(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 8.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 835, 17] NotebookDataLength[ 46576, 962] NotebookOptionsPosition[ 46367, 941] NotebookOutlinePosition[ 46927, 962] CellTagsIndexPosition[ 46884, 959] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Average Rate of Change", "Title", CellChangeTimes->{{3.5446077601088*^9, 3.5446077632149773`*^9}}, Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[{ "The ", StyleBox["Average Rate of Change", FontWeight->"Bold"], " of a function is defined to be the rate at which the ", StyleBox["y", FontSlant->"Italic"], "-values change with respect to ", StyleBox["x", FontSlant->"Italic"], " over a certain interval [", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], "]. \n\nIf ", StyleBox["a", FontSlant->"Italic"], " and ", StyleBox["b", FontSlant->"Italic"], ",", StyleBox[" a", FontSlant->"Italic"], " \[NotEqual] ", StyleBox["b", FontSlant->"Italic"], ", are in the domain of a function ", StyleBox["y", FontSlant->"Italic"], " = ", StyleBox["f", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "), the average rate of change of ", StyleBox["f", FontSlant->"Italic"], " from ", StyleBox["a", FontSlant->"Italic"], " to ", StyleBox["b", FontSlant->"Italic"], " is defined as\n\n", StyleBox["Average rate of change = ", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ FormBox[ RowBox[{ FractionBox["\[CapitalDelta]y", "\[CapitalDelta]x"], " ", "=", " ", FractionBox[ RowBox[{ RowBox[{"f", "(", "b", ")"}], " ", "-", " ", RowBox[{"f", "(", "a", ")"}]}], RowBox[{"b", " ", "-", " ", "a"}]]}], TraditionalForm]], FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[", where a \[NotEqual] b.", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], "\n\nNotice that this is the same formula for calculating the ", StyleBox["slope", FontWeight->"Bold"], " of the line containing the two points (a, f(a)) and (b, f(b))." }], "Subtitle", CellChangeTimes->{{3.5944064230987253`*^9, 3.5944064298781137`*^9}, { 3.5944064602498503`*^9, 3.594406565210854*^9}, {3.5944067195506816`*^9, 3.5944067761219177`*^9}, {3.5944120860606284`*^9, 3.5944122374682884`*^9}, { 3.5944126827187557`*^9, 3.594412726212243*^9}, {3.594417646490667*^9, 3.5944176467846837`*^9}, {3.6488300426471014`*^9, 3.648830043380143*^9}}], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Example 5:", FontSlant->"Italic", FontVariations->{"Underline"->True}], " Consider the function f(x) = -", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"4", "x"}], "+", "5"}], TraditionalForm]]], ". Calculate the average rate of change from x = -1 to x = 2.\n\nFirst, \ find the corresponding y-values at x = -1 and x = 2. Fill in the blanks: \n\ \nFor the function f, the coordinates (-1, ____) and (2, ____) are points on \ the graph.\n\nNow, find the slope between these two points: ", StyleBox["m = ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ RowBox[{"f", "(", "2", ")"}], " ", "-", " ", RowBox[{"f", "(", RowBox[{"-", "1"}], ")"}]}], RowBox[{"2", "-", RowBox[{"(", RowBox[{"-", "1"}], ")"}]}]], TraditionalForm]]], " = ________. This is the average rate of change on the interval [-1, 2]. \ \n\nIn the interactive frame below, use the sliders to set ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "0"], "=", RowBox[{"-", "1"}]}], TraditionalForm]]], " and \[CapitalDelta]x = 3. Notice that the average rate of change is \ represented by the slope of the secant line connecting those two points on \ the curve. Since the slope is positive, the secant line will be \ \[OpenCurlyDoubleQuote]uphill\[CloseCurlyDoubleQuote]. This means there is a \ ", StyleBox["positive", FontVariations->{"Underline"->True}], " average rate of change." }], "Subsection", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.594417763666369*^9, 3.5944177981423407`*^9}, { 3.594417838128628*^9, 3.594417841288809*^9}, {3.594419152697817*^9, 3.5944191808474274`*^9}, {3.594419223193849*^9, 3.5944192365036106`*^9}, { 3.5944200808629055`*^9, 3.59442017745043*^9}, {3.5944202573359985`*^9, 3.5944202831174736`*^9}, {3.595073228606969*^9, 3.5950732698170266`*^9}, { 3.5950733014520707`*^9, 