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In this worksheet we look more closely at Map le's plotting facility. We are going to use a lot of plotting in the f uture. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 46 "Fancier Plots. Plotting Families of Functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "There are many options th at can be added to the basic \"" }{TEXT 259 4 "plot" }{TEXT -1 45 "\" \+ command to enhance your plots. For example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "plot(t^2, t=0..3, color=blue,thickness=2,labels=[\"time in seconds\",\"velocity in feet \"]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The command \+ \"" }{TEXT 262 12 "labels=[...]" }{TEXT -1 60 "\" allows you to label \+ the axes on your graph. The commands \"" }{TEXT 260 10 "color=blue" } {TEXT -1 4 "\", \"" }{TEXT 261 11 "thickness=2" }{TEXT -1 270 "\" spec ify the color and the thickness of your graph. (The higher the number, the thicker the graph). These and many other options are especially h elpful when you want to graph several functions in one coordinate syst em. Suppose you want to study the family of functions " }{XPPEDIT 18 0 "y = sin(a*x);" "6#/%\"yG-%$sinG6#*&%\"aG\"\"\"%\"xGF*" }{TEXT -1 26 " for different values of " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 178 " . The simplest way is to graph a few of them and look at thei r graphs. We can graph several functions in one coordinate system by e ntering formulas for the functions under the \"" }{TEXT 270 4 "plot" } {TEXT -1 55 "\" command together with the common range. For example: \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(\{sin(x), sin(2*x) , sin(3*x)\}, x=-Pi..Pi);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "As you see, we put the functions we wanted graphed inside the c urly brackets " }{TEXT 266 5 "\{...\}" }{TEXT -1 253 ", which in Maple denotes the set of elements. Maple drew the three functions above. Bu t which is which? Without knowing that, we cannot really analyze the s ituation. One of the ways of distinguishing between your graphs is to \+ enter functions under the \"" }{TEXT 263 4 "plot" }{TEXT -1 19 "\" com mand not as a " }{TEXT 264 3 "set" }{TEXT -1 11 ", but as a " }{TEXT 265 4 "list" }{TEXT -1 54 ". A list in Maple is always denoted by squa re brackets" }{TEXT 267 7 " [....]" }{TEXT -1 155 ". The difference be tween a set and a list is that in the list the order of elements matte rs, while in the set it does not. We put our three functions as a " } {TEXT 268 4 "list" }{TEXT -1 14 " and assign a " }{TEXT 269 4 "list" } {TEXT -1 72 " of colors to them, so we know what color corresponds to \+ which function." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 68 "plot([sin(x),sin(2*x),sin(3*x)], x=-Pi..Pi, \+ color=[blue,red,black]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Now the graph of the first function " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 187 " on our list is in blue, the second \+ in red and so on. Instead of using colors, you could differentiate bet ween your graphs by assigning a list of thickness values to your funct ions, e.g. " }{TEXT 271 17 "thickness=[1,2,3]" }{TEXT -1 32 ", or diff erent line styles, e.g." }{TEXT 272 18 " linestyle=[1,2,3]" }{TEXT -1 75 ". The latter command will produce graphs drawn with different brok en lines." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 23 "Plotting Numerical Data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 182 "As we already \+ know, functions are often given numerically, through a table of values , rather than by formulas. Suppose we have the following data regardin g the population of a town, " }{XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT -1 16 ", in thousands, " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 87 " yea rs after Jan 1, 1990. The first coordinate stands for t, the second co ordinate for " }{XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 105 " (0,15.1), (1, 16), (2, 16. 9), (3, 18), (4, 18.9), (5, 20), (6, 21.4), (7, 22.7). " }}{PARA 0 "" 0 "" {TEXT -1 149 "Is the growth of the town's population approxi mately exponential? We know that to answer this question we have to ch eck the ratios between values of " }{XPPEDIT 18 0 "P;" "6#%\"PG" } {TEXT -1 43 " corresponding to equally spaced values of " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "16/15.1, 16.9/16, 18/16.9, 18.9/18, 20/18.9, 21.4/20, 22.7/21.4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)$\"+\\Egf5!