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It" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "achieves this by ex ecuting various tasks of considerable complexity quicky and accurately . We shall" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "describe below the elements of version V (Release 4) of this pr ogram; first, however, we need to go" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "through some basics." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 346 8 "Starting" }{TEXT -1 1 " " }{TEXT 258 5 "MAPLE" }{TEXT -1 78 " depends on the version of Windows your computer is equipped with \+ as well as a" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "few other factors. The lab instructor will give you direction s; for the moment we assume that the" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 92 "program is running. What you see on \+ the screen depends again to some extent on the Windows" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "version, but the fol lowing items are always present :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 13 "The title bar" }{TEXT -1 78 ". This is \+ the very first line at the top of the screen and contains the words" } }{PARA 0 "" 0 "" {TEXT -1 81 " MAPLE V RELEASE 4 - [....................]" }}{PARA 0 "" 0 "" {TEXT -1 97 "where the dots reprersent various possibilities . For example, un less you have brought up an old" }}{PARA 0 "" 0 "" {TEXT -1 98 "file, \+ instead of the dots you will see the words UNTITLED (n) where n \+ is a positive integer." }}{PARA 0 "" 0 "" {TEXT -1 79 "This will be fo llowed by a string of icons which is not relevant at this point." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 12 "The menu bar" }{TEXT -1 66 ". This sits right below the title bar and conta ins words like " }{TEXT 261 4 "File" }{TEXT -1 7 " and " }{TEXT 262 4 "Edit" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 263 12 "The tool bar" }{TEXT -1 85 ". This sits right below the menu bar and contains \" button \" controled shortcut s for" }}{PARA 0 "" 0 "" {TEXT -1 38 "operations such as 'save' and 'p rint'." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 15 "The context bar" }{TEXT -1 86 " . This sits right under the tool bar and also contains certain controls about which" }}{PARA 0 "" 0 " " {TEXT -1 44 "we shall talk later if it becomes necessary." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "In " }{TEXT 265 6 "MAPLE " }{TEXT -1 23 "you have the option of " }{TEXT 267 23 "performing calculations" }{TEXT -1 24 " an d also the option of " }{TEXT 268 12 "writing text" }{TEXT -1 6 ". Th e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "way to start a calculation run is to bring via the mouse the arrow \+ to the button [> and click" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 100 "once the left button of the mouse; on t he top left of the screen the prompt sign > will appear" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "followed by th e cursor (blinking vertical line | ) . To start a text run you clic k on the button T . No" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 76 "matter which option you choose, what you then ente r on the screen forms a " }{TEXT 266 9 "worksheet" }{TEXT -1 5 " . \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 269 21 "To retain a worksheet" }{TEXT -1 73 ". \+ After completing a certain amount of work you may wish to retain the " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "works heet. To achieve this you follow the steps:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "1.- Using the mouse move the arrow to the menu bar and click once the left button on the na me" }}{PARA 0 "" 0 "" {TEXT -1 8 " " }{TEXT 270 4 "File" } {TEXT -1 43 " . This will cause a menu to \"drop out\" ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "2.- In this men u select and click on the words " }{TEXT 271 13 "Save As... ; " } {TEXT -1 34 "this will cause a window to appear" }}{PARA 0 "" 0 "" {TEXT -1 49 " containing several boxes called 'fields' ." }} {PARA 0 "" 0 "" {TEXT -1 7 " " }}{PARA 0 "" 0 "" {TEXT -1 82 "3. - Choose a name for the worksheet you want to save and enter it in t he field " }{TEXT 274 9 "File Name" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "4.- Finally click on \+ the button labelled " }{TEXT 272 4 "OK ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Yo ur worksheet has now acquired the desired name which has been duly re corded. Also notice" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 77 "that in the title bar the word UNTITLED(n) has been rep laced by FILENAME.MWS." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "In case you wish to clear the screen for any reason \+ you can:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "5.- Click on " }{TEXT 278 4 "File" }{TEXT -1 63 " and in the men u that drops out select and click on the entry " }{TEXT 277 5 "Close " }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "You may wish to store you r worksheet on a disk rather than the computer itself. To achieve thi s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "you m ust:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "1 .- Insert a disquette ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 37 "2.- Follow steps 1 and 2 above." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "3.- Click on the downpointing arrow of the field " }{TEXT 275 6 "Drives" }{TEXT -1 29 ", causing a menu to drop out." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 27 "4.- Click on the entry " }{TEXT 276 3 "a :" }{TEXT -1 15 " of this menu." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "5.- Follow steps 3 and 4 above. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "Your \+ worksheet has now been recorded under the name you chose on the disque tte. You can close" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 37 "the file by following step 5 above." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 18 "To retrieve a file" }{TEXT -1 73 " (i.e. a recorded work sheet). You can bring on screen any such file by:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "1.- Click on " } {TEXT 280 4 "File" }{TEXT -1 23 " and select the option " }{TEXT 281 4 "Open" }{TEXT -1 26 " from the resulting menu. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "2.- In the window tha t opens click on the downpointing arrow of the field " }{TEXT 282 9 " File Name" }{TEXT -1 13 " ; this will" }}{PARA 0 "" 0 "" {TEXT -1 110 " cause the list of file names to scroll ; when the filenam e you are interested in appears, click on it," }}{PARA 0 "" 0 "" {TEXT -1 37 " and then click on the button " }{TEXT 283 4 "OK . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "The w orksheet you want will appear on the screen and you can now proceed t o modify or extend " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 49 "your worksheet. This procedure is also known as " } {TEXT 286 7 "opening" }{TEXT -1 10 " the file." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 4 "NOTE" }{TEXT -1 22 ". When wo rking on a " }{TEXT 347 5 "named" }{TEXT -1 63 " worksheet you can tra nsfer all additions or corrections to the" }}{PARA 0 "" 0 "" {TEXT -1 110 " copy on file by clicking on the icon of a disquett e showing on the toolbar. In fact it is wise" }}{PARA 0 "" 0 "" {TEXT -1 106 " to keep doing this regularly (maybe every five or ten minutes) to prevent possible loss of" }}{PARA 0 "" 0 "" {TEXT -1 55 " information due to the machine crashing." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 285 59 "To print the contents of a file or of an untitled worksheet" }{TEXT -1 38 ". Assuming a printer is connected \+ to " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "yo ur computer, follow the steps below:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 196 "1.- In dealing with a file ,open it ; this will bring it on the screen. In dealing with an untitled\n \+ worksheet (which necessarily is on the screen) g o to the next step." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 102 "2.- Click on the icon of a printer in the toolbar. \+ A window will then be opened containing various" }}{PARA 0 "" 0 "" {TEXT -1 16 " fields." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 101 "3.- If you want the whole file to be \+ printed enter the desired number of copies in the appropriate" }} {PARA 0 "" 0 "" {TEXT -1 42 " field. Then click on the button \+ " }{TEXT 287 6 "OK. " }{TEXT -1 59 "If you wish only part of the fi le to be printed, follow the" }}{PARA 0 "" 0 "" {TEXT -1 29 " s teps described next." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 288 43 "To see how a file is b roken down into pages" }{TEXT 292 1 "." }{TEXT -1 36 " This can be v ery useful at times." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 88 "1.- Open the file; if dealing with a worksheet alrea dy on the screen go to next step." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "2.- Click on " }{TEXT 290 4 "File" } {TEXT -1 25 " and then on the option " }{TEXT 289 20 "Print Preview.. . . " }{TEXT -1 33 "A window will appear in which the" }}{PARA 0 "" 0 "" {TEXT -1 108 " first page of the file is shown . By cl icking on the appropriate boxes you can see the remaining" }}{PARA 0 " " 0 "" {TEXT -1 102 " pages or enlarge the size of the materia l shown. After inspection is over click on the button" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{TEXT 291 7 "Cancel." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 20 "To terminate the run" }{TEXT -1 75 " . Again this is done in \+ various ways depending on the Windows version as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "well as certain other fac tors. The lab instructor will explain to you. " }}{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 257 "" 0 "" {TEXT 294 19 "CALCULATING WITH " }{TEXT 295 5 "MAPLE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 " " }}{PARA 0 "" 0 "" {TEXT 296 8 "COMMANDS" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 " O The way to tell the computer what to do is by me ans of commands . A " }{TEXT 297 7 "command" }{TEXT -1 15 " is a str ing of" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 7 "symbols" }{TEXT -1 89 ", the most common of which are listed further down. The program is supposed to have been" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "placed in calculating mode (by clicking, if necessary, on the button of the toolbar) and so \+ the" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "sc reen shows the prompt sign > followed by the cursor (blinking | ). \+ The command must be typed" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 101 "to the right of the prompt sign; it is OK if som e space in between is left blank. When we type any " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "symbol, it appears wher e the cursor was, which then moves one space to the right; when the en d of" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 " the line is reached the cursor will skip to the beginning of the next \+ line, and so on. The prompt sign" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "stays where it originally was." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 " O The end of a command is signalled by the symbol " }{TEXT 299 1 ":" }{TEXT -1 14 " (colon) or" }{TEXT 300 4 " ; \+ " }{TEXT -1 28 " (semicolon) . It must not" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "be omitted or else the program \+ will object. The command will be executed when the key " }{TEXT 301 5 "Enter" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "is pressed. If " }{TEXT 348 1 ":" }{TEXT -1 80 " has been use d, we see nothing new on the screen except that a new prompt sign" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "followed \+ by the cursor appear on the next line. If " }{TEXT 349 2 "; " } {TEXT -1 42 " has been used at the end of the command," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "then an output is p rinted on the next line(s) as well as a new prompt and a cursor on the line after" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "that." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 " O It is possible to make a \+ " }{TEXT 350 7 "mistake" }{TEXT -1 62 " while entering a command. The program will detect all errors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 104 "of syntax and sometimes pinpoint the loc ation of the error; it will certainly object in writing. Please" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "note that " }{TEXT 302 38 "it will not help to retype the command" }{TEXT -1 45 " correctly : we must go back and correct it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "This means we must move b ack the cursor at the proper location(s), erase the wrong entries and " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "ente r the correct ones. The cursor can be moved by using the four arrow \+ keys or by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "moving via the mouse the arrow to t he desired spot and then clicking the left button. The" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "key " }{TEXT 303 10 "Backspace " }{TEXT -1 79 " when pushed will move the cursor on e place to the left, erase what was in that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "spot and drag everything writt en to the right of the cursor with it. It is also useful to know that the" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "s pace bar moves the cursor to the right, one space for each depression. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 " O " }{TEXT 304 7 "Symbols" }{TEXT -1 58 ". Here is a list of the most common symbols, as promised." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 8 "Integers " }{TEXT -1 95 ". Type in from top line of keys rather than the side \+ pad to avoid complications related to the" }}{PARA 0 "" 0 "" {TEXT -1 15 " " }{TEXT 306 8 "Num Lock" }{TEXT -1 32 " key. Use the minus symbol " }{TEXT 351 1 "-" }{TEXT -1 22 " for negative sign." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 307 9 "Fractions" }{TEXT -1 91 ". Use slash / between numerator and de nominator. If denominator is negative, enclose" }}{PARA 0 "" 0 "" {TEXT -1 47 " in brackets. In many cases " }{TEXT 308 5 "MAPLE" }{TEXT -1 31 " will simplify to lowest terms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 309 8 "Decimals" } {TEXT -1 23 ". Enter the usual way." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 310 15 "Special numbers" }{TEXT -1 9 ". \+ " }{XPPEDIT 19 1 "Pi" "6#%#PiG" }{TEXT -1 30 " type \+ Pi" }}{PARA 0 "" 0 "" {TEXT -1 34 " \+ " }{XPPEDIT 19 1 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 44 " \+ type exp(1) " }}{PARA 0 "" 0 "" {TEXT -1 81 " \+ " }{TEXT 317 4 "Note" }{TEXT -1 42 ". The letter e wil l not do; it will be" }}{PARA 0 "" 0 "" {TEXT -1 131 ". \+ \+ interpreted as a letter. Also be aware" }}{PARA 0 "" 0 "" {TEXT -1 99 " \+ that " }{XPPEDIT 19 1 "pi" "6#%#p iG" }{TEXT -1 9 " and " }{XPPEDIT 19 1 "exp(1)" "6#-%$expG6#\"\"\" " }{TEXT -1 17 " are treated as" }}{PARA 0 "" 0 "" {TEXT -1 128 " \+ \+ symbols, not numbers; numerical eva-" }}{PARA 0 "" 0 "" {TEXT -1 127 " \+ luation requires use of the s pecial" }}{PARA 0 "" 0 "" {TEXT -1 102 " \+ command \+ " }{TEXT 311 5 "evalf" }{TEXT -1 26 " (see below). " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 " \+ " }{XPPEDIT 19 1 "infinity" "6#%)infinityG" } {TEXT -1 35 " type infinty" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 " \+ " }{XPPEDIT 19 1 "-infinity" "6#,$%)infinityG!\"\"" }{TEXT -1 37 " type - infinity" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 " \+ " }{TEXT 356 1 "i" }{TEXT -1 64 " type I (t his means, of course just " }{XPPEDIT 19 1 "sqrt(-1)" "6#-%%sqrtG6#, $\"\"\"!\"\"" }{TEXT -1 2 " )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 312 9 "Variables" }{TEXT -1 40 ". Any letter ca n be used as a variable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 313 21 "Non-numerical symbols" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 " ( left bracket" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 53 " [ left sq uare bracket" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 51 " \{ left curly bracket" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " \+ " }{TEXT 352 2 " ." }{TEXT -1 24 " period" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " \+ " }{TEXT 353 5 " , " }{TEXT -1 22 " comma" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " \+ " }{TEXT 354 2 " :" }{TEXT -1 23 " colon" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " \+ " }{TEXT 355 1 ";" }{TEXT -1 27 " semicolon" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " \+ )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 " ] right brackets" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " \}" }} {PARA 0 "" 0 "" {TEXT -1 15 " " }}{PARA 0 "" 0 "" {TEXT -1 85 " \" quote - has special meaning , will be defined later" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 64 " ' forward quote \+ - same comment" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 " @ functional composition - w ill be defined later" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 314 20 "Numerical operations" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 " + \+ addition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 " - subtraction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 " * \+ multiplication" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 39 " / division" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 " ^ exponentiation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 40 " ! factorial " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 315 19 "Nume rical relations" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 " = equality" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 " \+ < > inequality (i.e. " }{XPPEDIT 19 1 "<>" "6#0%#%? GF$" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 34 " < less" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 " > \+ greater" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 " < = less than or equal" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 " > = \+ greater than or equal" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 361 92 "Warning: To group together terms in an algeb raic expression use only round brackets, i.e." }}{PARA 0 "" 0 "" {TEXT 362 86 " ( and ) . The square and wiggly bra ckets have a special meaning. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 316 19 "Standard functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 " abs \+ absolute value" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " sqrt square root" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 " exp \+ exponential (base e )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 " log" }}{PARA 0 "" 0 "" {TEXT -1 50 " \+ logarithm to base e" }}{PARA 0 "" 0 "" {TEXT -1 13 " ln" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 46 " log[10] logarithm to base 10 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 " \+ sin " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " cos" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 12 " tan" }}{PARA 0 "" 0 "" {TEXT -1 76 " The standard trig and inverse trig functions" }}{PARA 0 "" 0 "" {TEXT -1 13 " arcsin" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 " arccos" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 " ar ctan" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 " \+ " }{TEXT 318 4 "Note" }{TEXT -1 69 ". All these functions can be evaluated at any point in their domain." }}{PARA 0 "" 0 "" {TEXT -1 117 " \+ Functional notation in required : for example abs(-3) is t he number" }}{PARA 0 "" 0 "" {TEXT -1 117 " \+ |-3| and so has value 3 . Trig functions assum e their arguments to" }}{PARA 0 "" 0 "" {TEXT -1 114 " \+ be in radians ; thus cos(56) is the co sine of an angle measuring" }}{PARA 0 "" 0 "" {TEXT -1 81 " \+ 56 radians, not 56 degrees; cos ( " }{XPPEDIT 19 1 "56^o" "6#)\"#c%\"oG" }{TEXT -1 29 " ) is meaning less. Inverse" }}{PARA 0 "" 0 "" {TEXT -1 104 " \+ trig functions return their values in radians, \+ not degrees." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " O " }{TEXT 319 11 "Expressio ns" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Any string of symbols is considered to be an expression . Not all expre ssions are useful. To" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "obtain a meaningful expression involving numerical c onstants and symbols the obvious rules of" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "operational sequencing must be obs erved . For example the fraction " }{XPPEDIT 19 1 "(a+b)/c" "6#*&,& %\"aG\"\"\"%\"bGF&F&%\"cG!\"\"" }{TEXT -1 19 " must be entered" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "(a+b)/c ; the expression a + b/c will be given the interpretation " } {XPPEDIT 19 1 "a+b/c" "6#,&%\"aG\"\"\"*&%\"bGF%%\"cG!\"\"F%" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " O " }{TEXT 320 8 "Commands" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "Again, an y string of symbols, but his time contained between a prompt and a (se mi)colon , can be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "considered to be a command; not all such \"commands\" ar e useful. If a command cannot be " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 97 "executed the program will say so, in vari ous different ways. A very common way is just to repeat" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "the command. A ty pical command has the format " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 258 "" 0 "" {TEXT -1 4 " " }{TEXT 321 54 "commandname(expres sion1,expression2,.....,expressionN)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "but there are, of course, many exceptio ns. For example just an expression can be considered as a" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "command: if we \+ enter, after the prompt > , the expression 2 + 3 the program wil l return 5 :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "2+3;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Here is an example of a response to a silly comma nd:" }}{PARA 0 "" 0 "" {TEXT -1 15 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "go fly a kite;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 322 16 "Special commands" }{TEXT -1 63 ". One of the most useful possibilities of this program is the " } {TEXT 328 6 "naming" }{TEXT -1 4 " or " }{TEXT 329 9 "assigning" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "command: \+ it allows you togive a name to anything at all so that in future comma nds you have great" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "flexibility. The format is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 " " } {TEXT 323 22 " name := expression" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Here the name " }{TEXT 324 4 "name" } {TEXT -1 39 " has been assigned to the expression " }{TEXT 325 10 "e xpression" }{TEXT -1 8 " . If " }{TEXT 326 10 "expression" }{TEXT -1 12 " is actually" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 15 "a command then " }{TEXT 327 4 "name" }{TEXT -1 80 " is \+ assigned to the result of executing this command. For the rest of the work-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "sheet the program will replace in every occurance " }{TEXT 331 4 "nam e" }{TEXT -1 4 " by " }{TEXT 330 10 "expression" }{TEXT -1 11 ". Exam ple:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a:=(x+1)/(x-1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 96 "The program has executed this command, and it tells \+ us that by simply repeating the command. Let" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "us see if it has really underst ood it; we shall ask it to execute a simple addition:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a+1;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "Yes, it \+ has replaced the variable a by the equivalent expression. True, it has not simplified, but" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "we did not ask it to do so." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Another useful general ty pe command is the quote \" . This actually means ditto in the sense that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 " it tells the program to do something to the last encountered expressio n. For example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "5+3;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Now we as k that this is added to 6:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "%+6 ;" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 69 "If we now use the symbol \" again, its meaning will be 14, not 8 :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "%+6;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "A third useful command is " }{TEXT 332 7 "restart" }{TEXT -1 63 ". Recall that when we assign a name to an expression, this is " } {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 " retained for the whole of the worksheet. But sometimes \+ we want to use the same name for some-" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 55 "thing else, or to start renaming thin gs. The command " }{TEXT 333 7 "restart" }{TEXT -1 37 " tells the p rogram to forget all the" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 54 "assignments made and start from scratch. For exam ple:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "b:=7;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "b;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "This clearly shows that the program replaces the variable b with the value 7. But observe:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "(The program has nothing to respond. So we try to find i f it remembers that b has the value 7 )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "b;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 98 "There! It returns the symbol b because it has erased the previous information that b had the " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "value 7 ; at this point no va lue has been assigned to b any more." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Here \+ are some useful general commands:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 " " }{TEXT 334 18 "factor(polynomial) " }{TEXT -1 23 " This returns " }{TEXT 335 10 "polynomial" } {TEXT -1 179 " written as a product of irreducible factors \+ \+ " }}{PARA 0 "" 0 "" {TEXT -1 72 " with int eger coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 336 18 "expand(expression)" }{TEXT -1 41 " \+ This returns an expanded form of " }{TEXT 337 10 "expression" } {TEXT -1 63 " using standard \+ " }}{PARA 0 "" 0 "" {TEXT -1 56 " \+ identities." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 5 " " }{TEXT 338 20 "simplify(expression)" }{TEXT -1 38 " This returns a simpler form of " }{TEXT 339 11 "expressi on." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " \+ " }{TEXT 340 28 "subs(expr1=expr2,expression)" }{TEXT -1 41 " This returns the result of replacing " }{TEXT 341 5 "expr1" }{TEXT -1 18 " , at each of its" }}{PARA 0 "" 0 "" {TEXT -1 72 " \+ occurances in " }{TEXT 342 10 "expression" }{TEXT -1 5 ", by " }{TEXT 343 5 "expr2" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Exampl es:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "factor(4*x^2-1);" }}}{PARA 0 "" 0 "" {TEXT -1 14 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "factor(2*x^2-1);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "It is obvious that the expression " }{XPPEDIT 19 1 "2*x^2-1" "6# ,&*&\"\"#\"\"\"*$%\"xG\"\"#F&F&\"\"\"!\"\"" }{TEXT -1 18 " factors i nto " }{XPPEDIT 19 1 "(sqrt(2)*x-1)*(sqrt(2)*x+1)" "6#*&,&*&-%%sqrtG 6#\"\"#\"\"\"%\"xGF*F*\"\"\"!\"\"F*,&*&-F'6#\"\"#F*F+F*F*\"\"\"F*F*" } {TEXT -1 10 " ; but the" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 83 "coefficients are not integers, and so the program refused to e xecute the command !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expand((a+b)^2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify((x^ 2-y^2)/(x+y));" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(x=u+1,x ^2-2);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 344 17 "evalf(expression)" }{TEXT -1 46 " This returns a numerical value for " }{TEXT 345 10 " expression" }{TEXT -1 156 " taking into account \+ \+ " }}{PARA 0 "" 0 "" {TEXT -1 112 " previously assigne d values of the variables and the known values of " }}{PARA 0 "" 0 "" {TEXT -1 63 " constants suc h as " }{XPPEDIT 19 1 "Pi" "6#%#PiG" }{TEXT -1 2 " ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf((2+Pi)/(1-3*6.2^2));" }}}{PARA 0 " " 0 "" {TEXT -1 23 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x:=3.4;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf((x*exp(1)+1));" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "A more sophisticated form of this command is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{TEXT 360 26 "evalf(expression, number) " }{TEXT -1 35 " This r eturns the value of " }{TEXT 359 10 "expression" }{TEXT -1 14 " giv en with " }{TEXT 358 7 "number " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 100 " \+ digits. As we see the default value of " }{TEXT 357 6 "number" } {TEXT -1 11 " is 10 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 8 "Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ev alf(Pi,20);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 363 30 "Solving Equations and S ystems" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 99 "In every area of mathematics we need \+ to be able to solve equations and /or systems . The following" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "commands \+ will give us enough solving power for this course." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 364 24 "solve(equation,variable) " }{TEXT -1 8 " Here " }{TEXT 365 9 "equation " }{TEXT -1 26 "is an \+ equation containing " }{TEXT 366 9 " variable" }{TEXT -1 22 " (or eve n the name of" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "the equation) which we want to solve. The programm will \+ produce " }{TEXT 367 27 "exact solutions if possible" }{TEXT -1 10 " . If the" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "equation has the form f(x) = 0, where f(x) is an expression \+ containing x, then we can also use" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 368 13 "solve(f(x),x)" }{TEXT -1 13 " instead o f " }{TEXT 369 15 "solve(f(x)=0,x)" }{TEXT -1 51 " , which sometimes is advantageous. Here are some" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 9 "examples." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "Let's clear Maple's memory again." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(x^2=3,x);" } }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Just c kecking out the alternative form:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(x^2-3,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 38 " Or, using the name of the expression:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "a:=x^2-3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(a,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 "How about something more complicated ?" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(x^3-2*x^2+2,x);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 103 "WOW! If we look carefully we shall see \+ one real root (the one printed first) and two complex roots." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Remember, " }{TEXT 370 1 "I" }{TEXT -1 9 " means " }{XPPEDIT 19 1 "sqrt(-1) " "6#-%%sqrtG6#,$\"\"\"!\"\"" }{TEXT -1 71 " . The above formulation \+ is quite common in MAPLE : if an expression" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "gets to be complicated, t he program introduces intermediate notation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }{TEXT -1 104 "So all this works as predicted ,so far. But what if there is no \+ exact solution? We have all heard that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "the roots of an an algebraic equatio n of degree five or more cannot always be expressed by means" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "of a formula." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(2*x^5+3*x^3-4,x);" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 105 "This answer does not seem very useful; \+ it literally tells us that what we want is a root of the equation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 19 1 "2*z^5+3*z^3-4=0" "6#/,(*&\"\"#\"\"\"*$%\"zG\"\"&F'F'*& \"\"$F'*$F)\"\"$F'F'\"\"%!