3.595073346467134*^9}, {3.595074898729503*^9, 3.5950751839599023`*^9}, {3.5950752418699837`*^9, 3.595075281385039*^9}, { 3.5950753426451244`*^9, 3.5950753433951254`*^9}, {3.5950753871551867`*^9, 3.595075491755333*^9}, {3.5950755310903883`*^9, 3.5950755341953926`*^9}, { 3.5950755676004395`*^9, 3.5950755970404806`*^9}, 3.5950756387705393`*^9, { 3.5967956690456886`*^9, 3.596795669354706*^9}, {3.5967957037966757`*^9, 3.5967957041636972`*^9}, {3.6472771849283113`*^9, 3.6472772164731154`*^9}}, FontSize->16, FontColor->GrayLevel[0]], Cell[TextData[{ "\n\n", StyleBox["Question 1:", FontWeight->"Bold", FontVariations->{"Underline"->True}], " Consider the function f(x) = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "-", " ", RowBox[{"6", SuperscriptBox["x", "2"]}], "+", RowBox[{"5", "x"}], "+", "12"}], TraditionalForm]]], ". Repeat the process above to calculate the average rate of change over \ the interval [-1, 1]. Is the function increasing or decreasing on that \ interval?\n\n", StyleBox["Question 2:", FontWeight->"Bold", FontVariations->{"Underline"->True}], " Consider the function f(x) = ", Cell[BoxData[ FormBox[ RadicalBox[ RowBox[{"x", "+", "1"}], "3"], TraditionalForm]]], " Repeat the process above to calculate the average rate of change over the \ interval [1, 2]. Is the function increasing or decreasing on that interval?\n\ ", StyleBox["\n", FontWeight->"Bold"], StyleBox["Question 3:", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" ", FontWeight->"Bold"], " Consider the function f(x) = ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"2", "x"}], "-", "1."}], TraditionalForm]]], " Repeat the process above to calculate the average rate of change over the \ interval [1, 4]. What do you notice about the secant line connecting the two \ points at (1, f(1)) and (4, f(4))?\n\n" }], "Subsection", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.594417763666369*^9, 3.5944177981423407`*^9}, { 3.594417838128628*^9, 3.594417841288809*^9}, {3.594419152697817*^9, 3.5944191808474274`*^9}, {3.594419223193849*^9, 3.5944192365036106`*^9}, { 3.5944200808629055`*^9, 3.59442017745043*^9}, {3.5944202573359985`*^9, 3.5944202831174736`*^9}, {3.595073228606969*^9, 3.5950732698170266`*^9}, { 3.5950733014520707`*^9, 3.595073346467134*^9}, {3.595074898729503*^9, 3.5950751839599023`*^9}, {3.5950752418699837`*^9, 3.595075281385039*^9}, { 3.5950753426451244`*^9, 3.5950753433951254`*^9}, {3.5950753871551867`*^9, 3.595075491755333*^9}, {3.5950755310903883`*^9, 3.5950755341953926`*^9}, { 3.5950755676004395`*^9, 3.5950755970404806`*^9}, 3.5950756387705393`*^9, { 3.595075698930623*^9, 3.5950756990656233`*^9}, {3.5950757569407043`*^9, 3.595075859555848*^9}}, 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Hold[$CellContext`deltax$$], 1, $CellContext`\[CapitalDelta]x}, Rational[1, 10000000], 4}}, Typeset`size$$ = {600., {190., 193.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`func$72167$$ = False, $CellContext`deltax$72168$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`deltax$$ = 1, $CellContext`func$$ = $CellContext`p2, $CellContext`x0$$ = 1}, "ControllerVariables" :> { Hold[$CellContext`func$$, $CellContext`func$72167$$, False], Hold[$CellContext`deltax$$, $CellContext`deltax$72168$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ((If[$CellContext`x0$$ < #, $CellContext`x0$$ = #]& )[ Switch[$CellContext`func$$, $CellContext`p2, -1, $CellContext`p3, \ -1, $CellContext`p4, -1]]; ( If[$CellContext`x0$$ > #, $CellContext`x0$$ = #]& )[ Switch[$CellContext`func$$, $CellContext`p2, 4.9, $CellContext`p3, 3.9, $CellContext`p4, 2.8]]; $CellContext`x1 = $CellContext`x0$$ + $CellContext`deltax$$; Switch[$CellContext`func$$, $CellContext`p2, {$CellContext`xstart, \ $CellContext`xstop, $CellContext`ystart, $CellContext`ystop, \ $CellContext`kchangesides, {$CellContext`koffsetleft, \ $CellContext`koffsetright}, $CellContext`deltaxpos, $CellContext`xyinset} = \ {-2, 6.1, -1, 11, -1, {1.6, -0.6}, -1, {4.5, 10}}; $CellContext`f = Function[$CellContext`x, -$CellContext`x^2 + 4 $CellContext`x + 5], $CellContext`p3, {$CellContext`xstart, $CellContext`xstop, \ $CellContext`ystart, $CellContext`ystop, $CellContext`kchangesides, \ {$CellContext`koffsetleft, $CellContext`koffsetright}, \ $CellContext`deltaxpos, $CellContext`xyinset} = {-2, 5, -5, 21, -1, { 1., -0.5}, -1, {3, 18}}; $CellContext`f = Function[$CellContext`x, $CellContext`x^3 - 6 $CellContext`x^2 + 5 $CellContext`x + 12], $CellContext`p4, {$CellContext`xstart, $CellContext`xstop, \ $CellContext`ystart, $CellContext`ystop, $CellContext`kchangesides, \ {$CellContext`koffsetleft, $CellContext`koffsetright}, \ $CellContext`deltaxpos, $CellContext`xyinset} = {-3, 4.5, -1.5, 1.5, -2, { 1., -0.6}, 0, {1.6, 1.1}}; $CellContext`f = Function[$CellContext`x, ($CellContext`x + 1)^(1/ 3)], $CellContext`p5, {$CellContext`xstart, $CellContext`xstop, \ $CellContext`ystart, $CellContext`ystop, $CellContext`kchangesides, \ {$CellContext`koffsetleft, $CellContext`koffsetright}, \ $CellContext`deltaxpos, $CellContext`xyinset} = {-2, 5, -5, 21, -1, { 1., -0.5}, -1, {3, 18}}; $CellContext`f = Function[$CellContext`x, 2 $CellContext`x - 1]]; $CellContext`xaddlength = ($CellContext`xstop - \ $CellContext`xstart)/10; $CellContext`secant = Function[$CellContext`x$, (($CellContext`f[$CellContext`x1] - \ $CellContext`f[$CellContext`x0$$])/$CellContext`deltax$$) $CellContext`x$ + \ ($CellContext`x1 $CellContext`f[$CellContext`x0$$] - $CellContext`x0$$ \ $CellContext`f[$CellContext`x1])/$CellContext`deltax$$]; $CellContext`m = \ ($CellContext`f[$CellContext`x1] - \ $CellContext`f[$CellContext`x0$$])/$CellContext`deltax$$; $CellContext`\ \[Lambda] = (($CellContext`ystop - $CellContext`ystart)/($CellContext`xstop - \ $CellContext`xstart))/(1/GoldenRatio); Plot[ $CellContext`f[$CellContext`x], {$CellContext`x, \ $CellContext`xstart, $CellContext`xstop}, PlotRange -> {{$CellContext`xstart, $CellContext`xstop}, \ {$CellContext`ystart, $CellContext`ystop}}, ImageSize -> 600, AxesLabel -> { Style["x ", 14, Italic], Style["y", 14, Italic]}, Background -> RGBColor[0.972549, 0.937255, 0.694118], PlotStyle -> { RGBColor[0.378912, 0.742199, 0.570321], Thickness[0.005]}, Epilog -> {Black, Arrowheads[0.025], Arrow[{{$CellContext`xstop - 0.02, 0}, {$CellContext`xstop, 0}}], Arrow[{{0, $CellContext`ystop - 0.02}, {0, $CellContext`ystop}}], Red, PointSize[0.015], Disk[{$CellContext`x0$$, $CellContext`f[$CellContext`x0$$]}, If[$CellContext`deltax$$ > 0.3, 0.2, 0.01] { 1, $CellContext`\[Lambda]}, $CellContext`arctan[$CellContext`m, $CellContext`\[Lambda]]], Black, Line[{{$CellContext`x0$$, $CellContext`f[$CellContext`x0$$]}, {$CellContext`x1, $CellContext`f[$CellContext`x0$$]}, {$CellContext`x1, $CellContext`f[$CellContext`x1]}}], Text[ Style[ "\[CapitalDelta]x", 14, Bold], {$CellContext`x0$$ + ($CellContext`x1 - \ $CellContext`x0$$)/ 2, $CellContext`f[$CellContext`x0$$] + $CellContext`deltaxpos}], Text[ Style[ "\[CapitalDelta]y", Medium], {$CellContext`x1 + 0.2, $CellContext`f[$CellContext`x0$$] + \ ($CellContext`f[$CellContext`x1] - $CellContext`f[$CellContext`x0$$])/2}], Blue, Thickness[0.005], Line[{{$CellContext`x0$$ - $CellContext`xaddlength, $CellContext`secant[$CellContext`x0$$ - \ $CellContext`xaddlength]}, {$CellContext`x1 + $CellContext`xaddlength, $CellContext`secant[$CellContext`x1 + \ $CellContext`xaddlength]}}], Red, Point[{$CellContext`x0$$, $CellContext`f[$CellContext`x0$$]}], Point[{$CellContext`x1, $CellContext`f[$CellContext`x1]}], Style[ Inset[ Row[{ Style["m", Italic], " = ", If[$CellContext`deltax$$ > 0.0001, ToString[ NumberForm[($CellContext`f[$CellContext`x1] - \ $CellContext`f[$CellContext`x0$$])/$CellContext`deltax$$, {5, 2}]], Style[ Text["undefined"], Red, 14]]}], If[$CellContext`x0$$ < $CellContext`kchangesides, \ {$CellContext`x0$$ + $CellContext`koffsetleft, $CellContext`f[$CellContext`x0$$]}, {$CellContext`x0$$ + \ $CellContext`koffsetright, $CellContext`f[$CellContext`x0$$]}]], 14], Style[ Inset[ ReplaceRepeated[ HoldForm[ Style["f(x)", Italic] = $CellContext`p], {$CellContext`p -> \ $CellContext`f[$CellContext`x]}], $CellContext`xyinset], RGBColor[0.378912, 0.742199, 0.570321], 20]}]), "Specifications" :> {{{$CellContext`func$$, $CellContext`p2, "functions"}, {$CellContext`p2 -> "- \!