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{EXCHG {PARA 239 "" 0 "" {TEXT 234 38 "Introduction to Maple \+ Versions 8 to 12" }}{PARA 235 "" 0 "" {TEXT 235 77 "Information in thi s document was compiled and updated by Angie Gardiner from:" }}{PARA 233 "" 0 "" {TEXT 236 95 "Multivariate Mathematics with Maple by James A. Carlson and Jennifer M. Johnson, copyright 1996" }}{PARA 217 "" 0 "" {TEXT 237 109 "Introduction to Maple V.4 in the Undergraduate Compu ter Lab by James A. Carlson and Jennifer M. Johnson, 1993" }}{PARA 218 "" 0 "" {TEXT 238 41 "Math 2250 Maple Tutorial by Nick Korevaar" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 220 "" 0 "" {TEXT 212 14 "Wha t is Maple?" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 681 "Maple is a gen eral purpose software package containing tools useful in the study of \+ a wide variety of problems. We can use Maple like a calculator to car ry out numerical computation, or we can use the programming features o f Maple to produce a more complicated sequence of computations. In ad dition to numerical computation, we can do symbolic or algebraic compu tations in Maple. This software package has built-in functions for so lving many kinds of equations, for working with matrices and vectors, \+ and for differentiation and integration. It also contains functions w hich allow us to create visual representations of curves and surfaces \+ from their mathematical descriptions." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 681 "Using a software package like \+ Maple gives new ways to think about problems. For example, we can use the computer to run numerical experiments. This means that we can no t only look at more realistic versions of standard problems, but we ca n easily look at many more examples, in the process developing a bette r intuitive feel for the essential features of a general problem and i ts solutions. Often these experiments suggest further questions or in terpretations that might otherwise go unnoticed. Patterns that appear in these experiments give insight into and motivation for the standar d approaches to these problems - and frequently highlight the limitati ons of these methods." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 237 "In many situations the graphics features of Map le are a useful tool for visualizing the geometric ideas behind import ant definitions and standard techniques. Again, this goes a long way \+ toward giving a better feeling for the essentials." }}{PARA 219 "" 0 " " {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 272 "Before you start l earning Maple, it would be helpful if you familiarize yourself with th e basics of the computer lab in the T. Benny Rushing Mathematics Stude nt Center (if you will be working here). Please refer to \"Introducti on to the Computer Lab\" for this information." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 222 "" 0 "" {TEXT 233 10 "The Basics" }}{PARA 209 "" 0 "" {TEXT 222 15 "Starting Maple." }{TEXT 239 100 " If you ar e using a UNIX station in our lab, either move the cursor into the loc al window and type " }{TEXT 222 8 "xmaple &" }{TEXT 239 88 ", use the \+ middle mouse button on the background to get a menu from which you can select " }{TEXT 222 11 "X Maple V12" }{TEXT 239 323 " (or another ver sion), or click on the maple leaf icon at the bottom of the screen. S oon the ghost of a window will appear. Move the mouse to position thi s window, then click with the left mouse button when satisfied. If yo u are using a Mac in our lab, click on the icon at the upper right cor ner of your screen labeled " }{TEXT 222 1 "/" }{TEXT 239 46 ". This w ill open a new window. Click on the " }{TEXT 222 13 "Applications " } {TEXT 239 39 "button in this window, and then select " }{TEXT 222 7 "X Darwin" }{TEXT 239 34 " from the applications and run it " }{TEXT 222 8 "rootless" }{TEXT 239 84 ". After a couple of minutes, you will get a terminal window. In this window, type " }{TEXT 222 11 "ssh sunfire " }{TEXT 239 20 ". Now you can type " }{TEXT 222 6 "xmaple" }{TEXT 239 149 " at the prompt to start Maple. Once Maple opens, you can use the buttons in the upper right corner of each window to resize the wi ndows if you like." }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }} {PARA 209 "" 0 "" {TEXT 222 14 "Saving a file." }{TEXT 239 83 " To sa ve a Maple session that you have been working on and give it a name, c hoose " }{TEXT 222 16 "File -> Save As " }{TEXT 239 44 "from the menus at the top of the screen. A " }{TEXT 222 7 "Save As" }{TEXT 239 67 " window will appear. If you want to save your file under the name " } {TEXT 222 5 "hwk1 " }{TEXT 239 21 "(for example), enter " }{TEXT 222 4 "hwk1" }{TEXT 239 8 " in the " }{TEXT 222 8 "Filename" }{TEXT 239 29 " box. If there is already a " }{TEXT 222 5 "*.mws" }{TEXT 239 43 " in the box, make sure that you delete the " }{TEXT 222 1 "*" }{TEXT 239 16 ". Then hit the " }{TEXT 222 6 "Return" }{TEXT 239 11 " (Macs) or " }{TEXT 222 5 "Enter" }{TEXT 239 25 " (UNIX) key, or click on " } {TEXT 222 2 "OK" }{TEXT 239 195 ". It is best to keep your file names simple, using only letters and numbers, without any special symbols. \+ In particular, it is a good idea to avoid using spaces or periods. M aple will add the " }{TEXT 222 5 ".mws " }{TEXT 211 2 "or" }{TEXT 222 4 " .mw" }{TEXT 239 43 " extension to identify the saved file as a " } {TEXT 222 1 "M" }{TEXT 239 5 "aple " }{TEXT 222 1 "w" }{TEXT 239 3 "or k" }{TEXT 222 1 "s" }{TEXT 239 118 "heet. Be careful about reusing a \+ file name from a previous session. For example, if last week you made a file named " }{TEXT 222 9 "Lines.mws" }{TEXT 239 56 " and this week you open a new Maple session and call it " }{TEXT 222 9 "Lines.mws" } {TEXT 239 72 " also, this will ERASE the file you created last week. \+ Notice that the " }{TEXT 222 7 "Save As" }{TEXT 239 157 " window lists the Maple files in your current directory, so you can see which names you have already used. Once you have given your file a name, you can do " }{TEXT 222 12 "File -> Save" }{TEXT 239 158 ", or use the save i con at the top of the window. It is a good idea to save your files fa irly often, so that you won't lose your work if something goes wrong." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 209 "" 0 "" {TEXT 222 10 " Copy/Paste" }{TEXT 239 365 ". Often you will want to re-enter a comma nd that is a small variation on an earlier command. In that case, ins tead of retyping you can copy an old command and then paste it onto a \+ new command line. Then you can move the cursor into the line and make modifications. If you are using a Mac, use the mouse to highlight th e command(s) you want to copy, then choose " }{TEXT 222 12 "Edit -> Co py" }{TEXT 239 87 " from the menus. Move the cursor to where you want to paste the command(s) and choose " }{TEXT 222 13 "Edit -> Paste" } {TEXT 239 481 ". If you are using a UNIX station, you can use the cop y/paste menu choices, or you can copy/paste with the mouse as follows. Press and hold down the left mouse button at the beginning of the te xt you want to copy. Still holding the button down, drag the mouse un til you have highlighted the desired text. Release the mouse button. \+ Move the cursor to the position where you want to paste. Click the l eft mouse button to mark this spot. Click the middle mouse button to \+ paste." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 209 "" 0 "" {TEXT 239 116 "You can paste repeatedly - the copied text stays in memory an d anytime you press the middle mouse button (or choose " }{TEXT 222 5 "Paste" }{TEXT 239 10 " from the " }{TEXT 222 4 "Edit" }{TEXT 239 221 " menu) it will be pasted at the current cursor position. (This can c ause some problems for you if you accidentally hit the middle mouse bu tton later on, so you may want to copy some blank space to clear the c opied text.)" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 209 "" 0 "" {TEXT 222 23 "Editing your Maple file" }{TEXT 239 456 ". It may take \+ more than one session at the computer to complete your analysis. You \+ will need to save your work, and come back to the lab and open your Ma ple file again to continue working. Often it will be useful to take w ith you a printed copy of your Maple session so you can think about wh at you have done so far, and how you want to continue when you come ba ck to the lab, or so that you can talk to your instructor about any qu estions you may have." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 209 "" 0 "" {TEXT 239 261 "Before you learn how to print your Maple sessio n, you should learn how to clean it up a bit. For example, to delete \+ a computation or a plot, just move the cursor into the command that pr oduced the computation or click on the plot. Then while holding down \+ the " }{TEXT 222 7 "Control" }{TEXT 239 16 " key, press the " }{TEXT 222 6 "Delete" }{TEXT 239 241 " key. You could also discover this by \+ looking at the Edit menu. You can learn the key commands for editing \+ from the menus. You can also delete text or commands by using the mou se to highlight the area you want to delete, then hitting the " }{TEXT 222 6 "Delete" }{TEXT 239 5 " key." }}{PARA 219 "" 0 "" {TEXT 249 0 " " }}{PARA 208 "" 0 "" {TEXT 208 191 "It is also a good idea to include some explanations with your work. There are two ways to do this. Fi rst, you can use comments. In any command line, Maple ignores anythin g typed after the " }{TEXT 209 1 "#" }{TEXT 208 34 " symbol. So inser ting a line like" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 57 "solve ( 3*x + 2*y - 5*z, z); # Get z i n terms of x and y" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 38 "makes yo ur work much easier to follow." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 173 "If you want to add explanatory comments between commands, move the curso r into the first command. Then click on the icon at the top of the sc reen containing a Maple prompt " }{TEXT 209 1 ">" }{TEXT 208 160 " . \+ A new command line should appear and the cursor will be on this new li ne. To enter text instead of Maple commands, click on the first toolb ar icon labeled " }{TEXT 205 1 "T" }{TEXT 208 241 ". Now you can type your explanation, and it will appear as ordinary text. Notice the pu ll-down menus and buttons at the top of the Maple window (when you are in text mode) that allow you to control the size and style of the tex t you enter." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 209 "" 0 "" {TEXT 222 9 "Printing." }{TEXT 239 58 " To print your Maple session d irectly from Maple, choose " }{TEXT 222 5 "Print" }{TEXT 239 10 " from the " }{TEXT 222 5 "File " }{TEXT 239 95 "menu (or click on the Print er icon). In the Printer Setup window that appears, make sure that " }{TEXT 222 13 "Print Command" }{TEXT 239 52 " is selected, rather than Output to File, and click " }{TEXT 222 5 "Print" }{TEXT 239 77 ". (I f the default is the printer, to print to a file, choose the box marke d " }{TEXT 222 13 "Print To File" }{TEXT 239 118 " on the General tab \+ of the print dialog box. Some maple versions don't have the feature.) \+ The default print command (" }{TEXT 242 3 "lpr" }{TEXT 239 144 ") sho uld send a printout of your Maple worksheet to the printer. If only p art of your worksheet is printing, try changing the print command to " }{TEXT 222 6 "lpr -l" }{TEXT 239 4 " or " }{TEXT 222 9 "lpr -oraw" } {TEXT 239 1 "." }}{PARA 209 "" 0 "" {TEXT 250 0 "" }}{PARA 209 "" 0 "" {TEXT 239 161 "Alternatively, you can save your Maple session to a po stscript file, then print that file from the local window. To do this , in the Printer Setup window choose " }{TEXT 222 14 "Output to File" }{TEXT 239 11 " and click " }{TEXT 222 5 "Print" }{TEXT 239 96 ". Thi s will create a postscript file from your Maple worksheet and will sto re it with the name " }{TEXT 222 7 "hwk1.ps" }{TEXT 239 141 " (if your Maple session had been named hwk1.mws as in the example above). To s end this file to the printer, go to the local window and type " }{TEXT 222 13 "print hwk1.ps" }{TEXT 239 98 ", then press Enter. If you are printing from the local window, make sure that you are printing a " } {TEXT 243 3 ".ps" }{TEXT 239 22 " file. The file types " }{TEXT 239 2 "a " }{TEXT 244 4 ".mws" }{TEXT 239 12 " file nor a " }{TEXT 245 4 ".p df" }{TEXT 239 1 " " }{TEXT 239 79 "can only be printed from their res pective applications, maple and acroread. Do " }{TEXT 239 1 " " }{TEXT 246 3 "NOT" }{TEXT 239 6 " use " }{TEXT 247 5 "print" }{TEXT 239 21 " on these file types." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 2 "**" }{TEXT 248 74 "If you are having trouble pri nting, please ask the lab assistant for help." }{TEXT 239 2 "**" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 209 "" 0 "" {TEXT 222 21 "Com ing back for more." }{TEXT 239 171 " When you come back to work on a \+ Maple file that you've already started, your task now is to reopen tha t file and get back to where you left off. To open a file, choose " } {TEXT 222 12 "File -> Open" }{TEXT 239 6 ". An " }{TEXT 222 9 "Open F ile" }{TEXT 239 130 " window will appear. In this window you will see a list of Maple files. Click on the file that you want to open, then press the " }{TEXT 222 5 "Enter" }{TEXT 239 14 " key or click " } {TEXT 222 2 "OK" }{TEXT 239 1 "." }}{PARA 219 "" 0 "" {TEXT 249 0 "" } }{PARA 209 "" 0 "" {TEXT 239 347 "Your Maple window should now look li ke it did the last time you saved your file. However, you are not qui te ready to continue working. Although Maple displays a record of you r last session, it does not remember what happened last time. One way to get Maple to remember what happened last time is to go to the begi nning of your file and hit the " }{TEXT 222 6 "Return" }{TEXT 239 288 " key until you get to the last line of the file. (You might want to \+ use the arrow keys to skip the lines that displayed plots or requests \+ for help files). This will ensure that Maple's memory of your last se ssion is restored. A shortcut for executing the entire worksheet is t o select " }{TEXT 222 28 "Edit -> Execute -> Worksheet" }{TEXT 239 1 " ." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 211 "" 0 "" {TEXT 213 11 "Using Maple" }}{PARA 208 "" 0 "" {TEXT 208 95 "The best way to lea rn Maple is by using it. But first, a few remarks about Maple. The s ymbol " }{TEXT 209 1 ">" }{TEXT 208 426 " is the command prompt, which Maple uses to signal you that it awaits your command. Commands norma lly end with a semicolon (if you want to see the output) or a colon (i f you don't want to see the output). Maple is much fussier about rule s of punctuation, grammar, and spelling than are humans. If something is not working right, check to see if you are following the rules. F or example, Maple will get confused if you say " }{TEXT 209 2 "2x" } {TEXT 208 12 " instead of " }{TEXT 209 3 "2*x" }{TEXT 208 189 ". Chec k for things like misspelled names or extra or missing parentheses. I f further thought doesn't clear things up, ask a human for help. You \+ will soon become an expert troubleshooter." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 252 "While learning Maple, you \+ will often have questions about how a particular command or function i s used. Fortunately, Maple can help you. To ask about a command whos e name you know, just type a question mark, followed by the name of th e command. Thus" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 207 "> " 0 "" {MPLTEXT 1 0 6 "?solve" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }} {PARA 208 "" 0 "" {TEXT 208 25 "gives information on the " }{TEXT 209 5 "solve" }{TEXT 208 303 " command. You will probably find the exampl es at the end more useful than the technical information at the beginn ing of the help file. If you don't know the name of the command you w ant to use, try doing a full text search or topic search of Maple's he lp files. You will find these options under the " }{TEXT 205 4 "Help" }{TEXT 208 101 " menu in the upper right corner of the Maple window. \+ You will also find other handy things, such as " }{TEXT 205 12 "Intro duction" }{TEXT 208 2 ", " }{TEXT 205 15 "New User's Tour" }{TEXT 208 6 ", and " }{TEXT 205 10 "Using Help" }{TEXT 208 1 "." }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 64 "Let 's get started! Go ahead and type along with these examples." }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 16 "2 + 2; 3*5; 6-2:" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 192 "All three computations were done, although only two results a re shown (the : at the end of a command suppresses the output). If yo u forget the semicolon, go ahead and put it on the next line:" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 3 "3*5" }}{PARA 207 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {EXCHG {PARA 221 "" 0 "" {TEXT 239 172 "Addition +, subtraction -, and division / are standard, and parentheses are used in the usual way. \+ An asterisk * indicates multiplication and a caret ^ is used for power s:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 28 "( 1 + 2) * (6 + 7) - 12 / 7;" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 8 "3^(2.1);" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }{TEXT 208 349 "Whenever possible, Maple tries to compute exa ct quantities. Our first command gives its answer as a fraction, rath er than as a decimal, contrary to what you might expect. The second c ommand gives a decimal, or \"floating point\" answer because we used t his form in our question. To force Maple to give results in floating \+ point (decimal) form, use " }{TEXT 209 5 "evalf" }{TEXT 208 1 ":" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 3 "Pi;" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 10 "ev alf(Pi);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 24 "exp(1); # t he number e" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 4 "The " }{TEXT 209 1 "%" }{TEXT 208 53 " sign stands for the most recently computed quantity." }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 76 "Be \+ aware that Maple distinguishes upper-case letters from lower-case. Th us " }{TEXT 209 9 "evalf(pi)" }{TEXT 208 20 " is not the same as " } {TEXT 209 9 "evalf(Pi)" }{TEXT 208 16 ". The function " }{TEXT 209 5 "evalf" }{TEXT 208 81 " can take a second (optional) input which deter mines the precision of the output." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 14 "evalf(Pi, 50);" }}} {EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 266 "For the most part, spacing is unimportant in Maple. In the comma nds above, spaces could be omitted or added without causing any proble ms. However, thoughtful use of spacing makes Maple code easier to rea d, and so easier to understand and, if necessary, to correct." }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 109 "St andard mathematical functions can be used in Maple so long as we know \+ their names. To compute the quantity" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 212 "" 0 "" {XPPEDIT 18 0 "Typesetting:-mrow(Type setting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mfenced(Typesetting: -mrow(Typesetting:-mo(\"&uminus0;\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesett ing:-mn(\"14\", mathvariant = \"normal\")), mathvariant = \"normal\", \+ Typesetting:-msemantics = \"abs\", open = \"|\", close = \"&ver bar;\", Typesetting:-msemantics = \"abs\"), Typesetting:-mo(\"+\", mat hvariant = \"normal\", fence = \"false\", separator = \"false\", stret chy = \"false\", symmetric = \"false\", largeop = \"false\", movableli mits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mi(\"sin\", italic = \"false\", mathvariant = \"normal\"), Typesetting:-mo(\"&ApplyFunct ion;\", mathvariant = \"normal\", fence = \"false\", separator = \"fal se\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\" , movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", r space = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting :-mn(\"1\", mathvariant = \"normal\")), mathvariant = \"normal\")), Ty pesetting:-mo(\"−\", mathvariant = \"normal\", fence = \"false\" , separator = \"false\", stretchy = \"false\", symmetric = \"false\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", ls pace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-msqrt( Typesetting:-mn(\"2\", mathvariant = \"normal\")), Typesetting:-mo(\"+ \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-msup(Typesetting:-mi(\"e\", i talic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Typeset ting:-mi(\"cos\", italic = \"false\", mathvariant = \"normal\"), Types etting:-mo(\"⁡\", mathvariant = \"normal\", fence = \"fa lse\", separator = \"false\", stretchy = \"false\", symmetric = \"fals e\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Type setting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mn (\"1.6\", mathvariant = \"normal\"), Typesetting:-mo(\"&InvisibleTimes ;\", mathvariant = \"normal\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rsp ace = \"0.0em\"), Typesetting:-mi(\"\1700\", italic = \"false\", mathv ariant = \"normal\")), Typesetting:-mi(\"\")), mathvariant = \"normal \")), superscriptshift = \"0\"), Typesetting:-mo(\"+\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222 222em\"), Typesetting:-mrow(Typesetting:-mi(\"arctan\", italic = \"fal se\", mathvariant = \"normal\"), Typesetting:-mo(\"⁡\", \+ mathvariant = \"normal\", fence = \"false\", separator = \"false\", st retchy = \"false\", symmetric = \"false\", largeop = \"false\", movabl elimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \+ \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mn(\"3 \", mathvariant = \"normal\")), mathvariant = \"normal\")), Typesettin g:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,T ypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6,-I(mfencedGF$6(-F#6$-I#moG F$6-Q*&uminus0;F'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*separat orGF?/%)stretchyGF?/%*symmetricGF?/%(largeopGF?/%.movablelimitsGF?/%'a ccentGF?/%'lspaceGQ,0.2222222emF'/%'rspaceGFN-I#mnGF$6$Q#14F'F:F:/I+ms emanticsGF$Q$absF'/%%openGQ)|F'/%&closeGFZFU-F76-Q\"+F'F:F=F@FB FDFFFHFJFLFO-F#6%-F,6%Q$sinF'/%'italicGF?F:-F76-Q0⁡F'F:F =F@FBFDFFFHFJ/FMQ&0.0emF'/FPFeo-F26$-F#6#-FR6$Q\"1F'F:F:-F76-Q(− F'F:F=F@FBFDFFFHFJFLFO-I&msqrtGF$6#-FR6$Q\"2F'F:Fgn-I%msupGF$6%-F,6%Q \"eF'/F`oQ%trueF'/F;Q'italicF'-F#6%-F,6%Q$cosF'F_oF:Fao-F26$-F#6%F+-F# 6%-FR6$Q$1.6F'F:-F76-Q1⁢F'F:F=F@FBFDFFFHFJFdoFfo-F,6%Q% πF'F_oF:F+F:/%1superscriptshiftGQ\"0F'Fgn-F#6%-F,6%Q'arctanF'F_oF:F ao-F26$-F#6#-FR6$Q\"3F'F:F:F+F+" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 3 "use" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 61 "abs (-14) + sin(1) - sqrt(2) + exp( cos(1.6*Pi) ) + arctan(3);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 141 "This example il lustrates another important point. The corrrect form of a Maple expre ssion can often be found by intelligent guessing. Thus " }{TEXT 209 7 "tan(45)" }{TEXT 208 38 " does indeed compute the tangent, and " } {TEXT 209 3 "20!" }{TEXT 208 115 " computes a factorial. If your firs t guess doesn't work, use one of Maple's help features to get more inf ormation." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 167 "In addition to the functions in the Maple library, ther e are specialized \"packages\" of functions that can be read into the \+ working memory as needed, using the command " }{TEXT 209 4 "with" } {TEXT 208 114 ". For example, to work with matrices and vectors you m ust load the linear algebra package. You do this by typing" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 14 "with (linalg);" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }} {PARA 221 "" 0 "" {TEXT 239 303 "This command produces a list of all t he functions in this package and gives you access to them in your curr ent Maple session. If you close Maple and reopen it later, you must r eload any special packages you want to use. Once you become familiar \+ with a package, you will prefer to load it using a colon" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 14 " with (linalg):" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 176 "rather than a semicolon. Again, this gives yo u access to the linear algebra commands, but without listing all their names. Some packages that might be of interest to you are " }{TEXT 209 5 "plots" }{TEXT 208 2 ", " }{TEXT 209 7 "DEtools" }{TEXT 208 6 ", and " }{TEXT 209 8 "student." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }} {PARA 216 "" 0 "" {TEXT 200 7 "Algebra" }}{PARA 209 "" 0 "" {TEXT 239 80 "Maple knows about variables and algebra. Consider, for example, t he expression " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\" ), Typesetting:-mrow(Typesetting:-msup(Typesetting:-mfenced(Typesettin g:-mrow(Typesetting:-mi(\"a\", italic = \"true\", mathvariant = \"ital ic\"), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"fals e\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mi (\"b\", italic = \"true\", mathvariant = \"italic\")), mathvariant = \+ \"normal\"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), supersc riptshift = \"0\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulename G6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6#-I%msupGF$6%-I(mfen cedGF$6$-F#6%-F,6%Q\"aF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I# moGF$6-Q\"+F'/F@Q'normalF'/%&fenceGQ&falseF'/%*separatorGFJ/%)stretchy GFJ/%*symmetricGFJ/%(largeopGFJ/%.movablelimitsGFJ/%'accentGFJ/%'lspac eGQ,0.2222222emF'/%'rspaceGFY-F,6%Q\"bF'F " 0 "" {MPLTEXT 1 0 10 "(a + b)^2;" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 11 "expand (%);" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 27 "give the expanded form, and" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 11 "factor (%);" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 136 "brings us back to our starting point. To make long computations \+ easier and more intelligible, we can assign values to variables using \+ \"" }{TEXT 209 2 ":=" }{TEXT 208 1 "\"" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 26 "p := (a + b)^2; b \+ := 1; p;" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 193 "In these examples , variables store an expression or a number. Variables can store almo st anything, for example, a list of points, an equation, a set, a piec e of text, or a function definition:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 24 "pts := [ [1,2], [3,4 ] ];" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 21 "eqn := 2*x - 3*y \+ = 5;" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 41 "eqns := \{ 2*x - \+ 3*y = 5, 5*x - 3*y = 1 \};" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 45 "tag := `The nth partial sum is`; #backquotes" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 54 "print (pts, eqn, eqns, tag); #checks \+ our assignments" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 43 "f := x -> x^2; #defines a function" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 12 "f(2); f(3);" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 102 "Anything we can define or compute in Maple can be assigned to a v ariable for future reference using \"" }{TEXT 209 2 ":=" }{TEXT 208 50 "\". Remember that it is different from the symbol " }{TEXT 209 1 "=" }{TEXT 208 80 ", which is used to test equality. Observe that no \+ space is allowed between the " }{TEXT 209 1 ":" }{TEXT 208 9 " and the " }{TEXT 209 1 "=" }{TEXT 208 28 " in an assignment statement." }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 86 "Var iables can be returned to their original symbolic (unassigned) state. \+ The commands" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 9 "b := 'b';" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 2 "p;" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }} {PARA 221 "" 0 "" {TEXT 239 109 "first remove the value 1 assigned to \+ the variable b above and then display the updated value of p. Similar ly" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 40 "unassign ( 'pts', 'eqn', 'eqns', 'tag');" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 28 "print (pts, eqn, eqns, tag);" }}} {EXCHG {PARA 221 "" 0 "" {TEXT 239 93 "clears the variables assigned a bove. A more drastic way of clearing Maple's memory is to say" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" } }{PARA 208 "" 0 "" {TEXT 208 113 "This command clears ALL variables an d unloads all packages. So, if you need one later, you must reload it using " }{TEXT 209 4 "with" }{TEXT 208 1 "." }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 102 "Pay special atte ntion to the kind of quotes used in examples. The possibilities are t he single quote " }{TEXT 209 1 "'" }{TEXT 208 16 ", the backquote " } {TEXT 209 1 "`" }{TEXT 208 23 ", and the double quote " }{TEXT 209 1 " \"" }{TEXT 208 33 ". They all play different roles." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 78 "Here is an extend ed example of how to use variables and assignment statements:" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 52 "F := m*a; # Newton's formula for force" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 55 "m := 2.1; a := 5; \+ # set the mass and acceleration" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 43 "F; # compute the force" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 48 "a := 21.9; # \+ reset the acceleration" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 41 "F; # recompute force" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 33 "a := 'a'; # clear a" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 37 "F; # recompu te F" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 4 "The " }{TEXT 209 4 "subs" }{TEXT 208 118 " command lets u s make temporary substitutions in an expression as opposed to assignin g values. For example, try this:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 35 "g := (a+1)^2 / (b-1)^3 \+ + a / (b-1);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 13 "simplify \+ (g);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 19 "subs( a=3, b=2, g );" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 7 "or this" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 27 "subs ( a = x+y, b = x+1, g);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 18 "simplify(%); a; b;" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 26 "Noti ce that the variables " }{TEXT 209 1 "a" }{TEXT 208 5 " and " }{TEXT 209 1 "b" }{TEXT 208 39 " were not permanently assigned a value." }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 213 "" 0 "" {TEXT 201 8 "Grap hing" }}{PARA 214 "" 0 "" {TEXT 223 20 "Standard Coordinates" }}{PARA 221 "" 0 "" {TEXT 239 130 "Maple can construct many kinds of graphs, a feature that you can use to visualize mathematical objects and proces ses. The command" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 30 "plot ( sin(3*x), x = -Pi..Pi );" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 239 37 " produces a plot containing the curve " }{XPPEDIT 18 0 "Typesetting:-mr ow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"y\", ita lic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", math variant = \"normal\", fence = \"false\", separator = \"false\", stretc hy = \"false\", symmetric = \"false\", largeop = \"false\", movablelim its = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace \+ = \"0.2777778em\"), Typesetting:-mrow(Typesetting:-mi(\"sin\", italic \+ = \"false\", mathvariant = \"normal\"), Typesetting:-mo(\"&ApplyFuncti on;\", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rs pace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting: -mi(\"\"), Typesetting:-mrow(Typesetting:-mn(\"3\", mathvariant = \"no rmal\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesettin g:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesettin g:-mi(\"\")), mathvariant = \"normal\")), Typesetting:-mi(\"\")), Type setting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sysl ibGF'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"yF'/%'italicGQ%trueF'/%,mathvariantG Q'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorG FB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'acce ntGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6%-F,6%Q$sinF'/F5FBF>-F;6 -Q0⁡F'F>F@FCFEFGFIFKFM/FPQ&0.0emF'/FSFhn-I(mfencedGF$6$- F#6%F+-F#6%-I#mnGF$6$Q\"3F'F>-F;6-Q1⁢F'F>F@FCFEFGFIFKFM FgnFin-F,6%Q\"xF'F4F7F+F>F+F+" }{TEXT 239 5 " for " }{XPPEDIT 18 0 "Ty pesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant \+ = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6#-I#miGF$6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 239 22 " in the interval from " }{XPPEDIT 18 0 "Typesetting:-mrow(Typ esetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mo(\"&uminus0;\", m athvariant = \"normal\", fence = \"false\", separator = \"false\", str etchy = \"false\", symmetric = \"false\", largeop = \"false\", movable limits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspa ce = \"0.2222222em\"), Typesetting:-mi(\"\1700\", italic = \"false\", \+ mathvariant = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+mod ulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6$-I#moGF$6-Q* &uminus0;F'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGF:/ %)stretchyGF:/%*symmetricGF:/%(largeopGF:/%.movablelimitsGF:/%'accentG F:/%'lspaceGQ,0.2222222emF'/%'rspaceGFI-F,6%Q%πF'/%'italicGF:F5F+" }{TEXT 239 4 " to " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi( \"\1700\", italic = \"false\", mathvariant = \"normal\"));" "-I%mrowG6 #/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q%πF'/%'ita licGQ&falseF'/%,mathvariantGQ'normalF'" }{TEXT 239 154 ". No space is allowed between the dots in a plot command. Notice that the scale on the x-axis is not the same as on the y-axis. To fix this you can say " }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 53 "plot( sin(3*x), x = -Pi..Pi, scaling = constrained ); " }}}{EXCHG {PARA 209 "" 0 "" {TEXT 239 56 "Sometimes it is useful to \+ restrict the range over which " }{XPPEDIT 18 0 "Typesetting:-mrow(Type setting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"));" "-I %mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"yF'/ %'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 239 40 " varies. W e get a misleading graph from" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 26 "plot( tan(x), x = -5..5 ); " }}}{EXCHG {PARA 209 "" 0 "" {TEXT 239 60 "It does not accurately rep resent the vertical asymptotes of " }{XPPEDIT 18 0 "Typesetting:-mrow( Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \+ \"0.2777778em\"), Typesetting:-mrow(Typesetting:-mi(\"tan\", italic = \+ \"false\", mathvariant = \"normal\"), Typesetting:-mo(\"&ApplyFunction ;\", mathvariant = \"normal\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rsp ace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:- mi(\"x\", italic = \"true\", mathvariant = \"italic\")), mathvariant = \"normal\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mr owG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&- F,6%Q\"yF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"=F'/ F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretchyGFB/%*symmetri cGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lspaceGQ,0.2777778e mF'/%'rspaceGFQ-F#6%-F,6%Q$tanF'/F5FBF>-F;6-Q0⁡F'F>F@FCF EFGFIFKFM/FPQ&0.0emF'/FSFhn-I(mfencedGF$6$-F#6#-F,6%Q\"xF'F4F7F>F+F+" }{TEXT 239 35 ". Better results are obtained with" }}{PARA 219 "" 0 " " {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 37 "plot( \+ tan(x), x = -5..5, y = -5..5 );" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 239 45 "We can plot several curves at once. To plot " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesettin g:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting :-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.277 7778em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Typesetting:-mi (\"sin\", italic = \"false\", mathvariant = \"normal\"), Typesetting:- mo(\"⁡\", mathvariant = \"normal\", fence = \"false\", s eparator = \"false\", stretchy = \"false\", symmetric = \"false\", lar geop = \"false\", movablelimits = \"false\", accent = \"false\", lspac e = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:- mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic \")), mathvariant = \"normal\")), Typesetting:-mi(\"\")), Typesetting: -mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%- I#miGF$6#Q!F'-F#6&-F,6%Q\"yF'/%'italicGQ%trueF'/%,mathvariantGQ'italic F'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)str etchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%' lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6%-F,6%Q$sinF'/F5FBF>-F;6-Q0&Appl yFunction;F'F>F@FCFEFGFIFKFM/FPQ&0.0emF'/FSFhn-I(mfencedGF$6$-F#6#-F,6 %Q\"xF'F4F7F>F+F+" }{TEXT 239 5 " and " }{XPPEDIT 18 0 "Typesetting:-m row(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"y\", it alic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mat hvariant = \"normal\", fence = \"false\", separator = \"false\", stret chy = \"false\", symmetric = \"false\", largeop = \"false\", movableli mits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Typesetting:-mi(\"sin\", italic = \"false\", mathvariant = \"normal\"), Typesetting:-mo(\"&ApplyFunct ion;\", mathvariant = \"normal\", fence = \"false\", separator = \"fal se\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\" , movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", r space = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting :-mi(\"\"), Typesetting:-mrow(Typesetting:-mn(\"3\", mathvariant = \"n ormal\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesettin g:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesettin g:-mi(\"\")), mathvariant = \"normal\")), Typesetting:-mi(\"\")), Type setting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sysl ibGF'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"yF'/%'italicGQ%trueF'/%,mathvariantG Q'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorG FB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'acce ntGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6%-F,6%Q$sinF'/F5FBF>-F;6 -Q0⁡F'F>F@FCFEFGFIFKFM/FPQ&0.0emF'/FSFhn-I(mfencedGF$6$- F#6%F+-F#6%-I#mnGF$6$Q\"3F'F>-F;6-Q1⁢F'F>F@FCFEFGFIFKFM FgnFin-F,6%Q\"xF'F4F7F+F>F+F+" }{TEXT 239 16 " together we say" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 42 "plot( \{ sin(x), sin(3*x) \+ \}, x = -Pi..Pi );" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 26 "Now the \+ first argument of " }{TEXT 209 4 "plot" }{TEXT 208 120 " is a set of e xpressions to be graphed. Sets are enclosed in curly braces and indiv idual items are separated by commas." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 22 "We could also use the " }{TEXT 209 7 "display" }{TEXT 208 22 " command found in the " }{TEXT 209 5 "p lots" }{TEXT 208 9 " package:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 12 "with(plots);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 31 "plot1 := plot( x^2, x = 0..4 ):" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 37 "plot2 := plot( 3*x^2 - \+ 2, x = 0..4 ):" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 28 "display ( \{ plot1, plot2 \} );" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 242 "" 0 "" {TEXT 224 20 "Par ametric Equations" }}{PARA 208 "" 0 "" {TEXT 208 85 "A curve in the pl ane can be described as the graph of a function, as in the graph of " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:- mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic \"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mfra c(Typesetting:-mrow(Typesetting:-mn(\"1\", mathvariant = \"normal\"), \+ Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fence \+ = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-msqrt(T ypesetting:-mrow(Typesetting:-mn(\"4\", mathvariant = \"normal\"), Typ esetting:-mo(\"−\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Ty pesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathvariant \+ = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), super scriptshift = \"0\")), Typesetting:-mi(\"\")))), Typesetting:-mrow(Typ esetting:-mn(\"2\", mathvariant = \"normal\")), linethickness = \"1\", denomalign = \"center\", numalign = \"center\", bevelled = \"false\") ), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingG I(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-F,6%Q\"yF'/%'italicGQ%trueF'/%,mathv ariantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*sep aratorGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB /%'accentGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-I&mfracGF$6(-F#6%-I# mnGF$6$Q\"1F'F>-F;6-Q1⁢F'F>F@FCFEFGFIFKFM/FPQ&0.0emF'/F SF[o-I&msqrtGF$6#-F#6&-FZ6$Q\"4F'F>-F;6-Q(−F'F>F@FCFEFGFIFKFM/FP Q,0.2222222emF'/FSFio-F#6#-I%msupGF$6%-F,6%Q\"xF'F4F7-FZ6$Q\"2F'F>/%1s uperscriptshiftGQ\"0F'F+-F#6#Fcp/%.linethicknessGQ\"1F'/%+denomalignGQ 'centerF'/%)numalignGF`q/%)bevelledGFBF+" }{TEXT 208 5 " for " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"tru e\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6#-I#miGF$6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ 'italicF'" }{TEXT 208 71 " in the interval from -1 to 1, or it can be \+ given parametrically as in " }{XPPEDIT 18 0 "Typesetting:-mrow(Typeset ting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mr ow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"x\", ita lic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&ApplyFun ction;\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetti ng:-mi(\"t\", italic = \"true\", mathvariant = \"italic\")), mathvaria nt = \"normal\")), Typesetting:-mo(\",\", mathvariant = \"normal\", fe nce = \"false\", separator = \"true\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.3333333em\"), Typesetting :-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"itali c\"), Typesetting:-mo(\"⁡\", mathvariant = \"normal\", f ence = \"false\", separator = \"false\", stretchy = \"false\", symmetr ic = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mf enced(Typesetting:-mrow(Typesetting:-mi(\"t\", italic = \"true\", math variant = \"italic\")), mathvariant = \"normal\")), Typesetting:-mi(\" \")), mathvariant = \"normal\"), Typesetting:-mo(\"=\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777 778em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"\") , Typesetting:-mrow(Typesetting:-mn(\"2\", mathvariant = \"normal\"), \+ Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fence \+ = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Ty pesetting:-mi(\"cos\", italic = \"false\", mathvariant = \"normal\"), \+ Typesetting:-mo(\"⁡\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced (Typesetting:-mrow(Typesetting:-mi(\"t\", italic = \"true\", mathvaria nt = \"italic\")), mathvariant = \"normal\")), Typesetting:-mi(\"\")), Typesetting:-mo(\",\", mathvariant = \"normal\", fence = \"false\", s eparator = \"true\", stretchy = \"false\", symmetric = \"false\", larg eop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.3333333em\"), Typesetting:-mrow(Typesetting :-mi(\"sin\", italic = \"false\", mathvariant = \"normal\"), Typesetti ng:-mo(\"⁡\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesett ing:-mrow(Typesetting:-mi(\"t\", italic = \"true\", mathvariant = \"it alic\")), mathvariant = \"normal\")), Typesetting:-mi(\"\")), mathvari ant = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG 6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-I(mfencedGF$6$-F#6' F+-F#6%-F,6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6 -Q0⁡F'/F?Q'normalF'/%&fenceGQ&falseF'/%*separatorGFI/%)s tretchyGFI/%*symmetricGFI/%(largeopGFI/%.movablelimitsGFI/%'accentGFI/ %'lspaceGQ&0.0emF'/%'rspaceGFX-F26$-F#6#-F,6%Q\"tF'F;F>FE-FB6-Q\",F'FE FG/FKF=FLFNFPFRFTFV/FZQ,0.3333333emF'-F#6%-F,6%Q\"yF'F;F>FAFenF+FE-FB6 -Q\"=F'FEFGFJFLFNFPFRFT/FWQ,0.2777778emF'/FZF[p-F26$-F#6'F+-F#6&-I#mnG F$6$Q\"2F'FE-FB6-Q1⁢F'FEFGFJFLFNFPFRFTFVFY-F#6%-F,6%Q$c osF'/F " 0 "" {MPLTEXT 1 0 38 "plot( [2*cos(t), sin(t), t = 0..Pi] );" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 147 "The resulting graph is somewhat distorted, because Maple did not use the same verti cal and horizontal scales. There are two ways to fix this. Use" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 52 "plot( [2*cos(t), sin(t), t = 0..Pi], -2..2, -2..2 );" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 18 "or use the option " }{TEXT 209 21 "scaling = constrained" }{TEXT 208 1 "." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 243 "" 0 "" {TEXT 225 17 "Polar Coordinates" }} {PARA 209 "" 0 "" {TEXT 239 75 "Polar plots are a special kind of para metric plot. The polar coordinates (" }{XPPEDIT 18 0 "Typesetting:-mr ow(Typesetting:-mi(\"r\", italic = \"true\", mathvariant = \"italic\") , Typesetting:-mo(\",\", mathvariant = \"normal\", fence = \"false\", \+ separator = \"true\", stretchy = \"false\", symmetric = \"false\", lar geop = \"false\", movablelimits = \"false\", accent = \"false\", lspac e = \"0.0em\", rspace = \"0.3333333em\"), Typesetting:-mi(\"\1670\", i talic = \"false\", mathvariant = \"normal\"));" "-I%mrowG6#/I+modulena meG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"rF'/%'italicGQ%trueF'/% ,mathvariantGQ'italicF'-I#moGF$6-Q\",F'/F3Q'normalF'/%&fenceGQ&falseF' /%*separatorGF1/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelim itsGF=/%'accentGF=/%'lspaceGQ&0.0emF'/%'rspaceGQ,0.3333333emF'-F,6%Q(& theta;F'/F0F=F9" }{TEXT 239 68 ") of points on a curve can be given as a function of some parameter " }{XPPEDIT 18 0 "Typesetting:-mrow(Type setting:-mi(\"t\", italic = \"true\", mathvariant = \"italic\"));" "-I %mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"tF'/ %'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 239 49 ". In many \+ cases the parameter is just the angle " }{XPPEDIT 18 0 "Typesetting:-m row(Typesetting:-mi(\"\1670\", italic = \"false\", mathvariant = \"nor mal\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#m iGF$6%Q(θF'/%'italicGQ&falseF'/%,mathvariantGQ'normalF'" }{TEXT 239 59 ". Consider the ellipse defined in standard coordinates by " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Ty pesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"tr ue\", mathvariant = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \+ \"normal\"), superscriptshift = \"0\")), Typesetting:-mo(\"+\", mathva riant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \+ \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mn(\"4\", mathvariant = \"normal\"), Typesetting:-mo(\"⁢\", mathvariant = \" normal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Type setting:-mrow(Typesetting:-msup(Typesetting:-mi(\"y\", italic = \"true \", mathvariant = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \" normal\"), superscriptshift = \"0\")), Typesetting:-mi(\"\")), Typeset ting:-mi(\"\")), Typesetting:-mo(\"=\", mathvariant = \"normal\", fenc e = \"false\", separator = \"false\", stretchy = \"false\", symmetric \+ = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typese tting:-mn(\"4\", mathvariant = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F' -F#6&F+-F#6'F+-F#6#-I%msupGF$6%-F,6%Q\"xF'/%'italicGQ%trueF'/%,mathvar iantGQ'italicF'-I#mnGF$6$Q\"2F'/F?Q'normalF'/%1superscriptshiftGQ\"0F' -I#moGF$6-Q\"+F'FE/%&fenceGQ&falseF'/%*separatorGFP/%)stretchyGFP/%*sy mmetricGFP/%(largeopGFP/%.movablelimitsGFP/%'accentGFP/%'lspaceGQ,0.22 22222emF'/%'rspaceGFin-F#6&-FB6$Q\"4F'FE-FK6-Q1⁢F'FEFNF QFSFUFWFYFen/FhnQ&0.0emF'/F[oFeo-F#6#-F66%-F,6%Q\"yF'F;F>FAFGF+F+-FK6- Q\"=F'FEFNFQFSFUFWFYFen/FhnQ,0.2777778emF'/F[oFbpF^oF+" }{TEXT 239 54 ". To find an equation relating the polar coordinates " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"r\", italic = \"true\", mathvar iant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sys libGF'6#-I#miGF$6%Q\"rF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" } {TEXT 239 5 " and " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi( \"\1670\", italic = \"false\", mathvariant = \"normal\"));" "-I%mrowG6 #/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q(θF'/%' italicGQ&falseF'/%,mathvariantGQ'normalF'" }{TEXT 239 62 " of a typica l point on this ellipse, we make the substitution " }{XPPEDIT 18 0 "Ty pesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:- mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-m o(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"fa lse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.277777 8em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Typesetting:-mi(\" r\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"& InvisibleTimes;\", mathvariant = \"normal\", fence = \"false\", separa tor = \"false\", stretchy = \"false\", symmetric = \"false\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:-mi(\"co s\", italic = \"false\", mathvariant = \"normal\"), Typesetting:-mo(\" ⁡\", mathvariant = \"normal\", fence = \"false\", separa tor = \"false\", stretchy = \"false\", symmetric = \"false\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow (Typesetting:-mi(\"t\", italic = \"true\", mathvariant = \"italic\")), mathvariant = \"normal\")), Typesetting:-mi(\"\")), Typesetting:-mi( \"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typeset tingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"xF'/%'italicGQ%trueF'/%, mathvariantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/ %*separatorGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimi tsGFB/%'accentGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6&-F,6%Q\"rF' F4F7-F;6-Q1⁢F'F>F@FCFEFGFIFKFM/FPQ&0.0emF'/FSFgn-F#6%-F ,6%Q$cosF'/F5FBF>-F;6-Q0⁡F'F>F@FCFEFGFIFKFMFfnFhn-I(mfen cedGF$6$-F#6#-F,6%Q\"tF'F4F7F>F+F+F+" }{TEXT 239 5 " and " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Types etting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Typese tting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separato r = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .2777778em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Typesetting :-mi(\"r\", italic = \"true\", mathvariant = \"italic\"), Typesetting: -mo(\"⁢\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:-m i(\"sin\", italic = \"false\", mathvariant = \"normal\"), Typesetting: -mo(\"⁡\", mathvariant = \"normal\", fence = \"false\", \+ separator = \"false\", stretchy = \"false\", symmetric = \"false\", la rgeop = \"false\", movablelimits = \"false\", accent = \"false\", lspa ce = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting: -mrow(Typesetting:-mi(\"t\", italic = \"true\", mathvariant = \"italic \")), mathvariant = \"normal\")), Typesetting:-mi(\"\")), Typesetting: -mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typ esettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"yF'/%'italicGQ%trueF '/%,mathvariantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&fals eF'/%*separatorGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movable limitsGFB/%'accentGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6&-F,6%Q \"rF'F4F7-F;6-Q1⁢F'F>F@FCFEFGFIFKFM/FPQ&0.0emF'/FSFgn-F #6%-F,6%Q$sinF'/F5FBF>-F;6-Q0⁡F'F>F@FCFEFGFIFKFMFfnFhn-I (mfencedGF$6$-F#6#-F,6%Q\"tF'F4F7F>F+F+F+" }{TEXT 239 8 ", where " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"t\", italic = \"true\", mathvariant = \"italic\" ), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mi(\"\1 670\", italic = \"false\", mathvariant = \"normal\")), Typesetting:-mi (\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#m iGF$6#Q!F'-F#6%-F,6%Q\"tF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'- I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretc hyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lsp aceGQ,0.2777778emF'/%'rspaceGFQ-F,6%Q(θF'/F5FBF>F+" }{TEXT 239 1 "." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 51 "subs( x = r*cos(t), y = r*sin(t), x^2 + 4*y^2 = 4);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 14 "simplify( % );" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 14 "solve( %, r );" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 239 79 "We find that the ellipse is the collection of points who se polar coordinates ( " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting :-mi(\"r\", italic = \"true\", mathvariant = \"italic\"), Typesetting: -mo(\",\", mathvariant = \"normal\", fence = \"false\", separator = \" true\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.3333333em\"), Typesetting:-mi(\"\1670\", italic = \"fals e\", mathvariant = \"normal\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6%-I#miGF$6%Q\"rF'/%'italicGQ%trueF'/%,mathvariantGQ 'italicF'-I#moGF$6-Q\",F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF 1/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accen tGF=/%'lspaceGQ&0.0emF'/%'rspaceGQ,0.3333333emF'-F,6%Q(θF'/F0F=F 9" }{TEXT 239 10 ") satisfy " }{XPPEDIT 18 0 "Typesetting:-mrow(Typese tting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting: -mrow(Typesetting:-msup(Typesetting:-mi(\"r\", italic = \"true\", math variant = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \"normal\" ), superscriptshift = \"0\")), Typesetting:-mo(\"=\", mathvariant = \" normal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.277777 8em\"), Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mn(\"4\", ma thvariant = \"normal\")), Typesetting:-mrow(Typesetting:-mi(\"\"), Typ esetting:-mrow(Typesetting:-mn(\"4\", mathvariant = \"normal\"), Types etting:-mo(\"−\", mathvariant = \"normal\", fence = \"false\", s eparator = \"false\", stretchy = \"false\", symmetric = \"false\", lar geop = \"false\", movablelimits = \"false\", accent = \"false\", lspac e = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Type setting:-mn(\"3\", mathvariant = \"normal\"), Typesetting:-mo(\"&Invis ibleTimes;\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"f alse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0e m\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:-msup(Typesett ing:-mrow(Typesetting:-mi(\"cos\", italic = \"false\", mathvariant = \+ \"normal\"), Typesetting:-mo(\"⁡\", mathvariant = \"norm al\", fence = \"false\", separator = \"false\", stretchy = \"false\", \+ symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesett ing:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"\1670\", italic = \"f alse\", mathvariant = \"normal\")), mathvariant = \"normal\")), Typese tting:-mn(\"2\", mathvariant = \"normal\"), superscriptshift = \"0\")) , Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\" )), linethickness = \"1\", denomalign = \"center\", numalign = \"cente r\", bevelled = \"false\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+mo dulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&F+-F#6#-I%m supGF$6%-F,6%Q\"rF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#mnGF$ 6$Q\"2F'/F=Q'normalF'/%1superscriptshiftGQ\"0F'-I#moGF$6-Q\"=F'FC/%&fe nceGQ&falseF'/%*separatorGFN/%)stretchyGFN/%*symmetricGFN/%(largeopGFN /%.movablelimitsGFN/%'accentGFN/%'lspaceGQ,0.2777778emF'/%'rspaceGFgn- I&mfracGF$6(-F#6#-F@6$Q\"4F'FC-F#6%F+-F#6&F_o-FI6-Q(−F'FCFLFOFQF SFUFWFY/FfnQ,0.2222222emF'/FinFjo-F#6&-F@6$Q\"3F'FC-FI6-Q1&InvisibleTi mes;F'FCFLFOFQFSFUFWFY/FfnQ&0.0emF'/FinFep-F#6#-F46%-F#6%-F,6%Q$cosF'/ F:FNFC-FI6-Q0⁡F'FCFLFOFQFSFUFWFYFdpFfp-I(mfencedGF$6$-F# 6#-F,6%Q(θF'F`qFCFCF?FEF+F+F+/%.linethicknessGQ\"1F'/%+denomalig nGQ'centerF'/%)numalignGFar/%)bevelledGFNF+" }{TEXT 239 64 ", and the \+ following commands draw the right half of the ellipse:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 30 "r \+ := 2/sqrt( 4 - 3*cos(t)^2 ):" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 50 "plot( [ r, t, t = -Pi/2..Pi/2 ], coords = polar );" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 63 "Here are some more of examples of pola r plots that you can try:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 43 "plot( [ 1, t, t = 0..2*Pi] , coords=polar ):" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 42 "plot ( [t, t, t = 0..2*Pi], coords=polar ):" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 50 "plot( [ sin(4*t), t, t = 0..2*Pi], coords=polar ):" } }}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 52 "As usual, a more realistic picture is obtained with " }{TEXT 209 21 "scaling = constrained" }{TEXT 208 1 "." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 241 "" 0 "" {TEXT 219 13 "Plotting Data" }} {PARA 209 "" 0 "" {TEXT 239 43 "Maple can plot data consisting of pair s of " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\" I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"xF'/%'italicGQ%trueF'/%,mathv ariantGQ'italicF'" }{TEXT 239 5 " and " }{XPPEDIT 18 0 "Typesetting:-m row(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\" ));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6 %Q\"yF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 239 32 " va lues. For example, if we say" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 55 "data := [ [0, 0.53], [1, 1 .14], [2, 1.84], [3, 4.12] ];" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 5 "then " }{TEXT 209 4 "data" }{TEXT 208 39 " refers to a sequence of \+ five points, (" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"x \", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\", \", mathvariant = \"normal\", fence = \"false\", separator = \"true\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mov ablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.3333333em\"), Typesetting:-mi(\"y\", italic = \"true\", mathvar iant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sys libGF'6%-I#miGF$6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I# moGF$6-Q\",F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF1/%)stretchy GF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%'lspac eGQ&0.0emF'/%'rspaceGQ,0.3333333emF'-F,6%Q\"yF'F/F2" }{TEXT 208 226 ") = (0, 0.53), etc. The result is a list: something enclosed in squar e brackets, with elements separated by commas. Lists are used for col lections of objects where the order matters. Individual items are acc essed this way:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 219 "" 0 " " {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 35 "data[1 ]; data[2]; data[3]; data[4];" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 170 "In this case, the list items are themselves lists - very short on es made up of the coordinates of a point. To access the second coordi nate of the third data point we say" }}{PARA 219 "" 0 "" {TEXT 249 0 " " }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 11 "data[3][2];" }}} {EXCHG {PARA 221 "" 0 "" {TEXT 239 51 "To plot the points in our list \+ we use commands like" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 59 "plot( data, style = point, symbol = circle, color = black);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 48 "plot( data, style = line, view = [0..4, 0..5] );" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 18 "You could also try" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 51 "plot( data, style = line, title = `Expe riment 1` ):" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 " " 0 "" {TEXT 208 19 "Here the backquote " }{TEXT 209 1 "`" }{TEXT 208 39 " is used to specify a plot title. See " }{TEXT 209 15 "?plot[opti ons] " }{TEXT 208 89 "for more information, e.g., about symbols and li ne styles available. You could also try " }{TEXT 209 9 "?readdata" } {TEXT 208 5 " and " }{TEXT 209 6 "?stats" }{TEXT 208 68 " to find out \+ how to read a file of data points into a Maple session." }}{PARA 219 " " 0 "" {TEXT 249 0 "" }}{PARA 240 "" 0 "" {TEXT 204 17 "Solving Equati ons" }}{PARA 221 "" 0 "" {TEXT 239 56 "Maple can also solve equations. Consider some examples:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 25 "solve( x^2 + 3*x = 2.1 );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 64 "solve( x^3 + x = 27 ): #This gives a complicated answer!" }{MPLTEXT 1 0 1 "\n" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 24 "solve( x^3 + x = 27.0 );" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 153 "The second command gives an exact, but complicated, answer. \+ Replacing 27 by 27.0 forces Maple to give decimal approximations inste ad, as do the commands" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 23 "fsolve( x^3 + x = 27 );" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 29 "fsolve( x^3 + x, x, comple x);" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 12 "In general, " }{TEXT 209 5 "solve" }{TEXT 208 58 " looks for exact answers using algebraic methods, whereas " }{TEXT 209 6 "fs olve" }{TEXT 208 87 " uses numberical methods to find approximate solu tions in floating-point form. Compare" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 24 "solve( tan(x) - x \+ = 2 );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 25 "fsolve( tan(x) \+ - x = 2 );" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 91 "Notice how Maple responds if it cannot find the sol ution you asked for. Also, notice that " }{TEXT 209 6 "fsolve" }{TEXT 208 85 " may not find all solutions. To understand why not, it is he lpful to look at a graph" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" } }}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 50 "plot( \{ tan(x) - x, 2 \+ \}, x = 0..10, y = -10..10 );" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 110 "This will give you an idea of how many solutions there are and wh at their approxmiate location is. Then give " }{TEXT 209 6 "fsolve" } {TEXT 208 12 " a range of " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesett ing:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"));" "-I%mro wG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"xF'/%'it alicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 208 27 "-values in whic h to search:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 " > " 0 "" {MPLTEXT 1 0 35 "fsolve( tan(x) - x = 2, x = 4..5 );" }}} {EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 124 "Often we need to use the solution of an equation in a later probl em. To do this, assign a name to it. Here is one example." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 33 "r := solve( x^2 + 3*x -2.1 = 0 );" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 209 "" 0 "" {TEXT 239 36 "Notice that the answe r has the form " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"r 1\", italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+mod ulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q#r1F'/%'italicGQ%tru eF'/%,mathvariantGQ'italicF'" }{TEXT 239 2 ", " }{XPPEDIT 18 0 "Typese tting:-mrow(Typesetting:-mi(\"r2\", italic = \"true\", mathvariant = \+ \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6 #-I#miGF$6%Q#r2F'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 239 8 ", where " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"r 1\", italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+mod ulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q#r1F'/%'italicGQ%tru eF'/%,mathvariantGQ'italicF'" }{TEXT 239 23 " is the first root and " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"r2\", italic = \"t rue\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Type settingGI(_syslibGF'6#-I#miGF$6%Q#r2F'/%'italicGQ%trueF'/%,mathvariant GQ'italicF'" }{TEXT 239 162 " is the second. Such an object - a bunch of items separated by commas, is called an expression sequence. One \+ picks out items of an expression sequence this way:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 5 "r[1];" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 5 "r[2];" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 66 "Here are some computations with items from a n expression sequence:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 46 "r[1] + r[2]; # sum of the roots" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 43 "r[1]*r[2 ]; # their product" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 52 "subs( x = r[1], 2*x + 3 ); # find 2(first soln) + 3" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 39 "We can also solve systems o f equations:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 " > " 0 "" {MPLTEXT 1 0 42 "solve( \{ 2*x + 3*y = 1, 5*x + 7*y = 2 \} ); " }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 5 "x; y;" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 36 "A system of equations is given as a " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"set\", italic = \" true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typ esettingGI(_syslibGF'6#-I#miGF$6%Q$setF'/%'italicGQ%trueF'/%,mathvaria ntGQ'italicF'" }{TEXT 208 156 " - a bunch of items enclosed in curly b rackets and separated by commas. Sets are often used when the order o f the objects is unimportant. In reply to the " }{TEXT 209 6 "solve " }{TEXT 208 44 "command above, Maple tells us how to choose " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"tru e\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6#-I#miGF$6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ 'italicF'" }{TEXT 208 5 " and " }{XPPEDIT 18 0 "Typesetting:-mrow(Type setting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"));" "-I %mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"yF'/ %'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 208 43 " to solve t he system, but it does not give " }{XPPEDIT 18 0 "Typesetting:-mrow(Ty pesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"));" " -I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"xF '/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 208 5 " and " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \"tru e\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6#-I#miGF$6%Q\"yF'/%'italicGQ%trueF'/%,mathvariantGQ 'italicF'" }{TEXT 208 74 " these particular values. To force it to as sign these values, we use the " }{TEXT 209 6 "assign" }{TEXT 208 10 " \+ function:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 47 "s := solve( \{ 2*x + 3*y = 1, 5*x + 7*y = 2 \} \+ );" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 12 "assign( s );" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "x; y; #check that it worked" }}} {EXCHG {PARA 209 "" 0 "" {TEXT 202 7 "A note:" }{TEXT 239 63 " To typ e multiple line commands, as above, use Shift-Return. " }{TEXT 202 14 "And a warning:" }{TEXT 239 85 " You may have trouble later if you leave numerical values assigned to the variables " }{XPPEDIT 18 0 "Ty pesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant \+ = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6#-I#miGF$6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 239 2 ", " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"y\", i talic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenam eG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"yF'/%'italicGQ%trueF'/%, mathvariantGQ'italicF'" }{TEXT 239 6 ", and " }{XPPEDIT 18 0 "Typesett ing:-mrow(Typesetting:-mi(\"r\", italic = \"true\", mathvariant = \"it alic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I# miGF$6%Q\"rF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 239 100 ". Maple will not forget these assigned values, even though you h ave gone on to a new problem where " }{XPPEDIT 18 0 "Typesetting:-mrow (Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")); " "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q \"xF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 239 146 " mea ns something different. It is a good idea to return variables to thei r unassigned state when you finish your problem. Recall how to do thi s:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 29 "x := 'x'; y := 'y'; r := 'r';" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 12 "Recall that " }{TEXT 209 7 "restart" }{TEXT 208 27 " also clears all variables." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }} {PARA 208 "" 0 "" {TEXT 208 54 "Finally, note that symbolic parameters are allowed in " }{TEXT 209 5 "solve" }{TEXT 208 129 " commands. How ever, in that case we have to tell Maple which ones to solve for and w hich ones to treat as unspecified constants:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 50 "solve( a *x^2 + b*x + c, x ); # solving for x" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 34 "solve( a*x^3 + b*x^2 + c*x +d, x):" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 86 "solve( \{ a*x \+ + b*y = h, c*x + d*y = k \}, \{ x,y \} ): # solving a system for x and y" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 238 "" 0 "" {TEXT 206 9 "Functions" }}{PARA 221 "" 0 "" {TEXT 239 115 "Although M aple has a large library of standard functions, we often need to defin e new ones. For example, to define" }}{PARA 237 "" 0 "" {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetti ng:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"p\", italic = \"true \", mathvariant = \"italic\"), Typesetting:-mo(\"⁡\", ma thvariant = \"normal\", fence = \"false\", separator = \"false\", stre tchy = \"false\", symmetric = \"false\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \" 0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\" , italic = \"true\", mathvariant = \"italic\")), mathvariant = \"norma l\")), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"fals e\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mr ow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mn(\"18\", ma thvariant = \"normal\"), Typesetting:-mo(\"⁢\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em \"), Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mn(\"4\", mathvar iant = \"normal\"), superscriptshift = \"0\")), Typesetting:-mi(\"\")) , Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", \+ separator = \"false\", stretchy = \"false\", symmetric = \"false\", la rgeop = \"false\", movablelimits = \"false\", accent = \"false\", lspa ce = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typ esetting:-mn(\"69\", mathvariant = \"normal\"), Typesetting:-mo(\"&Inv isibleTimes;\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:-msup(Types etting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typese tting:-mn(\"3\", mathvariant = \"normal\"), superscriptshift = \"0\")) , Typesetting:-mi(\"\")), Typesetting:-mo(\"−\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222 222em\"), Typesetting:-mrow(Typesetting:-mn(\"40\", mathvariant = \"no rmal\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesettin g:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", ma thvariant = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \"normal \"), superscriptshift = \"0\")), Typesetting:-mi(\"\")), Typesetting:- mo(\"−\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesetting :-mn(\"124\", mathvariant = \"normal\"), Typesetting:-mo(\"&InvisibleT imes;\", mathvariant = \"normal\", fence = \"false\", separator = \"fa lse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathva riant = \"italic\")), Typesetting:-mo(\"−\", mathvariant = \"nor mal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\" , accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em \"), Typesetting:-mn(\"48\", mathvariant = \"normal\")), Typesetting:- mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Type settingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6'F+-F#6%-F,6%Q\"pF'/%'italicGQ %trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q0⁡F'/F:Q'norm alF'/%&fenceGQ&falseF'/%*separatorGFD/%)stretchyGFD/%*symmetricGFD/%(l argeopGFD/%.movablelimitsGFD/%'accentGFD/%'lspaceGQ&0.0emF'/%'rspaceGF S-I(mfencedGF$6$-F#6#-F,6%Q\"xF'F6F9F@-F=6-Q\"=F'F@FBFEFGFIFKFMFO/FRQ, 0.2777778emF'/FUF\\o-F#6,F+-F#6&-I#mnGF$6$Q#18F'F@-F=6-Q1&InvisibleTim es;F'F@FBFEFGFIFKFMFOFQFT-F#6#-I%msupGF$6%Fen-Fco6$Q\"4F'F@/%1superscr iptshiftGQ\"0F'F+-F=6-Q\"+F'F@FBFEFGFIFKFMFO/FRQ,0.2222222emF'/FUFhp-F #6&-Fco6$Q#69F'F@Ffo-F#6#-F\\p6%Fen-Fco6$Q\"3F'F@FapF+-F=6-Q(−F' F@FBFEFGFIFKFMFOFgpFip-F#6&-Fco6$Q#40F'F@Ffo-F#6#-F\\p6%Fen-Fco6$Q\"2F 'F@FapF+Ffq-F#6%-Fco6$Q$124F'F@FfoFenFfq-Fco6$Q#48F'F@F+F+" }}{PARA 221 "" 0 "" {TEXT 239 6 "we say" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }} }{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 48 "p := x -> 18*x^4 + 69*x^3 - 40*x^2 - 124*x - 48;" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }} {PARA 208 "" 0 "" {TEXT 208 20 "Think of the symbol " }{TEXT 209 2 "-> " }{TEXT 208 50 " as an arrow: it tells what to do with the input " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"tru e\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6#-I#miGF$6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ 'italicF'" }{TEXT 208 29 ", namely, produce the output " }{TEXT 209 36 "18*x^4 + 69*x^3 - 40*x^2 -124*x - 48" }{TEXT 208 21 ". Once the f unction " }{TEXT 209 1 "p" }{TEXT 208 60 " is defined, we can do the u sual computations with it, e.g.," }}{PARA 219 "" 0 "" {TEXT 249 0 "" } }}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 6 "p(-2);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 9 "p( 1/2 );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 9 "p( a+b );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }} {PARA 221 "" 0 "" {TEXT 239 192 "It is important to keep in mind that \+ functions and expressions are different kinds of mathematical objects. Mathematicians know this, and so does Maple. Compare the results of the following:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 16 "p; # function" }}}{EXCHG {PARA 207 " > " 0 "" {MPLTEXT 1 0 18 "p(x); # expression" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 18 "p(y); # expression" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 18 "p(3); # expression" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 36 "As further proof, try the following:" }}{PARA 219 "" 0 " " {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 10 "factor (p);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 15 "factor( p(x) );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 17 "plot( p, -2..2 );" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 17 "plot( p, -2..2 );" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 24 "plot( p(x), x = -2..2 );" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 94 "Functions of several variabl es can be defined as easily as can functions of a single variable:" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 31 "f := (x,y) -> exp(-x) * sin(y);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 7 "f(1,2);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 39 "g := (x,y) -> alpha*exp(-k*x)*sin(w*y);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 7 "g(1,2);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 27 "alpha := 2; k := 3; g(1,2);" }{MPLTEXT 1 0 1 "\n " }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 17 "w := 3.5; g(1,2);" }} }{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 25 "alpha := 'alpha'; g(1,2); " }}}{EXCHG {PARA 236 "" 0 "" {TEXT 214 8 "Calculus" }}{PARA 234 "" 0 "" {TEXT 221 11 "Derivatives" }}{PARA 209 "" 0 "" {TEXT 239 44 "To com pute the derivative of the expression " }{XPPEDIT 18 0 "Typesetting:-m row(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Ty pesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"tr ue\", mathvariant = \"italic\"), Typesetting:-mn(\"3\", mathvariant = \+ \"normal\"), superscriptshift = \"0\")), Typesetting:-mo(\"−\", \+ mathvariant = \"normal\", fence = \"false\", separator = \"false\", st retchy = \"false\", symmetric = \"false\", largeop = \"false\", movabl elimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rsp ace = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mn(\"2\", mathv ariant = \"normal\"), Typesetting:-mo(\"⁢\", mathvarian t = \"normal\", fence = \"false\", separator = \"false\", stretchy = \+ \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\" ), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\") ), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mn(\"9 \", mathvariant = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I +modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6(F+-F#6#- I%msupGF$6%-F,6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#mn GF$6$Q\"3F'/F=Q'normalF'/%1superscriptshiftGQ\"0F'-I#moGF$6-Q(−F 'FC/%&fenceGQ&falseF'/%*separatorGFN/%)stretchyGFN/%*symmetricGFN/%(la rgeopGFN/%.movablelimitsGFN/%'accentGFN/%'lspaceGQ,0.2222222emF'/%'rsp aceGFgn-F#6%-F@6$Q\"2F'FC-FI6-Q1⁢F'FCFLFOFQFSFUFWFY/Ffn Q&0.0emF'/FinFcoF6-FI6-Q\"+F'FCFLFOFQFSFUFWFYFenFhn-F@6$Q\"9F'FCF+" } {TEXT 239 5 ", say" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 24 "diff( x^3 - 2*x + 9, x);" }}} {EXCHG {PARA 221 "" 0 "" {TEXT 239 37 "To compute the second derivativ e, say" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 28 "diff( x^3 - 2*x + 9, x, x );" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 17 "or alternatively," }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 27 "diff( x^3 - 2* x + 9, x$2 );" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 51 "This works be cause Maple translates the expression " }{TEXT 209 3 "x$2" }{TEXT 208 19 " into the sequence " }{TEXT 209 4 "x, x" }{TEXT 208 16 ". By anal ogy, " }{TEXT 209 3 "x$3" }{TEXT 208 131 " would give the third deriv ative. Thus one can easily compute derivatives of any order. Now sup pose that we are given a function " }{XPPEDIT 18 0 "Typesetting:-mrow( Typesetting:-mi(\"g\", italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\" gF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 208 11 " define d by" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 24 "g := x -> x^3 - 2*x + 9;" }}}{EXCHG {PARA 221 "" 0 " " {TEXT 239 23 "It seems natural to use" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 13 "diff( g, x );" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 239 25 "to get the derivative of " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"g\", italic = \"tr ue\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Types ettingGI(_syslibGF'6#-I#miGF$6%Q\"gF'/%'italicGQ%trueF'/%,mathvariantG Q'italicF'" }{TEXT 239 43 ". However, Maple expects an expression in " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \" true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typ esettingGI(_syslibGF'6#-I#miGF$6%Q\"xF'/%'italicGQ%trueF'/%,mathvarian tGQ'italicF'" }{TEXT 239 19 " and so interprets " }{XPPEDIT 18 0 "Type setting:-mrow(Typesetting:-mi(\"g\", italic = \"true\", mathvariant = \+ \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6 #-I#miGF$6%Q\"gF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 239 53 " as a constant, giving the wrong result. The command" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 16 "diff( g(x), x );" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 26 "which uses the expression " }{XPPEDIT 18 0 "Typesetting:-mrow( Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"g\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&ApplyFuncti on;\", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rs pace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting: -mi(\"x\", italic = \"true\", mathvariant = \"italic\")), mathvariant \+ = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I ,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-F,6%Q\"gF'/%'italicGQ%t rueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q0⁡F'/F8Q'normal F'/%&fenceGQ&falseF'/%*separatorGFB/%)stretchyGFB/%*symmetricGFB/%(lar geopGFB/%.movablelimitsGFB/%'accentGFB/%'lspaceGQ&0.0emF'/%'rspaceGFQ- I(mfencedGF$6$-F#6#-F,6%Q\"xF'F4F7F>F+" }{TEXT 208 60 ", works correct ly. The subtlety here is an important one: " }{TEXT 209 4 "diff" } {TEXT 208 45 " operates on expressions, not on functions - " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"g\", italic = \"tru e\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6#-I#miGF$6%Q\"gF'/%'italicGQ%trueF'/%,mathvariantGQ 'italicF'" }{TEXT 208 21 " is a function while " }{XPPEDIT 18 0 "Types etting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi( \"g\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo( \"⁡\", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mro w(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")) , mathvariant = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+m odulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-F,6%Q\"gF '/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q0&ApplyFunction ;F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretchyGFB/%*symm etricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lspaceGQ&0.0emF '/%'rspaceGFQ-I(mfencedGF$6$-F#6#-F,6%Q\"xF'F4F7F>F+" }{TEXT 208 72 " \+ is an expression. To define the derivative of a function, use Maple's " }{TEXT 209 1 "D" }{TEXT 208 10 " operator:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 11 "dg := D( g);" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 239 72 "The result is a functio n. You can work with it just as you worked with " }{XPPEDIT 18 0 "Typ esetting:-mrow(Typesetting:-mi(\"g\", italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF' 6#-I#miGF$6%Q\"gF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 239 55 ". Thus you can compute function values and make plots:" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 6 "dg(1);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 85 "plot( \{ g(x), dg(x) \}, x = -5..5, title = `g(x) = x^3 - 2x + 9 and \+ its derivative` );" }}}{EXCHG {PARA 232 "" 0 "" {TEXT 220 19 "Partial \+ Derivatives" }}{PARA 221 "" 0 "" {TEXT 239 38 "Maple can compute parti al derivatives:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 50 "q := sin(x*y); # expression with tw o variables" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 46 "diff( q, x ); # partial with respect to x" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 42 "diff( q, x, y ); # compute d/dy of dq/dx" }}} {EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 117 "As in the one variable case, there is an operator for computing d erivatives of functions (as opposed to expressions):" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 30 "k := (x,y) -> cos(x) + sin(y);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 43 "D[1](k); # partial with respect to x" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 43 "D[2](k); # partial with repsect to \+ y" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 50 "D[1,1](k); # se cond partial with respect to x" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 51 "D[1,2](k); # partial with respect to y, then x" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 32 "D[1](D[2](k)); # same as \+ above;" }}}{EXCHG {PARA 228 "" 0 "" {TEXT 216 9 "Integrals" }}{PARA 208 "" 0 "" {TEXT 208 26 "To compute integrals, use " }{TEXT 209 3 "in t" }{TEXT 208 44 ". The indefinite integral (antiderivative) " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mo(\"\21053\", foreground = \"[144,144,144]\", mathva riant = \"normal\", Typesetting:-msemantics = \"inert\", fence = \"fal se\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetti ng:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"ita lic\"), Typesetting:-mn(\"3\", mathvariant = \"normal\"), superscripts hift = \"0\")), Typesetting:-mi(\"\"), Typesetting:-mspace(height = \" 0.0ex\", width = \"0.3em\", depth = \"0.0ex\", linebreak = \"auto\"), \+ Typesetting:-mo(\"ⅆ\", foreground = \"[144,144,144]\", m athvariant = \"normal\", Typesetting:-msemantics = \"inert\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")) ;" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q !F'-F#6(-I#moGF$6/Q&∫F'/%+foregroundGQ.