(*^ ::[ 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, cellOutline, groupLikeTitle, center, M18, O486, R65535, e8, 24, "Calculus"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, bold, R21845, G21845, B21845, e6, 12, "Calculus"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, R21845, G21845, B21845, e6, 12, "Calculus"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Calculus"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, B65535, 12, "Calculus"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M36, N23, O486, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M36, N23, O486, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M18, O486, l34, w351, h314, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = leftheader, inactive, M18, O486, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M18, O486, 12, "Times"; fontset = leftfooter, inactive, M18, O486, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, whiteBox, M18, O486, bold, R21845, G21845, B21845, 12, "CalcMath"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, R21845, G21845, B21845, 12, "CalcMath"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, 10, "Courier"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 10, "Courier"; paletteColors = 256; currentKernel; ] :[font = text; inactive; Cclosed; preserveAspect; fontSize = 10; startGroup] Initialization stuff :[font = input; initialization; preserveAspect; endGroup] *) Needs["Graphics`Graphics`"] Needs["Graphics`Colors`"] (* :[font = title; inactive; preserveAspect] POLAR GRAPHING :[font = text; inactive; preserveAspect] The object of this lesson is to teach students to analyze basic polar graphs through experimentation with computer graphing. ;[s] 1:0,1;126,-1; 2:0,26,19,Calculus,0,12,0,0,0;1,26,19,Calculus,0,12,21845,21845,21845; :[font = input; preserveAspect] ;[s] 1:0,1;2,-1; 2:0,12,10,Courier,1,12,0,0,0;1,23,16,CalcMath,1,12,21845,21845,21845; :[font = input; preserveAspect; animationSpeed = 60] :[font = text; inactive; preserveAspect] This lesson will be looking at polar equations in the form Sin[a theta]. Place the mouse on the cell , click once, and push enter to see a display of five graphs of equations in this form. ;[s] 3:0,1;65,2;82,1;201,-1; 3:0,26,19,Calculus,0,12,0,0,0;2,26,19,Calculus,0,12,0,0,65535;1,26,19,Calculus,0,12,65535,0,0; :[font = input; preserveAspect] -- :[font = input; preserveAspect] :[font = input; preserveAspect] :[font = input; preserveAspect] :[font = input; preserveAspect] :[font = input; preserveAspect; startGroup; animationSpeed = 120] Clear[a,theta,lplot,pplot,t] Do[ lplot := Graphics[Text[Sin[a theta], {Cos[((.5 a a-5.5 a +17) (Pi/32)+.72)], Sin[(.5 a a-5.5 a +17) Pi/32]}]]; pplot := PolarPlot[Sin[a theta],{theta,0,2Pi}, PlotRange->{{-1,1},{-1,1}}, PlotStyle->RGBColor[a/5.01,1-a/5.01,0], DisplayFunction->Identity]; Show[pplot,lplot,DisplayFunction->$DisplayFunction] ,{a,1,5}] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 314; pictureHeight = 314; pictureID = 22476] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 314; pictureHeight = 314; pictureID = 22969] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 314; pictureHeight = 314; pictureID = 27558] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 314; pictureHeight = 314; pictureID = 27655] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 314; pictureHeight = 314; endGroup; pictureID = 8732] :[font = text; inactive; preserveAspect] When you reach this spot, return to the first graph displayed, group the cells--one click on second cell grouping -- and push command Y to see an animated display of all five graphs. Notice simiarities and differences in the graphs as the value of a changes. You will want to see more graphs of this type before you sketch the graphs on your worksheet;use the computer to graph equations by typing in the following: PolarPlot [Sin[(insert number) theta],{theta,0,nPi}] IMPORTANT: n must be chosen so that the graph displayed runs from -1 to 1 on the horizontal axis. ;[s] 7:0,0;130,1;142,0;258,2;261,0;503,3;618,0;629,-1; 4:4,26,19,Calculus,0,12,0,0,65535;1,26,19,Calculus,1,12,0,0,65535;1,28,20,Calculus,8,12,0,0,65535;1,26,19,Calculus,2,12,0,0,65535; :[font = input; preserveAspect] :[font = input; preserveAspect] :[font = text; inactive; preserveAspect] Now, I hope that you were curious enough to try many different numbers -- the world does not live by whole numbers alone! If you didn't try a variety of numbers, do so now -- I did while working late one night on the worksheet. Remember that with the aid of the computer, you can explore more than ever before. :[font = smalltext; inactive; preserveAspect] Now that you can quickly sketch equations in the form of Sin[ a theta], let's change the form to Cos[a theta]. Get started! You know what to do. ;[s] 5:0,0;75,2;90,0;120,1;134,0;184,-1; 3:3,26,19,Calculus,0,12,0,0,65535;1,26,19,Calculus,0,12,65535,0,0;1,26,19,Calculus,0,12,65535,0,65535; ^*)