(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 5297, 171] NotebookOptionsPosition[ 4700, 148] NotebookOutlinePosition[ 5165, 166] CellTagsIndexPosition[ 5122, 163] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Polar Graphs and Reciprocals", "Text", TextAlignment->Center, FontSize->36, FontColor->RGBColor[1, 0, 0]], Cell[CellGroupData[{ Cell["\<\ by Ruth Dover Illinois Mathematics and Science Academy\ \>", "Subsubtitle", CellChangeTimes->{{3.692576442902491*^9, 3.692576454860837*^9}}], Cell[TextData[{ "We'll look first at an animation of polar graphs. This will show the \ transformation from the circle ", StyleBox["r", FontSlant->"Italic"], " = 3 through lima\[CCedilla]ons and into the circle ", StyleBox["r", FontSlant->"Italic"], " = 3sin(\[Theta]). " }], "Text"], Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"PolarPlot", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"3", "-", "a"}], ")"}], "+", RowBox[{"a", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}], ",", " ", RowBox[{"{", RowBox[{"\[Theta]", ",", " ", "0", ",", " ", RowBox[{"2", "\[Pi]"}]}], "}"}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "3.2"}], ",", " ", "3.2"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "3.2"}], ",", " ", "3.2"}], "}"}]}], "}"}]}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Blue", ",", " ", RowBox[{"Thickness", "[", ".01", "]"}]}], "}"}]}], ",", " ", RowBox[{"ImageSize", "\[Rule]", RowBox[{"6", " ", "72"}]}], ",", " ", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"Style", "[", RowBox[{ RowBox[{"StringForm", "[", RowBox[{"\"\<\!\(\* StyleBox[\"r\", FontSlant->\"Italic\"]\) = `` + ``*sin(\!\(\* StyleBox[\"\[Theta]\", FontSlant->\"Italic\"]\))\>\"", ",", RowBox[{"NumberForm", "[", RowBox[{ RowBox[{"Chop", "[", RowBox[{"3", "-", "a"}], "]"}], ",", " ", "3"}], "]"}], ",", " ", RowBox[{"NumberForm", "[", RowBox[{ RowBox[{"Chop", "[", "a", "]"}], ",", " ", "3"}], "]"}]}], "]"}], ",", " ", RowBox[{"FontSize", "\[Rule]", "20"}]}], "]"}]}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"a", ",", " ", "0", ",", " ", "3", ",", " ", ".05"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.397776892878455*^9, 3.397776894244699*^9}, { 3.397777037776126*^9, 3.397777078698558*^9}, 3.397777137542409*^9, { 3.397777232367491*^9, 3.3977773274206467`*^9}, 3.3977773574262543`*^9}], Cell["\<\ Now, consider the reciprocals of these lima\[CCedilla]ons (shown in red). \ \>", "Text"], Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"PolarPlot", "[", " ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"3", "-", "a"}], ")"}], "+", RowBox[{"a", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}], ",", " ", RowBox[{"1", "/", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"3", "-", "a"}], ")"}], "+", RowBox[{"a", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}], ")"}]}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"\[Theta]", ",", " ", "0", ",", " ", RowBox[{"2", "\[Pi]"}]}], "}"}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "3.2"}], ",", " ", "3.2"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "3.2"}], ",", " ", "3.2"}], "}"}]}], "}"}]}], ",", " ", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"Blue", ",", "Thick"}], "}"}], ",", RowBox[{"{", " ", RowBox[{"Red", ",", " ", "Thick"}], "}"}]}], "}"}]}], ",", " ", RowBox[{"ImageSize", "->", " ", RowBox[{"6", " ", "72"}]}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"a", ",", " ", "0", ",", " ", "3", ",", " ", ".05"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.397776917565937*^9, 3.3977769338379507`*^9}}] }, Open ]] }, AutoGeneratedPackage->None, WindowSize->{778, 659}, WindowMargins->{{12, Automatic}, {Automatic, 18}}, PrivateNotebookOptions->{"VersionedStylesheet"->{"Default.nb"[8.] -> False}}, FrontEndVersion->"11.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (September \ 21, 2016)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[545, 20, 114, 3, 55, "Text"], Cell[CellGroupData[{ Cell[684, 27, 151, 4, 49, "Subsubtitle"], Cell[838, 33, 294, 9, 49, "Text"], Cell[1135, 44, 1930, 51, 117, "Input"], Cell[3068, 97, 98, 2, 30, "Text"], Cell[3169, 101, 1515, 44, 75, "Input"] }, Open ]] } ] *)