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          BaseStyle -> {CellBaseline -> Baseline}, FrameTicksStyle -> 
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          LabelStyle -> {FontFamily -> "Verdana", FontSize -> 10}, PlotLabel -> 
          Style["area of rectangle", 18], 
          PlotRange -> {{0, 2}, {0., 3.0792012269475255`}}, PlotRangeClipping -> 
          True, PlotRangePadding -> {
            Scaled[0.02], 
            Scaled[0.02]}, TicksStyle -> 
          Directive[FontFamily -> "Times", FontSize -> 10]}]}; 
     Typeset`initDone$$ = True),
    SynchronousInitialization->True,
    UnsavedVariables:>{Typeset`initDone$$},
    UntrackedVariables:>{Typeset`size$$}], "Manipulate",
   Deployed->True,
   StripOnInput->False],
  Manipulate`InterpretManipulate[1]]], "Output",
 CellID->2131044112]
}, {2}]]
}, Open  ]],

Cell[CellGroupData[{

Cell["", "ManipulateCaptionSection"],

Cell["\<\
This Demonstration illustrates a common type of max-min problem from a \
Calculus I course\[LongDash]that of finding the maximum area of a rectangle \
inscribed in the first quadrant under a given curve. A variety of curves are \
included.\
\>", "ManipulateCaption",
 CellChangeTimes->{
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   3.459605272421875*^9}, {3.45960566734375*^9, 3.4596056689375*^9}, {
   3.459789304013353*^9, 3.4597893107210236`*^9}}]
}, Open  ]],

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Cell["", "ThumbnailSection"],

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    Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = 
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       TraditionalForm[4 - $CellContext`x^2], $CellContext`f2 -> 
       TraditionalForm[2 + Rational[-1, 2] $CellContext`x], $CellContext`f3 -> 
       TraditionalForm[(16 - $CellContext`x^2)^
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       TraditionalForm[(4 - $CellContext`x)/(
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       TraditionalForm[4 Cos[Rational[1, 2] $CellContext`x]]}}, {{
       Hold[$CellContext`X$$], 1}, 0, 
      Dynamic[$CellContext`maxX$$]}, {{
       Hold[$CellContext`maxX$$], 2}}}, Typeset`size$$ = {560., {149., 154.}},
     Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = 
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$CellContext`maxX$$ = 2, $CellContext`X$$ = 1}, "ControllerVariables" :> {
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        Hold[$CellContext`X$$, $CellContext`X$302095$$, 0]}, 
      "OtherVariables" :> {
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        Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 
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      "Body" :> (
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         4, $CellContext`f4, $CellContext`maxX$$ = 4, $CellContext`f5, 
         If[$CellContext`X$$ > Pi, $CellContext`X$$ = 1]; $CellContext`maxX$$ = 
          Pi]; $CellContext`funcPlot = Plot[
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           0, $CellContext`maxX$$}, PlotStyle -> {
            Thickness[0.007], Black}, PlotRange -> {{0, 4}, {0, 4}}, 
          AspectRatio -> 1, ImageSize -> {300, 300}]; $CellContext`areaRect[
          Pattern[$CellContext`X$, 
           
           Blank[]]] := $CellContext`X$ $CellContext`func$$[$CellContext`X$]; \
$CellContext`pointOnCurve = {$CellContext`X$$, 
          $CellContext`func$$[$CellContext`X$$]}; \
$CellContext`pointOnAreaCurve = Graphics[{
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           RGBColor[0.6, 0.73, 0.36], 
           Point[{$CellContext`X$$, 
             $CellContext`areaRect[$CellContext`X$$]}]}]; \
$CellContext`rectangle = Graphics[{
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            Directive[Thick, Blue]], 
           RGBColor[0.6, 0.73, 0.36], 
           
           Rectangle[{0, 
            0}, $CellContext`pointOnCurve]}]; $CellContext`areaPlot = Plot[
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           0, $CellContext`maxX$$}, PlotLabel -> 
          Style["area of rectangle", 18], ImageSize -> 250]; Text[
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              Style[
               Row[{"area = ", 
                 $CellContext`areaRect[$CellContext`X$$]}], 24]}]}], 
          ImageSize -> {560, 300}]]), 
      "Specifications" :> {{{$CellContext`func$$, $CellContext`f1, 
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          TraditionalForm[4 - $CellContext`x^2], $CellContext`f2 -> 
          TraditionalForm[
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          TraditionalForm[(4 - $CellContext`x)/(
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          TraditionalForm[4 Cos[Rational[1, 2] $CellContext`x]]}, ControlType -> 
         PopupMenu}, {{$CellContext`X$$, 1}, 0, 
         Dynamic[$CellContext`maxX$$]}, {{$CellContext`maxX$$, 2}, 
         ControlType -> None}}, 
      "Options" :> {AutorunSequencing -> {1, 2}, TrackedSymbols -> True}, 
      "DefaultOptions" :> {ControllerLinking -> True}],
     ImageSizeCache->{603., {213.34375, 218.65625}},
     SingleEvaluation->True],
    Deinitialization:>None,
    DynamicModuleValues:>{},
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          Blank[]]] := 2 - $CellContext`X/2, $CellContext`f3[
         Pattern[$CellContext`X, 
          Blank[]]] := Sqrt[16 - $CellContext`X^2], $CellContext`f4[
         Pattern[$CellContext`X, 
          Blank[]]] := (4 - $CellContext`X)/(
        1 + $CellContext`X^2), $CellContext`f5[
         Pattern[$CellContext`X, 
          Blank[]]] := 4 Cos[$CellContext`X/2], $CellContext`funcPlot = 
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            Thickness[0.007], 
            GrayLevel[0], 
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          BaseStyle -> {CellBaseline -> Baseline}, FrameTicksStyle -> 
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          LabelStyle -> {FontFamily -> "Verdana", FontSize -> 10}, PlotLabel -> 
          Style["area of rectangle", 18], 
          PlotRange -> {{0, 2}, {0., 3.0792012269475255`}}, PlotRangeClipping -> 
          True, PlotRangePadding -> {
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            Scaled[0.02]}, TicksStyle -> 
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     Typeset`initDone$$ = True),
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    UntrackedVariables:>{Typeset`size$$}], "Manipulate",
   Deployed->True,
   StripOnInput->False],
  Manipulate`InterpretManipulate[1]]], "Output",
 CellID->726125718]
}, Open  ]],

Cell[CellGroupData[{

Cell["", "SnapshotsSection"],

Cell[BoxData[
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    Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", 
    Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = 
    "\"untitled\"", Typeset`specs$$ = {{{
       Hold[$CellContext`func$$], $CellContext`f1, "f(x)"}, {$CellContext`f1 -> 
       TraditionalForm[4 - $CellContext`x^2], $CellContext`f2 -> 
       TraditionalForm[2 + Rational[-1, 2] $CellContext`x], $CellContext`f3 -> 
       TraditionalForm[(16 - $CellContext`x^2)^
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       TraditionalForm[(4 - $CellContext`x)/(
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     Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = 
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    DynamicBox[Manipulate`ManipulateBoxes[
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$CellContext`maxX$$ = 2, $CellContext`X$$ = 1}, "ControllerVariables" :> {
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      "OtherVariables" :> {
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      "Body" :> (
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           Blank[]]] := $CellContext`X$ $CellContext`func$$[$CellContext`X$]; \
$CellContext`pointOnCurve = {$CellContext`X$$, 
          $CellContext`func$$[$CellContext`X$$]}; \
$CellContext`pointOnAreaCurve = Graphics[{
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$CellContext`rectangle = Graphics[{
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           RGBColor[0.6, 0.73, 0.36], 
           
           Rectangle[{0, 
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           0, $CellContext`maxX$$}, PlotLabel -> 
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            Column[{
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              Style[
               Row[{"area = ", 
                 $CellContext`areaRect[$CellContext`X$$]}], 24]}]}], 
          ImageSize -> {560, 300}]]), 
      "Specifications" :> {{{$CellContext`func$$, $CellContext`f1, 
          "f(x)"}, {$CellContext`f1 -> 
          TraditionalForm[4 - $CellContext`x^2], $CellContext`f2 -> 
          TraditionalForm[
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          TraditionalForm[(16 - $CellContext`x^2)^
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          TraditionalForm[(4 - $CellContext`x)/(
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          TraditionalForm[4 Cos[Rational[1, 2] $CellContext`x]]}, ControlType -> 
         PopupMenu}, {{$CellContext`X$$, 1}, 0, 
         Dynamic[$CellContext`maxX$$]}, {{$CellContext`maxX$$, 2}, 
         ControlType -> None}}, 
      "Options" :> {AutorunSequencing -> {1, 2}, TrackedSymbols -> True}, 
      "DefaultOptions" :> {ControllerLinking -> True}],
     ImageSizeCache->{603., {213.34375, 218.65625}},
     SingleEvaluation->True],
    Deinitialization:>None,
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          Blank[]]] := 2 - $CellContext`X/2, $CellContext`f3[
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          Blank[]]] := Sqrt[16 - $CellContext`X^2], $CellContext`f4[
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          Blank[]]] := (4 - $CellContext`X)/(
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         Pattern[$CellContext`X, 
          Blank[]]] := 4 Cos[$CellContext`X/2], $CellContext`funcPlot = 
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          LabelStyle -> {FontFamily -> "Verdana", FontSize -> 10}, 
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    "\"untitled\"", Typeset`specs$$ = {{{
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       TraditionalForm[4 - $CellContext`x^2], $CellContext`f2 -> 
       TraditionalForm[2 + Rational[-1, 2] $CellContext`x], $CellContext`f3 -> 
       TraditionalForm[(16 - $CellContext`x^2)^
         Rational[1, 2]], $CellContext`f4 -> 
       TraditionalForm[(4 - $CellContext`x)/(
         1 + $CellContext`x^2)], $CellContext`f5 -> 
       TraditionalForm[4 Cos[Rational[1, 2] $CellContext`x]]}}, {{
       Hold[$CellContext`X$$], 1}, 0, 
      Dynamic[$CellContext`maxX$$]}, {{
       Hold[$CellContext`maxX$$], 4}}}, Typeset`size$$ = {560., {149., 154.}},
     Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = 
    False, $CellContext`func$302212$$ = False, $CellContext`X$302213$$ = 0}, 
    DynamicBox[Manipulate`ManipulateBoxes[
     1, StandardForm, 
      "Variables" :> {$CellContext`func$$ = $CellContext`f2, \
$CellContext`maxX$$ = 4, $CellContext`X$$ = 1}, "ControllerVariables" :> {
        Hold[$CellContext`func$$, $CellContext`func$302212$$, False], 
        Hold[$CellContext`X$$, $CellContext`X$302213$$, 0]}, 
      "OtherVariables" :> {
       Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 
        Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 
        Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$,
         Typeset`skipInitDone$$}, 
      "Body" :> (
       Switch[$CellContext`func$$, $CellContext`f1, 
         If[$CellContext`X$$ > 2, $CellContext`X$$ = 1]; $CellContext`maxX$$ = 
          2, $CellContext`f2, $CellContext`maxX$$ = 
         4, $CellContext`f3, $CellContext`maxX$$ = 
         4, $CellContext`f4, $CellContext`maxX$$ = 4, $CellContext`f5, 
         If[$CellContext`X$$ > Pi, $CellContext`X$$ = 1]; $CellContext`maxX$$ = 
          Pi]; $CellContext`funcPlot = Plot[
          $CellContext`func$$[$CellContext`X$$], {$CellContext`X$$, 
           0, $CellContext`maxX$$}, PlotStyle -> {
            Thickness[0.007], Black}, PlotRange -> {{0, 4}, {0, 4}}, 
          AspectRatio -> 1, ImageSize -> {300, 300}]; $CellContext`areaRect[
          Pattern[$CellContext`X$, 
           
           Blank[]]] := $CellContext`X$ $CellContext`func$$[$CellContext`X$]; \
$CellContext`pointOnCurve = {$CellContext`X$$, 
          $CellContext`func$$[$CellContext`X$$]}; \
$CellContext`pointOnAreaCurve = Graphics[{
           PointSize[Large], 
           RGBColor[0.6, 0.73, 0.36], 
           Point[{$CellContext`X$$, 
             $CellContext`areaRect[$CellContext`X$$]}]}]; \
$CellContext`rectangle = Graphics[{
           EdgeForm[
            Directive[Thick, Blue]], 
           RGBColor[0.6, 0.73, 0.36], 
           
           Rectangle[{0, 
            0}, $CellContext`pointOnCurve]}]; $CellContext`areaPlot = Plot[
          $CellContext`areaRect[$CellContext`X$$], {$CellContext`X$$, 
           0, $CellContext`maxX$$}, PlotLabel -> 
          Style["area of rectangle", 18], ImageSize -> 250]; Text[
         Pane[
          Row[{
            Show[$CellContext`funcPlot, $CellContext`rectangle], 
            Column[{
              Show[$CellContext`areaPlot, $CellContext`pointOnAreaCurve], "", 
              Style[
               Row[{
                 Style["x", Italic], " = ", $CellContext`X$$}], 24], 
              Style[
               Row[{"area = ", 
                 $CellContext`areaRect[$CellContext`X$$]}], 24]}]}], 
          ImageSize -> {560, 300}]]), 
      "Specifications" :> {{{$CellContext`func$$, $CellContext`f2, 
          "f(x)"}, {$CellContext`f1 -> 
          TraditionalForm[4 - $CellContext`x^2], $CellContext`f2 -> 
          TraditionalForm[
           2 + Rational[-1, 2] $CellContext`x], $CellContext`f3 -> 
          TraditionalForm[(16 - $CellContext`x^2)^
            Rational[1, 2]], $CellContext`f4 -> 
          TraditionalForm[(4 - $CellContext`x)/(
            1 + $CellContext`x^2)], $CellContext`f5 -> 
          TraditionalForm[4 Cos[Rational[1, 2] $CellContext`x]]}, ControlType -> 
         PopupMenu}, {{$CellContext`X$$, 1}, 0, 
         Dynamic[$CellContext`maxX$$]}, {{$CellContext`maxX$$, 4}, 
         ControlType -> None}}, 
      "Options" :> {AutorunSequencing -> {1, 2}, TrackedSymbols -> True}, 
      "DefaultOptions" :> {ControllerLinking -> True}],
     ImageSizeCache->{603., {213.34375, 218.65625}},
     SingleEvaluation->True],
    Deinitialization:>None,
    DynamicModuleValues:>{},
    Initialization:>({$CellContext`f1[
         Pattern[$CellContext`X, 
          Blank[]]] := 4 - $CellContext`X^2, $CellContext`f2[
         Pattern[$CellContext`X, 
          Blank[]]] := 2 - $CellContext`X/2, $CellContext`f3[
         Pattern[$CellContext`X, 
          Blank[]]] := Sqrt[16 - $CellContext`X^2], $CellContext`f4[
         Pattern[$CellContext`X, 
          Blank[]]] := (4 - $CellContext`X)/(
        1 + $CellContext`X^2), $CellContext`f5[
         Pattern[$CellContext`X, 
          Blank[]]] := 4 Cos[$CellContext`X/2], $CellContext`funcPlot = 
       Graphics[{{{}, {}, {
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            Thickness[0.007], 
            GrayLevel[0], 
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         AspectRatio -> 1, Axes -> True, AxesOrigin -> {0, 0}, 
          BaseStyle -> {CellBaseline -> Baseline}, FrameTicksStyle -> 
          Directive[FontFamily -> "Times", FontSize -> 10], 
          ImageSize -> {300, 300}, 
          LabelStyle -> {FontFamily -> "Verdana", FontSize -> 10}, 
          PlotRange -> {{0, 4}, {0, 4}}, PlotRangeClipping -> True, 
          PlotRangePadding -> {Automatic, Automatic}, TicksStyle -> 
          Directive[FontFamily -> "Times", FontSize -> 10]}], 
       Attributes[PlotRange] = {ReadProtected}, $CellContext`areaRect[
         Pattern[$CellContext`X$, 
          Blank[]]] := $CellContext`X$ FE`func$$744[$CellContext`X$], 
       Attributes[$CellContext`X$] = {Temporary}, 
       FE`func$$744 = $CellContext`f1, $CellContext`pointOnCurve = {1, 
        3}, $CellContext`pointOnAreaCurve = Graphics[{
          PointSize[Large], 
          RGBColor[0.6, 0.73, 0.36], 
          Point[{1, 3}]}], $CellContext`rectangle = Graphics[{
          EdgeForm[
           Directive[
            Thickness[Large], 
            RGBColor[0, 0, 1]]], 
          RGBColor[0.6, 0.73, 0.36], 
          Rectangle[{0, 0}, {1, 3}]}], 
       Attributes[Thick] = {ReadProtected}, $CellContext`areaPlot = 
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          BaseStyle -> {CellBaseline -> Baseline}, FrameTicksStyle -> 
          Directive[FontFamily -> "Times", FontSize -> 10], ImageSize -> 250, 
          LabelStyle -> {FontFamily -> "Verdana", FontSize -> 10}, PlotLabel -> 
          Style["area of rectangle", 18], 
          PlotRange -> {{0, 2}, {0., 3.0792012269475255`}}, PlotRangeClipping -> 
          True, PlotRangePadding -> {
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            Scaled[0.02]}, TicksStyle -> 
          Directive[FontFamily -> "Times", FontSize -> 10]}]}; 
     Typeset`initDone$$ = True),
    SynchronousInitialization->True,
    UnsavedVariables:>{Typeset`initDone$$},
    UntrackedVariables:>{Typeset`size$$}], "Manipulate",
   Deployed->True,
   StripOnInput->False],
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    Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", 
    Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = 
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       TraditionalForm[4 - $CellContext`x^2], $CellContext`f2 -> 
       TraditionalForm[2 + Rational[-1, 2] $CellContext`x], $CellContext`f3 -> 
       TraditionalForm[(16 - $CellContext`x^2)^
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       Hold[$CellContext`X$$], 1}, 0, 
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$CellContext`maxX$$ = 4, $CellContext`X$$ = 1}, "ControllerVariables" :> {
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      "Body" :> (
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          AspectRatio -> 1, ImageSize -> {300, 300}]; $CellContext`areaRect[
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$CellContext`pointOnCurve = {$CellContext`X$$, 
          $CellContext`func$$[$CellContext`X$$]}; \
$CellContext`pointOnAreaCurve = Graphics[{
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$CellContext`rectangle = Graphics[{
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           RGBColor[0.6, 0.73, 0.36], 
           
           Rectangle[{0, 
            0}, $CellContext`pointOnCurve]}]; $CellContext`areaPlot = Plot[
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           0, $CellContext`maxX$$}, PlotLabel -> 
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              Style[
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      "Specifications" :> {{{$CellContext`func$$, $CellContext`f4, 
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          TraditionalForm[(4 - $CellContext`x)/(
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      "Options" :> {AutorunSequencing -> {1, 2}, TrackedSymbols -> True}, 
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          Blank[]]] := Sqrt[16 - $CellContext`X^2], $CellContext`f4[
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          Blank[]]] := (4 - $CellContext`X)/(
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          Blank[]]] := 4 Cos[$CellContext`X/2], $CellContext`funcPlot = 
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