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The work you are to complete as your firs t Maple assignment is the section 1.4 computing project, described on pages 43-44 in the text. The MAPLE commands you will need in order t o do that project are reviewed in this tutorial. A precise template \+ for your project answers will be available on Friday September 1, and will enlarge slightly on the Investigation a,b,c of page 44." }} {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 297 0 "" } {TEXT -1 0 "" }{TEXT 296 33 "0) A preliminary word of Advice: " } {TEXT -1 988 " Students often approach the task of reading mathematica l material as if they were reading a novel; they sort of skim along qu ickly. That approach is O.K. to get an overview, but in order to hav e a chance at real understanding you must be prepared to proceed much \+ more slowly, sentence by sentence and thought by thought. Otherwise y ou will almost certainly find yourself partly lost after several parag raphs and completely lost after several more. (This might happen anyw ay.) If you are working properly it can easily take half an hour to \+ read through one page of mathematical text. This takes a certain amou nt of discipline, patience, and practice. With the Math 2250 compute r projects there is the added temptation of having Maple execute co mmands in successive command sections by repeatedly hitting the ente r (return) key, without pausing to digest the interlaced text or the m eaning of the commands. There is a seductive appeal in having this c apability. Resist it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 " " 0 "" {TEXT -1 211 "We will assume you are working in the Math Depart ment computer lab. If you are in another lab (e.g. Engineering lab in EMCB or Library lab in Marriot) or on your own computer, you will ge t started differently." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 38 "1a) Logging in to a Math Lab machine:" }{TEXT -1 375 " The Math Department Computer Lab is located in the South Physic s 205, i.e. inside the 2-story brick building just north of the Math \+ building JWB, and south-east of the physics classrooms in the circular part of JFB. Room 205 is in the back (East) of the building, on th e second floor (from the west), which equals the level of the ground o n the uphill, east entrance." }}{PARA 0 "" 0 "" {TEXT -1 106 " \+ The following information about logging in and your initial password i s summarized from the handout " }{TEXT 257 111 "Introduction to the Un dergraduate Computer Lab Department of Mathematics, University of Utah , SLC, Utah 84112 ." }{TEXT -1 1 " " }{TEXT 258 82 " This and other us eful handouts are available on a table in the front of the lab. " }} {PARA 0 "" 0 "" {TEXT -1 479 " Everyone who is registered in Math 2250 should automatically have an account set up in our lab. These \+ accounts are created from University class lists so it sometimes happe ns that late-registering people don't have accounts yet. If you turn \+ out to be one of these people you will need to consult the lab assista nt about getting an account. Make sure to bring your student I.D. bec ause the first thing the assistant must do is verify that you are a Un iversity student. " }}{PARA 0 "" 0 "" {TEXT -1 778 " If your mach ine looks asleep jiggle the mouse or hit any key to wake it back up. \+ A window should appear which asks you to select a server. Since the \+ Math Department is sharing this lab with Physics, there are physics an d math server computers. You want to select one from the list of choi ces which has \".math\" in its name. If a \".physics\" computer is h ighlighted, use your cursor to choose one of the \".math\" computers, \+ and then O.K. your choice by clicking in the \"accept\" box. In a few seconds a login window should appear, asking for your login name and \+ password, and have \"Mathematics\" written in red on the top. (If t he login window says \"Physics\" you have accidently chosen a physics \+ server, and you should type \"control C\" to go back to the previous s tep.)" }}{PARA 0 "" 0 "" {TEXT -1 873 " Your default login name i s made out of your student I.D. number and your actual name, as follow s. All names from classes begin with ``c-''. If your name is Karl F red GausS, then your login name is c-gskf, following the recipe: c-(f irst letter of last name)(last letter of last name)(first letter of fi rst name)(middle initial). If there are multiple people registered th is term