(*^ ::[ 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, bold, e8, 24, "Calculus"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Calculus"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Calculus"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Calculus"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Calculus"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Calculus"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Calculus"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, B65535, 12, "Calculus"; 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, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; 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"; paletteColors = 128; showRuler; currentKernel; ] :[font = title; inactive; preserveAspect; fontColorRed = 65535] SECTION FIVE: Experimenting With Polar Graphs :[font = smalltext; inactive; preserveAspect] Mathematica allows you to experiment easily with a wide variety of polar graphs. Here are a few examples to help you get started. ;[s] 2:0,1;12,0;131,-1; 2:1,26,19,Calculus,0,12,0,0,65535;1,26,19,Calculus,2,12,0,0,65535; :[font = section; inactive; Cclosed; preserveAspect; startGroup] Graph r = 2Sinq + Cos2q. ;[s] 5:0,0;15,1;16,0;23,1;24,0;26,-1; 2:3,36,26,Calculus,1,18,0,0,0;2,25,18,Symbol,1,18,0,0,0; :[font = input; Cclosed; preserveAspect; startGroup] Clear[r,t,x,y,p, tex] r[t_] = (Sin[t] + Cos[2t]); x[t_] = r[t] Cos[t]; y[t_] = r[t] Sin[t]; p = ParametricPlot[{x[t],y[t]},{t,0, 2Pi}, AxesOrigin->{0,0},AxesLabel->{"x","y"}, PlotStyle->{RGBColor[1,0,0]}, AspectRatio->Automatic, DisplayFunction->Identity]; tex = Graphics[Text["r = Sin[t] + Cos[2t]", Scaled[{.85,.97}]]]; Show[p,tex, PlotRange->All, DisplayFunction->$DisplayFunction]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 277; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: .98307 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.433899 0.091868 0.433899 [ [(-1)] .0661 .09187 0 2 Msboxa [(-0.5)] .28305 .09187 0 2 Msboxa [(0.5)] .71695 .09187 0 2 Msboxa [(1)] .9339 .09187 0 2 Msboxa [(x)] 1.025 .09187 -1 0 Msboxa [(0.5)] .4875 .30882 1 0 Msboxa [(1)] .4875 .52577 1 0 Msboxa [(1.5)] .4875 .74272 1 0 Msboxa [(2)] .4875 .95967 1 0 Msboxa [(y)] .5 .98307 0 -4 Msboxa [(r = Sin[t] + Cos[2t])] .85 .95358 0 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .98407 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .0661 .09187 m .0661 .09812 L s P [(-1)] .0661 .09187 0 2 Mshowa p .002 w .28305 .09187 m .28305 .09812 L s P [(-0.5)] .28305 .09187 0 2 Mshowa p .002 w .71695 .09187 m .71695 .09812 L s P [(0.5)] .71695 .09187 0 2 Mshowa p .002 w .9339 .09187 m .9339 .09812 L s P [(1)] .9339 .09187 0 2 Mshowa p .001 w .10949 .09187 m .10949 .09562 L s P p .001 w .15288 .09187 m .15288 .09562 L s P p .001 w .19627 .09187 m .19627 .09562 L s P p .001 w .23966 .09187 m .23966 .09562 L s P p .001 w .32644 .09187 m .32644 .09562 L s P p .001 w .36983 .09187 m .36983 .09562 L s P p .001 w .41322 .09187 m .41322 .09562 L s P p .001 w .45661 .09187 m .45661 .09562 L s P p .001 w .54339 .09187 m .54339 .09562 L s P p .001 w .58678 .09187 m .58678 .09562 L s P p .001 w .63017 .09187 m .63017 .09562 L s P p .001 w .67356 .09187 m .67356 .09562 L s P p .001 w .76034 .09187 m .76034 .09562 L s P p .001 w .80373 .09187 m .80373 .09562 L s P p .001 w .84712 .09187 m .84712 .09562 L s P p .001 w .89051 .09187 m .89051 .09562 L s P p .001 w .02271 .09187 m .02271 .09562 L s P p .001 w .97729 .09187 m .97729 .09562 L s P [(x)] 1.025 .09187 -1 0 Mshowa p .002 w 0 .09187 m 1 .09187 L s P p .002 w .5 .30882 m .50625 .30882 L s P [(0.5)] .4875 .30882 1 0 Mshowa p .002 w .5 .52577 m .50625 .52577 L s P [(1)] .4875 .52577 1 0 Mshowa p .002 w .5 .74272 m .50625 .74272 L s P [(1.5)] .4875 .74272 1 0 Mshowa p .002 w .5 .95967 m .50625 .95967 L s P [(2)] .4875 .95967 1 0 Mshowa p .001 w .5 .13526 m .50375 .13526 L s P p .001 w .5 .17865 m .50375 .17865 L s P p .001 w .5 .22204 m .50375 .22204 L s P p .001 w .5 .26543 m .50375 .26543 L s P p .001 w .5 .35221 m .50375 .35221 L s P p .001 w .5 .3956 m .50375 .3956 L s P p .001 w .5 .43899 m .50375 .43899 L s P p .001 w .5 .48238 m .50375 .48238 L s P p .001 w .5 .56916 m .50375 .56916 L s P p .001 w .5 .61255 m .50375 .61255 L s P p .001 w .5 .65594 m .50375 .65594 L s P p .001 w .5 .69933 m .50375 .69933 L s P p .001 w .5 .78611 m .50375 .78611 L s P p .001 w .5 .8295 m .50375 .8295 L s P p .001 w .5 .87289 m .50375 .87289 L s P p .001 w .5 .91628 m .50375 .91628 L s P p .001 w .5 .04848 m .50375 .04848 L s P p .001 w .5 .00509 m .50375 .00509 L s P [(y)] .5 .98307 0 -4 Mshowa p .002 w .5 0 m .5 .98307 L s P P p p 1 0 0 r p .004 w .9339 .09187 m .94693 .1065 L .95758 .12186 L .96584 .13775 L .96906 .14583 L .97168 .15397 L .97276 .15805 L .9737 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.72526 .83444 L .70573 .85966 L .66123 .90242 L Mistroke .63667 .91964 L .61086 .9339 L .59756 .93989 L .58404 .94511 L .57033 .94954 L .55645 .95318 L .54245 .95601 L .53541 .95713 L .52836 .95804 L .52128 .95875 L .51419 .95926 L .5071 .95956 L .5 .95967 L .4929 .95956 L .48581 .95926 L .47872 .95875 L .47164 .95804 L .45755 .95601 L .44355 .95318 L .42967 .94954 L .41596 .94511 L .38914 .9339 L .36333 .91964 L .33877 .90242 L .31568 .88238 L .29427 .85966 L .25728 .8069 L .24204 .77724 L .22918 .74568 L .21882 .71245 L .21461 .69528 L .21106 .67779 L .20818 .66 L .20699 .651 L .20598 .64195 L .20513 .63283 L .20446 .62366 L .20397 .61444 L .20364 .60518 L .20349 .59587 L .20351 .58653 L .20371 .57715 L .20408 .56774 L .20463 .55831 L .20535 .54886 L .20731 .5299 L .20997 .51091 L .21332 .49192 L .22208 .45406 L .23352 .41658 L Mistroke .24756 .37972 L .28305 .30882 L .32749 .24316 L .37954 .1843 L .43763 .13354 L .5 .09187 L .56478 .05992 L .59748 .0477 L .63005 .038 L .64622 .03409 L .66227 .03079 L .67817 .02812 L .6939 .02605 L .70168 .02524 L .70942 .02458 L .71709 .02407 L .72471 .0237 L .73226 .02348 L .73974 .02341 L .74715 .02347 L .75449 .02368 L .76175 .02402 L .76893 .02451 L .77602 .02512 L .78303 .02587 L .79678 .02777 L .81014 .03018 L .83565 .03645 L .85938 .04456 L .88119 .05432 L .90095 .06559 L .9339 .09187 L Mfstroke P P [(r = Sin[t] + Cos[2t])] .85 .95358 0 0 Mshowa P 0 0 m 1 0 L 1 .98307 L 0 .98307 L closepath clip newpath % End of