MATLAB Graphics

MATLAB Graphics
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Published Date:03-08-2017
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Chapter5 MATLABGraphics In this chapter we describe more of MATLAB’s graphics commands and the most common ways of manipulating and customizing them. You can get a list of MATLAB graphics commands by typing help graphics (for general graphics commands), help graph2d (for two-dimensional graphing), help graph3d(forthree-dimensionalgraphing),orhelp specgraph(forspecial- izedgraphingcommands). WehavealreadydiscussedthecommandsplotandezplotinChapter2. Wewillbeginthischapterbydiscussingmoreusesofthesecommands,aswell astheothermostcommonlyusedplottingcommandsintwoandthreedimen- sions.Thenwewilldiscusssometechniquesforcustomizingandmanipulating graphics. Two-DimensionalPlots Oftenonewantstodrawacurveinthex-yplane,butwithynotgivenexplicitly as a function of x. There are two main techniques for plotting such curves: parametricplottingandcontourorimplicitplotting.Wediscusstheseinturn inthenexttwosubsections. ParametricPlots Sometimesxandyarebothgivenasfunctionsofsomeparameter.Forexample, thecircleofradius1centeredat(0,0)canbeexpressedinparametricformas x=cos(2πt),y=sin(2πt)wheretrunsfrom0to1.Thoughyisnotexpressed asafunctionofx,youcaneasilygraphthiscurvewith plot,asfollows: T = 0:0.01:1; 6768 Chapter5: MATLABGraphics plot(cos(2piT), sin(2piT)) axis square 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure5-1 TheoutputisshowninFigure5.1.Ifyouhadusedanincrementofonly0.1in theTvector,theresultwouldhavebeenapolygonwithclearlyvisiblecorners, an indication that you should repeat the process with a smaller increment untilyougetagraphthatlookssmooth. Ifyouhaveversion2.1orhigheroftheSymbolicMathToolbox(cor- respondingtoMATLABversion5.3orhigher),thenparametricplottingisalso possiblewithezplot.ThusonecanobtainalmostthesamepictureasFigure 5-1withthecommand ezplot(’cos(t)’, ’sin(t)’, 0 2pi); axis squareTwo-DimensionalPlots 69 ContourPlotsandImplicitPlots Acontourplotofafunctionoftwovariablesisaplotofthelevelcurvesofthe function, that is, sets of points in the x-y plane where the function assumes 2 2 aconstantvalue.Forexample,thelevelcurvesofx +y arecirclescentered attheorigin,andthelevelsarethesquaresoftheradiiofthecircles.Contour plotsareproducedinMATLABwithmeshgridandcontour.Thecommand meshgridproducesagridofpointsinaspecifiedrectangularregion,witha specified spacing. This grid is used by contour to produce a contour plot in thespecifiedregion. 2 2 Wecanmakeacontourplotofx +y asfollows: X Y = meshgrid(-3:0.1:3, -3:0.1:3); contour(X, Y, X.ˆ2 + Y.ˆ2) axis square The plot is shown in Figure 5-2. We have used MATLAB’s vector notation to 3 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3 Figure5-270 Chapter5: MATLABGraphics produceagridwithspacing0 .1inbothdirections.Wehavealsoused axis squaretoforcethesamescaleonbothaxes. You can specify particular level sets by including an additional vector ar- √ √ gument to contour. For example, to plot the circles of radii 1, 2, and 3, type contour(X, Y, X.ˆ2 + Y.ˆ2, 1 2 3) The vector argument must contain at least two elements, so if you want toplotasinglelevelset,youmustspecifythesameleveltwice.Thisisquite useful for implicit plotting of a curve given by an equation in x and y.For example,toplotthecircleofradius1abouttheorigin,type contour(X, Y, X.