[144,144,144]F'/%,mathvaria ntGQ'normalF'/I+msemanticsGF$Q&inertF'/%&fenceGQ&falseF'/%*separatorGF @/%)stretchyGF@/%*symmetricGF@/%(largeopGF@/%.movablelimitsGF@/%'accen tGF@/%'lspaceGQ&0.0emF'/%'rspaceGFO-F#6#-I%msupGF$6%-F,6%Q\"xF'/%'ital icGQ%trueF'/F9Q'italicF'-I#mnGF$6$Q\"3F'F8/%1superscriptshiftGQ\"0F'F+ -I'mspaceGF$6&/%'heightGQ&0.0exF'/%&widthGQ&0.3emF'/%&depthGFeo/%*line breakGQ%autoF'-F26/Q0ⅆF'F5F8F;F>FAFCFEFGFIFKFMFPFWF+" } {TEXT 208 25 " is given by " }{TEXT 209 12 "int( x^3, x)" }{TEXT 208 112 ". The following examples illustrate that Maple knows \+ integration by parts, substitution, and partial fractions:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 14 "int( 1/x, x );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 19 "int ( x*sin(x), x );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 23 "int( \+ sin(3*x + 1), x );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 29 "int ( x/ (x^2 - 5*x + 4), x );" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 39 " Nonetheless, Maple can't do everything:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 28 "int( sin( sqrt (1-x^3)), x );" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 209 "" 0 "" {TEXT 239 123 "The last response echoed back the indefinite i ntegral, a signal that Maple does not know how to find an antiderivati ve for " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Type setting:-mrow(Typesetting:-mi(\"sin\", italic = \"false\", mathvariant = \"normal\"), Typesetting:-mo(\"⁡\", mathvariant = \"n ormal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Type setting:-mfenced(Typesetting:-mrow(Typesetting:-msqrt(Typesetting:-mro w(Typesetting:-mn(\"1\", mathvariant = \"normal\"), Typesetting:-mo(\" −\", mathvariant = \"normal\", fence = \"false\", separator = \" false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", lspace = \"0.22222 22em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-msup (Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), \+ Typesetting:-mn(\"3\", mathvariant = \"normal\"), superscriptshift = \+ \"0\")), Typesetting:-mi(\"\")))), mathvariant = \"normal\")), Typeset ting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibG F'6%-I#miGF$6#Q!F'-F#6%-F,6%Q$sinF'/%'italicGQ&falseF'/%,mathvariantGQ 'normalF'-I#moGF$6-Q0⁡F'F7/%&fenceGF6/%*separatorGF6/%)s tretchyGF6/%*symmetricGF6/%(largeopGF6/%.movablelimitsGF6/%'accentGF6/ %'lspaceGQ&0.0emF'/%'rspaceGFN-I(mfencedGF$6$-F#6#-I&msqrtGF$6#-F#6&-I #mnGF$6$Q\"1F'F7-F;6-Q(−F'F7F>F@FBFDFFFHFJ/FMQ,0.2222222emF'/FPF ]o-F#6#-I%msupGF$6%-F,6%Q\"xF'/F5Q%trueF'/F8Q'italicF'-Ffn6$Q\"3F'F7/% 1superscriptshiftGQ\"0F'F+F7F+" }{TEXT 239 290 " in terms of elementar y functions (the ones built from addition, subtraction, multiplication , division, powers, roots, logarithms and exponentials, trig functions and their inverses). In fact, it can be proved that no such antideri vative exists. Thus, even a smarter Maple would not help." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 35 "To comput e definite integrals like " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesett ing:-mi(\"\"), Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mo( \"\21053\", foreground = \"[144,144,144]\", mathvariant = \"normal\", \+ Typesetting:-msemantics = \"inert\", fence = \"false\", separator = \" false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\" , rspace = \"0.0em\"), Typesetting:-mn(\"0\", mathvariant = \"normal\" ), Typesetting:-mn(\"1\", mathvariant = \"normal\"), superscriptshift \+ = \"2\", subscriptshift = \"0\"), Typesetting:-mi(\"\"), Typesetting:- mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathv ariant = \"italic\"), Typesetting:-mn(\"3\", mathvariant = \"normal\") , superscriptshift = \"0\")), Typesetting:-mi(\"\"), Typesetting:-mspa ce(height = \"0.0ex\", width = \"0.3em\", depth = \"0.0ex\", linebreak = \"auto\"), Typesetting:-mo(\"ⅆ\", foreground = \"[144 ,144,144]\", mathvariant = \"normal\", Typesetting:-msemantics = \"ine rt\", fence = \"false\", separator = \"false\", stretchy = \"false\", \+ symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesett ing:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesett ing:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6%-I#miGF$6#Q!F'-F#6)-I(msubsupGF$6'-I#moGF$6/Q&∫F'/%+foregroundG Q.[144,144,144]F'/%,mathvariantGQ'normalF'/I+msemanticsGF$Q&inertF'/%& fenceGQ&falseF'/%*separatorGFC/%)stretchyGFC/%*symmetricGFC/%(largeopG FC/%.movablelimitsGFC/%'accentGFC/%'lspaceGQ&0.0emF'/%'rspaceGFR-I#mnG F$6$Q\"0F'F;-FV6$Q\"1F'F;/%1superscriptshiftGQ\"2F'/%/subscriptshiftGQ \"0F'F+-F#6#-I%msupGF$6%-F,6%Q\"xF'/%'italicGQ%trueF'/FFAFDFFFH FJFLFNFPFSFaoF+" }{TEXT 208 9 " we say " }{TEXT 209 20 "int( x^3, x = 0..1 )" }{TEXT 208 108 ". Note that the only difference is that we g ive an interval of integration. Let us return to the integral " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mo(\"\21053\", foreground = \"[144,144,144]\", mathva riant = \"normal\", Typesetting:-msemantics = \"inert\", fence = \"fal se\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetti ng:-mi(\"sin\", italic = \"false\", mathvariant = \"normal\"), Typeset ting:-mo(\"⁡\", mathvariant = \"normal\", fence = \"fals e\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typese tting:-mrow(Typesetting:-msqrt(Typesetting:-mrow(Typesetting:-mn(\"1\" , mathvariant = \"normal\"), Typesetting:-mo(\"−\", mathvariant \+ = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"f alse\", symmetric = \"false\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.22 22222em\"), Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mn(\"3\", \+ mathvariant = \"normal\"), superscriptshift = \"0\")), Typesetting:-mi (\"\")))), mathvariant = \"normal\")), Typesetting:-mi(\"\"), Typesett ing:-mspace(height = \"0.0ex\", width = \"0.3em\", depth = \"0.0ex\", \+ linebreak = \"auto\"), Typesetting:-mo(\"ⅆ\", foreground = \"[144,144,144]\", mathvariant = \"normal\", Typesetting:-msemantic s = \"inert\", fence = \"false\", separator = \"false\", stretchy = \" false\", symmetric = \"false\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\") , Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")) , Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI (_syslibGF'6%-I#miGF$6#Q!F'-F#6(-I#moGF$6/Q&∫F'/%+foregroundGQ.[14 4,144,144]F'/%,mathvariantGQ'normalF'/I+msemanticsGF$Q&inertF'/%&fence GQ&falseF'/%*separatorGF@/%)stretchyGF@/%*symmetricGF@/%(largeopGF@/%. movablelimitsGF@/%'accentGF@/%'lspaceGQ&0.0emF'/%'rspaceGFO-F#6%-F,6%Q $sinF'/%'italicGF@F8-F26-Q0⁡F'F8F>FAFCFEFGFIFKFMFP-I(mfe ncedGF$6$-F#6#-I&msqrtGF$6#-F#6&-I#mnGF$6$Q\"1F'F8-F26-Q(−F'F8F> FAFCFEFGFIFK/FNQ,0.2222222emF'/FQFho-F#6#-I%msupGF$6%-F,6%Q\"xF'/FXQ%t rueF'/F9Q'italicF'-Fao6$Q\"3F'F8/%1superscriptshiftGQ\"0F'F+F8F+-I'msp aceGF$6&/%'heightGQ&0.0exF'/%&widthGQ&0.3emF'/%&depthGFaq/%*linebreakG Q%autoF'-F26/Q0ⅆF'F5F8F;F>FAFCFEFGFIFKFMFPF_pF+" }{TEXT 208 115 ", which Maple could not evaluate. We can, however, ask Maple to find a numerical value for the definite integral: " }}{PARA 221 "" 0 "" {TEXT 239 1 " " }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 38 "i nt( sin( sqrt(1 - x^3) ), x = 0..1 );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 140 " The second command forces Maple to apply a numerical method to evaluat e the integral. Of course, you could also put everything on one line: " }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 47 "evalf( int( sin( sqrt(1 - x^3) ), x = 0..1 ) );" }}} {EXCHG {PARA 229 "" 0 "" {TEXT 215 21 "Numerical Integration" }}{PARA 208 "" 0 "" {TEXT 208 55 "Another approach to numerical integration is to use the" }{TEXT 209 9 " student " }{TEXT 208 8 "package." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 46 "with( student ): # load the package" }{MPLTEXT 1 0 1 " \n" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 50 "j := sin( sqrt(1 - \+ x^3) ); # define the integrand" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 50 "trapezoid( j, x = 0..1 ); # apply tra pezoid rule" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 49 "evalf(%); # put in decimal form" } {MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 15 "By defau lt the " }{TEXT 209 9 "trapezoid" }{TEXT 208 50 " command approximates the area under the graph of " }{XPPEDIT 18 0 "Typesetting:-mrow(Types etting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"sin\", italic = \+ \"false\", mathvariant = \"normal\"), Typesetting:-mo(\"&ApplyFunction ;\", mathvariant = \"normal\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rsp ace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:- msqrt(Typesetting:-mrow(Typesetting:-mn(\"1\", mathvariant = \"normal \"), Typesetting:-mo(\"−\", mathvariant = \"normal\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting: -mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", math variant = \"italic\"), Typesetting:-mn(\"3\", mathvariant = \"normal\" ), superscriptshift = \"0\")), Typesetting:-mi(\"\")))), mathvariant = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I, TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-F,6%Q$sinF'/%'italicGQ&f alseF'/%,mathvariantGQ'normalF'-I#moGF$6-Q0⁡F'F7/%&fence GF6/%*separatorGF6/%)stretchyGF6/%*symmetricGF6/%(largeopGF6/%.movable limitsGF6/%'accentGF6/%'lspaceGQ&0.0emF'/%'rspaceGFN-I(mfencedGF$6$-F# 6#-I&msqrtGF$6#-F#6&-I#mnGF$6$Q\"1F'F7-F;6-Q(−F'F7F>F@FBFDFFFHFJ /FMQ,0.2222222emF'/FPF]o-F#6#-I%msupGF$6%-F,6%Q\"xF'/F5Q%trueF'/F8Q'it alicF'-Ffn6$Q\"3F'F7/%1superscriptshiftGQ\"0F'F+F7F+" }{TEXT 208 129 " with four equal trapezoids. For greater accuracy use more trapezoids , i.e., a finer subdivision of the interval of integration:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 38 "evalf( trapezoid( j, x = 0..1, 10 ) );" }}}{EXCHG {PARA 221 "" 0 " " {TEXT 239 74 "Better yet, use a more sophisticated numerical method \+ like Simpson's rule:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 31 "evalf( simpson( j, x = 0..1 ));" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 54 "Only an even number of subdi visions is allowed, as in " }{TEXT 209 26 "simpson( j, x = 0..1, 10 )" }{TEXT 208 1 "." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 4 "The " }{TEXT 209 7 "student" }{TEXT 208 131 " package \+ is well worth exploring. Among other things it has tools for displayi ng figures which explain the meaning of integration:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 27 "left box( j, x = 0..1, 10 );" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 36 "The area of the figure displayed by " }{TEXT 209 7 "leftbox" }{TEXT 208 16 " is computed by " }{TEXT 209 7 "leftsum" }{TEXT 208 1 ":" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 35 "evalf( leftsum( j, x = 0..1, 10 ));" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 29 "One can also experiment with " }{TEXT 209 8 "rightbox" } {TEXT 208 5 " and " }{TEXT 209 9 "middlebox" }{TEXT 208 31 " and their companion functions " }{TEXT 209 8 "rightsum" }{TEXT 208 5 " and " } {TEXT 209 9 "middlesum" }{TEXT 208 1 "." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 230 "" 0 "" {TEXT 207 18 "Multiple Integrals" }} {PARA 209 "" 0 "" {TEXT 239 4 "Let " }{XPPEDIT 18 0 "Typesetting:-mrow (Typesetting:-mi(\"R\", italic = \"true\", mathvariant = \"italic\")); " "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q \"RF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 239 38 " be t he rectangular region defined by " }{XPPEDIT 18 0 "Typesetting:-mrow(T ypesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"x F'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 239 22 " between 0 and 1, and " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"y \", italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modu lenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"yF'/%'italicGQ%true F'/%,mathvariantGQ'italicF'" }{TEXT 239 32 " between 0 and 1. To inte grate " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Types etting:-mrow(Typesetting:-mo(\"\21053\", foreground = \"[144,144,144] \", mathvariant = \"normal\", Typesetting:-msemantics = \"inert\", fen ce = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow (Typesetting:-mo(\"\21053\", foreground = \"[144,144,144]\", mathvaria nt = \"normal\", Typesetting:-msemantics = \"inert\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesett ing:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-msup(T ypesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Ty pesetting:-mn(\"2\", mathvariant = \"normal\"), superscriptshift = \"0 \")), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow (Typesetting:-msup(Typesetting:-mi(\"y\", italic = \"true\", mathvaria nt = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), su perscriptshift = \"0\")), Typesetting:-mi(\"\")), mathvariant = \"norm al\"), Typesetting:-mi(\"\"), Typesetting:-mspace(height = \"0.0ex\", \+ width = \"0.3em\", depth = \"0.0ex\", linebreak = \"auto\"), Typesetti ng:-mo(\"ⅆ\", foreground = \"[144,144,144]\", mathvarian t = \"normal\", Typesetting:-msemantics = \"inert\", fence = \"false\" , separator = \"false\", stretchy = \"false\", symmetric = \"false\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", ls pace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\"), Typesett ing:-mspace(height = \"0.0ex\", width = \"0.3em\", depth = \"0.0ex\", \+ linebreak = \"auto\"), Typesetting:-mo(\"ⅆ\", foreground = \"[144,144,144]\", mathvariant = \"normal\", Typesetting:-msemantic s = \"inert\", fence = \"false\", separator = \"false\", stretchy = \" false\", symmetric = \"false\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\") , Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\")) , Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI (_syslibGF'6%-I#miGF$6#Q!F'-F#6(-I#moGF$6/Q&∫F'/%+foregroundGQ.[14 4,144,144]F'/%,mathvariantGQ'normalF'/I+msemanticsGF$Q&inertF'/%&fence GQ&falseF'/%*separatorGF@/%)stretchyGF@/%*symmetricGF@/%(largeopGF@/%. movablelimitsGF@/%'accentGF@/%'lspaceGQ&0.0emF'/%'rspaceGFO-F#6(F1-I(m fencedGF$6$-F#6'F+-F#6#-I%msupGF$6%-F,6%Q\"xF'/%'italicGQ%trueF'/F9Q'i talicF'-I#mnGF$6$Q\"2F'F8/%1superscriptshiftGQ\"0F'-F26-Q\"+F'F8F>FAFC FEFGFIFK/FNQ,0.2222222emF'/FQF[p-F#6#-Ffn6%-F,6%Q\"yF'F[oF^oF`oFdoF+F8 F+-I'mspaceGF$6&/%'heightGQ&0.0exF'/%&widthGQ&0.3emF'/%&depthGFip/%*li nebreakGQ%autoF'-F26/Q0ⅆF'F5F8F;F>FAFCFEFGFIFKFMFPFhnF+F dpFbqFapF+" }{TEXT 239 6 " over " }{XPPEDIT 18 0 "Typesetting:-mrow(Ty pesetting:-mi(\"R\", italic = \"true\", mathvariant = \"italic\"));" " -I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"RF '/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 239 73 " in Maple , we compute the repeated integral. Here is one way to do this:" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 27 "int( x^2 + y^2, x = 0..1 );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 18 "int( %, y = 0..1);" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 239 49 "The first command integrates with respect to the " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"tru e\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6#-I#miGF$6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ 'italicF'" }{TEXT 239 38 " variable, producing an expression in " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \"tru e\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6#-I#miGF$6%Q\"yF'/%'italicGQ%trueF'/%,mathvariantGQ 'italicF'" }{TEXT 239 60 ". The second command integrates the result \+ with respect to " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\" y\", italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+mod ulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"yF'/%'italicGQ%tru eF'/%,mathvariantGQ'italicF'" }{TEXT 239 18 " to give a number." }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 231 "" 0 "" {TEXT 210 20 "Oth er Calculus Tools" }}{PARA 221 "" 0 "" {TEXT 239 7 "Limits:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 42 "g := x -> (x^3 - 2*x + 9)/(2*x^3 + x - 3);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 28 "limit( g(x), x = infinity );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 32 "limit( sin(x)/x, x = infinity );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 25 "limit( sin(x)/x, x = 0 );" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 27 "Taylor expansions and s ums:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 52 "taylor( exp(x), x = 0, 4 ); # expansion around x = 0 " }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 52 "sum( i^2, i = 1..100 \+ ); # be sure i is unassigned" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 53 "sum( x^n, n = 5..10 ); # needs n to be unassigned " }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 39 "sum( 1/d^5, d=1..infi nity ); evalf(%);" }}}{EXCHG {PARA 225 "" 0 "" {TEXT 228 22 "Differen tial Equations" }}{PARA 208 "" 0 "" {TEXT 208 67 "Maple can solve diff erential equations for you. One way is to use " }{TEXT 209 6 "dsolve" }{TEXT 208 1 ":" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 36 "deq := diff( y(x), x$2 ) + y(x) = 0;" } }}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 20 "dsolve( deq, y(x) );" }} }{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 4 "Try " }{TEXT 209 7 "?dsolve" }{TEXT 208 89 " for more informati on. We can specify initial conditions and experiment with parameters. " }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 38 "de := diff( y(x), x ) = y(x) - sin(x);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 39 "sol \+ := dsolve( \{de, y(0) = 0.1\}, y(x));" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 4 "sol;" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 59 "To gra ph the solution, we need only the right-hand side of " }{TEXT 209 3 "s ol" }{TEXT 208 21 ", which we can get by" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 27 "plot( rhs(sol) , x = 0..10);" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 103 "Now let's tr y to make the slope field and draw some trajectories. We can get the \+ slope field with the " }{TEXT 209 10 "dfieldplot" }{TEXT 208 18 " comm and from the " }{TEXT 209 7 "DEtools" }{TEXT 208 9 " package:" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 78 "dfieldplot( de, y(x), x = -3..3, y = -3..3, arrows = \+ line, dirgrid = [30,30]);" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 117 " Now let's use another command from the DEtools package to simultaneous ly plot the slope field and a few trajectories:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 138 "DEplot( de, y(x), x = -3..3, \{ [y(0)=0], [y(0)=1], [y(0)=3] \}, y = -3..3, c olor='black', linecolor='red', arrows='line', dirgrid=[30,30]);" }}} {EXCHG {PARA 227 "" 0 "" {TEXT 227 28 "Vector and Matrix Operations" } }{PARA 221 "" 0 "" {TEXT 239 82 "To work with vectors and matrices in \+ Maple, first load the linear algebra package:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 13 "with(lin alg):" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 38 "Define and display a vector like this:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 23 "v := vector( [1, -1] );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 16 "v[ 1]; v[2]; v;" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 9 "print(v) ;" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 108 "Note that Maple treats v ectors as columns, even though it displays them as rows. Now define a nother vector " }{TEXT 209 1 "w" }{TEXT 208 33 " and do some simple co mputations:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 22 "w := vector( [1, 1] );" }}}{EXCHG {PARA 207 " > " 0 "" {MPLTEXT 1 0 48 "matadd( v,w ); # vector or matrix addi tion" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 15 "dotprod( v,w );" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 28 "Next we define two matrices:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 30 "A := matrix([ [2,3], [1,2] ]);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 30 "B := matrix ([ [1,1], [0,1] ]);" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 64 "It is e asy to make new matrices (or vectors) from old ones as in" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 19 "augment( A, v, w );" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 48 "Next, \+ we do some computations with our matrices " }{TEXT 209 1 "A" }{TEXT 208 5 " and " }{TEXT 209 1 "B" }{TEXT 208 17 " and our vectors " } {TEXT 209 1 "v" }{TEXT 208 5 " and " }{TEXT 209 1 "w" }{TEXT 208 1 ":" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 42 "matadd( A, B ); # or use evalm( A+B );" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 42 "scalarmul( A, 2 ); # or u se evalm( 2*A );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 44 "multi ply( A,B ); # or use evalm( A &* B);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 49 "multiply( A, v+w ); # or use evalm( A &* (v+w) );" } }}{EXCHG {PARA 221 "" 0 "" {TEXT 239 77 "Finally, we can change the in dividual entries of a matrix with commands like:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 26 "A[1,1] : = 5; B[2,1] := 1;" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 29 "eva lm(A); evalm(B); # check" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 42 " Maple can compute inverses and transposes:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 39 "inverse(A); \+ #or use evalm( A^(-1) );" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 13 "transpose(A);" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 33 "It can al so compute determinants:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 17 "det(A); det(B);" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 27 "det( A+B); det(A) + det(B) ;" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 54 "Symbolic matrices are as \+ legitimate as numerical ones:" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}} {EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 30 "A := matrix([ [a,b], [c,d] ]);" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 7 "det(A);" }}} {EXCHG {PARA 221 "" 0 "" {TEXT 239 3 "Let" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 43 "C := matrix([ \+ [3,2,2], [3,1,2], [1,1,1] ]);" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 11 "We can put " }{TEXT 209 1 "C" }{TEXT 208 28 " into row-reduced for m using" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 13 "gausselim(C);" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 239 2 "or" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 13 "gaussjord(C);" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 85 "There are also commands for doing elementary row and col umn operations on a matrix: " }{TEXT 209 6 "addrow" }{TEXT 208 2 ", " }{TEXT 209 7 "swaprow" }{TEXT 208 2 ", " }{TEXT 209 6 "mulrow" }{TEXT 208 67 ", etc. You can find a list of functions in the help files fo r the " }{TEXT 209 6 "linalg" }{TEXT 208 9 " package." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 226 "" 0 "" {TEXT 230 15 "Troubleshooting" }}{PARA 221 "" 0 "" {TEXT 239 197 "Although errors in Maple can be qu ite confusing at first, you will quickly gain the experience needed to understand and fix them. Below is a list of common errors to conside r when troubleshooting." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 221 "" 0 "" {TEXT 239 50 "1. Do statements end with a semicolon or a \+ colon?" }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 66 "2. Are parentheses, braces, brackets, etc., balanced? Code l ike " }{TEXT 209 7 "\{ x,y \}" }{TEXT 208 15 " is good, but " }{TEXT 209 4 "x,y\}" }{TEXT 208 20 " will cause trouble." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 117 "3. Are you trying \+ to use something as a variable to which you have already assigned a va lue? To \"clear\" a variable " }{XPPEDIT 18 0 "Typesetting:-mrow(Type setting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"));" "-I %mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"xF'/ %'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 208 6 ", say " } {TEXT 209 13 "unassign('x')" }{TEXT 208 47 ". You can clear many vari ables at once, e.g., " }{TEXT 209 20 "unassign( 'x', 'y' )" }{TEXT 208 263 ". You will not have to reload any packages after using the u nassign command. Remember, a variable that has a value assigned to it no longer can function as a variable. To display the value of an ord inary variable, type its name, followed by a semicolon, e.g., " }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 0 2 "x;" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }} {PARA 208 "" 0 "" {TEXT 208 59 "4. If things seem hopelessly messed u p, issue the command " }{TEXT 209 8 "restart;" }{TEXT 208 57 " You wi ll have to reload any needed packages after this." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 88 "5. Are you using a \+ function when an expression is called for, or vice versa? Remember " }{TEXT 209 14 "g := x^2 + y^2" }{TEXT 208 29 " defines an expression w hile " }{TEXT 209 23 "f := (x,y) -> x^2 + y^2" }{TEXT 208 49 " defines a function. It makes good sense to say " }{TEXT 209 6 "f(1,3)" } {TEXT 208 30 " but not so much sense to say " }{TEXT 209 6 "g(1,2)" } {TEXT 208 1 "." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 18 "6. Are you using " }{TEXT 209 1 "=" }{TEXT 208 6 " whe n " }{TEXT 209 2 ":=" }{TEXT 208 103 " is called for? Remember, the f irst tests for equality while the second assigns a value to a variable ." }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 74 "7. Are you distinguishing between the three kinds of qu otes? They are: " }{TEXT 209 1 "\"" }{TEXT 208 19 " the double quote , " }{TEXT 209 1 "'" }{TEXT 208 23 " the single quote, and " }{TEXT 209 1 "`" }{TEXT 208 15 " the backquote." }}{PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 78 "8. If you use a function i n a package, load the package first. Thus, to use " }{TEXT 209 6 "mat rix" }{TEXT 208 7 ", load " }{TEXT 209 6 "linalg" }{TEXT 208 9 "; to u se " }{TEXT 209 7 "display" }{TEXT 208 7 ", load " }{TEXT 209 5 "plots " }{TEXT 208 12 ". You load " }{TEXT 209 6 "linalg" }{TEXT 208 16 " b y the command " }{TEXT 209 14 "with( linalg )" }{TEXT 208 1 "." }} {PARA 219 "" 0 "" {TEXT 249 0 "" }}{PARA 208 "" 0 "" {TEXT 208 20 "9. \+ Remember to use " }{TEXT 209 1 "?" }{TEXT 208 57 " to inquire about t he details of a Maple function, e.g., " }{TEXT 209 5 "?plot" }{TEXT 208 36 " for information about plot or just " }{TEXT 209 1 "?" }{TEXT 208 25 " for general information." }}{PARA 208 "" 0 "" {TEXT 251 0 "" }}{PARA 208 "" 0 "" {TEXT 208 193 "10. If all of the licenses for the current version of Maple are being used, you can use a different vers ion by calling it up by name, i.e. by typing xmapleV8 or xmapleV11 in \+ a terminal window." }{TEXT 208 1 "\n" }{TEXT 208 2 " " }}{PARA 219 "" 0 "" {TEXT 249 0 "" }{TEXT 239 123 "If you want to see more examples \+ of Maple code, there is an abundance of information on the web. A goo d place to start is " }{TEXT 222 17 "www.mapleapps.com" }{TEXT 239 261 ", but you will also find many good sites by using a search engine . We also have a Maple V Learning Guide that you can check out for us e in the Undergraduate Computer Lab. Even though it is for an older v ersion of Maple, it still has a lot of good information." }}}{EXCHG {PARA 219 "" 0 "" {TEXT 249 0 "" }}}{PARA 224 "" 0 "" {TEXT 252 0 "" } }{PARA 223 "" 0 "" {TEXT 253 0 "" }}{PARA 223 "" 0 "" {TEXT 253 0 "" } }{PARA 223 "" 0 "" {TEXT 253 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }