(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 57079, 1862]*) (*NotebookOutlinePosition[ 57876, 1890]*) (* CellTagsIndexPosition[ 57832, 1886]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["p. 519", "Subsection"], Cell[CellGroupData[{ Cell["87.", "Subsubsection"], Cell[BoxData[ \(f[x_]\ := \ ArcSin[\(x - 1\)\/\(x + 1\)]; \n g[x_]\ := \ 2 ArcTan[\@x]; \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[f[x], {x, 0, 10}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0952381 0.380186 0.232857 [ [.21429 .36769 -3 -9 ] [.21429 .36769 3 0 ] [.40476 .36769 -3 -9 ] [.40476 .36769 3 0 ] [.59524 .36769 -3 -9 ] [.59524 .36769 3 0 ] [.78571 .36769 -3 -9 ] [.78571 .36769 3 0 ] [.97619 .36769 -6 -9 ] [.97619 .36769 6 0 ] [.01131 .0309 -24 -4.5 ] [.01131 .0309 0 4.5 ] [.01131 .14733 -12 -4.5 ] [.01131 .14733 0 4.5 ] [.01131 .26376 -24 -4.5 ] [.01131 .26376 0 4.5 ] [.01131 .49661 -18 -4.5 ] [.01131 .49661 0 4.5 ] [.01131 .61304 -6 -4.5 ] [.01131 .61304 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .38019 m .21429 .38644 L s [(2)] .21429 .36769 0 1 Mshowa .40476 .38019 m .40476 .38644 L s [(4)] .40476 .36769 0 1 Mshowa .59524 .38019 m .59524 .38644 L s [(6)] .59524 .36769 0 1 Mshowa .78571 .38019 m .78571 .38644 L s [(8)] .78571 .36769 0 1 Mshowa .97619 .38019 m .97619 .38644 L s [(10)] .97619 .36769 0 1 Mshowa .125 Mabswid .07143 .38019 m .07143 .38394 L s .11905 .38019 m .11905 .38394 L s .16667 .38019 m .16667 .38394 L s .2619 .38019 m .2619 .38394 L s .30952 .38019 m .30952 .38394 L s .35714 .38019 m .35714 .38394 L s .45238 .38019 m .45238 .38394 L s .5 .38019 m .5 .38394 L s .54762 .38019 m .54762 .38394 L s .64286 .38019 m .64286 .38394 L s .69048 .38019 m .69048 .38394 L s .7381 .38019 m .7381 .38394 L s .83333 .38019 m .83333 .38394 L s .88095 .38019 m .88095 .38394 L s .92857 .38019 m .92857 .38394 L s .25 Mabswid 0 .38019 m 1 .38019 L s .02381 .0309 m .03006 .0309 L s [(-1.5)] .01131 .0309 1 0 Mshowa .02381 .14733 m .03006 .14733 L s [(-1)] .01131 .14733 1 0 Mshowa .02381 .26376 m .03006 .26376 L s [(-0.5)] .01131 .26376 1 0 Mshowa .02381 .49661 m .03006 .49661 L s [(0.5)] .01131 .49661 1 0 Mshowa .02381 .61304 m .03006 .61304 L s [(1)] .01131 .61304 1 0 Mshowa .125 Mabswid .02381 .05419 m .02756 .05419 L s .02381 .07747 m .02756 .07747 L s .02381 .10076 m .02756 .10076 L s .02381 .12404 m .02756 .12404 L s .02381 .17061 m .02756 .17061 L s .02381 .1939 m .02756 .1939 L s .02381 .21719 m .02756 .21719 L s .02381 .24047 m .02756 .24047 L s .02381 .28704 m .02756 .28704 L s .02381 .31033 m .02756 .31033 L s .02381 .33361 m .02756 .33361 L s .02381 .3569 m .02756 .3569 L s .02381 .40347 m .02756 .40347 L s .02381 .42676 m .02756 .42676 L s .02381 .45004 m .02756 .45004 L s .02381 .47333 m .02756 .47333 L s .02381 .5199 m .02756 .5199 L s .02381 .54319 m .02756 .54319 L s .02381 .56647 m .02756 .56647 L s .02381 .58976 m .02756 .58976 L s .02381 .00761 m .02756 .00761 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .01472 m .02499 .06595 L .02605 .08537 L .02729 .10238 L .02846 .11566 L .03279 .15314 L .0375 .18319 L .04262 .20917 L .05293 .24964 L .06244 .27853 L .0829 .32513 L .10458 .36102 L .14241 .40568 L .16182 .42313 L .1827 .43914 L .22393 .46472 L .26364 .48408 L .30183 .49935 L .34247 .51296 L .3816 .52413 L .42319 .5344 L .46325 .54305 L .5018 .55044 L .54281 .55747 L .58229 .56357 L .62027 .56891 L .66069 .57411 L .6996 .57871 L .74096 .58321 L .7808 .58722 L .81913 .59081 L .85992 .59437 L .89918 .59758 L .93693 .60048 L .97619 .60332 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgH0oooo000K0?ooo`030000003oool0oooo01T0oooo00<0 00000?ooo`3oool0iP3oool001/0oooo00<000000?ooo`3oool06P3oool00`000000oooo0?ooo`3U 0?ooo`006`3oool00`000000oooo0?ooo`0K0?ooo`030000003oool0oooo0>@0oooo000K0?ooo`80 00007@3oool00`000000oooo0?ooo`3S0?ooo`006`3oool00`000000oooo0?ooo`0M0?ooo`030000 003oool0oooo0>80oooo000K0?ooo`030000003oool0oooo01h0oooo00<000000?ooo`3oool0h@3o ool001/0oooo00<000000?ooo`3oool07`3oool200000>40oooo000K0?ooo`030000003oool0oooo 0240oooo00<000000?ooo`3oool0gP3oool001/0oooo00<000000?ooo`3oool08P3oool00`000000 oooo0?ooo`3M0?ooo`006`3oool2000002@0oooo00<000000?ooo`3oool0g03oool001/0oooo00<0 00000?ooo`3oool0903oool200000=`0oooo000K0?ooo`030000003oool0oooo02H0oooo00<00000 0?ooo`3oool0f@3oool001/0oooo00<000000?ooo`3oool09`3oool00`000000oooo0?ooo`3H0?oo o`006`3oool00`000000oooo0?ooo`0X0?ooo`800000f03oool001/0oooo00<000000?ooo`3oool0 :P3oool00`000000oooo0?ooo`3E0?ooo`006`3oool2000002`0oooo0P00003E0?ooo`006`3oool0 