(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 24596, 687] NotebookOptionsPosition[ 23159, 637] NotebookOutlinePosition[ 23554, 654] CellTagsIndexPosition[ 23511, 651] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Error of Taylor Polynomials: (to be done with a partner!)", \ "Subsubtitle", CellChangeTimes->{{3.4626169786217823`*^9, 3.462616995173403*^9}, { 3.4628038480590477`*^9, 3.4628038734592257`*^9}}], Cell["\<\ In this project you will experimentally investigate what happens to the error \ of Taylor approximations. For some of the questions I have included examples \ of the input/output that you should get.\ \>", "Text", CellChangeTimes->{{3.4626169996940193`*^9, 3.4626170276705017`*^9}, { 3.462618058016062*^9, 3.462618075457284*^9}}], Cell[TextData[{ StyleBox["1. Getting ", "Subsubsection"], StyleBox["Mathematica", "Subsubsection", FontSlant->"Italic"], StyleBox[" to define Taylor Polynomials", "Subsubsection"] }], "Text", CellChangeTimes->{{3.462617040382832*^9, 3.4626170584955463`*^9}}], Cell["\ta) Read about \"Series\" in the online documentation.", "Text", CellChangeTimes->{{3.462617065359824*^9, 3.462617075624022*^9}}], Cell[TextData[{ "\tb) Have ", StyleBox["Mathematica", FontSlant->"Italic"], " calculate the 15 Taylor polynomial for f(x) = ArcSin(x) based at 0." }], "Text", CellChangeTimes->{{3.462617709780718*^9, 3.462617734157792*^9}}], Cell[TextData[{ "\tWhat does the ", Cell[BoxData[ FormBox[ RowBox[{"O", "(", SuperscriptBox["x", "16"], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], "mean?" }], "Text", CellChangeTimes->{{3.462617758962038*^9, 3.462617762255145*^9}}], Cell[BoxData[""], "Input", CellChangeTimes->{{3.462616543443754*^9, 3.462616636932643*^9}, { 3.462616858403417*^9, 3.462616881291888*^9}, 3.462617061554534*^9, 3.462617705435623*^9, {3.462801234744927*^9, 3.462801235462092*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Series", "[", RowBox[{ RowBox[{"ArcSin", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "15"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.462616775210473*^9, 3.462616812448704*^9}, { 3.462616848945858*^9, 3.462616885739107*^9}}], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{"x", "+", FractionBox[ SuperscriptBox["x", "3"], "6"], "+", FractionBox[ RowBox[{"3", " ", SuperscriptBox["x", "5"]}], "40"], "+", FractionBox[ RowBox[{"5", " ", SuperscriptBox["x", "7"]}], "112"], "+", FractionBox[ RowBox[{"35", " ", SuperscriptBox["x", "9"]}], "1152"], "+", FractionBox[ RowBox[{"63", " ", SuperscriptBox["x", "11"]}], "2816"], "+", FractionBox[ RowBox[{"231", " ", SuperscriptBox["x", "13"]}], "13312"], "+", FractionBox[ RowBox[{"143", " ", SuperscriptBox["x", "15"]}], "10240"], "+", InterpretationBox[ RowBox[{"O", "(", SuperscriptBox["x", "16"], ")"}], SeriesData[$CellContext`x, 0, {}, 1, 16, 1], Editable->False]}], SeriesData[$CellContext`x, 0, {1, 0, Rational[1, 6], 0, Rational[3, 40], 0, Rational[5, 112], 0, Rational[35, 1152], 0, Rational[63, 2816], 0, Rational[231, 13312], 0, Rational[143, 10240]}, 1, 16, 1], Editable->False], TraditionalForm]], "Output", CellChangeTimes->{3.462616889236916*^9}] }, Open ]], Cell[TextData[{ "\n\n\tc) Have ", StyleBox["Mathematica", FontSlant->"Italic"], " calculate the 5th Taylor polynomial for g(x) = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["e", "x"], " ", StyleBox["based", "Text"], StyleBox[" ", "Text"], StyleBox["at", "Text"], StyleBox[" ", "Text"], StyleBox["0", "Text"]}], TraditionalForm]], FormatType->"TraditionalForm"], ". (Note that you need \tto use 'E' instead of 'e' in ", StyleBox["Mathematica", FontSlant->"Italic"], ".\n\td) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to add the answer from b) to the answer from c). Explain the ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"O", "(", SuperscriptBox["x", "5"], ")"}], StyleBox[" ", "Text"], StyleBox["term", "Text"]}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.462616895945506*^9, 3.46261694532368*^9}, 3.4626177487905893`*^9, {3.4626177821431437`*^9, 3.462617921173009*^9}, 3.462619278615943*^9}], Cell[TextData[{ "\te) To plug numbers in for x, we need to have ", StyleBox["Mathematica", FontSlant->"Italic"], " get rid of the ", Cell[BoxData[ FormBox[ RowBox[{"O", "(", SuperscriptBox["x", "5"], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " term. Use the Normal \tcommand to do this for your answers from b) and c).\ \n" }], "Text", CellChangeTimes->{{3.462617933063456*^9, 3.462618042912212*^9}, { 3.46261927440821*^9, 3.4626192770317297`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Normal", "[", RowBox[{"Series", "[", RowBox[{ RowBox[{"ArcSin", "[", "x", "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", "0", ",", "15"}], "}"}]}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.462618049005968*^9, 3.462618083346369*^9}}], Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{"143", " ", SuperscriptBox["x", "15"]}], "10240"], "+", FractionBox[ RowBox[{"231", " ", SuperscriptBox["x", "13"]}], "13312"], "+", FractionBox[ RowBox[{"63", " ", SuperscriptBox["x", "11"]}], "2816"], "+", FractionBox[ RowBox[{"35", " ", SuperscriptBox["x", "9"]}], "1152"], "+", FractionBox[ RowBox[{"5", " ", SuperscriptBox["x", "7"]}], "112"], "+", FractionBox[ RowBox[{"3", " ", SuperscriptBox["x", "5"]}], "40"], "+", FractionBox[ SuperscriptBox["x", "3"], "6"], "+", "x"}], TraditionalForm]], "Output", CellChangeTimes->{3.4626180841546993`*^9}] }, Open ]], Cell[TextData[{ "\tAccording to ", StyleBox["Mathematica", FontSlant->"Italic"], " we still don't have a function. In other words, we still can't plug in for \ x. \t\tUse the following command to do this for the ArcSin example" }], "Text", CellChangeTimes->{{3.462618091986897*^9, 3.462618129746223*^9}, { 3.4626192854161997`*^9, 3.462619287304302*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", "x_", "]"}], " ", ":=", " ", RowBox[{"ArcSin", "[", "x", "]"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"p", "[", "x_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"Normal", "[", RowBox[{"Series", "[", RowBox[{ RowBox[{"ArcSin", "[", "y", "]"}], ",", RowBox[{"{", RowBox[{"y", ",", "0", ",", "15"}], "}"}]}], "]"}], "]"}], "/.", RowBox[{"y", " ", "\[Rule]", " ", "x"}]}]}]}], "Input", CellChangeTimes->{{3.4626179292226353`*^9, 3.462617930004776*^9}, { 3.462618135283224*^9, 3.462618192486081*^9}}], Cell[TextData[{ "\tDefine similar functions for g(x) = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["e", "x"], " "}], TraditionalForm]], FormatType->"TraditionalForm"], "and the polynomials from 1 c." }], "Text", CellChangeTimes->{{3.462618200100424*^9, 3.462618215132882*^9}, { 3.462618301743922*^9, 3.462618318596575*^9}, {3.4626192428508577`*^9, 3.462619289168189*^9}}], Cell[TextData[{ "\tf) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to comput the result of plugging 3 into ArcSin[x], ", Cell[BoxData[ FormBox[ SuperscriptBox["e", "x"], TraditionalForm]], FormatType->"TraditionalForm"], ", and the Taylor polynomials." }], "Text", CellChangeTimes->{{3.462619308784358*^9, 3.462619387595065*^9}}], Cell[TextData[{ StyleBox["2. Graphing the Taylor polynomials and analyzing error.", "Subsubsection"], "\nHow good a Taylor approximation is, depends on the function being \ approximated, how many terms the Taylor polynomial has, and the point where \ the polynomial is based.\n\nRead about \"Manipulate\" in the documentation \ and study/play around with this example:" }], "Text", CellChangeTimes->{{3.462619403388802*^9, 3.462619418291979*^9}, { 3.462801267886297*^9, 3.462801363793743*^9}, {3.462802436590032*^9, 3.4628024596532307`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", "x_", "]"}], " ", ":=", " ", RowBox[{"Sin", "[", "x", "]"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"p", "[", RowBox[{"x_", ",", "n_", ",", "a_"}], "]"}], " ", ":=", " ", RowBox[{ RowBox[{"Normal", "[", RowBox[{"Series", "[", RowBox[{ RowBox[{"f", "[", "y", "]"}], ",", RowBox[{"{", RowBox[{"y", ",", "a", ",", "n"}], "}"}]}], "]"}], "]"}], "/.", RowBox[{"y", " ", "\[Rule]", " ", "x"}]}]}], "\[IndentingNewLine]", RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"f", "[", "x", "]"}], ",", RowBox[{"p", "[", RowBox[{"x", ",", "n", ",", "a"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{ RowBox[{"-", "2"}], " ", "Pi"}], ",", " ", RowBox[{"2", " ", "Pi"}]}], "}"}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}]}], ",", " ", RowBox[{"PlotStyle", " ", "\[Rule]", " ", "Thick"}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"a", ",", "0"}], "}"}], ",", RowBox[{ RowBox[{"-", "Pi"}], "/", "2"}], ",", " ", RowBox[{"Pi", "/", "2"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"n", ",", "1", ",", " ", "15", ",", "1"}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.462801367658453*^9, 3.462801639765648*^9}, { 3.4628016827709627`*^9, 3.462801683672423*^9}, {3.46280174730145*^9, 3.46280176634758*^9}, {3.462801814783251*^9, 3.4628018361173277`*^9}, { 3.462801880345621*^9, 3.462801925712036*^9}, {3.4628024123894*^9, 3.4628024167136517`*^9}, {3.462803401914513*^9, 3.462803434529607*^9}, { 3.462804284087036*^9, 3.46280428500851*^9}}], Cell[BoxData[ FormBox[ TagBox[ FormBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 0, $CellContext`n$$ = 1, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`a$$], 0}, Rational[-1, 2] Pi, Rational[1, 2] Pi}, { Hold[$CellContext`n$$], 1, 15, 1}}, Typeset`size$$ = { 360., {115., 119.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`a$334629$$ = 0, $CellContext`n$334630$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, TraditionalForm, "Variables" :> {$CellContext`a$$ = 0, $CellContext`n$$ = 1}, "ControllerVariables" :> { Hold[$CellContext`a$$, $CellContext`a$334629$$, 0], Hold[$CellContext`n$$, $CellContext`n$334630$$, 0]}, "OtherVariables" :> { Typeset`show$$, 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], $CellContext`p[$CellContext`x, $CellContext`n$$, \ $CellContext`a$$]}, {$CellContext`x, (-2) Pi, 2 Pi}, PlotRange -> {-1, 1}, PlotStyle -> Thick], "Specifications" :> {{{$CellContext`a$$, 0}, Rational[-1, 2] Pi, Rational[1, 2] Pi}, {$CellContext`n$$, 1, 15, 1}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{403., {200.40625, 206.59375}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], TraditionalForm], Manipulate`InterpretManipulate[1]], TraditionalForm]], "Output", CellChangeTimes->{3.462801645775313*^9, 3.4628016852183037`*^9, 3.4628017885694447`*^9, 3.4628018370601377`*^9, 3.46280189250356*^9, 3.462801938040551*^9, 3.462802418988482*^9, 3.462803435913164*^9, 3.462804287611375*^9}] }, Open ]], Cell[TextData[{ "To have the result display the values of ", StyleBox["a", FontSlant->"Italic"], " and ", StyleBox["n", FontSlant->"Italic"], " as pictured, click the little plus sign next to the slider.\n\nWe can have \ ", StyleBox["Mathematica", FontSlant->"Italic"], " plot the error of the approximation below the graphs if we add another \ plot command to \"Manipulate\". 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\ $CellContext`p[$CellContext`x, $CellContext`n$$, $CellContext`a$$], \ {$CellContext`x, -$CellContext`xRange, $CellContext`xRange}, PlotRange -> {-1, 1}, PlotStyle -> {Thick, Orange}]}], "Specifications" :> {{{$CellContext`a$$, 0}, Rational[-1, 2] Pi, Rational[1, 2] Pi}, {$CellContext`n$$, 1, 40, 1}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{403., {268.40625, 274.59375}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], TraditionalForm], Manipulate`InterpretManipulate[1]], TraditionalForm]], "Output", CellChangeTimes->{ 3.462806551782454*^9, 3.462806632475243*^9, 3.4628067391041718`*^9, { 3.462807222796249*^9, 3.4628072465578547`*^9}}] }, Open ]], Cell[TextData[{ "a) Spend some time playing with the plots above and analyze the example(s) \ above for the Taylor polynomials of f(x) = sin(x). Write 1 or 2 paragraphs \ describing the effects of changing the degree of the Taylor polynomial and \ changing the basepoint of the Taylor polynomial. How does the error behave \ with respect to ", StyleBox["n", FontSlant->"Italic"], ", ", StyleBox["x", FontSlant->"Italic"], ", and ", StyleBox["a", FontSlant->"Italic"], "?" }], "Text", CellChangeTimes->{{3.462803729151663*^9, 3.462803841953957*^9}}], Cell[TextData[{ "b) Choose two functions for which you'll do a similar analysis. You may \ choose any two functions as long as:\n\t\ti) Neither function is sin(x), \ cos(x), ln(1 + x), or ", Cell[BoxData[ FormBox[ SuperscriptBox["e", "x"], TraditionalForm]], FormatType->"TraditionalForm"], "\n\t\tii) The two functions are significantly different from each other. \n \ Remember that you can create functions by combining well known functions \ using basic arithmetic options, and function composition. For example, you \ could use the function ", Cell[BoxData[ FormBox[ SuperscriptBox["e", RowBox[{"sin", "(", "x", ")"}]], TraditionalForm]], FormatType->"TraditionalForm"], ".\n\nc) For each of your two functions, use Manipulate to create plots like \ those for sin(x) above. You will need to figure out what good choices for the \ x-Range and y-Range of your plots are. If you make your functions too \ complicated, it may take ", StyleBox["Mathematica", FontSlant->"Italic"], " a long time to compute the Taylor polynomials. Make sure that the results \ look good. You may find it helpful to refer frequently to the documentation.\n\ \nYour output does not have to look exactly like what I have. For example, \ perhaps you want to add a 2-D slider to Manipulate so that the user can \ adjust the dimensions of the plot for him/herself. Your output should allow \ for ", StyleBox["n", FontSlant->"Italic"], " to go up to at least 15, but depending on your function it may be useful \ to have ", StyleBox["n", FontSlant->"Italic"], " go considerably higher.\n\nd) For each of your two functions, use the \ output from your Manipulate command to analyze the dependence of the error on \ ", StyleBox["n", FontSlant->"Italic"], " and ", StyleBox["a", FontSlant->"Italic"], ". For each of your functions you should have one or two paragraphs \ summarizing your findings. Be sure to discuss any places where the \ approximation gets worse, the larger ", StyleBox["n", FontSlant->"Italic"], " gets as well as values of ", StyleBox["a", FontSlant->"Italic"], " that always result in better or worse approximations than other values of ", StyleBox["a", FontSlant->"Italic"], ".\n\ne) Be sure that your names are in the body of the ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook. Name your file using the last names of both people. For example, \ if Scott Taylor and Stephanie Arnold work together, the file would be called: \ TaylorArnold_Proj2.nb\n\nf) Email your file to sataylor@colby.edu" }], "Text", CellChangeTimes->{{3.462803886890957*^9, 3.4628042217817383`*^9}, { 3.4628052215683947`*^9, 3.4628052830482197`*^9}, {3.4628053703522253`*^9, 3.462805452661722*^9}, {3.4628068486862*^9, 3.462806895449054*^9}, { 3.462807127150228*^9, 3.4628071663892107`*^9}, {3.462807264882533*^9, 3.462807317610196*^9}}], Cell[BoxData[""], "Input", CellChangeTimes->{{3.4628038791089983`*^9, 3.4628038806191874`*^9}}] }, Open ]] }, WindowSize->{1136, 775}, WindowMargins->{{20, Automatic}, {37, Automatic}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, FrontEndVersion->"7.0 for Mac OS X x86 (32-bit) (November 10, 2008)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[567, 22, 204, 3, 26, "Subsubtitle"], Cell[774, 27, 342, 6, 26, "Text"], Cell[1119, 35, 264, 6, 26, "Text"], Cell[1386, 43, 137, 1, 26, "Text"], Cell[1526, 46, 229, 6, 26, "Text"], Cell[1758, 54, 262, 9, 32, "Text"], Cell[2023, 65, 236, 3, 27, "Input"], Cell[CellGroupData[{ Cell[2284, 72, 291, 7, 27, "Input"], Cell[2578, 81, 1175, 38, 52, "Output"] }, Open ]], Cell[3768, 122, 1046, 34, 78, "Text"], Cell[4817, 158, 486, 14, 48, "Text"], Cell[CellGroupData[{ Cell[5328, 176, 284, 7, 27, "Input"], Cell[5615, 185, 701, 23, 52, "Output"] }, Open ]], Cell[6331, 211, 361, 8, 26, "Text"], Cell[6695, 221, 593, 15, 43, "Input"], Cell[7291, 238, 397, 11, 26, "Text"], Cell[7691, 251, 354, 11, 26, "Text"], Cell[8048, 264, 552, 10, 71, "Text"], Cell[CellGroupData[{ Cell[8625, 278, 1853, 49, 73, "Input"], Cell[10481, 329, 2326, 47, 424, "Output"] }, Open ]], Cell[12822, 379, 594, 16, 56, "Text"], Cell[CellGroupData[{ Cell[13441, 399, 3466, 89, 328, "Input"], Cell[16910, 490, 2661, 55, 560, "Output"] }, Open ]], Cell[19586, 548, 562, 16, 41, "Text"], Cell[20151, 566, 2893, 65, 285, "Text"], Cell[23047, 633, 96, 1, 27, "Input"] }, Open ]] } ] *) (* End of internal cache information *)