(*********************************************************************** 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[ 22448, 697]*) (*NotebookOutlinePosition[ 23259, 725]*) (* CellTagsIndexPosition[ 23215, 721]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["p. 568", "Subsection"], Cell[CellGroupData[{ Cell["41.", "Subsubsection"], Cell[TextData[{ "The average value of a function ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], " on an interval ", Cell[BoxData[ \(TraditionalForm\`\([a, b]\)\)]], " is given by ", Cell[BoxData[ \(TraditionalForm \`avg(f)\ = \ \(1\/\(b - a\)\) \(\[Integral]\_a\%b\( f(x)\) \[DifferentialD]x\)\)]], ". So the average value of ", Cell[BoxData[ \(TraditionalForm\`f(t)\ = \ 2 \( e\^\(-t\)\) \(cos(t)\)\)]], " on ", Cell[BoxData[ \(TraditionalForm\`\([0, 2 \[Pi]]\)\)]], " is ", Cell[BoxData[ \(TraditionalForm \`\(1\/\(2 \[Pi]\)\) \(\[Integral]\ 2 \( e\^\(-t\)\) \(cos(t)\) \[DifferentialD]t\)\)]], ". Using integration by parts (similar to problem 21), we get ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[ RowBox[{ FormBox[\(avg(f)\ = \ 1\/\(2 \[Pi]\)\), "TraditionalForm"], \(\(e\^\(-t\)\)(sin(t)\ - \ cos(t))\)}], "TraditionalForm"], \(|\_0\%\(2 \[Pi]\)\)}], "=", " ", \(\(1\/\(2 \[Pi]\)\) \((1\ - \ e\^\(\(-2\) \[Pi]\))\)\)}], TraditionalForm]]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\ \(1\/\(2 \[Pi]\)\) \(\[Integral]\ 2 \( E\^\(-t\)\) Cos[t] \[DifferentialD]t\)\)\)], "Input"], Cell[BoxData[ \(\(-\(\(E\^\(-t\)\ \((Cos[t] - Sin[t])\)\)\/\(2\ \[Pi]\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\ \(1\/\(2 \[Pi]\)\) \(\[Integral]\_0\%\(2 \[Pi]\)\ 2 \( E\^\(-t\)\) Cos[t] \[DifferentialD]t\)\)\)], "Input"], Cell[BoxData[ \(\(1 - E\^\(\(-2\)\ \[Pi]\)\)\/\(2\ \[Pi]\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[\(1 - E\^\(\(-2\)\ \[Pi]\)\)\/\(2\ \[Pi]\)]\)], "Input"], Cell[BoxData[ \(0.158857730350202981`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{2 \( E\^\(-t\)\) Cos[t], \ \(1\/\(2 \[Pi]\)\) \((1\ - \ E\^\(\(-2\) \[Pi]\))\)}, {t, 0, 2 \[Pi]}, \n\tPlotStyle -> {RGBColor[0, 0, 1], \ RGBColor[1, 0, 0]}, \ PlotRange -> {\(-1\), 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 0.151576 0.206011 0.206011 [ [.17539 .19351 -3 -9 ] [.17539 .19351 3 0 ] [.32696 .19351 -3 -9 ] [.32696 .19351 3 0 ] [.47854 .19351 -3 -9 ] [.47854 .19351 3 0 ] [.63011 .19351 -3 -9 ] [.63011 .19351 3 0 ] [.78169 .19351 -3 -9 ] [.78169 .19351 3 0 ] [.93327 .19351 -3 -9 ] [.93327 .19351 3 0 ] [.01131 0 -12 -4.5 ] [.01131 0 0 4.5 ] [.01131 .10301 -24 -4.5 ] [.01131 .10301 0 4.5 ] [.01131 .30902 -18 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .41202 -6 -4.5 ] [.01131 .41202 0 4.5 ] [.01131 .51503 -18 -4.5 ] [.01131 .51503 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 .17539 .20601 m .17539 .21226 L s [(1)] .17539 .19351 0 1 Mshowa .32696 .20601 m .32696 .21226 L s [(2)] .32696 .19351 0 1 Mshowa .47854 .20601 m .47854 .21226 L s [(3)] .47854 .19351 0 1 Mshowa .63011 .20601 m .63011 .21226 L s [(4)] .63011 .19351 0 1 Mshowa .78169 .20601 m .78169 .21226 L s [(5)] .78169 .19351 0 1 Mshowa .93327 .20601 m .93327 .21226 L s [(6)] .93327 .19351 0 1 Mshowa .125 Mabswid .05412 .20601 m .05412 .20976 L s .08444 .20601 m .08444 .20976 L s .11476 .20601 m .11476 .20976 L s .14507 .20601 m .14507 .20976 L s .2057 .20601 m .2057 .20976 L s .23602 .20601 m .23602 .20976 L s .26633 .20601 m .26633 .20976 L s .29665 .20601 m .29665 .20976 L s .35728 .20601 m .35728 .20976 L s .38759 .20601 m .38759 .20976 L s .41791 .20601 m .41791 .20976 L s .44822 .20601 m .44822 .20976 L s .50885 .20601 m .50885 .20976 L s .53917 .20601 m .53917 .20976 L s .56948 .20601 m .56948 .20976 L s .5998 .20601 m .5998 .20976 L s .66043 .20601 m .66043 .20976 L s .69074 .20601 m .69074 .20976 L s .72106 .20601 m .72106 .20976 L s .75138 .20601 m .75138 .20976 L s .81201 .20601 m .81201 .20976 L s .84232 .20601 m .84232 .20976 L s .87264 .20601 m .87264 .20976 L s .90295 .20601 m .90295 .20976 L s .96358 .20601 m .96358 .20976 L s .9939 .20601 m .9939 .20976 L s .25 Mabswid 0 .20601 m 1 .20601 L s .02381 0 m .03006 0 L s [(-1)] .01131 0 1 0 Mshowa .02381 .10301 m .03006 .10301 L s [(-0.5)] .01131 .10301 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(0.5)] .01131 .30902 1 0 Mshowa .02381 .41202 m .03006 .41202 L s [(1)] .01131 .41202 1 0 Mshowa .02381 .51503 m .03006 .51503 L s [(1.5)] .01131 .51503 1 0 Mshowa .02381 .61803 m .03006 .61803 L s [(2)] .01131 .61803 1 0 Mshowa .125 Mabswid .02381 .0206 m .02756 .0206 L s .02381 .0412 m .02756 .0412 L s .02381 .0618 m .02756 .0618 L s .02381 .0824 m .02756 .0824 L s .02381 .12361 m .02756 .12361 L s .02381 .14421 m .02756 .14421 L s .02381 .16481 m .02756 .16481 L s .02381 .18541 m .02756 .18541 L s .02381 .22661 m .02756 .22661 L s .02381 .24721 m .02756 .24721 L s .02381 .26781 m .02756 .26781 L s .02381 .28842 m .02756 .28842 L s .02381 .32962 m .02756 .32962 L s .02381 .35022 m .02756 .35022 L s .02381 .37082 m .02756 .37082 L s .02381 .39142 m .02756 .39142 L s .02381 .43262 m .02756 .43262 L s .02381 .45322 m .02756 .45322 L s .02381 .47383 m .02756 .47383 L s .02381 .49443 m .02756 .49443 L s .02381 .53563 m .02756 .53563 L s .02381 .55623 m .02756 .55623 L s .02381 .57683 m .02756 .57683 L s .02381 .59743 m .02756 .59743 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 0 0 1 r .5 Mabswid .02381 .61803 m .06244 .51501 L .10458 .41431 L .14415 .33659 L .18221 .27874 L .20342 .25342 L .22272 .23437 L .24299 .21809 L .26171 .20612 L .28086 .19659 L .29159 .19231 L .30164 .18893 L .32037 .1841 L .3303 .18222 L .34099 .18068 L .3511 .17962 L .35681 .17918 L .36212 .17886 L .36475 .17874 L .36722 .17864 L .36947 .17856 L .37185 .1785 L .37308 .17847 L .37444 .17845 L .37569 .17843 L .37683 .17842 L .37747 .17841 L .37817 .17841 L .37891 .1784 L .37961 .1784 L .38027 .1784 L .38096 .1784 L .38221 .1784 L .38335 .1784 L .3846 .17841 L .38589 .17843 L .3871 .17844 L .38927 .17848 L .39158 .17853 L .39688 .17868 L .40169 .17887 L .41109 .17935 L .41997 .17993 L .45843 .18346 L .49935 .18813 L .53875 .19267 L .57664 .19662 L .61698 .20011 L .6558 .20272 L .67555 .20377 L Mistroke .69708 .20471 L .71795 .20545 L .73684 .20598 L .75655 .20641 L .76749 .2066 L .77754 .20675 L .79682 .20696 L .80696 .20704 L .81766 .20711 L .82716 .20715 L .83249 .20717 L .83734 .20718 L .84185 .20719 L .84424 .2072 L .84676 .2072 L .84897 .2072 L .85014 .2072 L .8514 .2072 L .85255 .2072 L .85361 .2072 L .85462 .2072 L .85568 .2072 L .85692 .2072 L .85762 .2072 L .85826 .2072 L .85945 .2072 L .8607 .2072 L .86182 .2072 L .86287 .2072 L .86525 .2072 L .86783 .2072 L .87018 .2072 L .87546 .20719 L .8849 .20717 L .89384 .20715 L .91373 .20708 L .93214 .207 L .97137 .20681 L .97619 .20678 L Mfstroke 1 0 0 r .02381 .23874 m .06244 .23874 L .10458 .23874 L .14415 .23874 L .18221 .23874 L .22272 .23874 L .26171 .23874 L .30316 .23874 L .34309 .23874 L .3815 .23874 L .42237 .23874 L .46172 .23874 L .49955 .23874 L .53984 .23874 L .57861 .23874 L .61984 .23874 L .65954 .23874 L .69774 .23874 L .73838 .23874 L .77751 .23874 L .81909 .23874 L .85916 .23874 L .89771 .23874 L .93871 .23874 L .97619 .23874 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{62, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg40oooo000K0?ooo`030000003oool0oooo01d0oooo00<0003o0?ooo`3oool0hP3o ool001/0oooo00<000000?ooo`3oool0703oool00`000?l0oooo0?ooo`3S0?ooo`006`3oool00`00 0000oooo0?ooo`0L0?ooo`030000o`3oool0oooo0><0oooo000K0?ooo`030000003oool0oooo01/0 oooo00<0003o0?ooo`3oool0i03oool001/0oooo0P00000L0?ooo`030000o`3oool0oooo0>@0oooo 000K0?ooo`030000003oool0oooo01X0oooo00<0003o0?ooo`3oool0i@3oool001/0oooo00<00000 0?ooo`3oool06P3oool00`000?l0oooo0?ooo`3U0?ooo`006`3oool00`000000oooo0?ooo`0I0?oo o`030000o`3oool0oooo0>H0oooo000K0?ooo`030000003oool0oooo01T0oooo00<0003o0?ooo`3o ool0iP3oool001/0oooo0P00000I0?ooo`030000o`3oool0oooo0>L0oooo000K0?ooo`030000003o ool0oooo01P0oooo00<0003o0?ooo`3oool0i`3oool001/0oooo00<000000?ooo`3oool05`3oool0 0`000?l0oooo0?ooo`3X0?ooo`006`3oool00`000000oooo0?ooo`0G0?ooo`030000o`3oool0oooo 0>P0oooo000K0?ooo`030000003oool0oooo01H0oooo00<0003o0?ooo`3oool0j@3oool001/0oooo 