\"*$\"+++Dc5F%$\"+d()3l5F%$\"++++]5F%$\"+e5?e5F%$\"++ ++q5F%$\"+kwug5F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 606 "The ratios are not exactly the sa me, but they are all relatively close to 1.06. This means that the po pulation of the town has been increasing at approximately 6% per year. The approximation is pretty good. The difference between the actual \+ ratios and 1.06 does not exceed 0.01=1%. In this context, 1% of the po pulation means a couple of hundred people. Hence, we can safely say th at the growth of the population has been approximately exponential wit h the growth factor 1.06. With the initial population of 15.1 thousand , the formula which describes approximately the growth of the town's p opulation is " }{XPPEDIT 18 0 "P = 15.1(1.06)^t;" "6#/%\"PG)--%&FloatG 6$\"$^\"!\"\"6#-F(6$\"$1\"!\"#%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "We would like to plot \+ the numerical data and compare it with the graph of the exponential fu nction. To plot the numerical data we first define a list of the conse cutive data points. As mentioned above, lists in Maple are built using the square brackets " }{TEXT 273 4 "[..]" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "s : = [[0,15.1],[1,16],[2,16.9],[3,18],[4,18.9],[5,20],[6,21.4],[7,22.7]]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG7*7$\"\"!$\"$^\"!\"\"7$\"\" \"\"#;7$\"\"#$\"$p\"F*7$\"\"$\"#=7$\"\"%$\"$*=F*7$\"\"&\"#?7$\"\"'$\"$ 9#F*7$\"\"($\"$F#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "The command to plot a list of poin ts is \"" }{TEXT 274 9 "pointplot" }{TEXT -1 29 "\". Similarly as the \+ command \"" }{TEXT 275 7 "display" }{TEXT -1 39 "\" that we use below, it is a part of a " }{TEXT 277 7 "package" }{TEXT -1 114 " of plottin g commands called \"plots\". When you start Maple it loads into the co mputer's memory only the so called " }{TEXT 287 6 "kernel" }{TEXT -1 196 ", which contains basic functions and commands. More advanced comm ands are contained in so called packages which have to be loaded when \+ you need them. The package \"plots\" is loaded by the command \"" } {TEXT 276 11 "with(plots)" }{TEXT -1 16 "\". If you type \"" }{TEXT 278 12 "with(plots);" }{TEXT -1 180 "\" ending with a semicolon, Maple will list all the commands in the package. If you want Maple to load \+ the package, but don't care to see its content end your command with a colon \"" }{TEXT 279 12 "with(plots):" }{TEXT -1 91 "\" Since this is our first time with the package \"plots\" we are curious to see its c ontent. " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "with(plots);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7U% (animateG%*animate3dG%-animatecurveG%-changecoordsG%,complexplotG%.com plexplot3dG%*conformalG%,contourplotG%.contourplot3dG%*coordplotG%,coo rdplot3dG%-cylinderplotG%,densityplotG%(displayG%*display3dG%*fieldplo tG%,fieldplot3dG%)gradplotG%+gradplot3dG%-implicitplotG%/implicitplot3 dG%(inequalG%-listcontplotG%/listcontplot3dG%0listdensityplotG%)listpl otG%+listplot3dG%+loglogplotG%(logplotG%+matrixplotG%(odeplotG%'pareto G%*pointplotG%,pointplot3dG%*polarplotG%,polygonplotG%.polygonplot3dG% 4polyhedra_supportedG%.polyhedraplotG%'replotG%*rootlocusG%,semilogplo tG%+setoptionsG%-setoptions3dG%+spacecurveG%1sparsematrixplotG%+sphere plotG%)surfdataG%)textplotG%+textplot3dG%)tubeplotG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Among the comma nds we see \"" }{TEXT 280 9 "pointplot" }{TEXT -1 58 "\" that is used \+ to plot a list of points, and the command \"" }{TEXT 281 7 "display" } {TEXT -1 168 "\", which allows you to plot graphs of different types o r over different ranges in one coordinate system. Having loaded the pa ckage, we can use commands contained in it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pointplot(s);" }} {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "In order to compare the graph with the graph of the corresponding exp onential function, we shall use the \"" }{TEXT 282 7 "display" }{TEXT -1 149 "\" command to plot the two graphs in one coordinate system. It is a good idea to first label the two graphs that we want plotted. Ob serve how we use \"" }{TEXT 283 2 ":=" }{TEXT -1 28 "\" to define the \+ meaning of \"" }{TEXT 284 1 "a" }{TEXT -1 7 "\" and \"" }{TEXT 285 1 " b" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a :=pointplot(s):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "b :=plot(15.1*(1.06)^x, x=0..7):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display([a,b]);" }}{PARA 13 "" 1 