who would have the same login name, say c-gskf, then they are \+ instead assigned login names as c-gskf1, c-gskf2, c-gskf3, etc. Mr. G auss would not know beforehand which case he fell into, so would proba bly try c-gskf first, followed by his password. In case of failure h e would then try c-gskf1, then c-gskf2, etc, through c-gskf4. Then he \+ would find a lab assistant. After entering your try at a login name, \+ type the ``return'' key and the cursor should be in the password box. " }}{PARA 0 "" 0 "" {TEXT -1 560 " Your initial password is just the c-gskf part of your login name followed by the last four digits o f your student I.D. number. If Mr. Gauss has ID number 000735421 the n his initial password is gskf5421, regardless whether his login name \+ was c-gskf or c-gskf3. If the login fails try again and then try th e different login names suggested above. Another possibility is that \+ your account was created using your social security number (which used to be used for student ID number). If failure continues find a lab a ssistant and he/she will help you. " }}{PARA 0 "" 0 "" {TEXT -1 332 " Once you are logged in successfully a ``local'' window should app ear. Notice that it has various parts: borders on the top (title bar) , borders on the side (scroll bar), etc. If you move your mouse on it s pad your pointer (called cursor) moves around the screen. If you wan t to work in a window, the cursor should be in it. " }}{PARA 0 "" 0 " " {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT 259 23 "1b) Changing passwo rd:" }{TEXT -1 197 " Sometime within the first several weeks of class es you must change your default password into a personal one. You do \+ this as follows: Get your cursor into a local window. Type the unix c ommand " }{TEXT 260 6 "passwd" }{TEXT -1 583 ", followed by return, an d follow the directions. Your new password should be exactly 8 charac ters long. Don't choose a word in the dictionary or a proper name. C omposites of dictionary words, like strawdog, are good. Even better i s to use one or two upper case letters, e.g. strAwdog. For still more security, use some digits, e.g. strAw4o9. Note that it takes about \+ 30 minutes for a new password to take effect. Also, you should be awa re that if a password is not changed within the first two weeks of cla ss, then your computer account will be disabled for security reasons. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 17 "1c) \+ Logging out:" }{TEXT -1 221 " Move the cursor out of all windows (int o the background), press the left mouse button and choose the last men u item: Exit X-Windows. (You probably don't want to do this now, but \+ at least locate the menu item for later.)" }}{PARA 0 "" 0 "" {TEXT -1 62 " At this point you are ready to get used to the X-windows:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 51 "2) X-wi ndows, opening netscape, maple, mail, more:" }{TEXT -1 26 " Go throug h the document " }{TEXT 263 35 "Introduction to Xwindows in the Lab" } {TEXT -1 472 ", which you should have a copy of. There should also be copies of this document in the front of the room. Xwindows are like most windows in most ways; your aim here is to experiment to see how \+ to open and close windows, resize them, move them about, and find the m if they happen to get hidden. When you get to the end of the documen t you should also have opened a NETSCAPE window and a MAPLE window. \+ Note: The command for the current (default) version of Maple is " } {TEXT 264 10 "xmaple &, " }{TEXT 299 60 "which you can type into a loc al window, followed by ." }{TEXT -1 196 " This is version 6, \+ and you can also find it as an option on one of your mouse buttons, it 's your choice. If you are using an earlier version of Maple (V5 or \+ V4, for example), there are slight" }}{PARA 0 "" 0 "" {TEXT -1 76 "dif ferences in commands and syntax, and they may confuse you once or twic e. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 21 "F urther information: " }{TEXT -1 123 " If you want more in-depth inform ation about the computing facilities in this lab, you might pick up a \+ copy of the handout " }{TEXT 266 32 "A Crash Course on CSC Facilities " }{TEXT -1 23 ", from the front table." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 177 " If \+ you are starting the tutorial at this point (because you're doing it o n your own at another location), you should have opened a mapleV6 wind ow and a web browser window. " }}{PARA 0 "" 0 "" {TEXT 267 31 "3a) Ma th Department resources:" }{TEXT -1 193 " There is introductory mate rial about Maple on our web pages. If you wish to see what's availabl e use the browser window you made in step (2) above, and go to the Mat h Department home page " }{TEXT 268 26 "http://www.math.utah.edu/." } {TEXT -1 253 " By following links from here you can find current and \+ future course offerings, faculty information, and much more. Since yo u are students, interested in Maple information, click on \"students \" (at the upper left of the page), and you are directed to " }{TEXT 300 31 "http://www.math.utah.edu/ugrad/" }{TEXT -1 46 ", the undergrad uate home page. If you choose " }{TEXT 303 1 "\"" }{TEXT -1 26 "Under graduate Computer Lab" }{TEXT 301 1 "\"" }{TEXT -1 14 ", followed by \+ " }{TEXT 302 29 "\"Selected software and tools\"" }{TEXT -1 49 ", you \+ will find some links to Maple information. " }}{PARA 0 "" 0 "" {TEXT 305 5 "3b) " }{TEXT 270 19 "More Information. " }{TEXT 304 79 "Maple has its own introduction, as well as a new user's tour, see item 5 be low." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 11 " 4) Maple: " }{TEXT -1 873 " Move your cursor into the \"Untitled\" (n ew) Maple window which you created in step (2). Maple is partly just \+ a very fancy calculator; it can do practically any undergraduate mathe matics computation or symbolic manipulation. You can write programs i n Maple and draw pictures as well. If you are doing a homework assign ment you can intersperse text with computations using the toolbar: to get a computation prompt click on the ``[>'' box near the top. To in sert text click on the ``T'' box. Or you can change command fields (s tarting with \"[>\") into text fields by putting the cursor into them \+ and then choosing \"T\". You can use the mouse to cut, paste, and edi t a document. You can change fonts, formats, and use other standard t ext editing tools by choosing appropriate menue items. This document \+ you are reading is a Maple document even though it is largely text." } }{PARA 0 "" 0 "" {TEXT -1 429 " To give you a flavor of what Mapl e can do, we will try a few commands. They should begin on a line hav ing a command prompt ``>'', and should be ended with either a semicol on ; or a colon : If you end with a semicolon you will see visible ou tput, if you end with a colon the output will be suppressed even thoug h the command is executed. Maple will not execute a command until you type the ``return'' or ``enter'' key. " }{TEXT 277 87 "If you have a multiline command use ``shift-return'' to change lines without execut ing." }{TEXT -1 644 " If you mess up your parentheses or brackets or d o something else which makes your command unexecutable you will get a \+ ``syntax error'' message and Maple will try to point out your mistake. After a while you will become good at fixing these mistakes but they can be annoying at first. Spaces are ignored in Maple, so you may us e them to make input easier to read. You can enter explanatory commen ts in a command line by inserting a ``#'' to the left of the comments; Maple ignores any text after the #. Sometimes this is more informati ve then entering nearby explanatory text, especially if you are explai ning various steps in a subroutine." }}{PARA 0 "" 0 "" {TEXT -1 222 " \+ Now, let's try some commands. (You try just the math commands, the editorial comments were only added to explain what the particular com mands are illustrating ! ) Check that you understand what each comman d is doing." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 202 "3+4; 4+5: 6 * 7; #one of these computations wi ll not be shown\n #even though all three will be done, illust rating the \n #difference between a semicolon and a colon\n \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 428 "(3+4)7; #if you want to multiply you must use *, so after\n #t rying the command as given, insert a * to fix the\n #resultin g syntax error. You can execute a line or\n #execution group (bracketed on the left) if\n #your cursor is anywhere in it. You can move the\n #cursor with the mouse or the arrow keys. Maple will\n #try to put it in a good place if it detects an error." }}{PARA 8 "" 1 "" {TEXT -1 25 "Error, unexpected number\n" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 290 "(3+4)^2/7; 3+4^2/7; evalf( 3+4^2/7); #the evalf command gives a \n #decimal approximat ion instead of an algebraic \n #expression. Notice that if gi ven a choice, Maple \n #computes powers first, then multiplie s and divides, \n #and finally adds or subtracts." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "diff(x^2,x); #``differentiate x^2 \+ with respect to x''" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "diff (exp(sin(x))*x^3,x); #a harder differentiation problem\n #you should get output:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(-%$cosG6#% \"xG\"\"\"-%$expG6#-%$sinGF'F))F(\"\"$F)F)*(F0F)F*F))F(\"\"#F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "f:= x-> exp(sin(x))*x^3; # this is the syntax for defining a \n #function, in this case t he function we just \n #differentiated" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "diff(f(x),x ); #should get the same answer as before. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "int(t^2*exp(t),t); #``integrate (t^2)* exp(t) with respect\n #to t'' (Maple doesn't put in the integ ration constant.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "int(t^ 3*exp(sin(t)),t); #this shows that Maple is not God, you \n \+ #will get" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$*&)%\"tG\"\"$\" \"\"-%$expG6#-%$sinG6#F(F*F(" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 " # since if Maple \+ can't find an elementary function \n #antiderivative it just \+ echos what you put in. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "evalf(int(t^3*exp(sin(t)),t=0..1)); #But you could do\n \+ #a definite integral (numerically) even if Maple\n #can't co mpute an elementary antiderivative" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "sum(3^(-n),n=1..100); #add a geometric series part \+ way,\n #this is the series 1/3 +1/9 +1/27 + ..." }}{PARA 11 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "eva lf(%); #get its decimal value. The symbol % refers to the \n \+ #last thing which Maple has computed, the command evalf \n \+ #gets its numerical value\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "Sum(3^(-n),n=1..infinity); evalf(%); #add the series\n # all the way to infinity. Sum with captial\n #S writes the su m but doesn't evaluate it,\n #but then evalf(%) does. \+ " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 275 "Sum((.001)*(n/1000)^2, n= 1..1000); evalf(%);\n #This is a Riemann sum for the integral \+ of x^2\n #from 0 to 1, with 1000 equal subdivisions.\n \+ #Sum with capital S writes the summation, but\n #doesn't eval uate it. evalf(%) gives its value. " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "int(x^2,x=0 ..1); #this is the exact value of the same integral\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Pi;exp(1);evalf(Pi);evalf(exp(1));i nfinity;\n #some important numbers" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 415 "It is always a good idea to sa ve your maple file periodically. Do this now using the tool bar, usin g the \"save\" option under the \"File\" menu item. The first time you save a new file, and any time you use the \"save as\" option, you wil l be asked to name your file and say where you want to keep it. You n ame it in the left part of the box, being careful to keep the suffix \+ \".mws\" so that Maple knows this file is a " }{TEXT 307 1 "M" }{TEXT -1 5 "aple " }{TEXT 308 1 "W" }{TEXT -1 4 "ork " }{TEXT 309 1 "S" } {TEXT -1 371 "heet. If your directory is new you probably haven't mad e any subdirectories yet (unix command mkdir, in a local window), but \+ as you create more files you may wish to organize where you save them \+ using the tree structure of Unix directories, which you can follow in the right side of your saving box. If you need more help saving you r file see the instructions in the " }{TEXT 306 59 "Introduction to Ma ple V.4 in the Undergraduate Computer Lab" }{TEXT -1 180 " handout at \+ the front of the lab, or ask an assistant. It will probably happen so me time that you will crash Maple long after your last save. This wil l not make you feel happy. " }}{PARA 0 "" 0 "" {TEXT -1 506 " Now we will see how to print a hard copy of our file. First, scroll to \+ somewhere in your worksheet and add some text with the ``T'' menu item . Maybe scroll to the top and put the title ``My first Maple workshee t'' (center it with the menu option on the right side of the toolbar), as well as your name and today's date. When you are doing your Maple \+ projects you will be expected to hand in more than a page of computati ons: You will be expected to add text explanations of what you've been doing. " }}{PARA 0 "" 0 "" {TEXT -1 20 " Now, go