially goes DOWN unless \n ", Cell[BoxData[ \(gradf[x\_0, y\_0] = {0, 0}\)]], ".\nThe upshot:\nIf ", Cell[BoxData[ \(gradf[x\_0, y\_0]\)]], " is not ", Cell[BoxData[ \({0, 0}\)]], ", then ", Cell[BoxData[ \({x\_0, y\_0, f[x\_0, y\_0]}\)]], " cannot sit at the bottom of a dip the surface ", Cell[BoxData[ \(z = f[x, y]\)]], ". In other words, if ", Cell[BoxData[ \({x\_0, y\_0, f[x\_0, y\_0]}\)]], " sits at the bottom of a dip on the surface ", Cell[BoxData[ \(z = f[x, y]\)]], ", then \n ", Cell[BoxData[ \(gradf[x\_0, y\_0] = {0, 0}\)]], "." }], "SmallText"] }, Closed]], Cell["T.3.b.i) Setting the gradient equal to 0", "Subsubsection"], Cell[TextData[{ "Find the maximizers and minimizers (if any) of\n ", Cell[BoxData[ \(f[x, y] = 2.1 x\^2 + \ 5.9\ y\^2 - \ 7.2 x + \ 3.3 y + \ 8.7\)]], ". " }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Look at the formula\n ", Cell[BoxData[ \(f[x, y] = 2.1 x\^2 + \ 5.9\ y\^2 - \ 7.2 x + \ 3.3 y + \ 8.7\)]], ". \nFor large ", Cell[BoxData[ \(\[LeftBracketingBar]x\[RightBracketingBar]\)]], " and ", Cell[BoxData[ \(\[LeftBracketingBar]y\[RightBracketingBar]\)]], ", the dominant terms \n ", Cell[BoxData[ \(2.1 x\^2 + \ 5.9\ y\^2\)]], "\nmake ", Cell[BoxData[ \(f[x, y]\)]], " really huge. This means ", Cell[BoxData[ \(f[x, y]\)]], " has no maximum value. It also means that there is no way for ", Cell[BoxData[ \(f[x, y]\)]], " to ever get near ", Cell[BoxData[ \(\(-\[Infinity]\)\)]], ". So ", Cell[BoxData[ \(f[x, y]\)]], " has a minimum (at the bottom of the deepest dip) and the minimizer must \ be a point at which ", Cell[BoxData[ \(gradf[x, y] = 0\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, y, f, gradf];\)\), "\n", \(\(f[x_, y_] = 2.1\ x\^2 + 5.9\ y\^2 - 7.2\ x + 3.3\ y + 8.7;\)\), "\n", \(\(gradf[x_, y_] = {D[f[x, y], \ x], D[f[x, y], \ y]};\)\n\), "\n", \(Solve[gradf[x, y] == {0, 0}, {x, y}]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \({{x \[Rule] 1.7142857142857142`, y \[Rule] \(-0.2796610169491525`\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "The minimizer is ", Cell[BoxData[ \({1.71429, \(-0.279661\)}\)]], " and the minimum value is ", Cell[BoxData[ \(f[1.71429, \(-0.279661\)]\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(f @@ {1.71429, \(-0.279661\)}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(2.0671307506439005`\)], "Output"] }, Open ]], Cell["Take a look:", "SmallText"], 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.27827 L .7175 .29953 L p .402 .669 .969 r F P 0 g s .37913 .28839 m .33965 .31906 L .31231 .30659 L p .781 .857 .907 r F P 0 g s .27257 .34345 m .31231 .30659 L .33965 .31906 L p .808 .852 .882 r F P 0 g s .69615 .27827 m .65372 .27384 L .63137 .25423 L p .376 .676 .976 r F P 0 g s .5896 .25594 m .63137 .25423 L .65372 .27384 L p .485 .73 .979 r F P 0 g s .23295 .38595 m .19333 .43414 L .16331 .4257 L p .85 .851 .843 r F P 0 g s .52489 .24587 m .48437 .25937 L .45931 .24374 L p .633 .823 .973 r F P 0 g s .41898 .26326 m .45931 .24374 L .48437 .25937 L p .696 .836 .95 r F P 0 g s .41898 .26326 m .37913 .28839 L .35228 .27533 L p .744 .858 .933 r F P 0 g s .31231 .30659 m .35228 .27533 L .37913 .28839 L p .781 .857 .907 r F P 0 g s .27257 .34345 m .23295 .38595 L .20352 .37667 L p .834 .86 .864 r F P 0 g s .16331 .4257 m .20352 .37667 L .23295 .38595 L p .85 .851 .843 r F P 0 g s .63137 .25423 m .5896 .25594 L .56617 .23808 L p .466 .742 .988 r F P 0 g s .52489 .24587 m .56617 .23808 L .5896 .25594 L p .563 .781 .981 r F P 0 g s .87129 .37373 m .82546 .3501 L .80541 .3272 L p .066 .442 .877 r F P 0 g s .82546 .3501 m .87129 .37373 L .8905 .39809 L p .115 .456 .886 r F P 0 g s .76077 .3102 m .80541 .3272 L .82546 .3501 L p .196 .529 .922 r F P 0 g s .9377 .4286 m .8905 .39809 L .87129 .37373 L p 0 .368 .835 r F P 0 g s .8905 .39809 m .9377 .4286 L .95612 .45437 L p .044 .386 .848 r F P 0 g s .31231 .30659 m .27257 .34345 L .24369 .33341 L p .813 .868 .887 r F P 0 g s .20352 .37667 m .24369 .33341 L .27257 .34345 L p .834 .86 .864 r F P 0 g s .80541 .3272 m .76077 .3102 L .7398 .28883 L p .152 .522 .917 r F P 0 g s .69615 .27827 m .7398 .28883 L .76077 .3102 L p .284 .604 .953 r F P 0 g s .45931 .24374 m .41898 .26326 L .3926 .2497 L p .694 .854 .959 r F P 0 g s .35228 .27533 m .3926 .2497 L .41898 .26326 L p .744 .858 .933 r F P 0 g s .56617 .23808 m .52489 .24587 L .50027 .22988 L p .551 .797 .99 r F P 0 g s .45931 .24374 m .50027 .22988 L .52489 .24587 L p .633 .823 .973 r F P 0 g s .7398 .28883 m .69615 .27827 L .67421 .2585 L p .246 .603 .953 r F P 0 g s .63137 .25423 m .67421 .2585 L .69615 .27827 L p .376 .676 .976 r F P 0 g s 1 0 0 r .01 w .52489 .49947 m .52829 .2432 L s .5 Mabswid .35228 .27533 m .31231 .30659 L .28396 .29586 L p .785 .874 .913 r F P 0 g s .24369 .33341 m .28396 .29586 L .31231 .30659 L p .813 .868 .887 r F P 0 g s .67421 .2585 m .63137 .25423 L .60835 .23615 L p .345 .681 .979 r F P 0 g s .56617 .23808 m .60835 .23615 L .63137 .25423 L p .466 .742 .988 r F P 0 g s .50027 .22988 m .45931 .24374 L .4334 .22973 L p .626 .84 .982 r F P 0 g s .3926 .2497 m .4334 .22973 L .45931 .24374 L p .694 .854 .959 r F P 0 g s .20352 .37667 m .16331 .4257 L .13213 .41918 L p .856 .864 .846 r F P 0 g s .3926 .2497 m .35228 .27533 L .32444 .264 L p .745 .876 .94 r F P 0 g s .28396 .29586 m .32444 .264 L .35228 .27533 L p .785 .874 .913 r F P 0 g s .60835 .23615 m .56617 .23808 L .54198 .2218 L p .443 .752 .994 r F P 0 g s .50027 .22988 m .54198 .2218 L .56617 .23808 L p .551 .797 .99 r F P 0 g s .24369 .33341 m .20352 .37667 L .17294 .36927 L p .84 .874 .868 r F P 0 g s .13213 .41918 m .17294 .36927 L .20352 .37667 L p .856 .864 .846 r F P 0 g s .91889 .4043 m .87129 .37373 L .85163 .35085 L p 0 .346 .816 r F P 0 g s .87129 .37373 m .91889 .4043 L .9377 .4286 L p 0 .368 .835 r F P 0 g s .80541 .3272 m .85163 .35085 L .87129 .37373 L p .066 .442 .877 r F P 0 g s .85163 .35085 m .80541 .3272 L .78485 .3058 L p .012 .425 .862 r F P 0 g s .7398 .28883 m .78485 .3058 L .80541 .3272 L p .152 .522 .917 r F P 0 g s .78485 .3058 m .7398 .28883 L .71826 .26897 L p .103 .511 .907 r F P 0 g s .67421 .2585 m .71826 .26897 L .7398 .28883 L p .246 .603 .953 r F P 0 g s .28396 .29586 m .24369 .33341 L .2137 .32521 L p .818 .883 .892 r F P 0 g s .17294 .36927 m .2137 .32521 L .24369 .33341 L p .84 .874 .868 r F P 0 g s .4334 .22973 m .3926 .2497 L .36527 .23784 L p .691 .872 .966 r F P 0 g s .32444 .264 m .36527 .23784 L .3926 .2497 L p .745 .876 .94 r F P 0 g s .54198 .2218 m .50027 .22988 L .47481 .21551 L p .535 .811 .997 r F P 0 g s .4334 .22973 m .47481 .21551 L .50027 .22988 L p .626 .84 .982 r F P 0 g s .71826 .26897 m .67421 .2585 L .6516 .24027 L p .204 .598 .947 r F P 0 g s .60835 .23615 m .6516 .24027 L .67421 .2585 L p .345 .681 .979 r F P 0 g s .32444 .264 m .28396 .29586 L .25453 .28695 L p .788 .891 .918 r F P 0 g s .2137 .32521 m .25453 .28695 L .28396 .29586 L p .818 .883 .892 r F P 0 g s .6516 .24027 m .60835 .23615 L .5846 .21965 L p .31 .683 .977 r F P 0 g s .54198 .2218 m .5846 .21965 L .60835 .23615 L p .443 .752 .994 r F P 0 g s .47481 .21551 m .4334 .22973 L .40656 .21743 L p .617 .857 .989 r F P 0 g s .36527 .23784 m .40656 .21743 L .4334 .22973 L p .691 .872 .966 r F P 0 g s .17294 .36927 m .13213 .41918 L .09969 .41469 L p .861 .877 .849 r F P 0 g s .36527 .23784 m .32444 .264 L .29556 .25447 L p .746 .894 .946 r F P 0 g s .25453 .28695 m .29556 .25447 L .32444 .264 L p .788 .891 .918 r F P 0 g s .5846 .21965 m .54198 .2218 L .51698 .20715 L p .417 .76 .995 r F P 0 g s .47481 .21551 m .51698 .20715 L .54198 .2218 L p .535 .811 .997 r F P 0 g s .89964 .38151 m .85163 .35085 L .83148 .32949 L p 0 .321 .79 r F P 0 g s .85163 .35085 m .89964 .38151 L .91889 .4043 L p 0 .346 .816 r F P 0 g s .78485 .3058 m .83148 .32949 L .85163 .35085 L p .012 .425 .862 r F P 0 g s .83148 .32949 m .78485 .3058 L .76372 .28593 L p 0 .404 .839 r F P 0 g s .71826 .26897 m .76372 .28593 L .78485 .3058 L p .103 .511 .907 r F P 0 g s .2137 .32521 m .17294 .36927 L .14114 .36385 L p .845 .889 .871 r F P 0 g s .09969 .41469 m .14114 .36385 L .17294 .36927 L p .861 .877 .849 r F P 0 g s .76372 .28593 m .71826 .26897 L .69609 .25067 L p .049 .495 .889 r F P 0 g s .6516 .24027 m .69609 .25067 L .71826 .26897 L p .204 .598 .947 r F P 0 g s .25453 .28695 m .2137 .32521 L .18251 .31896 L p .822 .899 .896 r F P 0 g s .14114 .36385 m .18251 .31896 L .2137 .32521 L p .845 .889 .871 r F P 0 g s .69609 .25067 m .6516 .24027 L .62829 .22364 L p .156 .589 .933 r F P 0 g s .5846 .21965 m .62829 .22364 L .6516 .24027 L p .31 .683 .977 r F P 0 g s .51698 .20715 m .47481 .21551 L .44845 .20284 L p .516 .823 1 r F P 0 g s .40656 .21743 m .44845 .20284 L .47481 .21551 L p .617 .857 .989 r F P 0 g s .40656 .21743 m .36527 .23784 L .3369 .22778 L p .687 .89 .972 r F P 0 g s .29556 .25447 m .3369 .22778 L .36527 .23784 L p .746 .894 .946 r F P 0 g s .62829 .22364 m .5846 .21965 L .56006 .2048 L p .27 .681 .969 r F P 0 g s .51698 .20715 m .56006 .2048 L .5846 .21965 L p .417 .76 .995 r F P 0 g s .29556 .25447 m .25453 .28695 L .22393 .27996 L p .79 .908 .922 r F P 0 g s .18251 .31896 m .22393 .27996 L .25453 .28695 L p .822 .899 .896 r F P 0 g s .44845 .20284 m .40656 .21743 L .37871 .20691 L p .605 .872 .992 r F P 0 g s .3369 .22778 m .37871 .20691 L .40656 .21743 L p .687 .89 .972 r F P 0 g s .87992 .36029 m .83148 .32949 L .81079 .30969 L p 0 .292 .756 r F P 0 g s .83148 .32949 m .87992 .36029 L .89964 .38151 L p 0 .321 .79 r F P 0 g s .76372 .28593 m .81079 .30969 L .83148 .32949 L p 0 .404 .839 r F P 0 g s .14114 .36385 m .09969 .41469 L .06587 .41232 L p .866 .89 .851 r F P 0 g s .56006 .2048 m .51698 .20715 L .4911 .1942 L p .385 .764 .991 r F P 0 g s .44845 .20284 m .4911 .1942 L .51698 .20715 L p .516 .823 1 r F P 0 g s .3369 .22778 m .29556 .25447 L .26552 .24683 L p .744 .911 .95 r F P 0 g s .22393 .27996 m .26552 .24683 L .29556 .25447 L p .79 .908 .922 r F P 0 g s .81079 .30969 m .76372 .28593 L .74199 .26765 L p 0 .378 .808 r F P 0 g s .69609 .25067 m .74199 .26765 L .76372 .28593 L p .049 .495 .889 r F P 0 g s .74199 .26765 m .69609 .25067 L .67322 .23399 L p 0 .474 .861 r F P 0 g s .62829 .22364 m .67322 .23399 L .69609 .25067 L p .156 .589 .933 r F P 0 g s .18251 .31896 m .14114 .36385 L .108 .36051 L p .849 .903 .873 r F P 0 g s .06587 .41232 m .108 .36051 L .14114 .36385 L p .866 .89 .851 r F P 0 g s .67322 .23399 m .62829 .22364 L .60421 .20867 L p .104 .575 .911 r F P 0 g s .56006 .2048 m .60421 .20867 L .62829 .22364 L p .27 .681 .969 r F P 0 g s .4911 .1942 m .44845 .20284 L .42111 .19195 L p .494 .833 .998 r F P 0 g s .37871 .20691 m .42111 .19195 L .44845 .20284 L p .605 .872 .992 r F P 0 g s .22393 .27996 m .18251 .31896 L .15002 .31475 L p .825 .915 .898 r F P 0 g s .108 .36051 m .15002 .31475 L .18251 .31896 L p .849 .903 .873 r F P 0 g s .37871 .20691 m .3369 .22778 L .30742 .21958 L p .68 .906 .975 r F P 0 g s .26552 .24683 m .30742 .21958 L .3369 .22778 L p .744 .911 .95 r F P 0 g s .60421 .20867 m .56006 .2048 L .53465 .19166 L p .226 .674 .952 r F P 0 g s .4911 .1942 m .53465 .19166 L .56006 .2048 L p .385 .764 .991 r F P 0 g s .26552 .24683 m .22393 .27996 L .19205 .27498 L p .79 .924 .925 r F P 0 g s .15002 .31475 m .19205 .27498 L .22393 .27996 L p .825 .915 .898 r F P 0 g s .85969 .34067 m .81079 .30969 L .78951 .29153 L p 0 .259 .714 r F P 0 g s .81079 .30969 m .85969 .34067 L .87992 .36029 L p 0 .292 .756 r F P 0 g s .74199 .26765 m .78951 .29153 L .81079 .30969 L p 0 .378 .808 r F P 0 g s .42111 .19195 m .37871 .20691 L .34976 .19825 L p .589 .885 .991 r F P 0 g s .30742 .21958 m .34976 .19825 L .37871 .20691 L p .68 .906 .975 r F P 0 g s .78951 .29153 m .74199 .26765 L .71958 .25102 L p 0 .348 .768 r F P 0 g s .67322 .23399 m .71958 .25102 L .74199 .26765 L p 0 .474 .861 r F P 0 g s .53465 .19166 m .4911 .1942 L .46425 .18303 L p .349 .764 .979 r F P 0 g s .42111 .19195 m .46425 .18303 L .4911 .1942 L p .494 .833 .998 r F P 0 g s .108 .36051 m .06587 .41232 L .03056 .41221 L p .871 .903 .852 r F P 0 g s .30742 .21958 m .26552 .24683 L .23424 .24118 L p .741 .927 .952 r F P 0 g s .19205 .27498 m .23424 .24118 L .26552 .24683 L p .79 .924 .925 r F P 0 g s .71958 .25102 m .67322 .23399 L .64961 .219 L p 0 .448 .825 r F P 0 g s .60421 .20867 m .64961 .219 L .67322 .23399 L p .104 .575 .911 r F P 0 g s .15002 .31475 m .108 .36051 L .0734 .35937 L p .853 .917 .875 r F P 0 g s .03056 .41221 m .0734 .35937 L .108 .36051 L p .871 .903 .852 r F P 0 g s .64961 .219 m .60421 .20867 L .57928 .19543 L p .048 .555 .879 r F P 0 g s .53465 .19166 m .57928 .19543 L .60421 .20867 L p .226 .674 .952 r F P 0 g s .46425 .18303 m .42111 .19195 L .39269 .18293 L p .466 .839 .99 r F P 0 g s .34976 .19825 m .39269 .18293 L .42111 .19195 L p .589 .885 .991 r F P 0 g s .19205 .27498 m .15002 .31475 L .1161 .3127 L p .827 .93 .9 r F P 0 g s .0734 .35937 m .1161 .3127 L .15002 .31475 L p .853 .917 .875 r F P 0 g s .34976 .19825 m .30742 .21958 L .27672 .21336 L p .67 .92 .975 r F P 0 g s .23424 .24118 m .27672 .21336 L .30742 .21958 L p .741 .927 .952 r F P 0 g s .57928 .19543 m .53465 .19166 L .50831 .18031 L p .176 .661 .926 r F P 0 g s .46425 .18303 m .50831 .18031 L .53465 .19166 L p .349 .764 .979 r F P 0 g s .23424 .24118 m .19205 .27498 L .15879 .27213 L p .789 .939 .926 r F P 0 g s .1161 .3127 m .15879 .27213 L .19205 .27498 L p .827 .93 .9 r F P 0 g s .78951 .29153 m .83889 .32273 L .85969 .34067 L p 0 .259 .714 r F P 0 g s .83889 .32273 m .78951 .29153 L .76759 .27505 L p 0 .224 .665 r F P 0 g s .71958 .25102 m .76759 .27505 L .78951 .29153 L p 0 .348 .768 r F P 0 g s .76759 .27505 m .71958 .25102 L .69645 .23611 L p 0 .314 .719 r F P 0 g s .64961 .219 m .69645 .23611 L .71958 .25102 L p 0 .448 .825 r F P 0 g s .39269 .18293 m .34976 .19825 L .31962 .19155 L p .57 .894 .986 r F P 0 g s .27672 .21336 m .31962 .19155 L .34976 .19825 L p .67 .92 .975 r F P 0 g s .50831 .18031 m .46425 .18303 L .43636 .17374 L p .308 .758 .959 r F P 0 g s .39269 .18293 m .43636 .17374 L .46425 .18303 L p .466 .839 .99 r F P 0 g s .69645 .23611 m .64961 .219 L .62517 .20577 L p 0 .417 .778 r F P 0 g s .57928 .19543 m .62517 .20577 L .64961 .219 L p .048 .555 .879 r F P 0 g s .27672 .21336 m .23424 .24118 L .20161 .23763 L p .735 .942 .951 r F P 0 g s .15879 .27213 m .20161 .23763 L .23424 .24118 L p .789 .939 .926 r F P 0 g s .62517 .20577 m .57928 .19543 L .55344 .18401 L p 0 .528 .837 r F P 0 g s .50831 .18031 m .55344 .18401 L .57928 .19543 L p .176 .661 .926 r F P 0 g s .43636 .17374 m .39269 .18293 L .36311 .17586 L p .434 .839 .974 r F P 0 g s .31962 .19155 m .36311 .17586 L .39269 .18293 L p .57 .894 .986 r F P 0 g s .31962 .19155 m .27672 .21336 L .24469 .20921 L p .657 .932 .97 r F P 0 g s .20161 .23763 m .24469 .20921 L .27672 .21336 L p .735 .942 .951 r F P 0 g s .55344 .18401 m .50831 .18031 L .48094 .17085 L p .124 .641 .889 r F P 0 g s .43636 .17374 m .48094 .17085 L .50831 .18031 L p .308 .758 .959 r F P 0 g s .76759 .27505 m .81747 .30653 L .83889 .32273 L p 0 .224 .665 r F P 0 g s .81747 .30653 m .76759 .27505 L .74496 .26034 L p 0 .187 .609 r F P 0 g s .69645 .23611 m .74496 .26034 L .76759 .27505 L p 0 .314 .719 r F P 0 g s .74496 .26034 m .69645 .23611 L .67252 .22299 L p 0 .278 .663 r F P 0 g s .62517 .20577 m .67252 .22299 L .69645 .23611 L p 0 .417 .778 r F P 0 g s .36311 .17586 m .31962 .19155 L .28819 .18693 L p .546 .899 .974 r F P 0 g s .24469 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x\)]], " \nFor large ", Cell[BoxData[ \(\[LeftBracketingBar]x\[RightBracketingBar]\)]], " and ", Cell[BoxData[ \(\[LeftBracketingBar]y\[RightBracketingBar]\)]], ", the dominant terms ", Cell[BoxData[ \(0.2 x\^4 + \ 0.1 y\^4\)]], " make ", Cell[BoxData[ \(f[x, y]\)]], " really huge. This means ", Cell[BoxData[ \(f[x, y]\)]], " has no maximum. It also means that there is no way for ", Cell[BoxData[ \(f[x, y]\)]], " to ever get near ", Cell[BoxData[ \(\(-\[Infinity]\)\)]], "; so ", Cell[BoxData[ \(f[x, y]\)]], " has a minimum and the minimizer must be a point at which \n ", Cell[BoxData[ \(gradf[x, y] = 0\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, y, f, gradf];\)\), "\n", \(\(f[x_, y_] = 0.2\ x\^4 + 0.1\ y\^4 + 8.2\ x\ y - 18.4\ x;\)\), "\n", \(\(gradf[x_, y_] = {D[f[x, y], \ x], D[f[x, y], \ y]};\)\n\), "\n", \(Solve[gradf[x, y] == {0, 0}, {x, y}]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \({{x \[Rule] \(-2.539056394131526`\) - 3.06993472367498`\ \[ImaginaryI], y \[Rule] \(-3.1628485309992334`\) + 2.969876688687498`\ \[ImaginaryI]}, {x \[Rule] \ \(-2.539056394131526`\) + 3.06993472367498`\ \[ImaginaryI], y \[Rule] \(-3.1628485309992334`\) - 2.969876688687498`\ \[ImaginaryI]}, {x \[Rule] \ \(-2.4586418554021536`\), y \[Rule] 3.6938822211491367`}, {x \[Rule] \(-0.6530389562491722`\) - 3.632281529065876`\ \[ImaginaryI], y \[Rule] \(-0.2506393141244161`\) - 4.2219831338616425`\ \[ImaginaryI]}, {x \[Rule] \ \(-0.6530389562491722`\) + 3.632281529065876`\ \[ImaginaryI], y \[Rule] \(-0.2506393141244161`\) + 4.2219831338616425`\ \[ImaginaryI]}, {x \[Rule] \ \(-0.5641428978789619`\), y \[Rule] 2.2614187848082286`}, {x \[Rule] \(\(2.6644764517609705`\)\(\ \[InvisibleSpace]\)\) - 1.6886255213598618`\ \[ImaginaryI], y \[Rule] \(\(2.622110884065714`\)\(\[InvisibleSpace]\)\) + 3.0390066876779054`\ \[ImaginaryI]}, {x \[Rule] \ \(\(2.6644764517609705`\)\(\[InvisibleSpace]\)\) + 1.6886255213598618`\ \[ImaginaryI], y \[Rule] \(\(2.622110884065714`\)\(\[InvisibleSpace]\)\) - 3.0390066876779054`\ \[ImaginaryI]}, {x \[Rule] 4.07802255052057`, y \[Rule] \(-4.372547083841495`\)}}\)], "Output"] }, Open ]], Cell["\<\ Tossing out the complex candidates gives three candidates for the \ minimizer: \ \>", "SmallText"], Cell[BoxData[{ \(\(candidate1 = {4.07802, \(-4.37255\)};\)\), "\n", \(\(candidate2 = {\(-0.564143\), 2.26142};\)\), "\n", \(\(candidate3 = {\(-2.45864\), 3.69388};\)\)}], "Input", AspectRatioFixed->True], Cell["Compare:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \({f @@ candidate1, f @@ candidate2, f @@ candidate3}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(-129.38522820248167`\), 2.554542269756692`, \(-3.3066694905755867`\)}\)], "Output"] }, Open ]], Cell["It's no contest. The minimizer is:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(candidate1\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({4.07802`, \(-4.37255`\)}\)], "Output"] }, Open ]], Cell["The minimum value is:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(f @@ candidate1\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-129.38522820248167`\)\)], "Output"] }, Open ]], Cell["Not much to it.", "SmallText"], Cell["T.3.c.i) Terrible gradients and the FindMinimum instruction", \ "Subsubsection"], Cell[TextData[{ "Find the maximizers and minimizers (if any) of\n ", Cell[BoxData[ \(f[x, y] = 1.2 e\^\(x\^2\) + \ 2.9 e\^\(y\^2\) - \ 8.2 x\ y\^2\)]], ".\nShow off your results with a plot." }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Look at the formula\n ", Cell[BoxData[ \(f[x, y] = 1.2 e\^\(x\^2\) + \ 2.9 e\^\(y\^2\) - \ 8.2 x\ y\^2\)]], ".\nFor large ", Cell[BoxData[ \(\[LeftBracketingBar]x\[RightBracketingBar]\)]], " and ", Cell[BoxData[ \(\[LeftBracketingBar]y\[RightBracketingBar]\)]], ", the dominant terms \n ", Cell[BoxData[ \(1.2 e\^\(x\^2\) + \ 2.9 e\^\(y\^2\)\)]], "\nmake ", Cell[BoxData[ \(f[x, y]\)]], " hugely positive. This means ", Cell[BoxData[ \(f[x, y]\)]], " has no maximum. It also means that there is no way for ", Cell[BoxData[ \(f[x, y]\)]], " to ever get near ", Cell[BoxData[ \(\(-\[Infinity]\)\)]], "; so ", Cell[BoxData[ \(f[x, y]\)]], " has a minimum value and the minimizer must be a point at which ", Cell[BoxData[ \(gradf[x, y] = 0\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, gradf, x, y];\)\), "\n", \(\(f[x_, y_] = 1.2\ E\^\(x\^2\) + 2.9\ E\^\(y\^2\) - 8.2\ x\ y\^2;\)\), "\n", \(gradf[x_, y_] = {D[f[x, y], \ x], D[f[x, y], \ y]}\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \({2.4`\ \[ExponentialE]\^\(x\^2\)\ x - 8.2`\ y\^2, 5.8`\ \[ExponentialE]\^\(y\^2\)\ y - 16.4`\ x\ y}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[gradf[x, y] == {0, 0}, {x, y}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(InverseFunction::"ifun" \(\(:\)\(\ \)\) "Inverse functions are being used. Values may be lost for multivalued \ inverses."\)], "Message"], Cell[BoxData[ \(InverseFunction::"ifun" \(\(:\)\(\ \)\) "Inverse functions are being used. Values may be lost for multivalued \ inverses."\)], "Message"], Cell[BoxData[ \(InverseFunction::"ifun" \(\(:\)\(\ \)\) "Inverse functions are being used. Values may be lost for multivalued \ inverses."\)], "Message"], Cell[BoxData[ \(General::"stop" \(\(:\)\(\ \)\) "Further output of \!\(InverseFunction :: \"ifun\"\) will be suppressed \ during this calculation."\)], "Message"], Cell[BoxData[ \(Solve::"tdep" \(\(:\)\(\ \)\) "The equations appear to involve the variables to be solved for in an \ essentially non-algebraic way."\)], "Message"], Cell[BoxData[ \(Solve[{2.4`\ \[ExponentialE]\^\(x\^2\)\ x - 8.2`\ y\^2, 5.8`\ \[ExponentialE]\^\(y\^2\)\ y - 16.4`\ x\ y} == {0, 0}, {x, y}]\)], "Output"] }, Open ]], Cell[TextData[{ "Dung. \n", StyleBox["Mathematica", FontSlant->"Italic"], " had to bail out.\nBut this is no reason for you to give up. Take another \ look at the formula:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(f[x, y]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1.2`\ \[ExponentialE]\^\(x\^2\) + 2.9`\ \[ExponentialE]\^\(y\^2\) - 8.2`\ x\ y\^2\)], "Output"] }, Open ]], Cell[TextData[{ "For the smaller values of ", Cell[BoxData[ \(\[LeftBracketingBar]x\[RightBracketingBar]\)]], " and ", Cell[BoxData[ \(\[LeftBracketingBar]y\[RightBracketingBar]\)]], ", the term ", Cell[BoxData[ \(8.2 x\ y\^2\)]], " has a lot of influence. 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