Graphics MathPictureEnd :[font = section; inactive; Cclosed; preserveAspect; startGroup] Graph r = Sinq + Sin2q. ;[s] 5:0,0;14,1;15,0;22,1;23,0;25,-1; 2:3,36,26,Calculus,1,18,0,0,0;2,25,18,Symbol,1,18,0,0,0; :[font = input; preserveAspect; startGroup] Clear[r,t,x,y,p, tex] r[t_] = (Sin[t] + Sin[2t]); x[t_] = r[t] Cos[t]; y[t_] = r[t] Sin[t]; p = ParametricPlot[{x[t],y[t]},{t,0, 2Pi}, AxesOrigin->{0,0},AxesLabel->{"x","y"}, PlotRange->{{-1.5,1.8},{-.4,2}}, PlotStyle->{RGBColor[1,0,0]}, AspectRatio->Automatic, DisplayFunction->Identity]; tex = Graphics[Text["r = Sin[t] + Sin[2t]", Scaled[{.7,.97}]]]; Show[p,tex, DisplayFunction->$DisplayFunction]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 205; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: .72727 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.454545 0.30303 0.121212 0.30303 [ [(-1.5)] 0 .12121 0 2 Msboxa [(-1)] .15152 .12121 0 2 Msboxa [(-0.5)] .30303 .12121 0 2 Msboxa [(0.5)] .60606 .12121 0 2 Msboxa [(1)] .75758 .12121 0 2 Msboxa [(1.5)] .90909 .12121 0 2 Msboxa [(x)] 1.025 .12121 -1 0 Msboxa [(0.5)] .44205 .27273 1 0 Msboxa [(1)] .44205 .42424 1 0 Msboxa [(1.5)] .44205 .57576 1 0 Msboxa [(2)] .44205 .72727 1 0 Msboxa [(y)] .45455 .72727 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .72827 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .12121 m 0 .12746 L s P [(-1.5)] 0 .12121 0 2 Mshowa p .002 w .15152 .12121 m .15152 .12746 L s P [(-1)] .15152 .12121 0 2 Mshowa p .002 w .30303 .12121 m .30303 .12746 L s P [(-0.5)] .30303 .12121 0 2 Mshowa p .002 w .60606 .12121 m .60606 .12746 L s P [(0.5)] .60606 .12121 0 2 Mshowa p .002 w .75758 .12121 m .75758 .12746 L s P [(1)] .75758 .12121 0 2 Mshowa p .002 w .90909 .12121 m .90909 .12746 L s P [(1.5)] .90909 .12121 0 2 Mshowa p .001 w .0303 .12121 m .0303 .12496 L s P p .001 w .06061 .12121 m .06061 .12496 L s P p .001 w .09091 .12121 m .09091 .12496 L s P p .001 w .12121 .12121 m .12121 .12496 L s P p .001 w .18182 .12121 m .18182 .12496 L s P p .001 w .21212 .12121 m .21212 .12496 L s P p .001 w .24242 .12121 m .24242 .12496 L s P p .001 w .27273 .12121 m .27273 .12496 L s P p .001 w .33333 .12121 m .33333 .12496 L s P p .001 w .36364 .12121 m .36364 .12496 L s P p .001 w .39394 .12121 m .39394 .12496 L s P p .001 w .42424 .12121 m .42424 .12496 L s P p .001 w .48485 .12121 m .48485 .12496 L s P p .001 w .51515 .12121 m .51515 .12496 L s P p .001 w .54545 .12121 m .54545 .12496 L s P p .001 w .57576 .12121 m .57576 .12496 L s P p .001 w .63636 .12121 m .63636 .12496 L s P p .001 w .66667 .12121 m .66667 .12496 L s P p .001 w .69697 .12121 m .69697 .12496 L s P p .001 w .72727 .12121 m .72727 .12496 L s P p .001 w .78788 .12121 m .78788 .12496 L s P p .001 w .81818 .12121 m .81818 .12496 L s P p .001 w .84848 .12121 m .84848 .12496 L s P p .001 w .87879 .12121 m .87879 .12496 L s P p .001 w .93939 .12121 m .93939 .12496 L s P p .001 w .9697 .12121 m .9697 .12496 L s P [(x)] 1.025 .12121 -1 0 Mshowa p .002 w 0 .12121 m 1 .12121 L s P p .002 w .45455 .27273 m .4608 .27273 L s P [(0.5)] .44205 .27273 1 0 Mshowa p .002 w .45455 .42424 m .4608 .42424 L s P [(1)] .44205 .42424 1 0 Mshowa p .002 w .45455 .57576 m .4608 .57576 L s P [(1.5)] .44205 .57576 1 0 Mshowa p .002 w .45455 .72727 m .4608 .72727 L s P [(2)] .44205 .72727 1 0 Mshowa p .001 w .45455 .15152 m .4583 .15152 L s P p .001 w .45455 .18182 m .4583 .18182 L s P p .001 w .45455 .21212 m .4583 .21212 L s P p .001 w .45455 .24242 m .4583 .24242 L s P p .001 w .45455 .30303 m .4583 .30303 L s P p .001 w .45455 .33333 m .4583 .33333 L s P p .001 w .45455 .36364 m .4583 .36364 L s P p .001 w .45455 .39394 m .4583 .39394 L s P p .001 w .45455 .45455 m .4583 .45455 L s P p .001 w .45455 .48485 m .4583 .48485 L s P p .001 w .45455 .51515 m .4583 .51515 L s P p .001 w .45455 .54545 m .4583 .54545 L s P p .001 w .45455 .60606 m .4583 .60606 L s P p .001 w .45455 .63636 m .4583 .63636 L s P p .001 w .45455 .66667 m .4583 .66667 L s P p .001 w .45455 .69697 m .4583 .69697 L s P p .001 w .45455 .09091 m .4583 .09091 L s P p .001 w .45455 .06061 m .4583 .06061 L s P p .001 w .45455 .0303 m .4583 .0303 L s P [(y)] .45455 .72727 0 -4 Mshowa p .002 w .45455 0 m .45455 .72727 L s P P 0 0 m 1 0 L 1 .72727 L 0 .72727 L closepath clip newpath p p 1 0 0 r p .004 w .45455 .12121 m .46198 .12127 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Mfstroke P P [(r = Sin[t] + Sin[2t])] .7 .70545 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = section; inactive; Cclosed; preserveAspect; startGroup] Graph r = SinÛq + Cos3q. ;[s] 5:0,0;14,1;15,0;22,1;23,0;25,-1; 2:3,36,26,Calculus,1,18,0,0,0;2,25,18,Symbol,1,18,0,0,0; :[font = input; preserveAspect; startGroup] Clear[r,t,x,y,p, tex] r[t_] = (Sin[t])^2 + Cos[3t]; x[t_] = r[t] Cos[t]; y[t_] = r[t] Sin[t]; p = ParametricPlot[{x[t],y[t]},{t,0, 2Pi}, AxesOrigin->{0,0},AxesLabel->{"x","y"}, PlotRange->All, PlotStyle->{RGBColor[1,0,0]}, AspectRatio->Automatic, DisplayFunction->Identity]; tex = Graphics[Text["r = (Sin[t])^2 + Cos[3t]", Scaled[{.85,.97}]]]; Show[p,tex,DisplayFunction->$DisplayFunction]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 169; pictureHeight = 287; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.69144 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % 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.83828 .74532 L .8527 .74765 L .86656 .75057 L .87978 .75407 L .90415 .76268 L .92544 .77328 L .94333 .7856 L .95091 .79232 L Mistroke .95754 .79935 L .96319 .80665 L .96785 .81418 L .97149 .8219 L .97292 .82582 L .9741 .82976 L .97501 .83373 L .97567 .83772 L .97606 .84171 L .97619 .84572 L Mfstroke P P [(r = \(Sin[t]\)^2 + Cos[3t])] .85 1.64069 0 0 Mshowa P 0 0 m 1 0 L 1 1.69144 L 0 1.69144 L closepath clip newpath % End of Graphics MathPictureEnd :[font = smalltext; inactive; preserveAspect] Try some of your own. One caution, however: many people find once they get started creating these graphs that it is very difficult to stop. Another note: to say you are really doing mathematics you should constantly be asking yourself why the graphs look the way they do. It might help to think about the same function graphed in rectangular coordinates. ;[s] 5:0,0;27,2;34,0;240,1;243,0;362,-1; 3:3,26,19,Calculus,0,12,0,0,65535;1,26,19,Calculus,2,12,0,0,65535;1,26,19,Calculus,0,12,65535,0,0; ^*)