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "Observe that the first two comman ds end with a colon and not a semicolon to prevent Maple from attempti ng to print each of the graphs separately before the \"" }{TEXT 286 7 "display" }{TEXT -1 10 "\" command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 4 "" 0 "" {TEXT -1 17 "Homework Problems" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 290 7 "Note. " } {TEXT 291 126 "Remember to give your explanations in complete sentence s and to type them in your homework worksheet using Maple's text mode ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 256 10 "Problem 1." }{TEXT -1 48 " Consider the family of expon ential functions " }{XPPEDIT 18 0 "y = 2*exp(k*x);" "6#/%\"yG*&\"\"# \"\"\"-%$expG6#*&%\"kGF'%\"xGF'F'" }{TEXT -1 23 " for various values o f " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 25 " . The proper syntax f or " }{XPPEDIT 18 0 "exp(k*x);" "6#-%$expG6#*&%\"kG\"\"\"%\"xGF(" } {TEXT -1 4 " is " }{TEXT 258 8 "exp(k*x)" }{TEXT -1 11 ". Again: " } {XPPEDIT 18 0 "exp(k*x);" "6#-%$expG6#*&%\"kG\"\"\"%\"xGF(" }{TEXT -1 35 " is entered on the command line as " }{TEXT 288 8 "exp(k*x)" } {TEXT -1 41 " and not anything else. Similarly, say, " }{XPPEDIT 18 0 "exp(a*b*t+5);" "6#-%$expG6#,&*(%\"aG\"\"\"%\"bGF)%\"tGF)F)\"\"&F)" }{TEXT -1 15 " is entered as " }{TEXT 289 12 "exp(a*b*t+5)" }{TEXT -1 36 " etc.. Follow the syntax carefully!" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 48 "(a) Plot a few funtions in the f amily, say for " }{XPPEDIT 18 0 "k = .7,1.5,-.7,-1.5;" "6&/%\"kG-%&Flo atG6$\"\"(!\"\"-F&6$\"#:F),$-F&6$F(F)F),$-F&6$F,F)F)" }{TEXT -1 55 " , in the same coordinate system to see the effects of " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 204 " on the exponential. Make sure you know \+ which graph is which by assigning a list of colors. In order to see th e graphs clearly, choose a range for x that is not too large. For exam ple, x between -2 and 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 50 "(b) Describe in words the effect of the parmeter \+ " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 14 " on the graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 12 "Problem 2. " }{TEXT -1 21 "Consider the family " }{XPPEDIT 18 0 "y \+ = exp(-(x-a)^2);" "6#/%\"yG-%$expG6#,$*$,&%\"xG\"\"\"%\"aG!\"\"\"\"#F. " }{TEXT -1 9 " , where " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 17 " is a parameter. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 206 "(a) Plot the functions for various values of a, e.g. a= 0 ,1, -1, and for x between -3 and 3, in the same coordinate system. D istinguish between the graphs by assigning a list of colors or thickne ss values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "(b) Descibe in words the effect of " }{XPPEDIT 18 0 "a;" "6#% \"aG" }{TEXT -1 15 " on the graph. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 11 "Problem 3. " }{TEXT -1 130 "The amount of a certain drug in a patient's blood t hours after t he injection, in micrograms per cc, is measured to be as follows:" }} {PARA 0 "" 0 "" {TEXT -1 63 " (0, 80.1), (2, 63), (4, 50.1), \+ (6, 40.6), (8, 32.4)," }}{PARA 0 "" 0 "" {TEXT -1 82 "where the first coordinate denotes time t, the second the amount of drug left, Q. " } }{PARA 0 "" 0 "" {TEXT -1 67 "(a) Is the drug decaying in an approxim ately exponential fashion? " }}{PARA 0 "" 0 "" {TEXT -1 67 "(b) Find t he approximate exponential formula for Q in terms of t. (" }{TEXT 292 5 "Hint:" }{TEXT -1 116 " Observe that the changes in t above are ever y 2 units. Make sure that you get the right base for your exponential. )" }}{PARA 0 "" 0 "" {TEXT -1 26 "(c) Plot the data points. " }}{PARA 0 "" 0 "" {TEXT -1 134 "(d) Plot in one coordinate system both the dat a points and the graph of the approximating exponential function. Do t hey appear close? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 293 6 "Note. " }{TEXT -1 19 "If you execute the \+ " }{TEXT 294 7 "restart" }{TEXT -1 65 " command in your homework works heet, do not forget to reload the " }{TEXT 295 12 "with(plots);" } {TEXT -1 26 " package before using the " }{TEXT 297 10 "pointplot " } {TEXT -1 8 "and the " }{TEXT 298 8 "display " }{TEXT -1 9 "commands!" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 76 "MTH 141 Maple Worksheets written by B. Kaskosz and L. Pakula, Copy right 1998" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 296 38 "Las t modified August 1999. Update 2006" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }