(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 9.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 9789, 293] NotebookOptionsPosition[ 9304, 272] NotebookOutlinePosition[ 9661, 288] CellTagsIndexPosition[ 9618, 285] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Intermediate Value Theorem", "Title", CellChangeTimes->{{3.573775627820302*^9, 3.573775633865159*^9}}, TextAlignment->Center], Cell["\<\ by Ruth Dover Illinois Mathematics and Science Academy\ \>", "Subsubtitle", CellChangeTimes->{{3.693181167679359*^9, 3.6931811802902737`*^9}}], Cell[TextData[{ "An example is given, but you may edit the function and the values for ", StyleBox["xmin", FontSlant->"Italic"], " and ", StyleBox["xmax", FontSlant->"Italic"], " as desired. Use the sliders to consider the interval [", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], "]. Does the function take on all ", StyleBox["y", FontSlant->"Italic"], "-values between \[Florin](", StyleBox["a", FontSlant->"Italic"], ") and \[Florin](", StyleBox["b", FontSlant->"Italic"], ")? That is, does \[Florin] take on all ", StyleBox["y", FontSlant->"Italic"], "-values in the rectangle?" }], "Text", CellChangeTimes->{{3.573775681938868*^9, 3.573775714474535*^9}, { 3.573775785894701*^9, 3.5737758302942867`*^9}, {3.573775866345705*^9, 3.573775876863049*^9}, {3.5737759662955093`*^9, 3.573775996134366*^9}, { 3.573776079861274*^9, 3.573776094506731*^9}, {3.573776163521154*^9, 3.5737761852130013`*^9}, {3.693181194454349*^9, 3.693181233222698*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", "x_", "]"}], ":=", RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"3", "x"}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"xmin", "=", RowBox[{"-", "2"}]}], ";", " ", RowBox[{"xmax", "=", "4"}], ";"}], "\[IndentingNewLine]", RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"f", "[", "x", "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "xmin", ",", " ", "xmax"}], "}"}], ",", " ", RowBox[{"PlotStyle", "\[Rule]", "Thick"}], ",", RowBox[{"Epilog", "\[Rule]", RowBox[{"{", " ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Hue", "[", ".55", "]"}], ",", RowBox[{"Opacity", "[", ".4", "]"}], ",", RowBox[{"Rectangle", "[", RowBox[{ RowBox[{"{", RowBox[{"a", ",", RowBox[{"f", "[", "a", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"b", ",", RowBox[{"f", "[", "b", "]"}]}], "}"}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "\[Rule]", "Large"}], ",", "Red", ",", " ", RowBox[{"Point", "[", RowBox[{"{", RowBox[{"a", ",", " ", RowBox[{"f", "[", "a", "]"}]}], "}"}], "]"}], ",", " ", RowBox[{"Point", "[", RowBox[{"{", RowBox[{"b", ",", " ", RowBox[{"f", "[", "b", "]"}]}], "}"}], "]"}]}], "}"}]}], "}"}]}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"a", ",", " ", RowBox[{"xmin", "+", RowBox[{".25", RowBox[{"(", RowBox[{"xmax", "-", "xmin"}], ")"}]}]}], ",", " ", "\"\<\!\(\* StyleBox[\"a\", FontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \", FontSlant->\"Italic\"]\)= left side\>\""}], "}"}], ",", " ", "xmin", ",", " ", "b"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"b", ",", RowBox[{"xmin", "+", RowBox[{".75", RowBox[{"(", RowBox[{"xmax", "-", "xmin"}], ")"}]}]}], ",", " ", "\"\<\!\(\* StyleBox[\"b\", FontSlant->\"Italic\"]\) = right side\>\""}], "}"}], ",", " ", "a", ",", " ", "xmax"}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.573774779748012*^9, 3.573774786606036*^9}, { 3.573774827090054*^9, 3.573775123307846*^9}, {3.573775158525597*^9, 3.5737752034876757`*^9}, {3.573775317884246*^9, 3.5737753228954077`*^9}, { 3.57377547622208*^9, 3.573775606439958*^9}, {3.573775837007244*^9, 3.5737758509328003`*^9}, {3.573775954216414*^9, 3.573775954738626*^9}}], Cell[TextData[{ "Another example is given below with a piecewise-defined function. Again, \ edit the function, ", StyleBox["xmin", FontSlant->"Italic"], ", and ", StyleBox["xmax", FontSlant->"Italic"], " as desired. (Big note: The \[OpenCurlyDoubleQuote]connector\ \[CloseCurlyDoubleQuote] between the pieces may show while using the sliders, \ but will then disappear!) " }], "Text", CellChangeTimes->{{3.573776202092066*^9, 3.573776265264866*^9}, { 3.57377640938678*^9, 3.5737764535142937`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"g", "[", "x_", "]"}], ":=", TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ {"x", RowBox[{"x", "\[LessEqual]", "1"}]}, { RowBox[{"3", "-", "x"}], RowBox[{"x", ">", "1"}]} }, AllowedDimensions->{2, Automatic}, Editable->True, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}, Selectable->True]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.35]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "Piecewise", DeleteWithContents->True, Editable->False, SelectWithContents->True, Selectable->False]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"xmin", "=", RowBox[{"-", "1"}]}], ";", " ", RowBox[{"xmax", "=", "3"}], ";"}], "\[IndentingNewLine]", RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"g", "[", "x", "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "xmin", ",", " ", "xmax"}], "}"}], ",", " ", RowBox[{"PlotStyle", "\[Rule]", "Thick"}], ",", RowBox[{"Epilog", "\[Rule]", RowBox[{"{", " ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Hue", "[", ".55", "]"}], ",", RowBox[{"Opacity", "[", ".4", "]"}], ",", RowBox[{"Rectangle", "[", RowBox[{ RowBox[{"{", RowBox[{"a", ",", RowBox[{"g", "[", "a", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"b", ",", RowBox[{"g", "[", "b", "]"}]}], "}"}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "\[Rule]", "Large"}], ",", "Red", ",", " ", RowBox[{"Point", "[", RowBox[{"{", RowBox[{"a", ",", " ", RowBox[{"g", "[", "a", "]"}]}], "}"}], "]"}], ",", " ", RowBox[{"Point", "[", RowBox[{"{", RowBox[{"b", ",", " ", RowBox[{"g", "[", "b", "]"}]}], "}"}], "]"}]}], "}"}]}], "}"}]}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"a", ",", " ", RowBox[{"xmin", "+", RowBox[{".25", RowBox[{"(", RowBox[{"xmax", "-", "xmin"}], ")"}]}]}], ",", " ", "\"\<\!\(\* StyleBox[\"a\", FontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \", FontSlant->\"Italic\"]\)= left side\>\""}], "}"}], ",", " ", "xmin", ",", " ", "b"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"b", ",", RowBox[{"xmin", "+", RowBox[{".75", RowBox[{"(", RowBox[{"xmax", "-", "xmin"}], ")"}]}]}], ",", " ", "\"\<\!\(\* StyleBox[\"b\", FontSlant->\"Italic\"]\) = right side\>\""}], "}"}], ",", " ", "a", ",", " ", "xmax"}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.573774779748012*^9, 3.573774786606036*^9}, { 3.573774827090054*^9, 3.573775123307846*^9}, {3.573775158525597*^9, 3.5737752034876757`*^9}, {3.573775317884246*^9, 3.5737753228954077`*^9}, { 3.57377547622208*^9, 3.573775606439958*^9}, {3.573775837007244*^9, 3.5737758509328003`*^9}, {3.573775954216414*^9, 3.573775954738626*^9}, { 3.573776278966992*^9, 3.5737763777654953`*^9}}] }, Open ]] }, WindowSize->{740, 720}, WindowMargins->{{4, Automatic}, {Automatic, 4}}, 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[CellGroupData[{ Cell[579, 22, 133, 2, 92, "Title"], Cell[715, 26, 153, 4, 49, "Subsubtitle"], Cell[871, 32, 1022, 31, 68, "Text"], Cell[1896, 65, 2695, 73, 159, "Input"], Cell[4594, 140, 512, 13, 68, "Text"], Cell[5109, 155, 4179, 114, 175, "Input"] }, Open ]] } ] *)