\(\*SuperscriptBox[\(x\), \(2\)]\) + 4 x + 5", $CellContext`p3 -> " \!\(\*SuperscriptBox[\(x\), \(3\)]\)- 6 \ \!\(\*SuperscriptBox[\(x\), \(2\)]\)+ 5 x + 12", $CellContext`p4 -> " (x + 1)^ (1/3)", $CellContext`p5 -> " 2x - 1"}}, Delimiter, {{$CellContext`x0$$, 1, Subscript[$CellContext`x, 0]}, Dynamic[ Switch[$CellContext`func$$, $CellContext`p2, Apply[Sequence, {-1, 4.9}], $CellContext`p3, Apply[Sequence, {-1, 3.9}], $CellContext`p4, Apply[Sequence, {-1, 2.8}], $CellContext`p5, Apply[Sequence, {1, 7.9}]]], ControlType -> LabeledSlider}, {{$CellContext`deltax$$, 1, $CellContext`\[CapitalDelta]x}, Rational[1, 10000000], 4, Appearance -> "Labeled"}}, "Options" :> { TrackedSymbols :> Manipulate, AutorunSequencing -> {{1, 25}}, FrameLabel -> Row[{ HoldForm[$CellContext`m], ", the average rate of change"}]}, "DefaultOptions" :> {}], ImageSizeCache->{649., {292., 299.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`x1 = 2.0100000000000002`, $CellContext`xstart = -2, $CellContext`xstop = 6.1, $CellContext`ystart = -1, $CellContext`ystop = 11, $CellContext`kchangesides = -1, $CellContext`koffsetleft = 1.6, $CellContext`koffsetright = -0.6, $CellContext`deltaxpos = -1, \ $CellContext`xyinset = {4.5, 10}, $CellContext`f = Function[$CellContext`x, -$CellContext`x^2 + 4 $CellContext`x + 5], $CellContext`x[ Pattern[$CellContext`t, Blank[]], Pattern[$CellContext`x0, Blank[]], Pattern[$CellContext`r, Blank[]]] := 1/(1 + (1/$CellContext`x0 - 1) Exp[(-$CellContext`r) $CellContext`t]), $CellContext`xaddlength = 0.81, $CellContext`secant = Function[$CellContext`x$, (($CellContext`f[$CellContext`x1] - \ $CellContext`f[FE`x0$$193])/ FE`deltax$$193) $CellContext`x$ + ($CellContext`x1 $CellContext`f[ FE`x0$$193] - FE`x0$$193 $CellContext`f[$CellContext`x1])/ FE`deltax$$193], Attributes[$CellContext`x$] = {Temporary}, FE`x0$$193 = 1.0100000000000002`, FE`deltax$$193 = 1, $CellContext`m = 0.9800000000000004, $CellContext`\[Lambda] = 2.397087390740585, Attributes[PlotRange] = {ReadProtected}, $CellContext`arctan[ Condition[ Pattern[$CellContext`\[Alpha], Blank[]], $CellContext`\[Alpha] >= 0], Pattern[$CellContext`b, Blank[]]] := {0, ArcTan[$CellContext`\[Alpha]/$CellContext`b]}, $CellContext`arctan[ Condition[ Pattern[$CellContext`\[Alpha], Blank[]], $CellContext`\[Alpha] < 0], Pattern[$CellContext`b, Blank[]]] := { ArcTan[$CellContext`\[Alpha]/$CellContext`b], 0}, Attributes[Subscript] = {NHoldRest}, Subscript[$CellContext`\[Phi], $CellContext`v][ Pattern[$CellContext`i, Blank[]]] := Exp[(Subscript[$CellContext`Z, $CellContext`v] - 1) ($CellContext`Bp[$CellContext`i]/ Subscript[$CellContext`B, $CellContext`v]) - Log[ Subscript[$CellContext`Z, $CellContext`v] - Subscript[$CellContext`B, $CellContext`v]] - (( Subscript[$CellContext`A, $CellContext`v]/((2 Sqrt[2]) Subscript[$CellContext`B, $CellContext`v])) ( 2 Sum[$CellContext`y[$CellContext`j] \ ($CellContext`Ab[$CellContext`i, $CellContext`j]/ Subscript[$CellContext`A, $CellContext`v]), {$CellContext`j, 1, 4}] - $CellContext`Bp[$CellContext`i]/ Subscript[$CellContext`B, $CellContext`v])) Log[(Subscript[$CellContext`Z, $CellContext`v] + (1 + Sqrt[2]) Subscript[$CellContext`B, $CellContext`v])/( Subscript[$CellContext`Z, $CellContext`v] + (1 - Sqrt[2]) Subscript[$CellContext`B, $CellContext`v])]], Subscript[$CellContext`\[Phi], $CellContext`l][ Pattern[$CellContext`i, Blank[]]] := Exp[(Subscript[$CellContext`Z, $CellContext`l] - 1) ($CellContext`Bp[$CellContext`i]/ Subscript[$CellContext`B, $CellContext`l]) - Log[ Subscript[$CellContext`Z, $CellContext`l] - Subscript[$CellContext`B, $CellContext`l]] - (( Subscript[$CellContext`A, $CellContext`l]/((2 Sqrt[2]) Subscript[$CellContext`B, $CellContext`l])) ( 2 Sum[$CellContext`x[$CellContext`j] \ ($CellContext`Ab[$CellContext`i, $CellContext`j]/ Subscript[$CellContext`A, $CellContext`l]), {$CellContext`j, 1, 4}] - $CellContext`Bp[$CellContext`i]/ Subscript[$CellContext`B, $CellContext`l])) Log[(Subscript[$CellContext`Z, $CellContext`l] + (1 + Sqrt[2]) Subscript[$CellContext`B, $CellContext`l])/( Subscript[$CellContext`Z, $CellContext`l] + (1 - Sqrt[2]) Subscript[$CellContext`B, $CellContext`l])]], Subscript[Notebook$$24$403756`\[Phi], Notebook$$24$403756`v][ Pattern[Notebook$$24$403756`i, Blank[]]] := Exp[(Subscript[Notebook$$24$403756`Z, Notebook$$24$403756`v] - 1) ( Notebook$$24$403756`Bp[Notebook$$24$403756`i]/Subscript[ Notebook$$24$403756`B, Notebook$$24$403756`v]) - Log[ Subscript[Notebook$$24$403756`Z, Notebook$$24$403756`v] - Subscript[ Notebook$$24$403756`B, Notebook$$24$403756`v]] - (( Subscript[Notebook$$24$403756`A, Notebook$$24$403756`v]/((2 Sqrt[2]) Subscript[Notebook$$24$403756`B, Notebook$$24$403756`v])) ( 2 Sum[Notebook$$24$403756`y[Notebook$$24$403756`j] ( Notebook$$24$403756`Ab[ Notebook$$24$403756`i, Notebook$$24$403756`j]/Subscript[ Notebook$$24$403756`A, Notebook$$24$403756`v]), { Notebook$$24$403756`j, 1, 4}] - Notebook$$24$403756`Bp[Notebook$$24$403756`i]/Subscript[ Notebook$$24$403756`B, Notebook$$24$403756`v])) Log[(Subscript[ Notebook$$24$403756`Z, Notebook$$24$403756`v] + (1 + Sqrt[2]) Subscript[Notebook$$24$403756`B, Notebook$$24$403756`v])/( Subscript[ Notebook$$24$403756`Z, Notebook$$24$403756`v] + (1 - Sqrt[2]) Subscript[Notebook$$24$403756`B, Notebook$$24$403756`v])]], Subscript[Notebook$$24$403756`\[Phi], Notebook$$24$403756`l][ Pattern[Notebook$$24$403756`i, Blank[]]] := Exp[(Subscript[Notebook$$24$403756`Z, Notebook$$24$403756`l] - 1) ( Notebook$$24$403756`Bp[Notebook$$24$403756`i]/Subscript[ Notebook$$24$403756`B, Notebook$$24$403756`l]) - Log[ Subscript[Notebook$$24$403756`Z, Notebook$$24$403756`l] - Subscript[ Notebook$$24$403756`B, Notebook$$24$403756`l]] - (( Subscript[Notebook$$24$403756`A, Notebook$$24$403756`l]/((2 Sqrt[2]) Subscript[Notebook$$24$403756`B, Notebook$$24$403756`l])) ( 2 Sum[Notebook$$24$403756`x[Notebook$$24$403756`j] ( Notebook$$24$403756`Ab[ Notebook$$24$403756`i, Notebook$$24$403756`j]/Subscript[ Notebook$$24$403756`A, Notebook$$24$403756`l]), { Notebook$$24$403756`j, 1, 4}] - Notebook$$24$403756`Bp[Notebook$$24$403756`i]/Subscript[ Notebook$$24$403756`B, Notebook$$24$403756`l])) Log[(Subscript[ Notebook$$24$403756`Z, Notebook$$24$403756`l] + (1 + Sqrt[2]) Subscript[Notebook$$24$403756`B, Notebook$$24$403756`l])/( Subscript[ Notebook$$24$403756`Z, Notebook$$24$403756`l] + (1 - Sqrt[2]) Subscript[Notebook$$24$403756`B, Notebook$$24$403756`l])]], Subscript[Notebook$$37$419844`\[Phi], Notebook$$37$419844`v][ Pattern[Notebook$$37$419844`i, Blank[]]] := Exp[(Subscript[Notebook$$37$419844`Z, Notebook$$37$419844`v] - 1) ( Notebook$$37$419844`Bp[Notebook$$37$419844`i]/Subscript[ Notebook$$37$419844`B, Notebook$$37$419844`v]) - Log[ Subscript[Notebook$$37$419844`Z, Notebook$$37$419844`v] - Subscript[ Notebook$$37$419844`B, Notebook$$37$419844`v]] - (( Subscript[Notebook$$37$419844`A, Notebook$$37$419844`v]/((2 Sqrt[2]) Subscript[Notebook$$37$419844`B, Notebook$$37$419844`v])) ( 2 Sum[Notebook$$37$419844`y[Notebook$$37$419844`j] ( Notebook$$37$419844`Ab[ Notebook$$37$419844`i, Notebook$$37$419844`j]/Subscript[ Notebook$$37$419844`A, Notebook$$37$419844`v]), { Notebook$$37$419844`j, 1, 4}] - Notebook$$37$419844`Bp[Notebook$$37$419844`i]/Subscript[ Notebook$$37$419844`B, Notebook$$37$419844`v])) Log[(Subscript[ Notebook$$37$419844`Z, Notebook$$37$419844`v] + (1 + Sqrt[2]) Subscript[Notebook$$37$419844`B, Notebook$$37$419844`v])/( Subscript[ Notebook$$37$419844`Z, Notebook$$37$419844`v] + (1 - Sqrt[2]) Subscript[Notebook$$37$419844`B, Notebook$$37$419844`v])]], Subscript[Notebook$$37$419844`\[Phi], Notebook$$37$419844`l][ Pattern[Notebook$$37$419844`i, Blank[]]] := Exp[(Subscript[Notebook$$37$419844`Z, Notebook$$37$419844`l] - 1) ( Notebook$$37$419844`Bp[Notebook$$37$419844`i]/Subscript[ Notebook$$37$419844`B, Notebook$$37$419844`l]) - Log[ Subscript[Notebook$$37$419844`Z, Notebook$$37$419844`l] - Subscript[ Notebook$$37$419844`B, Notebook$$37$419844`l]] - (( Subscript[Notebook$$37$419844`A, Notebook$$37$419844`l]/((2 Sqrt[2]) Subscript[Notebook$$37$419844`B, Notebook$$37$419844`l])) ( 2 Sum[Notebook$$37$419844`x[Notebook$$37$419844`j] ( Notebook$$37$419844`Ab[ Notebook$$37$419844`i, Notebook$$37$419844`j]/Subscript[ Notebook$$37$419844`A, Notebook$$37$419844`l]), { Notebook$$37$419844`j, 1, 4}] - Notebook$$37$419844`Bp[Notebook$$37$419844`i]/Subscript[ Notebook$$37$419844`B, Notebook$$37$419844`l])) Log[(Subscript[ Notebook$$37$419844`Z, Notebook$$37$419844`l] + (1 + Sqrt[2]) Subscript[Notebook$$37$419844`B, Notebook$$37$419844`l])/( Subscript[ Notebook$$37$419844`Z, Notebook$$37$419844`l] + (1 - Sqrt[2]) Subscript[Notebook$$37$419844`B, Notebook$$37$419844`l])]], Subscript[$CellContext`A, $CellContext`l] = 0.11899451239217573` $CellContext`x[1]^2 + ( 0.3258622941791799 $CellContext`x[1]) $CellContext`x[2] + 0.2239183443346985 $CellContext`x[2]^2 + ( 0.4130916755929901 $CellContext`x[1]) $CellContext`x[3] + ( 0.5685569625324797 $CellContext`x[2]) $CellContext`x[3] + 0.3618641784876383 $CellContext`x[3]^2 + ( 0.49961024025903095` $CellContext`x[1]) $CellContext`x[4] + ( 0.6887637621052163 $CellContext`x[2]) $CellContext`x[4] + ( 0.8775085005011348 $CellContext`x[3]) $CellContext`x[4] + 0.5330263931165605 $CellContext`x[4]^2, Subscript[$CellContext`A, $CellContext`v] = 0.12315418958000145` $CellContext`y[1]^2 + ( 0.33721549894635505` $CellContext`y[1]) $CellContext`y[2] + 0.23169371348661255` $CellContext`y[2]^2 + ( 0.4274534455075868 $CellContext`y[1]) $CellContext`y[3] + ( 0.5882575644139354 $CellContext`y[2]) $CellContext`y[3] + 0.3743760813515885 $CellContext`y[3]^2 + ( 0.5169875052836104 $CellContext`y[1]) $CellContext`y[4] + ( 0.7126399498841969 $CellContext`y[2]) $CellContext`y[4] + ( 0.9078626925373003 $CellContext`y[3]) $CellContext`y[4] + 0.5514725353873895 $CellContext`y[4]^2, Subscript[Notebook$$24$403756`A, Notebook$$24$403756`l] = 0.11899451239217573` Notebook$$24$403756`x[1]^2 + (0.3258622941791799 Notebook$$24$403756`x[1]) Notebook$$24$403756`x[2] + 0.2239183443346985 Notebook$$24$403756`x[2]^2 + (0.4130916755929901 Notebook$$24$403756`x[1]) Notebook$$24$403756`x[3] + (0.5685569625324797 Notebook$$24$403756`x[2]) Notebook$$24$403756`x[3] + 0.3618641784876383 Notebook$$24$403756`x[3]^2 + (0.49961024025903095` Notebook$$24$403756`x[1]) Notebook$$24$403756`x[4] + (0.6887637621052163 Notebook$$24$403756`x[2]) Notebook$$24$403756`x[4] + (0.8775085005011348 Notebook$$24$403756`x[3]) Notebook$$24$403756`x[4] + 0.5330263931165605 Notebook$$24$403756`x[4]^2, Subscript[Notebook$$24$403756`A, Notebook$$24$403756`v] = 0.12315418958000145` Notebook$$24$403756`y[1]^2 + (0.33721549894635505` Notebook$$24$403756`y[1]) Notebook$$24$403756`y[2] + 0.23169371348661255` Notebook$$24$403756`y[2]^2 + (0.4274534455075868 Notebook$$24$403756`y[1]) Notebook$$24$403756`y[3] + (0.5882575644139354 Notebook$$24$403756`y[2]) Notebook$$24$403756`y[3] + 0.3743760813515885 Notebook$$24$403756`y[3]^2 + (0.5169875052836104 Notebook$$24$403756`y[1]) Notebook$$24$403756`y[4] + (0.7126399498841969 Notebook$$24$403756`y[2]) Notebook$$24$403756`y[4] + (0.9078626925373003 Notebook$$24$403756`y[3]) Notebook$$24$403756`y[4] + 0.5514725353873895 Notebook$$24$403756`y[4]^2, Subscript[Notebook$$37$419844`A, Notebook$$37$419844`l] = 0.11899451239217573` Notebook$$37$419844`x[1]^2 + (0.3258622941791799 Notebook$$37$419844`x[1]) Notebook$$37$419844`x[2] + 0.2239183443346985 Notebook$$37$419844`x[2]^2 + (0.4130916755929901 Notebook$$37$419844`x[1]) Notebook$$37$419844`x[3] + (0.5685569625324797 Notebook$$37$419844`x[2]) Notebook$$37$419844`x[3] + 0.3618641784876383 Notebook$$37$419844`x[3]^2 + (0.49961024025903095` Notebook$$37$419844`x[1]) Notebook$$37$419844`x[4] + (0.6887637621052163 Notebook$$37$419844`x[2]) Notebook$$37$419844`x[4] + (0.8775085005011348 Notebook$$37$419844`x[3]) Notebook$$37$419844`x[4] + 0.5330263931165605 Notebook$$37$419844`x[4]^2, Subscript[Notebook$$37$419844`A, Notebook$$37$419844`v] = 0.12315418958000145` Notebook$$37$419844`y[1]^2 + (0.33721549894635505` Notebook$$37$419844`y[1]) Notebook$$37$419844`y[2] + 0.23169371348661255` Notebook$$37$419844`y[2]^2 + (0.4274534455075868 Notebook$$37$419844`y[1]) Notebook$$37$419844`y[3] + (0.5882575644139354 Notebook$$37$419844`y[2]) Notebook$$37$419844`y[3] + 0.3743760813515885 Notebook$$37$419844`y[3]^2 + (0.5169875052836104 Notebook$$37$419844`y[1]) Notebook$$37$419844`y[4] + (0.7126399498841969 Notebook$$37$419844`y[2]) Notebook$$37$419844`y[4] + (0.9078626925373003 Notebook$$37$419844`y[3]) Notebook$$37$419844`y[4] + 0.5514725353873895 Notebook$$37$419844`y[4]^2, Subscript[$CellContext`B, $CellContext`l] = 0.026892820713045433` $CellContext`x[1] + 0.0374327774133094 $CellContext`x[2] + 0.04823100360190885 $CellContext`x[3] + 0.05995452136531292 $CellContext`x[4], Subscript[$CellContext`B, $CellContext`v] = 0.026892820713045433` $CellContext`y[1] + 0.0374327774133094 $CellContext`y[2] + 0.04823100360190885 $CellContext`y[3] + 0.05995452136531292 $CellContext`y[4], Subscript[Notebook$$24$403756`B, Notebook$$24$403756`l] = 0.026892820713045433` Notebook$$24$403756`x[1] + 0.0374327774133094 Notebook$$24$403756`x[2] + 0.04823100360190885 Notebook$$24$403756`x[3] + 0.05995452136531292 Notebook$$24$403756`x[4], Subscript[Notebook$$24$403756`B, Notebook$$24$403756`v] = 0.026892820713045433` Notebook$$24$403756`y[1] + 0.0374327774133094 Notebook$$24$403756`y[2] + 0.04823100360190885 Notebook$$24$403756`y[3] + 0.05995452136531292 Notebook$$24$403756`y[4], Subscript[Notebook$$37$419844`B, Notebook$$37$419844`l] = 0.026892820713045433` Notebook$$37$419844`x[1] + 0.0374327774133094 Notebook$$37$419844`x[2] + 0.04823100360190885 Notebook$$37$419844`x[3] + 0.05995452136531292 Notebook$$37$419844`x[4], Subscript[Notebook$$37$419844`B, Notebook$$37$419844`v] = 0.026892820713045433` Notebook$$37$419844`y[1] + 0.0374327774133094 Notebook$$37$419844`y[2] + 0.04823100360190885 Notebook$$37$419844`y[3] + 0.05995452136531292 Notebook$$37$419844`y[4], $CellContext`Bp[ Pattern[$CellContext`i$, Blank[]]] := $CellContext`P \ ($CellContext`bp[$CellContext`i$]/($CellContext`R FE`T$$448)), Attributes[$CellContext`i$] = {Temporary}, $CellContext`P = 293.8, $CellContext`bp[ Pattern[$CellContext`i, Blank[]]] := ( 0.0778 $CellContext`R) \ ($CellContext`Tc[$CellContext`i]/$CellContext`Pc[$CellContext`i]), \ $CellContext`R = 1.987, $CellContext`Tc[1] = 550., $CellContext`Tc[2] = 665.9, $CellContext`Tc[3] = 765.3, $CellContext`Tc[4] = 845.6, $CellContext`Pc[1] = 709.8, $CellContext`Pc[2] = 617.4, $CellContext`Pc[3] = 550.7, $CellContext`Pc[4] = 489.5, FE`T$$448 = 658.6, $CellContext`Ab[ Pattern[$CellContext`i, Blank[]], Pattern[$CellContext`j, Blank[]]] = ( 0.0003097080935007487 ( 1 - $CellContext`k[$CellContext`i, $CellContext`j])) \ (((($CellContext`Tc[$CellContext`i]^2 $CellContext`Tc[$CellContext`j]^2) ( 1 + (1 - 25.663203229526903` \ ($CellContext`Tc[$CellContext`i]^(-1))^0.5) (0.37464 + 1.54226 $CellContext`\[Omega][$CellContext`i] - 0.26992 $CellContext`\[Omega][$CellContext`i]^2))^2) ( 1 + (1 - 25.663203229526903` \ ($CellContext`Tc[$CellContext`j]^(-1))^0.5) (0.37464 + 1.54226 $CellContext`\[Omega][$CellContext`j] - 0.26992 \ $CellContext`\[Omega][$CellContext`j]^2))^2)/($CellContext`Pc[$CellContext`i] \ $CellContext`Pc[$CellContext`j]))^0.5, $CellContext`k[1, 1] = 0, $CellContext`k[1, 2] = 0.00185, $CellContext`k[1, 3] = 0.00464, $CellContext`k[1, 4] = 0.00811, $CellContext`k[2, 1] = 0.00185, $CellContext`k[2, 2] = 0, $CellContext`k[2, 3] = 0.00132, $CellContext`k[2, 4] = 0.00317, $CellContext`k[3, 1] = 0.00464, $CellContext`k[3, 2] = 0.00132, $CellContext`k[3, 3] = 0, $CellContext`k[3, 4] = 0.00098, $CellContext`k[4, 1] = 0.00811, $CellContext`k[4, 2] = 0.00317, $CellContext`k[4, 3] = 0.00098, $CellContext`k[4, 4] = 0, $CellContext`\[Omega][1] = 0.1064, $CellContext`\[Omega][2] = 0.1538, $CellContext`\[Omega][3] = 0.1954, $CellContext`\[Omega][4] = 0.2387, Notebook$$24$403756`Bp[ Pattern[Notebook$$24$403756`i$, Blank[]]] := Notebook$$24$403756`P (Notebook$$24$403756`bp[Notebook$$24$403756`i$]/( Notebook$$24$403756`R FE`T$$448)), Attributes[Notebook$$24$403756`i$] = {Temporary}, Notebook$$24$403756`P = 293.8, Notebook$$24$403756`bp[ Pattern[Notebook$$24$403756`i, Blank[]]] := (0.0778 Notebook$$24$403756`R) ( Notebook$$24$403756`Tc[Notebook$$24$403756`i]/Notebook$$24$403756`Pc[ Notebook$$24$403756`i]), Notebook$$24$403756`R = 1.987, Notebook$$24$403756`Tc[1] = 550., Notebook$$24$403756`Tc[2] = 665.9, Notebook$$24$403756`Tc[3] = 765.3, Notebook$$24$403756`Tc[4] = 845.6, Notebook$$24$403756`Pc[1] = 709.8, Notebook$$24$403756`Pc[2] = 617.4, Notebook$$24$403756`Pc[3] = 550.7, Notebook$$24$403756`Pc[4] = 489.5, Notebook$$24$403756`Ab[ Pattern[Notebook$$24$403756`i, Blank[]], Pattern[Notebook$$24$403756`j, Blank[]]] = ( 0.0003097080935007487 (1 - Notebook$$24$403756`k[ Notebook$$24$403756`i, Notebook$$24$403756`j])) (((( Notebook$$24$403756`Tc[Notebook$$24$403756`i]^2 Notebook$$24$403756`Tc[Notebook$$24$403756`j]^2) ( 1 + (1 - 25.663203229526903` ( Notebook$$24$403756`Tc[Notebook$$24$403756`i]^(-1))^0.5) ( 0.37464 + 1.54226 Notebook$$24$403756`\[Omega][Notebook$$24$403756`i] - 0.26992 Notebook$$24$403756`\[Omega][Notebook$$24$403756`i]^2))^2) ( 1 + (1 - 25.663203229526903` ( Notebook$$24$403756`Tc[Notebook$$24$403756`j]^(-1))^0.5) ( 0.37464 + 1.54226 Notebook$$24$403756`\[Omega][Notebook$$24$403756`j] - 0.26992 Notebook$$24$403756`\[Omega][Notebook$$24$403756`j]^2))^2)/( Notebook$$24$403756`Pc[Notebook$$24$403756`i] Notebook$$24$403756`Pc[Notebook$$24$403756`j]))^0.5, Notebook$$24$403756`k[1, 1] = 0, Notebook$$24$403756`k[1, 2] = 0.00185, Notebook$$24$403756`k[1, 3] = 0.00464, Notebook$$24$403756`k[1, 4] = 0.00811, Notebook$$24$403756`k[2, 1] = 0.00185, Notebook$$24$403756`k[2, 2] = 0, Notebook$$24$403756`k[2, 3] = 0.00132, Notebook$$24$403756`k[2, 4] = 0.00317, Notebook$$24$403756`k[3, 1] = 0.00464, Notebook$$24$403756`k[3, 2] = 0.00132, Notebook$$24$403756`k[3, 3] = 0, Notebook$$24$403756`k[3, 4] = 0.00098, Notebook$$24$403756`k[4, 1] = 0.00811, Notebook$$24$403756`k[4, 2] = 0.00317, Notebook$$24$403756`k[4, 3] = 0.00098, Notebook$$24$403756`k[4, 4] = 0, Notebook$$24$403756`\[Omega][1] = 0.1064, Notebook$$24$403756`\[Omega][2] = 0.1538, Notebook$$24$403756`\[Omega][3] = 0.1954, Notebook$$24$403756`\[Omega][4] = 0.2387, Notebook$$37$419844`Bp[ Pattern[Notebook$$37$419844`i$, Blank[]]] := Notebook$$37$419844`P (Notebook$$37$419844`bp[Notebook$$37$419844`i$]/( Notebook$$37$419844`R FE`T$$448)), Attributes[Notebook$$37$419844`i$] = {Temporary}, Notebook$$37$419844`P = 293.8, Notebook$$37$419844`bp[ Pattern[Notebook$$37$419844`i, Blank[]]] := (0.0778 Notebook$$37$419844`R) ( Notebook$$37$419844`Tc[Notebook$$37$419844`i]/Notebook$$37$419844`Pc[ Notebook$$37$419844`i]), Notebook$$37$419844`R = 1.987, Notebook$$37$419844`Tc[1] = 550., Notebook$$37$419844`Tc[2] = 665.9, Notebook$$37$419844`Tc[3] = 765.3, Notebook$$37$419844`Tc[4] = 845.6, Notebook$$37$419844`Pc[1] = 709.8, Notebook$$37$419844`Pc[2] = 617.4, Notebook$$37$419844`Pc[3] = 550.7, Notebook$$37$419844`Pc[4] = 489.5, Notebook$$37$419844`Ab[ Pattern[Notebook$$37$419844`i, Blank[]], Pattern[Notebook$$37$419844`j, Blank[]]] = ( 0.0003097080935007487 (1 - Notebook$$37$419844`k[ Notebook$$37$419844`i, Notebook$$37$419844`j])) (((( Notebook$$37$419844`Tc[Notebook$$37$419844`i]^2 Notebook$$37$419844`Tc[Notebook$$37$419844`j]^2) ( 1 + (1 - 25.663203229526903` ( Notebook$$37$419844`Tc[Notebook$$37$419844`i]^(-1))^0.5) ( 0.37464 + 1.54226 Notebook$$37$419844`\[Omega][Notebook$$37$419844`i] - 0.26992 Notebook$$37$419844`\[Omega][Notebook$$37$419844`i]^2))^2) ( 1 + (1 - 25.663203229526903` ( Notebook$$37$419844`Tc[Notebook$$37$419844`j]^(-1))^0.5) ( 0.37464 + 1.54226 Notebook$$37$419844`\[Omega][Notebook$$37$419844`j] - 0.26992 Notebook$$37$419844`\[Omega][Notebook$$37$419844`j]^2))^2)/( Notebook$$37$419844`Pc[Notebook$$37$419844`i] Notebook$$37$419844`Pc[Notebook$$37$419844`j]))^0.5, Notebook$$37$419844`k[1, 1] = 0, Notebook$$37$419844`k[1, 2] = 0.00185, Notebook$$37$419844`k[1, 3] = 0.00464, Notebook$$37$419844`k[1, 4] = 0.00811, Notebook$$37$419844`k[2, 1] = 0.00185, Notebook$$37$419844`k[2, 2] = 0, Notebook$$37$419844`k[2, 3] = 0.00132, Notebook$$37$419844`k[2, 4] = 0.00317, Notebook$$37$419844`k[3, 1] = 0.00464, Notebook$$37$419844`k[3, 2] = 0.00132, Notebook$$37$419844`k[3, 3] = 0, Notebook$$37$419844`k[3, 4] = 0.00098, Notebook$$37$419844`k[4, 1] = 0.00811, Notebook$$37$419844`k[4, 2] = 0.00317, Notebook$$37$419844`k[4, 3] = 0.00098, Notebook$$37$419844`k[4, 4] = 0, Notebook$$37$419844`\[Omega][1] = 0.1064, Notebook$$37$419844`\[Omega][2] = 0.1538, Notebook$$37$419844`\[Omega][3] = 0.1954, Notebook$$37$419844`\[Omega][4] = 0.2387}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, 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