\"\"\"\"!" }{TEXT -1 78 " . Thank you very much ! However there is a way we can (most of the time)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "get out of this mess, and this is the command " }{TEXT 371 21 "allvalues(expression )" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "allvalues( RootOf(2*_Z^5+3*_Z^3-4));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Success! We get four complex and one real root. \+ But note : they are not given in exact form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "(probably because no such form exists, but possibly because it is inaccesible to the program). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 41 "Another command looks very much the same: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 372 25 "fsol ve(equation,variable)" }{TEXT -1 7 " or " }{TEXT 373 30 " fsolve (e xpression,variable)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 78 "the second one works as before; it computes the solutio n(s) to the equation " }{TEXT 374 16 "expression = 0" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Th is command produces the real roots and prints them in decimal form." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fsolve(2*x^5+3*x^3-4,x);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 " The comp lex roots are not present any more." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "fsolve(x^2-3,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "One of the problems \+ with this command is that usually, when dealing with a non-polynomial \+ equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "not all the roots are listed in the answer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "fsolve(2*sin(x)=x,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 " Here we see that two roots are missing : the root 0 and the roo t -1.895494267 . Obviously" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "these two numbers are roots. The fact that th ere are no others can easily be established by drawing" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "a good graph of both " }{XPPEDIT 19 1 "y=2*sin(x)" "6#/%\"yG*&\"\"#\"\"\"-%$sinG6#%\"xG F'" }{TEXT -1 9 " and " }{XPPEDIT 19 1 "y=x" "6#/%\"yG%\"xG" } {TEXT -1 54 " on the same set of axes. It is clear that since the" } }{PARA 13 "" 1 "" {GLPLOT2D 299 299 299 {PLOTDATA 2 "6'-%'CURVESG6#7S7 $$!\"$\"\"!F(7$$!1+++vq@pG!#:F,7$$!1++D^NUbFF.F07$$!1++]K3XFEF.F37$$!1 ++]F)H')\\#F.F67$$!1++D'3@/P#F.F97$$!1++Dr^b^AF.F<7$$!1++D,kZG@F.F?7$$ !1++Dh\")=,?F.FB7$$!1++DO\"3V(=F.FE7$$!1+++NkzViUC\"F.FT7$$!1++DhkaI6 F.FW7$$!1+++]XF`**!#;FZ7$$!1++++Az2))FfnFhn7$$!1++]7RKvuFfnF[o7$$!1-++ +P'eH'FfnF^o7$$!1****\\7*3=+&FfnFao7$$!1)***\\PFcpPFfnFdo7$$!1)****\\7 VQ[#FfnFgo7$$!1)***\\i6:.8FfnFjo7$$!1b+++v`hH!#=F]p7$$\"1++](QIKH\"Ffn Fap7$$\"1****\\7:xWCFfnFdp7$$\"1,++vuY)o$FfnFgp7$$\"1)******4FL(\\FfnF jp7$$\"1)****\\d6.B'FfnF]q7$$\"1++](o3lW(FfnF`q7$$\"1*****\\A))oz)FfnF cq7$$\"1+++Ik-,5F.Ffq7$$\"1+++D-eI6F.Fiq7$$\"1++v=_(zC\"F.F\\r7$$\"1++ +b*=jP\"F.F_r7$$\"1++v3/3(\\\"F.Fbr7$$\"1++vB4JB;F.Fer7$$\"1+++DVsYw7#F.Fas7$$\"1+ +v)Q?QD#F.Fds7$$\"1+++5jypBF.Fgs7$$\"1++]Ujp-DF.Fjs7$$\"1+++gEd@EF.F]t 7$$\"1++v3'>$[FF.F`t7$$\"1++D6EjpGF.Fct7$$\"\"$F*Fft-F$6#7Y7$F($!1Wt>h ,SAGFfn7$F,$!1!*4?3RS!Q&Ffn7$F0$!1ZQ>T![G`(Ffn7$F3$!1#pzz,cd$)*Ffn7$F6 $!1[q'QdQ\"*>\"F.7$F9$!1`Cw#o^RR\"F.7$F<$!13Gjx1>a:F.7$F?$!1#=/ZGqpp\" F.7$FB$!1HL6rYg<=F.7$FE$!1'eQ#zbe3>F.7$FH$!11'QP`X,(>F.7$$!1++DY!>jo\" F.$!1)=]lLpm)>F.7$FK$!1y+Z&oJm*>F.7$$!1++DJd8k:F.$!1(*4jOc&***>F.7$FN$ !1xr!4)*3\\*>F.7$$!1+++!>eWV\"F.$!1_Tq(oS9)>F.7$FQ$!1N8bf5hf>F.7$FT$!1 S'R$*\\>V*=F.7$FW$!13Hw9/H4=F.7$FZ$!1`CyUW(yn\"F.7$Fhn$!1uka^-ZU:F.7$F [o$!1_9WHBmf8F.7$F^o$!1I/Fm4ix6F.7$Fao$!1&zA`b&o\"f*Ffn7$Fdo$!1Os#R4V= O(Ffn7$Fgo$!1L'*fYNw;\\Ffn7$Fjo$!1Vy()o=$*)f#Ffn7$F]p$!1$GZ>$4=F.7$F\\r$\"1![\"QJ%)o'*=F. 7$F_r$\"1(4Rog(Hi>F.7$$\"1+](=o*pO9F.$\"17IV>]/#)>F.7$Fbr$\"1v0rB%oX*> F.7$$\"1++Dmc>g:F.$\"1M$[qi())**>F.7$Fer$\"1D*G3&GC(*>F.7$$\"1+]PCw,&o \"F.$\"1^gGtw'p)>F.7$Fhr$\"1)[]d3H\"p>F.7$F[s$\"1s)*Gxbh2>F.7$F^s$\"1) )=%\\z\"H==F.7$Fas$\"17%e<6xyp\"F.7$Fds$\"1_jJ9`L^:F.7$Fgs$\"1BL6\"zh[ R\"F.7$Fjs$\"1+tBF'>E>\"F.7$F]t$\"1GBIr(\\z$**Ffn7$F`t$\"1z.+UGFkwFfn7 $Fct$\"1bjk(4*Rs`Ffn7$Fft$\"1Wt>h,SAGFfn-%+AXESLABELSG6$%\"xG%!G-%'COL OURG6&%$RGBGF*F*F*-%%VIEWG6$;F(Fft%(DEFAULTG" 1 2 0 1 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 }}}{PARA 0 "" 0 "" {TEXT -1 97 " graph \+ of sin(x) cannot rise above 1 or drop below -1 , while the graph \+ of the line has gone" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 66 "beyond these values, the two curves cannot intersect a ny further." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "The way to avoid this problem is to first establish an interva l in which the desired root lies and then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "use a variation of this command:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 375 32 "fsolve (expression,variable,a..b)" }{TEXT -1 11 " where " }{TEXT 376 14 " expression = 0" }{TEXT -1 42 " is the equation we want to solve for t he" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "unkn own " }{TEXT 377 9 "variable " }{TEXT -1 6 " and " }{TEXT 378 7 " a \+ , b" }{TEXT -1 62 " are the endpoints of the interval in which we se ek the root." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "fsolve(2*sin(x)=x, x,-3..-1);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fsolve(2*sin(x)=x,x,-1..1); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fsolve(2*sin(x)=x,x,5.. 6);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "In the third case the program was unab le to arrive at a value (since no such root exists) and, to tell" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "us that, \+ it just repeated the command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 102 "It is important to keep in mind that it \+ the interval we enter contains more than one root, the program" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "may not g ive them all to us:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fsolve(2*s in(x)=x,x,-1..3);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "The \+ root 0 was not mentioned. The moral is that we must be careful to s elect the limits of search so" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "that only one root (if any) is contained \+ therein." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "For systems the commands are very \+ similar." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 379 50 "solve(\{expr1,expr2,...exprN\},\{var1,var2,...,varN\})" }{TEXT -1 49 " will produce, if possible, the symbolic (exact)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "solution of the system " }{TEXT 380 42 " exp1 = 0 , expr2 = 0 , ... , exprN = 0" }{TEXT -1 22 " for the unknowns " }{TEXT 381 16 "var1 , var2, ..." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 382 4 "varN" } {TEXT -1 39 " . For example (a very simple one!);" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(\{x^2-1,y^2-4\},\{x,y\});" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Making this a little more complicated:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(\{x^2-y,y^2-4\},\{x,y \});" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "allvalues(RootOf(_Z^2-2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "allvalues(RootOf(_Z^2+2));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "(We recall, of course, that the way to untangle a " }{TEXT 383 6 "RootOf" }{TEXT -1 24 " mess is to hit it with " }{TEXT 384 9 "allva lues" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "For the cases where exact solutions cannot be worked out \+ or are undesirable, we cal use " }{TEXT 385 7 "fsolve:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 386 41 "fsolve(\{expr1,...,exprN\},\{var1,...varN\}) " }{TEXT -1 59 " will give us solutions in decimal form, but not all of " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "them. Fo r example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "fsolve(\{x^2-y,y^2 -4\},\{x,y\});" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "has skipped one real and \+ two complex solutions. Now we know not to expect complex solutions" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "from " } {TEXT 387 6 "fsolve" }{TEXT -1 71 " and we also know that to get more \+ solutions we must specify intervals:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 388 68 "fsolve (\{expr1,...,exprN\},\{var1,...varN\},var1=a1..b1,...,varN=aN..bN)" } {TEXT -1 29 " . Try this in above case:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "fsolve(\{x^2-y,y^2-4\},\{x,y\},x=0..2,y=0..5);" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 101 "It worked. Of course in this simple cas e we know there are no more real solutions. In general, how-" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "ever, thi s can be a source of difficulties." }}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 509 10 "Functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Th e most important point to be made, without which a great deal of confu sion and error can be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "generated, is the distinction that MAPLE makes between a function and an expression. To under-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "stand this consider for example th e function given by the equation " }{XPPEDIT 19 1 "y=x^2" "6#/%\"yG* $%\"xG\"\"#" }{TEXT -1 23 " . Here, of course, x" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "is the independent variab le and y is the dependent variable. MAPLE insists that the functio n be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 " given a name of its own , for example f ; under these circumstance s the program denotes the value" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 75 "of the function by the symbol f(x) . T hus we have the identity f(x) = " }{XPPEDIT 19 1 "x^2" "6#*$%\"xG\" \"#" }{TEXT -1 28 " and it is incorrect to say" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "that we are dealing with \+ the function " }{XPPEDIT 19 1 "x^2" "6#*$%\"xG\"\"#" }{TEXT -1 62 " ( such a temptation is , of course, possible only if we are " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "dealing w ith a single formula function ) . Contrariwise, it is clear by the MA PLE conventions that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 19 1 "x^2" "6#*$%\"xG\"\"#" }{TEXT -1 29 " \+ is an expression ; it is " }{TEXT 423 7 "related" }{TEXT -1 40 " to the function we named f but is " }{TEXT 422 3 "not" }{TEXT -1 19 " the function f ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The way function names are created is described by t he command:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 389 33 "functioname : = var -> expression" }{TEXT -1 12 " Here \+ " }{TEXT 390 11 "functioname" }{TEXT -1 44 " is an expression (usually a single letter) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "which we choose as the name of the function to be defined , " }{TEXT 391 3 "var" }{TEXT -1 35 " is the independent variable , \+ and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 392 10 " expression" }{TEXT -1 89 " is an expression which is supposed to be t he value of the function when the