(*********************************************************************** 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). 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[ 349006, 19569]*) (*NotebookOutlinePosition[ 394441, 21143]*) (* CellTagsIndexPosition[ 394275, 21134]*) (*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[{ "Accumulation\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[{ "2.01 Integrals for Measuring Area\n", StyleBox["Samples", FontSize->16, FontSlant->"Italic"] }], "Title"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Initializations" }], "Special2"], Cell[BoxData[{ \(\(Off[General::spell]; \)\), \(\(Off[General::spell1]; \)\), \(\(Off[Plot::plnr]; \)\), \(\(Off[ParametricPlot3D::ppcom]; \)\), \(SetOptions[Limit, Analytic \[Rule] True]; \n Needs["\"]; \nNeeds["\"]; \n\n If[MemberQ[{"\", \ "\", \ "\"}, \n\ \ \ \ Context[Gray]], \ Remove["\<*`Gray\>"]]; \n<< "\"; \n Remove["\<*`Gray\>"]; \n<< "\"; \n Graphics`Colors`GosiaGreen = RGBColor[0, \ 0.392187, \ 0]; \)}], "Input", InitializationCell->True] }, Closed]], Cell["Basic Problems", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "B.1) ", Cell[BoxData[ \(\[Integral]\_a\%b f[x] \[DifferentialD]x\)]], " measures the signed area between \n the plot of ", Cell[BoxData[ \(f[x]\)]], " and the ", Cell[BoxData[ \(x\)]], "-axis for ", Cell[BoxData[ \(a \[LessEqual] x \[LessEqual] b\)]], ". " }], "Subsection", CellTags->"2.01.B1"], Cell[TextData[{ "Calculus&", StyleBox["Mathematica", FontSlant->"Italic"], " is pleased to acknowledge \nthe heavy influence of Emil Artin's book \n\ Calculus with Analytic Geometry (notes by G. B. Seligman), \nCommittee on the \ Undergraduate Program, \nMathematical Asociation of America, 1957. \nAlthough \ dated in spots, this short book remains a jewel." }], "Special2"], Cell[TextData[{ "Here is the idea of the integral:\nTake a function ", Cell[BoxData[ \(y = f[x]\)]], " and plot it on an interval ", Cell[BoxData[ \(\([a, b]\)\)]], ". \nIt might look something like this:" }], "Text"], Cell["\<\ If any picture is missing, then execute all the blank cells \ below.\ \>", "Special2"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, q];\)\), "\n", \(\(f[x_] = x\^2\/\(1 + x\^2\);\)\), "\n", \(\(q[x_] = \n\t ExpandAll[\((2.3 + 2.5\ Cos[2\ x]\ \ f[Cos[0.8\ x]])\)\ Sin[ x + Sin[0.8\ x]] + 1.85\ Exp[\(-\((7.85 - x)\)\^2\)]];\)\), "\n", \(\(a = 0.5;\)\), "\n", \(\(b = 10.5;\)\), "\n", \(\(x1 = x /. FindRoot[q[x] == 0, {x, 2}];\)\), "\n", \(\(x2 = x /. FindRoot[q[x] == 0, {x, 7}];\)\), "\n", \(\(x3 = x /. FindRoot[q[x] == 0, {x, 9}];\)\n\), "\n", \(\(graph = Plot[q[x], {x, a, b}, PlotStyle -> {{Thickness[0.01], Blue}}, PlotRange \[Rule] {{a - 1, b + 1}, All}, Ticks \[Rule] {{}, {}}, AxesOrigin \[Rule] {0, 0}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\n\), "\n", \(\(labels = Graphics[{Text["\", {a, \(- .3\)}], Text["\", {b, \(+ .3\)}], Text["\", {a + 3, 2}]}];\)\), "\n", \(\(ticks = Graphics[Line[{{a, 0}, {a, 0.1}}], Line[{{b, 0}, {b, 0.1}}]];\)\n\), "\n", \(\(Show[graph, ticks, labels, AxesLabel \[Rule] {"\", "\"}, AspectRatio \[Rule] Automatic, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\"}, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{a - 1, b + 1}, All}, Ticks \[Rule] None, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input", CellOpen->False, AspectRatioFixed->True, AnimationDisplayTime->1], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .45589 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 0.0416667 0.0833333 0.203362 0.0833333 [ [1.025 .20336 0 -6 ] [1.025 .20336 10 6 ] [.04167 .48089 -5 0 ] [.04167 .48089 5 12 ] [ 0 0 0 0 ] [ 1 .45589 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash 0 .20336 m 1 .20336 L s gsave 1.025 .20336 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .04167 0 m .04167 .45589 L s gsave .04167 .48089 -66 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (y) 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 0 0 m 1 0 L 1 .45589 L 0 .45589 L closepath clip newpath 0 0 1 r .01 w .08333 .39236 m .08524 .39496 L .08732 .39718 L .08837 .39805 L .0895 .3988 L .09056 .39935 L .09153 .39972 L .09208 .39987 L .09239 .39994 L .09267 .39999 L .09294 .40003 L .09318 .40005 L .09344 .40007 L .09359 .40008 L .09373 .40009 L .09388 .40009 L .09402 .40009 L .09418 .40009 L .09432 .40008 L .09459 .40007 L .09474 .40006 L .09488 .40004 L .09516 .40001 L .09548 .39996 L .09612 .39982 L .09669 .39965 L .09721 .39948 L .09837 .39897 L .10047 .39772 L .10257 .39606 L .10451 .3942 L .10892 .38905 L .11725 .37703 L .12526 .36455 L .13378 .35203 L .14199 .34143 L .15072 .33143 L .15912 .32201 L .16721 .31191 L .17581 .29893 L .18409 .28383 L .19289 .26535 L .20137 .24622 L .20953 .2278 L .21821 .2092 L .22657 .19267 L .23461 .17785 L .24317 .16261 L .2514 .1478 L .26016 .13139 L Mistroke .26859 .11485 L .2767 .0986 L .28534 .08165 L .29365 .06652 L .30248 .05273 L .30691 .04698 L .31099 .04251 L .31523 .03875 L .31918 .0361 L .32111 .03511 L .32318 .03428 L .32436 .03392 L .32547 .03364 L .32648 .03346 L .32704 .03337 L .32757 .03331 L .32782 .03329 L .32808 .03327 L .32831 .03326 L .32855 .03324 L .32881 .03323 L .32906 .03323 L .32921 .03323 L .32935 .03322 L .32961 .03323 L .32985 .03323 L .33009 .03324 L .3303 .03325 L .33053 .03326 L .33105 .0333 L .33153 .03335 L .33264 .03351 L .33364 .03372 L .3348 .03402 L .33591 .03437 L .33795 .03519 L .34012 .03629 L .34403 .03883 L .35203 .04608 L .36055 .05625 L .36875 .0676 L .37747 .08034 L .38587 .09227 L .39395 .10251 L .39845 .1074 L .40254 .11119 L .40685 .11442 L .41082 .11664 L .41279 .11745 L .41488 .11809 L Mistroke .41597 .11833 L .41698 .1185 L .41756 .11857 L .41809 .11862 L .41839 .11864 L .41854 .11865 L .41871 .11866 L .41887 .11866 L .41901 .11867 L .41916 .11867 L .41929 .11867 L .41954 .11868 L .4198 .11867 L .42005 .11867 L .42028 .11866 L .42055 .11864 L .42084 .11863 L .42137 .11858 L .42192 .11851 L .4225 .11842 L .42355 .11822 L .42542 .11769 L .42747 .11689 L .42962 .11579 L .43158 .11456 L .43601 .11101 L .44442 .10145 L .4525 .08927 L .46111 .07403 L .4694 .05837 L .4782 .04226 L .48669 .02882 L .49485 .01904 L .49907 .01545 L .5012 .01404 L .50354 .01281 L .5057 .01194 L .50672 .01163 L .50767 .01139 L .50875 .01117 L .50933 .01108 L .50994 .01101 L .51022 .01098 L .51052 .01096 L .51079 .01094 L .51107 .01092 L .51133 .01091 L .51161 .0109 L .51186 .0109 L .5121 .0109 L Mistroke .51237 .0109 L .5125 .0109 L .51265 .0109 L .51292 .01091 L .51317 .01092 L .51372 .01096 L .51432 .01101 L .51542 .01115 L .51643 .01132 L .51845 .01178 L .52037 .01235 L .52457 .01397 L .52847 .01584 L .53676 .0205 L .54558 .02611 L .55408 .03269 L .56225 .04153 L .56648 .04771 L .57095 .0558 L .579 .0753 L .58757 .10391 L .59582 .13906 L .60459 .18315 L .61304 .22959 L .62117 .27485 L .62982 .32029 L .63815 .35858 L .64615 .38859 L .6506 .4021 L .65468 .4125 L .65893 .4214 L .66288 .42804 L .66689 .43331 L .67128 .43762 L .67351 .43931 L .67562 .44063 L .67956 .44251 L .68162 .44323 L .68381 .44382 L .68567 .44422 L .68772 .44455 L .68882 .44469 L .68987 .4448 L .69083 .44488 L .69184 .44494 L .69242 .44497 L .69294 .44499 L .69347 .44501 L .69396 .44502 L .69424 .44503 L Mistroke .6944 .44503 L .69454 .44503 L .69479 .44503 L .69507 .44503 L .69523 .44503 L .69538 .44503 L .69565 .44503 L .69581 .44503 L .69595 .44503 L .69627 .44502 L .69652 .44502 L .69679 .44501 L .69726 .445 L .69777 .44498 L .69832 .44496 L .69931 .4449 L .70024 .44483 L .70235 .44462 L .70354 .44447 L .70461 .4443 L .70659 .44393 L .70869 .44342 L .71098 .44272 L .7131 .4419 L .71726 .43972 L .72115 .43686 L .72546 .43253 L .72939 .42729 L .73343 .42045 L .73783 .41113 L .74647 .38659 L .75479 .35495 L .76279 .31785 L .77131 .27304 L .77951 .22745 L .78822 .17976 L .79662 .13784 L .80469 .10357 L .81329 .07494 L .82156 .05506 L .82608 .04699 L .83035 .04088 L .8385 .03224 L .84717 .02566 L .85552 .0204 L .86355 .01592 L .86801 .01379 L .8721 .01224 L .87411 .01166 L .87518 .01141 L Mistroke .87632 .01119 L .87688 .0111 L .87741 .01103 L .87792 .01097 L .87839 .01093 L .87885 .0109 L .87908 .01088 L .87932 .01087 L .87957 .01086 L .8798 .01086 L .88007 .01085 L .88032 .01085 L .88057 .01086 L .88081 .01086 L .88107 .01087 L .88134 .01089 L .88162 .0109 L .88191 .01093 L .88243 .01098 L .88336 .0111 L .88437 .01129 L .88548 .01156 L .88666 .01192 L .88878 .01277 L .89095 .01391 L .89331 .01547 L .89758 .01912 L .9018 .02376 L .90626 .02969 L .9143 .04255 L .91667 .04674 L Mfstroke 0 g .5 Mabswid .08333 .20336 m .08333 .2117 L s gsave .08333 .17836 -66 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (a) 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 gsave .91667 .22836 -66 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (b) 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 gsave .33333 .37003 -87 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (y) show 75.000 13.000 moveto %%IncludeResource: font Math1Mono %%IncludeFont: Math1Mono /Math1Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (=) show 87.000 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 93.000 13.000 moveto (@) show 99.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 105.000 13.000 moveto (D) show 111.000 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, 131.25}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "Look at the area between the graph of ", Cell[BoxData[ \(y = f[x]\)]], " and the ", Cell[BoxData[ \(x\)]], "-axis:" }], "Text", AnimationDisplayTime->1], Cell[CellGroupData[{ Cell[BoxData[{ \(\(rad = 0.4;\)\), "\n", \(\(colordisks = \ Ivory;\)\), "\n", \(\(center1 := {\(1.5\ a + x1\)\/2.5, 1\/2\ q[\(a + x1\)\/2]};\)\), "\n", \(\(radius1 := rad;\)\), "\n", \(\(circle1 := Graphics[{colordisks, Disk[center1, radius1]}];\)\), "\n", \(\(center2 := {1\/4\ \((x1 + 3\ x2)\), 1\/2\ q[1\/4\ \((x1 + 3\ x2)\)]};\)\), "\n", \(\(radius2 := rad;\)\), "\n", \(\(circle2 := Graphics[{colordisks, Disk[center2, radius2]}];\)\), "\n", \(\(center3 := {\(x2 + x3\)\/2, 1\/2\ q[\(x2 + x3\)\/2]};\)\), "\n", \(\(radius3 := rad;\)\), "\n", \(\(circle3 := Graphics[{colordisks, Disk[center3, radius3]}];\)\), "\n", \(\(center4 := {\(1.5\ b + x3\)\/2.5, 1\/2\ q[\(1.5\ b + x3\)\/2.5]};\)\), "\n", \(\(radius4 := radius;\)\), "\n", \(\(circle4 := Graphics[{colordisks, Disk[center4, radius4]}];\)\), "\n", \(\(oncurve = Table[{t, q[t]}, {t, a, b, .1}];\)\), "\n", \(\(Clear[vertices];\)\), "\n", \(\(curveplusright = Join[oncurve, {{b, q[b]}, {b, 0}}];\)\), "\n", \(\(vertices = PrependTo[curveplusright, {a, 0}];\)\), "\n", \(\(shade = Graphics[{GrayLevel[ .75], Polygon[vertices]}];\)\), "\n", \(\(shadeColor = Graphics[{RGBColor[ .75, .75, 1], Polygon[vertices]}];\)\), "\n", \(\(dots = Graphics[{PointSize[ .02], Point /@ {{a, 0}, {x1, 0}, {x2, 0}, {x3, 0}, {b, 0}}}];\)\), "\n", \(\(line = Graphics[Line[{{a, 0}, {b, 0}}]];\)\), "\n", \(\(Clear[cross, thick];\)\), "\n", \(cross[{a_, b_}, radius_, thick_] := Polygon[{{a + radius, b + thick\ radius}, {a + thick\ radius, b + thick\ radius}, {a + thick\ radius, b + radius}, {a - thick\ radius, b + radius}, {a - thick\ radius, b + thick\ radius}, {a - radius, b + thick\ radius}, {a - radius, b - thick\ radius}, {a - thick\ radius, b - thick\ radius}, {a - thick\ radius, b - radius}, {a + thick\ radius, b - radius}, {a + thick\ radius, b - thick\ radius}, {a + radius, b - thick\ radius}}]\), "\n", \(\(Clear[dash];\)\), "\n", \(\(dash[{a_, b_}, radius_, thick_] := Polygon[{{a + radius, b + thick\ radius}, {a - radius, b + thick\ radius}, {a - radius, b - thick\ radius}, {a + radius, b - thick\ radius}}];\)\), "\n", \(\(plus[{a_, b_}, radius_, thick_] := Graphics[{RGBColor[1, 1, 1], Disk[{a, b}, 1.4\ radius], RGBColor[0, 0, 0], cross[{a, b}, radius, thick]}];\)\), "\n", \(\(minus[{a_, b_}, radius_, thick_] := Graphics[{RGBColor[1, 1, 1], Disk[{a, b}, 1.4\ radius], RGBColor[0, 0, 0], dash[{a, b}, radius, thick]}];\)\), "\n", \(\(signs1 := {plus[center1, 0.3, 0.15], minus[center2, 0.3, 0.15], plus[center3, 0.3, 0.15], minus[center4, 0.3, 0.15]};\)\), "\n", \(\(Show[shade, graph, line, ticks, labels, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\"}, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{a - 1, b + 1}, All}, Ticks \[Rule] None, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input", CellOpen->False, AspectRatioFixed->True, AnimationDisplayTime->1], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .45589 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 0.0416667 0.0833333 0.203362 0.0833333 [ [1.025 .20336 0 -6 ] [1.025 .20336 10 6 ] [.04167 .48089 -5 0 ] [.04167 .48089 5 12 ] [ 0 0 0 0 ] [ 1 .45589 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash 0 .20336 m 1 .20336 L s gsave 1.025 .20336 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .04167 0 m .04167 .45589 L s gsave .04167 .48089 -66 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (y) 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 0 0 m 1 0 L 1 .45589 L 0 .45589 L closepath clip newpath .75 g .08333 .20336 m .08333 .39236 L .09167 .39976 L .1 .39803 L .10833 .3898 L .11667 .37792 L .125 .36495 L .13333 .35265 L .14167 .34182 L .15 .33223 L .15833 .32293 L .16667 .31264 L .175 .30027 L .18333 .28531 L .19167 .26804 L .2 .24936 L .20833 .23047 L .21667 .21242 L .225 .19568 L .23333 .18015 L .24167 .16527 L .25 .15036 L .25833 .13487 L .26667 .11867 L .275 .10202 L .28333 .08551 L .29167 .06998 L .3 .05631 L .30833 .04534 L .31667 .03769 L .325 .03375 L .33333 .03365 L .34167 .03721 L .35 .04401 L .35833 .05341 L .36667 .06462 L .375 .07672 L .38333 .08876 L .39167 .09979 L .4 .10891 L .40833 .11534 L .41667 .11846 L .425 .11783 L .43333 .11329 L .44167 .10496 L .45 .09331 L .45833 .07913 L .46667 .06355 L .475 .04795 L .48333 .03379 L .49167 .02243 L .5 .0148 L .50833 .01125 L .51667 .01136 L .525 .01417 L .53333 .01849 L .54167 .02353 L .55 .02931 L .55833 .03687 L .56667 .04801 L .575 .06476 L .58333 .08872 L .59167 .12047 L .6 .15934 L .60833 .20341 L .61667 .24988 L .625 .29551 L .63333 .33723 L .64167 .37266 L .65 .40042 L .65833 .42028 L .66667 .43305 L .675 .44026 L .68333 .44371 L .69167 .44493 L .7 .44485 L .70833 .44352 L .71667 .44009 L .725 .43305 L .73333 .42064 L .74167 .40129 L .75 .37412 L .75833 .33926 L .76667 .29799 L .775 .25264 L .78333 .20621 L .79167 .16193 L .8 .12268 L .80833 .09042 L .81667 .06595 L .825 .04876 L .83333 .03733 L .84167 .02962 L .85 .02381 L .85833 .01876 L .86667 .01439 L .875 .01145 L .88333 .0111 L .89167 .01435 L .9 .02166 L .90833 .03276 L .91667 .04674 L .91667 .04674 L .91667 .20336 L F 0 0 1 r .01 w .08333 .39236 m .08524 .39496 L .08732 .39718 L .08837 .39805 L .0895 .3988 L .09056 .39935 L .09153 .39972 L .09208 .39987 L .09239 .39994 L .09267 .39999 L .09294 .40003 L .09318 .40005 L .09344 .40007 L .09359 .40008 L .09373 .40009 L .09388 .40009 L .09402 .40009 L .09418 .40009 L .09432 .40008 L .09459 .40007 L .09474 .40006 L .09488 .40004 L .09516 .40001 L .09548 .39996 L .09612 .39982 L .09669 .39965 L .09721 .39948 L .09837 .39897 L .10047 .39772 L .10257 .39606 L .10451 .3942 L .10892 .38905 L .11725 .37703 L .12526 .36455 L .13378 .35203 L .14199 .34143 L .15072 .33143 L .15912 .32201 L .16721 .31191 L .17581 .29893 L .18409 .28383 L .19289 .26535 L .20137 .24622 L .20953 .2278 L .21821 .2092 L .22657 .19267 L .23461 .17785 L .24317 .16261 L .2514 .1478 L .26016 .13139 L Mistroke .26859 .11485 L .2767 .0986 L .28534 .08165 L .29365 .06652 L .30248 .05273 L .30691 .04698 L .31099 .04251 L .31523 .03875 L .31918 .0361 L .32111 .03511 L .32318 .03428 L .32436 .03392 L .32547 .03364 L .32648 .03346 L .32704 .03337 L .32757 .03331 L .32782 .03329 L .32808 .03327 L .32831 .03326 L .32855 .03324 L .32881 .03323 L .32906 .03323 L .32921 .03323 L .32935 .03322 L .32961 .03323 L .32985 .03323 L .33009 .03324 L .3303 .03325 L .33053 .03326 L .33105 .0333 L .33153 .03335 L .33264 .03351 L .33364 .03372 L .3348 .03402 L .33591 .03437 L .33795 .03519 L .34012 .03629 L .34403 .03883 L .35203 .04608 L .36055 .05625 L .36875 .0676 L .37747 .08034 L .38587 .09227 L .39395 .10251 L .39845 .1074 L .40254 .11119 L .40685 .11442 L .41082 .11664 L .41279 .11745 L .41488 .11809 L Mistroke .41597 .11833 L .41698 .1185 L .41756 .11857 L .41809 .11862 L .41839 .11864 L .41854 .11865 L .41871 .11866 L .41887 .11866 L .41901 .11867 L .41916 .11867 L .41929 .11867 L .41954 .11868 L .4198 .11867 L .42005 .11867 L .42028 .11866 L .42055 .11864 L .42084 .11863 L .42137 .11858 L .42192 .11851 L .4225 .11842 L .42355 .11822 L .42542 .11769 L .42747 .11689 L .42962 .11579 L .43158 .11456 L .43601 .11101 L .44442 .10145 L .4525 .08927 L .46111 .07403 L .4694 .05837 L .4782 .04226 L .48669 .02882 L .49485 .01904 L .49907 .01545 L .5012 .01404 L .50354 .01281 L .5057 .01194 L .50672 .01163 L .50767 .01139 L .50875 .01117 L .50933 .01108 L .50994 .01101 L .51022 .01098 L .51052 .01096 L .51079 .01094 L .51107 .01092 L .51133 .01091 L .51161 .0109 L .51186 .0109 L .5121 .0109 L Mistroke .51237 .0109 L .5125 .0109 L .51265 .0109 L .51292 .01091 L .51317 .01092 L .51372 .01096 L .51432 .01101 L .51542 .01115 L .51643 .01132 L .51845 .01178 L .52037 .01235 L .52457 .01397 L .52847 .01584 L .53676 .0205 L .54558 .02611 L .55408 .03269 L .56225 .04153 L .56648 .04771 L .57095 .0558 L .579 .0753 L .58757 .10391 L .59582 .13906 L .60459 .18315 L .61304 .22959 L .62117 .27485 L .62982 .32029 L .63815 .35858 L .64615 .38859 L .6506 .4021 L .65468 .4125 L .65893 .4214 L .66288 .42804 L .66689 .43331 L .67128 .43762 L .67351 .43931 L .67562 .44063 L .67956 .44251 L .68162 .44323 L .68381 .44382 L .68567 .44422 L .68772 .44455 L .68882 .44469 L .68987 .4448 L .69083 .44488 L .69184 .44494 L .69242 .44497 L .69294 .44499 L .69347 .44501 L .69396 .44502 L .69424 .44503 L Mistroke .6944 .44503 L .69454 .44503 L .69479 .44503 L .69507 .44503 L .69523 .44503 L .69538 .44503 L .69565 .44503 L .69581 .44503 L .69595 .44503 L .69627 .44502 L .69652 .44502 L .69679 .44501 L .69726 .445 L .69777 .44498 L .69832 .44496 L .69931 .4449 L .70024 .44483 L .70235 .44462 L .70354 .44447 L .70461 .4443 L .70659 .44393 L .70869 .44342 L .71098 .44272 L .7131 .4419 L .71726 .43972 L .72115 .43686 L .72546 .43253 L .72939 .42729 L .73343 .42045 L .73783 .41113 L .74647 .38659 L .75479 .35495 L .76279 .31785 L .77131 .27304 L .77951 .22745 L .78822 .17976 L .79662 .13784 L .80469 .10357 L .81329 .07494 L .82156 .05506 L .82608 .04699 L .83035 .04088 L .8385 .03224 L .84717 .02566 L .85552 .0204 L .86355 .01592 L .86801 .01379 L .8721 .01224 L .87411 .01166 L .87518 .01141 L Mistroke .87632 .01119 L .87688 .0111 L .87741 .01103 L .87792 .01097 L .87839 .01093 L .87885 .0109 L .87908 .01088 L .87932 .01087 L .87957 .01086 L .8798 .01086 L .88007 .01085 L .88032 .01085 L .88057 .01086 L .88081 .01086 L .88107 .01087 L .88134 .01089 L .88162 .0109 L .88191 .01093 L .88243 .01098 L .88336 .0111 L .88437 .01129 L .88548 .01156 L .88666 .01192 L .88878 .01277 L .89095 .01391 L .89331 .01547 L .89758 .01912 L .9018 .02376 L .90626 .02969 L .9143 .04255 L .91667 .04674 L Mfstroke 0 g .5 Mabswid .08333 .20336 m .91667 .20336 L s .08333 .20336 m .08333 .2117 L s gsave .08333 .17836 -66 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (a) 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 gsave .91667 .22836 -66 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (b) 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 gsave .33333 .37003 -87 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (y) show 75.000 13.000 moveto %%IncludeResource: font Math1Mono %%IncludeFont: Math1Mono /Math1Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (=) show 87.000 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 93.000 13.000 moveto (@) show 99.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 105.000 13.000 moveto (D) show 111.000 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, 131.25}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "If a part of this area lies above the ", Cell[BoxData[ \(x\)]], "-axis, then call it positive. \nIf a part lies below the ", Cell[BoxData[ \(x\)]], "-axis, then call it negative. \nThis gives the signs as shown below:" }], "Text", AnimationDisplayTime->1], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[shade, graph, line, ticks, signs1, labels, AxesLabel \[Rule] {"\", "\"}, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\"}, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{a - 1, b + 1}, All}, Ticks \[Rule] None, DisplayFunction \[Rule] $DisplayFunction];\)\)], "Input", CellOpen->False, AspectRatioFixed->True, AnimationDisplayTime->1], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .45589 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 0.0416667 0.0833333 0.203362 0.0833333 [ [1.025 .20336 0 -6 ] [1.025 .20336 10 6 ] [.04167 .48089 -5 0 ] [.04167 .48089 5 12 ] [ 0 0 0 0 ] [ 1 .45589 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash 0 .20336 m 1 .20336 L s gsave 1.025 .20336 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .04167 0 m .04167 .45589 L s gsave .04167 .48089 -66 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (y) 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 0 0 m 1 0 L 1 .45589 L 0 .45589 L closepath clip newpath .75 g .08333 .20336 m .08333 .39236 L .09167 .39976 L .1 .39803 L .10833 .3898 L .11667 .37792 L .125 .36495 L .13333 .35265 L .14167 .34182 L .15 .33223 L .15833 .32293 L .16667 .31264 L .175 .30027 L .18333 .28531 L .19167 .26804 L .2 .24936 L .20833 .23047 L .21667 .21242 L .225 .19568 L .23333 .18015 L .24167 .16527 L .25 .15036 L .25833 .13487 L .26667 .11867 L .275 .10202 L .28333 .08551 L .29167 .06998 L .3 .05631 L .30833 .04534 L .31667 .03769 L .325 .03375 L .33333 .03365 L .34167 .03721 L .35 .04401 L .35833 .05341 L .36667 .06462 L .375 .07672 L .38333 .08876 L .39167 .09979 L .4 .10891 L .40833 .11534 L .41667 .11846 L .425 .11783 L .43333 .11329 L .44167 .10496 L .45 .09331 L .45833 .07913 L .46667 .06355 L .475 .04795 L .48333 .03379 L .49167 .02243 L .5 .0148 L .50833 .01125 L .51667 .01136 L .525 .01417 L .53333 .01849 L .54167 .02353 L .55 .02931 L .55833 .03687 L .56667 .04801 L .575 .06476 L .58333 .08872 L .59167 .12047 L .6 .15934 L .60833 .20341 L .61667 .24988 L .625 .29551 L .63333 .33723 L .64167 .37266 L .65 .40042 L .65833 .42028 L .66667 .43305 L .675 .44026 L .68333 .44371 L .69167 .44493 L .7 .44485 L .70833 .44352 L .71667 .44009 L .725 .43305 L .73333 .42064 L .74167 .40129 L .75 .37412 L .75833 .33926 L .76667 .29799 L .775 .25264 L .78333 .20621 L .79167 .16193 L .8 .12268 L .80833 .09042 L .81667 .06595 L .825 .04876 L .83333 .03733 L .84167 .02962 L .85 .02381 L .85833 .01876 L .86667 .01439 L .875 .01145 L .88333 .0111 L .89167 .01435 L .9 .02166 L .90833 .03276 L .91667 .04674 L .91667 .04674 L .91667 .20336 L F 0 0 1 r .01 w .08333 .39236 m .08524 .39496 L .08732 .39718 L .08837 .39805 L .0895 .3988 L .09056 .39935 L .09153 .39972 L .09208 .39987 L .09239 .39994 L .09267 .39999 L .09294 .40003 L .09318 .40005 L .09344 .40007 L .09359 .40008 L .09373 .40009 L .09388 .40009 L .09402 .40009 L .09418 .40009 L .09432 .40008 L .09459 .40007 L .09474 .40006 L .09488 .40004 L .09516 .40001 L .09548 .39996 L .09612 .39982 L .09669 .39965 L .09721 .39948 L .09837 .39897 L .10047 .39772 L .10257 .39606 L .10451 .3942 L .10892 .38905 L .11725 .37703 L .12526 .36455 L .13378 .35203 L .14199 .34143 L .15072 .33143 L .15912 .32201 L .16721 .31191 L .17581 .29893 L .18409 .28383 L .19289 .26535 L .20137 .24622 L .20953 .2278 L .21821 .2092 L .22657 .19267 L .23461 .17785 L .24317 .16261 L .2514 .1478 L .26016 .13139 L Mistroke .26859 .11485 L .2767 .0986 L .28534 .08165 L .29365 .06652 L .30248 .05273 L .30691 .04698 L .31099 .04251 L .31523 .03875 L .31918 .0361 L .32111 .03511 L .32318 .03428 L .32436 .03392 L .32547 .03364 L .32648 .03346 L .32704 .03337 L .32757 .03331 L .32782 .03329 L .32808 .03327 L .32831 .03326 L .32855 .03324 L .32881 .03323 L .32906 .03323 L .32921 .03323 L .32935 .03322 L .32961 .03323 L .32985 .03323 L .33009 .03324 L .3303 .03325 L .33053 .03326 L .33105 .0333 L .33153 .03335 L .33264 .03351 L .33364 .03372 L .3348 .03402 L .33591 .03437 L .33795 .03519 L .34012 .03629 L .34403 .03883 L .35203 .04608 L .36055 .05625 L .36875 .0676 L .37747 .08034 L .38587 .09227 L .39395 .10251 L .39845 .1074 L .40254 .11119 L .40685 .11442 L .41082 .11664 L .41279 .11745 L .41488 .11809 L Mistroke .41597 .11833 L .41698 .1185 L .41756 .11857 L .41809 .11862 L .41839 .11864 L .41854 .11865 L .41871 .11866 L .41887 .11866 L .41901 .11867 L .41916 .11867 L .41929 .11867 L .41954 .11868 L .4198 .11867 L .42005 .11867 L .42028 .11866 L .42055 .11864 L .42084 .11863 L .42137 .11858 L .42192 .11851 L .4225 .11842 L .42355 .11822 L .42542 .11769 L .42747 .11689 L .42962 .11579 L .43158 .11456 L .43601 .11101 L .44442 .10145 L .4525 .08927 L .46111 .07403 L .4694 .05837 L .4782 .04226 L .48669 .02882 L .49485 .01904 L .49907 .01545 L .5012 .01404 L .50354 .01281 L .5057 .01194 L .50672 .01163 L .50767 .01139 L .50875 .01117 L .50933 .01108 L .50994 .01101 L .51022 .01098 L .51052 .01096 L .51079 .01094 L .51107 .01092 L .51133 .01091 L .51161 .0109 L .51186 .0109 L .5121 .0109 L Mistroke .51237 .0109 L .5125 .0109 L .51265 .0109 L .51292 .01091 L .51317 .01092 L .51372 .01096 L .51432 .01101 L .51542 .01115 L .51643 .01132 L .51845 .01178 L .52037 .01235 L .52457 .01397 L .52847 .01584 L .53676 .0205 L .54558 .02611 L .55408 .03269 L .56225 .04153 L .56648 .04771 L .57095 .0558 L .579 .0753 L .58757 .10391 L .59582 .13906 L .60459 .18315 L .61304 .22959 L .62117 .27485 L .62982 .32029 L .63815 .35858 L .64615 .38859 L .6506 .4021 L .65468 .4125 L .65893 .4214 L .66288 .42804 L .66689 .43331 L .67128 .43762 L .67351 .43931 L .67562 .44063 L .67956 .44251 L .68162 .44323 L .68381 .44382 L .68567 .44422 L .68772 .44455 L .68882 .44469 L .68987 .4448 L .69083 .44488 L .69184 .44494 L .69242 .44497 L .69294 .44499 L .69347 .44501 L .69396 .44502 L .69424 .44503 L Mistroke .6944 .44503 L .69454 .44503 L .69479 .44503 L .69507 .44503 L .69523 .44503 L .69538 .44503 L .69565 .44503 L .69581 .44503 L .69595 .44503 L .69627 .44502 L .69652 .44502 L .69679 .44501 L .69726 .445 L .69777 .44498 L .69832 .44496 L .69931 .4449 L .70024 .44483 L .70235 .44462 L .70354 .44447 L .70461 .4443 L .70659 .44393 L .70869 .44342 L .71098 .44272 L .7131 .4419 L .71726 .43972 L .72115 .43686 L .72546 .43253 L .72939 .42729 L .73343 .42045 L .73783 .41113 L .74647 .38659 L .75479 .35495 L .76279 .31785 L .77131 .27304 L .77951 .22745 L .78822 .17976 L .79662 .13784 L .80469 .10357 L .81329 .07494 L .82156 .05506 L .82608 .04699 L .83035 .04088 L .8385 .03224 L .84717 .02566 L .85552 .0204 L .86355 .01592 L .86801 .01379 L .8721 .01224 L .87411 .01166 L .87518 .01141 L Mistroke .87632 .01119 L .87688 .0111 L .87741 .01103 L .87792 .01097 L .87839 .01093 L .87885 .0109 L .87908 .01088 L .87932 .01087 L .87957 .01086 L .8798 .01086 L .88007 .01085 L .88032 .01085 L .88057 .01086 L .88081 .01086 L .88107 .01087 L .88134 .01089 L .88162 .0109 L .88191 .01093 L .88243 .01098 L .88336 .0111 L .88437 .01129 L .88548 .01156 L .88666 .01192 L .88878 .01277 L .89095 .01391 L .89331 .01547 L .89758 .01912 L .9018 .02376 L .90626 .02969 L .9143 .04255 L .91667 .04674 L Mfstroke 0 g .5 Mabswid .08333 .20336 m .91667 .20336 L s .08333 .20336 m .08333 .2117 L s 1 1 1 r .13844 .26657 m .13844 .26657 .035 0 365.73 arc F 0 0 0 r .16344 .27032 m .14219 .27032 L .14219 .29157 L .13469 .29157 L .13469 .27032 L .11344 .27032 L .11344 .26282 L .13469 .26282 L .13469 .24157 L .14219 .24157 L .14219 .26282 L .16344 .26282 L F 1 1 1 r .51152 .10713 m .51152 .10713 .035 0 365.73 arc F 0 0 0 r .53652 .11088 m .48652 .11088 L .48652 .10338 L .53652 .10338 L F 1 1 1 r .69609 .32419 m .69609 .32419 .035 0 365.73 arc F 0 0 0 r .72109 .32794 m .69984 .32794 L .69984 .34919 L .69234 .34919 L .69234 .32794 L .67109 .32794 L .67109 .32044 L .69234 .32044 L .69234 .29919 L .69984 .29919 L .69984 .32044 L .72109 .32044 L F 1 1 1 r .86354 .10964 m .86354 .10964 .035 0 365.73 arc F 0 0 0 r .88854 .11339 m .83854 .11339 L .83854 .10589 L .88854 .10589 L F 0 g gsave .08333 .17836 -66 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (a) 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 gsave .91667 .22836 -66 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (b) 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 gsave .33333 .37003 -87 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (y) show 75.000 13.000 moveto %%IncludeResource: font Math1Mono %%IncludeFont: Math1Mono /Math1Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (=) show 87.000 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 93.000 13.000 moveto (@) show 99.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 105.000 13.000 moveto (D) show 111.000 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, 131.25}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "Now define a number that all folks call the integral of ", Cell[BoxData[ \(f[x]\)]], " from ", Cell[BoxData[ \(a\)]], " to ", Cell[BoxData[ \(b\)]], ".\nYou signify this number by writing\n ", Cell[BoxData[ \(\[Integral]\_a\%b f[x] \[DifferentialD]x\)]], "\nand you calculate its value by summing up the measurements of these \ areas taken with the corresponding signs.\nAccordingly, if ", Cell[BoxData[ \(x\_1 < x\_2 < x\_3\)]], " are the points between ", Cell[BoxData[ \(a\)]], " and ", Cell[BoxData[ \(b\)]], " at which the curve crosses the ", Cell[BoxData[ \(x\)]], "-axis as shown below:" }], "Text", AnimationDisplayTime->1], Cell[CellGroupData[{ Cell[BoxData[{ \(\(morelabels = Graphics[{Text["\", {x1, \(- .4\)}], Text["\", {x2 - .3, \(+ .3\)}], Text["\", {x3 + .3, \(+ .4\)}]}];\)\), "\n", \(\(Show[morelabels, shadeColor, graph, line, ticks, dots, signs1, labels, AxesLabel \[Rule] {"\", "\"}, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\"}, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{a - 1, b + 1}, All}, Ticks \[Rule] None, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input", CellOpen->False, AspectRatioFixed->True, AnimationDisplayTime->1], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .45589 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 0.0416667 0.0833333 0.203362 0.0833333 [ [1.025 .20336 0 -6 ] [1.025 .20336 10 6 ] [.04167 .48089 -5 0 ] [.04167 .48089 5 12 ] [ 0 0 0 0 ] [ 1 .45589 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash 0 .20336 m 1 .20336 L s gsave 1.025 .20336 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .04167 0 m .04167 .45589 L s gsave .04167 .48089 -66 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (y) 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 0 0 m 1 0 L 1 .45589 L 0 .45589 L closepath clip newpath gsave .22109 .17003 -69 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x1) show 75.000 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 gsave .58332 .22836 -69 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x2) show 75.000 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 gsave .80885 .2367 -69 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x3) show 75.000 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 .75 .75 1 r .08333 .20336 m .08333 .39236 L .09167 .39976 L .1 .39803 L .10833 .3898 L .11667 .37792 L .125 .36495 L .13333 .35265 L .14167 .34182 L .15 .33223 L .15833 .32293 L .16667 .31264 L .175 .30027 L .18333 .28531 L .19167 .26804 L .2 .24936 L .20833 .23047 L .21667 .21242 L .225 .19568 L .23333 .18015 L .24167 .16527 L .25 .15036 L .25833 .13487 L .26667 .11867 L .275 .10202 L .28333 .08551 L .29167 .06998 L .3 .05631 L .30833 .04534 L .31667 .03769 L .325 .03375 L .33333 .03365 L .34167 .03721 L .35 .04401 L .35833 .05341 L .36667 .06462 L .375 .07672 L .38333 .08876 L .39167 .09979 L .4 .10891 L .40833 .11534 L .41667 .11846 L .425 .11783 L .43333 .11329 L .44167 .10496 L .45 .09331 L .45833 .07913 L .46667 .06355 L .475 .04795 L .48333 .03379 L .49167 .02243 L .5 .0148 L .50833 .01125 L .51667 .01136 L .525 .01417 L .53333 .01849 L .54167 .02353 L .55 .02931 L .55833 .03687 L .56667 .04801 L .575 .06476 L .58333 .08872 L .59167 .12047 L .6 .15934 L .60833 .20341 L .61667 .24988 L .625 .29551 L .63333 .33723 L .64167 .37266 L .65 .40042 L .65833 .42028 L .66667 .43305 L .675 .44026 L .68333 .44371 L .69167 .44493 L .7 .44485 L .70833 .44352 L .71667 .44009 L .725 .43305 L .73333 .42064 L .74167 .40129 L .75 .37412 L .75833 .33926 L .76667 .29799 L .775 .25264 L .78333 .20621 L .79167 .16193 L .8 .12268 L .80833 .09042 L .81667 .06595 L .825 .04876 L .83333 .03733 L .84167 .02962 L .85 .02381 L .85833 .01876 L .86667 .01439 L .875 .01145 L .88333 .0111 L .89167 .01435 L .9 .02166 L .90833 .03276 L .91667 .04674 L .91667 .04674 L .91667 .20336 L F 0 0 1 r .01 w .08333 .39236 m .08524 .39496 L .08732 .39718 L .08837 .39805 L .0895 .3988 L .09056 .39935 L .09153 .39972 L .09208 .39987 L .09239 .39994 L .09267 .39999 L .09294 .40003 L .09318 .40005 L .09344 .40007 L .09359 .40008 L .09373 .40009 L .09388 .40009 L .09402 .40009 L .09418 .40009 L .09432 .40008 L .09459 .40007 L .09474 .40006 L .09488 .40004 L .09516 .40001 L .09548 .39996 L .09612 .39982 L .09669 .39965 L .09721 .39948 L .09837 .39897 L .10047 .39772 L .10257 .39606 L .10451 .3942 L .10892 .38905 L .11725 .37703 L .12526 .36455 L .13378 .35203 L .14199 .34143 L .15072 .33143 L .15912 .32201 L .16721 .31191 L .17581 .29893 L .18409 .28383 L .19289 .26535 L .20137 .24622 L .20953 .2278 L .21821 .2092 L .22657 .19267 L .23461 .17785 L .24317 .16261 L .2514 .1478 L .26016 .13139 L Mistroke .26859 .11485 L .2767 .0986 L .28534 .08165 L .29365 .06652 L .30248 .05273 L .30691 .04698 L .31099 .04251 L .31523 .03875 L .31918 .0361 L .32111 .03511 L .32318 .03428 L .32436 .03392 L .32547 .03364 L .32648 .03346 L .32704 .03337 L .32757 .03331 L .32782 .03329 L .32808 .03327 L .32831 .03326 L .32855 .03324 L .32881 .03323 L .32906 .03323 L .32921 .03323 L .32935 .03322 L .32961 .03323 L .32985 .03323 L .33009 .03324 L .3303 .03325 L .33053 .03326 L .33105 .0333 L .33153 .03335 L .33264 .03351 L .33364 .03372 L .3348 .03402 L .33591 .03437 L .33795 .03519 L .34012 .03629 L .34403 .03883 L .35203 .04608 L .36055 .05625 L .36875 .0676 L .37747 .08034 L .38587 .09227 L .39395 .10251 L .39845 .1074 L .40254 .11119 L .40685 .11442 L .41082 .11664 L .41279 .11745 L .41488 .11809 L Mistroke .41597 .11833 L .41698 .1185 L .41756 .11857 L .41809 .11862 L .41839 .11864 L .41854 .11865 L .41871 .11866 L .41887 .11866 L .41901 .11867 L .41916 .11867 L .41929 .11867 L .41954 .11868 L .4198 .11867 L .42005 .11867 L .42028 .11866 L .42055 .11864 L .42084 .11863 L .42137 .11858 L .42192 .11851 L .4225 .11842 L .42355 .11822 L .42542 .11769 L .42747 .11689 L .42962 .11579 L .43158 .11456 L .43601 .11101 L .44442 .10145 L .4525 .08927 L .46111 .07403 L .4694 .05837 L .4782 .04226 L .48669 .02882 L .49485 .01904 L .49907 .01545 L .5012 .01404 L .50354 .01281 L .5057 .01194 L .50672 .01163 L .50767 .01139 L .50875 .01117 L .50933 .01108 L .50994 .01101 L .51022 .01098 L .51052 .01096 L .51079 .01094 L .51107 .01092 L .51133 .01091 L .51161 .0109 L .51186 .0109 L .5121 .0109 L Mistroke .51237 .0109 L .5125 .0109 L .51265 .0109 L .51292 .01091 L .51317 .01092 L .51372 .01096 L .51432 .01101 L .51542 .01115 L .51643 .01132 L .51845 .01178 L .52037 .01235 L .52457 .01397 L .52847 .01584 L .53676 .0205 L .54558 .02611 L .55408 .03269 L .56225 .04153 L .56648 .04771 L .57095 .0558 L .579 .0753 L .58757 .10391 L .59582 .13906 L .60459 .18315 L .61304 .22959 L .62117 .27485 L .62982 .32029 L .63815 .35858 L .64615 .38859 L .6506 .4021 L .65468 .4125 L .65893 .4214 L .66288 .42804 L .66689 .43331 L .67128 .43762 L .67351 .43931 L .67562 .44063 L .67956 .44251 L .68162 .44323 L .68381 .44382 L .68567 .44422 L .68772 .44455 L .68882 .44469 L .68987 .4448 L .69083 .44488 L .69184 .44494 L .69242 .44497 L .69294 .44499 L .69347 .44501 L .69396 .44502 L .69424 .44503 L Mistroke .6944 .44503 L .69454 .44503 L .69479 .44503 L .69507 .44503 L .69523 .44503 L .69538 .44503 L .69565 .44503 L .69581 .44503 L .69595 .44503 L .69627 .44502 L .69652 .44502 L .69679 .44501 L .69726 .445 L .69777 .44498 L .69832 .44496 L .69931 .4449 L .70024 .44483 L .70235 .44462 L .70354 .44447 L .70461 .4443 L .70659 .44393 L .70869 .44342 L .71098 .44272 L .7131 .4419 L .71726 .43972 L .72115 .43686 L .72546 .43253 L .72939 .42729 L .73343 .42045 L .73783 .41113 L .74647 .38659 L .75479 .35495 L .76279 .31785 L .77131 .27304 L .77951 .22745 L .78822 .17976 L .79662 .13784 L .80469 .10357 L .81329 .07494 L .82156 .05506 L .82608 .04699 L .83035 .04088 L .8385 .03224 L .84717 .02566 L .85552 .0204 L .86355 .01592 L .86801 .01379 L .8721 .01224 L .87411 .01166 L .87518 .01141 L Mistroke .87632 .01119 L .87688 .0111 L .87741 .01103 L .87792 .01097 L .87839 .01093 L .87885 .0109 L .87908 .01088 L .87932 .01087 L .87957 .01086 L .8798 .01086 L .88007 .01085 L .88032 .01085 L .88057 .01086 L .88081 .01086 L .88107 .01087 L .88134 .01089 L .88162 .0109 L .88191 .01093 L .88243 .01098 L .88336 .0111 L .88437 .01129 L .88548 .01156 L .88666 .01192 L .88878 .01277 L .89095 .01391 L .89331 .01547 L .89758 .01912 L .9018 .02376 L .90626 .02969 L .9143 .04255 L .91667 .04674 L Mfstroke 0 g .5 Mabswid .08333 .20336 m .91667 .20336 L s .08333 .20336 m .08333 .2117 L s .02 w .08333 .20336 Mdot .22109 .20336 Mdot .60832 .20336 Mdot .78385 .20336 Mdot .91667 .20336 Mdot 1 1 1 r .13844 .26657 m .13844 .26657 .035 0 365.73 arc F 0 0 0 r .16344 .27032 m .14219 .27032 L .14219 .29157 L .13469 .29157 L .13469 .27032 L .11344 .27032 L .11344 .26282 L .13469 .26282 L .13469 .24157 L .14219 .24157 L .14219 .26282 L .16344 .26282 L F 1 1 1 r .51152 .10713 m .51152 .10713 .035 0 365.73 arc F 0 0 0 r .53652 .11088 m .48652 .11088 L .48652 .10338 L .53652 .10338 L F 1 1 1 r .69609 .32419 m .69609 .32419 .035 0 365.73 arc F 0 0 0 r .72109 .32794 m .69984 .32794 L .69984 .34919 L .69234 .34919 L .69234 .32794 L .67109 .32794 L .67109 .32044 L .69234 .32044 L .69234 .29919 L .69984 .29919 L .69984 .32044 L .72109 .32044 L F 1 1 1 r .86354 .10964 m .86354 .10964 .035 0 365.73 arc F 0 0 0 r .88854 .11339 m .83854 .11339 L .83854 .10589 L .88854 .10589 L F 0 g gsave .08333 .17836 -66 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (a) 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 gsave .91667 .22836 -66 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (b) 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 gsave .33333 .37003 -87 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (y) show 75.000 13.000 moveto %%IncludeResource: font Math1Mono %%IncludeFont: Math1Mono /Math1Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (=) show 87.000 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 93.000 13.000 moveto (@) show 99.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 105.000 13.000 moveto (D) show 111.000 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, 131.25}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell["\<\ To get a quick review of this, grab all four plots and everything \ between them, animate, and run forward at speed 2 or 3.\ \>", "Special2"], Cell[TextData[{ "Then you calculate \n ", Cell[BoxData[ \(\[Integral]\_a\%b f[x] \[DifferentialD]x\)]], " \nby taking the sum of four measurements:\n ", Cell[BoxData[ \(\(\[Integral]\_a\%x\)\_1 f[x] \[DifferentialD]x + \[Integral]\_\(\(x\_1\%x\)\_2\)f[x] \[DifferentialD]x + \[Integral]\_\(\(x\_2\%x\)\_3\)f[x] \[DifferentialD]x + \[Integral]\_\(x\_3\%b\)f[x] \[DifferentialD]x\)]], "\nwhere\n ", Cell[BoxData[ \(\(\[Integral]\_a\%x\)\_1 f[x] \[DifferentialD]x\)]], " and ", Cell[BoxData[ \(\[Integral]\_\(\(x\_2\%x\)\_3\)f[x] \[DifferentialD]x\)]], " are positive\nand\n ", Cell[BoxData[ \(\[Integral]\_\(\(x\_1\%x\)\_2\)f[x] \[DifferentialD]x\)]], " and ", Cell[BoxData[ \(\[Integral]\_\(x\_3\%b\)f[x] \[DifferentialD]x\)]], " are negative. " }], "Text"], Cell["B.1.a)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Given a specific function \n ", Cell[BoxData[ \(f[x]\)]], "\nand given specific numbers \n ", Cell[BoxData[ \(a\)]], " and ", Cell[BoxData[ \(b\)]], " with ", Cell[BoxData[ \(a \[LessEqual] b\)]], ", \nwhen you calculate \n ", Cell[BoxData[ \(\[Integral]\_a\%b f[x] \[DifferentialD]x\)]], ", \ndo you get a number or do you get another function? " }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "As a signed measurement of area, \n ", Cell[BoxData[ \(\[Integral]\_a\%b f[x] \[DifferentialD]x\)]], " is a number." }], "SmallText"] }, Closed]], Cell["B.1.b)", "Subsubsection"], Cell["Make the indicated area measurements:", "Text"], Cell["B.1.b.i)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\[Integral]\_1\%4\((x\/2 + 1)\) \[DifferentialD]x\)]] }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ " ", Cell[BoxData[ \(\[Integral]\_1\%4\((x\/2 + 1)\) \[DifferentialD]x\)]], "\nmeasures the area under the the plot of\n ", Cell[BoxData[ \(f[x] = \((x\/2 + 1)\)\)]], "\nand above the segment ", Cell[BoxData[ \(\([1, 4]\)\)]], " on the ", Cell[BoxData[ \(x\)]], "-axis. " }], "SmallText"], Cell[TextData[{ "Here's a picture of the area measured by ", Cell[BoxData[ \(\[Integral]\_1\%4\((x\/2 + 1)\) \[DifferentialD]x\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(\(f[x_] = x\/2 + 1;\)\n\), "\n", \(\(FilledPlot[f[x], {x, 1, 4}, \n\tAxesLabel \[Rule] {"\", "\"}, AspectRatio \[Rule] 1\/GoldenRatio];\)\)}], "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 1 0 0.618034 [ [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.293651 0.31746 0.0147151 0.196201 [ [.18254 .00222 -9 -9 ] [.18254 .00222 9 0 ] [.34127 .00222 -3 -9 ] [.34127 .00222 3 0 ] [.5 .00222 -9 -9 ] [.5 .00222 9 0 ] [.65873 .00222 -3 -9 ] [.65873 .00222 3 0 ] [.81746 .00222 -9 -9 ] [.81746 .00222 9 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [1.025 .01472 0 -6 ] [1.025 .01472 10 6 ] [.01131 .11282 -18 -4.5 ] [.01131 .11282 0 4.5 ] [.01131 .21092 -6 -4.5 ] [.01131 .21092 0 4.5 ] [.01131 .30902 -18 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .40712 -6 -4.5 ] [.01131 .40712 0 4.5 ] [.01131 .50522 -18 -4.5 ] [.01131 .50522 0 4.5 ] [.01131 .60332 -6 -4.5 ] [.01131 .60332 0 4.5 ] [.02381 .64303 -5 0 ] [.02381 .64303 5 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .18254 .01472 m .18254 .02097 L s [(1.5)] .18254 .00222 0 1 Mshowa .34127 .01472 m .34127 .02097 L s [(2)] .34127 .00222 0 1 Mshowa .5 .01472 m .5 .02097 L s [(2.5)] .5 .00222 0 1 Mshowa .65873 .01472 m .65873 .02097 L s [(3)] .65873 .00222 0 1 Mshowa .81746 .01472 m .81746 .02097 L s [(3.5)] .81746 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(4)] .97619 .00222 0 1 Mshowa .125 Mabswid .05556 .01472 m .05556 .01847 L s .0873 .01472 m .0873 .01847 L s .11905 .01472 m .11905 .01847 L s .15079 .01472 m .15079 .01847 L s .21429 .01472 m .21429 .01847 L s .24603 .01472 m .24603 .01847 L s .27778 .01472 m .27778 .01847 L s .30952 .01472 m .30952 .01847 L s .37302 .01472 m .37302 .01847 L s .40476 .01472 m .40476 .01847 L s .43651 .01472 m .43651 .01847 L s .46825 .01472 m .46825 .01847 L s .53175 .01472 m .53175 .01847 L s .56349 .01472 m .56349 .01847 L s .59524 .01472 m .59524 .01847 L s .62698 .01472 m .62698 .01847 L s .69048 .01472 m .69048 .01847 L s .72222 .01472 m .72222 .01847 L s .75397 .01472 m .75397 .01847 L s .78571 .01472 m .78571 .01847 L s .84921 .01472 m .84921 .01847 L s .88095 .01472 m .88095 .01847 L s .9127 .01472 m .9127 .01847 L s .94444 .01472 m .94444 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s gsave 1.025 .01472 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .11282 m .03006 .11282 L s [(0.5)] .01131 .11282 1 0 Mshowa .02381 .21092 m .03006 .21092 L s [(1)] .01131 .21092 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(1.5)] .01131 .30902 1 0 Mshowa .02381 .40712 m .03006 .40712 L s [(2)] .01131 .40712 1 0 Mshowa .02381 .50522 m .03006 .50522 L s [(2.5)] .01131 .50522 1 0 Mshowa .02381 .60332 m .03006 .60332 L s [(3)] .01131 .60332 1 0 Mshowa .125 Mabswid .02381 .03434 m .02756 .03434 L s .02381 .05396 m .02756 .05396 L s .02381 .07358 m .02756 .07358 L s .02381 .0932 m .02756 .0932 L s .02381 .13244 m .02756 .13244 L s .02381 .15206 m .02756 .15206 L s .02381 .17168 m .02756 .17168 L s .02381 .1913 m .02756 .1913 L s .02381 .23054 m .02756 .23054 L s .02381 .25016 m .02756 .25016 L s .02381 .26978 m .02756 .26978 L s .02381 .2894 m .02756 .2894 L s .02381 .32864 m .02756 .32864 L s .02381 .34826 m .02756 .34826 L s .02381 .36788 m .02756 .36788 L s .02381 .3875 m .02756 .3875 L s .02381 .42674 m .02756 .42674 L s .02381 .44636 m .02756 .44636 L s .02381 .46598 m .02756 .46598 L s .02381 .4856 m .02756 .4856 L s .02381 .52484 m .02756 .52484 L s .02381 .54446 m .02756 .54446 L s .02381 .56408 m .02756 .56408 L s .02381 .5837 m .02756 .5837 L s .25 Mabswid .02381 0 m .02381 .61803 L s gsave .02381 .64303 -66 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (y) 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 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .30902 m .06244 .32096 L .10458 .33398 L .14415 .3462 L .18221 .35796 L .22272 .37048 L .26171 .38253 L .30316 .39534 L .34309 .40768 L .3815 .41955 L .42237 .43218 L .46172 .44434 L .49955 .45603 L .53984 .46848 L .57861 .48046 L .61984 .4932 L .65954 .50547 L .69774 .51727 L .73838 .52983 L .77751 .54192 L .81909 .55477 L .85916 .56715 L .89771 .57907 L .93871 .59174 L .97619 .60332 L s .5 g .02381 .30902 m .06244 .32096 L .10458 .33398 L .14415 .3462 L .18221 .35796 L .22272 .37048 L .26171 .38253 L .30316 .39534 L .34309 .40768 L .3815 .41955 L .42237 .43218 L .46172 .44434 L .49955 .45603 L .53984 .46848 L .57861 .48046 L .61984 .4932 L .65954 .50547 L .69774 .51727 L .73838 .52983 L .77751 .54192 L .81909 .55477 L .85916 .56715 L .89771 .57907 L .93871 .59174 L .97619 .60332 L .97619 .01472 L .02381 .01472 L F 0 g .02381 .30902 m .06244 .32096 L .10458 .33398 L .14415 .3462 L .18221 .35796 L .22272 .37048 L .26171 .38253 L .30316 .39534 L .34309 .40768 L .3815 .41955 L .42237 .43218 L .46172 .44434 L .49955 .45603 L .53984 .46848 L .57861 .48046 L .61984 .4932 L .65954 .50547 L .69774 .51727 L .73838 .52983 L .77751 .54192 L .81909 .55477 L .85916 .56715 L .89771 .57907 L .93871 .59174 L .97619 .60332 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.293651 0.31746 0.0147151 0.196201 [ [.18254 .00222 -9 -9 ] [.18254 .00222 9 0 ] [.34127 .00222 -3 -9 ] [.34127 .00222 3 0 ] [.5 .00222 -9 -9 ] [.5 .00222 9 0 ] [.65873 .00222 -3 -9 ] [.65873 .00222 3 0 ] [.81746 .00222 -9 -9 ] [.81746 .00222 9 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [1.025 .01472 0 -6 ] [1.025 .01472 10 6 ] [.01131 .11282 -18 -4.5 ] [.01131 .11282 0 4.5 ] [.01131 .21092 -6 -4.5 ] [.01131 .21092 0 4.5 ] [.01131 .30902 -18 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .40712 -6 -4.5 ] [.01131 .40712 0 4.5 ] [.01131 .50522 -18 -4.5 ] [.01131 .50522 0 4.5 ] [.01131 .60332 -6 -4.5 ] [.01131 .60332 0 4.5 ] [.02381 .64303 -5 0 ] [.02381 .64303 5 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .18254 .01472 m .18254 .02097 L s [(1.5)] .18254 .00222 0 1 Mshowa .34127 .01472 m .34127 .02097 L s [(2)] .34127 .00222 0 1 Mshowa .5 .01472 m .5 .02097 L s [(2.5)] .5 .00222 0 1 Mshowa .65873 .01472 m .65873 .02097 L s [(3)] .65873 .00222 0 1 Mshowa .81746 .01472 m .81746 .02097 L s [(3.5)] .81746 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(4)] .97619 .00222 0 1 Mshowa .125 Mabswid .05556 .01472 m .05556 .01847 L s .0873 .01472 m .0873 .01847 L s .11905 .01472 m .11905 .01847 L s .15079 .01472 m .15079 .01847 L s .21429 .01472 m .21429 .01847 L s .24603 .01472 m .24603 .01847 L s .27778 .01472 m .27778 .01847 L s .30952 .01472 m .30952 .01847 L s .37302 .01472 m .37302 .01847 L s .40476 .01472 m .40476 .01847 L s .43651 .01472 m .43651 .01847 L s .46825 .01472 m .46825 .01847 L s .53175 .01472 m .53175 .01847 L s .56349 .01472 m .56349 .01847 L s .59524 .01472 m .59524 .01847 L s .62698 .01472 m .62698 .01847 L s .69048 .01472 m .69048 .01847 L s .72222 .01472 m .72222 .01847 L s .75397 .01472 m .75397 .01847 L s .78571 .01472 m .78571 .01847 L s .84921 .01472 m .84921 .01847 L s .88095 .01472 m .88095 .01847 L s .9127 .01472 m .9127 .01847 L s .94444 .01472 m .94444 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s gsave 1.025 .01472 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .11282 m .03006 .11282 L s [(0.5)] .01131 .11282 1 0 Mshowa .02381 .21092 m .03006 .21092 L s [(1)] .01131 .21092 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(1.5)] .01131 .30902 1 0 Mshowa .02381 .40712 m .03006 .40712 L s [(2)] .01131 .40712 1 0 Mshowa .02381 .50522 m .03006 .50522 L s [(2.5)] .01131 .50522 1 0 Mshowa .02381 .60332 m .03006 .60332 L s [(3)] .01131 .60332 1 0 Mshowa .125 Mabswid .02381 .03434 m .02756 .03434 L s .02381 .05396 m .02756 .05396 L s .02381 .07358 m .02756 .07358 L s .02381 .0932 m .02756 .0932 L s .02381 .13244 m .02756 .13244 L s .02381 .15206 m .02756 .15206 L s .02381 .17168 m .02756 .17168 L s .02381 .1913 m .02756 .1913 L s .02381 .23054 m .02756 .23054 L s .02381 .25016 m .02756 .25016 L s .02381 .26978 m .02756 .26978 L s .02381 .2894 m .02756 .2894 L s .02381 .32864 m .02756 .32864 L s .02381 .34826 m .02756 .34826 L s .02381 .36788 m .02756 .36788 L s .02381 .3875 m .02756 .3875 L s .02381 .42674 m .02756 .42674 L s .02381 .44636 m .02756 .44636 L s .02381 .46598 m .02756 .46598 L s .02381 .4856 m .02756 .4856 L s .02381 .52484 m .02756 .52484 L s .02381 .54446 m .02756 .54446 L s .02381 .56408 m .02756 .56408 L s .02381 .5837 m .02756 .5837 L s .25 Mabswid .02381 0 m .02381 .61803 L s gsave .02381 .64303 -66 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (y) 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 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath MathSubEnd P % End of sub-graphic % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell["This integral calculates the area of a trapezoid with ", "SmallText"], Cell[BoxData[{ \(\(shortHeight = f[1];\)\), "\n", \(\(tallHeight = f[4];\)\), "\n", \(\(width = 3;\)\)}], "Input"], Cell[TextData[{ "Consequently the area measurement\n ", Cell[BoxData[ \(\[Integral]\_1\%4 f[x] \[DifferentialD]x = \[Integral]\_1\%4\((x\/2 + 1)\) \[DifferentialD]x\)]], "\nin square units is given by:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(1\/2\ width\ \((tallHeight + shortHeight)\)\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(27\/4\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can also handle this measurement directly:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[\((x/2\ + \ 1)\), {x, 1, 4}]\)], "Input"], Cell[BoxData[ \(27\/4\)], "Output"] }, Open ]], Cell["Good going.", "SmallText"] }, Closed]], Cell["B.1.b.ii)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\[Integral]\_\(-4\)\%4\((x\/2 + 1)\) \[DifferentialD]x\)]] }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Here's a picture of the area measured by \n ", Cell[BoxData[ \(\[Integral]\_\(-4\)\%4\((x\/2 + 1)\) \[DifferentialD]x\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(\(f[x_] = x\/2 + 1;\)\n\), "\n", \(\(FilledPlot[f[x], {x, \(-4\), 4}, \n\t AxesLabel \[Rule] {"\", "\"}, AspectRatio \[Rule] 1\/GoldenRatio];\)\)}], "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 1 0 0.618034 [ [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.119048 0.161866 0.147151 [ [.02381 .14937 -6 -9 ] [.02381 .14937 6 0 ] [.2619 .14937 -6 -9 ] [.2619 .14937 6 0 ] [.7381 .14937 -3 -9 ] [.7381 .14937 3 0 ] [.97619 .14937 -3 -9 ] [.97619 .14937 3 0 ] [1.025 .16187 0 -6 ] [1.025 .16187 10 6 ] [.4875 .01472 -12 -4.5 ] [.4875 .01472 0 4.5 ] [.4875 .30902 -6 -4.5 ] [.4875 .30902 0 4.5 ] [.4875 .45617 -6 -4.5 ] [.4875 .45617 0 4.5 ] [.4875 .60332 -6 -4.5 ] [.4875 .60332 0 4.5 ] [.5 .64303 -5 0 ] [.5 .64303 5 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .16187 m .02381 .16812 L s [(-4)] .02381 .14937 0 1 Mshowa .2619 .16187 m .2619 .16812 L s [(-2)] .2619 .14937 0 1 Mshowa .7381 .16187 m .7381 .16812 L s [(2)] .7381 .14937 0 1 Mshowa .97619 .16187 m .97619 .16812 L s [(4)] .97619 .14937 0 1 Mshowa .125 Mabswid .08333 .16187 m .08333 .16562 L s .14286 .16187 m .14286 .16562 L s .20238 .16187 m .20238 .16562 L s .32143 .16187 m .32143 .16562 L s .38095 .16187 m .38095 .16562 L s .44048 .16187 m .44048 .16562 L s .55952 .16187 m .55952 .16562 L s .61905 .16187 m .61905 .16562 L s .67857 .16187 m .67857 .16562 L s .79762 .16187 m .79762 .16562 L s .85714 .16187 m .85714 .16562 L s .91667 .16187 m .91667 .16562 L s .25 Mabswid 0 .16187 m 1 .16187 L s gsave 1.025 .16187 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .5 .01472 m .50625 .01472 L s [(-1)] .4875 .01472 1 0 Mshowa .5 .30902 m .50625 .30902 L s [(1)] .4875 .30902 1 0 Mshowa .5 .45617 m .50625 .45617 L s [(2)] .4875 .45617 1 0 Mshowa .5 .60332 m .50625 .60332 L s [(3)] .4875 .60332 1 0 Mshowa .125 Mabswid .5 .04415 m .50375 .04415 L s .5 .07358 m .50375 .07358 L s .5 .10301 m .50375 .10301 L s .5 .13244 m .50375 .13244 L s .5 .1913 m .50375 .1913 L s .5 .22073 m .50375 .22073 L s .5 .25016 m .50375 .25016 L s .5 .27959 m .50375 .27959 L s .5 .33845 m .50375 .33845 L s .5 .36788 m .50375 .36788 L s .5 .39731 m .50375 .39731 L s .5 .42674 m .50375 .42674 L s .5 .4856 m .50375 .4856 L s .5 .51503 m .50375 .51503 L s .5 .54446 m .50375 .54446 L s .5 .57389 m .50375 .57389 L s .25 Mabswid .5 0 m .5 .61803 L s gsave .5 .64303 -66 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (y) 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 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .01472 m .06244 .03859 L .10458 .06463 L .14415 .08909 L .18221 .11261 L .22272 .13765 L .26171 .16175 L .30316 .18736 L .34309 .21204 L .3815 .23578 L .42237 .26104 L .46172 .28536 L .49955 .30874 L .53984 .33364 L .57861 .3576 L .61984 .38308 L .65954 .40762 L .69774 .43123 L .73838 .45635 L .77751 .48053 L .81909 .50623 L .85916 .53099 L .89771 .55482 L .93871 .58016 L .97619 .60332 L s .5 g .02381 .01472 m .06244 .03859 L .10458 .06463 L .14415 .08909 L .18221 .11261 L .22272 .13765 L .26171 .16175 L .30316 .18736 L .34309 .21204 L .3815 .23578 L .42237 .26104 L .46172 .28536 L .49955 .30874 L .53984 .33364 L .57861 .3576 L .61984 .38308 L .65954 .40762 L .69774 .43123 L .73838 .45635 L .77751 .48053 L .81909 .50623 L .85916 .53099 L .89771 .55482 L .93871 .58016 L .97619 .60332 L .97619 .16187 L .02381 .16187 L F 0 g .02381 .01472 m .06244 .03859 L .10458 .06463 L .14415 .08909 L .18221 .11261 L .22272 .13765 L .26171 .16175 L .30316 .18736 L .34309 .21204 L .3815 .23578 L .42237 .26104 L .46172 .28536 L .49955 .30874 L .53984 .33364 L .57861 .3576 L .61984 .38308 L .65954 .40762 L .69774 .43123 L .73838 .45635 L .77751 .48053 L .81909 .50623 L .85916 .53099 L .89771 .55482 L .93871 .58016 L .97619 .60332 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.119048 0.161866 0.147151 [ [.02381 .14937 -6 -9 ] [.02381 .14937 6 0 ] [.2619 .14937 -6 -9 ] [.2619 .14937 6 0 ] [.7381 .14937 -3 -9 ] [.7381 .14937 3 0 ] [.97619 .14937 -3 -9 ] [.97619 .14937 3 0 ] [1.025 .16187 0 -6 ] [1.025 .16187 10 6 ] [.4875 .01472 -12 -4.5 ] [.4875 .01472 0 4.5 ] [.4875 .30902 -6 -4.5 ] [.4875 .30902 0 4.5 ] [.4875 .45617 -6 -4.5 ] [.4875 .45617 0 4.5 ] [.4875 .60332 -6 -4.5 ] [.4875 .60332 0 4.5 ] [.5 .64303 -5 0 ] [.5 .64303 5 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .16187 m .02381 .16812 L s [(-4)] .02381 .14937 0 1 Mshowa .2619 .16187 m .2619 .16812 L s [(-2)] .2619 .14937 0 1 Mshowa .7381 .16187 m .7381 .16812 L s [(2)] .7381 .14937 0 1 Mshowa .97619 .16187 m .97619 .16812 L s [(4)] .97619 .14937 0 1 Mshowa .125 Mabswid .08333 .16187 m .08333 .16562 L s .14286 .16187 m .14286 .16562 L s .20238 .16187 m .20238 .16562 L s .32143 .16187 m .32143 .16562 L s .38095 .16187 m .38095 .16562 L s .44048 .16187 m .44048 .16562 L s .55952 .16187 m .55952 .16562 L s .61905 .16187 m .61905 .16562 L s .67857 .16187 m .67857 .16562 L s .79762 .16187 m .79762 .16562 L s .85714 .16187 m .85714 .16562 L s .91667 .16187 m .91667 .16562 L s .25 Mabswid 0 .16187 m 1 .16187 L s gsave 1.025 .16187 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .5 .01472 m .50625 .01472 L s [(-1)] .4875 .01472 1 0 Mshowa .5 .30902 m .50625 .30902 L s [(1)] .4875 .30902 1 0 Mshowa .5 .45617 m .50625 .45617 L s [(2)] .4875 .45617 1 0 Mshowa .5 .60332 m .50625 .60332 L s [(3)] .4875 .60332 1 0 Mshowa .125 Mabswid .5 .04415 m .50375 .04415 L s .5 .07358 m .50375 .07358 L s .5 .10301 m .50375 .10301 L s .5 .13244 m .50375 .13244 L s .5 .1913 m .50375 .1913 L s .5 .22073 m .50375 .22073 L s .5 .25016 m .50375 .25016 L s .5 .27959 m .50375 .27959 L s .5 .33845 m .50375 .33845 L s .5 .36788 m .50375 .36788 L s .5 .39731 m .50375 .39731 L s .5 .42674 m .50375 .42674 L s .5 .4856 m .50375 .4856 L s .5 .51503 m .50375 .51503 L s .5 .54446 m .50375 .54446 L s .5 .57389 m .50375 .57389 L s .25 Mabswid .5 0 m .5 .61803 L s gsave .5 .64303 -66 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (y) 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 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath MathSubEnd P % End of sub-graphic % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell["\<\ Better find out where the area goes from positive to negative:\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[f[x] == 0]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({{x \[Rule] \(-2\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "The measurement \n ", Cell[BoxData[ \(\[Integral]\_\(-4\)\%\(-2\)\((x\/2 + 1)\) \[DifferentialD]x\)]], "\nof the triangular area on the left is negative because it is below the ", Cell[BoxData[ \(x\)]], "-axis:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(height = f[\(-4\)];\)\), "\n", \(\(base = 2;\)\), "\n", \(triangleleft = \(base\ height\)\/2\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-1\)\)], "Output"] }, Open ]], Cell["Now look at the triangle on the right:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(FilledPlot[f[x], {x, \(-4\), 4}, \n\t AxesLabel \[Rule] {"\", "\"}, AspectRatio \[Rule] 1\/GoldenRatio];\)\)], "Input", AspectRatioFixed->True], 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 1 0 0.618034 [ [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.119048 0.161866 0.147151 [ [.02381 .14937 -6 -9 ] [.02381 .14937 6 0 ] [.2619 .14937 -6 -9 ] [.2619 .14937 6 0 ] [.7381 .14937 -3 -9 ] [.7381 .14937 3 0 ] [.97619 .14937 -3 -9 ] [.97619 .14937 3 0 ] [1.025 .16187 0 -6 ] [1.025 .16187 10 6 ] [.4875 .01472 -12 -4.5 ] [.4875 .01472 0 4.5 ] [.4875 .30902 -6 -4.5 ] [.4875 .30902 0 4.5 ] [.4875 .45617 -6 -4.5 ] [.4875 .45617 0 4.5 ] [.4875 .60332 -6 -4.5 ] [.4875 .60332 0 4.5 ] [.5 .64303 -5 0 ] [.5 .64303 5 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .16187 m .02381 .16812 L s [(-4)] .02381 .14937 0 1 Mshowa .2619 .16187 m .2619 .16812 L s [(-2)] .2619 .14937 0 1 Mshowa .7381 .16187 m .7381 .16812 L s [(2)] .7381 .14937 0 1 Mshowa .97619 .16187 m .97619 .16812 L s [(4)] .97619 .14937 0 1 Mshowa .125 Mabswid .08333 .16187 m .08333 .16562 L s .14286 .16187 m .14286 .16562 L s .20238 .16187 m .20238 .16562 L s .32143 .16187 m .32143 .16562 L s .38095 .16187 m .38095 .16562 L s .44048 .16187 m .44048 .16562 L s .55952 .16187 m .55952 .16562 L s .61905 .16187 m .61905 .16562 L s .67857 .16187 m .67857 .16562 L s .79762 .16187 m .79762 .16562 L s .85714 .16187 m .85714 .16562 L s .91667 .16187 m .91667 .16562 L s .25 Mabswid 0 .16187 m 1 .16187 L s gsave 1.025 .16187 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .5 .01472 m .50625 .01472 L s [(-1)] .4875 .01472 1 0 Mshowa .5 .30902 m .50625 .30902 L s [(1)] .4875 .30902 1 0 Mshowa .5 .45617 m .50625 .45617 L s [(2)] .4875 .45617 1 0 Mshowa .5 .60332 m .50625 .60332 L s [(3)] .4875 .60332 1 0 Mshowa .125 Mabswid .5 .04415 m .50375 .04415 L s .5 .07358 m .50375 .07358 L s .5 .10301 m .50375 .10301 L s .5 .13244 m .50375 .13244 L s .5 .1913 m .50375 .1913 L s .5 .22073 m .50375 .22073 L s .5 .25016 m .50375 .25016 L s .5 .27959 m .50375 .27959 L s .5 .33845 m .50375 .33845 L s .5 .36788 m .50375 .36788 L s .5 .39731 m .50375 .39731 L s .5 .42674 m .50375 .42674 L s .5 .4856 m .50375 .4856 L s .5 .51503 m .50375 .51503 L s .5 .54446 m .50375 .54446 L s .5 .57389 m .50375 .57389 L s .25 Mabswid .5 0 m .5 .61803 L s gsave .5 .64303 -66 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (y) 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 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .01472 m .06244 .03859 L .10458 .06463 L .14415 .08909 L .18221 .11261 L .22272 .13765 L .26171 .16175 L .30316 .18736 L .34309 .21204 L .3815 .23578 L .42237 .26104 L .46172 .28536 L .49955 .30874 L .53984 .33364 L .57861 .3576 L .61984 .38308 L .65954 .40762 L .69774 .43123 L .73838 .45635 L .77751 .48053 L .81909 .50623 L .85916 .53099 L .89771 .55482 L .93871 .58016 L .97619 .60332 L s .5 g .02381 .01472 m .06244 .03859 L .10458 .06463 L .14415 .08909 L .18221 .11261 L .22272 .13765 L .26171 .16175 L .30316 .18736 L .34309 .21204 L .3815 .23578 L .42237 .26104 L .46172 .28536 L .49955 .30874 L .53984 .33364 L .57861 .3576 L .61984 .38308 L .65954 .40762 L .69774 .43123 L .73838 .45635 L .77751 .48053 L .81909 .50623 L .85916 .53099 L .89771 .55482 L .93871 .58016 L .97619 .60332 L .97619 .16187 L .02381 .16187 L F 0 g .02381 .01472 m .06244 .03859 L .10458 .06463 L .14415 .08909 L .18221 .11261 L .22272 .13765 L .26171 .16175 L .30316 .18736 L .34309 .21204 L .3815 .23578 L .42237 .26104 L .46172 .28536 L .49955 .30874 L .53984 .33364 L .57861 .3576 L .61984 .38308 L .65954 .40762 L .69774 .43123 L .73838 .45635 L .77751 .48053 L .81909 .50623 L .85916 .53099 L .89771 .55482 L .93871 .58016 L .97619 .60332 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.119048 0.161866 0.147151 [ [.02381 .14937 -6 -9 ] [.02381 .14937 6 0 ] [.2619 .14937 -6 -9 ] [.2619 .14937 6 0 ] [.7381 .14937 -3 -9 ] [.7381 .14937 3 0 ] [.97619 .14937 -3 -9 ] [.97619 .14937 3 0 ] [1.025 .16187 0 -6 ] [1.025 .16187 10 6 ] [.4875 .01472 -12 -4.5 ] [.4875 .01472 0 4.5 ] [.4875 .30902 -6 -4.5 ] [.4875 .30902 0 4.5 ] [.4875 .45617 -6 -4.5 ] [.4875 .45617 0 4.5 ] [.4875 .60332 -6 -4.5 ] [.4875 .60332 0 4.5 ] [.5 .64303 -5 0 ] [.5 .64303 5 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .16187 m .02381 .16812 L s [(-4)] .02381 .14937 0 1 Mshowa .2619 .16187 m .2619 .16812 L s [(-2)] .2619 .14937 0 1 Mshowa .7381 .16187 m .7381 .16812 L s [(2)] .7381 .14937 0 1 Mshowa .97619 .16187 m .97619 .16812 L s [(4)] .97619 .14937 0 1 Mshowa .125 Mabswid .08333 .16187 m .08333 .16562 L s .14286 .16187 m .14286 .16562 L s .20238 .16187 m .20238 .16562 L s .32143 .16187 m .32143 .16562 L s .38095 .16187 m .38095 .16562 L s .44048 .16187 m .44048 .16562 L s .55952 .16187 m .55952 .16562 L s .61905 .16187 m .61905 .16562 L s .67857 .16187 m .67857 .16562 L s .79762 .16187 m .79762 .16562 L s .85714 .16187 m .85714 .16562 L s .91667 .16187 m .91667 .16562 L s .25 Mabswid 0 .16187 m 1 .16187 L s gsave 1.025 .16187 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .5 .01472 m .50625 .01472 L s [(-1)] .4875 .01472 1 0 Mshowa .5 .30902 m .50625 .30902 L s [(1)] .4875 .30902 1 0 Mshowa .5 .45617 m .50625 .45617 L s [(2)] .4875 .45617 1 0 Mshowa .5 .60332 m .50625 .60332 L s [(3)] .4875 .60332 1 0 Mshowa .125 Mabswid .5 .04415 m .50375 .04415 L s .5 .07358 m .50375 .07358 L s .5 .10301 m .50375 .10301 L s .5 .13244 m .50375 .13244 L s .5 .1913 m .50375 .1913 L s .5 .22073 m .50375 .22073 L s .5 .25016 m .50375 .25016 L s .5 .27959 m .50375 .27959 L s .5 .33845 m .50375 .33845 L s .5 .36788 m .50375 .36788 L s .5 .39731 m .50375 .39731 L s .5 .42674 m .50375 .42674 L s .5 .4856 m .50375 .4856 L s .5 .51503 m .50375 .51503 L s .5 .54446 m .50375 .54446 L s .5 .57389 m .50375 .57389 L s .25 Mabswid .5 0 m .5 .61803 L s gsave .5 .64303 -66 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (y) 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 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath MathSubEnd P % End of sub-graphic % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "The measurement\n ", Cell[BoxData[ \(\[Integral]\_\(-2\)\%4\((x\/2 + 1)\) \[DifferentialD]x\)]], "\nof the triangular area on the right is positive:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(height = f[4];\)\), "\n", \(\(base = 6;\)\), "\n", \(triangleright = \(base\ height\)\/2\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(9\)], "Output"] }, Open ]], Cell[TextData[{ "Consequently, \n ", Cell[BoxData[ \(\[Integral]\_\(-4\)\%4\((x\/2 + 1)\) \[DifferentialD]x = \[Integral]\_\(-4\)\%\(-2\)\((x\/2 + 1)\) \[DifferentialD]x + \[Integral]\_\(-2\)\%4\((x\/2 + 1)\) \[DifferentialD]x\)]], "\nis given by:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(triangleleft + triangleright\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(8\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can calculate\n ", Cell[BoxData[ \(\[Integral]\_\(-4\)\%4\((x\/2 + 1)\) \[DifferentialD]x\)]], "\ndirectly:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_\(-4\)\%4\((x\/2 + 1)\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(8\)], "Output"] }, Open ]], Cell["Got it, and it wasn't very hard.", "SmallText"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "B.3) Integration by approximation by trapezoids and \n the ", StyleBox["Mathematica", FontSlant->"Italic"], " instruction ", StyleBox["NIntegrate", "Input", FontColor->GrayLevel[0]] }], "Subsection", CellTags->"2.01.B3"], Cell["B.3.a) Trapezoids", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Plot on the same axes: \n ", Cell[BoxData[ \(f[x] = Sin[\[Pi]\ x]\)]], "\nfor ", Cell[BoxData[ \(0 \[LessEqual] x \[LessEqual] 1\)]], " and the broken line that joins the five points on the curve \n", Cell[BoxData[ \({0, f[0]}\)]], ", ", Cell[BoxData[ \({1\/4, f[1\/4]}\)]], ", ", Cell[BoxData[ \({1\/2, f[1\/2]}\)]], ", ", Cell[BoxData[ \({3\/4, f[3\/4]}\)]], " and ", Cell[BoxData[ \({1, f[1]}\)]], ".\nMeasure the area under the broken line segments over ", Cell[BoxData[ \(0 \[LessEqual] x \[LessEqual] 1\)]], " and discuss why this measurement gives a rough estimate of the area \ measurement \n ", Cell[BoxData[ \(\[Integral]\_0\%1 f[x] \[DifferentialD]x\)]], "." }], "Text"], Cell["Answer:", "Special1"], Cell["Define the function:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(f[x_] = Sin[\[Pi]\ x]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Sin[\[Pi]\ x]\)], "Output"] }, Open ]], Cell["Look at the plot:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(a\ = \ 0;\)\), "\n", \(\(b\ = \ 1;\)\), "\n", \(\(curveplot\ = \ Plot[f[x], {x, a, b}, \n\t\ \ AspectRatio -> 1\/GoldenRatio, \n\ \ \ \ \ \ PlotStyle -> {{Red, Thickness[ 0.01]}}, \n\ \ \ \ \ \ AxesLabel -> {"\", "\<\>"}, PlotRange -> {0, 1.2}, \n\ \ \ \ \ \ AxesOrigin -> {0, 0}];\)\)}], "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 0.952381 0 0.515028 [ [.21429 -0.0125 -9 -9 ] [.21429 -0.0125 9 0 ] [.40476 -0.0125 -9 -9 ] [.40476 -0.0125 9 0 ] [.59524 -0.0125 -9 -9 ] [.59524 -0.0125 9 0 ] [.78571 -0.0125 -9 -9 ] [.78571 -0.0125 9 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [1.025 0 0 -6 ] [1.025 0 10 6 ] [.01131 .10301 -18 -4.5 ] [.01131 .10301 0 4.5 ] [.01131 .20601 -18 -4.5 ] [.01131 .20601 0 4.5 ] [.01131 .30902 -18 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .41202 -18 -4.5 ] [.01131 .41202 0 4.5 ] [.01131 .51503 -6 -4.5 ] [.01131 .51503 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 [(0.2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(0.4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(0.6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(0.8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(1)] .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 gsave 1.025 0 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .10301 m .03006 .10301 L s [(0.2)] .01131 .10301 1 0 Mshowa .02381 .20601 m .03006 .20601 L s [(0.4)] .01131 .20601 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(0.6)] .01131 .30902 1 0 Mshowa .02381 .41202 m .03006 .41202 L s [(0.8)] .01131 .41202 1 0 Mshowa .02381 .51503 m .03006 .51503 L s [(1)] .01131 .51503 1 0 Mshowa .125 Mabswid .02381 .02575 m .02756 .02575 L s .02381 .0515 m .02756 .0515 L s .02381 .07725 m .02756 .07725 L s .02381 .12876 m .02756 .12876 L s .02381 .15451 m .02756 .15451 L s .02381 .18026 m .02756 .18026 L s .02381 .23176 m .02756 .23176 L s .02381 .25751 m .02756 .25751 L s .02381 .28327 m .02756 .28327 L s .02381 .33477 m .02756 .33477 L s .02381 .36052 m .02756 .36052 L s .02381 .38627 m .02756 .38627 L s .02381 .43777 m .02756 .43777 L s .02381 .46353 m .02756 .46353 L s .02381 .48928 m .02756 .48928 L s .02381 .54078 m .02756 .54078 L s .02381 .56653 m .02756 .56653 L s .02381 .59228 m .02756 .59228 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 1 0 0 r .01 w .02381 0 m .06244 .06546 L .10458 .1356 L .14415 .19912 L .18221 .25703 L .22272 .3142 L .26171 .36395 L .30316 .41022 L .34309 .44756 L .3815 .47618 L .40095 .48778 L .42237 .49823 L .44268 .50585 L .45178 .50853 L .46172 .51093 L .4671 .512 L .4721 .51285 L .47727 .51358 L .48196 .51412 L .48658 .51452 L .4887 .51467 L .49093 .5148 L .49332 .5149 L .49438 .51494 L .49552 .51497 L .49675 .515 L .49789 .51502 L .49859 .51502 L .49925 .51503 L .50049 .51503 L .50163 .51502 L .50286 .51501 L .50401 .51498 L .50508 .51496 L .50754 .51487 L .51014 .51474 L .51268 .51458 L .51504 .51439 L .5204 .51386 L .5293 .51262 L .53882 .51081 L .54906 .5083 L .56016 .50492 L .58032 .49706 L .62123 .47439 L .66064 .4444 L .69852 .40848 L .73886 .36325 L .77769 .31365 L .81897 .2553 L Mistroke .85873 .19461 L .89697 .13306 L .93767 .06527 L .97619 0 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ Cell[BoxData[ \(\[Integral]\_a\%b f[x] \[DifferentialD]x\)]], " measures the area under the curve and over the interval ", Cell[BoxData[ \(a \[LessEqual] x \[LessEqual] b\)]], " on the ", Cell[BoxData[ \(x\)]], "-axis. \nIn this plot, ", Cell[BoxData[ \(a = 0\)]], " and ", Cell[BoxData[ \(b = 1\)]], ".\nNow make a table of the points\n", Cell[BoxData[ \({0, f[0]}\)]], ", ", Cell[BoxData[ \({1\/4, f[1\/4]}\)]], ", ", Cell[BoxData[ \({1\/2, f[1\/2]}\)]], ", ", Cell[BoxData[ \({3\/4, f[3\/4]}\)]], ", and ", Cell[BoxData[ \({1, f[1]}\)]], ".\nThe jump in the ", Cell[BoxData[ \(x\)]], "-coordinate from point to point is:" }], "SmallText", AnimationDisplayTime->1], Cell[CellGroupData[{ Cell[BoxData[ \(jump = \(b - a\)\/4\)], "Input", AspectRatioFixed->True, AnimationDisplayTime->1], Cell[BoxData[ \(1\/4\)], "Output"] }, Open ]], Cell["So the points are given by:", "SmallText", AnimationDisplayTime->1], Cell[CellGroupData[{ Cell[BoxData[ \(points = Table[{x, f[x]}, {x, a, b, jump}]\)], "Input", AspectRatioFixed->True, AnimationDisplayTime->1], Cell[BoxData[ \({{0, 0}, {1\/4, 1\/\@2}, {1\/2, 1}, {3\/4, 1\/\@2}, {1, 0}}\)], "Output"] }, Open ]], Cell["Set up the plot of the points and the broken line segments:", "SmallText", AnimationDisplayTime->1], Cell[CellGroupData[{ Cell[BoxData[{ \(\(pointplot\ = \n\t ListPlot[points, \n\tAspectRatio -> 1\/GoldenRatio, \n\t PlotStyle -> {Red, PointSize[0.03]}, \n\t DisplayFunction -> \ Identity];\)\n\t\), "\n", \(\(brokenlineplot\ = \n\t Graphics[{Thickness[0.01], Line[points]}];\)\n\t\), "\n", \(\(Show[curveplot, pointplot, brokenlineplot, \n\ \ \ \ \ AxesLabel -> {"\", "\<\>"}, PlotRange -> {0, 1.2}, \n\ \ \ \ \ AxesOrigin -> {0, 0}];\)\)}], "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 0.952381 0 0.515028 [ [.21429 -0.0125 -9 -9 ] [.21429 -0.0125 9 0 ] [.40476 -0.0125 -9 -9 ] [.40476 -0.0125 9 0 ] [.59524 -0.0125 -9 -9 ] [.59524 -0.0125 9 0 ] [.78571 -0.0125 -9 -9 ] [.78571 -0.0125 9 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [1.025 0 0 -6 ] [1.025 0 10 6 ] [.01131 .10301 -18 -4.5 ] [.01131 .10301 0 4.5 ] [.01131 .20601 -18 -4.5 ] [.01131 .20601 0 4.5 ] [.01131 .30902 -18 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .41202 -18 -4.5 ] [.01131 .41202 0 4.5 ] [.01131 .51503 -6 -4.5 ] [.01131 .51503 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 [(0.2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(0.4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(0.6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(0.8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(1)] .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 gsave 1.025 0 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .10301 m .03006 .10301 L s [(0.2)] .01131 .10301 1 0 Mshowa .02381 .20601 m .03006 .20601 L s [(0.4)] .01131 .20601 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(0.6)] .01131 .30902 1 0 Mshowa .02381 .41202 m .03006 .41202 L s [(0.8)] .01131 .41202 1 0 Mshowa .02381 .51503 m .03006 .51503 L s [(1)] .01131 .51503 1 0 Mshowa .125 Mabswid .02381 .02575 m .02756 .02575 L s .02381 .0515 m .02756 .0515 L s .02381 .07725 m .02756 .07725 L s .02381 .12876 m .02756 .12876 L s .02381 .15451 m .02756 .15451 L s .02381 .18026 m .02756 .18026 L s .02381 .23176 m .02756 .23176 L s .02381 .25751 m .02756 .25751 L s .02381 .28327 m .02756 .28327 L s .02381 .33477 m .02756 .33477 L s .02381 .36052 m .02756 .36052 L s .02381 .38627 m .02756 .38627 L s .02381 .43777 m .02756 .43777 L s .02381 .46353 m .02756 .46353 L s .02381 .48928 m .02756 .48928 L s .02381 .54078 m .02756 .54078 L s .02381 .56653 m .02756 .56653 L s .02381 .59228 m .02756 .59228 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 1 0 0 r .01 w .02381 0 m .06244 .06546 L .10458 .1356 L .14415 .19912 L .18221 .25703 L .22272 .3142 L .26171 .36395 L .30316 .41022 L .34309 .44756 L .3815 .47618 L .40095 .48778 L .42237 .49823 L .44268 .50585 L .45178 .50853 L .46172 .51093 L .4671 .512 L .4721 .51285 L .47727 .51358 L .48196 .51412 L .48658 .51452 L .4887 .51467 L .49093 .5148 L .49332 .5149 L .49438 .51494 L .49552 .51497 L .49675 .515 L .49789 .51502 L .49859 .51502 L .49925 .51503 L .50049 .51503 L .50163 .51502 L .50286 .51501 L .50401 .51498 L .50508 .51496 L .50754 .51487 L .51014 .51474 L .51268 .51458 L .51504 .51439 L .5204 .51386 L .5293 .51262 L .53882 .51081 L .54906 .5083 L .56016 .50492 L .58032 .49706 L .62123 .47439 L .66064 .4444 L .69852 .40848 L .73886 .36325 L .77769 .31365 L .81897 .2553 L Mistroke .85873 .19461 L .89697 .13306 L .93767 .06527 L .97619 0 L Mfstroke .03 w .02381 0 Mdot .2619 .36418 Mdot .5 .51503 Mdot .7381 .36418 Mdot .97619 0 Mdot 0 g .01 w .02381 0 m .2619 .36418 L .5 .51503 L .7381 .36418 L .97619 0 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "The area underneath the broken line segments takes a very healthy bite of \ the area under the curve.\nThe (signed) area between the ", Cell[BoxData[ \(x\)]], "-axis and the broken line segments plotted above is the sum of the areas \ of consecutive trapezoids based on segments ", Cell[BoxData[ \(\([x, x + jump]\)\)]], " on the ", Cell[BoxData[ \(x\)]], "-axis:" }], "SmallText", AnimationDisplayTime->1], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[trapezoid, x];\)\), "\n", \(\(trapezoid[x_]\ := \ \n Graphics[{Thickness[ 0.01], \n\ \ \ \ \ \ \ \ \ \ Line[{{x, 0}, {x\ + \ jump, 0}, \ \n\ \ \ \ \ \ \ \ \ {x\ + \ jump, f[x\ + \ jump]}, {x, f[x]}, {x, 0}}]}];\)\n\), "\n", \(\(trapezoids\ = \ Table[trapezoid[x], {x, a, b - jump, jump}];\)\n\), "\n", \(\(Show[curveplot, pointplot, trapezoids, \n\ \ \ \ \ AxesLabel -> {"\", "\<\>"}, PlotRange -> {0, 1.2}, \n\ \ \ \ \ AspectRatio -> 1\/GoldenRatio, \n\ \ \ \ \ AxesOrigin -> {0, 0}];\)\)}], "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 0.952381 0 0.515028 [ [.21429 -0.0125 -9 -9 ] [.21429 -0.0125 9 0 ] [.40476 -0.0125 -9 -9 ] [.40476 -0.0125 9 0 ] [.59524 -0.0125 -9 -9 ] [.59524 -0.0125 9 0 ] [.78571 -0.0125 -9 -9 ] [.78571 -0.0125 9 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [1.025 0 0 -6 ] [1.025 0 10 6 ] [.01131 .10301 -18 -4.5 ] [.01131 .10301 0 4.5 ] [.01131 .20601 -18 -4.5 ] [.01131 .20601 0 4.5 ] [.01131 .30902 -18 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .41202 -18 -4.5 ] [.01131 .41202 0 4.5 ] [.01131 .51503 -6 -4.5 ] [.01131 .51503 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 [(0.2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(0.4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(0.6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(0.8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(1)] .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 gsave 1.025 0 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .10301 m .03006 .10301 L s [(0.2)] .01131 .10301 1 0 Mshowa .02381 .20601 m .03006 .20601 L s [(0.4)] .01131 .20601 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(0.6)] .01131 .30902 1 0 Mshowa .02381 .41202 m .03006 .41202 L s [(0.8)] .01131 .41202 1 0 Mshowa .02381 .51503 m .03006 .51503 L s [(1)] .01131 .51503 1 0 Mshowa .125 Mabswid .02381 .02575 m .02756 .02575 L s .02381 .0515 m .02756 .0515 L s .02381 .07725 m .02756 .07725 L s .02381 .12876 m .02756 .12876 L s .02381 .15451 m .02756 .15451 L s .02381 .18026 m .02756 .18026 L s .02381 .23176 m .02756 .23176 L s .02381 .25751 m .02756 .25751 L s .02381 .28327 m .02756 .28327 L s .02381 .33477 m .02756 .33477 L s .02381 .36052 m .02756 .36052 L s .02381 .38627 m .02756 .38627 L s .02381 .43777 m .02756 .43777 L s .02381 .46353 m .02756 .46353 L s .02381 .48928 m .02756 .48928 L s .02381 .54078 m .02756 .54078 L s .02381 .56653 m .02756 .56653 L s .02381 .59228 m .02756 .59228 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 1 0 0 r .01 w .02381 0 m .06244 .06546 L .10458 .1356 L .14415 .19912 L .18221 .25703 L .22272 .3142 L .26171 .36395 L .30316 .41022 L .34309 .44756 L .3815 .47618 L .40095 .48778 L .42237 .49823 L .44268 .50585 L .45178 .50853 L .46172 .51093 L .4671 .512 L .4721 .51285 L .47727 .51358 L .48196 .51412 L .48658 .51452 L .4887 .51467 L .49093 .5148 L .49332 .5149 L .49438 .51494 L .49552 .51497 L .49675 .515 L .49789 .51502 L .49859 .51502 L .49925 .51503 L .50049 .51503 L .50163 .51502 L .50286 .51501 L .50401 .51498 L .50508 .51496 L .50754 .51487 L .51014 .51474 L .51268 .51458 L .51504 .51439 L .5204 .51386 L .5293 .51262 L .53882 .51081 L .54906 .5083 L .56016 .50492 L .58032 .49706 L .62123 .47439 L .66064 .4444 L .69852 .40848 L .73886 .36325 L .77769 .31365 L .81897 .2553 L Mistroke .85873 .19461 L .89697 .13306 L .93767 .06527 L .97619 0 L Mfstroke .03 w .02381 0 Mdot .2619 .36418 Mdot .5 .51503 Mdot .7381 .36418 Mdot .97619 0 Mdot 0 g .01 w .02381 0 m .2619 0 L .2619 .36418 L .02381 0 L .02381 0 L s .2619 0 m .5 0 L .5 .51503 L .2619 .36418 L .2619 0 L s .5 0 m .7381 0 L .7381 .36418 L .5 .51503 L .5 0 L s .7381 0 m .97619 0 L .97619 0 L .7381 .36418 L .7381 0 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell["\<\ To get the full effect, grab all the plots above and everything between them, animate and run forward at speed 2.\ \>", "Special2"], Cell[TextData[{ "Each trapezoid has a base whose ", Cell[BoxData[ \(length = jump\)]], ".\nEach trapezoid has two heights: ", Cell[BoxData[ \(f[x]\)]], " and ", Cell[BoxData[ \(f[x + jump]\)]], ".\nSo the area of each of the trapezoids is\n ", Cell[BoxData[ \(jump[f[x] + \ f[x + jump]]\/2\)], FontSize->16], "\nand you start at \n ", Cell[BoxData[ \(x = \(0 = a\)\)]], " \nand finish at \n ", Cell[BoxData[ \(x = \(1 - jump = b - jump\)\)]], ". \nThe sum of the area of these four trapezoids is:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Area[4]\ = \ N[Sum[1\/2\ jump\ \((f[x] + f[x + jump])\), \ \n\ \ \ \ {x, a, b - jump, jump}]]\)], "Input"], Cell[BoxData[ \(0.6035533905932737`\)], "Output"] }, Open ]], Cell["\<\ This sum measures the sum of areas of the following four \ trapezoids:\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Table[ Show[curveplot, pointplot, trapezoid[x], AxesLabel \[Rule] {"\", "\<\>"}, AspectRatio \[Rule] 1\/GoldenRatio], {x, a, b - jump, jump}];\)\)], "Input", AspectRatioFixed->True, AnimationDisplayTime->1], 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 0.952381 0 0.515028 [ [.21429 -0.0125 -9 -9 ] [.21429 -0.0125 9 0 ] [.40476 -0.0125 -9 -9 ] [.40476 -0.0125 9 0 ] [.59524 -0.0125 -9 -9 ] [.59524 -0.0125 9 0 ] [.78571 -0.0125 -9 -9 ] [.78571 -0.0125 9 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [1.025 0 0 -6 ] [1.025 0 10 6 ] [.01131 .10301 -18 -4.5 ] [.01131 .10301 0 4.5 ] [.01131 .20601 -18 -4.5 ] [.01131 .20601 0 4.5 ] [.01131 .30902 -18 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .41202 -18 -4.5 ] [.01131 .41202 0 4.5 ] [.01131 .51503 -6 -4.5 ] [.01131 .51503 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 [(0.2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(0.4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(0.6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(0.8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(1)] .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 gsave 1.025 0 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .10301 m .03006 .10301 L s [(0.2)] .01131 .10301 1 0 Mshowa .02381 .20601 m .03006 .20601 L s [(0.4)] .01131 .20601 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(0.6)] .01131 .30902 1 0 Mshowa .02381 .41202 m .03006 .41202 L s [(0.8)] .01131 .41202 1 0 Mshowa .02381 .51503 m .03006 .51503 L s [(1)] .01131 .51503 1 0 Mshowa .125 Mabswid .02381 .02575 m .02756 .02575 L s .02381 .0515 m .02756 .0515 L s .02381 .07725 m .02756 .07725 L s .02381 .12876 m .02756 .12876 L s .02381 .15451 m .02756 .15451 L s .02381 .18026 m .02756 .18026 L s .02381 .23176 m .02756 .23176 L s .02381 .25751 m .02756 .25751 L s .02381 .28327 m .02756 .28327 L s .02381 .33477 m .02756 .33477 L s .02381 .36052 m .02756 .36052 L s .02381 .38627 m .02756 .38627 L s .02381 .43777 m .02756 .43777 L s .02381 .46353 m .02756 .46353 L s .02381 .48928 m .02756 .48928 L s .02381 .54078 m .02756 .54078 L s .02381 .56653 m .02756 .56653 L s .02381 .59228 m .02756 .59228 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 1 0 0 r .01 w .02381 0 m .06244 .06546 L .10458 .1356 L .14415 .19912 L .18221 .25703 L .22272 .3142 L .26171 .36395 L .30316 .41022 L .34309 .44756 L .3815 .47618 L .40095 .48778 L .42237 .49823 L .44268 .50585 L .45178 .50853 L .46172 .51093 L .4671 .512 L .4721 .51285 L .47727 .51358 L .48196 .51412 L .48658 .51452 L .4887 .51467 L .49093 .5148 L .49332 .5149 L .49438 .51494 L .49552 .51497 L .49675 .515 L .49789 .51502 L .49859 .51502 L .49925 .51503 L .50049 .51503 L .50163 .51502 L .50286 .51501 L .50401 .51498 L .50508 .51496 L .50754 .51487 L .51014 .51474 L .51268 .51458 L .51504 .51439 L .5204 .51386 L .5293 .51262 L .53882 .51081 L .54906 .5083 L .56016 .50492 L .58032 .49706 L .62123 .47439 L .66064 .4444 L .69852 .40848 L .73886 .36325 L .77769 .31365 L .81897 .2553 L Mistroke .85873 .19461 L .89697 .13306 L .93767 .06527 L .97619 0 L Mfstroke .03 w .02381 0 Mdot .2619 .36418 Mdot .5 .51503 Mdot .7381 .36418 Mdot .97619 0 Mdot 0 g .01 w .02381 0 m .2619 0 L .2619 .36418 L .02381 0 L .02381 0 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False], 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 0.952381 0 0.515028 [ [.21429 -0.0125 -9 -9 ] [.21429 -0.0125 9 0 ] [.40476 -0.0125 -9 -9 ] [.40476 -0.0125 9 0 ] [.59524 -0.0125 -9 -9 ] [.59524 -0.0125 9 0 ] [.78571 -0.0125 -9 -9 ] [.78571 -0.0125 9 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [1.025 0 0 -6 ] [1.025 0 10 6 ] [.01131 .10301 -18 -4.5 ] [.01131 .10301 0 4.5 ] [.01131 .20601 -18 -4.5 ] [.01131 .20601 0 4.5 ] [.01131 .30902 -18 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .41202 -18 -4.5 ] [.01131 .41202 0 4.5 ] [.01131 .51503 -6 -4.5 ] [.01131 .51503 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 [(0.2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(0.4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(0.6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(0.8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(1)] .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 gsave 1.025 0 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .10301 m .03006 .10301 L s [(0.2)] .01131 .10301 1 0 Mshowa .02381 .20601 m .03006 .20601 L s [(0.4)] .01131 .20601 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(0.6)] .01131 .30902 1 0 Mshowa .02381 .41202 m .03006 .41202 L s [(0.8)] .01131 .41202 1 0 Mshowa .02381 .51503 m .03006 .51503 L s [(1)] .01131 .51503 1 0 Mshowa .125 Mabswid .02381 .02575 m .02756 .02575 L s .02381 .0515 m .02756 .0515 L s .02381 .07725 m .02756 .07725 L s .02381 .12876 m .02756 .12876 L s .02381 .15451 m .02756 .15451 L s .02381 .18026 m .02756 .18026 L s .02381 .23176 m .02756 .23176 L s .02381 .25751 m .02756 .25751 L s .02381 .28327 m .02756 .28327 L s .02381 .33477 m .02756 .33477 L s .02381 .36052 m .02756 .36052 L s .02381 .38627 m .02756 .38627 L s .02381 .43777 m .02756 .43777 L s .02381 .46353 m .02756 .46353 L s .02381 .48928 m .02756 .48928 L s .02381 .54078 m .02756 .54078 L s .02381 .56653 m .02756 .56653 L s .02381 .59228 m .02756 .59228 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 1 0 0 r .01 w .02381 0 m .06244 .06546 L .10458 .1356 L .14415 .19912 L .18221 .25703 L .22272 .3142 L .26171 .36395 L .30316 .41022 L .34309 .44756 L .3815 .47618 L .40095 .48778 L .42237 .49823 L .44268 .50585 L .45178 .50853 L .46172 .51093 L .4671 .512 L .4721 .51285 L .47727 .51358 L .48196 .51412 L .48658 .51452 L .4887 .51467 L .49093 .5148 L .49332 .5149 L .49438 .51494 L .49552 .51497 L .49675 .515 L .49789 .51502 L .49859 .51502 L .49925 .51503 L .50049 .51503 L .50163 .51502 L .50286 .51501 L .50401 .51498 L .50508 .51496 L .50754 .51487 L .51014 .51474 L .51268 .51458 L .51504 .51439 L .5204 .51386 L .5293 .51262 L .53882 .51081 L .54906 .5083 L .56016 .50492 L .58032 .49706 L .62123 .47439 L .66064 .4444 L .69852 .40848 L .73886 .36325 L .77769 .31365 L .81897 .2553 L Mistroke .85873 .19461 L .89697 .13306 L .93767 .06527 L .97619 0 L Mfstroke .03 w .02381 0 Mdot .2619 .36418 Mdot .5 .51503 Mdot .7381 .36418 Mdot .97619 0 Mdot 0 g .01 w .2619 0 m .5 0 L .5 .51503 L .2619 .36418 L .2619 0 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False], 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 0.952381 0 0.515028 [ [.21429 -0.0125 -9 -9 ] [.21429 -0.0125 9 0 ] [.40476 -0.0125 -9 -9 ] [.40476 -0.0125 9 0 ] [.59524 -0.0125 -9 -9 ] [.59524 -0.0125 9 0 ] [.78571 -0.0125 -9 -9 ] [.78571 -0.0125 9 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [1.025 0 0 -6 ] [1.025 0 10 6 ] [.01131 .10301 -18 -4.5 ] [.01131 .10301 0 4.5 ] [.01131 .20601 -18 -4.5 ] [.01131 .20601 0 4.5 ] [.01131 .30902 -18 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .41202 -18 -4.5 ] [.01131 .41202 0 4.5 ] [.01131 .51503 -6 -4.5 ] [.01131 .51503 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 [(0.2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(0.4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(0.6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(0.8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(1)] .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 gsave 1.025 0 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .10301 m .03006 .10301 L s [(0.2)] .01131 .10301 1 0 Mshowa .02381 .20601 m .03006 .20601 L s [(0.4)] .01131 .20601 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(0.6)] .01131 .30902 1 0 Mshowa .02381 .41202 m .03006 .41202 L s [(0.8)] .01131 .41202 1 0 Mshowa .02381 .51503 m .03006 .51503 L s [(1)] .01131 .51503 1 0 Mshowa .125 Mabswid .02381 .02575 m .02756 .02575 L s .02381 .0515 m .02756 .0515 L s .02381 .07725 m .02756 .07725 L s .02381 .12876 m .02756 .12876 L s .02381 .15451 m .02756 .15451 L s .02381 .18026 m .02756 .18026 L s .02381 .23176 m .02756 .23176 L s .02381 .25751 m .02756 .25751 L s .02381 .28327 m .02756 .28327 L s .02381 .33477 m .02756 .33477 L s .02381 .36052 m .02756 .36052 L s .02381 .38627 m .02756 .38627 L s .02381 .43777 m .02756 .43777 L s .02381 .46353 m .02756 .46353 L s .02381 .48928 m .02756 .48928 L s .02381 .54078 m .02756 .54078 L s .02381 .56653 m .02756 .56653 L s .02381 .59228 m .02756 .59228 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 1 0 0 r .01 w .02381 0 m .06244 .06546 L .10458 .1356 L .14415 .19912 L .18221 .25703 L .22272 .3142 L .26171 .36395 L .30316 .41022 L .34309 .44756 L .3815 .47618 L .40095 .48778 L .42237 .49823 L .44268 .50585 L .45178 .50853 L .46172 .51093 L .4671 .512 L .4721 .51285 L .47727 .51358 L .48196 .51412 L .48658 .51452 L .4887 .51467 L .49093 .5148 L .49332 .5149 L .49438 .51494 L .49552 .51497 L .49675 .515 L .49789 .51502 L .49859 .51502 L .49925 .51503 L .50049 .51503 L .50163 .51502 L .50286 .51501 L .50401 .51498 L .50508 .51496 L .50754 .51487 L .51014 .51474 L .51268 .51458 L .51504 .51439 L .5204 .51386 L .5293 .51262 L .53882 .51081 L .54906 .5083 L .56016 .50492 L .58032 .49706 L .62123 .47439 L .66064 .4444 L .69852 .40848 L .73886 .36325 L .77769 .31365 L .81897 .2553 L Mistroke .85873 .19461 L .89697 .13306 L .93767 .06527 L .97619 0 L Mfstroke .03 w .02381 0 Mdot .2619 .36418 Mdot .5 .51503 Mdot .7381 .36418 Mdot .97619 0 Mdot 0 g .01 w .5 0 m .7381 0 L .7381 .36418 L .5 .51503 L .5 0 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False], 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 0.952381 0 0.515028 [ [.21429 -0.0125 -9 -9 ] [.21429 -0.0125 9 0 ] [.40476 -0.0125 -9 -9 ] [.40476 -0.0125 9 0 ] [.59524 -0.0125 -9 -9 ] [.59524 -0.0125 9 0 ] [.78571 -0.0125 -9 -9 ] [.78571 -0.0125 9 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [1.025 0 0 -6 ] [1.025 0 10 6 ] [.01131 .10301 -18 -4.5 ] [.01131 .10301 0 4.5 ] [.01131 .20601 -18 -4.5 ] [.01131 .20601 0 4.5 ] [.01131 .30902 -18 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .41202 -18 -4.5 ] [.01131 .41202 0 4.5 ] [.01131 .51503 -6 -4.5 ] [.01131 .51503 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 [(0.2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(0.4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(0.6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(0.8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(1)] .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 gsave 1.025 0 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .10301 m .03006 .10301 L s [(0.2)] .01131 .10301 1 0 Mshowa .02381 .20601 m .03006 .20601 L s [(0.4)] .01131 .20601 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(0.6)] .01131 .30902 1 0 Mshowa .02381 .41202 m .03006 .41202 L s [(0.8)] .01131 .41202 1 0 Mshowa .02381 .51503 m .03006 .51503 L s [(1)] .01131 .51503 1 0 Mshowa .125 Mabswid .02381 .02575 m .02756 .02575 L s .02381 .0515 m .02756 .0515 L s .02381 .07725 m .02756 .07725 L s .02381 .12876 m .02756 .12876 L s .02381 .15451 m .02756 .15451 L s .02381 .18026 m .02756 .18026 L s .02381 .23176 m .02756 .23176 L s .02381 .25751 m .02756 .25751 L s .02381 .28327 m .02756 .28327 L s .02381 .33477 m .02756 .33477 L s .02381 .36052 m .02756 .36052 L s .02381 .38627 m .02756 .38627 L s .02381 .43777 m .02756 .43777 L s .02381 .46353 m .02756 .46353 L s .02381 .48928 m .02756 .48928 L s .02381 .54078 m .02756 .54078 L s .02381 .56653 m .02756 .56653 L s .02381 .59228 m .02756 .59228 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 1 0 0 r .01 w .02381 0 m .06244 .06546 L .10458 .1356 L .14415 .19912 L .18221 .25703 L .22272 .3142 L .26171 .36395 L .30316 .41022 L .34309 .44756 L .3815 .47618 L .40095 .48778 L .42237 .49823 L .44268 .50585 L .45178 .50853 L .46172 .51093 L .4671 .512 L .4721 .51285 L .47727 .51358 L .48196 .51412 L .48658 .51452 L .4887 .51467 L .49093 .5148 L .49332 .5149 L .49438 .51494 L .49552 .51497 L .49675 .515 L .49789 .51502 L .49859 .51502 L .49925 .51503 L .50049 .51503 L .50163 .51502 L .50286 .51501 L .50401 .51498 L .50508 .51496 L .50754 .51487 L .51014 .51474 L .51268 .51458 L .51504 .51439 L .5204 .51386 L .5293 .51262 L .53882 .51081 L .54906 .5083 L .56016 .50492 L .58032 .49706 L .62123 .47439 L .66064 .4444 L .69852 .40848 L .73886 .36325 L .77769 .31365 L .81897 .2553 L Mistroke .85873 .19461 L .89697 .13306 L .93767 .06527 L .97619 0 L Mfstroke .03 w .02381 0 Mdot .2619 .36418 Mdot .5 .51503 Mdot .7381 .36418 Mdot .97619 0 Mdot 0 g .01 w .7381 0 m .97619 0 L .97619 0 L .7381 .36418 L .7381 0 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell["\<\ To get the full effect, select all four plots, animate and run \ forward at speed 2 or 3.\ \>", "Special2"], Cell[TextData[{ "The total area under the broken line segments over ", Cell[BoxData[ \(0 \[LessEqual] x \[LessEqual] 1\)]], " is the sum of the areas of the trapezoids:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Area[4]\ = \ \n\t N[Sum[1\/2\ jump\ \((f[x] + f[x + jump])\), {x, a, b - jump, jump}]]\)], "Input"], Cell[BoxData[ \(0.6035533905932737`\)], "Output"] }, Open ]], Cell[TextData[{ "This is a fairly reasonable rough estimate of the true value of the area \ measurement ", Cell[BoxData[ \(\[Integral]\_a\%b f[x] \[DifferentialD]x\)]], ".\nAnd it wasn't all that hard to get!" }], "SmallText"] }, Closed]], Cell["B.3.a.ii)", "Subsubsection"], Cell[CellGroupData[{ Cell["\<\ How do you get a better estimate than the estimate in part i)?\ \>", "Text"], Cell["Answer:", "Special1"], Cell["This assumes that the cells in part i) are activated.", "Special2"], Cell[TextData[{ "Use a smaller jump; the old jump was ", Cell[BoxData[ RowBox[{ StyleBox[\(\((b - a)\)\/4\), FontSize->16], "=", \(1\/4\)}]]], ".\nGo with ", Cell[BoxData[ RowBox[{"jump", "=", RowBox[{ StyleBox[\(\((b - a)\)\/10\), FontSize->16], "=", \(1\/10\)}]}]]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(jump = \(b - a\)\/10;\)\), "\n", \(\(points = Table[{x, f[x]}, {x, a, b, jump}];\)\n\), "\n", \(\(pointplot = ListPlot[points, PlotStyle \[Rule] {Red, PointSize[0.03]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(brokenlineplot = Graphics[{Thickness[0.01], Line[points]}];\)\n\), "\n", \(\(Show[curveplot, pointplot, brokenlineplot, \n\t AspectRatio \[Rule] 1\/GoldenRatio, AxesLabel \[Rule] {"\", "\<\>"}, PlotRange \[Rule] {0, 1.2}, AxesOrigin \[Rule] {0, 0}];\)\)}], "Input", AspectRatioFixed->True], 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 0.952381 0 0.515028 [ [.21429 -0.0125 -9 -9 ] [.21429 -0.0125 9 0 ] [.40476 -0.0125 -9 -9 ] [.40476 -0.0125 9 0 ] [.59524 -0.0125 -9 -9 ] [.59524 -0.0125 9 0 ] [.78571 -0.0125 -9 -9 ] [.78571 -0.0125 9 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [1.025 0 0 -6 ] [1.025 0 10 6 ] [.01131 .10301 -18 -4.5 ] [.01131 .10301 0 4.5 ] [.01131 .20601 -18 -4.5 ] [.01131 .20601 0 4.5 ] [.01131 .30902 -18 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .41202 -18 -4.5 ] [.01131 .41202 0 4.5 ] [.01131 .51503 -6 -4.5 ] [.01131 .51503 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 [(0.2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(0.4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(0.6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(0.8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(1)] .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 gsave 1.025 0 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .10301 m .03006 .10301 L s [(0.2)] .01131 .10301 1 0 Mshowa .02381 .20601 m .03006 .20601 L s [(0.4)] .01131 .20601 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(0.6)] .01131 .30902 1 0 Mshowa .02381 .41202 m .03006 .41202 L s [(0.8)] .01131 .41202 1 0 Mshowa .02381 .51503 m .03006 .51503 L s [(1)] .01131 .51503 1 0 Mshowa .125 Mabswid .02381 .02575 m .02756 .02575 L s .02381 .0515 m .02756 .0515 L s .02381 .07725 m .02756 .07725 L s .02381 .12876 m .02756 .12876 L s .02381 .15451 m .02756 .15451 L s .02381 .18026 m .02756 .18026 L s .02381 .23176 m .02756 .23176 L s .02381 .25751 m .02756 .25751 L s .02381 .28327 m .02756 .28327 L s .02381 .33477 m .02756 .33477 L s .02381 .36052 m .02756 .36052 L s .02381 .38627 m .02756 .38627 L s .02381 .43777 m .02756 .43777 L s .02381 .46353 m .02756 .46353 L s .02381 .48928 m .02756 .48928 L s .02381 .54078 m .02756 .54078 L s .02381 .56653 m .02756 .56653 L s .02381 .59228 m .02756 .59228 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 1 0 0 r .01 w .02381 0 m .06244 .06546 L .10458 .1356 L .14415 .19912 L .18221 .25703 L .22272 .3142 L .26171 .36395 L .30316 .41022 L .34309 .44756 L .3815 .47618 L .40095 .48778 L .42237 .49823 L .44268 .50585 L .45178 .50853 L .46172 .51093 L .4671 .512 L .4721 .51285 L .47727 .51358 L .48196 .51412 L .48658 .51452 L .4887 .51467 L .49093 .5148 L .49332 .5149 L .49438 .51494 L .49552 .51497 L .49675 .515 L .49789 .51502 L .49859 .51502 L .49925 .51503 L .50049 .51503 L .50163 .51502 L .50286 .51501 L .50401 .51498 L .50508 .51496 L .50754 .51487 L .51014 .51474 L .51268 .51458 L .51504 .51439 L .5204 .51386 L .5293 .51262 L .53882 .51081 L .54906 .5083 L .56016 .50492 L .58032 .49706 L .62123 .47439 L .66064 .4444 L .69852 .40848 L .73886 .36325 L .77769 .31365 L .81897 .2553 L Mistroke .85873 .19461 L .89697 .13306 L .93767 .06527 L .97619 0 L Mfstroke .03 w .02381 0 Mdot .11905 .15915 Mdot .21429 .30273 Mdot .30952 .41667 Mdot .40476 .48982 Mdot .5 .51503 Mdot .59524 .48982 Mdot .69048 .41667 Mdot .78571 .30273 Mdot .88095 .15915 Mdot .97619 0 Mdot 0 g .01 w .02381 0 m .11905 .15915 L .21429 .30273 L .30952 .41667 L .40476 .48982 L .5 .51503 L .59524 .48982 L .69048 .41667 L .78571 .30273 L .88095 .15915 L .97619 0 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "This time it's really hard to tell the difference between the area \ underneath the broken line segments over the ", Cell[BoxData[ \(x\)]], "-axis and the true area \n ", Cell[BoxData[ \(\[Integral]\_a\%b f[x] \[DifferentialD]x\)]], "\nunder the curve and over the ", Cell[BoxData[ \(x\)]], "-axis. " }], "SmallText"], Cell["Here are the corresponding trapezoids:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[trapezoid, x];\)\), "\n", \(\(trapezoid[ x_]\ := \ \n\ \ \ Graphics[{Thickness[ 0.01], \n\ \ \ \ \ \ \ Line[{{x, 0}, {x\ + \ jump, 0}, \n\ \ \ \ \ \ \ {x\ + \ jump, f[x\ + \ jump]}, {x, f[x]}, {x, 0}}]}];\)\n\), "\n", \(\(trapezoids\ = \ Table[trapezoid[x], {x, a, b - jump, jump}];\)\n\), "\n", \(\(Show[curveplot, pointplot, trapezoids, \n\ \ \ \ \ AxesLabel -> {"\", "\<\>"}, \n\ \ \ \ \ \ AspectRatio -> 1/GoldenRatio, \n\ \ \ \ \ PlotRange -> {0, 1.2}, \n\ \ \ \ \ AxesOrigin -> {0, 0}];\)\)}], "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 0.952381 0 0.515028 [ [.21429 -0.0125 -9 -9 ] [.21429 -0.0125 9 0 ] [.40476 -0.0125 -9 -9 ] [.40476 -0.0125 9 0 ] [.59524 -0.0125 -9 -9 ] [.59524 -0.0125 9 0 ] [.78571 -0.0125 -9 -9 ] [.78571 -0.0125 9 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [1.025 0 0 -6 ] [1.025 0 10 6 ] [.01131 .10301 -18 -4.5 ] [.01131 .10301 0 4.5 ] [.01131 .20601 -18 -4.5 ] [.01131 .20601 0 4.5 ] [.01131 .30902 -18 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .41202 -18 -4.5 ] [.01131 .41202 0 4.5 ] [.01131 .51503 -6 -4.5 ] [.01131 .51503 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 [(0.2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(0.4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(0.6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(0.8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(1)] .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 gsave 1.025 0 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .10301 m .03006 .10301 L s [(0.2)] .01131 .10301 1 0 Mshowa .02381 .20601 m .03006 .20601 L s [(0.4)] .01131 .20601 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(0.6)] .01131 .30902 1 0 Mshowa .02381 .41202 m .03006 .41202 L s [(0.8)] .01131 .41202 1 0 Mshowa .02381 .51503 m .03006 .51503 L s [(1)] .01131 .51503 1 0 Mshowa .125 Mabswid .02381 .02575 m .02756 .02575 L s .02381 .0515 m .02756 .0515 L s .02381 .07725 m .02756 .07725 L s .02381 .12876 m .02756 .12876 L s .02381 .15451 m .02756 .15451 L s .02381 .18026 m .02756 .18026 L s .02381 .23176 m .02756 .23176 L s .02381 .25751 m .02756 .25751 L s .02381 .28327 m .02756 .28327 L s .02381 .33477 m .02756 .33477 L s .02381 .36052 m .02756 .36052 L s .02381 .38627 m .02756 .38627 L s .02381 .43777 m .02756 .43777 L s .02381 .46353 m .02756 .46353 L s .02381 .48928 m .02756 .48928 L s .02381 .54078 m .02756 .54078 L s .02381 .56653 m .02756 .56653 L s .02381 .59228 m .02756 .59228 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 1 0 0 r .01 w .02381 0 m .06244 .06546 L .10458 .1356 L .14415 .19912 L .18221 .25703 L .22272 .3142 L .26171 .36395 L .30316 .41022 L .34309 .44756 L .3815 .47618 L .40095 .48778 L .42237 .49823 L .44268 .50585 L .45178 .50853 L .46172 .51093 L .4671 .512 L .4721 .51285 L .47727 .51358 L .48196 .51412 L .48658 .51452 L .4887 .51467 L .49093 .5148 L .49332 .5149 L .49438 .51494 L .49552 .51497 L .49675 .515 L .49789 .51502 L .49859 .51502 L .49925 .51503 L .50049 .51503 L .50163 .51502 L .50286 .51501 L .50401 .51498 L .50508 .51496 L .50754 .51487 L .51014 .51474 L .51268 .51458 L .51504 .51439 L .5204 .51386 L .5293 .51262 L .53882 .51081 L .54906 .5083 L .56016 .50492 L .58032 .49706 L .62123 .47439 L .66064 .4444 L .69852 .40848 L .73886 .36325 L .77769 .31365 L .81897 .2553 L Mistroke .85873 .19461 L .89697 .13306 L .93767 .06527 L .97619 0 L Mfstroke .03 w .02381 0 Mdot .11905 .15915 Mdot .21429 .30273 Mdot .30952 .41667 Mdot .40476 .48982 Mdot .5 .51503 Mdot .59524 .48982 Mdot .69048 .41667 Mdot .78571 .30273 Mdot .88095 .15915 Mdot .97619 0 Mdot 0 g .01 w .02381 0 m .11905 0 L .11905 .15915 L .02381 0 L .02381 0 L s .11905 0 m .21429 0 L .21429 .30273 L .11905 .15915 L .11905 0 L s .21429 0 m .30952 0 L .30952 .41667 L .21429 .30273 L .21429 0 L s .30952 0 m .40476 0 L .40476 .48982 L .30952 .41667 L .30952 0 L s .40476 0 m .5 0 L .5 .51503 L .40476 .48982 L .40476 0 L s .5 0 m .59524 0 L .59524 .48982 L .5 .51503 L .5 0 L s .59524 0 m .69048 0 L .69048 .41667 L .59524 .48982 L .59524 0 L s .69048 0 m .78571 0 L .78571 .30273 L .69048 .41667 L .69048 0 L s .78571 0 m .88095 0 L .88095 .15915 L .78571 .30273 L .78571 0 L s .88095 0 m .97619 0 L .97619 0 L .88095 .15915 L .88095 0 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "The sum of the area inside these ", Cell[BoxData[ \(10\)]], " trapezoids is:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Area[10]\ = \ \n N[Sum[1\/2\ jump\ \((f[x] + f[x + jump])\), {x, a, b - jump, jump}]]\)], "Input"], Cell[BoxData[ \(0.6313751514675043`\)], "Output"] }, Open ]], Cell["Compare the old estimate with the new estimate:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \({Area[4], Area[10]}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({0.6035533905932737`, 0.6313751514675043`}\)], "Output"] }, Open ]], Cell["\<\ Of the two, the second estimate is the more trustworthy, and the \ plots tell you why. To go for even more accuracy, decrease the jump size \ again:\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(jump\ = \ \(b - a\)\/50;\)\), "\n", \(\(points\ = \ Table[{x, f[x]}, {x, a, b, jump}];\)\n\), "\n", \(\(pointplot\ = \n\t ListPlot[points, \n\tPlotStyle -> {Red, PointSize[0.03]}, \n\t AspectRatio -> 1\/GoldenRatio, \n\t DisplayFunction -> \ Identity];\)\n\t\), "\n", \(\(brokenlineplot\ = \n\t Graphics[{Thickness[0.01], Line[points]}];\)\n\t\), "\n", \(\(Show[curveplot, pointplot, brokenlineplot, \n\ \ \ \ \ AxesLabel -> {"\", "\<\>"}, PlotRange -> {0, 1.2}, \n\ \ \ \ \ AxesOrigin -> {0, 0}];\)\)}], "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 0.952381 0 0.515028 [ [.21429 -0.0125 -9 -9 ] [.21429 -0.0125 9 0 ] [.40476 -0.0125 -9 -9 ] [.40476 -0.0125 9 0 ] [.59524 -0.0125 -9 -9 ] [.59524 -0.0125 9 0 ] [.78571 -0.0125 -9 -9 ] [.78571 -0.0125 9 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [1.025 0 0 -6 ] [1.025 0 10 6 ] [.01131 .10301 -18 -4.5 ] [.01131 .10301 0 4.5 ] [.01131 .20601 -18 -4.5 ] [.01131 .20601 0 4.5 ] [.01131 .30902 -18 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .41202 -18 -4.5 ] [.01131 .41202 0 4.5 ] [.01131 .51503 -6 -4.5 ] [.01131 .51503 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 [(0.2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(0.4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(0.6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(0.8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(1)] .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 gsave 1.025 0 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .10301 m .03006 .10301 L s [(0.2)] .01131 .10301 1 0 Mshowa .02381 .20601 m .03006 .20601 L s [(0.4)] .01131 .20601 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(0.6)] .01131 .30902 1 0 Mshowa .02381 .41202 m .03006 .41202 L s [(0.8)] .01131 .41202 1 0 Mshowa .02381 .51503 m .03006 .51503 L s [(1)] .01131 .51503 1 0 Mshowa .125 Mabswid .02381 .02575 m .02756 .02575 L s .02381 .0515 m .02756 .0515 L s .02381 .07725 m .02756 .07725 L s .02381 .12876 m .02756 .12876 L s .02381 .15451 m .02756 .15451 L s .02381 .18026 m .02756 .18026 L s .02381 .23176 m .02756 .23176 L s .02381 .25751 m .02756 .25751 L s .02381 .28327 m .02756 .28327 L s .02381 .33477 m .02756 .33477 L s .02381 .36052 m .02756 .36052 L s .02381 .38627 m .02756 .38627 L s .02381 .43777 m .02756 .43777 L s .02381 .46353 m .02756 .46353 L s .02381 .48928 m .02756 .48928 L s .02381 .54078 m .02756 .54078 L s .02381 .56653 m .02756 .56653 L s .02381 .59228 m .02756 .59228 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 1 0 0 r .01 w .02381 0 m .06244 .06546 L .10458 .1356 L .14415 .19912 L .18221 .25703 L .22272 .3142 L .26171 .36395 L .30316 .41022 L .34309 .44756 L .3815 .47618 L .40095 .48778 L .42237 .49823 L .44268 .50585 L .45178 .50853 L .46172 .51093 L .4671 .512 L .4721 .51285 L .47727 .51358 L .48196 .51412 L .48658 .51452 L .4887 .51467 L .49093 .5148 L .49332 .5149 L .49438 .51494 L .49552 .51497 L .49675 .515 L .49789 .51502 L .49859 .51502 L .49925 .51503 L .50049 .51503 L .50163 .51502 L .50286 .51501 L .50401 .51498 L .50508 .51496 L .50754 .51487 L .51014 .51474 L .51268 .51458 L .51504 .51439 L .5204 .51386 L .5293 .51262 L .53882 .51081 L .54906 .5083 L .56016 .50492 L .58032 .49706 L .62123 .47439 L .66064 .4444 L .69852 .40848 L .73886 .36325 L .77769 .31365 L .81897 .2553 L Mistroke .85873 .19461 L .89697 .13306 L .93767 .06527 L .97619 0 L Mfstroke .03 w .02381 0 Mdot .04286 .03234 Mdot .0619 .06455 Mdot .08095 .09651 Mdot .1 .12808 Mdot .11905 .15915 Mdot .1381 .18959 Mdot .15714 .21929 Mdot .17619 .24812 Mdot .19524 .27597 Mdot .21429 .30273 Mdot .23333 .32829 Mdot .25238 .35256 Mdot .27143 .37544 Mdot .29048 .39684 Mdot .30952 .41667 Mdot .32857 .43485 Mdot .34762 .45132 Mdot .36667 .46601 Mdot .38571 .47886 Mdot .40476 .48982 Mdot .42381 .49885 Mdot .44286 .50591 Mdot .4619 .51097 Mdot .48095 .51401 Mdot .5 .51503 Mdot .51905 .51401 Mdot .5381 .51097 Mdot .55714 .50591 Mdot .57619 .49885 Mdot .59524 .48982 Mdot .61429 .47886 Mdot .63333 .46601 Mdot .65238 .45132 Mdot .67143 .43485 Mdot .69048 .41667 Mdot .70952 .39684 Mdot .72857 .37544 Mdot .74762 .35256 Mdot .76667 .32829 Mdot .78571 .30273 Mdot .80476 .27597 Mdot .82381 .24812 Mdot .84286 .21929 Mdot .8619 .18959 Mdot .88095 .15915 Mdot .9 .12808 Mdot .91905 .09651 Mdot .9381 .06455 Mdot .95714 .03234 Mdot .97619 0 Mdot 0 g .01 w .02381 0 m .04286 .03234 L .0619 .06455 L .08095 .09651 L .1 .12808 L .11905 .15915 L .1381 .18959 L .15714 .21929 L .17619 .24812 L .19524 .27597 L .21429 .30273 L .23333 .32829 L .25238 .35256 L .27143 .37544 L .29048 .39684 L .30952 .41667 L .32857 .43485 L .34762 .45132 L .36667 .46601 L .38571 .47886 L .40476 .48982 L .42381 .49885 L .44286 .50591 L .4619 .51097 L .48095 .51401 L .5 .51503 L .51905 .51401 L .5381 .51097 L .55714 .50591 L .57619 .49885 L .59524 .48982 L .61429 .47886 L .63333 .46601 L .65238 .45132 L .67143 .43485 L .69048 .41667 L .70952 .39684 L .72857 .37544 L .74762 .35256 L .76667 .32829 L .78571 .30273 L .80476 .27597 L .82381 .24812 L .84286 .21929 L .8619 .18959 L .88095 .15915 L .9 .12808 L .91905 .09651 L .9381 .06455 L .95714 .03234 L Mistroke .97619 0 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "This time the points are packed so densely that it's really hard to tell \ the difference between the area underneath the broken line segments over the \ ", Cell[BoxData[ \(x\)]], "-axis and the true area ", Cell[BoxData[ \(\[Integral]\_a\%b f[x] \[DifferentialD]x\)]], " under the curve over the ", Cell[BoxData[ \(x\)]], "-axis. " }], "SmallText"], Cell["Here are the trapezoids:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(trapezoids = Table[trapezoid[x], {x, a, b - jump, jump}];\)\n\), "\n", \(\(Show[curveplot, pointplot, trapezoids, \n\t AxesLabel \[Rule] {"\", "\<\>"}, PlotRange \[Rule] {0, 1.2}, AspectRatio \[Rule] 1\/GoldenRatio, AxesOrigin \[Rule] {0, 0}];\)\)}], "Input", AspectRatioFixed->True], 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 0.952381 0 0.515028 [ [.21429 -0.0125 -9 -9 ] [.21429 -0.0125 9 0 ] [.40476 -0.0125 -9 -9 ] [.40476 -0.0125 9 0 ] [.59524 -0.0125 -9 -9 ] [.59524 -0.0125 9 0 ] [.78571 -0.0125 -9 -9 ] [.78571 -0.0125 9 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [1.025 0 0 -6 ] [1.025 0 10 6 ] [.01131 .10301 -18 -4.5 ] [.01131 .10301 0 4.5 ] [.01131 .20601 -18 -4.5 ] [.01131 .20601 0 4.5 ] [.01131 .30902 -18 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .41202 -18 -4.5 ] [.01131 .41202 0 4.5 ] [.01131 .51503 -6 -4.5 ] [.01131 .51503 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 [(0.2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(0.4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(0.6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(0.8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(1)] .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 gsave 1.025 0 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .10301 m .03006 .10301 L s [(0.2)] .01131 .10301 1 0 Mshowa .02381 .20601 m .03006 .20601 L s [(0.4)] .01131 .20601 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(0.6)] .01131 .30902 1 0 Mshowa .02381 .41202 m .03006 .41202 L s [(0.8)] .01131 .41202 1 0 Mshowa .02381 .51503 m .03006 .51503 L s [(1)] .01131 .51503 1 0 Mshowa .125 Mabswid .02381 .02575 m .02756 .02575 L s .02381 .0515 m .02756 .0515 L s .02381 .07725 m .02756 .07725 L s .02381 .12876 m .02756 .12876 L s .02381 .15451 m .02756 .15451 L s .02381 .18026 m .02756 .18026 L s .02381 .23176 m .02756 .23176 L s .02381 .25751 m .02756 .25751 L s .02381 .28327 m .02756 .28327 L s .02381 .33477 m .02756 .33477 L s .02381 .36052 m .02756 .36052 L s .02381 .38627 m .02756 .38627 L s .02381 .43777 m .02756 .43777 L s .02381 .46353 m .02756 .46353 L s .02381 .48928 m .02756 .48928 L s .02381 .54078 m .02756 .54078 L s .02381 .56653 m .02756 .56653 L s .02381 .59228 m .02756 .59228 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 1 0 0 r .01 w .02381 0 m .06244 .06546 L .10458 .1356 L .14415 .19912 L .18221 .25703 L .22272 .3142 L .26171 .36395 L .30316 .41022 L .34309 .44756 L .3815 .47618 L .40095 .48778 L .42237 .49823 L .44268 .50585 L .45178 .50853 L .46172 .51093 L .4671 .512 L .4721 .51285 L .47727 .51358 L .48196 .51412 L .48658 .51452 L .4887 .51467 L .49093 .5148 L .49332 .5149 L .49438 .51494 L .49552 .51497 L .49675 .515 L .49789 .51502 L .49859 .51502 L .49925 .51503 L .50049 .51503 L .50163 .51502 L .50286 .51501 L .50401 .51498 L .50508 .51496 L .50754 .51487 L .51014 .51474 L .51268 .51458 L .51504 .51439 L .5204 .51386 L .5293 .51262 L .53882 .51081 L .54906 .5083 L .56016 .50492 L .58032 .49706 L .62123 .47439 L .66064 .4444 L .69852 .40848 L .73886 .36325 L .77769 .31365 L .81897 .2553 L Mistroke .85873 .19461 L .89697 .13306 L .93767 .06527 L .97619 0 L Mfstroke .03 w .02381 0 Mdot .04286 .03234 Mdot .0619 .06455 Mdot .08095 .09651 Mdot .1 .12808 Mdot .11905 .15915 Mdot .1381 .18959 Mdot .15714 .21929 Mdot .17619 .24812 Mdot .19524 .27597 Mdot .21429 .30273 Mdot .23333 .32829 Mdot .25238 .35256 Mdot .27143 .37544 Mdot .29048 .39684 Mdot .30952 .41667 Mdot .32857 .43485 Mdot .34762 .45132 Mdot .36667 .46601 Mdot .38571 .47886 Mdot .40476 .48982 Mdot .42381 .49885 Mdot .44286 .50591 Mdot .4619 .51097 Mdot .48095 .51401 Mdot .5 .51503 Mdot .51905 .51401 Mdot .5381 .51097 Mdot .55714 .50591 Mdot .57619 .49885 Mdot .59524 .48982 Mdot .61429 .47886 Mdot .63333 .46601 Mdot .65238 .45132 Mdot .67143 .43485 Mdot .69048 .41667 Mdot .70952 .39684 Mdot .72857 .37544 Mdot .74762 .35256 Mdot .76667 .32829 Mdot .78571 .30273 Mdot .80476 .27597 Mdot .82381 .24812 Mdot .84286 .21929 Mdot .8619 .18959 Mdot .88095 .15915 Mdot .9 .12808 Mdot .91905 .09651 Mdot .9381 .06455 Mdot .95714 .03234 Mdot .97619 0 Mdot 0 g .01 w .02381 0 m .04286 0 L .04286 .03234 L .02381 0 L .02381 0 L s .04286 0 m .0619 0 L .0619 .06455 L .04286 .03234 L .04286 0 L s .0619 0 m .08095 0 L .08095 .09651 L .0619 .06455 L .0619 0 L s .08095 0 m .1 0 L .1 .12808 L .08095 .09651 L .08095 0 L s .1 0 m .11905 0 L .11905 .15915 L .1 .12808 L .1 0 L s .11905 0 m .1381 0 L .1381 .18959 L .11905 .15915 L .11905 0 L s .1381 0 m .15714 0 L .15714 .21929 L .1381 .18959 L .1381 0 L s .15714 0 m .17619 0 L .17619 .24812 L .15714 .21929 L .15714 0 L s .17619 0 m .19524 0 L .19524 .27597 L .17619 .24812 L .17619 0 L s .19524 0 m .21429 0 L .21429 .30273 L .19524 .27597 L .19524 0 L s .21429 0 m .23333 0 L .23333 .32829 L .21429 .30273 L .21429 0 L s .23333 0 m .25238 0 L .25238 .35256 L .23333 .32829 L .23333 0 L s .25238 0 m .27143 0 L .27143 .37544 L .25238 .35256 L .25238 0 L s .27143 0 m .29048 0 L .29048 .39684 L .27143 .37544 L .27143 0 L s .29048 0 m .30952 0 L .30952 .41667 L .29048 .39684 L .29048 0 L s .30952 0 m .32857 0 L .32857 .43485 L .30952 .41667 L .30952 0 L s .32857 0 m .34762 0 L .34762 .45132 L .32857 .43485 L .32857 0 L s .34762 0 m .36667 0 L .36667 .46601 L .34762 .45132 L .34762 0 L s .36667 0 m .38571 0 L .38571 .47886 L .36667 .46601 L .36667 0 L s .38571 0 m .40476 0 L .40476 .48982 L .38571 .47886 L .38571 0 L s .40476 0 m .42381 0 L .42381 .49885 L .40476 .48982 L .40476 0 L s .42381 0 m .44286 0 L .44286 .50591 L .42381 .49885 L .42381 0 L s .44286 0 m .4619 0 L .4619 .51097 L .44286 .50591 L .44286 0 L s .4619 0 m .48095 0 L .48095 .51401 L .4619 .51097 L .4619 0 L s .48095 0 m .5 0 L .5 .51503 L .48095 .51401 L .48095 0 L s .5 0 m .51905 0 L .51905 .51401 L .5 .51503 L .5 0 L s .51905 0 m .5381 0 L .5381 .51097 L .51905 .51401 L .51905 0 L s .5381 0 m .55714 0 L .55714 .50591 L .5381 .51097 L .5381 0 L s .55714 0 m .57619 0 L .57619 .49885 L .55714 .50591 L .55714 0 L s .57619 0 m .59524 0 L .59524 .48982 L .57619 .49885 L .57619 0 L s .59524 0 m .61429 0 L .61429 .47886 L .59524 .48982 L .59524 0 L s .61429 0 m .63333 0 L .63333 .46601 L .61429 .47886 L .61429 0 L s .63333 0 m .65238 0 L .65238 .45132 L .63333 .46601 L .63333 0 L s .65238 0 m .67143 0 L .67143 .43485 L .65238 .45132 L .65238 0 L s .67143 0 m .69048 0 L .69048 .41667 L .67143 .43485 L .67143 0 L s .69048 0 m .70952 0 L .70952 .39684 L .69048 .41667 L .69048 0 L s .70952 0 m .72857 0 L .72857 .37544 L .70952 .39684 L .70952 0 L s .72857 0 m .74762 0 L .74762 .35256 L .72857 .37544 L .72857 0 L s .74762 0 m .76667 0 L .76667 .32829 L .74762 .35256 L .74762 0 L s .76667 0 m .78571 0 L .78571 .30273 L .76667 .32829 L .76667 0 L s .78571 0 m .80476 0 L .80476 .27597 L .78571 .30273 L .78571 0 L s .80476 0 m .82381 0 L .82381 .24812 L .80476 .27597 L .80476 0 L s .82381 0 m .84286 0 L .84286 .21929 L .82381 .24812 L .82381 0 L s .84286 0 m .8619 0 L .8619 .18959 L .84286 .21929 L .84286 0 L s .8619 0 m .88095 0 L .88095 .15915 L .8619 .18959 L .8619 0 L s .88095 0 m .9 0 L .9 .12808 L .88095 .15915 L .88095 0 L s .9 0 m .91905 0 L .91905 .09651 L .9 .12808 L .9 0 L s .91905 0 m .9381 0 L .9381 .06455 L .91905 .09651 L .91905 0 L s .9381 0 m .95714 0 L .95714 .03234 L .9381 .06455 L .9381 0 L s .95714 0 m .97619 0 L .97619 0 L .95714 .03234 L .95714 0 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "The sum of the area measurements of these ", Cell[BoxData[ \(50\)]], " trapezoids is:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Area[50]\ = \ \n\t N[Sum[1\/2\ jump\ \((f[x] + f[x + jump])\), {x, a, b - jump, jump}]]\)], "Input"], Cell[BoxData[ \(0.6364103190754791`\)], "Output"] }, Open ]], Cell["Compare the old estimates with the new estimate:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \({Area[4], Area[10], Area[50]}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({0.6035533905932737`, 0.6313751514675043`, 0.6364103190754791`}\)], "Output"] }, Open ]], Cell[TextData[{ "Of the three, the last estimate is the most trustworthy, and again the \ plots tell you why.\nYou can say with some confidence that\n ", Cell[BoxData[ \(\[Integral]\_a\%b f[x] \[DifferentialD]x = 0.6\)]], " \nto one accurate decimal.\nIf you want a better estimate, go with:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(jump\ = \ \(b - a\)\/200;\)\), "\n", \(Area[200]\ = \ \n\t N[Sum[1\/2\ jump\ \((f[x] + f[x + jump])\), {x, a, b - jump, jump}]]\)}], "Input"], Cell[BoxData[ \(0.6366066823443611`\)], "Output"] }, Open ]], Cell[TextData[{ "This is even more trustworthy than the estimate immediately above. You can \ say with even more confidence that\n ", Cell[BoxData[ \(\[Integral]\_a\%b f[x] \[DifferentialD]x = 0.64\)]], "\nto two accurate decimals.\nSee what ", StyleBox["Mathematica", FontSlant->"Italic"], " thinks:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(N[\[Integral]\_a\%b f[x] \[DifferentialD]x]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.6366197723675814`\)], "Output"] }, Open ]], Cell["That last estimate was really close!", "SmallText"] }, Closed]], Cell["B.3.a.iii) NIntegrate", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "How are these calculations related to the ", StyleBox["Mathematica", FontSlant->"Italic"], " instruction ", StyleBox["NIntegrate", "Input", FontColor->GrayLevel[0]], "?" }], "Text"], Cell["Answer:", "Subsubsection"], Cell["This assumes that the cells in part i) are activated.", "Special2"], Cell[TextData[{ "The ", StyleBox["Mathematica", FontSlant->"Italic"], " instruction ", Cell[BoxData[ \(NIntegrate\)]], " uses a professionally written approximation procedure similar in spirit \ to what you just did with the trapezoids. Let's try this instruction to test \ our estimates above:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(MathematicaEstimate = NIntegrate[f[x], {x, a, b}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.6366197723675814`\)], "Output"] }, Open ]], Cell["Compare this to the estimates you got above:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \({Area[4], Area[10], Area[50], Area[200], MathematicaEstimate}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({0.6035533905932737`, 0.6313751514675043`, 0.6364103190754791`, 0.6366066823443611`, 0.6366197723675814`}\)], "Output"] }, Open ]], Cell[TextData[{ "The estimate coming from ", Cell[BoxData[ \(200\)]], " trapezoids was pretty darned good.\nThe advantages of the ", StyleBox["Mathematica", FontSlant->"Italic"], " instruction are:\n\[RightArrow] i) Little typing.\n\[RightArrow] ii) \ Lots of accuracy." }], "SmallText"] }, Closed]], Cell["B.3.b)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Use trapezoids to get a reasonable idea of the value of the area \ measurement\n ", Cell[BoxData[ \(\[Integral]\_1\%4\(\((x\^3 - 2\ x\^2)\)\ E\^\(-x\)\) \[DifferentialD]x\)]], "\nCheck your estimate with the ", StyleBox["Mathematica", FontSlant->"Italic"], " instruction ", StyleBox["NIntegrate", "Input", FontColor->GrayLevel[0]], ". " }], "Text"], Cell["Answer:", "Special1"], Cell["Define the function:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(a = 1;\)\), "\n", \(\(b = 4;\)\), "\n", \(\(Clear[f, x];\)\), "\n", \(f[x_] = \((x\^3 - 2\ x\^2)\)\ E\^\(-x\)\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\[ExponentialE]\^\(-x\)\ \((\(-2\)\ x\^2 + x\^3)\)\)], "Output"] }, Open ]], Cell["Look at the plot:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(curveplot\ = \ Plot[f[x], {x, a, b}, \n\ \ \ \ \ \ PlotStyle -> {{Red, Thickness[0.01]}}, \n\ \ \ \ \ \ AspectRatio -> 1\/GoldenRatio, \n\ \ \ \ \ \ AxesLabel -> {"\", "\<\>"}, PlotRange -> All, \n\ \ \ \ \ \ AxesOrigin -> {a, 0}];\)\)], "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 -0.293651 0.31746 0.241696 0.616998 [ [.18254 .2292 -9 -9 ] [.18254 .2292 9 0 ] [.34127 .2292 -3 -9 ] [.34127 .2292 3 0 ] [.5 .2292 -9 -9 ] [.5 .2292 9 0 ] [.65873 .2292 -3 -9 ] [.65873 .2292 3 0 ] [.81746 .2292 -9 -9 ] [.81746 .2292 9 0 ] [.97619 .2292 -3 -9 ] [.97619 .2292 3 0 ] [1.025 .2417 0 -6 ] [1.025 .2417 10 6 ] [.01131 .1183 -24 -4.5 ] [.01131 .1183 0 4.5 ] [.01131 .3651 -18 -4.5 ] [.01131 .3651 0 4.5 ] [.01131 .4885 -18 -4.5 ] [.01131 .4885 0 4.5 ] [.01131 .61189 -18 -4.5 ] [.01131 .61189 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 .18254 .2417 m .18254 .24795 L s [(1.5)] .18254 .2292 0 1 Mshowa .34127 .2417 m .34127 .24795 L s [(2)] .34127 .2292 0 1 Mshowa .5 .2417 m .5 .24795 L s [(2.5)] .5 .2292 0 1 Mshowa .65873 .2417 m .65873 .24795 L s [(3)] .65873 .2292 0 1 Mshowa .81746 .2417 m .81746 .24795 L s [(3.5)] .81746 .2292 0 1 Mshowa .97619 .2417 m .97619 .24795 L s [(4)] .97619 .2292 0 1 Mshowa .125 Mabswid .05556 .2417 m .05556 .24545 L s .0873 .2417 m .0873 .24545 L s .11905 .2417 m .11905 .24545 L s .15079 .2417 m .15079 .24545 L s .21429 .2417 m .21429 .24545 L s .24603 .2417 m .24603 .24545 L s .27778 .2417 m .27778 .24545 L s .30952 .2417 m .30952 .24545 L s .37302 .2417 m .37302 .24545 L s .40476 .2417 m .40476 .24545 L s .43651 .2417 m .43651 .24545 L s .46825 .2417 m .46825 .24545 L s .53175 .2417 m .53175 .24545 L s .56349 .2417 m .56349 .24545 L s .59524 .2417 m .59524 .24545 L s .62698 .2417 m .62698 .24545 L s .69048 .2417 m .69048 .24545 L s .72222 .2417 m .72222 .24545 L s .75397 .2417 m .75397 .24545 L s .78571 .2417 m .78571 .24545 L s .84921 .2417 m .84921 .24545 L s .88095 .2417 m .88095 .24545 L s .9127 .2417 m .9127 .24545 L s .94444 .2417 m .94444 .24545 L s .25 Mabswid 0 .2417 m 1 .2417 L s gsave 1.025 .2417 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .1183 m .03006 .1183 L s [(-0.2)] .01131 .1183 1 0 Mshowa .02381 .3651 m .03006 .3651 L s [(0.2)] .01131 .3651 1 0 Mshowa .02381 .4885 m .03006 .4885 L s [(0.4)] .01131 .4885 1 0 Mshowa .02381 .61189 m .03006 .61189 L s [(0.6)] .01131 .61189 1 0 Mshowa .125 Mabswid .02381 .02575 m .02756 .02575 L s .02381 .0566 m .02756 .0566 L s .02381 .08745 m .02756 .08745 L s .02381 .14915 m .02756 .14915 L s .02381 .18 m .02756 .18 L s .02381 .21085 m .02756 .21085 L s .02381 .27255 m .02756 .27255 L s .02381 .3034 m .02756 .3034 L s .02381 .33425 m .02756 .33425 L s .02381 .39595 m .02756 .39595 L s .02381 .4268 m .02756 .4268 L s .02381 .45765 m .02756 .45765 L s .02381 .51935 m .02756 .51935 L s .02381 .5502 m .02756 .5502 L s .02381 .58104 m .02756 .58104 L s .25 Mabswid .02381 0 m .02381 .61803 L s 1 0 0 r .01 w .02381 .01472 m .02499 .01472 L .02605 .01473 L .02729 .01476 L .02846 .01479 L .03053 .01487 L .03279 .01499 L .03527 .01515 L .0379 .01538 L .04262 .01589 L .05205 .01735 L .06244 .0196 L .07293 .02254 L .0842 .02642 L .10458 .03522 L .1458 .05934 L .18551 .08925 L .22371 .12276 L .26435 .16205 L .30348 .20208 L .34506 .24569 L .38513 .28763 L .42368 .32707 L .46468 .36726 L .50417 .40369 L .54214 .43621 L .58257 .4678 L .62148 .49506 L .66284 .52053 L .70268 .54162 L .74101 .55876 L .78179 .57368 L .82106 .58497 L .85881 .59311 L .87816 .5963 L .89901 .59904 L .91905 .60101 L .92913 .60177 L .94015 .60242 L .94928 .60282 L .95758 .60308 L .96217 .60319 L .96476 .60323 L .96715 .60326 L .96924 .60329 L .97044 .6033 L .97154 .6033 L .9727 .60331 L .97394 .60332 L .9751 .60332 L Mistroke .97619 .60332 L Mfstroke 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "Here are the trapezoids for ", Cell[BoxData[ RowBox[{"jump", "=", StyleBox[\(\((b - a)\)\/4\), FontSize->16]}]]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(jump\ = \ \(b - a\)\/4;\)\), "\n", \(\(points\ = \ Table[{x, f[x]}, {x, a, b, jump}];\)\n\), "\n", \(\(pointplot\ = \n\t ListPlot[points, \n\tPlotStyle -> {Red, PointSize[0.03]}, \n\t DisplayFunction -> \ Identity];\)\n\t\), "\n", \(\(Clear[trapezoid, x];\)\), "\n", \(\(trapezoid[ x_]\ := \ \n\ \ \ Graphics[{Thickness[ 0.01], \n\ \ \ \ \ \ \ Line[{{x, 0}, {x\ + \ jump, 0}, \ \n\ \ \ \ \ {x\ + \ jump, f[x\ + \ jump]}, {x, f[x]}, {x, 0}}]}];\)\n\), "\n", \(\(trapezoids\ = \ Table[trapezoid[x], {x, a, b - jump, jump}];\)\n\), "\n", \(\(Show[curveplot, trapezoids, pointplot, \n\t\ AspectRatio -> 1\/GoldenRatio, \n\ \ \ \ \ AxesLabel -> {"\", "\<\>"}, PlotRange -> All, \n\ \ \ \ \ AxesOrigin -> {a, 0}];\)\)}], "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 -0.293651 0.31746 0.241696 0.616998 [ [.18254 .2292 -9 -9 ] [.18254 .2292 9 0 ] [.34127 .2292 -3 -9 ] [.34127 .2292 3 0 ] [.5 .2292 -9 -9 ] [.5 .2292 9 0 ] [.65873 .2292 -3 -9 ] [.65873 .2292 3 0 ] [.81746 .2292 -9 -9 ] [.81746 .2292 9 0 ] [.97619 .2292 -3 -9 ] [.97619 .2292 3 0 ] [1.025 .2417 0 -6 ] [1.025 .2417 10 6 ] [.01131 .1183 -24 -4.5 ] [.01131 .1183 0 4.5 ] [.01131 .3651 -18 -4.5 ] [.01131 .3651 0 4.5 ] [.01131 .4885 -18 -4.5 ] [.01131 .4885 0 4.5 ] [.01131 .61189 -18 -4.5 ] [.01131 .61189 0 4.5 ] [ 0 -0.00028 0 0 ] [ 1 .61832 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .18254 .2417 m .18254 .24795 L s [(1.5)] .18254 .2292 0 1 Mshowa .34127 .2417 m .34127 .24795 L s [(2)] .34127 .2292 0 1 Mshowa .5 .2417 m .5 .24795 L s [(2.5)] .5 .2292 0 1 Mshowa .65873 .2417 m .65873 .24795 L s [(3)] .65873 .2292 0 1 Mshowa .81746 .2417 m .81746 .24795 L s [(3.5)] .81746 .2292 0 1 Mshowa .97619 .2417 m .97619 .24795 L s [(4)] .97619 .2292 0 1 Mshowa .125 Mabswid .05556 .2417 m .05556 .24545 L s .0873 .2417 m .0873 .24545 L s .11905 .2417 m .11905 .24545 L s .15079 .2417 m .15079 .24545 L s .21429 .2417 m .21429 .24545 L s .24603 .2417 m .24603 .24545 L s .27778 .2417 m .27778 .24545 L s .30952 .2417 m .30952 .24545 L s .37302 .2417 m .37302 .24545 L s .40476 .2417 m .40476 .24545 L s .43651 .2417 m .43651 .24545 L s .46825 .2417 m .46825 .24545 L s .53175 .2417 m .53175 .24545 L s .56349 .2417 m .56349 .24545 L s .59524 .2417 m .59524 .24545 L s .62698 .2417 m .62698 .24545 L s .69048 .2417 m .69048 .24545 L s .72222 .2417 m .72222 .24545 L s .75397 .2417 m .75397 .24545 L s .78571 .2417 m .78571 .24545 L s .84921 .2417 m .84921 .24545 L s .88095 .2417 m .88095 .24545 L s .9127 .2417 m .9127 .24545 L s .94444 .2417 m .94444 .24545 L s .25 Mabswid 0 .2417 m 1 .2417 L s gsave 1.025 .2417 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .1183 m .03006 .1183 L s [(-0.2)] .01131 .1183 1 0 Mshowa .02381 .3651 m .03006 .3651 L s [(0.2)] .01131 .3651 1 0 Mshowa .02381 .4885 m .03006 .4885 L s [(0.4)] .01131 .4885 1 0 Mshowa .02381 .61189 m .03006 .61189 L s [(0.6)] .01131 .61189 1 0 Mshowa .125 Mabswid .02381 .02575 m .02756 .02575 L s .02381 .0566 m .02756 .0566 L s .02381 .08745 m .02756 .08745 L s .02381 .14915 m .02756 .14915 L s .02381 .18 m .02756 .18 L s .02381 .21085 m .02756 .21085 L s .02381 .27255 m .02756 .27255 L s .02381 .3034 m .02756 .3034 L s .02381 .33425 m .02756 .33425 L s .02381 .39595 m .02756 .39595 L s .02381 .4268 m .02756 .4268 L s .02381 .45765 m .02756 .45765 L s .02381 .51935 m .02756 .51935 L s .02381 .5502 m .02756 .5502 L s .02381 .58104 m .02756 .58104 L s .25 Mabswid .02381 0 m .02381 .61803 L s 1 0 0 r .01 w .02381 .01472 m .02499 .01472 L .02605 .01473 L .02729 .01476 L .02846 .01479 L .03053 .01487 L .03279 .01499 L .03527 .01515 L .0379 .01538 L .04262 .01589 L .05205 .01735 L .06244 .0196 L .07293 .02254 L .0842 .02642 L .10458 .03522 L .1458 .05934 L .18551 .08925 L .22371 .12276 L .26435 .16205 L .30348 .20208 L .34506 .24569 L .38513 .28763 L .42368 .32707 L .46468 .36726 L .50417 .40369 L .54214 .43621 L .58257 .4678 L .62148 .49506 L .66284 .52053 L .70268 .54162 L .74101 .55876 L .78179 .57368 L .82106 .58497 L .85881 .59311 L .87816 .5963 L .89901 .59904 L .91905 .60101 L .92913 .60177 L .94015 .60242 L .94928 .60282 L .95758 .60308 L .96217 .60319 L .96476 .60323 L .96715 .60326 L .96924 .60329 L .97044 .6033 L .97154 .6033 L .9727 .60331 L .97394 .60332 L .9751 .60332 L Mistroke .97619 .60332 L Mfstroke 0 g .02381 .2417 m .2619 .2417 L .2619 .15961 L .02381 .01472 L .02381 .2417 L s .2619 .2417 m .5 .2417 L .5 .39997 L .2619 .15961 L .2619 .2417 L s .5 .2417 m .7381 .2417 L .7381 .55756 L .5 .39997 L .5 .2417 L s .7381 .2417 m .97619 .2417 L .97619 .60332 L .7381 .55756 L .7381 .2417 L s 1 0 0 r .03 w .02381 .01472 Mdot .2619 .15961 Mdot .5 .39997 Mdot .7381 .55756 Mdot .97619 .60332 Mdot 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "The measurement\n ", Cell[BoxData[ \(\[Integral]\_a\%b f[x] \[DifferentialD]x\)]], "\nis roughly the sum of the (signed) measurements of areas within the \ consecutive trapezoids. Each trapezoid is based on on interval ", Cell[BoxData[ \(\([x, x + jump]\)\)]], " on the ", Cell[BoxData[ \(x\)]], "-axis. \nSo each trapezoid has ", Cell[BoxData[ \(base = jump\)]], ", and each trapezoid has two heights: f", Cell[BoxData[ \(\([x]\)\)]], " and ", Cell[BoxData[ \(f[x + jump]\)]], ".\nSo the area of each of the trapezoids is\n ", Cell[BoxData[ StyleBox[\(jump[f[x] + f[x + jump]]\/2\), FontSize->16]]], "\nThe sum of these areas is: " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Area[4]\ = \ \n\t N[Sum[1\/2\ jump\ \((f[x] + f[x + jump])\), {x, a, b\ - \ jump, jump}]]\)], "Input"], Cell[BoxData[ \(0.5583908697432504`\)], "Output"] }, Open ]], Cell["Note how the leftmost trapezoid carries a negative area:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(N[1\/2\ jump\ \((f[a] + f[a + jump])\)]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-0.18784691873463352`\)\)], "Output"] }, Open ]], Cell[TextData[{ "This is the way it should be because that trapezoid runs below the ", Cell[BoxData[ \(x\)]], "-axis:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[curveplot, pointplot, trapezoid[a], \n\t AxesLabel \[Rule] {"\", "\<\>"}, PlotRange \[Rule] All, AxesOrigin \[Rule] {a, 0}];\)\)], "Input", AspectRatioFixed->True], 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 -0.293651 0.31746 0.241696 0.616998 [ [.18254 .2292 -9 -9 ] [.18254 .2292 9 0 ] [.34127 .2292 -3 -9 ] [.34127 .2292 3 0 ] [.5 .2292 -9 -9 ] [.5 .2292 9 0 ] [.65873 .2292 -3 -9 ] [.65873 .2292 3 0 ] [.81746 .2292 -9 -9 ] [.81746 .2292 9 0 ] [.97619 .2292 -3 -9 ] [.97619 .2292 3 0 ] [1.025 .2417 0 -6 ] [1.025 .2417 10 6 ] [.01131 .1183 -24 -4.5 ] [.01131 .1183 0 4.5 ] [.01131 .3651 -18 -4.5 ] [.01131 .3651 0 4.5 ] [.01131 .4885 -18 -4.5 ] [.01131 .4885 0 4.5 ] [.01131 .61189 -18 -4.5 ] [.01131 .61189 0 4.5 ] [ 0 -0.00028 0 0 ] [ 1 .61832 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .18254 .2417 m .18254 .24795 L s [(1.5)] .18254 .2292 0 1 Mshowa .34127 .2417 m .34127 .24795 L s [(2)] .34127 .2292 0 1 Mshowa .5 .2417 m .5 .24795 L s [(2.5)] .5 .2292 0 1 Mshowa .65873 .2417 m .65873 .24795 L s [(3)] .65873 .2292 0 1 Mshowa .81746 .2417 m .81746 .24795 L s [(3.5)] .81746 .2292 0 1 Mshowa .97619 .2417 m .97619 .24795 L s [(4)] .97619 .2292 0 1 Mshowa .125 Mabswid .05556 .2417 m .05556 .24545 L s .0873 .2417 m .0873 .24545 L s .11905 .2417 m .11905 .24545 L s .15079 .2417 m .15079 .24545 L s .21429 .2417 m .21429 .24545 L s .24603 .2417 m .24603 .24545 L s .27778 .2417 m .27778 .24545 L s .30952 .2417 m .30952 .24545 L s .37302 .2417 m .37302 .24545 L s .40476 .2417 m .40476 .24545 L s .43651 .2417 m .43651 .24545 L s .46825 .2417 m .46825 .24545 L s .53175 .2417 m .53175 .24545 L s .56349 .2417 m .56349 .24545 L s .59524 .2417 m .59524 .24545 L s .62698 .2417 m .62698 .24545 L s .69048 .2417 m .69048 .24545 L s .72222 .2417 m .72222 .24545 L s .75397 .2417 m .75397 .24545 L s .78571 .2417 m .78571 .24545 L s .84921 .2417 m .84921 .24545 L s .88095 .2417 m .88095 .24545 L s .9127 .2417 m .9127 .24545 L s .94444 .2417 m .94444 .24545 L s .25 Mabswid 0 .2417 m 1 .2417 L s gsave 1.025 .2417 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .1183 m .03006 .1183 L s [(-0.2)] .01131 .1183 1 0 Mshowa .02381 .3651 m .03006 .3651 L s [(0.2)] .01131 .3651 1 0 Mshowa .02381 .4885 m .03006 .4885 L s [(0.4)] .01131 .4885 1 0 Mshowa .02381 .61189 m .03006 .61189 L s [(0.6)] .01131 .61189 1 0 Mshowa .125 Mabswid .02381 .02575 m .02756 .02575 L s .02381 .0566 m .02756 .0566 L s .02381 .08745 m .02756 .08745 L s .02381 .14915 m .02756 .14915 L s .02381 .18 m .02756 .18 L s .02381 .21085 m .02756 .21085 L s .02381 .27255 m .02756 .27255 L s .02381 .3034 m .02756 .3034 L s .02381 .33425 m .02756 .33425 L s .02381 .39595 m .02756 .39595 L s .02381 .4268 m .02756 .4268 L s .02381 .45765 m .02756 .45765 L s .02381 .51935 m .02756 .51935 L s .02381 .5502 m .02756 .5502 L s .02381 .58104 m .02756 .58104 L s .25 Mabswid .02381 0 m .02381 .61803 L s 1 0 0 r .01 w .02381 .01472 m .02499 .01472 L .02605 .01473 L .02729 .01476 L .02846 .01479 L .03053 .01487 L .03279 .01499 L .03527 .01515 L .0379 .01538 L .04262 .01589 L .05205 .01735 L .06244 .0196 L .07293 .02254 L .0842 .02642 L .10458 .03522 L .1458 .05934 L .18551 .08925 L .22371 .12276 L .26435 .16205 L .30348 .20208 L .34506 .24569 L .38513 .28763 L .42368 .32707 L .46468 .36726 L .50417 .40369 L .54214 .43621 L .58257 .4678 L .62148 .49506 L .66284 .52053 L .70268 .54162 L .74101 .55876 L .78179 .57368 L .82106 .58497 L .85881 .59311 L .87816 .5963 L .89901 .59904 L .91905 .60101 L .92913 .60177 L .94015 .60242 L .94928 .60282 L .95758 .60308 L .96217 .60319 L .96476 .60323 L .96715 .60326 L .96924 .60329 L .97044 .6033 L .97154 .6033 L .9727 .60331 L .97394 .60332 L .9751 .60332 L Mistroke .97619 .60332 L Mfstroke .03 w .02381 .01472 Mdot .2619 .15961 Mdot .5 .39997 Mdot .7381 .55756 Mdot .97619 .60332 Mdot 0 g .01 w .02381 .2417 m .2619 .2417 L .2619 .15961 L .02381 .01472 L .02381 .2417 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell["To get more trustworthy estimates, decrease the jump:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(jump\ = \ \(b - a\)\/10;\)\), "\n", \(Area[10]\ = \ \n\t N[Sum[1\/2\ jump\ \((f[x] + f[x + jump])\), \n\t\ \ \ \ \ {x, a, b\ - \ jump, jump}]]\)}], "Input"], Cell[BoxData[ \(0.5588584879119315`\)], "Output"] }, Open ]], Cell["Again and again:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(jump\ = \ \(b - a\)\/100;\)\), "\n", \(Area[100]\ = \ \n\t N[Sum[1\/2\ jump\ \((f[x] + f[x + jump])\), \n\t\ \ \ \ \ \ \ {x, a, b\ - \ jump, jump}]]\)}], "Input"], Cell[BoxData[ \(0.5588691460084739`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(jump\ = \ \(b - a\)\/200;\)\), "\n", \(Area[200]\ = \ \n\t N[Sum[1\/2\ jump\ \((f[x] + f[x + jump])\), \n\t\ \ \ \ \ \ \ {x, a, b\ - \ jump, jump}]]\)}], "Input"], Cell[BoxData[ \(0.5588691469779612`\)], "Output"] }, Open ]], Cell["Compare the estimates:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \({Area[4], Area[10], Area[100], Area[200]}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({0.5583908697432504`, 0.5588584879119315`, 0.5588691460084739`, 0.5588691469779612`}\)], "Output"] }, Open ]], Cell[TextData[{ "This is pretty convincing. \nYou can bet a pitcher of golden liquid \ refreshment on the estimate that\n ", Cell[BoxData[ \(\[Integral]\_a\%b f[x] \[DifferentialD]x = 0.558869\)]], "\nto ", Cell[BoxData[ \(6\)]], " accurate decimals.\nCheck it out: " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(MathematicaEstimate = NIntegrate[f[x], {x, a, b}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.5588691470425777`\)], "Output"] }, Open ]], Cell["\<\ That was a safe bet. Get out of here and collect your winnings.\ \>", "SmallText"] }, Closed]] }, Closed]], Cell["Tutorial Problem", "Subsubsection"], Cell[CellGroupData[{ Cell["T.1) Calculation of integrals for area measurements", "Subsection", CellTags->"2.01.T1"], Cell[" Make the indicated area measurements:", "Text"], Cell["T.1.a)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\[Integral]\_\(-1\)\%5\((5.89 - 1.9 x)\) \[DifferentialD]x\)]] }], "Text"], Cell["Answer:", "Special1"], Cell["To see what's happening, plot:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(\(f[x_] = 5.89 - 1.9\ x;\)\n\), "\n", \(\(Plot[f[x], {x, \(-1\), 5}, \n\tAspectRatio \[Rule] 1\/GoldenRatio, PlotStyle \[Rule] {{Thickness[0.01], Red}}, AxesLabel \[Rule] {"\", "\"}];\)\)}], "Input", AspectRatioFixed->True], 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 0.18254 0.15873 0.201106 0.0516319 [ [.02381 .18861 -6 -9 ] [.02381 .18861 6 0 ] [.34127 .18861 -3 -9 ] [.34127 .18861 3 0 ] [.5 .18861 -3 -9 ] [.5 .18861 3 0 ] [.65873 .18861 -3 -9 ] [.65873 .18861 3 0 ] [.81746 .18861 -3 -9 ] [.81746 .18861 3 0 ] [.97619 .18861 -3 -9 ] [.97619 .18861 3 0 ] [1.025 .20111 0 -6 ] [1.025 .20111 10 6 ] [.17004 .09784 -12 -4.5 ] [.17004 .09784 0 4.5 ] [.17004 .30437 -6 -4.5 ] [.17004 .30437 0 4.5 ] [.17004 .40763 -6 -4.5 ] [.17004 .40763 0 4.5 ] [.17004 .5109 -6 -4.5 ] [.17004 .5109 0 4.5 ] [.17004 .61416 -6 -4.5 ] [.17004 .61416 0 4.5 ] [.18254 .64303 -14 0 ] [.18254 .64303 14 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .20111 m .02381 .20736 L s [(-1)] .02381 .18861 0 1 Mshowa .34127 .20111 m .34127 .20736 L s [(1)] .34127 .18861 0 1 Mshowa .5 .20111 m .5 .20736 L s [(2)] .5 .18861 0 1 Mshowa .65873 .20111 m .65873 .20736 L s [(3)] .65873 .18861 0 1 Mshowa .81746 .20111 m .81746 .20736 L s [(4)] .81746 .18861 0 1 Mshowa .97619 .20111 m .97619 .20736 L s [(5)] .97619 .18861 0 1 Mshowa .125 Mabswid .05556 .20111 m .05556 .20486 L s .0873 .20111 m .0873 .20486 L s .11905 .20111 m .11905 .20486 L s .15079 .20111 m .15079 .20486 L s .21429 .20111 m .21429 .20486 L s .24603 .20111 m .24603 .20486 L s .27778 .20111 m .27778 .20486 L s .30952 .20111 m .30952 .20486 L s .37302 .20111 m .37302 .20486 L s .40476 .20111 m .40476 .20486 L s .43651 .20111 m .43651 .20486 L s .46825 .20111 m .46825 .20486 L s .53175 .20111 m .53175 .20486 L s .56349 .20111 m .56349 .20486 L s .59524 .20111 m .59524 .20486 L s .62698 .20111 m .62698 .20486 L s .69048 .20111 m .69048 .20486 L s .72222 .20111 m .72222 .20486 L s .75397 .20111 m .75397 .20486 L s .78571 .20111 m .78571 .20486 L s .84921 .20111 m .84921 .20486 L s .88095 .20111 m .88095 .20486 L s .9127 .20111 m .9127 .20486 L s .94444 .20111 m .94444 .20486 L s .25 Mabswid 0 .20111 m 1 .20111 L s gsave 1.025 .20111 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .18254 .09784 m .18879 .09784 L s [(-2)] .17004 .09784 1 0 Mshowa .18254 .30437 m .18879 .30437 L s [(2)] .17004 .30437 1 0 Mshowa .18254 .40763 m .18879 .40763 L s [(4)] .17004 .40763 1 0 Mshowa .18254 .5109 m .18879 .5109 L s [(6)] .17004 .5109 1 0 Mshowa .18254 .61416 m .18879 .61416 L s [(8)] .17004 .61416 1 0 Mshowa .125 Mabswid .18254 .12366 m .18629 .12366 L s .18254 .14947 m .18629 .14947 L s .18254 .17529 m .18629 .17529 L s .18254 .22692 m .18629 .22692 L s .18254 .25274 m .18629 .25274 L s .18254 .27855 m .18629 .27855 L s .18254 .33019 m .18629 .33019 L s .18254 .356 m .18629 .356 L s .18254 .38182 m .18629 .38182 L s .18254 .43345 m .18629 .43345 L s .18254 .45927 m .18629 .45927 L s .18254 .48508 m .18629 .48508 L s .18254 .53671 m .18629 .53671 L s .18254 .56253 m .18629 .56253 L s .18254 .58835 m .18629 .58835 L s .18254 .07203 m .18629 .07203 L s .18254 .04621 m .18629 .04621 L s .18254 .02039 m .18629 .02039 L s .25 Mabswid .18254 0 m .18254 .61803 L s gsave .18254 .64303 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 69.000 13.000 moveto (@) show 75.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 81.000 13.000 moveto (D) show 87.000 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 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 1 0 0 r .01 w .02381 .60332 m .06244 .57944 L .10458 .5534 L .14415 .52894 L .18221 .50542 L .22272 .48039 L .26171 .45629 L .30316 .43067 L .34309 .406 L .3815 .38225 L .42237 .357 L .46172 .33268 L .49955 .30929 L .53984 .2844 L .57861 .26043 L .61984 .23495 L .65954 .21041 L .69774 .18681 L .73838 .16169 L .77751 .13751 L .81909 .11181 L .85916 .08704 L .89771 .06322 L .93871 .03788 L .97619 .01472 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "Here is a picture of the signed area measured by \n ", Cell[BoxData[ \(\[Integral]\_\(-1\)\%5 f[x] \[DifferentialD]x\)]], ":" }], "SmallText"], Cell[TextData[{ "This instruction makes use of the ", StyleBox["Mathematica", FontSlant->"Italic"], " package \nGraphics`FilledPlot found in the initialization cells." }], "Special2"], Cell[CellGroupData[{ Cell[BoxData[ \(\(FilledPlot[f[x], {x, \(-1\), 5}, AxesLabel \[Rule] {"\", "\"}];\)\)], "Input", AspectRatioFixed->True], 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 1 0 0.618034 [ [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.18254 0.15873 0.201106 0.0516319 [ [.02381 .18861 -6 -9 ] [.02381 .18861 6 0 ] [.34127 .18861 -3 -9 ] [.34127 .18861 3 0 ] [.5 .18861 -3 -9 ] [.5 .18861 3 0 ] [.65873 .18861 -3 -9 ] [.65873 .18861 3 0 ] [.81746 .18861 -3 -9 ] [.81746 .18861 3 0 ] [.97619 .18861 -3 -9 ] [.97619 .18861 3 0 ] [1.025 .20111 0 -6 ] [1.025 .20111 10 6 ] [.17004 .09784 -12 -4.5 ] [.17004 .09784 0 4.5 ] [.17004 .30437 -6 -4.5 ] [.17004 .30437 0 4.5 ] [.17004 .40763 -6 -4.5 ] [.17004 .40763 0 4.5 ] [.17004 .5109 -6 -4.5 ] [.17004 .5109 0 4.5 ] [.17004 .61416 -6 -4.5 ] [.17004 .61416 0 4.5 ] [.18254 .64303 -14 0 ] [.18254 .64303 14 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .20111 m .02381 .20736 L s [(-1)] .02381 .18861 0 1 Mshowa .34127 .20111 m .34127 .20736 L s [(1)] .34127 .18861 0 1 Mshowa .5 .20111 m .5 .20736 L s [(2)] .5 .18861 0 1 Mshowa .65873 .20111 m .65873 .20736 L s [(3)] .65873 .18861 0 1 Mshowa .81746 .20111 m .81746 .20736 L s [(4)] .81746 .18861 0 1 Mshowa .97619 .20111 m .97619 .20736 L s [(5)] .97619 .18861 0 1 Mshowa .125 Mabswid .05556 .20111 m .05556 .20486 L s .0873 .20111 m .0873 .20486 L s .11905 .20111 m .11905 .20486 L s .15079 .20111 m .15079 .20486 L s .21429 .20111 m .21429 .20486 L s .24603 .20111 m .24603 .20486 L s .27778 .20111 m .27778 .20486 L s .30952 .20111 m .30952 .20486 L s .37302 .20111 m .37302 .20486 L s .40476 .20111 m .40476 .20486 L s .43651 .20111 m .43651 .20486 L s .46825 .20111 m .46825 .20486 L s .53175 .20111 m .53175 .20486 L s .56349 .20111 m .56349 .20486 L s .59524 .20111 m .59524 .20486 L s .62698 .20111 m .62698 .20486 L s .69048 .20111 m .69048 .20486 L s .72222 .20111 m .72222 .20486 L s .75397 .20111 m .75397 .20486 L s .78571 .20111 m .78571 .20486 L s .84921 .20111 m .84921 .20486 L s .88095 .20111 m .88095 .20486 L s .9127 .20111 m .9127 .20486 L s .94444 .20111 m .94444 .20486 L s .25 Mabswid 0 .20111 m 1 .20111 L s gsave 1.025 .20111 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .18254 .09784 m .18879 .09784 L s [(-2)] .17004 .09784 1 0 Mshowa .18254 .30437 m .18879 .30437 L s [(2)] .17004 .30437 1 0 Mshowa .18254 .40763 m .18879 .40763 L s [(4)] .17004 .40763 1 0 Mshowa .18254 .5109 m .18879 .5109 L s [(6)] .17004 .5109 1 0 Mshowa .18254 .61416 m .18879 .61416 L s [(8)] .17004 .61416 1 0 Mshowa .125 Mabswid .18254 .12366 m .18629 .12366 L s .18254 .14947 m .18629 .14947 L s .18254 .17529 m .18629 .17529 L s .18254 .22692 m .18629 .22692 L s .18254 .25274 m .18629 .25274 L s .18254 .27855 m .18629 .27855 L s .18254 .33019 m .18629 .33019 L s .18254 .356 m .18629 .356 L s .18254 .38182 m .18629 .38182 L s .18254 .43345 m .18629 .43345 L s .18254 .45927 m .18629 .45927 L s .18254 .48508 m .18629 .48508 L s .18254 .53671 m .18629 .53671 L s .18254 .56253 m .18629 .56253 L s .18254 .58835 m .18629 .58835 L s .18254 .07203 m .18629 .07203 L s .18254 .04621 m .18629 .04621 L s .18254 .02039 m .18629 .02039 L s .25 Mabswid .18254 0 m .18254 .61803 L s gsave .18254 .64303 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 69.000 13.000 moveto (@) show 75.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 81.000 13.000 moveto (D) show 87.000 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 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .60332 m .06244 .57944 L .10458 .5534 L .14415 .52894 L .18221 .50542 L .22272 .48039 L .26171 .45629 L .30316 .43067 L .34309 .406 L .3815 .38225 L .42237 .357 L .46172 .33268 L .49955 .30929 L .53984 .2844 L .57861 .26043 L .61984 .23495 L .65954 .21041 L .69774 .18681 L .73838 .16169 L .77751 .13751 L .81909 .11181 L .85916 .08704 L .89771 .06322 L .93871 .03788 L .97619 .01472 L s .5 g .02381 .60332 m .06244 .57944 L .10458 .5534 L .14415 .52894 L .18221 .50542 L .22272 .48039 L .26171 .45629 L .30316 .43067 L .34309 .406 L .3815 .38225 L .42237 .357 L .46172 .33268 L .49955 .30929 L .53984 .2844 L .57861 .26043 L .61984 .23495 L .65954 .21041 L .69774 .18681 L .73838 .16169 L .77751 .13751 L .81909 .11181 L .85916 .08704 L .89771 .06322 L .93871 .03788 L .97619 .01472 L .97619 .20111 L .02381 .20111 L F 0 g .02381 .60332 m .06244 .57944 L .10458 .5534 L .14415 .52894 L .18221 .50542 L .22272 .48039 L .26171 .45629 L .30316 .43067 L .34309 .406 L .3815 .38225 L .42237 .357 L .46172 .33268 L .49955 .30929 L .53984 .2844 L .57861 .26043 L .61984 .23495 L .65954 .21041 L .69774 .18681 L .73838 .16169 L .77751 .13751 L .81909 .11181 L .85916 .08704 L .89771 .06322 L .93871 .03788 L .97619 .01472 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.18254 0.15873 0.201106 0.0516319 [ [.02381 .18861 -6 -9 ] [.02381 .18861 6 0 ] [.34127 .18861 -3 -9 ] [.34127 .18861 3 0 ] [.5 .18861 -3 -9 ] [.5 .18861 3 0 ] [.65873 .18861 -3 -9 ] [.65873 .18861 3 0 ] [.81746 .18861 -3 -9 ] [.81746 .18861 3 0 ] [.97619 .18861 -3 -9 ] [.97619 .18861 3 0 ] [1.025 .20111 0 -6 ] [1.025 .20111 10 6 ] [.17004 .09784 -12 -4.5 ] [.17004 .09784 0 4.5 ] [.17004 .30437 -6 -4.5 ] [.17004 .30437 0 4.5 ] [.17004 .40763 -6 -4.5 ] [.17004 .40763 0 4.5 ] [.17004 .5109 -6 -4.5 ] [.17004 .5109 0 4.5 ] [.17004 .61416 -6 -4.5 ] [.17004 .61416 0 4.5 ] [.18254 .64303 -14 0 ] [.18254 .64303 14 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .20111 m .02381 .20736 L s [(-1)] .02381 .18861 0 1 Mshowa .34127 .20111 m .34127 .20736 L s [(1)] .34127 .18861 0 1 Mshowa .5 .20111 m .5 .20736 L s [(2)] .5 .18861 0 1 Mshowa .65873 .20111 m .65873 .20736 L s [(3)] .65873 .18861 0 1 Mshowa .81746 .20111 m .81746 .20736 L s [(4)] .81746 .18861 0 1 Mshowa .97619 .20111 m .97619 .20736 L s [(5)] .97619 .18861 0 1 Mshowa .125 Mabswid .05556 .20111 m .05556 .20486 L s .0873 .20111 m .0873 .20486 L s .11905 .20111 m .11905 .20486 L s .15079 .20111 m .15079 .20486 L s .21429 .20111 m .21429 .20486 L s .24603 .20111 m .24603 .20486 L s .27778 .20111 m .27778 .20486 L s .30952 .20111 m .30952 .20486 L s .37302 .20111 m .37302 .20486 L s .40476 .20111 m .40476 .20486 L s .43651 .20111 m .43651 .20486 L s .46825 .20111 m .46825 .20486 L s .53175 .20111 m .53175 .20486 L s .56349 .20111 m .56349 .20486 L s .59524 .20111 m .59524 .20486 L s .62698 .20111 m .62698 .20486 L s .69048 .20111 m .69048 .20486 L s .72222 .20111 m .72222 .20486 L s .75397 .20111 m .75397 .20486 L s .78571 .20111 m .78571 .20486 L s .84921 .20111 m .84921 .20486 L s .88095 .20111 m .88095 .20486 L s .9127 .20111 m .9127 .20486 L s .94444 .20111 m .94444 .20486 L s .25 Mabswid 0 .20111 m 1 .20111 L s gsave 1.025 .20111 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .18254 .09784 m .18879 .09784 L s [(-2)] .17004 .09784 1 0 Mshowa .18254 .30437 m .18879 .30437 L s [(2)] .17004 .30437 1 0 Mshowa .18254 .40763 m .18879 .40763 L s [(4)] .17004 .40763 1 0 Mshowa .18254 .5109 m .18879 .5109 L s [(6)] .17004 .5109 1 0 Mshowa .18254 .61416 m .18879 .61416 L s [(8)] .17004 .61416 1 0 Mshowa .125 Mabswid .18254 .12366 m .18629 .12366 L s .18254 .14947 m .18629 .14947 L s .18254 .17529 m .18629 .17529 L s .18254 .22692 m .18629 .22692 L s .18254 .25274 m .18629 .25274 L s .18254 .27855 m .18629 .27855 L s .18254 .33019 m .18629 .33019 L s .18254 .356 m .18629 .356 L s .18254 .38182 m .18629 .38182 L s .18254 .43345 m .18629 .43345 L s .18254 .45927 m .18629 .45927 L s .18254 .48508 m .18629 .48508 L s .18254 .53671 m .18629 .53671 L s .18254 .56253 m .18629 .56253 L s .18254 .58835 m .18629 .58835 L s .18254 .07203 m .18629 .07203 L s .18254 .04621 m .18629 .04621 L s .18254 .02039 m .18629 .02039 L s .25 Mabswid .18254 0 m .18254 .61803 L s gsave .18254 .64303 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 69.000 13.000 moveto (@) show 75.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 81.000 13.000 moveto (D) show 87.000 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 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath MathSubEnd P % End of sub-graphic % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell["\<\ Big positive area on the left and a small negative area on the \ right. Better find out where the area goes from positive to negative:\ \>", \ "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[f[x] == 0]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({{x \[Rule] 3.0999999999999996`}}\)], "Output"] }, Open ]], Cell[TextData[{ "The measurement \n ", Cell[BoxData[ \(\[Integral]\_\(-1\)\%3.1 f[x] \[DifferentialD]x\)]], " \nof the triangular area on the left is positive because it is above the \ ", Cell[BoxData[ \(x\)]], "-axis:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(height = f[\(-1\)];\)\), "\n", \(\(base = 3.1 - \(-1\);\)\), "\n", \(triangleleft = \(base\ height\)\/2\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(15.969499999999996`\)], "Output"] }, Open ]], Cell[TextData[{ "The measure ", Cell[BoxData[ \(\[Integral]\_3.1\%5 f[x] \[DifferentialD]x\)]], " of the triangular area on the right is negative because it is below the \ ", Cell[BoxData[ \(x\)]], "-axis:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(height = f[5];\)\), "\n", \(\(base = 5 - 3.1;\)\), "\n", \(triangleright = \(base\ height\)\/2\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-3.4295`\)\)], "Output"] }, Open ]], Cell[TextData[{ "Consequently, the area measurement\n ", Cell[BoxData[ \(\[Integral]\_\(-1\)\%5 f[ x] \[DifferentialD]x = \[Integral]\_\(-1\)\%3.1 f[ x] \[DifferentialD]x + \[Integral]\_3.1\%5 f[ x] \[DifferentialD]x\)]], "\nis given by:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(triangleleft + triangleright\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(12.539999999999996`\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can make the area measurement \n ", Cell[BoxData[ \(\[Integral]\_\(-1\)\%5 f[x] \[DifferentialD]x\)]], "\ndirectly:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(NIntegrate[f[x], {x, \(-1\), 5}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(12.54`\)], "Output"] }, Open ]], Cell["Piece of cake.", "SmallText"] }, Closed]], Cell["T.1.b)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\[Integral]\_\(-3\)\%3\(\@\( 9 - x\^2\)\) \[DifferentialD]x\)]] }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "This integral measures the area of the top half of the circle \n \ ", Cell[BoxData[ \(x\^2 + y\^2 = 9\)]], ".\nIf you don't believe this, then just solve ", Cell[BoxData[ \(x\^2 + y\^2 = 9\)]], " for ", Cell[BoxData[ \(y\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, y];\)\), "\n", \(ysolved = Solve[x\^2 + y\^2 == 9, y]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \({{y \[Rule] \(-\@\(9 - x\^2\)\)}, {y \[Rule] \@\(9 - x\^2\)}}\)], \ "Output"] }, Open ]], Cell[TextData[{ "The function you are integrating is ", Cell[BoxData[ \(\@\(9 - x\^2\)\)]], ". \nHere's a true scale plot:" }], "SmallText"], Cell[TextData[{ "When you are plotting circles, you should always plot them in true scale \ so they don't look like ellipses.\nThe option ", Cell[BoxData[ \(AspectRatio \[Rule] Automatic\)]], " guarantees a true scale plot." }], "Special2"], Cell[CellGroupData[{ Cell[BoxData[ \(\(FilledPlot[\@\(9 - x\^2\), {x, \(-3\), 3}, AspectRatio \[Rule] Automatic];\)\)], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .5 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 1 0 0.5 [ [ 0 0 0 0 ] [ 1 .5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .5 L 0 .5 L closepath clip newpath % Start of sub-graphic p 0 0 1 0.5 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.15873 0.0119048 0.15873 [ [.02381 -0.0006 -6 -9 ] [.02381 -0.0006 6 0 ] [.18254 -0.0006 -6 -9 ] [.18254 -0.0006 6 0 ] [.34127 -0.0006 -6 -9 ] [.34127 -0.0006 6 0 ] [.65873 -0.0006 -3 -9 ] [.65873 -0.0006 3 0 ] [.81746 -0.0006 -3 -9 ] [.81746 -0.0006 3 0 ] [.97619 -0.0006 -3 -9 ] [.97619 -0.0006 3 0 ] [.4875 .09127 -18 -4.5 ] [.4875 .09127 0 4.5 ] [.4875 .17063 -6 -4.5 ] [.4875 .17063 0 4.5 ] [.4875 .25 -18 -4.5 ] [.4875 .25 0 4.5 ] [.4875 .32937 -6 -4.5 ] [.4875 .32937 0 4.5 ] [.4875 .40873 -18 -4.5 ] [.4875 .40873 0 4.5 ] [.4875 .4881 -6 -4.5 ] [.4875 .4881 0 4.5 ] [ 0 0 0 0 ] [ 1 .5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .0119 m .02381 .01815 L s [(-3)] .02381 -0.0006 0 1 Mshowa .18254 .0119 m .18254 .01815 L s [(-2)] .18254 -0.0006 0 1 Mshowa .34127 .0119 m .34127 .01815 L s [(-1)] .34127 -0.0006 0 1 Mshowa .65873 .0119 m .65873 .01815 L s [(1)] .65873 -0.0006 0 1 Mshowa .81746 .0119 m .81746 .01815 L s [(2)] .81746 -0.0006 0 1 Mshowa .97619 .0119 m .97619 .01815 L s [(3)] .97619 -0.0006 0 1 Mshowa .125 Mabswid .05556 .0119 m .05556 .01565 L s .0873 .0119 m .0873 .01565 L s .11905 .0119 m .11905 .01565 L s .15079 .0119 m .15079 .01565 L s .21429 .0119 m .21429 .01565 L s .24603 .0119 m .24603 .01565 L s .27778 .0119 m .27778 .01565 L s .30952 .0119 m .30952 .01565 L s .37302 .0119 m .37302 .01565 L s .40476 .0119 m .40476 .01565 L s .43651 .0119 m .43651 .01565 L s .46825 .0119 m .46825 .01565 L s .53175 .0119 m .53175 .01565 L s .56349 .0119 m .56349 .01565 L s .59524 .0119 m .59524 .01565 L s .62698 .0119 m .62698 .01565 L s .69048 .0119 m .69048 .01565 L s .72222 .0119 m .72222 .01565 L s .75397 .0119 m .75397 .01565 L s .78571 .0119 m .78571 .01565 L s .84921 .0119 m .84921 .01565 L s .88095 .0119 m .88095 .01565 L s .9127 .0119 m .9127 .01565 L s .94444 .0119 m .94444 .01565 L s .25 Mabswid 0 .0119 m 1 .0119 L s .5 .09127 m .50625 .09127 L s [(0.5)] .4875 .09127 1 0 Mshowa .5 .17063 m .50625 .17063 L s [(1)] .4875 .17063 1 0 Mshowa .5 .25 m .50625 .25 L s [(1.5)] .4875 .25 1 0 Mshowa .5 .32937 m .50625 .32937 L s [(2)] .4875 .32937 1 0 Mshowa .5 .40873 m .50625 .40873 L s [(2.5)] .4875 .40873 1 0 Mshowa .5 .4881 m .50625 .4881 L s [(3)] .4875 .4881 1 0 Mshowa .125 Mabswid .5 .02778 m .50375 .02778 L s .5 .04365 m .50375 .04365 L s .5 .05952 m .50375 .05952 L s .5 .0754 m .50375 .0754 L s .5 .10714 m .50375 .10714 L s .5 .12302 m .50375 .12302 L s .5 .13889 m .50375 .13889 L s .5 .15476 m .50375 .15476 L s .5 .18651 m .50375 .18651 L s .5 .20238 m .50375 .20238 L s .5 .21825 m .50375 .21825 L s .5 .23413 m .50375 .23413 L s .5 .26587 m .50375 .26587 L s .5 .28175 m .50375 .28175 L s .5 .29762 m .50375 .29762 L s .5 .31349 m .50375 .31349 L s .5 .34524 m .50375 .34524 L s .5 .36111 m .50375 .36111 L s .5 .37698 m .50375 .37698 L s .5 .39286 m .50375 .39286 L s .5 .4246 m .50375 .4246 L s .5 .44048 m .50375 .44048 L s .5 .45635 m .50375 .45635 L s .5 .47222 m .50375 .47222 L s .25 Mabswid .5 0 m .5 .5 L s 0 0 m 1 0 L 1 .5 L 0 .5 L closepath clip newpath .5 Mabswid .02381 .0121 m .02499 .04535 L .02605 .05809 L .02846 .07827 L .03279 .10393 L .03784 .12664 L .04262 .14441 L .06244 .1998 L .08265 .2412 L .10458 .27724 L .14545 .32979 L .1848 .36884 L .22263 .39898 L .26292 .42488 L .30169 .44484 L .34292 .46144 L .36351 .46811 L .38262 .4734 L .40229 .47796 L .42082 .48147 L .44134 .48447 L .45995 .48641 L .46971 .48713 L .47505 .48744 L .48003 .48768 L .48428 .48784 L .48883 .48796 L .49143 .48802 L .49271 .48804 L .49389 .48806 L .49494 .48807 L .4961 .48808 L .49726 .48809 L .4985 .48809 L .49971 .4881 L .50085 .48809 L .50209 .48809 L .50278 .48809 L .50341 .48808 L .50469 .48807 L .50608 .48806 L .50859 .48802 L .51093 .48797 L .51341 .48791 L .51785 .48776 L .52298 .48754 L .52778 .48728 L .5387 .48652 L .54911 .48556 L .55899 .48443 L Mistroke .57753 .48174 L .59737 .47803 L .61577 .47381 L .65495 .46218 L .69659 .44562 L .73671 .4251 L .77531 .40044 L .81636 .36781 L .8559 .32828 L .89393 .27945 L .91511 .24523 L .9344 .20697 L .944 .18401 L .95442 .15423 L .9592 .13797 L .96427 .11779 L .9686 .09662 L .97089 .08274 L .97336 .06371 L .97408 .05665 L .97484 .04775 L .97549 .03781 L .97619 .0121 L Mfstroke .5 g .02381 .0121 m .02499 .04535 L .02605 .05809 L .02846 .07827 L .03279 .10393 L .03784 .12664 L .04262 .14441 L .06244 .1998 L .08265 .2412 L .10458 .27724 L .14545 .32979 L .1848 .36884 L .22263 .39898 L .26292 .42488 L .30169 .44484 L .34292 .46144 L .36351 .46811 L .38262 .4734 L .40229 .47796 L .42082 .48147 L .44134 .48447 L .45995 .48641 L .46971 .48713 L .47505 .48744 L .48003 .48768 L .48428 .48784 L .48883 .48796 L .49143 .48802 L .49271 .48804 L .49389 .48806 L .49494 .48807 L .4961 .48808 L .49726 .48809 L .4985 .48809 L .49971 .4881 L .50085 .48809 L .50209 .48809 L .50278 .48809 L .50341 .48808 L .50469 .48807 L .50608 .48806 L .50859 .48802 L .51093 .48797 L .51341 .48791 L .51785 .48776 L .52298 .48754 L .52778 .48728 L .5387 .48652 L .54911 .48556 L .55899 .48443 L .57753 .48174 L .59737 .47803 L .61577 .47381 L .65495 .46218 L .69659 .44562 L .73671 .4251 L .77531 .40044 L .81636 .36781 L .8559 .32828 L .89393 .27945 L .91511 .24523 L .9344 .20697 L .944 .18401 L .95442 .15423 L .9592 .13797 L .96427 .11779 L .9686 .09662 L .97089 .08274 L .97336 .06371 L .97408 .05665 L .97484 .04775 L .97549 .03781 L .97619 .0121 L .97619 .0119 L .02381 .0119 L F 0 g .02381 .0121 m .02499 .04535 L .02605 .05809 L .02846 .07827 L .03279 .10393 L .03784 .12664 L .04262 .14441 L .06244 .1998 L .08265 .2412 L .10458 .27724 L .14545 .32979 L .1848 .36884 L .22263 .39898 L .26292 .42488 L .30169 .44484 L .34292 .46144 L .36351 .46811 L .38262 .4734 L .40229 .47796 L .42082 .48147 L .44134 .48447 L .45995 .48641 L .46971 .48713 L .47505 .48744 L .48003 .48768 L .48428 .48784 L .48883 .48796 L .49143 .48802 L .49271 .48804 L .49389 .48806 L .49494 .48807 L .4961 .48808 L .49726 .48809 L .4985 .48809 L .49971 .4881 L .50085 .48809 L .50209 .48809 L .50278 .48809 L .50341 .48808 L .50469 .48807 L .50608 .48806 L .50859 .48802 L .51093 .48797 L .51341 .48791 L .51785 .48776 L .52298 .48754 L .52778 .48728 L .5387 .48652 L .54911 .48556 L .55899 .48443 L Mistroke .57753 .48174 L .59737 .47803 L .61577 .47381 L .65495 .46218 L .69659 .44562 L .73671 .4251 L .77531 .40044 L .81636 .36781 L .8559 .32828 L .89393 .27945 L .91511 .24523 L .9344 .20697 L .944 .18401 L .95442 .15423 L .9592 .13797 L .96427 .11779 L .9686 .09662 L .97089 .08274 L .97336 .06371 L .97408 .05665 L .97484 .04775 L .97549 .03781 L .97619 .0121 L Mfstroke MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0 0 1 0.5 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.15873 0.0119048 0.15873 [ [.02381 -0.0006 -6 -9 ] [.02381 -0.0006 6 0 ] [.18254 -0.0006 -6 -9 ] [.18254 -0.0006 6 0 ] [.34127 -0.0006 -6 -9 ] [.34127 -0.0006 6 0 ] [.65873 -0.0006 -3 -9 ] [.65873 -0.0006 3 0 ] [.81746 -0.0006 -3 -9 ] [.81746 -0.0006 3 0 ] [.97619 -0.0006 -3 -9 ] [.97619 -0.0006 3 0 ] [.4875 .09127 -18 -4.5 ] [.4875 .09127 0 4.5 ] [.4875 .17063 -6 -4.5 ] [.4875 .17063 0 4.5 ] [.4875 .25 -18 -4.5 ] [.4875 .25 0 4.5 ] [.4875 .32937 -6 -4.5 ] [.4875 .32937 0 4.5 ] [.4875 .40873 -18 -4.5 ] [.4875 .40873 0 4.5 ] [.4875 .4881 -6 -4.5 ] [.4875 .4881 0 4.5 ] [ 0 0 0 0 ] [ 1 .5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .0119 m .02381 .01815 L s [(-3)] .02381 -0.0006 0 1 Mshowa .18254 .0119 m .18254 .01815 L s [(-2)] .18254 -0.0006 0 1 Mshowa .34127 .0119 m .34127 .01815 L s [(-1)] .34127 -0.0006 0 1 Mshowa .65873 .0119 m .65873 .01815 L s [(1)] .65873 -0.0006 0 1 Mshowa .81746 .0119 m .81746 .01815 L s [(2)] .81746 -0.0006 0 1 Mshowa .97619 .0119 m .97619 .01815 L s [(3)] .97619 -0.0006 0 1 Mshowa .125 Mabswid .05556 .0119 m .05556 .01565 L s .0873 .0119 m .0873 .01565 L s .11905 .0119 m .11905 .01565 L s .15079 .0119 m .15079 .01565 L s .21429 .0119 m .21429 .01565 L s .24603 .0119 m .24603 .01565 L s .27778 .0119 m .27778 .01565 L s .30952 .0119 m .30952 .01565 L s .37302 .0119 m .37302 .01565 L s .40476 .0119 m .40476 .01565 L s .43651 .0119 m .43651 .01565 L s .46825 .0119 m .46825 .01565 L s .53175 .0119 m .53175 .01565 L s .56349 .0119 m .56349 .01565 L s .59524 .0119 m .59524 .01565 L s .62698 .0119 m .62698 .01565 L s .69048 .0119 m .69048 .01565 L s .72222 .0119 m .72222 .01565 L s .75397 .0119 m .75397 .01565 L s .78571 .0119 m .78571 .01565 L s .84921 .0119 m .84921 .01565 L s .88095 .0119 m .88095 .01565 L s .9127 .0119 m .9127 .01565 L s .94444 .0119 m .94444 .01565 L s .25 Mabswid 0 .0119 m 1 .0119 L s .5 .09127 m .50625 .09127 L s [(0.5)] .4875 .09127 1 0 Mshowa .5 .17063 m .50625 .17063 L s [(1)] .4875 .17063 1 0 Mshowa .5 .25 m .50625 .25 L s [(1.5)] .4875 .25 1 0 Mshowa .5 .32937 m .50625 .32937 L s [(2)] .4875 .32937 1 0 Mshowa .5 .40873 m .50625 .40873 L s [(2.5)] .4875 .40873 1 0 Mshowa .5 .4881 m .50625 .4881 L s [(3)] .4875 .4881 1 0 Mshowa .125 Mabswid .5 .02778 m .50375 .02778 L s .5 .04365 m .50375 .04365 L s .5 .05952 m .50375 .05952 L s .5 .0754 m .50375 .0754 L s .5 .10714 m .50375 .10714 L s .5 .12302 m .50375 .12302 L s .5 .13889 m .50375 .13889 L s .5 .15476 m .50375 .15476 L s .5 .18651 m .50375 .18651 L s .5 .20238 m .50375 .20238 L s .5 .21825 m .50375 .21825 L s .5 .23413 m .50375 .23413 L s .5 .26587 m .50375 .26587 L s .5 .28175 m .50375 .28175 L s .5 .29762 m .50375 .29762 L s .5 .31349 m .50375 .31349 L s .5 .34524 m .50375 .34524 L s .5 .36111 m .50375 .36111 L s .5 .37698 m .50375 .37698 L s .5 .39286 m .50375 .39286 L s .5 .4246 m .50375 .4246 L s .5 .44048 m .50375 .44048 L s .5 .45635 m .50375 .45635 L s .5 .47222 m .50375 .47222 L s .25 Mabswid .5 0 m .5 .5 L s 0 0 m 1 0 L 1 .5 L 0 .5 L closepath clip newpath MathSubEnd P % End of sub-graphic % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 144}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "Now you know that\n ", Cell[BoxData[ \(\[Integral]\_\(-3\)\%3\(\@\( 9 - x\^2\)\) \[DifferentialD]x\)]], "\nmeasures half the area of a circle of radius ", Cell[BoxData[ \(3\)]], ". \nSo\n ", Cell[BoxData[ \(\[Integral]\_\(-3\)\%3\(\@\( 9 - x\^2\)\) \[DifferentialD]x\)]], "\nis given by:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(r = 3;\)\), "\n", \(\(\[Pi]\ r\^2\)\/2\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(9\ \[Pi]\)\/2\)], "Output"] }, Open ]], Cell[TextData[{ "See what ", StyleBox["Mathematica", FontSlant->"Italic"], " does with this integral:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_\(-3\)\%3\(\@\( 9 - x\^2\)\) \[DifferentialD]x\)], "Input",\ AspectRatioFixed->True], Cell[BoxData[ \(\(9\ \[Pi]\)\/2\)], "Output"] }, Open ]], Cell["Got it.", "SmallText"] }, Closed]], Cell["T.1.c)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\[Integral]\_\(-\[Pi]\)\%\[Pi]\( x\^2\) Sin[3 x] \[DifferentialD]x\)]] }], "Text"], Cell["Answer:", "Special1"], Cell["To see what's happening, plot:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(\(f[x_] = x\^2\ Sin[3\ x];\)\n\), "\n", \(\(Plot[f[x], {x, \(-\[Pi]\), \[Pi]}, \n\t AspectRatio \[Rule] 1\/GoldenRatio, PlotStyle \[Rule] {{Thickness[0.01], Red}}, AxesLabel \[Rule] {"\", "\"}];\)\)}], "Input", AspectRatioFixed->True], 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 0.5 0.151576 0.309016 0.0416256 [ [.04527 .29652 -6 -9 ] [.04527 .29652 6 0 ] [.19685 .29652 -6 -9 ] [.19685 .29652 6 0 ] [.34842 .29652 -6 -9 ] [.34842 .29652 6 0 ] [.65158 .29652 -3 -9 ] [.65158 .29652 3 0 ] [.80315 .29652 -3 -9 ] [.80315 .29652 3 0 ] [.95473 .29652 -3 -9 ] [.95473 .29652 3 0 ] [1.025 .30902 0 -6 ] [1.025 .30902 10 6 ] [.4875 .05926 -12 -4.5 ] [.4875 .05926 0 4.5 ] [.4875 .14251 -12 -4.5 ] [.4875 .14251 0 4.5 ] [.4875 .22576 -12 -4.5 ] [.4875 .22576 0 4.5 ] [.4875 .39227 -6 -4.5 ] [.4875 .39227 0 4.5 ] [.4875 .47552 -6 -4.5 ] [.4875 .47552 0 4.5 ] [.4875 .55877 -6 -4.5 ] [.4875 .55877 0 4.5 ] [.5 .64303 -14 0 ] [.5 .64303 14 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .04527 .30902 m .04527 .31527 L s [(-3)] .04527 .29652 0 1 Mshowa .19685 .30902 m .19685 .31527 L s [(-2)] .19685 .29652 0 1 Mshowa .34842 .30902 m .34842 .31527 L s [(-1)] .34842 .29652 0 1 Mshowa .65158 .30902 m .65158 .31527 L s [(1)] .65158 .29652 0 1 Mshowa .80315 .30902 m .80315 .31527 L s [(2)] .80315 .29652 0 1 Mshowa .95473 .30902 m .95473 .31527 L s [(3)] .95473 .29652 0 1 Mshowa .125 Mabswid .07559 .30902 m .07559 .31277 L s .1059 .30902 m .1059 .31277 L s .13622 .30902 m .13622 .31277 L s .16653 .30902 m .16653 .31277 L s .22716 .30902 m .22716 .31277 L s .25748 .30902 m .25748 .31277 L s .28779 .30902 m .28779 .31277 L s .31811 .30902 m .31811 .31277 L s .37874 .30902 m .37874 .31277 L s .40905 .30902 m .40905 .31277 L s .43937 .30902 m .43937 .31277 L s .46968 .30902 m .46968 .31277 L s .53032 .30902 m .53032 .31277 L s .56063 .30902 m .56063 .31277 L s .59095 .30902 m .59095 .31277 L s .62126 .30902 m .62126 .31277 L s .68189 .30902 m .68189 .31277 L s .71221 .30902 m .71221 .31277 L s .74252 .30902 m .74252 .31277 L s .77284 .30902 m .77284 .31277 L s .83347 .30902 m .83347 .31277 L s .86378 .30902 m .86378 .31277 L s .8941 .30902 m .8941 .31277 L s .92441 .30902 m .92441 .31277 L s .01496 .30902 m .01496 .31277 L s .98504 .30902 m .98504 .31277 L s .25 Mabswid 0 .30902 m 1 .30902 L s gsave 1.025 .30902 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .5 .05926 m .50625 .05926 L s [(-6)] .4875 .05926 1 0 Mshowa .5 .14251 m .50625 .14251 L s [(-4)] .4875 .14251 1 0 Mshowa .5 .22576 m .50625 .22576 L s [(-2)] .4875 .22576 1 0 Mshowa .5 .39227 m .50625 .39227 L s [(2)] .4875 .39227 1 0 Mshowa .5 .47552 m .50625 .47552 L s [(4)] .4875 .47552 1 0 Mshowa .5 .55877 m .50625 .55877 L s [(6)] .4875 .55877 1 0 Mshowa .125 Mabswid .5 .08008 m .50375 .08008 L s .5 .10089 m .50375 .10089 L s .5 .1217 m .50375 .1217 L s .5 .16333 m .50375 .16333 L s .5 .18414 m .50375 .18414 L s .5 .20495 m .50375 .20495 L s .5 .24658 m .50375 .24658 L s .5 .26739 m .50375 .26739 L s .5 .2882 m .50375 .2882 L s .5 .32983 m .50375 .32983 L s .5 .35064 m .50375 .35064 L s .5 .37145 m .50375 .37145 L s .5 .41308 m .50375 .41308 L s .5 .43389 m .50375 .43389 L s .5 .45471 m .50375 .45471 L s .5 .49633 m .50375 .49633 L s .5 .51714 m .50375 .51714 L s .5 .53796 m .50375 .53796 L s .5 .03845 m .50375 .03845 L s .5 .01764 m .50375 .01764 L s .5 .57958 m .50375 .57958 L s .5 .6004 m .50375 .6004 L s .25 Mabswid .5 0 m .5 .61803 L s gsave .5 .64303 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 69.000 13.000 moveto (@) show 75.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 81.000 13.000 moveto (D) show 87.000 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 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 1 0 0 r .01 w .02381 .30902 m .03279 .23912 L .04262 .17117 L .053 .11132 L .06244 .06888 L .06786 .05001 L .07298 .0359 L .07839 .02494 L .08144 .02053 L .0828 .01897 L .08426 .01756 L .08552 .01657 L .08668 .01585 L .08785 .01531 L .08895 .01495 L .09023 .01473 L .09142 .01472 L .09273 .0149 L .09346 .0151 L .09412 .01533 L .09547 .01597 L .09671 .01676 L .09951 .01919 L .10197 .02207 L .10458 .02585 L .10954 .03498 L .11406 .04539 L .12422 .07505 L .14255 .14434 L .16325 .23246 L .1824 .3085 L .20098 .36684 L .21037 .38855 L .21541 .39789 L .22073 .40593 L .22555 .41165 L .2281 .41408 L .23082 .41622 L .23333 .41779 L .23561 .4189 L .23671 .41932 L .23791 .41969 L .23903 .41998 L .24007 .42017 L .24127 .42032 L .24257 .4204 L .24378 .42038 L .2449 .4203 L .2462 .42013 L .24694 .41999 L Mistroke .24762 .41984 L .2502 .41905 L .25293 .41787 L .2554 .41652 L .26093 .41255 L .27098 .40251 L .28021 .39085 L .30105 .36026 L .32151 .33102 L .3315 .31891 L .34059 .30964 L .35002 .30199 L .35895 .29666 L .36385 .2945 L .36837 .29297 L .37094 .2923 L .37334 .29178 L .37585 .29136 L .37719 .29119 L .37861 .29103 L .37989 .29093 L .38105 .29085 L .38222 .2908 L .38332 .29078 L .38461 .29077 L .38579 .29078 L .38709 .29083 L .38775 .29085 L .38847 .29089 L .39105 .29109 L .3925 .29124 L .39383 .2914 L .39883 .29217 L .40834 .29424 L .41735 .29667 L .43555 .30182 L .44591 .30437 L .45565 .30627 L .46027 .30699 L .46515 .30762 L .47049 .30815 L .4754 .3085 L .47791 .30864 L .48059 .30876 L .48312 .30885 L .48542 .30891 L .48813 .30896 L .48966 .30898 L .49107 .30899 L .4923 .309 L Mistroke .49366 .30901 L .49434 .30901 L .49507 .30901 L .49577 .30901 L .4964 .30901 L .49769 .30902 L .4989 .30902 L .49999 .30902 L .50117 .30902 L .50246 .30902 L .50311 .30902 L .50382 .30902 L .50508 .30902 L .50625 .30902 L .50744 .30903 L .50856 .30904 L .50957 .30905 L .51067 .30906 L .51186 .30908 L .51313 .3091 L .5154 .30914 L .51785 .30922 L .52048 .30932 L .52525 .30957 L .52802 .30976 L .53093 .31001 L .53613 .31057 L .54568 .31199 L .55588 .31407 L .57426 .31895 L .58394 .32173 L .59418 .3244 L .59953 .32555 L .60237 .32607 L .60542 .32653 L .60822 .32687 L .61077 .32709 L .612 .32716 L .61265 .3272 L .61334 .32722 L .6146 .32725 L .61577 .32726 L .61709 .32725 L .61776 .32723 L .61849 .3272 L .61978 .32713 L .62098 .32704 L .6224 .32689 L .62369 .32673 L .6266 .32626 L Mistroke .6291 .32573 L .63183 .325 L .63677 .32329 L .64157 .32111 L .64665 .31824 L .65578 .31159 L .66518 .30272 L .6753 .29108 L .6936 .26579 L .7133 .23632 L .72371 .222 L .72942 .21505 L .73481 .20933 L .73999 .20474 L .74272 .20273 L .7456 .20093 L .74803 .19971 L .74932 .19916 L .7507 .19867 L .75197 .1983 L .75313 .19803 L .75429 .19783 L .75539 .1977 L .75649 .19764 L .7577 .19764 L .75896 .19773 L .76015 .1979 L .76125 .19812 L .76226 .19839 L .76455 .1992 L .76686 .20035 L .7694 .20197 L .77451 .20644 L .77979 .2128 L .78477 .22046 L .79411 .23913 L .80416 .26539 L .81506 .30046 L .85596 .4675 L .86577 .50704 L .87631 .54471 L .88626 .57343 L .89066 .58347 L .89535 .59208 L .89788 .59577 L .90017 .59851 L .90248 .60066 L .90351 .60143 L .90465 .60212 L .90588 .6027 L Mistroke .90718 .60311 L .90839 .6033 L .90952 .60332 L .9102 .60325 L .91085 .60313 L .91206 .60276 L .91336 .60215 L .91478 .60124 L .91699 .59929 L .91943 .59637 L .92437 .58801 L .92894 .57726 L .93323 .56456 L .94308 .52578 L .95344 .47114 L .97195 .34281 L .97619 .30902 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "Here is a picture of the signed area measured by \n ", Cell[BoxData[ \(\[Integral]\_\(-\[Pi]\)\%\[Pi]\( x\^2\) Sin[3 x] \[DifferentialD]x\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(FilledPlot[f[x], {x, \(-\[Pi]\), \[Pi]}, \n\t AspectRatio \[Rule] 1\/GoldenRatio, AxesLabel \[Rule] {"\", "\"}];\)\)], "Input", AspectRatioFixed->True], 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 1 0 0.618034 [ [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.151576 0.309016 0.0416256 [ [.04527 .29652 -6 -9 ] [.04527 .29652 6 0 ] [.19685 .29652 -6 -9 ] [.19685 .29652 6 0 ] [.34842 .29652 -6 -9 ] [.34842 .29652 6 0 ] [.65158 .29652 -3 -9 ] [.65158 .29652 3 0 ] [.80315 .29652 -3 -9 ] [.80315 .29652 3 0 ] [.95473 .29652 -3 -9 ] [.95473 .29652 3 0 ] [1.025 .30902 0 -6 ] [1.025 .30902 10 6 ] [.4875 .05926 -12 -4.5 ] [.4875 .05926 0 4.5 ] [.4875 .14251 -12 -4.5 ] [.4875 .14251 0 4.5 ] [.4875 .22576 -12 -4.5 ] [.4875 .22576 0 4.5 ] [.4875 .39227 -6 -4.5 ] [.4875 .39227 0 4.5 ] [.4875 .47552 -6 -4.5 ] [.4875 .47552 0 4.5 ] [.4875 .55877 -6 -4.5 ] [.4875 .55877 0 4.5 ] [.5 .64303 -14 0 ] [.5 .64303 14 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .04527 .30902 m .04527 .31527 L s [(-3)] .04527 .29652 0 1 Mshowa .19685 .30902 m .19685 .31527 L s [(-2)] .19685 .29652 0 1 Mshowa .34842 .30902 m .34842 .31527 L s [(-1)] .34842 .29652 0 1 Mshowa .65158 .30902 m .65158 .31527 L s [(1)] .65158 .29652 0 1 Mshowa .80315 .30902 m .80315 .31527 L s [(2)] .80315 .29652 0 1 Mshowa .95473 .30902 m .95473 .31527 L s [(3)] .95473 .29652 0 1 Mshowa .125 Mabswid .07559 .30902 m .07559 .31277 L s .1059 .30902 m .1059 .31277 L s .13622 .30902 m .13622 .31277 L s .16653 .30902 m .16653 .31277 L s .22716 .30902 m .22716 .31277 L s .25748 .30902 m .25748 .31277 L s .28779 .30902 m .28779 .31277 L s .31811 .30902 m .31811 .31277 L s .37874 .30902 m .37874 .31277 L s .40905 .30902 m .40905 .31277 L s .43937 .30902 m .43937 .31277 L s .46968 .30902 m .46968 .31277 L s .53032 .30902 m .53032 .31277 L s .56063 .30902 m .56063 .31277 L s .59095 .30902 m .59095 .31277 L s .62126 .30902 m .62126 .31277 L s .68189 .30902 m .68189 .31277 L s .71221 .30902 m .71221 .31277 L s .74252 .30902 m .74252 .31277 L s .77284 .30902 m .77284 .31277 L s .83347 .30902 m .83347 .31277 L s .86378 .30902 m .86378 .31277 L s .8941 .30902 m .8941 .31277 L s .92441 .30902 m .92441 .31277 L s .01496 .30902 m .01496 .31277 L s .98504 .30902 m .98504 .31277 L s .25 Mabswid 0 .30902 m 1 .30902 L s gsave 1.025 .30902 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .5 .05926 m .50625 .05926 L s [(-6)] .4875 .05926 1 0 Mshowa .5 .14251 m .50625 .14251 L s [(-4)] .4875 .14251 1 0 Mshowa .5 .22576 m .50625 .22576 L s [(-2)] .4875 .22576 1 0 Mshowa .5 .39227 m .50625 .39227 L s [(2)] .4875 .39227 1 0 Mshowa .5 .47552 m .50625 .47552 L s [(4)] .4875 .47552 1 0 Mshowa .5 .55877 m .50625 .55877 L s [(6)] .4875 .55877 1 0 Mshowa .125 Mabswid .5 .08008 m .50375 .08008 L s .5 .10089 m .50375 .10089 L s .5 .1217 m .50375 .1217 L s .5 .16333 m .50375 .16333 L s .5 .18414 m .50375 .18414 L s .5 .20495 m .50375 .20495 L s .5 .24658 m .50375 .24658 L s .5 .26739 m .50375 .26739 L s .5 .2882 m .50375 .2882 L s .5 .32983 m .50375 .32983 L s .5 .35064 m .50375 .35064 L s .5 .37145 m .50375 .37145 L s .5 .41308 m .50375 .41308 L s .5 .43389 m .50375 .43389 L s .5 .45471 m .50375 .45471 L s .5 .49633 m .50375 .49633 L s .5 .51714 m .50375 .51714 L s .5 .53796 m .50375 .53796 L s .5 .03845 m .50375 .03845 L s .5 .01764 m .50375 .01764 L s .5 .57958 m .50375 .57958 L s .5 .6004 m .50375 .6004 L s .25 Mabswid .5 0 m .5 .61803 L s gsave .5 .64303 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 69.000 13.000 moveto (@) show 75.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 81.000 13.000 moveto (D) show 87.000 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 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .30902 m .03279 .23912 L .04262 .17117 L .053 .11132 L .06244 .06888 L .06786 .05001 L .07298 .0359 L .07839 .02494 L .08144 .02053 L .0828 .01897 L .08426 .01756 L .08552 .01657 L .08668 .01585 L .08785 .01531 L .08895 .01495 L .09023 .01473 L .09142 .01472 L .09273 .0149 L .09346 .0151 L .09412 .01533 L .09547 .01597 L .09671 .01676 L .09951 .01919 L .10197 .02207 L .10458 .02585 L .10954 .03498 L .11406 .04539 L .12422 .07505 L .14255 .14434 L .16325 .23246 L .1824 .3085 L .20098 .36684 L .21037 .38855 L .21541 .39789 L .22073 .40593 L .22555 .41165 L .2281 .41408 L .23082 .41622 L .23333 .41779 L .23561 .4189 L .23671 .41932 L .23791 .41969 L .23903 .41998 L .24007 .42017 L .24127 .42032 L .24257 .4204 L .24378 .42038 L .2449 .4203 L .2462 .42013 L .24694 .41999 L Mistroke .24762 .41984 L .2502 .41905 L .25293 .41787 L .2554 .41652 L .26093 .41255 L .27098 .40251 L .28021 .39085 L .30105 .36026 L .32151 .33102 L .3315 .31891 L .34059 .30964 L .35002 .30199 L .35895 .29666 L .36385 .2945 L .36837 .29297 L .37094 .2923 L .37334 .29178 L .37585 .29136 L .37719 .29119 L .37861 .29103 L .37989 .29093 L .38105 .29085 L .38222 .2908 L .38332 .29078 L .38461 .29077 L .38579 .29078 L .38709 .29083 L .38775 .29085 L .38847 .29089 L .39105 .29109 L .3925 .29124 L .39383 .2914 L .39883 .29217 L .40834 .29424 L .41735 .29667 L .43555 .30182 L .44591 .30437 L .45565 .30627 L .46027 .30699 L .46515 .30762 L .47049 .30815 L .4754 .3085 L .47791 .30864 L .48059 .30876 L .48312 .30885 L .48542 .30891 L .48813 .30896 L .48966 .30898 L .49107 .30899 L .4923 .309 L Mistroke .49366 .30901 L .49434 .30901 L .49507 .30901 L .49577 .30901 L .4964 .30901 L .49769 .30902 L .4989 .30902 L .49999 .30902 L .50117 .30902 L .50246 .30902 L .50311 .30902 L .50382 .30902 L .50508 .30902 L .50625 .30902 L .50744 .30903 L .50856 .30904 L .50957 .30905 L .51067 .30906 L .51186 .30908 L .51313 .3091 L .5154 .30914 L .51785 .30922 L .52048 .30932 L .52525 .30957 L .52802 .30976 L .53093 .31001 L .53613 .31057 L .54568 .31199 L .55588 .31407 L .57426 .31895 L .58394 .32173 L .59418 .3244 L .59953 .32555 L .60237 .32607 L .60542 .32653 L .60822 .32687 L .61077 .32709 L .612 .32716 L .61265 .3272 L .61334 .32722 L .6146 .32725 L .61577 .32726 L .61709 .32725 L .61776 .32723 L .61849 .3272 L .61978 .32713 L .62098 .32704 L .6224 .32689 L .62369 .32673 L .6266 .32626 L Mistroke .6291 .32573 L .63183 .325 L .63677 .32329 L .64157 .32111 L .64665 .31824 L .65578 .31159 L .66518 .30272 L .6753 .29108 L .6936 .26579 L .7133 .23632 L .72371 .222 L .72942 .21505 L .73481 .20933 L .73999 .20474 L .74272 .20273 L .7456 .20093 L .74803 .19971 L .74932 .19916 L .7507 .19867 L .75197 .1983 L .75313 .19803 L .75429 .19783 L .75539 .1977 L .75649 .19764 L .7577 .19764 L .75896 .19773 L .76015 .1979 L .76125 .19812 L .76226 .19839 L .76455 .1992 L .76686 .20035 L .7694 .20197 L .77451 .20644 L .77979 .2128 L .78477 .22046 L .79411 .23913 L .80416 .26539 L .81506 .30046 L .85596 .4675 L .86577 .50704 L .87631 .54471 L .88626 .57343 L .89066 .58347 L .89535 .59208 L .89788 .59577 L .90017 .59851 L .90248 .60066 L .90351 .60143 L .90465 .60212 L .90588 .6027 L Mistroke .90718 .60311 L .90839 .6033 L .90952 .60332 L .9102 .60325 L .91085 .60313 L .91206 .60276 L .91336 .60215 L .91478 .60124 L .91699 .59929 L .91943 .59637 L .92437 .58801 L .92894 .57726 L .93323 .56456 L .94308 .52578 L .95344 .47114 L .97195 .34281 L .97619 .30902 L Mfstroke .5 g .02381 .30902 m .03279 .23912 L .04262 .17117 L .053 .11132 L .06244 .06888 L .06786 .05001 L .07298 .0359 L .07839 .02494 L .08144 .02053 L .0828 .01897 L .08426 .01756 L .08552 .01657 L .08668 .01585 L .08785 .01531 L .08895 .01495 L .09023 .01473 L .09142 .01472 L .09273 .0149 L .09346 .0151 L .09412 .01533 L .09547 .01597 L .09671 .01676 L .09951 .01919 L .10197 .02207 L .10458 .02585 L .10954 .03498 L .11406 .04539 L .12422 .07505 L .14255 .14434 L .16325 .23246 L .1824 .3085 L .20098 .36684 L .21037 .38855 L .21541 .39789 L .22073 .40593 L .22555 .41165 L .2281 .41408 L .23082 .41622 L .23333 .41779 L .23561 .4189 L .23671 .41932 L .23791 .41969 L .23903 .41998 L .24007 .42017 L .24127 .42032 L .24257 .4204 L .24378 .42038 L .2449 .4203 L .2462 .42013 L .24694 .41999 L .24762 .41984 L .2502 .41905 L .25293 .41787 L .2554 .41652 L .26093 .41255 L .27098 .40251 L .28021 .39085 L .30105 .36026 L .32151 .33102 L .3315 .31891 L .34059 .30964 L .35002 .30199 L .35895 .29666 L .36385 .2945 L .36837 .29297 L .37094 .2923 L .37334 .29178 L .37585 .29136 L .37719 .29119 L .37861 .29103 L .37989 .29093 L .38105 .29085 L .38222 .2908 L .38332 .29078 L .38461 .29077 L .38579 .29078 L .38709 .29083 L .38775 .29085 L .38847 .29089 L .39105 .29109 L .3925 .29124 L .39383 .2914 L .39883 .29217 L .40834 .29424 L .41735 .29667 L .43555 .30182 L .44591 .30437 L .45565 .30627 L .46027 .30699 L .46515 .30762 L .47049 .30815 L .4754 .3085 L .47791 .30864 L .48059 .30876 L .48312 .30885 L .48542 .30891 L .48813 .30896 L .48966 .30898 L .49107 .30899 L .4923 .309 L .49366 .30901 L .49434 .30901 L .49507 .30901 L .49577 .30901 L .4964 .30901 L .49769 .30902 L .4989 .30902 L .49999 .30902 L .50117 .30902 L .50246 .30902 L .50311 .30902 L .50382 .30902 L .50508 .30902 L .50625 .30902 L .50744 .30903 L .50856 .30904 L .50957 .30905 L .51067 .30906 L .51186 .30908 L .51313 .3091 L .5154 .30914 L .51785 .30922 L .52048 .30932 L .52525 .30957 L .52802 .30976 L .53093 .31001 L .53613 .31057 L .54568 .31199 L .55588 .31407 L .57426 .31895 L .58394 .32173 L .59418 .3244 L .59953 .32555 L .60237 .32607 L .60542 .32653 L .60822 .32687 L .61077 .32709 L .612 .32716 L .61265 .3272 L .61334 .32722 L .6146 .32725 L .61577 .32726 L .61709 .32725 L .61776 .32723 L .61849 .3272 L .61978 .32713 L .62098 .32704 L .6224 .32689 L .62369 .32673 L .6266 .32626 L .6291 .32573 L .63183 .325 L .63677 .32329 L .64157 .32111 L .64665 .31824 L .65578 .31159 L .66518 .30272 L .6753 .29108 L .6936 .26579 L .7133 .23632 L .72371 .222 L .72942 .21505 L .73481 .20933 L .73999 .20474 L .74272 .20273 L .7456 .20093 L .74803 .19971 L .74932 .19916 L .7507 .19867 L .75197 .1983 L .75313 .19803 L .75429 .19783 L .75539 .1977 L .75649 .19764 L .7577 .19764 L .75896 .19773 L .76015 .1979 L .76125 .19812 L .76226 .19839 L .76455 .1992 L .76686 .20035 L .7694 .20197 L .77451 .20644 L .77979 .2128 L .78477 .22046 L .79411 .23913 L .80416 .26539 L .81506 .30046 L .85596 .4675 L .86577 .50704 L .87631 .54471 L .88626 .57343 L .89066 .58347 L .89535 .59208 L .89788 .59577 L .90017 .59851 L .90248 .60066 L .90351 .60143 L .90465 .60212 L .90588 .6027 L .90718 .60311 L .90839 .6033 L .90952 .60332 L .9102 .60325 L .91085 .60313 L .91206 .60276 L .91336 .60215 L .91478 .60124 L .91699 .59929 L .91943 .59637 L .92437 .58801 L .92894 .57726 L .93323 .56456 L .94308 .52578 L .95344 .47114 L .97195 .34281 L .97619 .30902 L .97619 .30902 L .02381 .30902 L F 0 g .02381 .30902 m .03279 .23912 L .04262 .17117 L .053 .11132 L .06244 .06888 L .06786 .05001 L .07298 .0359 L .07839 .02494 L .08144 .02053 L .0828 .01897 L .08426 .01756 L .08552 .01657 L .08668 .01585 L .08785 .01531 L .08895 .01495 L .09023 .01473 L .09142 .01472 L .09273 .0149 L .09346 .0151 L .09412 .01533 L .09547 .01597 L .09671 .01676 L .09951 .01919 L .10197 .02207 L .10458 .02585 L .10954 .03498 L .11406 .04539 L .12422 .07505 L .14255 .14434 L .16325 .23246 L .1824 .3085 L .20098 .36684 L .21037 .38855 L .21541 .39789 L .22073 .40593 L .22555 .41165 L .2281 .41408 L .23082 .41622 L .23333 .41779 L .23561 .4189 L .23671 .41932 L .23791 .41969 L .23903 .41998 L .24007 .42017 L .24127 .42032 L .24257 .4204 L .24378 .42038 L .2449 .4203 L .2462 .42013 L .24694 .41999 L Mistroke .24762 .41984 L .2502 .41905 L .25293 .41787 L .2554 .41652 L .26093 .41255 L .27098 .40251 L .28021 .39085 L .30105 .36026 L .32151 .33102 L .3315 .31891 L .34059 .30964 L .35002 .30199 L .35895 .29666 L .36385 .2945 L .36837 .29297 L .37094 .2923 L .37334 .29178 L .37585 .29136 L .37719 .29119 L .37861 .29103 L .37989 .29093 L .38105 .29085 L .38222 .2908 L .38332 .29078 L .38461 .29077 L .38579 .29078 L .38709 .29083 L .38775 .29085 L .38847 .29089 L .39105 .29109 L .3925 .29124 L .39383 .2914 L .39883 .29217 L .40834 .29424 L .41735 .29667 L .43555 .30182 L .44591 .30437 L .45565 .30627 L .46027 .30699 L .46515 .30762 L .47049 .30815 L .4754 .3085 L .47791 .30864 L .48059 .30876 L .48312 .30885 L .48542 .30891 L .48813 .30896 L .48966 .30898 L .49107 .30899 L .4923 .309 L Mistroke .49366 .30901 L .49434 .30901 L .49507 .30901 L .49577 .30901 L .4964 .30901 L .49769 .30902 L .4989 .30902 L .49999 .30902 L .50117 .30902 L .50246 .30902 L .50311 .30902 L .50382 .30902 L .50508 .30902 L .50625 .30902 L .50744 .30903 L .50856 .30904 L .50957 .30905 L .51067 .30906 L .51186 .30908 L .51313 .3091 L .5154 .30914 L .51785 .30922 L .52048 .30932 L .52525 .30957 L .52802 .30976 L .53093 .31001 L .53613 .31057 L .54568 .31199 L .55588 .31407 L .57426 .31895 L .58394 .32173 L .59418 .3244 L .59953 .32555 L .60237 .32607 L .60542 .32653 L .60822 .32687 L .61077 .32709 L .612 .32716 L .61265 .3272 L .61334 .32722 L .6146 .32725 L .61577 .32726 L .61709 .32725 L .61776 .32723 L .61849 .3272 L .61978 .32713 L .62098 .32704 L .6224 .32689 L .62369 .32673 L .6266 .32626 L Mistroke .6291 .32573 L .63183 .325 L .63677 .32329 L .64157 .32111 L .64665 .31824 L .65578 .31159 L .66518 .30272 L .6753 .29108 L .6936 .26579 L .7133 .23632 L .72371 .222 L .72942 .21505 L .73481 .20933 L .73999 .20474 L .74272 .20273 L .7456 .20093 L .74803 .19971 L .74932 .19916 L .7507 .19867 L .75197 .1983 L .75313 .19803 L .75429 .19783 L .75539 .1977 L .75649 .19764 L .7577 .19764 L .75896 .19773 L .76015 .1979 L .76125 .19812 L .76226 .19839 L .76455 .1992 L .76686 .20035 L .7694 .20197 L .77451 .20644 L .77979 .2128 L .78477 .22046 L .79411 .23913 L .80416 .26539 L .81506 .30046 L .85596 .4675 L .86577 .50704 L .87631 .54471 L .88626 .57343 L .89066 .58347 L .89535 .59208 L .89788 .59577 L .90017 .59851 L .90248 .60066 L .90351 .60143 L .90465 .60212 L .90588 .6027 L Mistroke .90718 .60311 L .90839 .6033 L .90952 .60332 L .9102 .60325 L .91085 .60313 L .91206 .60276 L .91336 .60215 L .91478 .60124 L .91699 .59929 L .91943 .59637 L .92437 .58801 L .92894 .57726 L .93323 .56456 L .94308 .52578 L .95344 .47114 L .97195 .34281 L .97619 .30902 L Mfstroke MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.151576 0.309016 0.0416256 [ [.04527 .29652 -6 -9 ] [.04527 .29652 6 0 ] [.19685 .29652 -6 -9 ] [.19685 .29652 6 0 ] [.34842 .29652 -6 -9 ] [.34842 .29652 6 0 ] [.65158 .29652 -3 -9 ] [.65158 .29652 3 0 ] [.80315 .29652 -3 -9 ] [.80315 .29652 3 0 ] [.95473 .29652 -3 -9 ] [.95473 .29652 3 0 ] [1.025 .30902 0 -6 ] [1.025 .30902 10 6 ] [.4875 .05926 -12 -4.5 ] [.4875 .05926 0 4.5 ] [.4875 .14251 -12 -4.5 ] [.4875 .14251 0 4.5 ] [.4875 .22576 -12 -4.5 ] [.4875 .22576 0 4.5 ] [.4875 .39227 -6 -4.5 ] [.4875 .39227 0 4.5 ] [.4875 .47552 -6 -4.5 ] [.4875 .47552 0 4.5 ] [.4875 .55877 -6 -4.5 ] [.4875 .55877 0 4.5 ] [.5 .64303 -14 0 ] [.5 .64303 14 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .04527 .30902 m .04527 .31527 L s [(-3)] .04527 .29652 0 1 Mshowa .19685 .30902 m .19685 .31527 L s [(-2)] .19685 .29652 0 1 Mshowa .34842 .30902 m .34842 .31527 L s [(-1)] .34842 .29652 0 1 Mshowa .65158 .30902 m .65158 .31527 L s [(1)] .65158 .29652 0 1 Mshowa .80315 .30902 m .80315 .31527 L s [(2)] .80315 .29652 0 1 Mshowa .95473 .30902 m .95473 .31527 L s [(3)] .95473 .29652 0 1 Mshowa .125 Mabswid .07559 .30902 m .07559 .31277 L s .1059 .30902 m .1059 .31277 L s .13622 .30902 m .13622 .31277 L s .16653 .30902 m .16653 .31277 L s .22716 .30902 m .22716 .31277 L s .25748 .30902 m .25748 .31277 L s .28779 .30902 m .28779 .31277 L s .31811 .30902 m .31811 .31277 L s .37874 .30902 m .37874 .31277 L s .40905 .30902 m .40905 .31277 L s .43937 .30902 m .43937 .31277 L s .46968 .30902 m .46968 .31277 L s .53032 .30902 m .53032 .31277 L s .56063 .30902 m .56063 .31277 L s .59095 .30902 m .59095 .31277 L s .62126 .30902 m .62126 .31277 L s .68189 .30902 m .68189 .31277 L s .71221 .30902 m .71221 .31277 L s .74252 .30902 m .74252 .31277 L s .77284 .30902 m .77284 .31277 L s .83347 .30902 m .83347 .31277 L s .86378 .30902 m .86378 .31277 L s .8941 .30902 m .8941 .31277 L s .92441 .30902 m .92441 .31277 L s .01496 .30902 m .01496 .31277 L s .98504 .30902 m .98504 .31277 L s .25 Mabswid 0 .30902 m 1 .30902 L s gsave 1.025 .30902 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .5 .05926 m .50625 .05926 L s [(-6)] .4875 .05926 1 0 Mshowa .5 .14251 m .50625 .14251 L s [(-4)] .4875 .14251 1 0 Mshowa .5 .22576 m .50625 .22576 L s [(-2)] .4875 .22576 1 0 Mshowa .5 .39227 m .50625 .39227 L s [(2)] .4875 .39227 1 0 Mshowa .5 .47552 m .50625 .47552 L s [(4)] .4875 .47552 1 0 Mshowa .5 .55877 m .50625 .55877 L s [(6)] .4875 .55877 1 0 Mshowa .125 Mabswid .5 .08008 m .50375 .08008 L s .5 .10089 m .50375 .10089 L s .5 .1217 m .50375 .1217 L s .5 .16333 m .50375 .16333 L s .5 .18414 m .50375 .18414 L s .5 .20495 m .50375 .20495 L s .5 .24658 m .50375 .24658 L s .5 .26739 m .50375 .26739 L s .5 .2882 m .50375 .2882 L s .5 .32983 m .50375 .32983 L s .5 .35064 m .50375 .35064 L s .5 .37145 m .50375 .37145 L s .5 .41308 m .50375 .41308 L s .5 .43389 m .50375 .43389 L s .5 .45471 m .50375 .45471 L s .5 .49633 m .50375 .49633 L s .5 .51714 m .50375 .51714 L s .5 .53796 m .50375 .53796 L s .5 .03845 m .50375 .03845 L s .5 .01764 m .50375 .01764 L s .5 .57958 m .50375 .57958 L s .5 .6004 m .50375 .6004 L s .25 Mabswid .5 0 m .5 .61803 L s gsave .5 .64303 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 69.000 13.000 moveto (@) show 75.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 81.000 13.000 moveto (D) show 87.000 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 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath MathSubEnd P % End of sub-graphic % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell["\<\ The positive and negative areas cancel each other! This happens because:\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(f[\(-x\)] == \(-f[x]\)\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "Because the positive and negative areas cancel each other, you can say \ with complete confidence that\n ", Cell[BoxData[ \(\[Integral]\_\(-\[Pi]\)\%\[Pi]\( x\^2\) Sin[3 x] \[DifferentialD]x = 0\)]], ".\nCheck: " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_\(-\[Pi]\)\%\[Pi]\( x\^2\ Sin[ 3\ x]\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["There was never a doubt.", "SmallText"] }, Closed]] }, Closed]], Cell["Give It a Try Problem", "Subsubsection"], Cell[CellGroupData[{ Cell["G.3) Calculating some integrals*", "Subsection", CellTags->"2.01.G3"], Cell["G.3.a)", "Subsubsection"], Cell[TextData[{ "Look at this plot of ", Cell[BoxData[ \(f[x] = 2 x - 3\)]], " for ", Cell[BoxData[ \(3 \[LessEqual] x \[LessEqual] 5\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(\(f[x_] = 2\ x - 3;\)\n\), "\n", \(\(Plot[f[x], {x, 3, 5}, \n\t PlotStyle \[Rule] {{Thickness[0.01], DarkGreen}}, \n\t AspectRatio \[Rule] 1\/GoldenRatio, PlotRange \[Rule] {0, f[5]}, AxesLabel \[Rule] {"\", "\"}];\)\)}], "Input", AspectRatioFixed->True], 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 -1.40476 0.476191 0 0.0882906 [ [.2619 -0.0125 -9 -9 ] [.2619 -0.0125 9 0 ] [.5 -0.0125 -3 -9 ] [.5 -0.0125 3 0 ] [.7381 -0.0125 -9 -9 ] [.7381 -0.0125 9 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [1.025 0 0 -6 ] [1.025 0 10 6 ] [.01131 .08829 -6 -4.5 ] [.01131 .08829 0 4.5 ] [.01131 .17658 -6 -4.5 ] [.01131 .17658 0 4.5 ] [.01131 .26487 -6 -4.5 ] [.01131 .26487 0 4.5 ] [.01131 .35316 -6 -4.5 ] [.01131 .35316 0 4.5 ] [.01131 .44145 -6 -4.5 ] [.01131 .44145 0 4.5 ] [.01131 .52974 -6 -4.5 ] [.01131 .52974 0 4.5 ] [.01131 .61803 -6 -4.5 ] [.01131 .61803 0 4.5 ] [.02381 .64303 -14 0 ] [.02381 .64303 14 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .2619 0 m .2619 .00625 L s [(3.5)] .2619 -0.0125 0 1 Mshowa .5 0 m .5 .00625 L s [(4)] .5 -0.0125 0 1 Mshowa .7381 0 m .7381 .00625 L s [(4.5)] .7381 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(5)] .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 .21429 0 m .21429 .00375 L s .30952 0 m .30952 .00375 L s .35714 0 m .35714 .00375 L s .40476 0 m .40476 .00375 L s .45238 0 m .45238 .00375 L s .54762 0 m .54762 .00375 L s .59524 0 m .59524 .00375 L s .64286 0 m .64286 .00375 L s .69048 0 m .69048 .00375 L s .78571 0 m .78571 .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 gsave 1.025 0 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .08829 m .03006 .08829 L s [(1)] .01131 .08829 1 0 Mshowa .02381 .17658 m .03006 .17658 L s [(2)] .01131 .17658 1 0 Mshowa .02381 .26487 m .03006 .26487 L s [(3)] .01131 .26487 1 0 Mshowa .02381 .35316 m .03006 .35316 L s [(4)] .01131 .35316 1 0 Mshowa .02381 .44145 m .03006 .44145 L s [(5)] .01131 .44145 1 0 Mshowa .02381 .52974 m .03006 .52974 L s [(6)] .01131 .52974 1 0 Mshowa .02381 .61803 m .03006 .61803 L s [(7)] .01131 .61803 1 0 Mshowa .125 Mabswid .02381 .01766 m .02756 .01766 L s .02381 .03532 m .02756 .03532 L s .02381 .05297 m .02756 .05297 L s .02381 .07063 m .02756 .07063 L s .02381 .10595 m .02756 .10595 L s .02381 .12361 m .02756 .12361 L s .02381 .14126 m .02756 .14126 L s .02381 .15892 m .02756 .15892 L s .02381 .19424 m .02756 .19424 L s .02381 .2119 m .02756 .2119 L s .02381 .22956 m .02756 .22956 L s .02381 .24721 m .02756 .24721 L s .02381 .28253 m .02756 .28253 L s .02381 .30019 m .02756 .30019 L s .02381 .31785 m .02756 .31785 L s .02381 .3355 m .02756 .3355 L s .02381 .37082 m .02756 .37082 L s .02381 .38848 m .02756 .38848 L s .02381 .40614 m .02756 .40614 L s .02381 .42379 m .02756 .42379 L s .02381 .45911 m .02756 .45911 L s .02381 .47677 m .02756 .47677 L s .02381 .49443 m .02756 .49443 L s .02381 .51209 m .02756 .51209 L s .02381 .5474 m .02756 .5474 L s .02381 .56506 m .02756 .56506 L s .02381 .58272 m .02756 .58272 L s .02381 .60038 m .02756 .60038 L s .25 Mabswid .02381 0 m .02381 .61803 L s gsave .02381 .64303 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 69.000 13.000 moveto (@) show 75.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 81.000 13.000 moveto (D) show 87.000 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 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 .392 0 r .01 w .02381 .26487 m .06244 .2792 L .10458 .29482 L .14415 .3095 L .18221 .32361 L .22272 .33863 L .26171 .35309 L .30316 .36846 L .34309 .38327 L .3815 .39751 L .42237 .41266 L .46172 .42726 L .49955 .44129 L .53984 .45623 L .57861 .4706 L .61984 .48589 L .65954 .50062 L .69774 .51478 L .73838 .52985 L .77751 .54436 L .81909 .55978 L .85916 .57464 L .89771 .58893 L .93871 .60414 L .97619 .61803 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "Use what you know about areas inside trapezoids to calculate the value of \ \n ", Cell[BoxData[ \(\[Integral]\_3\%5 f[x] \[DifferentialD]x\)]], ". \nCheck with ", StyleBox["Mathematica", FontSlant->"Italic"], ". " }], "Text"], Cell["G.3.b.i)", "Subsubsection"], Cell[TextData[{ "Look at this plot of ", Cell[BoxData[ \(f[x] = 1 - x\)]], " for ", Cell[BoxData[ \(\(-1\) \[LessEqual] x \[LessEqual] \(\(1\)\(:\)\)\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(\(f[x_] = 1 - x;\)\n\), "\n", \(\(Plot[f[x], {x, \(-1\), 1}, \n\t PlotStyle \[Rule] {{Thickness[0.01], Blue}}, AspectRatio \[Rule] 1\/GoldenRatio, PlotRange \[Rule] All, AxesLabel \[Rule] {"\", "\"}];\)\)}], "Input", AspectRatioFixed->True], 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 0.5 0.476191 0.0147151 0.294302 [ [.02381 .00222 -6 -9 ] [.02381 .00222 6 0 ] [.2619 .00222 -12 -9 ] [.2619 .00222 12 0 ] [.7381 .00222 -9 -9 ] [.7381 .00222 9 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [1.025 .01472 0 -6 ] [1.025 .01472 10 6 ] [.4875 .16187 -18 -4.5 ] [.4875 .16187 0 4.5 ] [.4875 .30902 -6 -4.5 ] [.4875 .30902 0 4.5 ] [.4875 .45617 -18 -4.5 ] [.4875 .45617 0 4.5 ] [.4875 .60332 -6 -4.5 ] [.4875 .60332 0 4.5 ] [.5 .64303 -14 0 ] [.5 .64303 14 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .01472 m .02381 .02097 L s [(-1)] .02381 .00222 0 1 Mshowa .2619 .01472 m .2619 .02097 L s [(-0.5)] .2619 .00222 0 1 Mshowa .7381 .01472 m .7381 .02097 L s [(0.5)] .7381 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(1)] .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 .21429 .01472 m .21429 .01847 L s .30952 .01472 m .30952 .01847 L s .35714 .01472 m .35714 .01847 L s .40476 .01472 m .40476 .01847 L s .45238 .01472 m .45238 .01847 L s .54762 .01472 m .54762 .01847 L s .59524 .01472 m .59524 .01847 L s .64286 .01472 m .64286 .01847 L s .69048 .01472 m .69048 .01847 L s .78571 .01472 m .78571 .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 gsave 1.025 .01472 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .5 .16187 m .50625 .16187 L s [(0.5)] .4875 .16187 1 0 Mshowa .5 .30902 m .50625 .30902 L s [(1)] .4875 .30902 1 0 Mshowa .5 .45617 m .50625 .45617 L s [(1.5)] .4875 .45617 1 0 Mshowa .5 .60332 m .50625 .60332 L s [(2)] .4875 .60332 1 0 Mshowa .125 Mabswid .5 .04415 m .50375 .04415 L s .5 .07358 m .50375 .07358 L s .5 .10301 m .50375 .10301 L s .5 .13244 m .50375 .13244 L s .5 .1913 m .50375 .1913 L s .5 .22073 m .50375 .22073 L s .5 .25016 m .50375 .25016 L s .5 .27959 m .50375 .27959 L s .5 .33845 m .50375 .33845 L s .5 .36788 m .50375 .36788 L s .5 .39731 m .50375 .39731 L s .5 .42674 m .50375 .42674 L s .5 .4856 m .50375 .4856 L s .5 .51503 m .50375 .51503 L s .5 .54446 m .50375 .54446 L s .5 .57389 m .50375 .57389 L s .25 Mabswid .5 0 m .5 .61803 L s gsave .5 .64303 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 69.000 13.000 moveto (@) show 75.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 81.000 13.000 moveto (D) show 87.000 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 0 0 1 r .01 w .02381 .60332 m .06244 .57944 L .10458 .5534 L .14415 .52894 L .18221 .50542 L .22272 .48039 L .26171 .45629 L .30316 .43067 L .34309 .406 L .3815 .38225 L .42237 .357 L .46172 .33268 L .49955 .30929 L .53984 .2844 L .57861 .26043 L .61984 .23495 L .65954 .21041 L .69774 .18681 L .73838 .16169 L .77751 .13751 L .81909 .11181 L .85916 .08704 L .89771 .06322 L .93871 .03788 L .97619 .01472 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "Use what you know about areas inside triangles to calculate the value of \n\ ", Cell[BoxData[ \(\[Integral]\_\(-1\)\%1 f[x] \[DifferentialD]x\)]], ". \nCheck with ", StyleBox["Mathematica", FontSlant->"Italic"], ". " }], "Text"], Cell["G.3.b.ii)", "Subsubsection"], Cell[TextData[{ "Look at this plot of ", Cell[BoxData[ \(f[x] = 1 - x\)]], " for ", Cell[BoxData[ \(\(-1\) \[LessEqual] x \[LessEqual] 2\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(\(f[x_] = 1 - x;\)\), "\n", \(\(Plot[f[x], {x, \(-1\), 2}, \n\t PlotStyle \[Rule] {{Thickness[0.01], Blue}}, AspectRatio \[Rule] 1\/GoldenRatio, PlotRange \[Rule] All, AxesLabel \[Rule] {"\", "\"}];\)\)}], "Input", AspectRatioFixed->True], 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 0.34127 0.31746 0.210916 0.196201 [ [.02381 .19842 -6 -9 ] [.02381 .19842 6 0 ] [.18254 .19842 -12 -9 ] [.18254 .19842 12 0 ] [.5 .19842 -9 -9 ] [.5 .19842 9 0 ] [.65873 .19842 -3 -9 ] [.65873 .19842 3 0 ] [.81746 .19842 -9 -9 ] [.81746 .19842 9 0 ] [.97619 .19842 -3 -9 ] [.97619 .19842 3 0 ] [1.025 .21092 0 -6 ] [1.025 .21092 10 6 ] [.32877 .01472 -12 -4.5 ] [.32877 .01472 0 4.5 ] [.32877 .11282 -24 -4.5 ] [.32877 .11282 0 4.5 ] [.32877 .30902 -18 -4.5 ] [.32877 .30902 0 4.5 ] [.32877 .40712 -6 -4.5 ] [.32877 .40712 0 4.5 ] [.32877 .50522 -18 -4.5 ] [.32877 .50522 0 4.5 ] [.32877 .60332 -6 -4.5 ] [.32877 .60332 0 4.5 ] [.34127 .64303 -14 0 ] [.34127 .64303 14 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .21092 m .02381 .21717 L s [(-1)] .02381 .19842 0 1 Mshowa .18254 .21092 m .18254 .21717 L s [(-0.5)] .18254 .19842 0 1 Mshowa .5 .21092 m .5 .21717 L s [(0.5)] .5 .19842 0 1 Mshowa .65873 .21092 m .65873 .21717 L s [(1)] .65873 .19842 0 1 Mshowa .81746 .21092 m .81746 .21717 L s [(1.5)] .81746 .19842 0 1 Mshowa .97619 .21092 m .97619 .21717 L s [(2)] .97619 .19842 0 1 Mshowa .125 Mabswid .05556 .21092 m .05556 .21467 L s .0873 .21092 m .0873 .21467 L s .11905 .21092 m .11905 .21467 L s .15079 .21092 m .15079 .21467 L s .21429 .21092 m .21429 .21467 L s .24603 .21092 m .24603 .21467 L s .27778 .21092 m .27778 .21467 L s .30952 .21092 m .30952 .21467 L s .37302 .21092 m .37302 .21467 L s .40476 .21092 m .40476 .21467 L s .43651 .21092 m .43651 .21467 L s .46825 .21092 m .46825 .21467 L s .53175 .21092 m .53175 .21467 L s .56349 .21092 m .56349 .21467 L s .59524 .21092 m .59524 .21467 L s .62698 .21092 m .62698 .21467 L s .69048 .21092 m .69048 .21467 L s .72222 .21092 m .72222 .21467 L s .75397 .21092 m .75397 .21467 L s .78571 .21092 m .78571 .21467 L s .84921 .21092 m .84921 .21467 L s .88095 .21092 m .88095 .21467 L s .9127 .21092 m .9127 .21467 L s .94444 .21092 m .94444 .21467 L s .25 Mabswid 0 .21092 m 1 .21092 L s gsave 1.025 .21092 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .34127 .01472 m .34752 .01472 L s [(-1)] .32877 .01472 1 0 Mshowa .34127 .11282 m .34752 .11282 L s [(-0.5)] .32877 .11282 1 0 Mshowa .34127 .30902 m .34752 .30902 L s [(0.5)] .32877 .30902 1 0 Mshowa .34127 .40712 m .34752 .40712 L s [(1)] .32877 .40712 1 0 Mshowa .34127 .50522 m .34752 .50522 L s [(1.5)] .32877 .50522 1 0 Mshowa .34127 .60332 m .34752 .60332 L s [(2)] .32877 .60332 1 0 Mshowa .125 Mabswid .34127 .03434 m .34502 .03434 L s .34127 .05396 m .34502 .05396 L s .34127 .07358 m .34502 .07358 L s .34127 .0932 m .34502 .0932 L s .34127 .13244 m .34502 .13244 L s .34127 .15206 m .34502 .15206 L s .34127 .17168 m .34502 .17168 L s .34127 .1913 m .34502 .1913 L s .34127 .23054 m .34502 .23054 L s .34127 .25016 m .34502 .25016 L s .34127 .26978 m .34502 .26978 L s .34127 .2894 m .34502 .2894 L s .34127 .32864 m .34502 .32864 L s .34127 .34826 m .34502 .34826 L s .34127 .36788 m .34502 .36788 L s .34127 .3875 m .34502 .3875 L s .34127 .42674 m .34502 .42674 L s .34127 .44636 m .34502 .44636 L s .34127 .46598 m .34502 .46598 L s .34127 .4856 m .34502 .4856 L s .34127 .52484 m .34502 .52484 L s .34127 .54446 m .34502 .54446 L s .34127 .56408 m .34502 .56408 L s .34127 .5837 m .34502 .5837 L s .25 Mabswid .34127 0 m .34127 .61803 L s gsave .34127 .64303 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 69.000 13.000 moveto (@) show 75.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 81.000 13.000 moveto (D) show 87.000 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 0 0 1 r .01 w .02381 .60332 m .06244 .57944 L .10458 .5534 L .14415 .52894 L .18221 .50542 L .22272 .48039 L .26171 .45629 L .30316 .43067 L .34309 .406 L .3815 .38225 L .42237 .357 L .46172 .33268 L .49955 .30929 L .53984 .2844 L .57861 .26043 L .61984 .23495 L .65954 .21041 L .69774 .18681 L .73838 .16169 L .77751 .13751 L .81909 .11181 L .85916 .08704 L .89771 .06322 L .93871 .03788 L .97619 .01472 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "Use what you know about areas inside triangles to calculate the value of \n\ ", Cell[BoxData[ \(\[Integral]\_\(-1\)\%2 f[x] \[DifferentialD]x\)]], ". \nCheck with ", StyleBox["Mathematica", FontSlant->"Italic"], ". " }], "Text"], Cell["G.3.c)", "Subsubsection"], Cell[TextData[{ "Look at this plot of ", Cell[BoxData[ \(f[x] = \@\(16 - x\^2\)\)]], " for ", Cell[BoxData[ \(0 \[LessEqual] x \[LessEqual] 4\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(\(f[x_] = \@\(16 - x\^2\);\)\n\), "\n", \(\(Plot[f[x], {x, 0, 4}, \n\t PlotStyle \[Rule] {{Thickness[0.01], Blue}}, AspectRatio \[Rule] Automatic, PlotRange \[Rule] All, AxesLabel \[Rule] {"\", "\"}];\)\)}], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .99971 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 0.238095 0.0235277 0.238095 [ [.2619 .01103 -3 -9 ] [.2619 .01103 3 0 ] [.5 .01103 -3 -9 ] [.5 .01103 3 0 ] [.7381 .01103 -3 -9 ] [.7381 .01103 3 0 ] [.97619 .01103 -3 -9 ] [.97619 .01103 3 0 ] [1.025 .02353 0 -6 ] [1.025 .02353 10 6 ] [.01131 .26162 -6 -4.5 ] [.01131 .26162 0 4.5 ] [.01131 .49972 -6 -4.5 ] [.01131 .49972 0 4.5 ] [.01131 .73781 -6 -4.5 ] [.01131 .73781 0 4.5 ] [.01131 .97591 -6 -4.5 ] [.01131 .97591 0 4.5 ] [.02381 1.02471 -14 0 ] [.02381 1.02471 14 12 ] [ 0 0 0 0 ] [ 1 .99971 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .2619 .02353 m .2619 .02978 L s [(1)] .2619 .01103 0 1 Mshowa .5 .02353 m .5 .02978 L s [(2)] .5 .01103 0 1 Mshowa .7381 .02353 m .7381 .02978 L s [(3)] .7381 .01103 0 1 Mshowa .97619 .02353 m .97619 .02978 L s [(4)] .97619 .01103 0 1 Mshowa .125 Mabswid .07143 .02353 m .07143 .02728 L s .11905 .02353 m .11905 .02728 L s .16667 .02353 m .16667 .02728 L s .21429 .02353 m .21429 .02728 L s .30952 .02353 m .30952 .02728 L s .35714 .02353 m .35714 .02728 L s .40476 .02353 m .40476 .02728 L s .45238 .02353 m .45238 .02728 L s .54762 .02353 m .54762 .02728 L s .59524 .02353 m .59524 .02728 L s .64286 .02353 m .64286 .02728 L s .69048 .02353 m .69048 .02728 L s .78571 .02353 m .78571 .02728 L s .83333 .02353 m .83333 .02728 L s .88095 .02353 m .88095 .02728 L s .92857 .02353 m .92857 .02728 L s .25 Mabswid 0 .02353 m 1 .02353 L s gsave 1.025 .02353 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .02381 .26162 m .03006 .26162 L s [(1)] .01131 .26162 1 0 Mshowa .02381 .49972 m .03006 .49972 L s [(2)] .01131 .49972 1 0 Mshowa .02381 .73781 m .03006 .73781 L s [(3)] .01131 .73781 1 0 Mshowa .02381 .97591 m .03006 .97591 L s [(4)] .01131 .97591 1 0 Mshowa .125 Mabswid .02381 .07115 m .02756 .07115 L s .02381 .11877 m .02756 .11877 L s .02381 .16638 m .02756 .16638 L s .02381 .214 m .02756 .214 L s .02381 .30924 m .02756 .30924 L s .02381 .35686 m .02756 .35686 L s .02381 .40448 m .02756 .40448 L s .02381 .4521 m .02756 .4521 L s .02381 .54734 m .02756 .54734 L s .02381 .59496 m .02756 .59496 L s .02381 .64258 m .02756 .64258 L s .02381 .69019 m .02756 .69019 L s .02381 .78543 m .02756 .78543 L s .02381 .83305 m .02756 .83305 L s .02381 .88067 m .02756 .88067 L s .02381 .92829 m .02756 .92829 L s .25 Mabswid .02381 0 m .02381 .99971 L s gsave .02381 1.02471 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 69.000 13.000 moveto (@) show 75.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 81.000 13.000 moveto (D) show 87.000 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 0 0 1 r .01 w .02381 .97591 m .02499 .97591 L .02605 .97591 L .02729 .9759 L .02846 .9759 L .03053 .97589 L .03279 .97587 L .03527 .97584 L .0379 .9758 L .04262 .97572 L .04749 .97561 L .05205 .97549 L .06244 .97512 L .07305 .97464 L .08274 .97408 L .10458 .97248 L .12357 .97067 L .14429 .96826 L .18493 .96218 L .22406 .95462 L .26565 .94469 L .30571 .93323 L .34426 .92038 L .38527 .90465 L .42475 .8874 L .46273 .86874 L .50315 .84649 L .54206 .82256 L .58342 .79416 L .62326 .76358 L .66159 .73082 L .70238 .69179 L .74164 .64942 L .77939 .60329 L .8196 .54674 L .85828 .48253 L .89942 .39813 L .91995 .34596 L .93905 .28691 L .94829 .25238 L .95701 .21372 L .96183 .1883 L .96706 .15511 L .96937 .13734 L .97181 .11478 L .9739 .0896 L .97508 .06948 L .97619 .0238 L s 0 0 m 1 0 L 1 .99971 L 0 .99971 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 287.875}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "Use what you know about areas inside circles to calculate the value of \n \ ", Cell[BoxData[ \(\[Integral]\_0\%4\(\@\( 16 - x\^2\)\) \[DifferentialD]x\)]], ". \nCheck with ", StyleBox["Mathematica", FontSlant->"Italic"], ". " }], "Text"], Cell["G.3.d)", "Subsubsection"], Cell[TextData[{ "Look at this plot of ", Cell[BoxData[ \(f[x] = x \@\( 1 - x\^2\)\)]], " for ", Cell[BoxData[ \(\(-1\) \[LessEqual] x \[LessEqual] 1\)]], ": " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(\(f[x_] = x\ \@\(1 - x\^2\);\)\n\), "\n", \(\(Plot[f[x], {x, \(-1\), 1}, \n\t PlotStyle \[Rule] {{Thickness[0.01], DarkGreen}}, AspectRatio \[Rule] 1\/GoldenRatio, AxesLabel \[Rule] {"\", "\"}];\)\)}], "Input", AspectRatioFixed->True], 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 0.5 0.476191 0.309018 0.588605 [ [.02381 .29652 -6 -9 ] [.02381 .29652 6 0 ] [.2619 .29652 -12 -9 ] [.2619 .29652 12 0 ] [.7381 .29652 -9 -9 ] [.7381 .29652 9 0 ] [.97619 .29652 -3 -9 ] [.97619 .29652 3 0 ] [1.025 .30902 0 -6 ] [1.025 .30902 10 6 ] [.4875 .07358 -24 -4.5 ] [.4875 .07358 0 4.5 ] [.4875 .1913 -24 -4.5 ] [.4875 .1913 0 4.5 ] [.4875 .42674 -18 -4.5 ] [.4875 .42674 0 4.5 ] [.4875 .54446 -18 -4.5 ] [.4875 .54446 0 4.5 ] [.5 .64303 -14 0 ] [.5 .64303 14 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .30902 m .02381 .31527 L s [(-1)] .02381 .29652 0 1 Mshowa .2619 .30902 m .2619 .31527 L s [(-0.5)] .2619 .29652 0 1 Mshowa .7381 .30902 m .7381 .31527 L s [(0.5)] .7381 .29652 0 1 Mshowa .97619 .30902 m .97619 .31527 L s [(1)] .97619 .29652 0 1 Mshowa .125 Mabswid .07143 .30902 m .07143 .31277 L s .11905 .30902 m .11905 .31277 L s .16667 .30902 m .16667 .31277 L s .21429 .30902 m .21429 .31277 L s .30952 .30902 m .30952 .31277 L s .35714 .30902 m .35714 .31277 L s .40476 .30902 m .40476 .31277 L s .45238 .30902 m .45238 .31277 L s .54762 .30902 m .54762 .31277 L s .59524 .30902 m .59524 .31277 L s .64286 .30902 m .64286 .31277 L s .69048 .30902 m .69048 .31277 L s .78571 .30902 m .78571 .31277 L s .83333 .30902 m .83333 .31277 L s .88095 .30902 m .88095 .31277 L s .92857 .30902 m .92857 .31277 L s .25 Mabswid 0 .30902 m 1 .30902 L s gsave 1.025 .30902 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .5 .07358 m .50625 .07358 L s [(-0.4)] .4875 .07358 1 0 Mshowa .5 .1913 m .50625 .1913 L s [(-0.2)] .4875 .1913 1 0 Mshowa .5 .42674 m .50625 .42674 L s [(0.2)] .4875 .42674 1 0 Mshowa .5 .54446 m .50625 .54446 L s [(0.4)] .4875 .54446 1 0 Mshowa .125 Mabswid .5 .10301 m .50375 .10301 L s .5 .13244 m .50375 .13244 L s .5 .16187 m .50375 .16187 L s .5 .22073 m .50375 .22073 L s .5 .25016 m .50375 .25016 L s .5 .27959 m .50375 .27959 L s .5 .33845 m .50375 .33845 L s .5 .36788 m .50375 .36788 L s .5 .39731 m .50375 .39731 L s .5 .45617 m .50375 .45617 L s .5 .4856 m .50375 .4856 L s .5 .51503 m .50375 .51503 L s .5 .04415 m .50375 .04415 L s .5 .01471 m .50375 .01471 L s .5 .57389 m .50375 .57389 L s .5 .60332 m .50375 .60332 L s .25 Mabswid .5 0 m .5 .61803 L s gsave .5 .64303 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 69.000 13.000 moveto (@) show 75.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 81.000 13.000 moveto (D) show 87.000 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 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 .392 0 r .01 w .02381 .30878 m .02499 .26778 L .02605 .2522 L .02729 .23851 L .02846 .22779 L .03279 .19742 L .0375 .17292 L .04262 .1517 L .05293 .11874 L .06244 .09561 L .0829 .06027 L .09322 .04762 L .10458 .03668 L .11472 .02914 L .12399 .02383 L .12911 .02149 L .13469 .01936 L .13958 .01786 L .14487 .01658 L .14763 .01605 L .15018 .01564 L .15257 .01533 L .15519 .01506 L .15632 .01497 L .15754 .01489 L .15859 .01483 L .15973 .01478 L .16103 .01474 L .16225 .01472 L .16344 .01472 L .16454 .01472 L .16574 .01475 L .16703 .01479 L .16825 .01484 L .16937 .0149 L .17197 .0151 L .17471 .01537 L .17987 .01608 L .18551 .01712 L .19579 .01973 L .20512 .02282 L .22607 .03206 L .24526 .04298 L .26606 .05715 L .30697 .09091 L .34638 .12928 L .38426 .17025 L .4246 .217 L .46343 .26395 L .50471 .31483 L Mistroke .54447 .36374 L .58271 .4097 L .62341 .45635 L .66259 .49791 L .70423 .53706 L .74434 .56825 L .76276 .57989 L .78295 .59032 L .7926 .59436 L .80292 .59792 L .80863 .59954 L .81383 .60077 L .81877 .60173 L .82147 .60216 L .824 .60251 L .82647 .60279 L .82873 .603 L .83124 .60317 L .83256 .60323 L .83328 .60326 L .83394 .60328 L .83504 .60331 L .83622 .60332 L .8373 .60332 L .83833 .60331 L .83953 .60328 L .84063 .60324 L .84184 .60318 L .84313 .6031 L .84559 .6029 L .84824 .60261 L .85302 .60187 L .85871 .60063 L .8639 .59914 L .8738 .59526 L .88296 .59036 L .89302 .58332 L .90371 .57366 L .91307 .56305 L .92316 .5489 L .93188 .53389 L .94143 .51366 L .9502 .49036 L .95836 .46262 L .96279 .44378 L .96698 .42202 L .96929 .40741 L .9718 .38801 L .97291 .37752 L .97408 .36407 L Mistroke .97509 .34898 L .97619 .30926 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "Use this plot to read off the value of\n ", Cell[BoxData[ \(\[Integral]\_\(-1\)\%1 x \(\@\( 1 - x\^2\)\) \[DifferentialD]x\)]], "." }], "Text"], Cell["G.3.e)", "Subsubsection"], Cell[TextData[{ "Look at this plot of ", Cell[BoxData[ \(f[x] = x\)]], " for ", Cell[BoxData[ \(\(-1\) \[LessEqual] x \[LessEqual] 1\)]], ". " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(\(f[x_] = x;\)\n\), "\n", \(\(Plot[f[x], {x, \(-1\), 1}, \n\t PlotStyle \[Rule] {{Thickness[0.01], Blue}}, AspectRatio \[Rule] Automatic, PlotRange \[Rule] All, AxesLabel \[Rule] {"\", "\"}];\)\)}], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.476191 0.5 0.476191 [ [.02381 .4875 -6 -9 ] [.02381 .4875 6 0 ] [.2619 .4875 -12 -9 ] [.2619 .4875 12 0 ] [.7381 .4875 -9 -9 ] [.7381 .4875 9 0 ] [.97619 .4875 -3 -9 ] [.97619 .4875 3 0 ] [1.025 .5 0 -6 ] [1.025 .5 10 6 ] [.4875 .02381 -12 -4.5 ] [.4875 .02381 0 4.5 ] [.4875 .2619 -24 -4.5 ] [.4875 .2619 0 4.5 ] [.4875 .7381 -18 -4.5 ] [.4875 .7381 0 4.5 ] [.4875 .97619 -6 -4.5 ] [.4875 .97619 0 4.5 ] [.5 1.025 -14 0 ] [.5 1.025 14 12 ] [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .5 m .02381 .50625 L s [(-1)] .02381 .4875 0 1 Mshowa .2619 .5 m .2619 .50625 L s [(-0.5)] .2619 .4875 0 1 Mshowa .7381 .5 m .7381 .50625 L s [(0.5)] .7381 .4875 0 1 Mshowa .97619 .5 m .97619 .50625 L s [(1)] .97619 .4875 0 1 Mshowa .125 Mabswid .07143 .5 m .07143 .50375 L s .11905 .5 m .11905 .50375 L s .16667 .5 m .16667 .50375 L s .21429 .5 m .21429 .50375 L s .30952 .5 m .30952 .50375 L s .35714 .5 m .35714 .50375 L s .40476 .5 m .40476 .50375 L s .45238 .5 m .45238 .50375 L s .54762 .5 m .54762 .50375 L s .59524 .5 m .59524 .50375 L s .64286 .5 m .64286 .50375 L s .69048 .5 m .69048 .50375 L s .78571 .5 m .78571 .50375 L s .83333 .5 m .83333 .50375 L s .88095 .5 m .88095 .50375 L s .92857 .5 m .92857 .50375 L s .25 Mabswid 0 .5 m 1 .5 L s gsave 1.025 .5 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .5 .02381 m .50625 .02381 L s [(-1)] .4875 .02381 1 0 Mshowa .5 .2619 m .50625 .2619 L s [(-0.5)] .4875 .2619 1 0 Mshowa .5 .7381 m .50625 .7381 L s [(0.5)] .4875 .7381 1 0 Mshowa .5 .97619 m .50625 .97619 L s [(1)] .4875 .97619 1 0 Mshowa .125 Mabswid .5 .07143 m .50375 .07143 L s .5 .11905 m .50375 .11905 L s .5 .16667 m .50375 .16667 L s .5 .21429 m .50375 .21429 L s .5 .30952 m .50375 .30952 L s .5 .35714 m .50375 .35714 L s .5 .40476 m .50375 .40476 L s .5 .45238 m .50375 .45238 L s .5 .54762 m .50375 .54762 L s .5 .59524 m .50375 .59524 L s .5 .64286 m .50375 .64286 L s .5 .69048 m .50375 .69048 L s .5 .78571 m .50375 .78571 L s .5 .83333 m .50375 .83333 L s .5 .88095 m .50375 .88095 L s .5 .92857 m .50375 .92857 L s .25 Mabswid .5 0 m .5 1 L s gsave .5 1.025 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 69.000 13.000 moveto (@) show 75.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 81.000 13.000 moveto (D) show 87.000 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 0 0 1 r .01 w .02381 .02381 m .06244 .06244 L .10458 .10458 L .14415 .14415 L .18221 .18221 L .22272 .22272 L .26171 .26171 L .30316 .30316 L .34309 .34309 L .3815 .3815 L .42237 .42237 L .46172 .46172 L .49955 .49955 L .53984 .53984 L .57861 .57861 L .61984 .61984 L .65954 .65954 L .69774 .69774 L .73838 .73838 L .77751 .77751 L .81909 .81909 L .85916 .85916 L .89771 .89771 L .93871 .93871 L .97619 .97619 L s 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 288}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "Look at this plot of ", Cell[BoxData[ \(g[x] = \(\(\(|\)\(x\)\)\(|\)\)\)]], " for ", Cell[BoxData[ \(\(-1\) \[LessEqual] x \[LessEqual] 1\)]], ". " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[g, x];\)\), "\n", \(\(g[x_] = Abs[x];\)\n\), "\n", \(\(Plot[g[x], {x, \(-1\), 1}, \n\t PlotStyle \[Rule] {{Thickness[0.01], Blue}}, AspectRatio \[Rule] Automatic, PlotRange \[Rule] All, AxesLabel \[Rule] {"\", "\"}];\)\)}], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .49953 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.476191 0.0114451 0.476191 [ [.02381 -0.00105 -6 -9 ] [.02381 -0.00105 6 0 ] [.2619 -0.00105 -12 -9 ] [.2619 -0.00105 12 0 ] [.7381 -0.00105 -9 -9 ] [.7381 -0.00105 9 0 ] [.97619 -0.00105 -3 -9 ] [.97619 -0.00105 3 0 ] [1.025 .01145 0 -6 ] [1.025 .01145 10 6 ] [.4875 .10668 -18 -4.5 ] [.4875 .10668 0 4.5 ] [.4875 .20192 -18 -4.5 ] [.4875 .20192 0 4.5 ] [.4875 .29716 -18 -4.5 ] [.4875 .29716 0 4.5 ] [.4875 .3924 -18 -4.5 ] [.4875 .3924 0 4.5 ] [.4875 .48764 -6 -4.5 ] [.4875 .48764 0 4.5 ] [.5 .52453 -14 0 ] [.5 .52453 14 12 ] [ 0 0 0 0 ] [ 1 .49953 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .01145 m .02381 .0177 L s [(-1)] .02381 -0.00105 0 1 Mshowa .2619 .01145 m .2619 .0177 L s [(-0.5)] .2619 -0.00105 0 1 Mshowa .7381 .01145 m .7381 .0177 L s [(0.5)] .7381 -0.00105 0 1 Mshowa .97619 .01145 m .97619 .0177 L s [(1)] .97619 -0.00105 0 1 Mshowa .125 Mabswid .07143 .01145 m .07143 .0152 L s .11905 .01145 m .11905 .0152 L s .16667 .01145 m .16667 .0152 L s .21429 .01145 m .21429 .0152 L s .30952 .01145 m .30952 .0152 L s .35714 .01145 m .35714 .0152 L s .40476 .01145 m .40476 .0152 L s .45238 .01145 m .45238 .0152 L s .54762 .01145 m .54762 .0152 L s .59524 .01145 m .59524 .0152 L s .64286 .01145 m .64286 .0152 L s .69048 .01145 m .69048 .0152 L s .78571 .01145 m .78571 .0152 L s .83333 .01145 m .83333 .0152 L s .88095 .01145 m .88095 .0152 L s .92857 .01145 m .92857 .0152 L s .25 Mabswid 0 .01145 m 1 .01145 L s gsave 1.025 .01145 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .5 .10668 m .50625 .10668 L s [(0.2)] .4875 .10668 1 0 Mshowa .5 .20192 m .50625 .20192 L s [(0.4)] .4875 .20192 1 0 Mshowa .5 .29716 m .50625 .29716 L s [(0.6)] .4875 .29716 1 0 Mshowa .5 .3924 m .50625 .3924 L s [(0.8)] .4875 .3924 1 0 Mshowa .5 .48764 m .50625 .48764 L s [(1)] .4875 .48764 1 0 Mshowa .125 Mabswid .5 .03525 m .50375 .03525 L s .5 .05906 m .50375 .05906 L s .5 .08287 m .50375 .08287 L s .5 .13049 m .50375 .13049 L s .5 .1543 m .50375 .1543 L s .5 .17811 m .50375 .17811 L s .5 .22573 m .50375 .22573 L s .5 .24954 m .50375 .24954 L s .5 .27335 m .50375 .27335 L s .5 .32097 m .50375 .32097 L s .5 .34478 m .50375 .34478 L s .5 .36859 m .50375 .36859 L s .5 .41621 m .50375 .41621 L s .5 .44002 m .50375 .44002 L s .5 .46383 m .50375 .46383 L s .25 Mabswid .5 0 m .5 .49953 L s gsave .5 .52453 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 69.000 13.000 moveto (@) show 75.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 81.000 13.000 moveto (D) show 87.000 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 0 0 1 r .01 w .02381 .48764 m .06244 .449 L .10458 .40687 L .14415 .36729 L .18221 .32924 L .22272 .28873 L .26171 .24974 L .30316 .20829 L .34309 .16836 L .3815 .12995 L .42237 .08908 L .46172 .04973 L .48147 .02997 L .49012 .02133 L .49468 .01676 L .49719 .01425 L .49842 .01303 L .49955 .01189 L .50085 .0123 L .50154 .01298 L .50226 .0137 L .50471 .01616 L .5095 .02095 L .51896 .03041 L .53984 .05128 L .57781 .08926 L .61824 .12968 L .65714 .16859 L .6985 .20995 L .73835 .2498 L .77668 .28812 L .81746 .32891 L .85673 .36817 L .89448 .40592 L .93468 .44613 L .97337 .48481 L .97619 .48764 L s 0 0 m 1 0 L 1 .49953 L 0 .49953 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 143.812}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "Use whatever methods you like to calculate the values of\n ", Cell[BoxData[ \(\(\(\(|\)\(\[Integral]\_\(-1\)\%1 x \[DifferentialD]x\)\)\(|\)\)\)]], " and ", Cell[BoxData[ \(\[Integral]\_\(-1\)\%1\(\(\(|\)\(x\)\) | \[DifferentialD]x\)\)]], "\nTry to explain the results in terms of the plots." }], "Text"] }, Closed]], Cell["Literacy Problem", "Subsubsection"], Cell[TextData[{ "Here's a plot of ", Cell[BoxData[ \(f[x] = 2 x\ e\^\(0.5\ x\)\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x];\)\), "\n", \(\(Plot[x\ E\^\(x/2\), {x, \(-3\), 3}, PlotStyle \[Rule] Thickness[0.01], PlotRange \[Rule] All, AxesLabel \[Rule] {"\", "\"}];\)\)}], "Input", CellOpen->False, AspectRatioFixed->True], 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 0.5 0.15873 0.0452543 0.041507 [ [.02381 .03275 -6 -9 ] [.02381 .03275 6 0 ] [.18254 .03275 -6 -9 ] [.18254 .03275 6 0 ] [.34127 .03275 -6 -9 ] [.34127 .03275 6 0 ] [.65873 .03275 -3 -9 ] [.65873 .03275 3 0 ] [.81746 .03275 -3 -9 ] [.81746 .03275 3 0 ] [.97619 .03275 -3 -9 ] [.97619 .03275 3 0 ] [1.025 .04525 0 -6 ] [1.025 .04525 10 6 ] [.4875 .12827 -6 -4.5 ] [.4875 .12827 0 4.5 ] [.4875 .21128 -6 -4.5 ] [.4875 .21128 0 4.5 ] [.4875 .2943 -6 -4.5 ] [.4875 .2943 0 4.5 ] [.4875 .37731 -6 -4.5 ] [.4875 .37731 0 4.5 ] [.4875 .46032 -12 -4.5 ] [.4875 .46032 0 4.5 ] [.4875 .54334 -12 -4.5 ] [.4875 .54334 0 4.5 ] [.5 .64303 -14 0 ] [.5 .64303 14 12 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .04525 m .02381 .0515 L s [(-3)] .02381 .03275 0 1 Mshowa .18254 .04525 m .18254 .0515 L s [(-2)] .18254 .03275 0 1 Mshowa .34127 .04525 m .34127 .0515 L s [(-1)] .34127 .03275 0 1 Mshowa .65873 .04525 m .65873 .0515 L s [(1)] .65873 .03275 0 1 Mshowa .81746 .04525 m .81746 .0515 L s [(2)] .81746 .03275 0 1 Mshowa .97619 .04525 m .97619 .0515 L s [(3)] .97619 .03275 0 1 Mshowa .125 Mabswid .05556 .04525 m .05556 .049 L s .0873 .04525 m .0873 .049 L s .11905 .04525 m .11905 .049 L s .15079 .04525 m .15079 .049 L s .21429 .04525 m .21429 .049 L s .24603 .04525 m .24603 .049 L s .27778 .04525 m .27778 .049 L s .30952 .04525 m .30952 .049 L s .37302 .04525 m .37302 .049 L s .40476 .04525 m .40476 .049 L s .43651 .04525 m .43651 .049 L s .46825 .04525 m .46825 .049 L s .53175 .04525 m .53175 .049 L s .56349 .04525 m .56349 .049 L s .59524 .04525 m .59524 .049 L s .62698 .04525 m .62698 .049 L s .69048 .04525 m .69048 .049 L s .72222 .04525 m .72222 .049 L s .75397 .04525 m .75397 .049 L s .78571 .04525 m .78571 .049 L s .84921 .04525 m .84921 .049 L s .88095 .04525 m .88095 .049 L s .9127 .04525 m .9127 .049 L s .94444 .04525 m .94444 .049 L s .25 Mabswid 0 .04525 m 1 .04525 L s gsave 1.025 .04525 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (x) 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 .5 .12827 m .50625 .12827 L s [(2)] .4875 .12827 1 0 Mshowa .5 .21128 m .50625 .21128 L s [(4)] .4875 .21128 1 0 Mshowa .5 .2943 m .50625 .2943 L s [(6)] .4875 .2943 1 0 Mshowa .5 .37731 m .50625 .37731 L s [(8)] .4875 .37731 1 0 Mshowa .5 .46032 m .50625 .46032 L s [(10)] .4875 .46032 1 0 Mshowa .5 .54334 m .50625 .54334 L s [(12)] .4875 .54334 1 0 Mshowa .125 Mabswid .5 .06601 m .50375 .06601 L s .5 .08676 m .50375 .08676 L s .5 .10751 m .50375 .10751 L s .5 .14902 m .50375 .14902 L s .5 .16978 m .50375 .16978 L s .5 .19053 m .50375 .19053 L s .5 .23204 m .50375 .23204 L s .5 .25279 m .50375 .25279 L s .5 .27354 m .50375 .27354 L s .5 .31505 m .50375 .31505 L s .5 .3358 m .50375 .3358 L s .5 .35656 m .50375 .35656 L s .5 .39806 m .50375 .39806 L s .5 .41882 m .50375 .41882 L s .5 .43957 m .50375 .43957 L s .5 .48108 m .50375 .48108 L s .5 .50183 m .50375 .50183 L s .5 .52259 m .50375 .52259 L s .5 .0245 m .50375 .0245 L s .5 .00375 m .50375 .00375 L s .5 .56409 m .50375 .56409 L s .5 .58485 m .50375 .58485 L s .5 .6056 m .50375 .6056 L s .25 Mabswid .5 0 m .5 .61803 L s gsave .5 .64303 -75 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 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 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 (f) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 69.000 13.000 moveto (@) show 75.000 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 (x) show %%IncludeResource: font Math2Mono %%IncludeFont: Math2Mono /Math2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 81.000 13.000 moveto (D) show 87.000 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 .01 w .02381 .01747 m .06244 .01642 L .08255 .01595 L .10458 .0155 L .1253 .01516 L .14415 .01492 L .15421 .01483 L .16372 .01477 L .16626 .01475 L .16894 .01474 L .17127 .01473 L .17376 .01473 L .17516 .01472 L .17644 .01472 L .17774 .01472 L .17895 .01472 L .17963 .01472 L .18035 .01472 L .18115 .01472 L .18188 .01472 L .18257 .01472 L .18331 .01472 L .18401 .01472 L .18466 .01472 L .18592 .01472 L .18712 .01472 L .18819 .01472 L .18936 .01472 L .19198 .01473 L .1944 .01474 L .1988 .01476 L .20349 .01478 L .20855 .01482 L .21331 .01487 L .22415 .015 L .23398 .01516 L .24433 .01537 L .26284 .01587 L .28258 .01659 L .30094 .01745 L .33999 .01998 L .36133 .02183 L .38149 .02392 L .41995 .02899 L .46087 .03621 L .50027 .04533 L .53816 .05651 L .5785 .07154 L .61733 .08965 L .6586 .11361 L Mistroke .69836 .14215 L .73661 .17562 L .77731 .21895 L .81649 .26953 L .85812 .3346 L .89824 .41036 L .93684 .49753 L .97619 .60332 L Mfstroke 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{202.812, 125.375}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgH0 00l0oo3V0`00000G0?o`iPD000005P3ol>H300000300oo3V0`00000G0?o`iPD000005`3ol>H30000 01X0oo3V000>0?o`iP050000003ol>H0oo3V0?o`iP0000005`3ol>H010000000oo3V0?o`iP00000G 0?o`iP030000003ol>H0oo3V0300oo3V00<000000?o`iP3ol>H05`3ol>H010000000oo3V0?o`iP00 000F0?o`iP050000003ol>H0oo3V0?o`iP0000006@3ol>H000P0oo3V1@0000050?o`iP030000003o l>H0oo3V00h0oo3V1@0000030?o`iP030000003ol>H0oo3V00l0oo3V1@0000030?o`iP030000003o l>H0oo3V0300oo3V00<000000?o`iP3ol>H0603ol>H00`000000oo3V0?o`iP0J0?o`iP030000003o l>H0oo3V01L0oo3V000@0?o`iP8000006P3ol>H00`000000oo3V0?o`iP0F0?o`iP030000003ol>H0 oo3V01H0oo3V0P00000H0?o`iP030000003ol>H0oo3V01T0oo3V00<000000?o`iP3ol>H05`3ol>H2 000001X0oo3V000B0?o`iP030000003ol>H0oo3V01@0oo3V00D000000?o`iP3ol>H0oo3V0000000G 0?o`iP030000003ol>H0oo3V01H0oo3V00<000000?o`iP3ol>H05`3ol>H00`000000oo3V0?o`iP0F 0?o`iP050000003ol>H0oo3V0?o`iP0000006P3ol>H00`000000oo3V0?o`iP0G0?o`iP003P3ol>H0 0`000000oo3V0?o`iP0[000000P0oo3V0P00000H0?o`iP030000003ol>H0oo3V01H0oo3V0P00000H 0?o`iP050000003ol>H0oo3V0?o`iP0000005P3ol>H01@000000oo3V0?o`iP3ol>H0000001T0oo3V 000>0?o`iP@00000603ol>H3000000l0oo3V3P00000D0?o`iP030000003ol>H0oo3V01L0oo3V00<0 00000?o`iP3ol>H05`3ol>H3000001P0oo3V0`00000J0?o`iP00BP3ol>H:000000X0oo3V0P00001Z 0?o`iP00E03ol>H3000000L0oo3V00<000000?o`iP3ol>H0F03ol>H2000000030?o`iP0000000000 00`0oo3V001G0?o`iPD000000P3ol>H00`000000oo3V0?o`iP1I0?o`iP030000003ol>H0000000d0 oo3V000:0?o`iZX000001`3ol>H00`000000oo3V0?o`iP0<0?o`iP003P3ol>H00`000000oo3V0?o` iP020?o`iP030000003ol>H0oo3V0080oo3V00<000000?o`iP3ol>H00`3ol>H00`000000oo3V0?o` iP020?o`iP030000003ol>H0oo3V00<0oo3V00<000000?o`iP3ol>H00P3ol>H00`000000oo3V0?o` iP020?o`iP030000003ol>H0oo3V00<0oo3V00<000000?o`iP3ol>H00P3ol>H00`000000oo3V0?o` iP020?o`iP030000003ol>H0oo3V00<0oo3V00<000000?o`iP3ol>H00P3ol>H00`000000oo3V0?o` iP030?o`iP030000003ol>H0oo3V0080oo3V00<000000?o`iP3ol>H00P3ol>H00`000000oo3V0?o` iP04000000@0oo3V00<000000?o`iP3ol>H00P3ol>H00`000000oo3V0?o`iP030?o`iP030000003o l>H0oo3V0080oo3V00<000000?o`iP3ol>H00`3ol>H00`000000oo3V0?o`iP020?o`iP030000003o l>H0oo3V0080oo3V00<000000?o`iP3ol>H00`3ol>H00`000000oo3V0?o`iP020?o`iP030000003o l>H0oo3V0080oo3V00<000000?o`iP3ol>H00`3ol>H00`000000oo3V0?o`iP020?o`iP030000003o l>H0oo3V00<0oo3V00<000000?o`iP3ol>H00P3ol>H00`000000oo3V0?o`iP080?o`iP030000003o l>H0000000d0oo3V001N0?o`iP030000003ol>H0oo3V00<0oo3V0`00001B0?o`iP80000000<0oo3V 000000000000303ol>H005h0oo3V00<000000?o`iP3ol>H01P3ol>H300000600oo3V001N0?o`iP80 00002P3ol>H2000005h0oo3V001N0?o`iP030000003ol>H0oo3V00/0oo3V0P00001L0?o`iP00GP3o l>H00`000000oo3V0?o`iP0=0?o`iP800000FP3ol>H005h0oo3V0P00000@0?o`iP800000F03ol>H0 05h0oo3V00<000000?o`iP3ol>H04@3ol>H2000005H0oo3V001N0?o`iP030000003ol>H0oo3V01<0 oo3V0P00001D0?o`iP00GP3ol>H00`000000oo3V0?o`iP0E0?o`iP030000003ol>H0oo3V0540oo3V 001N0?o`iP8000005`3ol>H200000540oo3V001F0?o`iPD000000`3ol>H00`000000oo3V0?o`iP0H 0?o`iP800000C`3ol>H005L0oo3V00@000000?o`iP3ol>H000000`3ol>H00`000000oo3V0?o`iP0J 0?o`iP030000003ol>H0oo3V04`0oo3V001H0?o`iP030000003ol>H0oo3V00<0oo3V0P00000L0?o` iP800000C03ol>H005T0oo3V00<000000?o`iP3ol>H00P3ol>H00`000000oo3V0?o`iP0M0?o`iP03 0000003ol>H0oo3V04T0oo3V001F0?o`iP050000003ol>H0oo3V0?o`iP0000000`3ol>H00`000000 oo3V0?o`iP0N0?o`iP030000003ol>H0oo3V04P0oo3V001F0?o`iP050000003ol>H0oo3V0?o`iP00 00000`3ol>H00`000000oo3V0?o`iP0O0?o`iP800000B03ol>H005L0oo3V0`0000040?o`iP800000 8P3ol>H00`000000oo3V0?o`iP150?o`iP00GP3ol>H00`000000oo3V0?o`iP0R0?o`iP030000003o l>H0oo3V04@0oo3V001N0?o`iP030000003ol>H0oo3V02<0oo3V00<000000?o`iP3ol>H0@`3ol>H0 05h0oo3V0P00000U0?o`iP030000003ol>H0oo3V0480oo3V001N0?o`iP030000003ol>H0oo3V02D0 oo3V00<000000?o`iP3ol>H0@@3ol>H005h0oo3V00<000000?o`iP3ol>H09@3ol>H00`000000oo3V 0?o`iP110?o`iP00GP3ol>H00`000000oo3V0?o`iP0V0?o`iP030000003ol>H0oo3V0400oo3V001N 0?o`iP800000:03ol>H00`000000oo3V0?o`iP0o0?o`iP00F03ol>H3000000<0oo3V00<000000?o` iP3ol>H0:03ol>H00`000000oo3V0?o`iP0n0?o`iP00F@3ol>H00`000000oo3V0?o`iP020?o`iP03 0000003ol>H0oo3V02T0oo3V00<000000?o`iP3ol>H0?@3ol>H005H0oo3V1@0000030?o`iP800000 :`3ol>H00`000000oo3V0?o`iP0l0?o`iP00EP3ol>H010000000oo3V0?o`iP0000040?o`iP030000 003ol>H0oo3V02/0oo3V00<000000?o`iP3ol>H0>`3ol>H005L0oo3V00<000000?o`iP000000103o l>H00`000000oo3V0?o`iP0/0?o`iP030000003ol>H0oo3V03X0oo3V001H0?o`iP800000103ol>H0 0`000000oo3V0?o`iP0/0?o`iP030000003ol>H0oo3V03X0oo3V001I0?o`iP030000003ol>H0oo3V 0080oo3V0P00000^0?o`iP030000003ol>H0oo3V03T0oo3V001N0?o`iP030000003ol>H0oo3V02h0 oo3V00<000000?o`iP3ol>H0>03ol>H005h0oo3V00<000000?o`iP3ol>H0;`3ol>H00`000000oo3V 0?o`iP0g0?o`iP00GP3ol>H200000340oo3V00<000000?o`iP3ol>H0=P3ol>H005h0oo3V00<00000 0?o`iP3ol>H0<@3ol>H00`000000oo3V0?o`iP0e0?o`iP00GP3ol>H00`000000oo3V0?o`iP0b0?o` iP030000003ol>H0oo3V03@0oo3V001N0?o`iP030000003ol>H0oo3V03<0oo3V00<000000?o`iP3o l>H0<`3ol>H005h0oo3V0P00000d0?o`iP030000003ol>H0oo3V03<0oo3V001G0?o`iP<00000103o l>H00`000000oo3V0?o`iP0d0?o`iP030000003ol>H0oo3V0380oo3V001F0?o`iP050000003ol>H0 oo3V0?o`iP0000000`3ol>H00`000000oo3V0?o`iP0e0?o`iP030000003ol>H0oo3V0340oo3V001F 0?o`iP050000003ol>H0oo3V0?o`iP0000000`3ol>H2000003H0oo3V00<000000?o`iP3ol>H0<@3o l>H005H0oo3V100000040?o`iP030000003ol>H0oo3V03H0oo3V00<000000?o`iP3ol>H0<03ol>H0 05H0oo3V00<000000?o`iP3ol>H01@3ol>H00`000000oo3V0?o`iP0g0?o`iP030000003ol>H0oo3V 02l0oo3V001G0?o`iP030000003ol>H0oo3V00@0oo3V00<000000?o`iP3ol>H0=`3ol>H00`000000 oo3V0?o`iP0_0?o`iP00F03ol>H2000000@0oo3V0P00000i0?o`iP030000003ol>H0oo3V02h0oo3V 001N0?o`iP030000003ol>H0oo3V03P0oo3V00<000000?o`iP3ol>H0;P3ol>H005h0oo3V00<00000 0?o`iP3ol>H0>@3ol>H00`000000oo3V0?o`iP0]0?o`iP00GP3ol>H2000003/0oo3V00<000000?o` iP3ol>H0;03ol>H005h0oo3V00<000000?o`iP3ol>H0>P3ol>H00`000000oo3V0?o`iP0/0?o`iP00 GP3ol>H00`000000oo3V0?o`iP0k0?o`iP030000003ol>H0oo3V02/0oo3V001N0?o`iP030000003o l>H0oo3V03/0oo3V00<000000?o`iP3ol>H0:`3ol>H005h0oo3V0P00000m0?o`iP030000003ol>H0 oo3V02X0oo3V001G0?o`iP<00000103ol>H00`000000oo3V0?o`iP0l0?o`iP030000003ol>H0oo3V 02X0oo3V001F0?o`iP050000003ol>H0oo3V0?o`iP0000000`3ol>H00`000000oo3V0?o`iP0m0?o` iP030000003ol>H0oo3V02T0oo3V001F0?o`iP050000003ol>H0oo3V0?o`iP0000000`3ol>H20000 03l0oo3V00<000000?o`iP3ol>H0:03ol>H005L0oo3V0`0000040?o`iP030000003ol>H0oo3V03h0 oo3V00<000000?o`iP3ol>H0:03ol>H005H0oo3V00D000000?o`iP3ol>H0oo3V000000030?o`iP03 0000003ol>H0oo3V03l0oo3V00<000000?o`iP3ol>H09`3ol>H005H0oo3V00D000000?o`iP3ol>H0 oo3V000000030?o`iP030000003ol>H0oo3V03l0oo3V00<000000?o`iP3ol>H09`3ol>H005L0oo3V 0`0000040?o`iP800000@@3ol>H00`000000oo3V0?o`iP0V0?o`iP00GP3ol>H00`000000oo3V0?o` iP100?o`iP030000003ol>H0oo3V02H0oo3V001N0?o`iP030000003ol>H0oo3V0440oo3V00<00000 0?o`iP3ol>H09@3ol>H005h0oo3V00<000000?o`iP3ol>H0@@3ol>H00`000000oo3V0?o`iP0U0?o` iP00GP3ol>H2000004<0oo3V00<000000?o`iP3ol>H0903ol>H005h0oo3V00<000000?o`iP3ol>H0 @P3ol>H00`000000oo3V0?o`iP0T0?o`iP00GP3ol>H00`000000oo3V0?o`iP120?o`iP030000003o l>H0oo3V02@0oo3V001N0?o`iP800000A03ol>H00`000000oo3V0?o`iP0S0?o`iP00D@3ol>H30000 00@0oo3V0P0000040?o`iP030000003ol>H0oo3V04<0oo3V00<000000?o`iP3ol>H08`3ol>H00580 oo3V00<000000?o`iP3ol>H00P3ol>H010000000oo3V0?o`iP0000030?o`iP030000003ol>H0oo3V 04@0oo3V00<000000?o`iP3ol>H08P3ol>H00580oo3V00<000000?o`iP3ol>H00P3ol>H010000000 oo3V0?o`iP0000030?o`iP030000003ol>H0oo3V04@0oo3V00<000000?o`iP3ol>H08P3ol>H00580 oo3V00<000000?o`iP3ol>H00P3ol>H010000000oo3V0?o`iP0000030?o`iP800000A@3ol>H00`00 0000oo3V0?o`iP0R0?o`iP00DP3ol>H00`000000oo3V0?o`iP020?o`iP040000003ol>H0oo3V0000 00<0oo3V00<000000?o`iP3ol>H0A@3ol>H00`000000oo3V0?o`iP0Q0?o`iP00D@3ol>H2000000@0 oo3V00@000000?o`iP3ol>H000000`3ol>H00`000000oo3V0?o`iP150?o`iP030000003ol>H0oo3V 0240oo3V001B0?o`iP030000003ol>H0oo3V00<0oo3V0P0000040?o`iP800000A`3ol>H00`000000 oo3V0?o`iP0P0?o`iP00GP3ol>H00`000000oo3V0?o`iP160?o`iP030000003ol>H0oo3V0200oo3V 001N0?o`iP030000003ol>H0oo3V04H0oo3V00<000000?o`iP3ol>H0803ol>H005h0oo3V00<00000 0?o`iP3ol>H0A`3ol>H00`000000oo3V0?o`iP0O0?o`iP00GP3ol>H2000004P0oo3V00<000000?o` iP3ol>H07`3ol>H005h0oo3V00<000000?o`iP3ol>H0B03ol>H00`000000oo3V0?o`iP0N0?o`iP00 GP3ol>H00`000000oo3V0?o`iP180?o`iP030000003ol>H0oo3V01h0oo3V001N0?o`iP800000B@3o l>H00`000000oo3V0?o`iP0N0?o`iP00D@3ol>H300000080oo3V1@0000030?o`iP030000003ol>H0 oo3V04T0oo3V00<000000?o`iP3ol>H07@3ol>H00580oo3V00<000000?o`iP3ol>H00P3ol>H01000 0000oo3V0?o`iP0000030?o`iP030000003ol>H0oo3V04T0oo3V00<000000?o`iP3ol>H07@3ol>H0 0580oo3V00<000000?o`iP3ol>H00`3ol>H00`000000oo3V0?o`iP030?o`iP030000003ol>H0oo3V 04X0oo3V00<000000?o`iP3ol>H0703ol>H00580oo3V00<000000?o`iP3ol>H0103ol>H00`000000 oo3V0?o`iP020?o`iP800000B`3ol>H00`000000oo3V0?o`iP0L0?o`iP00DP3ol>H01@000000oo3V 0?o`iP3ol>H0000000<0oo3V00D000000?o`iP3ol>H0oo3V0000001<0?o`iP030000003ol>H0oo3V 01`0oo3V001A0?o`iP8000000`3ol>H01@000000oo3V0?o`iP3ol>H0000000<0oo3V00<000000?o` iP3ol>H0B`3ol>H00`000000oo3V0?o`iP0K0?o`iP00DP3ol>H00`000000oo3V0?o`iP020?o`iP<0 0000103ol>H2000004`0oo3V00<000000?o`iP3ol>H06`3ol>H005h0oo3V00<000000?o`iP3ol>H0 B`3ol>H00`000000oo3V0?o`iP0K0?o`iP00GP3ol>H00`000000oo3V0?o`iP1<0?o`iP030000003o l>H0oo3V01X0oo3V001N0?o`iP030000003ol>H0oo3V04`0oo3V00<000000?o`iP3ol>H06P3ol>H0 05h0oo3V0P00001>0?o`iP030000003ol>H0oo3V01T0oo3V001N0?o`iP030000003ol>H0oo3V04d0 oo3V00<000000?o`iP3ol>H06@3ol>H005h0oo3V00<000000?o`iP3ol>H0C@3ol>H00`000000oo3V 0?o`iP0I0?o`iP00GP3ol>H2000004l0oo3V00<000000?o`iP3ol>H0603ol>H005h0oo3V00<00000 0?o`iP3ol>H0J@3ol>H005h0oo3V00<000000?o`iP3ol>H0J@3ol>H00H00H2000000P0oo3V0P00001S0?o`iP00F`3ol>H00`000000oo3V 0?o`iP080?o`iP030000003ol>H0oo3V0640oo3V001B0?o`iPD00000103ol>H00`000000oo3V0?o` iP02000000030?o`iP000000000000<0oo3V00<000000?o`iP3ol>H0H@3ol>H005@0oo3V00<00000 0?o`iP3ol>H0103ol>H01`000000oo3V0?o`iP3ol>H000000?o`iP000000103ol>H00`000000oo3V 0?o`iP1Q0?o`iP00E03ol>H00`000000oo3V0?o`iP040?o`iP030000003ol>H0oo3V0080oo3V00<0 00000?o`iP3ol>H00`3ol>H00`000000oo3V0?o`iP1Q0?o`iP00E03ol>H00`000000oo3V0?o`iP04 0?o`iP070000003ol>H0oo3V0?o`iP000000oo3V000000040?o`iP030000003ol>H0oo3V0640oo3V 001C0?o`iP@00000103ol>H00`000000oo3V0?o`iP02000000030?o`iP000000000000<0oo3V00<0 00000?o`iP3ol>H0H@3ol>H005@0oo3V00<000000?o`iP3ol>H0103ol>H2000000P0oo3V0P00001S 0?o`iP00E@3ol>H2000007<0oo3V003:0?o`iP00bP3ol>H00001\ \>"], ImageRangeCache->{{{0, 201.812}, {124.375, 0}} -> {-3.52485, -1.58142, \ 0.0371928, 0.142232}}] }, Open ]], Cell[TextData[{ "Look at the plot and then use the words positive, negative, or zero to \ complete the blanks: \n ", Cell[BoxData[ \(\[Integral]\_\(-3\)\%3 f[x] \[DifferentialD]x\)]], "_____________________.\n\t\n \t ", Cell[BoxData[ \(\[Integral]\_0\%3 f[x] \[DifferentialD]x\)]], " _____________________.\n \t \n \t ", Cell[BoxData[ \(\[Integral]\_\(-3\)\%0 f[x] \[DifferentialD]x\)]], "_____________________.\n \t \n \t ", Cell[BoxData[ \(\[Integral]\_\(-1\)\%2 f[x] \[DifferentialD]x\)]], "_____________________." }], "Text"] }, FrontEndVersion->"4.0 for Macintosh", ScreenRectangle->{{0, 640}, {0, 460}}, AutoGeneratedPackage->None, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{627, 431}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ The following notebook is copyright 1999 by Math \ Everywhere,Inc.and may not be reproduced, copied, or distributed, in whole or \ in part, through any means electronic or otherwise, without written consent \ of the copyright holder.\ \>", "Text"], Cell[CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[StyleData[All, "Working"], ScriptMinSize->9], Cell[StyleData[All, "Presentation"], ScriptMinSize->12, FontSize->18], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, PrivateFontOptions->{"FontType"->"Outline"}], Cell[StyleData[All, "TwoColumn"], PageWidth->PaperWidth, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell["Notebook Options", "Section"], Cell["\<\ The options defined for the style below will be used at the \ Notebook level.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Notebook"], CellGrouping->Manual, ShowClosedCellArea->True, StyleMenuListing->None, Background->RGBColor[1, 0.940002, 0.900008]], Cell[StyleData["Notebook", "Printout"], Background->GrayLevel[1]], Cell[StyleData["Notebook", "TwoColumn"], PageHeaders->{{None, None, None}, {None, None, None}}, PageHeaderLines->{False, False}, PrintingOptions->{"PrintingMargins"->{{55, 25}, {40, 45}}}, Background->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["Title"], CellFrame->{{1, 1}, {1, 5}}, ShowCellBracket->False, CellMargins->{{24, 24}, {12, 5}}, CellGroupingRules->{"TitleGrouping", 0}, PageBreakBelow->False, CellFrameMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, LineSpacing->{1.1, 1}, CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontFamily->"Times", FontSize->24, FontWeight->"Bold", FontColor->GrayLevel[0], Background->RGBColor[0.925002, 0.854978, 0.774975]], Cell[StyleData["Title", "Presentation"], CellMargins->{{24, Inherited}, {60, Inherited}}, TextAlignment->Center, FontSize->24, FontColor->GrayLevel[1], Background->RGBColor[0.294118, 0.521569, 0.0941176]], Cell[StyleData["Title", "Printout"], CellMargins->{{0, Inherited}, {0, 0}}, TextAlignment->Center, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["Title", "TwoColumn"], CellFrame->{{1, 1}, {5, 0}}, CellMargins->{{0, Inherited}, {0, 0}}, TextAlignment->Center, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], ShowCellBracket->False, ShowClosedCellArea->True, CellMargins->{{6, Inherited}, {0, 0}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, FontFamily->"Times", FontSize->14], Cell[StyleData["Subtitle", "Presentation"], CellFrame->False, CellMargins->{{24, Inherited}, {6, Inherited}}], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{14, Inherited}, {2, 2}}], Cell[StyleData["Subtitle", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 2}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubtitle"], CellFrame->True, ShowClosedCellArea->True, CellMargins->{{6, Inherited}, {6, Inherited}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Times", FontSize->24, FontColor->RGBColor[1, 0, 0]], Cell[StyleData["Subsubtitle", "Presentation"], CellFrame->True, CellMargins->{{24, Inherited}, {6, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{14, Inherited}, {2, 2}}, FontColor->GrayLevel[0]], Cell[StyleData["Subsubtitle", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 2}}, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellDingbat->"\[GraySquare]", ShowCellBracket->True, ShowGroupOpenCloseIcon->True, CellMargins->{{22, Inherited}, {Inherited, 20}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily->"Times", FontSize->16], Cell[StyleData["Section", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, 20}}, LineSpacing->{1.5, 0}, FontSize->18], Cell[StyleData["Section", "Printout"], CellMargins->{{14, Inherited}, {2, 10}}], Cell[StyleData["Section", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 10}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"", ShowCellBracket->True, ShowGroupOpenCloseIcon->True, CellMargins->{{19, Inherited}, {Inherited, 18}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontFamily->"Times", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0, 0.392187, 0]], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, 15}}], Cell[StyleData["Subsection", "Printout"], CellMargins->{{14, Inherited}, {2, 5}}, FontSize->12, FontColor->GrayLevel[0]], Cell[StyleData["Subsection", "TwoColumn"], CellFrame->{{0, 0}, {0, 1}}, CellMargins->{{14, Inherited}, {2, 10}}, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[EmptySquare]", ShowClosedCellArea->True, CellMargins->{{18, Inherited}, {Inherited, 12}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, CounterIncrements->"Subsubsection", FontFamily->"Times", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.689998, 0.0899977, 0.119997]], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, 12}}, LineSpacing->{1, 0}], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{14, Inherited}, {2, 3}}, FontSize->12, FontColor->GrayLevel[0]], Cell[StyleData["Subsubsection", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 3}}, FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["PrefaceText"], CellMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1, 1}, LimitsPositioningTokens->{}, StyleMenuListing->None, FontFamily->"Times", FontSize->10, FontWeight->"Plain"], Cell[StyleData["PrefaceText", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, LineSpacing->{2, 0}, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["PrefaceText", "Printout"], CellMargins->{{0, Inherited}, {2, 2}}, LineSpacing->{1, 1}, FontSize->12, FontColor->GrayLevel[0], Background->None], Cell[StyleData["PrefaceText", "TwoColumn"], CellMargins->{{0, Inherited}, {2, 2}}, LineSpacing->{1, 1}, FontSize->12, FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[StyleData["PrefaceHyperlink"], CellMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1, 1}, LimitsPositioningTokens->{}, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Times", FontSize->10, FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[CellGroupData[{ Cell[StyleData["Text"], ShowClosedCellArea->True, CellMargins->{{15, 10}, {Inherited, Inherited}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, LimitsPositioningTokens->{}, FontFamily->"Times", FontSize->16, FontColor->RGBColor[0, 0, 0.500008]], Cell[StyleData["Text", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, LineSpacing->{2, 0}, FontSize->16, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Text", "Printout"], CellMargins->{{14, Inherited}, {3, 1}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 2}, FontColor->GrayLevel[0], Background->None], Cell[StyleData["Text", "TwoColumn"], CellMargins->{{14, Inherited}, {3, 1}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 2}, FontSize->14, FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], ShowClosedCellArea->True, CellMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, LimitsPositioningTokens->{}, FontFamily->"Times", FontSize->16], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["SmallText", "Printout"], CellMargins->{{14, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True], Cell[StyleData["SmallText", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output \ to the kernel. Be careful when modifying, renaming, or removing these \ styles, because the front end associates special meanings with these style \ names.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{45, Inherited}, {Inherited, Inherited}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, GroupPageBreakWithin->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", FontWeight->"Bold"], Cell[StyleData["Input", "Presentation"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{45, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Input", "Printout"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->9], Cell[StyleData["Input", "TwoColumn"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1.5, 0}, FormatType->StandardForm, FontFamily->"Courier", FontSize->14, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Output", "Presentation"], CellMargins->{{45, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Output", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 0}, FontSize->10, FontColor->GrayLevel[0]], Cell[StyleData["Output", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 0}, FontSize->12, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Message", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Message", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9, FontColor->GrayLevel[0]], Cell[StyleData["Message", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Print", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9], Cell[StyleData["Print", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Info"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None], Cell[StyleData["Info", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Info", "Printout"], CellMargins->{{30, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["Info", "TwoColumn"], CellMargins->{{30, Inherited}, {Inherited, Inherited}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{15, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, ImageMargins->{{35, Inherited}, {Inherited, 0}}, AnimationDisplayTime->0.2, StyleMenuListing->None, FontSize->14], Cell[StyleData["Graphics", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Graphics", "Printout"], CellMargins->{{30, Inherited}, {0, 0}}, CellFrameMargins->False, ImageSize->{Inherited, 150}, ImageMargins->{{45, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, FontSize->9], Cell[StyleData["Graphics", "TwoColumn"], CellMargins->{{20, Inherited}, {0, 0}}, CellFrameMargins->False, ImageSize->{Inherited, 150}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Times", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Presentation"], FontSize->14], Cell[StyleData["CellLabel", "Printout"], FontColor->GrayLevel[1]], Cell[StyleData["CellLabel", "TwoColumn"], FontColor->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Special Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["PrefaceTitle"], CellFrame->{{1, 1}, {1, 5}}, ShowCellBracket->False, CellMargins->{{24, 24}, {0, 10}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, CellFrameMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, LineSpacing->{1.4, 1}, FontFamily->"Times", FontSize->24, FontWeight->"Bold", FontColor->GrayLevel[1], Background->RGBColor[0, 0.392187, 0]], Cell[StyleData["PrefaceTitle", "Presentation"], CellMargins->{{24, Inherited}, {60, Inherited}}, TextAlignment->Center, FontSize->38, FontColor->GrayLevel[1], Background->RGBColor[0.596078, 0.65098, 0.0196078]], Cell[StyleData["PrefaceTitle", "Printout"], CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["PrefaceTitle", "TwoColumn"], CellFrame->{{1, 1}, {0, 5}}, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Special Body Text and Index", "Section"], Cell[CellGroupData[{ Cell[StyleData["Accident"], CellFrame->3, ShowCellBracket->False, CellMargins->{{24, 24}, {0, 10}}, CellFrameMargins->{{15, Inherited}, {Inherited, Inherited}}, TextAlignment->Center, LineSpacing->{1.4, 1}, FontFamily->"Times", FontSize->24, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell[StyleData["Accident", "Presentation"], CellMargins->{{24, Inherited}, {60, Inherited}}, TextAlignment->Center, FontSize->36], Cell[StyleData["Accident", "Printout"], CellFrame->2, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["Accident", "TwoColumn"], CellFrame->2, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ContentsText"], CellMargins->{{50, 10}, {5, 5}}, FontFamily->"Times", FontSize->16], Cell[StyleData["ContentsText", "Presentation"]], Cell[StyleData["ContentsText", "Printout"], FontColor->GrayLevel[0], Background->None], Cell[StyleData["ContentsText", "TwoColumn"], FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special1"], CellDingbat->"\[EmptySquare]", ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, FontFamily->"Times", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.689998, 0.0899977, 0.119997]], Cell[StyleData["Special1", "Presentation"], FontSize->16], Cell[StyleData["Special1", "Printout"], FontSize->12, FontColor->GrayLevel[0]], Cell[StyleData["Special1", "TwoColumn"], FontSize->12, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special2"], CellMargins->{{6, 0}, {0, 0}}, CellGroupingRules->{"SectionGrouping", 40}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special2", "Presentation"], FontSize->12], Cell[StyleData["Special2", "Printout"], FontSize->10, FontColor->GrayLevel[0]], Cell[StyleData["Special2", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special3"], CellDingbat->"\[GraySquare]", ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special3", "Presentation"]], Cell[StyleData["Special3", "Printout"], FontColor->GrayLevel[0]], Cell[StyleData["Special3", "TwoColumn"], FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special4"], ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special4", "Presentation"], FontSize->12], Cell[StyleData["Special4", "Printout"], FontColor->GrayLevel[0]], Cell[StyleData["Special4", "TwoColumn"], FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special5"], ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special5", "Presentation"], FontSize->12], Cell[StyleData["Special5", "Printout"]], Cell[StyleData["Special5", "TwoColumn"]] }, Closed]], Cell[StyleData["IndexEntry"], ShowCellBracket->False, CellMargins->{{15, 5}, {0, 5}}, PageBreakBelow->False, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, ParagraphIndent->-40, StyleMenuListing->None, FontSize->16], Cell[StyleData["IndexSubEntry"], ShowCellBracket->False, CellMargins->{{45, 5}, {0, 0}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, ParagraphIndent->-40, StyleMenuListing->None, FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell["Styles for License Agreement", "Section"], Cell[CellGroupData[{ Cell[StyleData["LicenseHeading"], ShowCellBracket->True, ShowGroupOpenCloseIcon->True, CellMargins->{{24, 24}, {-1, 2}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0.4, 0.300008, 0.6]], Cell[StyleData["LicenseHeading", "Presentation"], FontSize->12], Cell[StyleData["LicenseHeading", "Printout"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["LicenseHeading", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["LicenseText"], CellFrame->True, ShowCellBracket->False, CellMargins->{{24, 24}, {5, -1}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->12, Background->RGBColor[1, 0.537743, 0.509071]], Cell[StyleData["LicenseText", "Presentation"], FontSize->18], Cell[StyleData["LicenseText", "Printout"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["LicenseText", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Automatic Numbering", "Section"], Cell["\<\ The following styles are useful for numbered equations, figures, \ etc. They automatically give the cell a FrameLabel containing a reference to \ a particular counter, and also increment that counter.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["NumberedEquation"], CounterIncrements->"NumberedEquation"], Cell[StyleData["NumberedEquation", "Presentation"]], Cell[StyleData["NumberedEquation", "Printout"]], Cell[StyleData["NumberedEquation", "TwoColumn"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedFigure"], CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Figure ", CounterBox[ "NumberedFigure"]}]], None}}, CounterIncrements->"NumberedFigure", FormatTypeAutoConvert->False, FontFamily->"Times"], Cell[StyleData["NumberedFigure", "Presentation"], CellMargins->{{24, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["NumberedFigure", "Printout"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["NumberedFigure", "TwoColumn"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedTable"], CellMargins->{{6, Inherited}, {Inherited, Inherited}}, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Table ", CounterBox[ "NumberedTable"]}]], None}}, CounterIncrements->"NumberedTable", FormatTypeAutoConvert->False, FontFamily->"Times"], Cell[StyleData["NumberedTable", "Presentation"], CellMargins->{{24, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["NumberedTable", "Printout"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["NumberedTable", "TwoColumn"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontSize->10], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, StyleMenuListing->None, FontSize->9], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontSize->14, FontColor->RGBColor[0, 0.392187, 0], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}], Cell[StyleData["Hyperlink", "TwoColumn"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["MEIHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.650004, 0.680003, 0.0800031], Background->RGBColor[0, 0.392187, 0], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Come visit us!"}], Cell[StyleData["MEIHyperlink", "Printout"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}], Cell[StyleData["MEIHyperlink", "TwoColumn"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["BasicsHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.848096, 0.171878, 0.228321], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Go to Basics"}], Cell[StyleData["BasicsHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["BasicsHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["BasicsIndexHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontColor->RGBColor[0.848096, 0.171878, 0.228321], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEnd`SelectionMove[ FrontEnd`SelectedNotebook[ ], Next, CellGroup], FrontEndToken[ "SelectionCloseAllGroups"], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Link into Basics"}], Cell[StyleData["BasicsIndexHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["BasicsIndexHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["TutorialsHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.0199588, 0.346716, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Go to Tutorials"}], Cell[StyleData["TutorialsHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["TutorialsHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["TutorialsIndexHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontColor->RGBColor[0.0199588, 0.346716, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEnd`SelectionMove[ FrontEnd`SelectedNotebook[ ], Next, CellGroup], FrontEndToken[ "SelectionCloseAllGroups"], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Link into Tutorials"}], Cell[StyleData["TutorialsIndexHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["TutorialsIndexHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GiveItaTryHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.497459, 0.196094, 0.543877], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Go to GiveItaTry"}], Cell[StyleData["GiveItaTryHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["GiveItaTryHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GiveItaTryIndexHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontColor->RGBColor[0.497459, 0.196094, 0.543877], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEnd`SelectionMove[ FrontEnd`SelectedNotebook[ ], Next, CellGroup], FrontEndToken[ "SelectionCloseAllGroups"], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Link into GiveItaTry"}], Cell[StyleData["GiveItaTryIndexHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["GiveItaTryIndexHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["LiteracyHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[1, 0.433326, 0], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Go to Literacy"}], Cell[StyleData["LiteracyHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["LiteracyHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["LiteracyIndexHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontColor->RGBColor[1, 0.433326, 0], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEnd`SelectionMove[ FrontEnd`SelectedNotebook[ ], Next, CellGroup], FrontEndToken[ "SelectionCloseAllGroups"], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Link into Literacy"}], Cell[StyleData["LiteracyIndexHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["LiteracyIndexHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["PreviewHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->GrayLevel[0.250004], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Preview of Lesson"}], Cell[StyleData["PreviewHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["PreviewHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuideLink", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"]], Cell[StyleData["RefGuideLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["RefGuideLink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"]], Cell[StyleData["GettingStartedLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["GettingStartedLink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"]], Cell[StyleData["OtherInformationLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["OtherInformationLink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder \ objects in palette templates.\ \>", "Text"], Cell[StyleData["Placeholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->True, StyleMenuListing->None], Cell[StyleData["SelectionPlaceholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->PrimaryPlaceholder, StyleMenuListing->None, DrawHighlighted->True] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles \ of most cells. If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier", FontSize->12], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{1, -5}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["InlineCell"], DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, ScriptLevel->1, LimitsPositioningTokens->{}, StyleMenuListing->None, FontFamily->"Times", FontSize->16, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["InlineCell", "Presentation"], DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, ScriptLevel->1, LimitsPositioningTokens->{}, StyleMenuListing->None, FontFamily->"Times", FontSize->16, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["InlineCell", "Printout"], DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, ScriptLevel->1, LimitsPositioningTokens->{}, StyleMenuListing->None, FontColor->GrayLevel[0]], Cell[StyleData["InlineCell", "TwoColumn"], DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, ScriptLevel->1, LimitsPositioningTokens->{}, StyleMenuListing->None, FontSize->14, FontColor->GrayLevel[0]] }, Closed]] }, Closed]] }, Open ]] }], MacintoshSystemPageSetup->"\<\ 00000000000000000Z^LD0000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000\>" ] (*********************************************************************** 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->{ "2.01.B1"->{ Cell[5892, 143, 365, 15, 86, "Subsection", CellTags->"2.01.B1"]}, "2.01.B3"->{ Cell[110054, 5747, 264, 8, 72, "Subsection", CellTags->"2.01.B3"]}, "2.01.T1"->{ Cell[209338, 11752, 97, 1, 60, "Subsection", CellTags->"2.01.T1"]}, "2.01.G3"->{ Cell[280930, 16125, 78, 1, 60, "Subsection", CellTags->"2.01.G3"]} } *) (*CellTagsIndex CellTagsIndex->{ {"2.01.B1", 393895, 21118}, {"2.01.B3", 393986, 21121}, {"2.01.T1", 394079, 21124}, {"2.01.G3", 394172, 21127} } *) (*NotebookFileOutline Notebook[{ Cell[1717, 49, 2361, 33, 34, 2210, 28, "GraphicsData", "Bitmap", "Graphics"], Cell[4081, 84, 812, 23, 134, "PrefaceTitle"], Cell[4896, 109, 134, 5, 91, "Title"], Cell[CellGroupData[{ Cell[5055, 118, 104, 4, 29, "Special2"], Cell[5162, 124, 651, 12, 247, "Input", InitializationCell->True] }, Closed]], Cell[5828, 139, 39, 0, 34, "Subsubsection"], Cell[CellGroupData[{ Cell[5892, 143, 365, 15, 86, "Subsection", CellTags->"2.01.B1"], Cell[6260, 160, 386, 8, 78, "Special2"], Cell[6649, 170, 239, 8, 86, "Text"], Cell[6891, 180, 96, 3, 13, "Special2"], Cell[CellGroupData[{ Cell[7012, 187, 1493, 30, 19, "Input", CellOpen->False], Cell[8508, 219, 11292, 597, 70, 11170, 593, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[19815, 819, 193, 9, 34, "Text"], Cell[CellGroupData[{ Cell[20033, 832, 3356, 64, 19, "Input", CellOpen->False], Cell[23392, 898, 12903, 706, 70, 12781, 702, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[36310, 1607, 296, 9, 86, "Text"], Cell[CellGroupData[{ Cell[36631, 1620, 422, 8, 19, "Input", CellOpen->False], Cell[37056, 1630, 13691, 763, 70, 13569, 759, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[50762, 2396, 739, 28, 191, "Text"], Cell[CellGroupData[{ Cell[51526, 2428, 635, 12, 19, "Input", CellOpen->False], Cell[52164, 2442, 16749, 898, 70, 16627, 894, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[68928, 3343, 151, 3, 26, "Special2"], Cell[69082, 3348, 869, 23, 223, "Text"], Cell[69954, 3373, 31, 0, 42, "Subsubsection"], Cell[CellGroupData[{ Cell[70010, 3377, 446, 17, 203, "Text"], Cell[70459, 3396, 27, 0, 26, "Special1"], Cell[70489, 3398, 168, 5, 52, "SmallText"] }, Closed]], Cell[70672, 3406, 31, 0, 34, "Subsubsection"], Cell[70706, 3408, 53, 0, 34, "Text"], Cell[70762, 3410, 33, 0, 42, "Subsubsection"], Cell[CellGroupData[{ Cell[70820, 3414, 118, 4, 35, "Text"], Cell[70941, 3420, 27, 0, 26, "Special1"], Cell[70971, 3422, 358, 14, 113, "SmallText"], Cell[71332, 3438, 164, 5, 35, "SmallText"], Cell[CellGroupData[{ Cell[71521, 3447, 227, 4, 122, "Input"], Cell[71751, 3453, 13019, 813, 70, 12896, 809, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[84785, 4269, 76, 0, 34, "SmallText"], Cell[84864, 4271, 128, 3, 55, "Input"], Cell[84995, 4276, 246, 6, 87, "SmallText"], Cell[CellGroupData[{ Cell[85266, 4286, 102, 2, 38, "Input"], Cell[85371, 4290, 39, 1, 70, "Output"] }, Open ]], Cell[85425, 4294, 132, 4, 34, "SmallText"], Cell[CellGroupData[{ Cell[85582, 4302, 72, 1, 23, "Input"], Cell[85657, 4305, 39, 1, 70, "Output"] }, Open ]], Cell[85711, 4309, 32, 0, 34, "SmallText"] }, Closed]], Cell[85758, 4312, 34, 0, 34, "Subsubsection"], Cell[CellGroupData[{ Cell[85817, 4316, 123, 4, 47, "Text"], Cell[85943, 4322, 27, 0, 26, "Special1"], Cell[85973, 4324, 178, 5, 53, "SmallText"], Cell[CellGroupData[{ Cell[86176, 4333, 241, 5, 122, "Input"], Cell[86420, 4340, 10560, 645, 70, 10437, 641, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[96995, 4988, 94, 3, 30, "SmallText"], Cell[CellGroupData[{ Cell[97114, 4995, 75, 2, 23, "Input"], Cell[97192, 4999, 54, 1, 70, "Output"] }, Open ]], Cell[97261, 5003, 276, 9, 75, "SmallText"], Cell[CellGroupData[{ Cell[97562, 5016, 165, 4, 72, "Input"], Cell[97730, 5022, 40, 1, 70, "Output"] }, Open ]], Cell[97785, 5026, 59, 0, 34, "SmallText"], Cell[CellGroupData[{ Cell[97869, 5030, 192, 4, 58, "Input"], Cell[98064, 5036, 10560, 645, 70, 10437, 641, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[108639, 5684, 201, 5, 87, "SmallText"], Cell[CellGroupData[{ Cell[108865, 5693, 161, 4, 72, "Input"], Cell[109029, 5699, 35, 1, 70, "Output"] }, Open ]], Cell[109079, 5703, 304, 7, 87, "SmallText"], Cell[CellGroupData[{ Cell[109408, 5714, 87, 2, 23, "Input"], Cell[109498, 5718, 35, 1, 70, "Output"] }, Open ]], Cell[109548, 5722, 213, 7, 87, "SmallText"], Cell[CellGroupData[{ Cell[109786, 5733, 113, 2, 39, "Input"], Cell[109902, 5737, 35, 1, 70, "Output"] }, Open ]], Cell[109952, 5741, 53, 0, 34, "SmallText"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[110054, 5747, 264, 8, 72, "Subsection", CellTags->"2.01.B3"], Cell[110321, 5757, 42, 0, 42, "Subsubsection"], Cell[CellGroupData[{ Cell[110388, 5761, 802, 30, 203, "Text"], Cell[111193, 5793, 27, 0, 26, "Special1"], Cell[111223, 5795, 41, 0, 30, "SmallText"], Cell[CellGroupData[{ Cell[111289, 5799, 115, 3, 39, "Input"], Cell[111407, 5804, 47, 1, 70, "Output"] }, Open ]], Cell[111469, 5808, 38, 0, 30, "SmallText"], Cell[CellGroupData[{ Cell[111532, 5812, 420, 9, 144, "Input"], Cell[111955, 5823, 4867, 315, 70, 4744, 311, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[116837, 6141, 794, 35, 120, "SmallText"], Cell[CellGroupData[{ Cell[117656, 6180, 105, 3, 38, "Input"], Cell[117764, 6185, 38, 1, 70, "Output"] }, Open ]], Cell[117817, 6189, 75, 1, 30, "SmallText"], Cell[CellGroupData[{ Cell[117917, 6194, 128, 3, 23, "Input"], Cell[118048, 6199, 93, 1, 70, "Output"] }, Open ]], Cell[118156, 6203, 110, 2, 30, "SmallText"], Cell[CellGroupData[{ Cell[118291, 6209, 515, 10, 225, "Input"], Cell[118809, 6221, 5028, 329, 70, 4905, 325, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[123852, 6553, 457, 14, 74, "SmallText"], Cell[CellGroupData[{ Cell[124334, 6571, 652, 12, 224, "Input"], Cell[124989, 6585, 5190, 347, 70, 5067, 343, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[130194, 6935, 143, 4, 39, "Special2"], Cell[130340, 6941, 601, 21, 206, "SmallText"], Cell[CellGroupData[{ Cell[130966, 6966, 147, 3, 58, "Input"], Cell[131116, 6971, 53, 1, 70, "Output"] }, Open ]], Cell[131184, 6975, 99, 3, 30, "SmallText"], Cell[CellGroupData[{ Cell[131308, 6982, 277, 7, 78, "Input"], Cell[131588, 6991, 5022, 329, 70, 4899, 325, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False], Cell[136613, 7322, 5018, 329, 70, 4895, 325, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False], Cell[141634, 7653, 5015, 329, 70, 4892, 325, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False], Cell[146652, 7984, 5021, 329, 70, 4898, 325, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[151688, 8316, 117, 3, 13, "Special2"], Cell[151808, 8321, 198, 5, 52, "SmallText"], Cell[CellGroupData[{ Cell[152031, 8330, 139, 3, 56, "Input"], Cell[152173, 8335, 53, 1, 70, "Output"] }, Open ]], Cell[152241, 8339, 238, 6, 53, "SmallText"] }, Closed]], Cell[152494, 8348, 34, 0, 34, "Subsubsection"], Cell[CellGroupData[{ Cell[152553, 8352, 89, 3, 46, "Text"], Cell[152645, 8357, 27, 0, 26, "Special1"], Cell[152675, 8359, 73, 0, 13, "Special2"], Cell[152751, 8361, 360, 13, 54, "SmallText"], Cell[CellGroupData[{ Cell[153136, 8378, 612, 12, 208, "Input"], Cell[153751, 8392, 5242, 341, 70, 5119, 337, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[159008, 8736, 372, 12, 96, "SmallText"], Cell[159383, 8750, 59, 0, 30, "SmallText"], Cell[CellGroupData[{ Cell[159467, 8754, 665, 13, 215, "Input"], Cell[160135, 8769, 5718, 389, 70, 5595, 385, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[165868, 9161, 123, 5, 30, "SmallText"], Cell[CellGroupData[{ Cell[166016, 9170, 138, 3, 56, "Input"], Cell[166157, 9175, 53, 1, 70, "Output"] }, Open ]], Cell[166225, 9179, 68, 0, 30, "SmallText"], Cell[CellGroupData[{ Cell[166318, 9183, 78, 2, 23, "Input"], Cell[166399, 9187, 76, 1, 70, "Output"] }, Open ]], Cell[166490, 9191, 177, 4, 52, "SmallText"], Cell[CellGroupData[{ Cell[166692, 9199, 631, 12, 290, "Input"], Cell[167326, 9213, 6630, 422, 70, 6507, 418, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[173971, 9638, 399, 13, 75, "SmallText"], Cell[174373, 9653, 45, 0, 30, "SmallText"], Cell[CellGroupData[{ Cell[174443, 9657, 352, 7, 108, "Input"], Cell[174798, 9666, 9074, 669, 70, 8951, 665, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[183887, 10338, 132, 5, 34, "SmallText"], Cell[CellGroupData[{ Cell[184044, 10347, 140, 3, 56, "Input"], Cell[184187, 10352, 53, 1, 70, "Output"] }, Open ]], Cell[184255, 10356, 69, 0, 34, "SmallText"], Cell[CellGroupData[{ Cell[184349, 10360, 88, 2, 23, "Input"], Cell[184440, 10364, 104, 2, 70, "Output"] }, Open ]], Cell[184559, 10369, 327, 6, 139, "SmallText"], Cell[CellGroupData[{ Cell[184911, 10379, 190, 4, 89, "Input"], Cell[185104, 10385, 53, 1, 70, "Output"] }, Open ]], Cell[185172, 10389, 344, 9, 139, "SmallText"], Cell[CellGroupData[{ Cell[185541, 10402, 102, 2, 39, "Input"], Cell[185646, 10406, 53, 1, 70, "Output"] }, Open ]], Cell[185714, 10410, 57, 0, 34, "SmallText"] }, Closed]], Cell[185786, 10413, 46, 0, 42, "Subsubsection"], Cell[CellGroupData[{ Cell[185857, 10417, 216, 8, 34, "Text"], Cell[186076, 10427, 32, 0, 42, "Subsubsection"], Cell[186111, 10429, 73, 0, 13, "Special2"], Cell[186187, 10431, 332, 10, 86, "SmallText"], Cell[CellGroupData[{ Cell[186544, 10445, 108, 2, 23, "Input"], Cell[186655, 10449, 53, 1, 70, "Output"] }, Open ]], Cell[186723, 10453, 65, 0, 34, "SmallText"], Cell[CellGroupData[{ Cell[186813, 10457, 127, 3, 23, "Input"], Cell[186943, 10462, 146, 2, 70, "Output"] }, Open ]], Cell[187104, 10467, 312, 9, 112, "SmallText"] }, Closed]], Cell[187431, 10479, 31, 0, 42, "Subsubsection"], Cell[CellGroupData[{ Cell[187487, 10483, 418, 13, 87, "Text"], Cell[187908, 10498, 27, 0, 26, "Special1"], Cell[187938, 10500, 41, 0, 34, "SmallText"], Cell[CellGroupData[{ Cell[188004, 10504, 187, 5, 75, "Input"], Cell[188194, 10511, 84, 1, 70, "Output"] }, Open ]], Cell[188293, 10515, 38, 0, 34, "SmallText"], Cell[CellGroupData[{ Cell[188356, 10519, 336, 6, 112, "Input"], Cell[188695, 10527, 5059, 330, 70, 4936, 326, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[193769, 10860, 182, 7, 34, "SmallText"], Cell[CellGroupData[{ Cell[193976, 10871, 906, 17, 369, "Input"], Cell[194885, 10890, 5459, 362, 70, 5336, 358, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[200359, 11255, 765, 25, 185, "SmallText"], Cell[CellGroupData[{ Cell[201149, 11284, 143, 3, 56, "Input"], Cell[201295, 11289, 53, 1, 70, "Output"] }, Open ]], Cell[201363, 11293, 77, 0, 30, "SmallText"], Cell[CellGroupData[{ Cell[201465, 11297, 98, 2, 38, "Input"], Cell[201566, 11301, 59, 1, 70, "Output"] }, Open ]], Cell[201640, 11305, 147, 5, 30, "SmallText"], Cell[CellGroupData[{ Cell[201812, 11314, 213, 4, 39, "Input"], Cell[202028, 11320, 5248, 344, 70, 5125, 340, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[207291, 11667, 74, 0, 30, "SmallText"], Cell[CellGroupData[{ Cell[207390, 11671, 206, 4, 109, "Input"], Cell[207599, 11677, 53, 1, 70, "Output"] }, Open ]], Cell[207667, 11681, 37, 0, 34, "SmallText"], Cell[CellGroupData[{ Cell[207729, 11685, 212, 4, 109, "Input"], Cell[207944, 11691, 53, 1, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[208034, 11697, 212, 4, 109, "Input"], Cell[208249, 11703, 53, 1, 70, "Output"] }, Open ]], Cell[208317, 11707, 43, 0, 34, "SmallText"], Cell[CellGroupData[{ Cell[208385, 11711, 100, 2, 23, "Input"], Cell[208488, 11715, 125, 2, 70, "Output"] }, Open ]], Cell[208628, 11720, 321, 9, 139, "SmallText"], Cell[CellGroupData[{ Cell[208974, 11733, 108, 2, 23, "Input"], Cell[209085, 11737, 53, 1, 70, "Output"] }, Open ]], Cell[209153, 11741, 92, 3, 60, "SmallText"] }, Closed]] }, Closed]], Cell[209272, 11748, 41, 0, 34, "Subsubsection"], Cell[CellGroupData[{ Cell[209338, 11752, 97, 1, 60, "Subsection", CellTags->"2.01.T1"], Cell[209438, 11755, 54, 0, 34, "Text"], Cell[209495, 11757, 31, 0, 42, "Subsubsection"], Cell[CellGroupData[{ Cell[209551, 11761, 128, 4, 35, "Text"], Cell[209682, 11767, 27, 0, 26, "Special1"], Cell[209712, 11769, 51, 0, 34, "SmallText"], Cell[CellGroupData[{ Cell[209788, 11773, 313, 6, 127, "Input"], Cell[210104, 11781, 6490, 384, 70, 6367, 380, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[216609, 12168, 177, 5, 61, "SmallText"], Cell[216789, 12175, 193, 5, 26, "Special2"], Cell[CellGroupData[{ Cell[217007, 12184, 146, 3, 23, "Input"], Cell[217156, 12189, 13477, 807, 70, 13354, 803, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[230648, 12999, 165, 5, 60, "SmallText"], Cell[CellGroupData[{ Cell[230838, 13008, 75, 2, 23, "Input"], Cell[230916, 13012, 67, 1, 70, "Output"] }, Open ]], Cell[230998, 13016, 264, 9, 87, "SmallText"], Cell[CellGroupData[{ Cell[231287, 13029, 176, 4, 72, "Input"], Cell[231466, 13035, 53, 1, 70, "Output"] }, Open ]], Cell[231534, 13039, 245, 9, 61, "SmallText"], Cell[CellGroupData[{ Cell[231804, 13052, 167, 4, 72, "Input"], Cell[231974, 13058, 46, 1, 70, "Output"] }, Open ]], Cell[232035, 13062, 319, 8, 87, "SmallText"], Cell[CellGroupData[{ Cell[232379, 13074, 87, 2, 23, "Input"], Cell[232469, 13078, 53, 1, 70, "Output"] }, Open ]], Cell[232537, 13082, 221, 7, 87, "SmallText"], Cell[CellGroupData[{ Cell[232783, 13093, 91, 2, 23, "Input"], Cell[232877, 13097, 40, 1, 70, "Output"] }, Open ]], Cell[232932, 13101, 35, 0, 34, "SmallText"] }, Closed]], Cell[232982, 13104, 31, 0, 42, "Subsubsection"], Cell[CellGroupData[{ Cell[233038, 13108, 128, 4, 47, "Text"], Cell[233169, 13114, 27, 0, 26, "Special1"], Cell[233199, 13116, 304, 12, 86, "SmallText"], Cell[CellGroupData[{ Cell[233528, 13132, 130, 3, 43, "Input"], Cell[233661, 13137, 97, 2, 70, "Output"] }, Open ]], Cell[233773, 13142, 152, 5, 60, "SmallText"], Cell[233928, 13149, 252, 6, 42, "Special2"], Cell[CellGroupData[{ Cell[234205, 13159, 146, 3, 32, "Input"], Cell[234354, 13164, 10820, 779, 70, 10701, 775, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[245189, 13946, 366, 11, 165, "SmallText"], Cell[CellGroupData[{ Cell[245580, 13961, 106, 3, 59, "Input"], Cell[245689, 13966, 49, 1, 70, "Output"] }, Open ]], Cell[245753, 13970, 129, 5, 34, "SmallText"], Cell[CellGroupData[{ Cell[245907, 13979, 120, 3, 39, "Input"], Cell[246030, 13984, 49, 1, 70, "Output"] }, Open ]], Cell[246094, 13988, 28, 0, 34, "SmallText"] }, Closed]], Cell[246137, 13991, 31, 0, 34, "Subsubsection"], Cell[CellGroupData[{ Cell[246193, 13995, 147, 5, 46, "Text"], Cell[246343, 14002, 27, 0, 26, "Special1"], Cell[246373, 14004, 51, 0, 30, "SmallText"], Cell[CellGroupData[{ Cell[246449, 14008, 332, 7, 129, "Input"], Cell[246784, 14017, 9745, 604, 70, 9622, 600, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[256544, 14624, 210, 6, 53, "SmallText"], Cell[CellGroupData[{ Cell[256779, 14634, 203, 4, 58, "Input"], Cell[256985, 14640, 23006, 1439, 70, 22883, 1435, "GraphicsData", \ "PostScript", "Graphics", ImageCacheValid->False] }, Open ]], Cell[280006, 16082, 102, 3, 52, "SmallText"], Cell[CellGroupData[{ Cell[280133, 16089, 81, 2, 23, "Input"], Cell[280217, 16093, 38, 1, 70, "Output"] }, Open ]], Cell[280270, 16097, 299, 7, 97, "SmallText"], Cell[CellGroupData[{ Cell[280594, 16108, 140, 3, 38, "Input"], Cell[280737, 16113, 35, 1, 70, "Output"] }, Open ]], Cell[280787, 16117, 45, 0, 30, "SmallText"] }, Closed]] }, Closed]], Cell[280859, 16121, 46, 0, 34, "Subsubsection"], Cell[CellGroupData[{ Cell[280930, 16125, 78, 1, 60, "Subsection", CellTags->"2.01.G3"], Cell[281011, 16128, 31, 0, 42, "Subsubsection"], Cell[281045, 16130, 177, 8, 34, "Text"], Cell[CellGroupData[{ Cell[281247, 16142, 351, 7, 144, "Input"], Cell[281601, 16151, 6452, 390, 70, 6329, 386, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[288068, 16544, 268, 9, 87, "Text"], Cell[288339, 16555, 33, 0, 42, "Subsubsection"], Cell[288375, 16557, 185, 7, 34, "Text"], Cell[CellGroupData[{ Cell[288585, 16568, 338, 7, 144, "Input"], Cell[288926, 16577, 5734, 336, 70, 5611, 332, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[294675, 16916, 272, 9, 87, "Text"], Cell[294950, 16927, 34, 0, 42, "Subsubsection"], Cell[294987, 16929, 179, 8, 34, "Text"], Cell[CellGroupData[{ Cell[295191, 16941, 336, 7, 128, "Input"], Cell[295530, 16950, 6822, 408, 70, 6699, 404, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[302367, 17361, 272, 9, 87, "Text"], Cell[302642, 17372, 31, 0, 42, "Subsubsection"], Cell[302676, 17374, 184, 8, 34, "Text"], Cell[CellGroupData[{ Cell[302885, 17386, 338, 7, 113, "Input"], Cell[303226, 17395, 6184, 359, 70, 6061, 355, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[309425, 17757, 280, 9, 87, "Text"], Cell[309708, 17768, 31, 0, 42, "Subsubsection"], Cell[309742, 17770, 192, 8, 34, "Text"], Cell[CellGroupData[{ Cell[309959, 17782, 332, 7, 134, "Input"], Cell[310294, 17791, 6982, 415, 70, 6859, 411, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[317291, 18209, 178, 5, 61, "Text"], Cell[317472, 18216, 31, 0, 42, "Subsubsection"], Cell[317506, 18218, 176, 8, 34, "Text"], Cell[CellGroupData[{ Cell[317707, 18230, 329, 7, 103, "Input"], Cell[318039, 18239, 5586, 336, 70, 5467, 332, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[323640, 18578, 198, 8, 34, "Text"], Cell[CellGroupData[{ Cell[323863, 18590, 334, 7, 103, "Input"], Cell[324200, 18599, 6008, 351, 70, 5885, 347, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[330223, 18953, 354, 8, 87, "Text"] }, Closed]], Cell[330592, 18964, 41, 0, 34, "Subsubsection"], Cell[330636, 18966, 112, 5, 34, "Text"], Cell[CellGroupData[{ Cell[330773, 18975, 267, 6, 19, "Input", CellOpen->False], Cell[331043, 18983, 17339, 566, 134, 6994, 434, "GraphicsData", "PostScript", \ "Graphics"] }, Open ]], Cell[348397, 19552, 605, 15, 220, "Text"] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)