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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 55174, 1625]*) (*NotebookOutlinePosition[ 56034, 1652]*) (* CellTagsIndexPosition[ 55990, 1648]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ Cell[BoxData[{ StyleBox[\(MATH\ 257\ Calculus\ III\t\t\t\tWeek\ of\ April\ 28, \ 2003\), FontSize->14], "\n", StyleBox[\(Lab\ 5 : \ More\ Taylor\ Fun\), "Title"], "\n", StyleBox[\(Name\ 1\), "Section"], "\n", StyleBox[\(Name\ 2\), "Section"], "\n", StyleBox[\(\(Section\)\(:\)\), "Section"]}], "Input"], "\n" }], "Text"], Cell[TextData[{ "\nLast time, we used the ", StyleBox["Normal[Series[f[x], {x, a, n}]]", "Input"], " command to make ", StyleBox["Mathematica", FontSlant->"Italic"], " generate the ", StyleBox["n", "Input"], "th degree Taylor Polynomial for the function of ", StyleBox["f[x] ", "Input"], "centered at the point ", StyleBox["a", "Input"], ". \n\nIn this lab you'll practice doing this some more, this time with \ functions we'd never want to do using only pencil and paper...\n\nFor \ example, let's look at the function ", StyleBox["Sin[", "Input"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^x\)], "Input"], "Input"], StyleBox["]", "Input"], ". 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Let's see what the 4th Taylor polynomial centered at \ ", StyleBox["a=0", "Input"], " looks like:" }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(T4[x_] = Normal[Series[Sin[\[ExponentialE]\^x], \ {x, 0, \ 4}]]\)], "Input"], Cell[BoxData[ \(x\ Cos[1] + x\^2\ \((Cos[1]\/2 - Sin[1]\/2)\) + x\^4\ \((\(-\(\(5\ Cos[1]\)\/24\)\) - Sin[1]\/4)\) + Sin[1] - 1\/2\ x\^3\ Sin[1]\)], "Output"] }, Open ]], Cell[TextData[{ "Blech! That doesn't look like something I would have wanted to do by \ hand! \n\nLet's plot ", StyleBox["T4[x]", "Input"], " together with ", StyleBox["Sin[", "Input"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^x\)], "Input"], "Input"], StyleBox["]", "Input"], " to see how good a job ", StyleBox["T4[x]", "Input"], " does approximating it. I'll use the ", StyleBox["Thickness[]", "Input"], " command to make the original function a thick line, and the ", StyleBox["RGBColor[]", "Input"], " command to make ", StyleBox["T4[x]", "Input"], " a different color:" }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", StyleBox[ RowBox[{ RowBox[{"Sin", "[", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^x\)], "Input"], "]"}], ",", " ", \(T4[x]\)}], "Input"], StyleBox["}", "Input"]}], StyleBox[",", "Input"], StyleBox[" ", "Input"], RowBox[{ StyleBox["{", "Input"], RowBox[{ StyleBox["x", "Input"], StyleBox[",", "Input"], RowBox[{ StyleBox["-", "Input"], "\[Pi]"}], ",", " ", "\[Pi]"}], "}"}], ",", " ", \(PlotRange \[Rule] \ {\(-5\), 5}\), ",", " ", \(PlotStyle \[Rule] \ {Thickness[ .01], \ RGBColor[1, 0, 0]}\)}], "]"}]], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } 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Editable->False]], "Output"] }, Open ]], Cell["\<\ Not too bad for only the 4th Taylor polynomial. In the questions \ below you'll be asked to do better with higher-order Taylor \ polynomials.\ \>", "Text", FontSize->14], Cell[TextData[{ StyleBox["Question 1", FontWeight->"Bold"], " Find the 10th, 20th, and 30th order Taylor polynomials centered at the \ origin for", StyleBox[" Sin[", "Input"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^x\)], "Input"], "Input"], StyleBox["]", "Input"], ". Give them the names T10[x], T20[x], and T30[x]." }], "Text", FontSize->14, FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[TextData[{ StyleBox["Question 2", FontWeight->"Bold"], " Plot the original function ", StyleBox["Sin[", "Input"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^x\)], "Input"], "Input"], StyleBox["]", "Input"], " along with the 10th, 20th, and 30th degree Taylor polynomials. Make sure \ the original function is thick and the Taylor polynomials different colors. \ ", StyleBox["Explain", FontWeight->"Bold"], " which colors are which polynomial!" }], "Text", FontSize->14, FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[TextData[{ StyleBox["Question 3 ", FontWeight->"Bold"], " Give the approximate interval on which the Taylor polynomial of degree 30 \ approximates ", StyleBox["Sin[", "Input"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^x\)], "Input"], "Input"], StyleBox["]", "Input"], ". What do you think the interval of convergence for the whole, infinite \ Taylor series is for this function? (Hint: what it the interval of \ convergence for Sin[x] and ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^x\)]], " by themselves?)" }], "Text", FontSize->14, FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[TextData[{ StyleBox["Question 4", FontWeight->"Bold"], " Find the 5th, 10th, and 15th degree Taylor polynomials for ", StyleBox["Tan[x]", "Input"], " centered at the origin." }], "Text", FontSize->14, FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[TextData[{ StyleBox["Question 5 ", FontWeight->"Bold"], "Plot ", StyleBox["Tan[x]", "Input"], " along with the 5th, 10th, and 15th degree Taylor polynomials found in #4 \ on the interval (-", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], "). Make ", StyleBox["Tan[x]", "Input"], " a thick graph and the polynomials different colors, and ", StyleBox["explain", FontWeight->"Bold"], " which colors correspond to which polynomial." }], "Text", FontSize->14, FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[TextData[{ StyleBox["Question 6a ", FontWeight->"Bold"], "Find the 9th order Taylor polynomial for ", StyleBox["f[x]=2x", "Input"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^x\)], "Input"], "Input"], ". " }], "Text", FontSize->14, FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input", FontSize->14], Cell[TextData[{ StyleBox["Question 6b", FontWeight->"Bold"], " Find the 9th order Taylor polynomial for ", StyleBox["f[x]=", "Input"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^x\)], "Input"], "Input"], " and then multiply it by ", StyleBox["2x", "Input"], ".\n" }], "Text", FontSize->14, FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[TextData[{ StyleBox["Question 6c", FontWeight->"Bold"], " Did you get the same answers in the above two questions? Explain why \ this happened." }], "Text", FontSize->14, FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[TextData[{ StyleBox["Now that you're done, ", FontSize->18], "\n\t(1) \t", StyleBox["Clean up your work", FontWeight->"Bold"], " by deleting everything that's not needed. \n\t\tThere should only be the \ title, your names, section number, and the questions and your answers\n\t\t\ (with any needed explanations, graphs, etc.).\n\t(2) \tSave this to your \ disk.\n\t(3)\tGo back to Blackboard and upload your final lab report to the \ ", StyleBox["Digital Drop Box", FontWeight->"Bold"], ", located\n\t\tin the ", StyleBox["User Tools", FontWeight->"Bold"], " area. Remember that you have to first ", StyleBox["ADD", FontWeight->"Bold"], " your file to your digital drop box and \n\t\tthen ", StyleBox["SEND it to me!", FontWeight->"Bold"] }], "Text", CellFrame->{{0, 0}, {0, 2}}, FontSize->12] }, FrontEndVersion->"4.2 for Macintosh", ScreenRectangle->{{0, 800}, {0, 580}}, WindowToolbars->"EditBar", WindowSize->{659, 311}, WindowMargins->{{22, Automatic}, {0, Automatic}}, MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@00000000000000060801T1T00000000000 00000000000000000000000000000000\>" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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