00<000000?ooo`3oool05P3oool00`000?l0oooo0?ooo`3Y0?ooo`006`3oool2000001H0oooo00<0 003o0?ooo`3oool0jP3oool001/0oooo00<000000?ooo`3oool05@3oool00`000?l0oooo0?ooo`3Z 0?ooo`006`3oool00`000000oooo0?ooo`0D0?ooo`030000o`3oool0oooo0>/0oooo000K0?ooo`03 0000003oool0oooo01@0oooo00<0003o0?ooo`3oool0j`3oool001<0oooo100000040?ooo`030000 003oool0oooo01<0oooo00<0003o0?ooo`3oool0k03oool001D0oooo00<000000?ooo`3oool00`3o ool2000001@0oooo00<0003o0?ooo`3oool0k03oool001D0oooo00<000000?ooo`3oool00`3oool0 0`000000oooo0?ooo`0B0?ooo`030000o`3oool0oooo0>d0oooo000E0?ooo`030000003oool0oooo 00<0oooo00<000000?ooo`3oool04P3oool00`000?l0oooo0?ooo`3]0?ooo`005@3oool00`000000 oooo0?ooo`030?ooo`030000003oool0oooo0140oooo00<0003o0?ooo`3oool0kP3oool001<0oooo 0`0000050?ooo`030000003oool0oooo0140oooo00<0003o0?ooo`3oool0kP3oool001D0oooo00<0 00000?ooo`3oool00`3oool00`000000oooo0?ooo`0@0?ooo`030000o`3oool0oooo0>l0oooo000K 0?ooo`8000004@3oool00`000?l0oooo0?ooo`3_0?ooo`006`3oool00`000000oooo0?ooo`0@0?oo o`030000o`3oool0oooo0>l0oooo000K0?ooo`030000003oool0oooo00l0oooo00<0003o0?ooo`3o ool0l03oool001/0oooo00<000000?ooo`3oool03`3oool00`000?l0oooo0?ooo`3`0?ooo`006`3o ool00`000000oooo0?ooo`0>0?ooo`030000o`3oool0oooo0?40oooo000K0?ooo`8000003`3oool0 0`000?l0oooo0?ooo`3a0?ooo`006`3oool00`000000oooo0?ooo`0>0?ooo`030000o`3oool0oooo 0?40oooo000K0?ooo`030000003oool0oooo00d0oooo00<0003o0?ooo`3oool0lP3oool001/0oooo 00<000000?ooo`3oool03@3oool00`000?l0oooo0?ooo`3b0?ooo`006`3oool00`000000oooo0?oo o`0<0?ooo`030000o`3oool0oooo0?<0oooo000K0?ooo`030000003oool0oooo00`0oooo00<0003o 0?ooo`3oool0l`3oool001/0oooo0P00000<0?ooo`030000o`3oool0oooo0?@0oooo000K0?ooo`03 0000003oool0oooo00/0oooo00<0003o0?ooo`3oool0m03oool001/0oooo00<000000?ooo`3oool0 2`3oool00`000?l0oooo0?ooo`3d0?ooo`006`3oool00`000000oooo0?ooo`0:0?ooo`030000o`3o ool0oooo0?D0oooo000K0?ooo`030000003oool0oooo00X0oooo00<0003o0?ooo`3oool0m@3oool0 01/0oooo0P00000:0?ooo`030000o`3oool0oooo0?H0oooo000K0?ooo`030000003oool0oooo00T0 oooo00<0003o0?ooo`3oool0mP3oool001/0oooo00<000000?ooo`3oool02@3oool00`000?l0oooo 0?ooo`3f0?ooo`006`3oool00`000000oooo0?ooo`080?ooo`030000o`3oool0oooo0?L0oooo000K 0?ooo`030000003oool0oooo00P0oooo00<0003o0?ooo`3oool0m`3oool000L0oooo100000030?oo o`030000003oool0oooo0080oooo0`0000050?ooo`030000003oool0oooo00L0oooo00<0003o0?oo o`3oool0n03oool000T0oooo00<000000?ooo`3oool02P3oool00`000000oooo0?ooo`020?ooo`80 0000203oool00`000?l0oooo0?ooo`3h0?ooo`002@3oool00`000000oooo0?ooo`0:0?ooo`030000 003oool0oooo0080oooo00<000000?ooo`3oool01`3oool00`000?l0oooo0?ooo`3h0?ooo`002@3o ool00`000000oooo0?ooo`070?ooo`<000001@3oool00`000000oooo0?ooo`060?ooo`030000o`3o ool0oooo0?T0oooo00090?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool01@3oool0 0`000000oooo0?ooo`060?ooo`030000o`3oool0oooo0?T0oooo00070?ooo`<000002@3oool00`00 0000oooo0?ooo`050?ooo`030000003oool0oooo00H0oooo00<0003o0?ooo`3oool0n@3oool000T0 oooo00<000000?ooo`3oool01`3oool4000000@0oooo0P0000060?ooo`030000o`3oool0oooo0?X0 oooo000K0?ooo`030000003oool0oooo00D0oooo00<0003o0?ooo`3oool0nP3oool001/0oooo00<0 00000?ooo`3oool0103oool00`000?l0oooo0?ooo`3k0?ooo`006`3oool00`000000oooo0?ooo`04 0?ooo`030000o`3oool0oooo0?/0oooo000K0?ooo`030000003oool0oooo00@0oooo00<0003o0?oo o`3oool0n`3oool001/0oooo00<000000?ooo`3oool00`3oool00`000?l0oooo0?ooo`3l0?ooo`00 6`3oool2000000@0oooo00<0003o0?ooo`3oool0o03oool001/0oooo00<000000?ooo`3oool00`3o ool00`000?l0oooo0?ooo`3l0?ooo`006`3oool00`000000oooo0?ooo`020?ooo`030000o`3oool0 oooo0?d0oooo000K0?ooo`030000003oool0oooo0080oooo00<0003o0?ooo`3oool0o@3oool001/0 oooo00D000000?ooo`3oool0oooo0000o`3o0?ooo`40oooo000K0?ooo`8000000P3oool00`000?l0 oooo0?ooo`3n0?ooo`006`3oool01@000000oooo0?ooo`3oool0003o0?l0oooo0@3oool001/0oooo 00@000000?ooo`3oool0003oo`3oool20?ooo`006`3oool010000000oooo0?ooo`000?oo0?ooo`80 oooo000K0?ooo`040000003oool0oooo0000ool0oooo0P3oool001/0oooo00<000000?ooo`000?l0 o`3oool30?ooo`006`3oool2000000030000o`3oool0oooo0?l0oooo0@3oool001/0oooo00<00000 0000o`3oool0o`3oool30?ooo`006`3oool00`000000003o0?ooo`3o0?ooo`<0oooo000K0?ooo`03 000000000?l0oooo0?l0oooo0`3oool001<0oooo100000040?ooo`030000o`3oool0oooo0?l0oooo 0`3oool001<0oooo00<000000?ooo`3oool01@3oool00`000?l000000?ooo`3o0?ooo`<0oooo000D 0?ooo`030000003oool0oooo0?l0oooo2P3oool001D0oooo00<000000?ooo`3oool0o`3oool90?oo o`005P3oool00`000000oooo0?ooo`3o0?ooo`P0oooo000C0?ooo`040000003oool0oooo00000?l0 oooo2P3oool001@0oooo0P00003o0?ooo`/0oooo003o0?ooob40oooo0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.677733, -1.1144, 0.0248015, 0.0182481}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1024}, {0, 768}}, ScreenStyleEnvironment->"Working", WindowSize->{520, 600}, WindowMargins->{{178, Automatic}, {-15, Automatic}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "Magnification"->1} ] (*********************************************************************** 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, 1222, 38, 94, "Text"], Cell[CellGroupData[{ Cell[3062, 99, 124, 2, 43, "Input"], Cell[3189, 103, 93, 1, 44, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3319, 109, 150, 3, 44, "Input"], Cell[3472, 114, 76, 1, 45, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3585, 120, 78, 1, 45, "Input"], Cell[3666, 123, 55, 1, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3758, 129, 251, 4, 81, "Input"], Cell[4012, 135, 18263, 552, 186, 5567, 391, "GraphicsData", "PostScript", "Graphics"], Cell[22278, 689, 130, 3, 22, "Output"] }, Open ]] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)