(*^ ::[ 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, e6, 18, "B Garamond Bold"; 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, M18, N55, 12, "Garamond"; 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 162 Laboratory 13 Week of April 26, 1993 Name: _____________________________ Lab Partner: ___________________________ Consulted with: ____________________________________________________________ :[font = smalltext; inactive; preserveAspect; right] © Lafayette College, 1994 :[font = title; inactive; preserveAspect] Polar Coordinates :[font = text; inactive; preserveAspect] In today's laboratory, you will become familiar with plotting in polar coordinates, and compute the areas of some regions which are bounded by the graphs of polar equations. Before we begin the lab, enter the following command to load some special commands we will use: :[font = input; preserveAspect] <Red, AspectRatio->Automatic] :[font = text; inactive; preserveAspect] This graph can be described in polar coordinates by r = ¹/q, and Mathematica can graph it from this equation using the PolarPlot command. ;[s] 9:0,0;52,1;53,0;58,2;59,0;65,1;76,0;119,3;128,0;138,-1; 4:5,14,10,AGaramond,0,12,0,0,0;2,14,10,AGaramond,2,12,0,0,0;1,18,13,Symbol,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] otherplot = PolarPlot[Pi/t, {t, Pi/6, 5Pi/2}, PlotStyle->Blue] :[font = text; inactive; preserveAspect] We can see that these are really the same graph by entering :[font = input; preserveAspect] Show[oneplot, otherplot] :[font = text; inactive; preserveAspect] Here, you can only see the blue curve, because the red one is directly underneath it. Switch the order of the plots (otherplot, oneplot), and reenter to check that they are both really there. Print out one of these plots (they are all basically the same!) and indicate on it: 1) the points where q = ¹/6, ¹, and 2¹; and 2) the direction along the curve in which q increases. ;[s] 7:0,0;118,1;136,0;298,2;299,0;364,2;365,0;377,-1; 3:4,14,10,AGaramond,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0;2,18,13,Symbol,0,12,0,0,0; :[font = text; inactive; preserveAspect] The command PolarPlot can be used to plot any curve described by a polar equation of the form r = f (q). ;[s] 9:0,0;12,1;21,0;94,2;95,0;98,2;99,0;101,3;102,0;105,-1; 4:5,14,10,AGaramond,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0;2,14,10,AGaramond,2,12,0,0,0;1,18,13,Symbol,0,12,0,0,0; :[font = subsection; inactive; preserveAspect] Examples :[font = text; inactive; preserveAspect] Enter these commands, and examine the resulting curves. :[font = input; preserveAspect] PolarPlot[1 - Sin[t], {t, 0, 2Pi}, PlotStyle->Red] :[font = text; inactive; preserveAspect] This plots r = 1 Ð sinq; the curve is called a CARDIOID (heart-shaped thing). ;[s] 5:0,0;11,1;12,0;22,2;23,0;78,-1; 3:3,14,10,AGaramond,0,12,0,0,0;1,14,10,AGaramond,2,12,0,0,0;1,18,13,Symbol,0,12,0,0,0; :[font = input; preserveAspect] PolarPlot[Sin[t], {t,0,2Pi}, PlotStyle->Blue] :[font = text; inactive; preserveAspect] What is this curve called? :[font = text; inactive; preserveAspect] :[font = input; preserveAspect] PolarPlot[Sqrt[Sin[2 t]], {t, 0, 2Pi}] :[font = text; inactive; preserveAspect] This plots a LEMNISCATE (infinity symbol). Why does Mathematica complain "does not evaluate to a pair of real numbers at t = 1.8326," etc? ;[s] 7:0,0;53,1;64,0;75,2;132,0;135,1;138,0;140,-1; 3:4,14,10,AGaramond,0,12,0,0,0;2,14,10,AGaramond,2,12,0,0,0;1,13,10,Courier,0,12,0,0,0; :[font = text; inactive; preserveAspect] :[font = section; inactive; preserveAspect] Part 2: Hypothesizing about Roses :[font = text; inactive; preserveAspect] Graphs of equations of the form r = sinnq, where n is an integer, are called "roses," because they have "petals" that resemble a flower's. The command below defines a function that will plot roses for any value of n. ;[s] 10:0,0;32,1;33,0;39,1;40,2;41,0;49,1;50,0;215,1;216,0;218,-1; 3:5,14,10,AGaramond,0,12,0,0,0;4,14,10,AGaramond,2,12,0,0,0;1,18,13,Symbol,0,12,0,0,0; :[font = input; preserveAspect] rose[n_]:=PolarPlot[Sin[n t], {t, 0, 2Pi}]/;IntegerQ[n] :[font = text; inactive; preserveAspect] Enter the definition and then use the command below to create a rose with n = 2. How many petals does it have? ;[s] 3:0,0;74,1;75,0;112,-1; 2:2,14,10,AGaramond,0,12,0,0,0;1,14,10,AGaramond,2,12,0,0,0; :[font = text; inactive; preserveAspect] :[font = input; preserveAspect] rose[2] :[font = text; inactive; preserveAspect] Create plots of roses with other values of n (try 1, 3, 4, and 5, for instance). It is possible to create roses with only a few petals, or with many, but there are certain numbers of petals it is impossible to have. Which numbers are those? (Hint: Express the number of petals as a function of n when n is even. What if n is odd?) ;[s] 9:0,0;43,1;44,0;299,1;300,0;306,1;307,0;326,1;327,0;337,-1; 2:5,14,10,AGaramond,0,12,0,0,0;4,14,10,AGaramond,2,12,0,0,0; :[font = text; inactive; preserveAspect] :[font = section; inactive; preserveAspect] Part 3: Area in Polar Coordinates :[font = text; inactive; preserveAspect] To visualize how one integrates in polar coordinates, we will use a special package your intrepid instructor has developed for this lab. The command below will load the package. :[font = input; preserveAspect] <