(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, e8, 24, "B Univers 65 Bold"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Garamond"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, e6, 14, "I Garamond LightItalic"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, a20, 18, "B Univers 65 Bold"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, a15, 14, "AGaramond Semibold"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Garamond"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "AGaramond"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "AGaramond"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Garamond"; fontset = leftheader, inactive, L2, 12, "Garamond"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Garamond"; fontset = leftfooter, inactive, L2, 12, "Garamond"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; showRuler; currentKernel; ] :[font = text; inactive; preserveAspect; leftWrapOffset = 17; leftNameWrapOffset = 1] Mathematics 161 Laboratory 9 October 29, 1992 Name: _____________________________ Lab Partner: ___________________________ Consulted with: ____________________________________________________________ :[font = smalltext; inactive; preserveAspect; right] © Lafayette College, 1994 :[font = title; inactive; preserveAspect; fontSize = 23] Rational Functions :[font = text; inactive; preserveAspect] In this lab we will investigate graphs of rational functions. Recall, a function that can be written R(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial in x, is called rational. ;[s] 23:0,0;102,1;103,0;104,1;105,0;109,1;110,0;111,1;112,0;114,1;115,0;116,1;117,0;127,1;128,0;129,1;130,0;137,1;138,0;139,1;140,0;160,1;161,0;183,-1; 2:12,14,10,AGaramond,0,12,0,0,0;11,14,10,AGaramond,2,12,0,0,0; :[font = section; inactive; preserveAspect] Part 0: Graphs from Functions :[font = subsection; inactive; preserveAspect] Preliminaries :[font = text; inactive; preserveAspect] As in preceding labs, we will use some special commands. They are in a new package called "RationalGraphics." We input the package as follows: :[font = input; preserveAspect] < Infinity] :[font = text; inactive; preserveAspect] Can you compute it faster in your head? What is this function's horizontal asymptote? :[font = subsubsection; inactive; preserveAspect] Roots of the function :[font = text; inactive; preserveAspect] Rational functions are zero only when their numerator is zero. Where will the graph of f cross the x-axis? ;[s] 5:0,0;88,1;89,0;100,1;101,0;109,-1; 2:3,14,10,AGaramond,0,12,0,0,0;2,14,10,AGaramond,2,12,0,0,0; :[font = text; inactive; preserveAspect] Now that we have an idea of what to expect, we plot the function, using the command :[font = input; preserveAspect] Plot[f[x], {x, -5, 5}] :[font = text; inactive; preserveAspect] This interval contains the vertical asymptotes and roots, with some room at the sides so we can see if the graph is approaching the horizontal asymptote; it is a good choice for this function. Notice that the graph has the attributes we anticipated before plotting it. :[font = subsection; inactive; preserveAspect] Another Way of Finding Horizontal Asymptotes :[font = text; inactive; preserveAspect] Another way of finding asymptotes of a rational function is to use polynomial long division to express the function as a polynomial (called the quotient) plus a rational remainder with the degree of the numerator strictly less than that of the denominator. The RationalGraphics package has a built-in command that performs the division, it is called ProperForm, since the remainder is called a proper rational function (sort of like a proper fraction). The command ;[s] 3:0,0;351,1;361,0;467,-1; 2:2,14,10,AGaramond,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] ProperForm[f[x],x] :[font = text; inactive; preserveAspect] shows that for our function f the polynomial quotient is a constant. This must be the case if the graph is to have a horizontal asymptote. ;[s] 3:0,0;28,1;29,0;141,-1; 2:2,14,10,AGaramond,0,12,0,0,0;1,14,10,AGaramond,2,12,0,0,0; :[font = section; inactive; preserveAspect] Part 1: Oblique Asymptotes :[font = text; inactive; preserveAspect] Now consider the rational function :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 133; pictureWidth = 114; pictureHeight = 32] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.280702 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.280702 scale 1 string 114 32 1 [114 0 0 32 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FC7FFFFFFFFF FFFFFF1FFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FF8FFFFFFFFF FFFFFF1FFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1C7FFFFFFFFF FFFFFF1FFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7E00FFF000 FFFFFF1FFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE381FF1FFFFF FFFFFF1FFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FE3F1FFFFF FFFFFF1FFFFFFFFFFFFFFFFFFC FFFE00FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FE00FFFFFF FFFFFF1FFFFFFFFFFFFFFFFFFC FFF1FF1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFFF FFFFF81FFFFFFFFFFFFFFFFFFC FFFFFFE3F1FFFFFE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFF FFFFFF1FFFFFFFFFFFFFFFFFFC FFFFC0038FF8FF8FC7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0001FFFFF FFFFFFFFFFFFFFFFFFFFFFFFFC FFFFC7FC0FFF1FF1C7FFF0001FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFC FFFFC7FC7FFFE38FF8FFFFFFFFFE00000000000000000000000000000000 00000000000000000000000000 FFFFF8FC0FFFFF8FF8FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFC FFFFF8FC0FFFFC71F8FFF0001FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFF1C71FFFC7FC0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFE001FFFC7FC0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFF8FFFFFFFC7FFFFFFFFFFFF1FFFFFFFFFFFFFFFFFFFFFFF1FFFC7 FFFFFFFFFFFFFFFFFFFFFFC7FC FFFFFFFF8FFFFFFFC7FFFFFFFFFFF8FF8FF8FFFFFFFFFFFFFC0000E3FE3F E3FE3FFFFFFFFFFFFFE001F8FC FFFFFFFFF1FFFFFE3FFFFFFFFFFFF8FFF1FF1FFFFFFFFFFFFF8FFFE3FE3F FC7FC7FFFFFFFFFFFF1FFE38FC FFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFE38FFFFFFFFFFFFFFF1FFFC71FF FF8E3FFFFFFFFFFFFFFFFE3F1C FFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFF8FFFFFFE001FFFFFE3FFC71FF FFFE3FFFFFF8007FFFFFFE3F1C FFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFC700003FFFFFFFFFFFC7FC71FF FFF1C0000FFFFFFFFFFFFE3F1C FFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFC7FC7FFFFFFFFFFFFFF8FC71FF FFF1FF1FFFFFFFFFFF0001FF1C FFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFC7FC0FFFFFFFFFFC7FF8FC71FF FFF1FF03FFFFFFFFFF1FFFFF1C FFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFF1FFFFFFFFFF8FF8E3FE3F FFFFFFFC7FFFFFFFFF1FFFF8FC FFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFE3FFFFFFFFFF007E3FE3F FFFFFFFF8FFFFFFFFF000038FC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1FFFFFE3FE3FFFFFFFFFFFFF1FFFC7 FFFFF8FF8FFFFFFFFFFFFFC7FC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC01FFFFFFFFFFFFFFFFFFFF FFFFFF007FFFFFFFFFFFFFFFFC pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] Following our work above, give the equations of any vertical asymptotes, horizontal asymptotes, and the locations of any roots of the function below. :[font = subsubsection; inactive; preserveAspect] Vertical asymptotes :[font = subsubsection; inactive; preserveAspect] Horizontal asymptotes :[font = subsubsection; inactive; preserveAspect] Roots of the function :[font = text; inactive; preserveAspect] Based on your work, what is a good interval on which to graph this function? (Be sure to include room at the sides to verify the asymptotic behavior of the function.) :[font = text; inactive; preserveAspect] Plot the graph of g. Does your plot show the asymptotes and zeros you anticipated? ;[s] 3:0,0;18,1;19,0;84,-1; 2:2,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,2,12,0,0,0; :[font = text; inactive; preserveAspect] Notice the graph of g is approximately linear for large values of x or Ðx. Plot the function on a longer interval if this is not apparent yet. Plot the graph of g and the line y = x together to verify that this is the asymptote for this function. Asymptotes which are not horizontal or vertical are called oblique; so this is the oblique asymptote for the graph of g. Apply the ProperForm command to g. Explain why the result proves the existence of the oblique asymptote. (Hint: you will need to evaluate a limit.) ;[s] 21:0,0;20,1;21,0;66,1;67,0;72,1;73,0;163,1;164,0;178,1;179,0;182,1;183,0;309,2;316,0;368,1;369,0;382,3;392,0;404,1;405,0;524,-1; 4:11,16,12,Chicago,0,12,0,0,0;8,16,12,Chicago,2,12,0,0,0;1,16,12,Chicago,1,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = section; inactive; preserveAspect] Part 2: Functions from Graphs :[font = subsection; inactive; preserveAspect] The First Mystery Function :[font = text; inactive; preserveAspect] The RationalGraphics package contains information about two rational functions whose definitions you must guess. To find out about the first function enter the command :[font = input; preserveAspect] Mystery1 :[font = text; inactive; preserveAspect] This will generate a list of x and y coordinates of points on the graph, and a dotted outline of the graph of the function. Using what you know about asymptotes and zeros, define a function r that you believe has the same graph and table of values. ;[s] 7:0,0;29,1;30,0;35,1;36,0;191,1;192,0;250,-1; 2:4,16,12,Chicago,0,12,0,0,0;3,16,12,Chicago,2,12,0,0,0; :[font = input; preserveAspect] r[x_] := :[font = text; inactive; preserveAspect] To compare the graph of your guess with the actual graph, use the command :[font = input; preserveAspect] Compare[r] :[font = text; inactive; preserveAspect] To compare the values of your function with the actual ones, use the command :[font = input; preserveAspect] Score[r] :[font = text; inactive; preserveAspect] (This command may take a minute to execute, be patient.) The result is a "grade" for your function, where 100 is a perfect match. :[font = text; inactive; preserveAspect; center] (SCRATCH WORK) ;[s] 1:0,1;35,-1; 2:0,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,1,12,0,0,0; :[font = text; inactive; preserveAspect] Refine your guess until you find the definition of the actual function. DO NOT GO ON TO MYSTERY2 UNTIL YOU ARE FINISHED WITH MYSTERY1! :[font = subsection; inactive; preserveAspect] The Second Mystery Function :[font = text; inactive; preserveAspect] A second mystery function has information provided by the command :[font = input; preserveAspect] Mystery2 :[font = text; inactive; preserveAspect] Use the technique you used above to find the definition of this function. The Compare and Score commands will compare your guess with the function defined by Mystery2 after you enter this command (unless you reenter Mystery1 again), so only work on one mystery function at a time on any one computer. ;[s] 9:0,0;79,1;86,0;91,1;96,0;159,1;167,0;217,1;225,0;302,-1; 2:5,16,12,Chicago,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] Once you have defined this function, give the equations of all its asymptotes, including the oblique asymptote. (The ProperForm command may be handy here.) ;[s] 3:0,0;118,1;128,0;157,-1; 2:2,16,12,Chicago,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect; center] (SCRATCH WORK) ;[s] 1:0,1;35,-1; 2:0,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,1,12,0,0,0; ^*)