ˆ2 + Y.ˆ2, 1 1) 2 2 2 2 2 Ortoplotthelemniscatex −y =(x +y ) ,rewritetheequationas 2 2 2 2 2 (x +y ) −x +y =0 andtype X Y = meshgrid(-1.1:0.01:1.1, -1.1:0.01:1.1); contour(X, Y, (X.ˆ2 + Y.ˆ2).ˆ2 - X.ˆ2 + Y.ˆ2, 0 0) axis square title(’The lemniscate xˆ2-yˆ2=(xˆ2+yˆ2)ˆ2’) Thecommandtitlelabelstheplotwiththeindicatedstring.(Inthedefault string interpreter, ˆ is used for inserting an exponent and is used for sub- scripts.)TheresultisshowninFigure5-3. IfyouhavetheSymbolicMathToolbox,contourplottingcanalsobe donewiththecommand ezcontour,andimplicitplottingofacurve f(x,y)=0 can also be done with ezplot. One can obtain almost the same picture as Figure5-2withthecommand ezcontour(’xˆ2 + yˆ2’, -3, 3, -3, 3); axis square andalmostthesamepictureasFigure5-3withthecommand ezplot(’(xˆ2 + yˆ2)ˆ2 - xˆ2 + yˆ2’, ... -1.1, 1.1, -1.1, 1.1); axis squareTwo-DimensionalPlots 71 2 2 2 2 2 The lemniscate x -y =(x +y ) 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure5-3 FieldPlots TheMATLABroutinequiverisusedtoplotvectorfieldsorarraysofarrows. The arrows can be located at equally spaced points in the plane (if x and y coordinates are not given explicitly), or they can be placed at specified loca- tions. Sometimes some fiddling is required to scale the arrows so that they don’tcomeoutlookingtoobigortoosmall.Forthispurpose,quivertakesan optionalscalefactorargument.Thefollowingcode,forexample,plotsavector fieldwitha“saddlepoint,”correspondingtoacombinationofanattractive forcepointingtowardthexaxisandarepulsiveforcepointingawayfromthe yaxis: x, y = meshgrid(-1.1:.2:1.1, -1.1:.2:1.1); quiver(x, -y); axis equal; axis off TheoutputisshowninFigure5-4.72 Chapter5: MATLABGraphics Figure5-4 Three-DimensionalPlots MATLABhasseveralroutinesforproducingthree-dimensionalplots. CurvesinThree-DimensionalSpace Forplottingcurvesin3-space,thebasiccommandisplot3,anditworkslike plot, except that it takes three vectors instead of two, one for the x coordi- nates, one for the y coordinates, and one for thez coordinates. For example, wecanplotahelix(seeFigure5-5)with T = -2:0.01:2; plot3(cos(2piT), sin(2piT), T) Again, if you have the Symbolic Math Toolbox, there is a shortcut usingezplot3;youcaninsteadplotthehelixwith ezplot3(’cos(2pit)’, ’sin(2pit)’, ’t’, -2, 2)Three-DimensionalPlots 73 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 1 0.5 1 0.5 0 0 -0.5 -0.5 -1 -1 Figure5-5 SurfacesinThree-DimensionalSpace There are two basic commands for plotting surfaces in 3-space: mesh and surf.Theformerproducesatransparent“mesh”surface;thelatterproduces anopaqueshadedone.Therearetwodifferentwaysofusingeachcommand, oneforplottingsurfacesinwhichthezcoordinateisgivenasafunctionofx and y, and one forparametricsurfaces in which x, y, and z are all given as functionsoftwootherparameters.Letusillustratetheformerwithmeshand thelatterwithsurf. Toplotz= f(x, y),onebeginswitha meshgridcommandasinthecaseof 2 2 contour.Forexample,the“saddlesurface”z=x −y canbeplottedwith X,Y = meshgrid(-2:.1:2, -2:.1:2); Z = X.ˆ2 - Y.ˆ2; mesh(X, Y, Z) TheresultisshowninFigure5-6,althoughitlooksmuchbetteronthescreen since MATLAB shades the surface with a color scheme depending on the z coordinate.Wecouldhavegottenanopaquesurfaceinsteadbyreplacingmesh withsurf.74 Chapter5: MATLABGraphics 4 3 2 1 0 -1 -2 -3 -4 2 1 2 1 0 0 -1 -1 -2 -2 Figure5-6 WiththeSymbolicMathToolbox,thereisashortcutcommandezmesh, andyoucanobtainaresultverysimilartoFigure5-6with ezmesh(’xˆ2 - yˆ2’, -2, 2, -2, 2) If one wants to plot a surface that cannot be represented by an equation 2 2 2 oftheformz= f(x,y),forexamplethespherex +y +z =1,thenitisbet- ter to parameterize the surface using a suitable coordinate system, in this casecylindricalorsphericalcoordinates.Forexample,wecantakeasparam- eters the vertical coordinatezand the polar coordinate θ in thex-y plane. If r denotesthedistancetothezaxis,thentheequationofthespherebecomes √ √ √ 2 2 2 2 2 r +z =1, or r= 1−z , and so x= 1−z cosθ, y= 1−z sinθ. Thus wecanproduceourplotwith theta, Z = meshgrid((0:0.1:2)pi, (-1:0.1:1)); X = sqrt(1 - Z.ˆ2).cos(theta);SpecialEffects 75 1 0.5 0 -0.5 -1 1 0.5 1 0.5 0 0 -0.5 -0.5 -1 -1 Figure5-7 Y = sqrt(1 - Z.ˆ2).sin(theta); surf(X, Y, Z); axis square TheresultisshowninFigure5-7. WiththeSymbolicMathToolbox,parametricplottingofsurfaceshas beengreatlysimplifiedwiththecommands ezsurfandezmesh,andyoucan obtainaresultverysimilartoFigure5-7with ezsurf(’sqrt(1-sˆ2)cos(t)’, ’sqrt(1-sˆ2)sin(t)’, ... ’s’, -1, 1, 0, 2pi); axis equal SpecialEffects So far we have only discussed graphics commands that produce or modify a singlestaticfigurewindow.ButMATLABisalsocapableofcombiningseveral76 Chapter5: MATLABGraphics figures in one window, or of producing animated graphics that change with time. CombiningFiguresinOneWindow The command subplot divides the figure window into an array of smaller figures. The first two arguments give the dimensions of the array of sub- plots, and the last argument gives the number of the subplot (counting left torightacrossthefirstrow,thenlefttorightacrossthenextrow,andsoon) inwhichtoputthenextfigure.Thefollowingexample,whoseoutputappears asFigure5-8,producesa2×2arrayofplotsofthefirstfourBesselfunctions J,0≤n≤3: n x = 0:0.05:40; for j = 1:4, subplot(2,2,j) plot(x, besselj(jones(size(x)), x)) end 1 0.6 0.4 0.5 0.2 0 0 -0.2 -0.5 -0.4 0 10 20 30 40 0 10 20 30 40 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 0 10 20 30 40 0 10 20 30 40 Figure5-8SpecialEffects 77 Animations The simplest way to produce an animated picture is with comet, which pro- duces a parametric plot of a curve (the way plot does), except that you can seethecurvebeingtracedoutintime.Forexample, t = 0:0.01pi:2pi; figure; axis equal; axis(-1 1 -1 1); hold on comet(cos(t), sin(t)) displaysuniformcircularmotion. Formorecomplicatedanimations,youcanuse getframeand movie.The command getframe captures the active figure window for one frame of the movie, and movie then plays back the result. For example, the following (in MATLAB5.3orlater—earlierversionsofthesoftwareusedaslightlydiffer- entsyntax)producesamovieofavibratingstring: x = 0:0.01:1; for j = 0:50 plot(x, sin(jpi/5)sin(pix)), axis(0, 1, -2, 2) M(j+1) = getframe; end movie(M) It is worth noting that the axis command here is important, to ensure that each frame of the movie is drawn with the same coordinate axes. (Other- wise the scale of the axes will be different in each frame and the result- ing movie will be totally misleading.) The semicolon after the getframe command is also important; it prevents the spewing forth of a lot of nu- merical data with each frame of the movie. Finally, make sure that while MATLAB executes the loop that generates the frames, you do not cover the activefigurewindowwithanotherwindow(suchastheCommandWindow). Ifyoudo,thecontentsoftheotherwindowwillbestoredintheframesofthe movie.  MATLAB6hasanewcommandmovieviewthatyoucanuseinplaceof movietoviewtheanimationinaseparatewindow,withabuttontoreplay themoviewhenitisdone.78 Chapter5: MATLABGraphics CustomizingandManipulating Graphics  Thisisamoreadvancedtopic;ifyouwishyoucanskipitonafirstreading. So far in this chapter, we have discussed the most commonly used MATLAB routinesforgeneratingplots.Butoften,togettheresultsonewants,oneneeds to customize or manipulate the graphics these commands produce. Knowing how to do this requires understanding a few basic principles concerning the wayMATLABstoresanddisplaysgraphics.Formostpurposes,thediscussion herewillbesufficient.Butifyouneedmoreinformation,youmighteventually want to consult one of the books devoted exclusively to MATLAB graphics, suchas UsingMATLABGraphics , which comes free (in PDF format) with the software and can be accessed in the “MATLAB Manuals” subsection of the “Printable Documentation” section in the Help Browser (or under “Full DocumentationSet”fromthehelpdeskinMATLAB5.3andearlierversions), orGraphicsandGUIswithMATLAB, 2nd ed., by P. Marchand, CRC Press, BocaRaton,FL,1999. In a typical MATLAB session, one may have many figure windows open at once. However, only one of these can be “active” at any one time. One can findoutwhichfigureisactivewiththecommand gcf, short for “get current figure,”andonecanchangetheactivefigureto,say,figurenumber5withthe commandfigure(5),orelsebyclickingonfigurewindow5withthemouse. The command figure(withnoarguments)createsablankfigurewindow. (Thisissometimesusefulifyouwanttoavoidoverwritinganexistingplot.) Onceafigurehasbeencreatedandmadeactive,therearetwobasicwaysto manipulateit.TheactivefigurecanbemodifiedbyMATLABcommandsinthe commandwindow,suchasthecommands title and axis square that we have already encountered. Or one can modify the figure by using the menus and/ortoolsinthefigurewindowitself.Let’sconsiderafewexamples.Toinsert labelsortextintoaplot,onemayusethecommandstext,xlabel,ylabel, zlabel, and legend, in addition to title. As the names suggest, xlabel, ylabel, and zlabel add text next to the coordinate axes, legend puts a “legend”ontheplot,andtextaddstextataspecificpoint.Thesecommands take various optional arguments that can be used to change the font family andfontsizeofthetext.Asanexample,let’sillustratehowtomodifyourplot ofthelemniscate(Figure5-3)byaddingandmodifyingtext: figure(3) title(’The lemniscate xˆ2-yˆ2=(xˆ2+yˆ2)ˆ2’,...CustomizingandManipulatingGraphics 79 2 2 2 2 2 The lemniscate x -y =(x +y ) 1 0.8 0.6 0.4 0.2 0 ← a node, also an inflection point for each branch -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x Figure5-9 ’FontSize’, 16, ’FontName’, ’Helvetica’,... ’FontWeight’, ’bold’) text(0, 0, ’\leftarrow a node, also an inflection’) text(0.2, -0.1, ’point for each branch’) xlabel(’x’); ylabel(’y’) TheresultisshowninFigure5-9.Notethatmanysymbols(anarrowpointing to the left in this case) can be inserted into a text string by calling them withnamesstartingwith \.(Ifyou’veusedthescientifictypesettingprogram T X, you’ll recognize the convention here.) In most cases the names are self- E explanatory.Forexample,yougetaGreekπ bytyping\pi,asummationsign bytypingeither\Sigma(foracapitalsigma)or\sum,andarrowspointing invariousdirectionswith\leftarrow,\uparrow,andsoon.Formoredetails and a complete list of available symbols, see the listing for “Text Properties” intheHelpBrowser. Analternativeistomakeuseofthetoolbaratthetopofthefigurewindow. Thebuttonindicatedbytheletter“A”addstexttoafigure,andthemenuitem y80 Chapter5: MATLABGraphics Text Properties... in the Tools menu (in MATLAB 5.3), or else the menu itemFigureProperties...intheEditmenu(inMATLAB6),canbeusedto changethefontstyleandfontsize. ChangeofViewpoint Anothercommonandimportantwaytovaryagraphicistochangetheview- pointin3-space.Thiscanbedonewiththecommandview,andalso(atleast inMATLAB5.3andhigher)byusingtheRotate3DoptionintheToolsmenu atthetopofthefigurewindow.Thecommandview(2)projectsafigureinto the x-y plane (by looking down on it from the positive z axis), and the com- mandview(3)viewsitfromthedefaultdirectionin3-space,whichisinthe direction looking toward the origin from a point far out on the ray z=0.5t, x=−0.5272t,y=−0.3044t,t 0. ➱ InMATLAB,anytwo-dimensionalplotcanbe“viewedin3D,”and anythree-dimensionalplotcanbeprojectedintotheplane.Thus Figure5-5above(thehelix),iffollowedbythecommandview(2), producesacircle. ChangeofPlotStyle Anotherimportantwaytochangethestyleofgraphicsistomodifythecoloror linestyleinaplotortochangethescaleontheaxes.Withinaplotcommand, one can change the color of a graph, or plot with a dashed or dotted line, or marktheplottedpointswithspecialsymbols,simplybyaddingastringasa thirdargumentforeveryx-ypair.Symbolsforcolorsare ’y’foryellow, ’m’ for magenta, ’c’ for cyan, ’r’ for red, ’g’ for green, ’b’ for blue, ’w’ for white,and’k’forblack.Symbolsforpointmarkersinclude’o’foracircle, ’x’ for an X-mark, ’+’ for a plus sign, and ’’ for a star. Symbols for line stylesinclude’-’forasolidline,’:’foradottedline,and’’foradashed line.Ifapointstyleisgivenbutnolinestyle,thenthepointsareplottedbut no curve is drawn connecting them. The same methods work with plot3 in placeofplot.Forexample,onecanproduceasolidredsinecurvealongwitha dottedbluecosinecurve,markingallthelocalmaximumpointsoneachcurve withadistinctivesymbolofthesamecolorastheplot,asfollows: X = (-2:0.02:2)pi; Y1 = sin(X); Y2 = cos(X); plot(X, Y1, ’r-’, X, Y2, ’b:’); hold on X1 = -3pi/2 pi/2; Y3 = 1 1; plot(X1, Y3, ’r+’) X2 = -2pi 0 2pi; Y4 = 1 1 1; plot(X2, Y4, ’b’)CustomizingandManipulatingGraphics 81 Herewewouldprobablywantthetickmarksonthex axislocatedatmul- tiplesofπ.Thiscanbedonewiththesetcommandappliedtotheproperties of the axes (and/or by selecting Edit :Axes Properties... in MATLAB 6, or Tools :Axes Properties... in MATLAB 5.3). The command set is used to change various properties of graphics. To apply it to “Axes”, it has to be combinedwiththecommand gca, which stands for “get current axes”. The code set(gca, ’XTick’, (-2:2)pi, ’XTickLabel’,... ’-2pi-pi0pi2pi’) in combination with the code above gets the current axes, sets the ticks on thex axistogofrom−2π to2π inmultiplesofπ,andthenlabelstheseticks thewayonewouldwant(ratherthanindecimalnotation,whichisuglyhere). TheresultisshowninFigure5-10.Incidentally,youmightwonderhowtolabel theticksas−2π,−π,etc.,insteadof-2pi,-pi,andsoon.Thisistrickierbut youcandoitbytyping 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -2pi -pi 0 pi 2pi Figure5-1082 Chapter5: MATLABGraphics set(gca, ’FontName’, ’Symbol’) set(gca, ’XTickLabel’, ’-2p-p0p2p’) sinceintheSymbolfont,π occupiestheslotheldbypintextfonts. Full-FledgedCustomization Whataboutchangestootheraspectsofaplot?Theusefulcommandsgetand setcanbeusedtoobtainacompletelistofthepropertiesofagraphicswindow, and then to modify them. These properties are arranged in a hierarchical structure, identified by markers (which are simply numbers) calledhandles. If you type get(gcf), you will “get” a (rather long) list of properties of the currentfigure(whosenumberisreturnedbythefunctiongcf).Someofthese mightread Color = 0.8 0.8 0.8 CurrentAxes = 72.0009 PaperSize = 8.5 11 Children = 72.0009 HerePaperSizeisself-explanatory;Colorgivesthebackgroundcolorofthe plot in RGB (red-green-blue) coordinates, where 0 0 0 is black and 1 1 1 is white. (0.8 0.8 0.8 is light gray.) Note that CurrentAxes and Children in this example have the same value, the one-element vector containing the funny-looking number 72.0009. This number would also be returned by the command gca (“get current axes”); it is the handle to the axis properties of theplot.ThefactthatthisalsoshowsupunderChildrenindicatesthatthe axispropertiesare“children”ofthefigure,thisis,theylieoneleveldowninthe hierarchicalstructure.Typingget(gca)orget(72.0009)wouldthengive youalistofaxisproperties,includingfurtherChildrensuchas Lineobjects, withinwhichyouwouldfindtheXDataandYDataencodingtheactualplot. Once you have located the properties you’re interested in, they can be changedwithset.Forexample, set(gcf, ’Color’, 1 0 0) changesthebackgroundcoloroftheborderofthefigurewindowtored,and set(gca, ’Color’, 1 1 0) changes the background color of the plot itself (a child of the figure window) toyellow(whichintheRGBschemeishalfred,halfgreen).CustomizingandManipulatingGraphics 83 This “one at a time” method for locating and modifying figure properties can be speeded up using the command findobj to locate the handles of all the descendents (the main figure window, its children, children of children, etc.)ofthecurrentfigure.Onecanalsolimitthesearchtohandlescontaining elementsofaspecifictype.Forexample,findobj(’Type’, ’Line’)hunts for all handles of objects containing a Line element. Once one has located these, set can be used to change the LineStyle from solid to dashed, etc. In addition, the low-level graphics commands line, rectangle, fill, surface, and image can be used to create new graphics elements within a figurewindow. Asanexampleofthesetechniques,thefollowingcodecreatesachessboard onawhitebackground,asshowninFigure5-11: white = 1 1 1; gray = 0.7white; a=0110;b=0011;c=1111; Figure5-1184 Chapter5: MATLABGraphics figure; hold on for k = 0:1, for j = 0:2:6 fill(a’c + c’(0:2:6) + k, b’c+j+k, gray) end, end plot(8a’, 8b’, ’k’) set(gca, ’XTickLabel’, , ’YTickLabel’, ) set(gcf, ’Color’, white); axis square Here white and gray are the RGB codings for white and gray. The double for loop draws the 32 dark squares on the chessboard, using fill, with j indexingthedarksquaresinasingleverticalcolumn,withk=0givingthe odd-numbered rows, and withk=1 giving the even-numbered rows. Note thatfillheretakesthreearguments:amatrix,eachofwhosecolumnsgives thexcoordinatesoftheverticesofapolygontobefilled(inthiscaseasquare), a second matrix whose corresponding columns give the y coordinates of the vertices, and a color. We’ve constructed the matrices with four columns, one for each of the solid squares in a single horizontal row. The plot command draws the solid black line around the outside of the board. Finally, the first set command removes the printed labels on the axes, and the second set commandresetsthebackgroundcolortowhite. QuickPlotEditingintheFigureWindow Almostallofthecommand-linechangesonecanmakeinafigurehavecoun- terparts that can be executed using the menus in the figure window. So why botherlearningbothtechniques?Thereasonisthateditinginthefigurewin- dowisoftenmoreconvenient,especiallywhenonewishesto“experiment”with various changes, while editing a figure with MATLAB code is often required whenwritingM-files.SothetrueMATLABexpertusesbothtechniques.The figurewindowmenusareabitdifferentinMATLAB6thaninMATLAB5.3. InMATLAB6,youcanzoominandoutandrotatethefigureusingtheTools menu,youcaninsertlabelsandtextwiththe Insertmenu,andyoucanview and edit the figure properties (just as you would with set)withthe Edit menu. For example you can change the ticks and labels on the axes by se- lectingEdit:EditAxes....InMATLAB5.3,editingofthefigurepropertiesis donewiththe Property Editor, located under the File menu of the figure window.Bydefaultthisopenstothefigureproperties,anddouble-clickingon “Children”thenenablesyoutoaccesstheaxesproperties,etc.Sound 85 Sound You can use sound to generate sound on your computer (provided that your computer is suitably equipped). Although, strictly speaking, sound is not a graphicscommand,wehaveplaceditinthischaptersincewethinkof“sight” and“sound”asbeingalliedfeatures.Thecommandsoundtakesavector,views itasthewaveformofasound,and“plays”it.Thelengthofthevector,dividedby 8192,isthelengthofthesoundinseconds.A“sinusoidal”vectorcorresponds toapuretone,andthefrequencyofthesinusoidalsignaldeterminesthepitch. ThusthefollowingexampleplaysthemottofromBeethoven’s5thSymphony: x=0:0.1pi:250pi; y=zeros(1,200); z=0:0.1pi:1000pi; sound(sin(x),y,sin(x),y,sin(x),y,sin(z4/5),y,... sin(8/9x),y,sin(8/9x),y,sin(8/9x),y,sin(z3/4)); Notethatthezerovectoryinthisexamplecreatesaveryshortpausebetween successivenotes.PracticeSetB Calculus,Graphics, andLinearAlgebra Problems2,3,5–7,andpartsof10–12requiretheSymbolicMathTool- box.Theothersdonot. 1. Usecontourtodothefollowing: 3 3 (a) Plot the level curves of the function f(x,y)=3y+y −x in the regionwherex andyarebetween−1and1(togetanideaofwhat thecurveslooklikeneartheorigin),andinsomelargerregions(to getthebigpicture). 3 3 (b) Plotthecurve3y+y −x =5. (c) Plotthelevelcurveofthefunction f(x,y)=ylnx+xlnythatcon- tainsthepoint(1,1). 2. Find the derivatives of the following functions. If possible, simplify each answer. 3 2 (a) f(x)=6x −5x +2x−3. 2x−1 (b) f(x)= . 2 x +1 2 (c) f(x)=sin(3x +2). (d) f(x)=arcsin(2x+3). √ 4 (e) f(x)= 1+x . r (f) f(x)=x . 2 (g) f(x)=arctan(x +1). 3. SeeifMATLABcandothefollowingintegralssymbolically.Fortheindef- initeintegrals,checktheresultsbydifferentiating. π/2 (a) cosxdx. 0 2 (b) xsin(x )dx. √ (c) sin(3x) 1−cos(3x)dx. 86

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