independent" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "variable is given the val ue " }{TEXT 393 3 "var" }{TEXT -1 69 " . The \"arrow\" , which we cou ld interpret to mean \"produces thevalue\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "is obtained by typing the minus sign - and the inequality sign > in succession. For example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->x^2;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "produces the function we mentioned above. To verify this we try:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 " As we see the progr am responds by giving us the correct value of f for x = 3 , which is 9 ;" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 75 "but also note that if we ask the value for x = a we get what w e should:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(a);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Or even for more complicated things:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(x+1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "As already ment ioned, it is very important to see that a very plausible alternative \+ " }{TEXT 424 8 "does not" }{TEXT -1 7 " define" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "a function. Suppose tha t we want to define (or name, if you wish) a function g such that \+ g(x) =" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 19 1 "x^2+2x" "6#,&*$%\"xG\"\"#\"\"\"*&\"\"#F'F%F'F'" } {TEXT -1 15 " and we try :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "g( x):=x^2+2*x;" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The program has given the expression " }{XPPEDIT 19 1 "x^2+2x" "6#,&*$%\"xG\"\"#\"\"\"*&\" \"#F'F%F'F'" }{TEXT -1 50 " the name g(x) ; but if we attempt to f ind the" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "value of this hypothetical function g at the point x = 3 for e xample , we end up nowhere:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(3 );" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 102 " You see the progr am does not recognize that a function g exists, but only that an ex pression g(x)" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 65 "exists. We can find what numerical value g(3) has as \+ follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(x=3,g(x));" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 58 " but still the program will not recognize a function g :" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(5);" }}}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Of course, the moment we use the convention with the arro w, the function emerges:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "g:=x- >x^2+2*x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(3);" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 " The name of the independent variable does not make any d ifference:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(y);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Or perhaps more st rikingly, let us define a function h :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "h:=z->z^2+2*z;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 43 "Clearly h does to z whatever g woul d:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(z);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Thus we see that g and h are different names for the same function, since no matter what \+ value" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 " z has the numbers g(z) and h(z) are the same." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 394 12 "To su mmarize" }{TEXT -1 1 ":" }{MPLTEXT 1 0 2 " " }{TEXT -1 84 "f is the \+ name of a function while f(x) is the name of the expression which is the" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 " value of the function for the value x of the independent variable. \+ The name used for the independent" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 98 "variable is not relevant. Also please no te that the dependent variable need not be given a name !" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 395 4 "Note" }{TEXT -1 87 ". The domain of functions defined this way is what we have calle d the natural domain." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Another command that we may wish to use is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 418 39 "functioname : \+ = unapply(expression,var)" }{TEXT -1 19 " Here as always, " }{TEXT 419 11 "functioname" }{TEXT -1 20 " is the name of the " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "function, " }{TEXT 420 3 "var" }{TEXT -1 35 " is the independent variable and " }{TEXT 421 10 "expression" }{TEXT -1 41 " is the functional value. It does ex actly" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 " the same thing as the previous command, and in some cases may be consi dered advantageous." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g:=unapply (x^2/(x-1),x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(3);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Oops ! We should have been more careful." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Now what if we need to work wit h functions not having a single formula representation? In such a " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "case thi s is the command we use:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 81 "functioname : = \+ var - > piecewise(cond1,expr1,cond2,expr2,...,condN,exprN,expr)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "This is rather tricky, so we shall explain in d etail. Here, as before," }{TEXT 396 12 " functioname" }{TEXT -1 21 " \+ is the name (symbol)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 30 "of the desired function, and " }{TEXT 397 3 "var" } {TEXT -1 57 " is the name of the independent variable. The terms \+ " }{TEXT 398 5 "cond1" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 401 6 "cond2 " }{TEXT -1 74 "etc are possibl e constraints on the independent variable, and the terms " }{TEXT 400 6 "expr1 " }{TEXT -1 2 ", " }{TEXT 399 5 "expr2" }{TEXT -1 6 " , e tc" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "are expressions (usually but not necessarily containing the term " } {TEXT 402 3 "var" }{TEXT -1 35 " ) . The meaning of the righthand" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "side is \+ the following: if " }{TEXT 403 3 "var" }{TEXT -1 13 " satisfies \+ " }{TEXT 404 5 "cond1" }{TEXT -1 37 " then the value of the function \+ is " }{TEXT 405 5 "expr1" }{TEXT -1 1 ";" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " if " }{TEXT 406 3 "var" } {TEXT -1 20 " does not satisfy " }{TEXT 407 5 "cond1" }{TEXT -1 15 " but satisfies " }{TEXT 408 5 "cond2" }{TEXT -1 38 " , then the value \+ of the function is " }{TEXT 409 5 "expr2" }{TEXT -1 1 ";" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " if " } {TEXT 410 3 "var" }{TEXT -1 20 " does nor satisfy " }{TEXT 411 5 "co nd1" }{TEXT -1 4 " or " }{TEXT 412 5 "cond2" }{TEXT -1 16 " but satis fies " }{TEXT 413 6 "cond3," }{TEXT -1 19 " then the value is " } {TEXT 414 5 "expr3" }{TEXT -1 1 ";" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 62 " \+ etc" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "moving from left to right and checking these constrai nts one by one we can figure out the value of the" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "function ; the last prov ision is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " if " }{TEXT 417 3 "var" }{TEXT -1 36 " satisfies none of \+ the constraints " }{TEXT 416 20 "con1,cond2,...,condN" }{TEXT -1 20 " \+ then the value is " }{TEXT 415 4 "expr" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "These constraints usually have the f orm of inequalities. Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f:=x->piecewise(x<0,x^2-1,x> 0,2*x,5);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 28 " Let us \+ compute some values:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(-3);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 5 "f(0);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "Another example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f:=x->piecewise(x<-2,x-1,x>-2 and x<3, x^2,2*x-5); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(-5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }{TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(-2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(1);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 100 "Note that at the point -2 the \+ program returned the value -9 , which is computed according to the" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "default section (last expression) 2x-5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 29 "Actually the default value " }{TEXT 425 5 "expr " }{TEXT -1 58 " is optional. If left out the defauly va lue becomes 0 :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g:=x->piecew ise(x>0,x^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "g(-2);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 99 " MAPLE allows us to compute limits of functions, either o ne sided or double sided. If the limit we" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 1 " " }}{PARA 0 "" 0 "" {TEXT -1 99 "are lookin g for cannot be computed by the program or does not exist we will be t old this; sometimes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 37 "we are given estimates for the limit." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 426 35 "limit(expression,variable = a) " }{TEXT -1 34 " \+ for the two-sided limit" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT 427 36 "limit(expression,variable = a, left)" } {TEXT -1 35 " for the limit from the left" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 428 37 "limit(expression,variabl e = a, right)" }{TEXT -1 33 " for the limit from the right" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The notation is self-evident. It is interestin g to note that we do not enter " }{TEXT 429 2 " " }{XPPEDIT 19 1 "v ar->a" "6#R6#%$varG7\"6$%)operatorG%&arrowG6\"%\"aGF*F*F*" }{TEXT -1 15 " but var = a" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 103 "instead, in spite of the fact that in limits the varia ble never takes the limiting value on. Examples:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "limit(x^2-1,x=0); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "limit(x^2-a,x=0);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "limit((x^2-1)/(x-1),x=1);" } }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 96 " In this last limit we have both numerato r and denominator approach 0 , but the program worked" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "fine." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "limit(x/abs( x),x=0,left);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit(x/ab s(x),x=0,right);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(x /abs(x),x=0);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit(sqrt(x^2-1),x=0);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 57 "Here we got a complex nu mber as an answer (recall that " }{TEXT 430 1 "I" }{TEXT -1 9 " mea ns " }{XPPEDIT 19 1 "sqrt(-1)" "6#-%%sqrtG6#,$\"\"\"!\"\"" }{TEXT -1 4 " ) ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "limit(1/x,x=0);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(1/x,x=0,left);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(1/x,x=0,right);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 101 "Note hoe the program separates the three cases and responds to the first correctly (since the limits" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "from the \+ left and right are not equal)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "limit(x/(x^2+1),x=infinity); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "limit(x/(1-x),x=-infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "limit(x^2/(1-x),x=infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(sin(1+x^2),x=0);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "If you really want to know how much this is, you musy use evalf:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{PARA 0 " " 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "limit(1/(sin(x)-1),x=Pi/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "limit(x^2/cos(x)-1,x=0);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Again we have here both numerator and denominator approac hing zero ; still, the program over-" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "came the difficulty. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Here is an example w ith a multiformula function:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "f:=x->piecewise(x<-2,x^2-1,x>-2and \+ x<3,sin(x),x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "limit(f(x),x =-2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(f(x),x=-2,left); " }}}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit(f(x),x=-2,right);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Th is next example is a little trickier:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(sin(1/x),x=0);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 111 "The program is not telling us that from \+ the left the limit is -1 while from the right it is 1 ; in such a " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "case it would respond " }{TEXT 431 9 "undefined" }{TEXT -1 72 " . It \+ is telling us that the values of the function as x approaches" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "0 oscill ate between -1 and 1 (which is a lot more information than just " }{TEXT 432 9 "undefined" }{TEXT -1 17 " !) . Something" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "similar:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "limit(( 1+x)/(2+cos(1/x)),x=0);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "T here are, of course, several other things we can do with functions, fo r example differentiation and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 104 "integration. We shall see these later o n. At this time it is very useful to deal with a few aspects of" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "graphing. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 433 8 "Plotting " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 95 "There is a large number of commands relat ed to graphing (which is in MAPLEese called plotting)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "We shall now discuss only a few, and as we proceed we shall enlarge our repertory." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The very \+ first thing to see is how to plot the graph of a single formula functi on. Saay we want to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 21 "plot the function " }{XPPEDIT 19 1 "y=x^2" "6#/%\"yG *$%\"xG\"\"#" }{TEXT -1 11 " . We try:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "plot(x^2);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "The prog ram insists that we made an error resulting in no graph at all. Which is quite reasonable if" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "we notice that we asked for the whole graph of the \+ function ! There is not enough room to draw it !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "So we must specify not on ly what expression we are graphing but also the desired range of the " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "indep endent variable. The correct format is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 434 33 "p lot(expression , var = a..b) " }{TEXT -1 32 "Here we want to plot a graph of " }{TEXT 436 12 " expression" }{MPLTEXT 1 0 1 " " }{TEXT -1 17 "which may or may" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 26 "not contain the variable " }{TEXT 435 3 "var" } {TEXT -1 54 " ; the plot will be shown between the stated values " } {TEXT 437 2 "a " }{TEXT -1 6 " and " }{TEXT 438 1 "b" }{TEXT -1 11 " \+ . It is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "essential to keep in mind that expression cannot contain and \+ other variables besides var . Of " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 45 "course letters representing numbers lik e " }{XPPEDIT 19 1 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 10 " and " }{XPPEDIT 19 1 "Pi" "6#%#PiG" }{TEXT -1 15 " are allowed." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "pl ot(x^2,x=-1..3);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 100 " It is important to not ice that the two axes do not have the same length units. We can contr ol that" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 70 " but will postpone the discussion in order to see more examples fi rst." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "plot(3,-1..2);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 91 "This time not only the axes have unequal units, but \+ the origin is different for each axis !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot((x^2-1)/(1+x^3),x= -5..5);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Again note that the axes have uneq ual units. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 " Sometimes we want to use the name of the function rather than \+ the expression." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->x^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "plot(f,-1..3);" }}}{PARA 0 "" 0 "" {TEXT -1 65 " This is of course the exact graph we had befor e. The format is:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT 447 24 " plot(functioname,a..b)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 99 " There \+ are cases where we have to use the name of the function, because it is not given by a single" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "formula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "f:=x->piecewise(x<-2,x^2-1,x>-2and x<3,sin( x),x);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "plot(f,-4..4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "The vertical segments are there because the formula for f(x) changes at exactl.y those point s" }}{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 84 "Many times we need to have several graphs on th e same axes in order to compare them." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 439 40 "plot (\{expr1,expr2,...,exprN\},var = a..b)" }{TEXT -1 11 " Again, " } {TEXT 440 5 "expr1" }{TEXT -1 5 ",...," }{TEXT 441 5 "exprN" }{TEXT -1 32 " are expressions containing the" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "variable " }{TEXT 442 3 "var" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(\{x^2,2*x+1,sin(x)\},x=-2..2);" }}}{PARA 0 " " 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "As a final example we attempt:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot((x^2-6)/(x-3),x=-2..10) ;" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 98 " T his is, of course, quite ridiculous. It happened because (due to the d enominator x - 3 ) the" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "program computed large values for the y-coordina tes of several points and so had to accomodate" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "these points in a reasona bly small space. Thus we must control the vertical coordinate which w e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "can \+ do with the command:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 443 38 "plot(expression,var1= a..b,var2= c..d)" }{TEXT -1 9 " \+ Here " }{TEXT 444 4 "var1" }{TEXT -1 40 " is the independent varia ble showing in" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 445 10 "expression" }{TEXT -1 7 " and " }{TEXT 446 4 "var2" } {TEXT -1 29 " is the dependent variable ." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot((x^2-6)/(x-3),x =-2..10,y=-15..15);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "This is much more reas onable ! The vertical line (the asymptote) is just a bonus." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 448 15 "Differentiation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "There are sever al commands that can be used to compute formulas for the derivatives o f given" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "First suppose that we have estabished a name for the func tion we want to differentiate, say " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 449 12 "functioname " }{TEXT -1 21 ". Then t he command:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 450 15 "D(functioname) " }{TEXT -1 81 " will produce a name for the d erivative function. For example suppose that we " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "define a function f a nd then we differentiate:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f:=x- >x*sin(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(f);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "We see that D(f) is the name of the derivati ve (which was computed according to the product rule!). " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We shall verify th is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "D(f)(Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "D(f)(2.73);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We can also give anothe \+ name to the derivative if we do not want to carry the symbol D(f) t hrough:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "g:=D(f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "g(2.73);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "It is also quite common that we need to differentiate an " }{TEXT 451 10 "expression" }{TEXT -1 32 " which has not been designated" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "as a func tional value. Then we use another command:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 452 26 "diff(expression,variable) " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Note that we must enter the expression itself and also the variable we want to \+ differentiate with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "respect to:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "diff( (x^2-3*x+1)/(x-sqrt(x)),x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "If we omit the \+ variable with respect we are differentiating we get an error message: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "diff((x^2-3*x+1)/(x-sqrt(x)) );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 56 "If we enter a different variable we get \+ what we deserve:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "diff((x^2-3* x+1)/(x-sqrt(x)),y);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "This shows that the program is recognizing only the s tated variable as a true variable, treating all other" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "letters as constants. \+ Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "diff(x^2*y-y^3*x,y); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Here , we see, x is treated as a constant, and y as a variable." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 463 7 "However" }{TEXT -1 85 ", we can bypass this problem by telling the program whic h letters are supposed to be " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 59 "functions of the given independent variab le. For example, " }{TEXT 464 35 "if we want the program to take into " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 93 "acc ount that x is actually a function of y , we enter it as x(y ), not as plain x ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "diff(x(y)^ 2*y-y^3*x(y),y);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We may even wish in this result to eliminate the cumbersome x(y) and replace it by x ; \+ we just" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "use the command for substitution:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "subs(x(y)=x,2*x(y)*y*diff(x(y),y)+x(y)^2-3*y^2*x(y)-y^3*diff(x (y),y));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Sometimes, even though the origina l function has not been given a name, we want a name for its" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "derivativ e. We can try for example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "g:=x ->diff(x^2-3*x,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(x); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(x=1,g(x));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "As we see, the attempt to define a function g as the deriva tive failed ; all we achieved was to " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "define g(x) as an expression co ntaining the variable x . This is clear by our attempt to evaluate " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "g(1) which gave an error message ; but note that our attempt to evaluate g(1) by substitution actually" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 90 "worked. But not all is lost. We can get the derivative as a function using the command " }{TEXT 454 8 "unapp ly " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "wh ich we have seen earlier:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "h:=un apply(diff(x^2-3*x,x),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 9 "Success !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "There are many cases where we are dealing with the e quality of two expressions, and we need to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "differentiate both and set the \+ results equal. We can do this as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 455 22 "diff(equation,variable" } {TEXT -1 13 ") Here, " }{TEXT 456 8 "equation" }{TEXT -1 57 " is \+ the equality relating the two given expressions and " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 " \+ " }{TEXT 457 8 "variable" }{TEXT -1 64 " is the v ariable of differentiation; for example if we wish to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 " \+ differentiate both sides of the relation " } {XPPEDIT 19 1 "x^3+2/x=-sin(x)" "6#/,&*$%\"xG\"\"$\"\"\"*&\"\"#F(F&!\" \"F(,$-%$sinG6#F&F+" }{TEXT -1 11 " we write:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "diff(x^3+2/x =-sin(x),x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 95 "The commands we saw will work even if our function is not a single formula function. Consider :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f:=x->piecewise(x<-1,x^2,x>2 ,x^3-x,3*x+1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(f);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The command " }{TEXT 458 4 "diff" }{TEXT -1 77 " can also work here if written the right w ay, even though there is no single" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 26 "expression to apply it to:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f(x),x);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 97 "One final case, that of implicitely defined funct ions . Recall, this means that we are given an" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "equation connecting two \+ variables say x and y and, without actually solving the equation, \+ we are " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "considering one of these , say y , to be a function of the ot her; implicit differentiation allows us" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "to compute the derivative " } {XPPEDIT 19 1 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 70 " dir ectly. The price we pay for this convenience is that the result" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "will cont ain both variables x and y . The command is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 453 34 "implicitdiff(equation, v ar2,var1) " }{TEXT -1 18 " Here " }{TEXT 459 9 "equation \+ " }{TEXT -1 39 " is the equation that connects the two " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "variables, " } {TEXT 460 4 "var1" }{TEXT -1 59 " is the variable which we consider t o be independent and " }{TEXT 461 5 "var2 " }{TEXT -1 21 " is the var iable that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "we consider to be a function of " }{TEXT 462 19 "var1. For ex ample:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "implicitdiff(x^2*y^3= sq rt(x-2*y),y,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "This is the derivative o f y with respect to x . But we may wish to think of x as a funct ion of y" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "in which case the command for " }{XPPEDIT 19 1 "dx/dy" "6#*&%#d xG\"\"\"%#dyG!\"\"" }{TEXT -1 12 " would be:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "implicitdiff(x^2*y^3=sqrt(x-2*y),x,y);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 465 13 " Integration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 " We first treat indefinite integrals. The command is simple." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 466 13 "int(expr ,var)" }{TEXT -1 14 " Here " }{TEXT 467 5 "expr " }{TEXT -1 52 " is the function expression we want to integrate ; " }{TEXT 468 4 "var " }{TEXT -1 16 " is the variable" }}{PARA 0 "" 0 "" {TEXT -1 49 " (usually contained in " }{TEXT 469 4 "e xpr" }{TEXT -1 45 " ) with respect to which we are integrating." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "int(x^2,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(x^2);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 98 " Observe that in the last command we did not indicate t he independent variable over which we are" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 " integrating \+ - hence the objection." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "int(x^2*y+x*y^2,y); " }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 96 " Al so observe that the program does not add the constant of integration . We are assumed to do" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "this ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Now for definite integrals. The command is equally simple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 470 20 "int(expr,var = a..b)" }{TEXT -1 85 " As we \+ see this is identical to the previous command except that the range of " }}{PARA 0 "" 0 "" {TEXT -1 83 " \+ the variable of integration is now indicated." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "int(sin(x),x=1..3) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "int(x^2/(x^2+1),x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 261 "" 0 "" {TEXT 471 11 "PROGRAMMING" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 " There is a large number of programming commands in MAPLE . We s hall now discuss what is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 98 "many times called a \"do loop \", which facilitat es executing the same command over and over again." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Later we shall encounter \+ others also." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 472 9 "Do loop." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The command has the following format:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 53 "for var1 from var 2 by var3 to var4 do " }{XPPEDIT 18 0 "P[var1]" "6#&%\"PG6#%% var1G" }{TEXT -1 4 " od" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 7 "Here " }{TEXT 474 28 "var1, var2, var3, var 4" }{TEXT -1 24 " are variables and " }{TEXT 475 1 " " }{XPPEDIT 18 0 "P[var1]" "6#&%\"PG6#%%var1G" }{TEXT -1 34 " an executionable c ommand which " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "contains var1 . To facilitate the explanation we sha ll use the abbreviations : k for var1" }}{PARA 0 "" 0 "" {TEXT -1 135 " \+ m f or var2" }}{PARA 0 "" 0 "" {TEXT -1 136 " \+ \+ q for var3" }}{PARA 0 "" 0 "" {TEXT -1 136 " \+ n for var 4" }}{PARA 0 "" 0 "" {TEXT -1 57 "Under these conditions the program w ill do the following:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "First it executes " }{XPPEDIT 18 0 "P[m]" "6#&%\"PG6 #%\"mG" }{TEXT -1 86 " since we indicate that the smallest value of \+ k is m . Then it checks whether" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "m + q < n ; if so it executes " }{XPPEDIT 18 0 "P[m+q]" "6#&%\"PG6#,&%\"mG\"\"\"%\"qGF(" }{TEXT -1 68 " , otherwise it stops . Then it checks if m + 2q < n in which " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "case \+ it executes " }{XPPEDIT 18 0 "P[m+2*q]" "6#&%\"PG6#,&%\"mG\"\"\"*& \"\"#F(%\"qGF(F(" }{TEXT -1 81 " ; otherwise it stops. It continues increasing the value of k by q till" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "we reach the largest value m + tq which is < n ; at that point it executes " }{XPPEDIT 18 0 "P[m+tq]" "6#&%\"PG6#,&%\"mG\"\"\"%#tqGF(" }{TEXT -1 12 " and stops ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Examp le:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "for k from 1 by 1 to 6 do k^2+k od; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "for k fro m 1 by 2 to 8 do k^2+k od;" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 103 " This is fine. We got answers except that we have no record of the values of k used, only the" }}{PARA 0 " " 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "values of the command " }{XPPEDIT 18 0 "P[k]" "6#&%\"PG6#%\"kG " }{TEXT 476 1 " " }{TEXT -1 75 " . In case of a very large number \+ of such values it is useful (and some-" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 71 "times essential) to give a name to e ach result showing the value of " }{TEXT 473 4 "var1" }{TEXT -1 30 " that was used to obtain it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 95 "This name must contain the variable var 1 (in our case the letter k ) . A standard MAPLE " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "procedure is to us e square brackets, which MAPLE translates into subscripts. Thus if we want to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "obtain " }{XPPEDIT 18 0 "P[k]" "6#&%\"PG6#%\"kG" }{TEXT -1 22 " we write P[k] . " }{TEXT 477 60 "Please note that P(k) for e xample or Pk will not work." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "P[k]:=k^2+k;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "for k f rom 1 by 1 to 9 do P[k]:=k^2+k od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for k from 1 by 1 to 6 do P[k]:=P[k-1]+k od;" }}} {PARA 0 "" 0 "" {TEXT -1 6 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 " Oops! we forgot to tell the program what the initial value " }{XPPEDIT 18 0 "P[0]" "6#&%\"PG6#\"\"!" }{TEXT -1 12 " is . So:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "P[0]:=3;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "for k from 1 by 1 to 6 do P[k]:=P[k-1]+k od;" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Here is a nother example. We want to keep relacing the variable, say x , in so me expression by " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "several times over . We can do it thusly:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Q[0]:=x ^3-x+1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for k from 1 by \+ 1 to 5 do Q[k]:=subs(x=x^2,Q[k-1]) od;" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 " A equally useful command is the so \+ called " }{TEXT 478 7 "if/then" }{TEXT -1 47 ", which allows you do e xecute different actions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 43 "depending on the value(s) of some variable." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 479 7 "if/then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The com mand has four versions, but we present only two." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 33 "if cond1 then expr1 els e expr2 fi" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Here " }{TEXT 480 6 "cond1 " }{TEXT -1 71 "is a relation (equali ty or inequality) containing some variable , and " }{TEXT 481 5 "expr 1" }{TEXT -1 3 " , " }{TEXT 482 6 " expr2" }{TEXT -1 5 " are" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "expressio ns which may or may not contain this variable. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 483 19 "Very important note" } {TEXT -1 31 " : the variable showing in " }{TEXT 484 5 "cond1" } {TEXT -1 50 " must first be given a value using the assignment" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "symbol \+ " }{TEXT 485 4 ": = " }{TEXT -1 29 " ; then we enter the above " } {TEXT 486 8 "if...fi " }{TEXT -1 58 " statement . In case we do not \+ specify the value of that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 98 "variable an error message will be returned. The p rogram tests the value of the variable against " }{TEXT 487 5 "cond1 " }{TEXT -1 1 ";" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "if this condition is satisfied then the program executes \+ " }{TEXT 488 5 "expr1" }{TEXT -1 9 " . If " }{TEXT 490 5 "cond1" } {TEXT -1 24 " fails then the program" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "executes " }{TEXT 489 5 "expr2" } {TEXT -1 27 " . Here are some examples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "if x>0 then x^2 fi;" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "We forgot to give the value of x ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x:=5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "if x>0 then x^2 fi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "x:=-3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "if x>0 then x^2 fi;" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 99 "As we se e the command was not executed ; this is because we forgot the second \+ part of the statement" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "which tells the program what to do in case the value o f our variable violates the condition." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x:= -3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "if x>0 then x^2 else x^3 fi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "if x>0 then x^3 el se x^2 fi;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 " This is just about all one can say about the first level " } {TEXT 491 8 " if/then" }{TEXT -1 35 " command. There is a more sophi s-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "tic ated one which allows several mutually exclusive conditions to enter \+ the calculation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "This is the format:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 492 84 "if cond1 then expr1 elif cond2 then expr2 elif cond3 then expr3 ........else e xpr fi" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The dots represent any number of intermediate statements of the form \+ " }{TEXT 493 23 "elif condk then exprk" }{TEXT -1 2 " ;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "as before " } {TEXT 494 6 "condk " }{TEXT -1 48 " is a relation containing some vari able , and " }{TEXT 495 5 "exprk" }{TEXT -1 31 " is an expression wh ich may or " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "may not contain this variable. Keep in mind that this variabl e must be the same in all " }{TEXT 496 6 "condk " }{TEXT -1 9 " and t hat" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 497 27 "c ond1 , cond2 , ... , condn" }{TEXT -1 31 " must be mutually exclusiv e ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "x:= 4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "if x<0 then x+1 elif x>3 then x^2 else x^3 fi;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "x:=-1;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "if x<0 then x+1 elif x>3 then x^2 else x^3 fi; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "if x<0 then x+1 elif x>3 then x^2 else x^ 3 fi;" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 498 22 "The addition command ." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 93 "There are many cases, such as the est imate of areas or volumes, where we need to write out a " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "compact formula fo r a very long sum . For example, think of " }{XPPEDIT 18 0 "1/(1+ 1)+2/(4+1)+3/(9+1)+5/(25+1)" "6#,**&\"\"\"\"\"\",&\"\"\"F&\"\"\"F&!\" \"F&*&\"\"#F&,&\"\"%F&\"\"\"F&F*F&*&\"\"$F&,&\"\"*F&\"\"\"F&F*F&*&\"\" &F&,&\"#DF&\"\"\"F&F*F&" }{TEXT -1 2 " +" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "....+ " }{XPPEDIT 18 0 "36/(1296+1 )" "6#*&\"#O\"\"\",&\"%'H\"F%\"\"\"F%!\"\"" }{TEXT -1 74 " as such a sum albeit rather simple. Hand writing does this as follows:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " } {XPPEDIT 18 0 "sum(k/(k^2+1),k=1..36" "6#-%$sumG6$*&%\"kG\"\"\",&*$F' \"\"#F(\"\"\"F(!\"\"/F';\"\"\"\"#O" }{MPLTEXT 1 0 2 " " }{TEXT -1 75 ". For use in a MAPLE calculation we need a special command . Here \+ it is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 21 "sum (expr, var = a..b)" }{TEXT 500 17 " . Here " }{TEXT -1 4 "exp r" }{TEXT 501 46 " is an expression containing the variable " } {TEXT -1 3 "var" }{TEXT 502 244 " whose range \+ \+ \+ " } {TEXT -1 15 " " }}{PARA 0 "" 0 "" {TEXT -1 46 " \+ " }{TEXT 504 10 "is from " } {TEXT -1 1 "a" }{TEXT 505 8 " to " }{TEXT -1 44 "b . The program calculates the values of " }{TEXT 499 6 " expr " }{TEXT -1 6 " for \+ " }}{PARA 0 "" 0 "" {TEXT -1 44 " \+ " }{TEXT 506 7 "var = a" }{TEXT -1 12 " , then for " }{TEXT 507 10 "var = a+1" }{TEXT -1 12 ", then for " }{TEXT 503 9 "var = a+ 2" }{TEXT -1 26 " etc until it reaches the " }}{PARA 0 "" 0 "" {TEXT -1 54 " value " }{TEXT 508 7 "var = b" }{TEXT -1 47 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 113 " \+ Then it adds up all these values and returnes the res ult. Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "sum(k,k=1..5 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "sum(k^2,k=1..3);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sum(1/n,n=2..30);" }}}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 31 " Can you verify that??" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 " A slightly mo re complicated case:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f:=x->x/(x +1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 10 " " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "for k from 1 by 1 to 13 do x[k]:=k-1 od;" }}}{PARA 0 "" 0 "" {TEXT -1 8 " " }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sum(f(x[j]),j=1..13);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 103 " No te that the variable k we used in the previous command , the variab le that counts the terms of" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 " the sum, has been replaced in the last comma nd by j , because in the course of executing the" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "previous commands the p rogram assigned the value 13 to the letter k , and so we cannoy us e it " }}{PARA 0 "" 0 "" {TEXT -1 23 "again as a variable. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 " As as \+ final example we compute a Riemann sum " }{XPPEDIT 18 0 "sum(f(c[k] )*xxx[k],k=1..n)" "6#-%$sumG6$*&-%\"fG6#&%\"cG6#%\"kG\"\"\"&%$xxxG6#F- F./F-;\"\"\"%\"nG" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f:=x->x*sin(x);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "x[0]:= 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "for k from 1 by 1 to \+ 10 do x[k]:=x[k-1]+1/10 od; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "c[0]:=(x[0]+x[1])/2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for j from 1 by 1 to 9 do c[j]:=(x[j]+x[j+1])/2 od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sum(f(c[n])*1/10,n=0..9);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 " This is of course an approximate value for the integra l " }{XPPEDIT 18 0 "int(x*sin(x),x=1..2)" "6#-%$intG6$*&%\"xG\"\"\"- %$sinG6#F'F(/F';\"\"\"\"\"#" }{TEXT -1 31 " . We compute its exact va lue." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "int(x*sin(x),x=1..2);" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 " As we see the approxim ate value is not that bad." }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{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 0 "" }}}{MARK "1565" 0 } {VIEWOPTS 1 1 0 1 1 1803 }