0`000000oooo0?ooo`0]0?ooo`800000d`3oool001/0oooo00<000000?ooo`3oool0;`3oool00`00 0000oooo0?ooo`3@0?ooo`006`3oool00`000000oooo0?ooo`0`0?ooo`800000d03oool001/0oooo 00<000000?ooo`3oool0P3oool200000/0oooo2P00 000=0?ooo`006`3oool00`000000oooo0?ooo`3e0?ooo`H000001`3oool001<0oooo100000040?oo o`030000003oool0oooo0?l0oooo0`3oool001D0oooo00<000000?ooo`3oool00`3oool200000?l0 oooo103oool001D0oooo00<000000?ooo`3oool00`3oool00`000000oooo0?ooo`3o0?ooo`<0oooo 000E0?ooo`030000003oool0oooo00<0oooo00<000000?ooo`3oool0o`3oool30?ooo`005@3oool0 0`000000oooo0?ooo`3o0?ooo`T0oooo000C0?ooo`<00000o`3oool;0?ooo`005@3oool00`000000 oooo0?ooo`3o0?ooo`T0oooo003o0?ooob40oooo003o0?ooob40oooo003o0?ooob40oooo003o0?oo ob40oooo0000\ \>"], ImageRangeCache->{{{91.5625, 320.938}, {328.312, 187}} -> {-5.60228, 2.07038, 0.0395151, 0.0161616}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[g[x], {x, 0, 10}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0952381 0.0147151 0.232738 [ [.21429 .00222 -3 -9 ] [.21429 .00222 3 0 ] [.40476 .00222 -3 -9 ] [.40476 .00222 3 0 ] [.59524 .00222 -3 -9 ] [.59524 .00222 3 0 ] [.78571 .00222 -3 -9 ] [.78571 .00222 3 0 ] [.97619 .00222 -6 -9 ] [.97619 .00222 6 0 ] [.01131 .13108 -18 -4.5 ] [.01131 .13108 0 4.5 ] [.01131 .24745 -6 -4.5 ] [.01131 .24745 0 4.5 ] [.01131 .36382 -18 -4.5 ] [.01131 .36382 0 4.5 ] [.01131 .48019 -6 -4.5 ] [.01131 .48019 0 4.5 ] [.01131 .59656 -18 -4.5 ] [.01131 .59656 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .01472 m .21429 .02097 L s [(2)] .21429 .00222 0 1 Mshowa .40476 .01472 m .40476 .02097 L s [(4)] .40476 .00222 0 1 Mshowa .59524 .01472 m .59524 .02097 L s [(6)] .59524 .00222 0 1 Mshowa .78571 .01472 m .78571 .02097 L s [(8)] .78571 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(10)] .97619 .00222 0 1 Mshowa .125 Mabswid .07143 .01472 m .07143 .01847 L s .11905 .01472 m .11905 .01847 L s .16667 .01472 m .16667 .01847 L s .2619 .01472 m .2619 .01847 L s .30952 .01472 m .30952 .01847 L s .35714 .01472 m .35714 .01847 L s .45238 .01472 m .45238 .01847 L s .5 .01472 m .5 .01847 L s .54762 .01472 m .54762 .01847 L s .64286 .01472 m .64286 .01847 L s .69048 .01472 m .69048 .01847 L s .7381 .01472 m .7381 .01847 L s .83333 .01472 m .83333 .01847 L s .88095 .01472 m .88095 .01847 L s .92857 .01472 m .92857 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .02381 .13108 m .03006 .13108 L s [(0.5)] .01131 .13108 1 0 Mshowa .02381 .24745 m .03006 .24745 L s [(1)] .01131 .24745 1 0 Mshowa .02381 .36382 m .03006 .36382 L s [(1.5)] .01131 .36382 1 0 Mshowa .02381 .48019 m .03006 .48019 L s [(2)] .01131 .48019 1 0 Mshowa .02381 .59656 m .03006 .59656 L s [(2.5)] .01131 .59656 1 0 Mshowa .125 Mabswid .02381 .03799 m .02756 .03799 L s .02381 .06126 m .02756 .06126 L s .02381 .08454 m .02756 .08454 L s .02381 .10781 m .02756 .10781 L s .02381 .15436 m .02756 .15436 L s .02381 .17763 m .02756 .17763 L s .02381 .20091 m .02756 .20091 L s .02381 .22418 m .02756 .22418 L s .02381 .27073 m .02756 .27073 L s .02381 .294 m .02756 .294 L s .02381 .31727 m .02756 .31727 L s .02381 .34055 m .02756 .34055 L s .02381 .3871 m .02756 .3871 L s .02381 .41037 m .02756 .41037 L s .02381 .43364 m .02756 .43364 L s .02381 .45692 m .02756 .45692 L s .02381 .50347 m .02756 .50347 L s .02381 .52674 m .02756 .52674 L s .02381 .55001 m .02756 .55001 L s .02381 .57329 m .02756 .57329 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .01502 m .02499 .06622 L .02605 .08563 L .02729 .10263 L .02846 .11591 L .03279 .15337 L .0375 .1834 L .04262 .20937 L .05293 .24982 L .06244 .2787 L .0829 .32527 L .10458 .36115 L .14241 .40578 L .16182 .42322 L .1827 .43922 L .22393 .46479 L .26364 .48414 L .30183 .4994 L .34247 .51301 L .3816 .52417 L .42319 .53443 L .46325 .54308 L .5018 .55046 L .54281 .55749 L .58229 .56359 L .62027 .56893 L .66069 .57413 L .6996 .57872 L .74096 .58322 L .7808 .58722 L .81913 .59081 L .85992 .59437 L .89918 .59758 L .93693 .60048 L .97619 .60332 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgh0oooo00030?ooo`030000003oool0oooo00L0oooo10000004 0?ooo`030000003oool0oooo01P0oooo00<000000?ooo`3oool0k@3oool001D0oooo0P00000I0?oo o`030000003oool0oooo0>d0oooo000E0?ooo`030000003oool0oooo01T0oooo00<000000?ooo`3o ool0k03oool001D0oooo00<000000?ooo`3oool06P3oool00`000000oooo0?ooo`3[0?ooo`005@3o ool00`000000oooo0?ooo`0K0?ooo`030000003oool0oooo0>X0oooo000E0?ooo`030000003oool0 oooo01/0oooo00<000000?ooo`3oool0jP3oool001D0oooo00<000000?ooo`3oool0703oool00`00 0000oooo0?ooo`3Y0?ooo`005@3oool00`000000oooo0?ooo`0M0?ooo`030000003oool0oooo0>P0 oooo000E0?ooo`8000007`3oool00`000000oooo0?ooo`3W0?ooo`005@3oool00`000000oooo0?oo o`0O0?ooo`800000i`3oool001D0oooo00<000000?ooo`3oool08@3oool00`000000oooo0?ooo`3T 0?ooo`005@3oool00`000000oooo0?ooo`0R0?ooo`030000003oool0oooo0><0oooo000E0?ooo`03 0000003oool0oooo02<0oooo00<000000?ooo`3oool0hP3oool001D0oooo00<000000?ooo`3oool0 903oool00`000000oooo0?ooo`3Q0?ooo`005@3oool2000002H0oooo0P00003Q0?ooo`005@3oool0 0`000000oooo0?ooo`0W0?ooo`030000003oool0oooo0=h0oooo000E0?ooo`030000003oool0oooo 02P0oooo00<000000?ooo`3oool0g@3oool001D0oooo00<000000?ooo`3oool0:@3oool200000=d0 oooo000E0?ooo`030000003oool0oooo02/0oooo00<000000?ooo`3oool0fP3oool001D0oooo00<0 00000?ooo`3oool0;03oool200000=X0oooo000E0?ooo`800000;`3oool200000=P0oooo000E0?oo o`030000003oool0oooo0300oooo00<000000?ooo`3oool0e@3oool001D0oooo00<000000?ooo`3o ool0<@3oool200000=D0oooo000E0?ooo`030000003oool0oooo03<0oooo0P00003C0?ooo`005@3o ool00`000000oooo0?ooo`0e0?ooo`800000d@3oool000d0oooo100000040?ooo`030000003oool0 oooo03L0oooo0P00003?0?ooo`003@3oool00`000000oooo0?ooo`050?ooo`030000003oool0oooo 03T0oooo0P00003=0?ooo`003P3oool00`000000oooo0?ooo`040?ooo`<00000>`3oool200000 0?ooo`8000001@3oool00`000000oooo0?ooo`150?ooo`<00000`03oool001D0oooo00<000000?oo o`3oool0B03oool300000;d0oooo000E0?ooo`800000C03oool300000;X0oooo000E0?ooo`030000 003oool0oooo04h0oooo0P00002h0?ooo`005@3oool00`000000oooo0?ooo`1@0?ooo`<00000]@3o ool001D0oooo00<000000?ooo`3oool0D`3oool300000;80oooo000E0?ooo`030000003oool0oooo 05H0oooo0`00002_0?ooo`005@3oool00`000000oooo0?ooo`1I0?ooo`@00000Z`3oool001D0oooo 0P00001N0?ooo`<00000Z03oool001D0oooo00<000000?ooo`3oool0H03oool400000:@0oooo000E 0?ooo`030000003oool0oooo06@0oooo1000002P0?ooo`005@3oool00`000000oooo0?ooo`1X0?oo o`D00000V`3oool001D0oooo00<000000?ooo`3oool0K@3oool5000009H0oooo000E0?ooo`030000 003oool0oooo0780oooo1@00002A0?ooo`005@3oool00`000000oooo0?ooo`1g0?ooo`H00000R`3o ool001D0oooo0P00001n0?ooo`D00000QP3oool001D0oooo00<000000?ooo`3oool0PP3oool60000 0800oooo000E0?ooo`030000003oool0oooo08P0oooo2000001h0?ooo`005@3oool00`000000oooo 0?ooo`2@0?ooo`P00000L03oool001D0oooo00<000000?ooo`3oool0V03oool5000006/0oooo000E 0?ooo`030000003oool0oooo09d0oooo2000001S0?ooo`005@3oool200000:H0oooo2000001K0?oo o`005@3oool00`000000oooo0?ooo`2]0?ooo`H00000E@3oool001D0oooo00<000000?ooo`3oool0 /`3oool8000004d0oooo000E0?ooo`030000003oool0oooo0;/0oooo2`0000120?ooo`005@3oool0 0`000000oooo0?ooo`360?ooo`X00000>03oool000030?ooo`0000000000008000000`3oool00`00 0000oooo0?ooo`020?ooo`<000001@3oool00`000000oooo0?ooo`3@0?ooo`/00000;@3oool00003 0?ooo`000000oooo00d0oooo00<000000?ooo`3oool00P3oool300000=/0oooo2`00000R0?ooo`00 0P3oool00`000000oooo0?ooo`0;0?ooo`030000003oool0oooo0080oooo00<000000?ooo`3oool0 iP3oool:000001P0oooo00030?ooo`030000003oool0oooo00L0oooo0`0000050?ooo`030000003o ool0oooo0?00oooo4@0000070?ooo`00103oool00`000000oooo0?ooo`060?ooo`030000003oool0 oooo00D0oooo00<000000?ooo`3oool0o`3oool90?ooo`0000D0oooo0000003oool0oooo00000008 0?ooo`030000003oool0oooo00D0oooo00<000000?ooo`3oool0o`3oool90?ooo`000P3oool20000 00T0oooo100000040?ooo`030000003oool0oooo0?l0oooo2@3oool001D0oooo00<000000?ooo`3o ool0o`3oool90?ooo`00\ \>"], ImageRangeCache->{{{91.5625, 320.938}, {521, 379.688}} -> {-5.24833, 7.30491, 0.0386328, 0.0158088}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ "The plot of ", Cell[BoxData[ \(TraditionalForm\`g(x)\)]], " looks like the plot of ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], " shifted up. Plotting the difference indicates it is a constant equal to \ \[Pi]/2." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[g[x]\ - f[x], {x, 0, 10}, \ PlotRange -> {0, 2}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0952381 0 0.309017 [ [.21429 -0.0125 -3 -9 ] [.21429 -0.0125 3 0 ] [.40476 -0.0125 -3 -9 ] [.40476 -0.0125 3 0 ] [.59524 -0.0125 -3 -9 ] [.59524 -0.0125 3 0 ] [.78571 -0.0125 -3 -9 ] [.78571 -0.0125 3 0 ] [.97619 -0.0125 -6 -9 ] [.97619 -0.0125 6 0 ] [.01131 .07725 -24 -4.5 ] [.01131 .07725 0 4.5 ] [.01131 .15451 -18 -4.5 ] [.01131 .15451 0 4.5 ] [.01131 .23176 -24 -4.5 ] [.01131 .23176 0 4.5 ] [.01131 .30902 -6 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .38627 -24 -4.5 ] [.01131 .38627 0 4.5 ] [.01131 .46353 -18 -4.5 ] [.01131 .46353 0 4.5 ] [.01131 .54078 -24 -4.5 ] [.01131 .54078 0 4.5 ] [.01131 .61803 -6 -4.5 ] [.01131 .61803 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 0 m .21429 .00625 L s [(2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(10)] .97619 -0.0125 0 1 Mshowa .125 Mabswid .07143 0 m .07143 .00375 L s .11905 0 m .11905 .00375 L s .16667 0 m .16667 .00375 L s .2619 0 m .2619 .00375 L s .30952 0 m .30952 .00375 L s .35714 0 m .35714 .00375 L s .45238 0 m .45238 .00375 L s .5 0 m .5 .00375 L s .54762 0 m .54762 .00375 L s .64286 0 m .64286 .00375 L s .69048 0 m .69048 .00375 L s .7381 0 m .7381 .00375 L s .83333 0 m .83333 .00375 L s .88095 0 m .88095 .00375 L s .92857 0 m .92857 .00375 L s .25 Mabswid 0 0 m 1 0 L s .02381 .07725 m .03006 .07725 L s [(0.25)] .01131 .07725 1 0 Mshowa .02381 .15451 m .03006 .15451 L s [(0.5)] .01131 .15451 1 0 Mshowa .02381 .23176 m .03006 .23176 L s [(0.75)] .01131 .23176 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(1)] .01131 .30902 1 0 Mshowa .02381 .38627 m .03006 .38627 L s [(1.25)] .01131 .38627 1 0 Mshowa .02381 .46353 m .03006 .46353 L s [(1.5)] .01131 .46353 1 0 Mshowa .02381 .54078 m .03006 .54078 L s [(1.75)] .01131 .54078 1 0 Mshowa .02381 .61803 m .03006 .61803 L s [(2)] .01131 .61803 1 0 Mshowa .125 Mabswid .02381 .01545 m .02756 .01545 L s .02381 .0309 m .02756 .0309 L s .02381 .04635 m .02756 .04635 L s .02381 .0618 m .02756 .0618 L s .02381 .09271 m .02756 .09271 L s .02381 .10816 m .02756 .10816 L s .02381 .12361 m .02756 .12361 L s .02381 .13906 m .02756 .13906 L s .02381 .16996 m .02756 .16996 L s .02381 .18541 m .02756 .18541 L s .02381 .20086 m .02756 .20086 L s .02381 .21631 m .02756 .21631 L s .02381 .24721 m .02756 .24721 L s .02381 .26266 m .02756 .26266 L s .02381 .27812 m .02756 .27812 L s .02381 .29357 m .02756 .29357 L s .02381 .32447 m .02756 .32447 L s .02381 .33992 m .02756 .33992 L s .02381 .35537 m .02756 .35537 L s .02381 .37082 m .02756 .37082 L s .02381 .40172 m .02756 .40172 L s .02381 .41717 m .02756 .41717 L s .02381 .43262 m .02756 .43262 L s .02381 .44807 m .02756 .44807 L s .02381 .47898 m .02756 .47898 L s .02381 .49443 m .02756 .49443 L s .02381 .50988 m .02756 .50988 L s .02381 .52533 m .02756 .52533 L s .02381 .55623 m .02756 .55623 L s .02381 .57168 m .02756 .57168 L s .02381 .58713 m .02756 .58713 L s .02381 .60258 m .02756 .60258 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .4854 m .02499 .4854 L .02605 .4854 L .02729 .4854 L .02846 .4854 L .02954 .4854 L .03053 .4854 L .03163 .4854 L .03279 .4854 L .03395 .4854 L .0352 .4854 L .03638 .4854 L .03746 .4854 L .04262 .4854 L .04748 .4854 L .05278 .4854 L .05521 .4854 L .05779 .4854 L .05905 .4854 L .06023 .4854 L .0613 .4854 L .06244 .4854 L .06371 .4854 L .06443 .4854 L .06508 .4854 L .06753 .4854 L .07312 .4854 L .08439 .4854 L .08936 .4854 L .09001 .4854 L .09071 .4854 L .09198 .4854 L .0933 .4854 L .09406 .4854 L .09476 .4854 L .09594 .4854 L .09724 .4854 L .0979 .4854 L .0986 .4854 L .09986 .4854 L .10113 .4854 L .10233 .4854 L .10341 .4854 L .10458 .4854 L .10581 .4854 L .10694 .4854 L .10947 .4854 L .11404 .4854 L .11666 .4854 L .11741 .4854 L Mistroke .11812 .4854 L .11876 .4854 L .11945 .4854 L .12072 .4854 L .12136 .4854 L .12206 .4854 L .12446 .4854 L .12575 .4854 L .12647 .4854 L .12713 .4854 L .12841 .4854 L .12962 .4854 L .1303 .4854 L .13102 .4854 L .13232 .4854 L .13299 .4854 L .13371 .4854 L .13451 .4854 L .13525 .4854 L .1365 .4854 L .1372 .4854 L .13786 .4854 L .1406 .4854 L .14297 .4854 L .14365 .4854 L .1443 .4854 L .1455 .4854 L .14666 .4854 L .1479 .4854 L .15014 .4854 L .15525 .4854 L .16555 .4854 L .18402 .4854 L .2038 .4854 L .22535 .4854 L .23536 .4854 L .24086 .4854 L .24355 .4854 L .24602 .4854 L .24815 .4854 L .24922 .4854 L .25039 .4854 L .25157 .4854 L .25284 .4854 L .25403 .4854 L .25511 .4854 L .2564 .4854 L .25759 .4854 L .2589 .4854 L .25955 .4854 L Mistroke .26028 .4854 L .26281 .4854 L .26404 .4854 L .26516 .4854 L .26636 .4854 L .26765 .4854 L .27036 .4854 L .2753 .4854 L .28458 .4854 L .28975 .4854 L .29534 .4854 L .2979 .4854 L .30061 .4854 L .30293 .4854 L .30424 .4854 L .30547 .4854 L .30673 .4854 L .30737 .4854 L .30807 .4854 L .30931 .4854 L .31048 .4854 L .31115 .4854 L .31185 .4854 L .3131 .4854 L .3138 .4854 L .31445 .4854 L .31517 .4854 L .31593 .4854 L .31721 .4854 L .31793 .4854 L .3186 .4854 L .31982 .4854 L .32112 .4854 L .32583 .4854 L .32804 .4854 L .3304 .4854 L .33155 .4854 L .33264 .4854 L .33361 .4854 L .33467 .4854 L .33596 .4854 L .33712 .4854 L .33776 .4854 L .33846 .4854 L .33972 .4854 L .34091 .4854 L .34216 .4854 L .34322 .4854 L .34439 .4854 L .34571 .4854 L Mistroke .34692 .4854 L .34755 .4854 L .34825 .4854 L .34966 .4854 L .35464 .4854 L .36396 .4854 L .38489 .4854 L .39484 .4854 L .39974 .4854 L .40429 .4854 L .40559 .4854 L .40681 .4854 L .40749 .4854 L .40822 .4854 L .4095 .4854 L .41019 .4854 L .41094 .4854 L .41173 .4854 L .41246 .4854 L .41378 .4854 L .41452 .4854 L .41519 .4854 L .41646 .4854 L .41764 .4854 L .42031 .4854 L .42267 .4854 L .42385 .4854 L .42517 .4854 L .42638 .4854 L .42751 .4854 L .42875 .4854 L .42945 .4854 L .43008 .4854 L .43525 .4854 L .4445 .4854 L .447 .4854 L .44766 .4854 L .44837 .4854 L .44965 .4854 L .45037 .4854 L .45103 .4854 L .45181 .4854 L .45255 .4854 L .45383 .4854 L .45456 .4854 L .45523 .4854 L .45587 .4854 L .45656 .4854 L .4578 .4854 L .45902 .4854 L Mistroke .46013 .4854 L .46135 .4854 L .46203 .4854 L .46268 .4854 L .46397 .4854 L .4647 .4854 L .46536 .4854 L .46654 .4854 L .46765 .4854 L .46887 .4854 L .47018 .4854 L .47148 .4854 L .47213 .4854 L .47285 .4854 L .47411 .4854 L .47529 .4854 L .48447 .4854 L .49452 .4854 L .49741 .4854 L .50014 .4854 L .50133 .4854 L .50259 .4854 L .50325 .4854 L .50397 .4854 L .50524 .4854 L .50774 .4854 L .50906 .4854 L .50973 .4854 L .51047 .4854 L .51111 .4854 L .5118 .4854 L .51304 .4854 L .51543 .4854 L .52472 .4854 L .54562 .4854 L .56524 .4854 L .58672 .4854 L .58916 .4854 L .59181 .4854 L .59302 .4854 L .59431 .4854 L .59541 .4854 L .59662 .4854 L .59734 .4854 L .598 .4854 L .59876 .4854 L .59948 .4854 L .60077 .4854 L .60217 .4854 L .60721 .4854 L Mistroke .612 .4854 L .61711 .4854 L .61959 .4854 L .62081 .4854 L .62193 .4854 L .62295 .4854 L .62403 .4854 L .62511 .4854 L .6263 .4854 L .6276 .4854 L .62881 .4854 L .62944 .4854 L .63013 .4854 L .63087 .4854 L .63154 .4854 L .63277 .4854 L .63342 .4854 L .6341 .4854 L .63525 .4854 L .63651 .4854 L .63773 .4854 L .63884 .4854 L .64012 .4854 L .64133 .4854 L .64346 .4854 L .64581 .4854 L .64712 .4854 L .64835 .4854 L .64964 .4854 L .65031 .4854 L .65104 .4854 L .65233 .4854 L .65355 .4854 L .65474 .4854 L .65585 .4854 L .65654 .4854 L .65728 .4854 L .65797 .4854 L .65862 .4854 L .65983 .4854 L .66112 .4854 L .66179 .4854 L .6625 .4854 L .6638 .4854 L .66459 .4854 L .66533 .4854 L .666 .4854 L .66673 .4854 L .66795 .4854 L .66907 .4854 L Mistroke .67037 .4854 L .6716 .4854 L .67616 .4854 L .67739 .4854 L .67856 .4854 L .67921 .4854 L .6799 .4854 L .68112 .4854 L .68184 .4854 L .68259 .4854 L .68323 .4854 L .68394 .4854 L .68459 .4854 L .68529 .4854 L .68655 .4854 L .68723 .4854 L .68795 .4854 L .68925 .4854 L .68997 .4854 L .69065 .4854 L .69139 .4854 L .69219 .4854 L .69754 .4854 L .70757 .4854 L .74869 .4854 L .76919 .4854 L .77039 .4854 L .77166 .4854 L .77232 .4854 L .77305 .4854 L .77434 .4854 L .77555 .4854 L .77685 .4854 L .77807 .4854 L .77919 .4854 L .78035 .4854 L .78142 .4854 L .7826 .4854 L .78386 .4854 L .78829 .4854 L .78945 .4854 L .79068 .4854 L .79132 .4854 L .79202 .4854 L .79326 .4854 L .79443 .4854 L .79568 .4854 L .79794 .4854 L .79907 .4854 L .80029 .4854 L Mistroke .80145 .4854 L .80252 .4854 L .80454 .4854 L .80557 .4854 L .80669 .4854 L .80794 .4854 L .80924 .4854 L .81036 .4854 L .81158 .4854 L .8129 .4854 L .81431 .4854 L .81499 .4854 L .81564 .4854 L .81685 .4854 L .818 .4854 L .81922 .4854 L .82027 .4854 L .82142 .4854 L .82265 .4854 L .82333 .4854 L .82398 .4854 L .82523 .4854 L .82637 .4854 L .86464 .4854 L .86529 .4854 L .86598 .4854 L .86722 .4854 L .86852 .4854 L .86926 .4854 L .86994 .4854 L .87068 .4854 L .87147 .4854 L .87223 .4854 L .87292 .4854 L .87566 .4854 L .87804 .4854 L .87872 .4854 L .87937 .4854 L .88059 .4854 L .88129 .4854 L .88193 .4854 L .88264 .4854 L .8834 .4854 L .8841 .4854 L .88477 .4854 L .88602 .4854 L .89053 .4854 L .89177 .4854 L .89307 .4854 L .89419 .4854 L Mistroke .89541 .4854 L .89671 .4854 L .8979 .4854 L .89855 .4854 L .89926 .4854 L .90056 .4854 L .90177 .4854 L .90306 .4854 L .90416 .4854 L .90537 .4854 L .90655 .4854 L .90784 .4854 L .90849 .4854 L .90919 .4854 L .91044 .4854 L .9117 .4854 L .91288 .4854 L .91395 .4854 L .9151 .4854 L .92559 .4854 L .94458 .4854 L .95251 .4854 L .95993 .4854 L .96366 .4854 L .96474 .4854 L .96576 .4854 L .96667 .4854 L .96767 .4854 L .9688 .4854 L .96981 .4854 L .97099 .4854 L .97209 .4854 L .97312 .4854 L .97406 .4854 L .9751 .4854 L .97619 .4854 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg0?ooo`030000003oool0oooo02d0oooo1@00000]0?ooo`040000003oool0oooo0000 02h0oooo00@000000?ooo`3oool00000;03oool01@000000oooo0?ooo`3oool000000080oooo00<0 00000?ooo`3oool0103oool004l0oooo00<000000?ooo`3oool0;03oool010000000oooo0?ooo`00 000^0?ooo`<00000<03oool2000002d0oooo00D000000?ooo`3oool0oooo000000020?ooo`030000 003oool0oooo00@0oooo001@0?ooo`030000003oool0oooo02`0oooo00<000000?ooo`000000;P3o ool00`000000oooo0?ooo`0_0?ooo`040000003oool0oooo000002`0oooo00D000000?ooo`3oool0 oooo000000020?ooo`030000003oool0oooo00@0oooo001=0?ooo`040000003oool0oooo000002l0 oooo0P00000^0?ooo`030000003oool0oooo02l0oooo00@000000?ooo`3oool00000:P3oool30000 00<0oooo00@000000?ooo`3oool000001P3oool004h0oooo0P00000a0?ooo`030000003oool0oooo 02d0oooo0`00000_0?ooo`800000;@3oool00`000000oooo0?ooo`020?ooo`8000001`3oool00?l0 oooo8@3oool00?l0oooo8@3oool001P0ooooo`000005000000@0oooo000N0?ooo`030000003oool0 oooo00T0oooo00<000000?ooo`3oool02P3oool00`000000oooo0?ooo`090?ooo`030000003oool0 oooo00T0oooo00<000000?ooo`3oool02P3oool00`000000oooo0?ooo`090?ooo`030000003oool0 oooo00X0oooo00<000000?ooo`3oool02@3oool00`000000oooo0?ooo`090?ooo`030000003oool0 oooo00X0oooo00<000000?ooo`3oool02@3oool00`000000oooo0?ooo`090?ooo`030000003oool0 oooo00X0oooo00<000000?ooo`3oool02@3oool00`000000oooo0?ooo`090?ooo`030000003oool0 oooo00X0oooo00<000000?ooo`3oool02@3oool00`000000oooo0?ooo`090?ooo`030000003oool0 oooo00X0oooo00<000000?ooo`3oool02@3oool00`000000oooo0?ooo`080?ooo`007P3oool00`00 0000oooo0?ooo`3o0?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`007P3oool200000?l0oooo 0@3oool001h0oooo00<000000?ooo`3oool0o`3oool001h0oooo00<000000?ooo`3oool0o`3oool0 01h0oooo00<000000?ooo`3oool0o`3oool001h0oooo0P00003o0?ooo`40oooo000N0?ooo`030000 003oool0oooo0?l0oooo000N0?ooo`030000003oool0oooo0?l0oooo000N0?ooo`030000003oool0 oooo0?l0oooo000N0?ooo`800000o`3oool10?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`00 7P3oool00`000000oooo0?ooo`3o0?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`007P3oool2 00000?l0oooo0@3oool001h0oooo00<000000?ooo`3oool0o`3oool001h0oooo00<000000?ooo`3o ool0o`3oool000D0oooo0P0000040?ooo`030000003oool0oooo0080oooo100000020?ooo`<00000 1@3oool00`000000oooo0?ooo`3o0?ooo`00103oool010000000oooo0?ooo`0000080?ooo`030000 003oool0oooo00H0oooo00<000000?ooo`3oool00P3oool300000?l0oooo00040?ooo`040000003o ool0oooo000000T0oooo00<000000?ooo`3oool01@3oool00`000000oooo0?ooo`020?ooo`030000 003oool0oooo0?l0oooo00040?ooo`040000003oool0oooo000000X0oooo00@000000?ooo`3oool0 oooo0`0000050?ooo`030000003oool0oooo0?l0oooo00040?ooo`040000003oool0oooo000000/0 oooo00@000000?ooo`3oool000001`3oool00`000000oooo0?ooo`3o0?ooo`00103oool010000000 oooo0?ooo`0000080?ooo`040000003oool0oooo00000080oooo00<000000?ooo`3oool01@3oool2 00000?l0oooo0@3oool000D0oooo0P00000:0?ooo`8000000`3oool4000000@0oooo00<000000?oo o`3oool0o`3oool001h0oooo00<000000?ooo`3oool0o`3oool001h0oooo00<000000?ooo`3oool0 o`3oool001h0oooo0P00003o0?ooo`40oooo000N0?ooo`030000003oool0oooo0?l0oooo000N0?oo o`030000003oool0oooo0?l0oooo000N0?ooo`030000003oool0oooo0?l0oooo000N0?ooo`800000 o`3oool10?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`007P3oool00`000000oooo0?ooo`3o 0?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`007P3oool200000?l0oooo0@3oool001h0oooo 00<000000?ooo`3oool0o`3oool001h0oooo00<000000?ooo`3oool0o`3oool000/0oooo0P000004 0?ooo`030000003oool0oooo0080oooo0`0000050?ooo`030000003oool0oooo0?l0oooo000:0?oo o`040000003oool0oooo000000/0oooo00<000000?ooo`3oool00P3oool300000?l0oooo000:0?oo o`040000003oool0oooo000000/0oooo00<000000?ooo`3oool00P3oool00`000000oooo0?ooo`3o 0?ooo`002P3oool010000000oooo0?ooo`0000080?ooo`<000001@3oool00`000000oooo0?ooo`3o 0?ooo`002P3oool010000000oooo0?ooo`0000080?ooo`030000003oool0oooo00D0oooo00<00000 0?ooo`3oool0o`3oool000X0oooo00@000000?ooo`3oool00000203oool00`000000oooo0?ooo`05 0?ooo`800000o`3oool10?ooo`002`3oool2000000T0oooo100000040?ooo`030000003oool0oooo 0?l0oooo000N0?ooo`030000003oool0oooo0?l0oooo000N0?ooo`030000003oool0oooo0?l0oooo 000N0?ooo`800000o`3oool10?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`007P3oool00`00 0000oooo0?ooo`3o0?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`007P3oool200000?l0oooo 0@3oool001h0oooo00<000000?ooo`3oool0o`3oool001h0oooo00<000000?ooo`3oool0o`3oool0 01h0oooo00<000000?ooo`3oool0o`3oool001h0oooo0P00003o0?ooo`40oooo000N0?ooo`030000 003oool0oooo0?l0oooo000N0?ooo`030000003oool0oooo0?l0oooo00050?ooo`800000103oool0 0`000000oooo0?ooo`030?ooo`030000003oool0oooo0080oooo0`0000050?ooo`030000003oool0 oooo0?l0oooo00040?ooo`040000003oool0oooo000000T0oooo00<000000?ooo`3oool01@3oool0 0`000000oooo0?ooo`020?ooo`<00000o`3oool000@0oooo00@000000?ooo`3oool000002P3oool0 0`000000oooo0?ooo`040?ooo`030000003oool0oooo0080oooo00<000000?ooo`3oool0o`3oool0 00@0oooo00@000000?ooo`3oool000002P3oool010000000oooo0?ooo`3oool3000000D0oooo00<0 00000?ooo`3oool0o`3oool000@0oooo00@000000?ooo`3oool000002`3oool010000000oooo0?oo o`0000070?ooo`030000003oool0oooo0?l0oooo00040?ooo`040000003oool0oooo000000P0oooo 00@000000?ooo`3oool000000P3oool00`000000oooo0?ooo`050?ooo`800000o`3oool10?ooo`00 1@3oool2000000T0oooo100000020?ooo`@00000103oool00`000000oooo0?ooo`3o0?ooo`007P3o ool00`000000oooo0?ooo`3o0?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`007P3oool20000 0?l0oooo0@3oool001h0oooo00<000000?ooo`3oool0o`3oool001h0oooo00<000000?ooo`3oool0 o`3oool001h0oooo00<000000?ooo`3oool0o`3oool001h0oooo0P00003o0?ooo`40oooo000N0?oo o`030000003oool0oooo0?l0oooo000N0?ooo`030000003oool0oooo0?l0oooo000N0?ooo`030000 003oool0oooo0?l0oooo000N0?ooo`800000o`3oool10?ooo`007P3oool00`000000oooo0?ooo`3o 0?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`005P3oool4000000@0oooo00<000000?ooo`3o ool0o`3oool001P0oooo00<000000?ooo`3oool00`3oool300000?l0oooo000H0?ooo`030000003o ool0oooo00<0oooo00<000000?ooo`3oool0o`3oool001P0oooo00<000000?ooo`3oool00`3oool0 0`000000oooo0?ooo`3o0?ooo`00603oool00`000000oooo0?ooo`030?ooo`030000003oool0oooo 0?l0oooo000F0?ooo`<000001@3oool200000?l0oooo0@3oool001P0oooo00<000000?ooo`3oool0 0`3oool00`000000oooo0?ooo`3o0?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`007P3oool0 0`000000oooo0?ooo`3o0?ooo`007P3oool200000?l0oooo0@3oool001h0oooo00<000000?ooo`3o ool0o`3oool001h0oooo00<000000?ooo`3oool0o`3oool001h0oooo00<000000?ooo`3oool0o`3o ool001h0oooo0P00003o0?ooo`40oooo000N0?ooo`030000003oool0oooo0?l0oooo000N0?ooo`03 0000003oool0oooo0?l0oooo000N0?ooo`030000003oool0oooo0?l0oooo000N0?ooo`800000o`3o ool10?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`007P3oool00`000000oooo0?ooo`3o0?oo o`00103oool4000000<0oooo00<000000?ooo`3oool00P3oool400000080oooo0`0000050?ooo`03 0000003oool0oooo0?l0oooo00060?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool0 1P3oool00`000000oooo0?ooo`020?ooo`<00000o`3oool000H0oooo00<000000?ooo`3oool0203o ool00`000000oooo0?ooo`050?ooo`030000003oool0oooo0080oooo00<000000?ooo`3oool0o`3o ool000H0oooo00<000000?ooo`3oool02@3oool010000000oooo0?ooo`3oool3000000D0oooo00<0 00000?ooo`3oool0o`3oool000H0oooo00<000000?ooo`3oool02P3oool010000000oooo0?ooo`00 00070?ooo`030000003oool0oooo0?l0oooo00040?ooo`<000002@3oool010000000oooo0?ooo`00 00020?ooo`030000003oool0oooo00D0oooo0P00003o0?ooo`40oooo00060?ooo`030000003oool0 oooo00P0oooo0P0000030?ooo`@00000103oool00`000000oooo0?ooo`3o0?ooo`007P3oool00`00 0000oooo0?ooo`3o0?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`007P3oool200000?l0oooo 0@3oool001h0oooo00<000000?ooo`3oool0o`3oool001h0oooo00<000000?ooo`3oool0o`3oool0 01h0oooo00<000000?ooo`3oool0o`3oool001h0oooo0P00003o0?ooo`40oooo000N0?ooo`030000 003oool0oooo0?l0oooo000N0?ooo`030000003oool0oooo0?l0oooo000N0?ooo`030000003oool0 oooo0?l0oooo000N0?ooo`800000o`3oool10?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`00 7P3oool00`000000oooo0?ooo`3o0?ooo`002P3oool4000000<0oooo00<000000?ooo`3oool00P3o ool3000000D0oooo00<000000?ooo`3oool0o`3oool000`0oooo00<000000?ooo`3oool02P3oool0 0`000000oooo0?ooo`020?ooo`<00000o`3oool000`0oooo00<000000?ooo`3oool02P3oool00`00 0000oooo0?ooo`020?ooo`030000003oool0oooo0?l0oooo000<0?ooo`030000003oool0oooo00L0 oooo0`0000050?ooo`030000003oool0oooo0?l0oooo000<0?ooo`030000003oool0oooo00L0oooo 00<000000?ooo`3oool01@3oool00`000000oooo0?ooo`3o0?ooo`002P3oool3000000T0oooo00<0 00000?ooo`3oool01@3oool200000?l0oooo0@3oool000`0oooo00<000000?ooo`3oool01`3oool4 000000@0oooo00<000000?ooo`3oool0o`3oool001h0oooon000000:0?ooo`007P3oool00`000000 oooo0?ooo`3o0?ooo`007P3oool200000?l0oooo0@3oool001h0oooo00<000000?ooo`3oool0o`3o ool001h0oooo00<000000?ooo`3oool0o`3oool001h0oooo00<000000?ooo`3oool0o`3oool001h0 oooo0P00003o0?ooo`40oooo000N0?ooo`030000003oool0oooo0?l0oooo000N0?ooo`030000003o ool0oooo0?l0oooo000N0?ooo`030000003oool0oooo0?l0oooo000N0?ooo`800000o`3oool10?oo o`007P3oool00`000000oooo0?ooo`3o0?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`00103o ool4000000<0oooo00<000000?ooo`3oool00`3oool00`000000oooo0?ooo`020?ooo`<000001@3o ool00`000000oooo0?ooo`3o0?ooo`001P3oool00`000000oooo0?ooo`080?ooo`030000003oool0 oooo00D0oooo00<000000?ooo`3oool00P3oool300000?l0oooo00060?ooo`030000003oool0oooo 00T0oooo00<000000?ooo`3oool0103oool00`000000oooo0?ooo`020?ooo`030000003oool0oooo 0?l0oooo00060?ooo`030000003oool0oooo00T0oooo00@000000?ooo`3oool0oooo0`0000050?oo o`030000003oool0oooo0?l0oooo00060?ooo`030000003oool0oooo00X0oooo00@000000?ooo`3o ool000001`3oool00`000000oooo0?ooo`3o0?ooo`00103oool3000000T0oooo00@000000?ooo`3o ool000000P3oool00`000000oooo0?ooo`050?ooo`800000o`3oool10?ooo`001P3oool00`000000 oooo0?ooo`070?ooo`@000000P3oool4000000@0oooo00<000000?ooo`3oool0o`3oool001h0oooo 00<000000?ooo`3oool0o`3oool001h0oooo00<000000?ooo`3oool0o`3oool001h0oooo0P00003o 0?ooo`40oooo000N0?ooo`030000003oool0oooo0?l0oooo000N0?ooo`030000003oool0oooo0?l0 oooo000N0?ooo`030000003oool0oooo0?l0oooo000N0?ooo`800000o`3oool10?ooo`007P3oool0 0`000000oooo0?ooo`3o0?ooo`007P3oool00`000000oooo0?ooo`3o0?ooo`007P3oool00`000000 oooo0?ooo`3o0?ooo`007P3oool200000?l0oooo0@3oool001h0oooo00<000000?ooo`3oool0o`3o ool001h0oooo00<000000?ooo`3oool0o`3oool001H0oooo100000040?ooo`030000003oool0oooo 0?l0oooo000F0?ooo`030000003oool0oooo00D0oooo0`00003o0?ooo`005`3oool00`000000oooo 0?ooo`3o0?ooo`L0oooo000H0?ooo`030000003oool0oooo0?l0oooo1P3oool001T0oooo00<00000 0?ooo`3oool0o`3oool50?ooo`005P3oool010000000oooo0?ooo`00003o0?ooo`L0oooo0000\ \>"], ImageRangeCache->{{{91.5625, 320.938}, {238.5, 97.1875}} -> {-5.89025, 1.36642, 0.0405824, 0.0125074}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[0] - f[0]\)], "Input"], Cell[BoxData[ \(\[Pi]\/2\)], "Output"] }, Open ]], Cell["\<\ To prove that the functions differ by a constant we take their \ derivatives:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(f'\)[x]\)], "Input"], Cell[BoxData[ \(\(\(-\(\(\(-1\) + x\)\/\((1 + x)\)\^2\)\) + 1\/\(1 + x\)\)\/\@\(1 - \((\(-1\) + x)\)\^2\/\((1 + x)\)\^2\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\(f'\)[x]]\)], "Input"], Cell[BoxData[ \(\@\(x\/\((1 + x)\)\^2\)\/x\)], "Output"] }, Open ]], Cell[TextData[{ "I had trouble getting Mathematica to simplify this further (the program \ does not know that ", Cell[BoxData[ \(TraditionalForm\`x\)]], " is a real number or ", Cell[BoxData[ \(TraditionalForm\`x\ > \ 0\)]], ") , so I looked up ", StyleBox["Simplify[]", "Input"], " in the Help Browser and found some other functions to try. ", StyleBox["PowerExpand[]", "Input"], " did the simplification I was looking for (Of course, getting ", StyleBox["Mathematica", FontSlant->"Italic"], " to simplify this expression is purely optional. You can always do the \ derivatives and simplifications by hand.)" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(PowerExpand[Simplify[\(f'\)[x]]]\)], "Input"], Cell[BoxData[ \(1\/\(\@x\ \((1 + x)\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "Now we see that ", Cell[BoxData[ \(TraditionalForm\`g(x)\)]], " has the same derivative." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(g'\)[x]\)], "Input"], Cell[BoxData[ \(1\/\(\@x\ \((1 + x)\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "So the two functions do indeed differ by a constant. This constant is \ given by ", Cell[BoxData[ \(TraditionalForm\`g(0)\ - \ f(0)\ = \ \[Pi]/2\)]], " as determined above, so ", Cell[BoxData[ \(TraditionalForm\`g(x)\ = \ f(x)\ + \ \[Pi]/2\)]], "." }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1024}, {0, 768}}, WindowSize->{520, 600}, WindowMargins->{{Automatic, 175}, {-7, Automatic}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "Magnification"->1}, ShowCellLabel->False ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1731, 51, 28, 0, 45, "Subsection"], Cell[CellGroupData[{ Cell[1784, 55, 28, 0, 42, "Subsubsection"], Cell[1815, 57, 112, 2, 68, "Input"], Cell[CellGroupData[{ Cell[1952, 63, 55, 1, 27, "Input"], Cell[2010, 66, 15551, 406, 186, 3621, 254, "GraphicsData", "PostScript", "Graphics"], Cell[17564, 474, 130, 3, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17731, 482, 55, 1, 27, "Input"], Cell[17789, 485, 15687, 405, 186, 3587, 251, "GraphicsData", "PostScript", "Graphics"], Cell[33479, 892, 130, 3, 27, "Output"] }, Open ]], Cell[33624, 898, 265, 9, 50, "Text"], Cell[CellGroupData[{ Cell[33914, 911, 87, 1, 22, "Input"], Cell[34004, 914, 20882, 849, 148, 10263, 714, "GraphicsData", "PostScript", "Graphics"], Cell[54889, 1765, 130, 3, 22, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[55056, 1773, 44, 1, 22, "Input"], Cell[55103, 1776, 42, 1, 26, "Output"] }, Open ]], Cell[55160, 1780, 101, 3, 24, "Text"], Cell[CellGroupData[{ Cell[55286, 1787, 42, 1, 22, "Input"], Cell[55331, 1790, 151, 3, 41, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[55519, 1798, 52, 1, 22, "Input"], Cell[55574, 1801, 60, 1, 35, "Output"] }, Open ]], Cell[55649, 1805, 662, 17, 59, "Text"], Cell[CellGroupData[{ Cell[56336, 1826, 65, 1, 22, "Input"], Cell[56404, 1829, 57, 1, 31, "Output"] }, Open ]], Cell[56476, 1833, 130, 5, 24, "Text"], Cell[CellGroupData[{ Cell[56631, 1842, 42, 1, 22, "Input"], Cell[56676, 1845, 57, 1, 31, "Output"] }, Open ]], Cell[56748, 1849, 303, 9, 35, "Text"] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)