to the " }{TEXT 278 4 "file" }{TEXT -1 28 " menu option and choose the " }{TEXT 279 5 "print" }{TEXT -1 72 " option. You get a little printer setup box. I f you then click on the " }{TEXT 281 14 "print command " }{TEXT -1 52 "diamond, followed by ``enter'' or by a click on the " }{TEXT 280 5 "p rint" }{TEXT -1 192 " box at the bottom of the window, a paper copy wi ll come out of one of the printers at the side of the lab. Do this no w. Alternately, if you want to use a different printer, you can use t he " }{TEXT 282 14 "output to file" }{TEXT -1 558 " diamond to create \+ a postscript file which you can then print anywhere, using the appropr iate unix commands. For example, to print a postscript file to the la b printers from a local window, the command would be \"lpr -P b129lab1 \", or \"lpr -P b129lab2\", followed by the return key. You do not p ut in the quote marks, but you are careful to leave spaces exactly as \+ indicated. The lpr stands for line printer, the -P stands for print, \+ and the b129lab1 or b129lab2 are the names of the two printers. If yo u have trouble printing ask a lab assistant for help." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 59 "5) Differential Equa tions, and using Maple's help windows:" }{TEXT -1 367 " So, it looks \+ like Maple might be interesting to use in Calculus, but how do we find out what it can do for us in that subject, or in another subject, say differential equations? An answer for DE's can be obtained by perusi ng your Computing Projects textbook (which you can pick up for free a t the main Math office JWB 233) , but it is also instructive to use th e " }{TEXT 291 4 "Help" }{TEXT -1 107 " directory located at the upper right-hand corner of the maple window. That's what you're going to d o now." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 284 54 "5a) In mapleV6 (and V5) there's an online tutori al: " }{TEXT -1 856 "Click on the ``Help'' box, and then on the choi ce ``New User's Tour''. Probably this tour will superimpose onto your current Maple session. Or maybe you can't see the new tour because i ts hidden behind your current window. In the latter case use the ``wi ndow''menu option to change windows. The tutorial gives examples from many areas of mathematics, including differential equations, which y ou can peruse at your leisure. In this tour you will be able to put yo ur cursor onto any command line, type return, and see what the command does. If you wish you can explore now, or you can continue with the \+ Math 2250 notes below and come back to the tour later. To close the n ew tour (or any other top window), use the ``close'' option inside the ``file'' menu item. To keep the tour open but bring another window t o the front, use ``window'' menu item." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 285 56 "5b) Getting an on-li ne copy of this Math 2250 tutorial: " }{TEXT -1 138 " (This is also \+ the process you will go through to get your project solution templat es.) This xeroxed tutorial is available as either " }{TEXT 286 51 "ht tp://www.math.utah.edu/~korevaar/2250tutorial.txt" }{TEXT -1 6 " or \+ " }{TEXT 310 53 "http://www.math.utah.edu/~korevaar/2250tutorial.mws \+ " }{TEXT 312 630 "The first file is in \"Maple text\" format, and will be easier to read from your browser. The second file is in \"Maple w ork sheet\" format, and will look ugly in your browser, but if you're \+ lucky your own Maple will be able to open it directly once you save it from the internet. The advantage of the second format is that plots, formatting and output are reproduced exactly as in your xerox copy. \+ The disadvantage is that some computers (not the ones in our lab) may \+ not be able to understand it. The advantage of the first format is th at it is more universal. The disadvantage is that Maple output and fo rmatting may be lost. " }{TEXT -1 367 " For now, go to the first web \+ address using your browser window. If you're using netscape as your browser, pick the ``file'' menu option, choosing ``save as''. Unles s you give another name for it you will now save this document with th e same name it has on the web, namely ``2250tutorial.txt''. It creat es a copy of this file in your home directory. Do it. " }}{PARA 0 " " 0 "" {TEXT -1 173 " Now return to your Maple window and use the \+ ``file'' menu item to open``2250tutorial.txt''. In order to open a `` Maple text'' document, which this is, you must chose " }{TEXT 292 4 " open" }{TEXT -1 10 " from the " }{TEXT 293 4 "file" }{TEXT -1 32 " men u option. In the resulting " }{TEXT 294 9 "open file" }{TEXT -1 22 " \+ dialog box go to the " }{TEXT 295 9 " filetype" }{TEXT -1 674 " box at the bottom, click on the triangle to see the list of choices, and use your mouse to choose ``Maple text.'' At this point ``2250tutorial.tx t should appear as a choice in the central box. Click on it with the \+ mouse to highlight it and then click ``OK'' or type ``return''. A cop y of this tutorial should then appear in your Maple window, as a Maple document that you can work in. The copy is not as pretty as your xer ox (the execution groups are all single lines, and the text formatting is not as neat, and some output may be lost), but it is O.K. It has \+ text and it has Maple input. (If you had saved the \".mws\" version y ou could open it as a maple worksheet.)" }}{PARA 0 "" 0 "" {TEXT -1 373 " You can modify the text and input using the toolbar and menu options. You will notice many brackets on the left of the document. \+ These are execution groups. Maple will execute everything in one exe cution group at once, and then move the cursor to the next execution g roup. You can create large execution groups by highlighting sections \+ of a document, going to the " }{TEXT 287 4 "Edit" }{TEXT -1 20 " optio n and picking " }{TEXT 288 24 "join execution groups. " }{TEXT -1 231 "You can remove brackets by highlighting them with the mouse and d eleting them with the delete key or the menu option. And you can inse rt new prompts or new text wherever your cursor is, by using the > or \+ T buttons on your toolbar." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 289 56 "5c) An example of help, related to your first p roject :" }{TEXT -1 117 " You may proceed now whether or not you crea ted a copy of tutorial.txt in your own directory and opened from Maple . " }}{PARA 0 "" 0 "" {TEXT -1 196 " Reading the material on page s 43-44 you notice that you will need Maple to draw a slope field for you. Recall this concept: First order differential equations have t he standard form " }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%# dyG\"\"\"%#dxG!\"\"-%\"fG6$%\"xG%\"yG" }}}{PARA 0 "" 0 "" {TEXT -1 720 "and their solutions are functions y(x) which make this equatio n true. Geometrically that means that if y(x) is a solution, and if y ou plot its graph in the x-y plane, then the slope of the graph at eac h point (x,y(x)) is given by the formula f(x,y(x)). One can therefore work backwards, and this is what we did in section 1.3 of the text, \+ to see what graphs of solutions look like: Construct slope (directions ) fields: at a representative number of points (x,y) in the plane plo t short segments having slopes f(x,y). If you plot enough slopes you \+ will be able to sketch in solution curves to the DE. The resulting pic ture is called a slope field or a direction field. Let's use Maple t o illustrate these ideas." }}{PARA 0 "" 0 "" {TEXT -1 162 " Does M aple have commands to draw slope fields, to solve general differential equations, to graph the solutions?????? OF COURSE!!!!! Let's try to find them." }}{PARA 0 "" 0 "" {TEXT -1 727 " Click on the Help op tion at the upper right corner of your Maple window. A little window \+ opens with further choices. Pick the ``using help'' option. Thus so \+ far you have done help/using help. You may need to change windows (me nu item) to see the help window. Then in the grey box at the top left , click on ``Mathematics''. Then make successive choices so that you \+ create the chain Mathematics/Differential Equations/DEtools/plotting/d fieldplot. This final word is actually a Maple command to make pictur es of direction fields. For many queries, this table of contents appr oach to getting help ususally works best, but you can also use the ``t opics search'' or ``full text search'' options at the top level of hel p." }}{PARA 0 "" 0 "" {TEXT -1 655 " If you now read about the com mand dfieldplot, you will see that it seems to plot direction fields f or first order DE's, among other things. Often you can get an idea of how a command works by skipping to the end of the help file and copyi ng one of the sample commands into your own worksheet. For example, u se your mouse and menu options now to copy the block of commands from the end of the dfieldplot help file and paste them into your workshee t. Then execute the entire block by getting your cursor anywhere into \+ the block and typing ``return''.: The block you are getting should lo ok something like (you don't want to type this in by hand!): " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 395 "with(DEtools):\ndfieldplot([diff(x(t),t)=x(t)*(1-y(t )),diff(y(t),t)=.3*y(t)*(x(t)-1)],\n### WARNING: incomplete quoted nam e; use ` to end the name\n[x(t),y(t)],t=-2..2,x=-1..2,y=-1..2,arrows= LARGE,title=`Lotka-Volterra\nmodel`, color=[.3*y(t)*(x(t)-1),x(t)*(1-y (t)),.1]);\ndfieldplot(diff(y(x),x)=1/2*(-x-(x^2+4*y(x))^(1/2)),y(x),x =-3..3,y=-3..2,\ntitle=`Restricted domain`,color=1/2*(-x-(x^2+4*y))); " }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }} }{PARA 0 "" 0 "" {TEXT -1 833 "You should execute this block. You wil l find that there were three commands in it: the first one, ``with(DEt ools):'' loaded a library of tools for solving differential equations. If you had ended this command with a semicolon instead of a colon Ma ple would have listed all the commands it was loading from this librar y. You can find out more about all of these tools by using the help f iles if you wish. Then there are two plotting commands. The first one creates a ``velocity'' vector field (with fat arrows) in the plane fo r a system of two first order differential equations, apparently calle d the Lotka-Volterra model. This is also sometimes called a predator \+ -prey system, see chapter 6.3, page 360 for a similar picture. The se cond picture is closer to what we want. It is a direction field for t he differential equation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " diff(y(x),x)=1/2*(-x-(x^2+4*y(x))^(1/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xGF*,&F*#!\"\"\"\"#*$-%%sqrtG6#,& *$)F*F.\"\"\"F6F'\"\"%F6F," }}}{PARA 0 "" 0 "" {TEXT -1 515 "The reaso n Maple only draws slopes above a certain parabola, is that the slopes are not defined as real numbers below it, as you can see from the rig ht side of the above formula. Looking at the commands gives you an id ea of the syntax which is used. More specific information can be foun d in the complete help file. For example, the command which draws the last plot clearly has a place to put the DE, a place to put the x and y range of the picture, and there seem to be various other options av ailable as well." }}{PARA 0 "" 0 "" {TEXT -1 294 " To close the he lp files after you've used them, use the``file/close '' sequence in th e toolbar, or the equivalent key stroke given next to it, which is sim ultaneous ``control-F4'' on my workstation. Or you can keep them arou nd and return to your worksheet with the ``window'' menu option." }} {PARA 0 "" 0 "" {TEXT -1 196 " Having played with these commands, \+ could you figure out how to make the slope field part of the picture o n page 44? Of course, the other resource you have is your computing p rojects book. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 290 43 "6) Commands for your actual first proje ct:" }{TEXT -1 160 " Let's try to make the slope field on page 44. U sing my resources I come up with the following commands (see also the \+ Computing Projects manual, pages 3-5.) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "with(DEtools):\ndeqtn:=diff(x(t),t)=.01*x(t) - .0001 *x(t)^2;\ndfieldplot(deqtn,x(t), t=0..1000, x=0..200);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 125 "If you want to inc lude some solution graphs, you can use the command DEplot, and get a p icture like the one on the top of 44:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "DEplot(deqtn,x(t),t=0.. 1000,\{[x(0)=.1],[x(0)=1],[x(0)=10],\n[x(0)=100],[x(300)=150],[x(600)= 150]\},x=0..200,\n colour=`black`,linecolour=`black`,\n \+ arrows=`line`, dirgrid=[30,30]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6acn7$7$$!+Jz8C\"F?F]t7$7$$\"+@#\\^@\"F?Fgs7$$\"+p9ZV:F?F]t7$7$$\"+2o(*f:F? 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You can use the steps out lined on page 43, or you can directly use dsolve: The following comma nds may prove useful for the actual project:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dsolve(deqtn,x(t)); #general solution" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "dsolve(\{deqtn,x(0)=50\},x(t)); #in itial value problem" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "int( 1/(x*(100-x)),x); #compute integrals as part of solving\n #separab le equations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "convert(1/( x*(100-x)),parfrac,x); #do partial fractions." }}}}{MARK "90 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }