(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 27241, 890]*) (*NotebookOutlinePosition[ 28296, 926]*) (* CellTagsIndexPosition[ 28252, 922]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[ "Mathematics 161 Laboratory 9 \ October 29, 1992\n\t \nName: _____________________________ \ Lab Partner: ___________________________\n\t \nConsulted with: \ ____________________________________________________________"], "Text", CellMargins->{{17, Inherited}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{1, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["\[Copyright] Lafayette College, 1994"], "SmallText", Evaluatable->False, TextAlignment->Right, AspectRatioFixed->True], Cell[TextData["Rational Functions"], "Title", Evaluatable->False, AspectRatioFixed->True, FontSize->23], Cell[TextData[{ StyleBox[ "In this lab we will investigate graphs of rational functions. 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", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" can compute the limit using the command", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Limit[f[x], x -> Infinity]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Can you compute it faster in your head? What is this function's horizontal \ asymptote?\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Roots of the function"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Rational functions are zero only when their numerator is zero. 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Does your plot show the asymptotes and zeros you anticipated?", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Notice the graph of ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" is approximately linear for large values of ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" or \[Dash]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ ". Plot the function on a longer interval if this is not apparent yet. \ Plot the graph of ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" and the line ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" = ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ " together to verify that this is the asymptote for this function. \ Asymptotes which are not horizontal or vertical are called ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["oblique", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontWeight->"Bold"], StyleBox["; so this is the oblique asymptote for the graph of ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[". Apply the ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["ProperForm", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" command to ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ ". Explain why the result proves the existence of the oblique asymptote. \ (Hint: you will need to evaluate a limit.)", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Part 2: Functions from Graphs"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["The First Mystery Function"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The RationalGraphics package contains information about two rational \ functions whose definitions you must guess. 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The result is a \ \"grade\" for your function, where 100 is a perfect match."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "(SCRATCH WORK)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True, FontFamily->"Chicago", FontWeight->"Bold"]], "Text", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True], Cell[TextData[ "Refine your guess until you find the definition of the actual function. DO \ NOT GO ON TO MYSTERY2 UNTIL YOU ARE FINISHED WITH MYSTERY1!"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["The Second Mystery Function"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "A second mystery function has information provided by the command"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Mystery2"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Use the technique you used above to find the definition of this function. \ The ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["Compare", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["Score", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" commands will compare your guess with the function defined by ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["Mystery2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" after you enter this command (unless you reenter ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["Mystery1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " again), so only work on one mystery function at a time on any one \ computer.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Once you have defined this function, give the equations of all its \ asymptotes, including the oblique asymptote. 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