(* 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[ 16020, 418] NotebookOptionsPosition[ 15354, 390] NotebookOutlinePosition[ 15693, 405] CellTagsIndexPosition[ 15650, 402] WindowFrame->Normal ContainsDynamic->True *) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["L'Hopital's Rule", "Title", CellChangeTimes->{{3.4308696379639664`*^9, 3.4308696409510803`*^9}, { 3.4342749987450185`*^9, 3.4342750018543935`*^9}}], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"L", "'"}], RowBox[{"Hopital", "'"}], "s", " ", "rule", " ", "says", " ", "that", " ", "if", " ", RowBox[{"f", "[", "0", "]"}]}], " ", "=", " ", RowBox[{ RowBox[{"g", "[", "0", "]"}], " ", "=", " ", "0"}]}], ",", " ", RowBox[{ RowBox[{"then", " ", "lim_", RowBox[{"{", RowBox[{ RowBox[{"x", " ", "--"}], ">", " ", "0"}], "}"}], " ", "f", RowBox[{ RowBox[{"(", "x", ")"}], "/", "g"}], RowBox[{"(", "x", ")"}], " ", "equals", " ", RowBox[{"f", "'"}], RowBox[{ RowBox[{"(", "0", ")"}], "/", RowBox[{"g", "'"}]}], RowBox[{ RowBox[{"(", "0", ")"}], ".", " ", "Below"}], " ", "we", " ", "investigate", " ", "a", " ", "few", " ", "special", " ", RowBox[{"cases", ".", " ", "If"}], " ", RowBox[{"f", "'"}], RowBox[{"(", "0", ")"}]}], " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"g", "'"}], RowBox[{"(", "0", ")"}]}], " ", "=", " ", "0"}]}], ",", " ", RowBox[{"then", " ", "one", " ", "repeats", " ", "the", " ", "process"}], ",", " ", RowBox[{"and", " ", "the", " ", "limit", " ", "equals", " ", RowBox[{"f", "''"}], RowBox[{ RowBox[{"(", "0", ")"}], "/", RowBox[{"g", "''"}]}], RowBox[{"(", "0", ")"}]}], ",", " ", RowBox[{"and", " ", "so", " ", RowBox[{"on", "."}]}]}], "Subsubsection"]], "Input", CellChangeTimes->{{3.4342750056043935`*^9, 3.4342750810575185`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"cosminus1", "[", "x_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], " ", "-", " ", "1"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"expminus1", "[", "x_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"Exp", "[", "x", "]"}], " ", "-", " ", "1"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"f", "[", "x", "]"}], "/", "x"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"f", "'"}], "[", "x", "]"}], "/", "1"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "c"}], ",", " ", "c"}], "}"}], ",", RowBox[{"PlotRange", " ", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "r"}], ",", "r"}], "}"}]}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"c", ",", " ", RowBox[{"2", " ", "Pi"}], ",", " ", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"r", ",", "2", ",", ".2"}], "}"}], ",", RowBox[{"{", RowBox[{"f", ",", " ", RowBox[{"{", RowBox[{"Sin", ",", "cosminus1", ",", "expminus1"}], "}"}]}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.430869648912484*^9, 3.430869651423066*^9}, { 3.4308697359752717`*^9, 3.4308698715007772`*^9}, {3.430869926464056*^9, 3.430869969330453*^9}, {3.430870020198291*^9, 3.4308700250759087`*^9}, { 3.4308700954776525`*^9, 3.430870104835452*^9}, 3.430870151433056*^9, { 3.430870186867311*^9, 3.43087019036904*^9}, {3.4308702256079955`*^9, 3.4308703634644065`*^9}, {3.4308704170185957`*^9, 3.430870694146366*^9}, { 3.4312722811044655`*^9, 3.431272294081174*^9}, {3.4342742044793935`*^9, 3.4342743794637685`*^9}, {3.4342744139168935`*^9, 3.4342744143387685`*^9}, {3.4342744994793935`*^9, 3.4342745073231435`*^9}, {3.4342746071512685`*^9, 3.4342746270887685`*^9}, {3.4342749102606435`*^9, 3.4342749190887685`*^9}}],\ Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`c$$ = 3.9458403729087803`, $CellContext`f$$ = $CellContext`cosminus1, \ $CellContext`r$$ = 2., Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`c$$], 2 Pi, 0}, { Hold[$CellContext`r$$], 2, 0.2}, { Hold[$CellContext`f$$], { Sin, $CellContext`cosminus1, $CellContext`expminus1}}}, Typeset`size$$ = {360., {115., 119.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`c$13592$$ = 0, $CellContext`r$13593$$ = 0, $CellContext`f$13594$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`c$$ = 2 Pi, $CellContext`f$$ = Sin, $CellContext`r$$ = 2}, "ControllerVariables" :> { Hold[$CellContext`c$$, $CellContext`c$13592$$, 0], Hold[$CellContext`r$$, $CellContext`r$13593$$, 0], Hold[$CellContext`f$$, $CellContext`f$13594$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Plot[{$CellContext`f$$[$CellContext`x]/$CellContext`x, Derivative[1][$CellContext`f$$][$CellContext`x]/ 1}, {$CellContext`x, -$CellContext`c$$, $CellContext`c$$}, PlotRange -> {-$CellContext`r$$, $CellContext`r$$}], "Specifications" :> {{$CellContext`c$$, 2 Pi, 0}, {$CellContext`r$$, 2, 0.2}, {$CellContext`f$$, { Sin, $CellContext`cosminus1, $CellContext`expminus1}}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{408., {184., 189.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.4308701917839885`*^9, {3.430870227450651*^9, 3.430870296817305*^9}, { 3.430870329565209*^9, 3.4308703637964163`*^9}, {3.4308704380533824`*^9, 3.4308705637601805`*^9}, 3.4308706631844807`*^9, 3.4308706946551228`*^9, 3.43107658999644*^9, {3.4312722740443707`*^9, 3.4312722947071104`*^9}, { 3.4342742085731435`*^9, 3.4342742613231435`*^9}, {3.4342742986512685`*^9, 3.4342743805887685`*^9}, 3.4342744153387685`*^9, {3.4342745025575185`*^9, 3.4342745082606435`*^9}, {3.4342746132918935`*^9, 3.4342746275418935`*^9}, 3.4342749198856435`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"sinminus", "[", "x_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], " ", "-", " ", "x"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"cosminus", "[", "x_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], " ", "-", " ", "1"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"expminus", "[", "x_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"Exp", "[", "x", "]"}], " ", "-", " ", "1", "-", "x"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"f", "[", "x", "]"}], "/", RowBox[{"x", "^", "2"}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"f", "'"}], "[", "x", "]"}], "/", RowBox[{"(", RowBox[{"2", "x"}], ")"}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"f", "''"}], "[", "x", "]"}], "/", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "c"}], ",", " ", "c"}], "}"}], ",", RowBox[{"PlotRange", " ", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "r"}], ",", "r"}], "}"}]}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"c", ",", " ", RowBox[{"2", " ", "Pi"}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"r", ",", "2", ",", " ", ".2"}], "}"}], ",", RowBox[{"{", RowBox[{"f", ",", " ", RowBox[{"{", RowBox[{"sinminus", ",", "cosminus", ",", "expminus"}], "}"}]}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.4342746907450185`*^9, 3.4342747596200185`*^9}, { 3.4342748176043935`*^9, 3.4342748873075185`*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`c$$ = 0.9801769079200157, $CellContext`f$$ = $CellContext`sinminus, \ $CellContext`r$$ = 0.366, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`c$$], 2 Pi, 0}, { Hold[$CellContext`r$$], 2, 0.2}, { Hold[$CellContext`f$$], {$CellContext`sinminus, $CellContext`cosminus, \ $CellContext`expminus}}}, Typeset`size$$ = {360., {106., 111.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`c$10806$$ = 0, $CellContext`r$10807$$ = 0, $CellContext`f$10808$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`c$$ = 2 Pi, $CellContext`f$$ = $CellContext`sinminus, $CellContext`r$$ = 2}, "ControllerVariables" :> { Hold[$CellContext`c$$, $CellContext`c$10806$$, 0], Hold[$CellContext`r$$, $CellContext`r$10807$$, 0], Hold[$CellContext`f$$, $CellContext`f$10808$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Plot[{$CellContext`f$$[$CellContext`x]/$CellContext`x^2, Derivative[1][$CellContext`f$$][$CellContext`x]/(2 $CellContext`x), Derivative[2][$CellContext`f$$][$CellContext`x]/ 2}, {$CellContext`x, -$CellContext`c$$, $CellContext`c$$}, PlotRange -> {-$CellContext`r$$, $CellContext`r$$}], "Specifications" :> {{$CellContext`c$$, 2 Pi, 0}, {$CellContext`r$$, 2, 0.2}, {$CellContext`f$$, {$CellContext`sinminus, \ $CellContext`cosminus, $CellContext`expminus}}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{408., {175., 180.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.4342747138387685`*^9, 3.4342747601668935`*^9, {3.4342748195887685`*^9, 3.4342748406981435`*^9}, {3.4342748753543935`*^9, 3.4342748878231435`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "you", " ", "can", " ", "input", " ", "your", " ", "functions", " ", "f", " ", "and", " ", "g", " ", "here", " ", "and", " ", "compare"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"f", "[", "x_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"x", " ", RowBox[{"Cos", "[", "x", "]"}]}], " ", "-", " ", RowBox[{"Exp", "[", "x", "]"}], "+", "1"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"g", "[", "x_", "]"}], " ", ":=", " ", RowBox[{"x", "^", "2"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"f", "[", "x", "]"}], "/", RowBox[{"g", "[", "x", "]"}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"f", "'"}], "[", "x", "]"}], "/", RowBox[{ RowBox[{"g", "'"}], "[", "x", "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "c"}], ",", " ", "c"}], "}"}], ",", RowBox[{"PlotRange", " ", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "r"}], ",", "r"}], "}"}]}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"c", ",", " ", RowBox[{"2", " ", "Pi"}], ",", " ", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"r", ",", "2", ",", ".2"}], "}"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.4342753078700185`*^9, 3.4342755675887685`*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`c$$ = 6.283185307179586, $CellContext`r$$ = 2., Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`c$$], 2 Pi, 0}, { Hold[$CellContext`r$$], 2, 0.2}}, Typeset`size$$ = {360., {115., 120.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`c$25297$$ = 0, $CellContext`r$25298$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`c$$ = 2 Pi, $CellContext`r$$ = 2}, "ControllerVariables" :> { Hold[$CellContext`c$$, $CellContext`c$25297$$, 0], Hold[$CellContext`r$$, $CellContext`r$25298$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Plot[{$CellContext`f[$CellContext`x]/$CellContext`g[$CellContext`x], Derivative[1][$CellContext`f][$CellContext`x]/Derivative[ 1][$CellContext`g][$CellContext`x]}, {$CellContext`x, \ -$CellContext`c$$, $CellContext`c$$}, PlotRange -> {-$CellContext`r$$, $CellContext`r$$}], "Specifications" :> {{$CellContext`c$$, 2 Pi, 0}, {$CellContext`r$$, 2, 0.2}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{408., {173., 178.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.4342753584325185`*^9, 3.4342753899325185`*^9, {3.4342754220106435`*^9, 3.4342755414012685`*^9}}] }, Open ]] }, Open ]] }, WindowSize->{791, 670}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, FrontEndVersion->"6.0 for Microsoft Windows (32-bit) (June 19, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[590, 23, 156, 2, 83, "Title"], Cell[749, 27, 1538, 43, 92, "Input"], Cell[CellGroupData[{ Cell[2312, 74, 2034, 51, 92, "Input"], Cell[4351, 128, 2847, 54, 390, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[7235, 187, 1790, 54, 112, "Input"], Cell[9028, 243, 2556, 49, 372, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[11621, 297, 1616, 46, 112, "Input"], Cell[13240, 345, 2086, 41, 368, "Output"] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)