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1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 256 1 {CSTYLE "" -1 -1 "Time s" 1 24 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 37 "Mathematics Department M aple Tutorial" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "The aim of this worksheet is to introduce you to the \+ basic commands and uses of Maple. The emphasis is placed on " }{TEXT 299 20 "learning through use" }{TEXT -1 43 ", so explanatory text is k ept to a minimum." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 58 "Command lines in Maple are in red and are precede d by a '" }{TEXT 369 1 ">" }{TEXT -1 12 "' symbol. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 219 "Read carefully throug h the text, then place the cursor anywhere in the next red Maple comma nd line and press Enter to execute that command.. The cursor will t hen be automatically moved to the next Maple command line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "Contents" {TEXT -1 8 "Co ntents" }}{PARA 0 "" 0 "" {HYPERLNK 17 "Help in Maple" 1 "" "Help in M aple" }}{PARA 0 "" 0 "" {HYPERLNK 17 "Basic commands" 1 "" "Basic comm ands" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "" 1 "" "" }{HYPERLNK 17 "Plotting functions" 1 "" "Plotting functions" } }{PARA 0 "" 0 "" {HYPERLNK 17 "Calculus of functions" 1 "" "Calculus o f functions" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {HYPERLNK 17 "The SOLVE command" 1 "" "The SOLVE command" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "" 1 "" "Calculus of functions" } {HYPERLNK 17 "Manipulating expressions" 1 "" "Manipulating expressions " }}{PARA 0 "" 0 "" {HYPERLNK 17 "Defining functions" 1 "" "Defining f unctions" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "Help in Maple" {TEXT -1 13 "Help in Maple" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 423 "You need to remember virtually nothing a bout Maple commands -- all this information is contained in the on-li ne Help. Should you require more information on a topic there are thr ee ways to open a Help window on it. (Close or minimise the Help windo w after you have finished.) You can go to Help->Topic Search (from th e Help menu at the top of the window). Equivalently type a question mark followed by the topic, e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "?entering;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "or highlight a " }{TEXT 315 5 "word " }{TEXT -1 44 "with your cursor, th en go to Help->\"Help on " }{TEXT 316 4 "word" }{TEXT -1 22 "\" in the top menu bar." }}{PARA 0 "" 0 "" {TEXT -1 220 "Here you will find det ailed explanations of most of Maple's commands, including optional par ameters. Most helpful are the examples at the bottom of each Help win dow, which you can copy and paste into your own worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "At the beginning of each section you will see. \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 175 "This clears Maple's internal memory as i f we had just started Maple. This is to ensure our calculations are n ot being affected by previous workings with the same variables. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "Basic commands" {TEXT -1 0 "" }{TEXT 256 14 "Basic commands" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "W ith the cursor placed anywhere in the red Maple command line, press En ter to execute the command. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 76 " Do this with each of the command lines w hich follow and observe the result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 8 "Addition" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "1 + 1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 14 "Multiplication" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "4 * 2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 8 "Division" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "10 / 3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Maple " } {TEXT 286 19 "preserves fractions" }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 38 "To express this as a decimal, use the " }{TEXT 287 5 "e valf" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Here, the " } {TEXT 288 1 "%" }{TEXT -1 69 " symbol indicates that the previous expr ession must be used by Maple." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 309 5 "Note " }{TEXT -1 5 "th at " }{TEXT 308 0 "" }{TEXT 300 44 "each Maple code line ends with a s emi-colon." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "**Your cursor should now be in the next Maple input line which is blank. Type in the Maple command line to add together the f ractions " }{XPPEDIT 18 0 "671/12+289/15;" "6#,&*&\"$r'\"\"\"\"#7!\"\" F&*&\"$*GF&\"#:F(F&" }{TEXT -1 87 " and observe what happens. Don't \+ forget the semi-colon at the end of your input line." }}{PARA 0 "" 0 " " {TEXT -1 3 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Maple has given the answer as a fr action with the lowest common denominator. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "**Now insert a new " }{TEXT 372 15 "Execution Group" }{TEXT -1 107 " after this paragraph as follo ws. Place the cursor anywhere in this paragraph and click on the icon \"[>\" " }{TEXT 357 0 "" }{TEXT -1 94 "which appears roughly in the \+ middle of the toolbar at the top of the screen (or hold down the " } {TEXT 303 4 "Ctrl" }{TEXT -1 15 " key and press " }{TEXT 304 1 "j" } {TEXT -1 90 "). Use this new Maple input line to obtain a decimal app roximation of the fraction with " }{TEXT 302 9 "evalf. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 305 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 23 "Trigonometric functions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Evaluate the following \+ trigonmetric expressions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sin(2*Pi/3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "cos(45); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Maple will not evaluate " }{TEXT 354 10 "cos(45), " }{TEXT -1 11 "because it " }{TEXT 353 0 "" }{TEXT -1 33 "expects angle arguments to be in " }{TEXT 289 9 "radians " }{TEXT -1 45 "(so it will accept dec imals or multiples of " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 4 ") . " }}{PARA 0 "" 0 "" {TEXT -1 87 "**Edit the last Maple input line b y replacing 45 degrees with its radian equivalent " }{XPPEDIT 18 0 "Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 43 " , and re-evaluate \+ the cosine function. " }}{PARA 0 "" 0 "" {TEXT -1 72 "What happens i f you replace 45 by 45.0? How did Maple interpret 45.0?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "tan (Pi/2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Why is there an error message after " }{TEXT 306 9 "tan( Pi/2)" }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 35 "Exponent ial and logarithm functions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The exponential function is given \+ by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "exp(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The constant " }{TEXT 307 1 "e" }{TEXT -1 15 " is obtained as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ex p(1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "exp(1.0);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 262 5 "Note:" }{TEXT -1 186 " The use of \+ decimal numbers (as opposed to integers) forces Maple to calculate the decimal approximation of the expression. You can force a numerical evaluation where necessary using " }{TEXT 310 9 "evalf. " }{TEXT -1 49 "Furthermore, you can specify the desired accuracy" }{TEXT 311 3 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(exp(1),20); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The inverse to the exponential is the natural logarithm:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ln(2.0);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 11 "ln(exp(1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Also denoted as " }{TEXT 312 3 "log" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "log(2.0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Thus the " }{TEXT 270 4 "log " } {TEXT -1 13 " function is " }{TEXT 263 3 "not" }{TEXT -1 102 " to base 10,.(which you might be accustomed to on calculators). The base 10 l ogarithm is obtained by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "log[10](2.0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 23 "The infinite quanit ity " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "1/infinity;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 264 15 "Complex numbers" }}{PARA 0 "" 0 "" {TEXT -1 57 "M aple assigns the square root of negative one the letter " }{TEXT 317 1 "I" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "(3 + 4*I) + (5 - 9*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "sqrt(-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "I*I;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 9 "Variables" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 66 "You can assign names to specific values or expressions \+ as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a:=2;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "c:=a*3+I;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "What happens if \+ you omit the * in the above expression? **Try this (edit the above l ine) and see what happens. " }}{PARA 0 "" 0 "" {TEXT -1 19 "(Maple in terprets " }{TEXT 375 2 "a3" }{TEXT -1 32 " as a new unassigned, vari able.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "circle1:=x^2+y^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Note the freedom you have in ch oosing names, as long as your name is not one already used by Maple." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 313 20 "Vectors and Matrices" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "You can also assign names to a vector or a matrix. These can be defined as examples of " } {TEXT 267 9 "arrays. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "A vector is given as an " }{TEXT 355 5 "array" } {TEXT -1 6 " of a " }{TEXT 356 5 "list " }{TEXT -1 33 "of 3 numbers. \+ Note the brackets." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "v:=array([0, 1, 0]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "A " }{TEXT 314 3 "3x 3" }{TEXT -1 62 " matrix is given as a list of lists. Notice all the \+ brackets." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 46 "A:=array([ [1, 1, 1],[0, 2, 1],[-1, 1, -1] ]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "T hese obey the familiar rules of matrix and vector arithmetic. However , note that matrix multiplication is denoted by " }{TEXT 268 2 "&*" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "b:=A &* v;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(b);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Note the use of " }{TEXT 269 6 "evalm " }{TEXT -1 34 " to obtain a numeric al evaluation." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Maple has a linear algebra package with routines for dealing with matrices, the " }{TEXT 271 6 "linalg" }{TEXT -1 32 " pac kage. Open it by evaluating" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Do not wor ry about the Warning message, you will see a few others as you call up on new packages. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 318 4 "Note" }{TEXT -1 95 " the use of a colon instead of a semi -colon here. This tells Maple not to display the output. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 86 "**Replac e the colon above with a semi-colon to see what functions are included in the " }{TEXT 272 6 "linalg" }{TEXT -1 37 " package. Can you find \+ the commands " }{TEXT 290 6 "matrix" }{TEXT -1 5 " and " }{TEXT 291 6 "vector" }{TEXT -1 62 "? These provide another way of defining vector s and matrices." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Of particular interest to us are the matrix inverse \+ command," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "invA:=inverse(A );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "which we can check satisfie s the required property," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalm(A &* invA);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "and the vec tor inner product (scalar or dot product). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dotprod(b, v);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {HYPERLNK 17 "Return to contents." 1 "" "C ontents" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "Plott ing functions" {TEXT -1 0 "" }{TEXT 273 18 "Plotting functions" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "W ith the cursor placed anywhere in the red Maple command line, press En ter to execute the command. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 76 " Do this with each of the command lines w hich follow and observe the result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 "A graph can pr ove invaluable in providing insight into a problem. The plot syntax i s tremendously simple: tell Maple the function and its domain. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot( t^2 - 4*t + 5 , t=0..5 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 6 "Remark" }{TEXT -1 15 ". The variable " }{TEXT 275 1 "t" }{TEXT -1 27 " in the command line is a " }{TEXT 276 5 "dummy" }{TEXT -1 86 " variable in that it holds no special significance: it can therefore \+ be replaced with " }{TEXT 292 1 "s" }{TEXT -1 2 ", " }{TEXT 293 1 "x" }{TEXT -1 61 " or any other variable name. As long as you are consist ent, " }{TEXT 277 24 "within each command line" }{TEXT -1 41 ", there \+ should be no problems with Maple." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "You may wish to plot more than one function simultaneously. This is done by specifying a " }{TEXT 294 4 "list" }{TEXT -1 13 " of functions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot([sin(x)/x, cos(x)], x=0..Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "You may wish to restrict the vertical range of the output," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot( [sin(x)/x, cos(x)], x=0..Pi, y=0..1);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "an d specify the colours:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "p lot([sin(x)/x, cos(x)], x=0..Pi, y=0..1, colour=[black,blue]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Th e transition to 3-dimensional graphs is not too difficult either." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot3d(sin(s^2*t),s=-1..1,t= 0..Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 249 "Now, using the left m ouse button, click and drag the pointer on the graph. You are able to rotate the picture to get a better view of the surface. Right-clicki ng on the plot gives you a list of options to add, or remove, informat ion from the plot." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 201 "**Look up \"plot\" in Help by using your cursor \+ to highlight the word \"plot\" in any of the above command lines and t hen clicking on \"Help\" in the menu bar. The second line should read \"Help on \"plot\". " }{TEXT 373 4 "WAIT" }{TEXT -1 471 " until you' ve finished reading this paragraph, then choose this. When the Help w indow appears scroll down to see examples of plot commands. Copy the \+ \"Multiple plots (in a set or list)\" example and paste it into the bl ank input line which follows to see what it does. Close the Help wind ow as you would any other window (e.g. click on the X box of the inner upper left corner), or toggle between windows (e.g. using Window in t he menu bar) to return to this worksheet.." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {HYPERLNK 17 "Return to contents." 1 "" "Contents" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "Calculus of func tions" {TEXT -1 21 "Calculus of functions" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "We \+ can use Maple to perform integration and differentiation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 297 11 "Integration" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 78 "Integration can be done between numerical limits, yielding a numerical answer:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "int( 3*x^2 + 6*x + 3, x=0..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "Or symbolically to g ive an expression, with the variable of integration supplied as an arg ument." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "int( 3*x^2 + 6*x \+ + 3, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "**See what happens if you replace only the final \"" }{TEXT 319 1 "x" }{TEXT -1 1 "\"" } {TEXT 374 1 " " }{TEXT -1 87 "in the above command line with a differe nt letter. Why does Maple's answer make sense?" }}{PARA 0 "" 0 "" {TEXT -1 68 "Note that Maple's answer does not include a constant of i ntegration." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 48 "You can use symbolic integration to define the " } {TEXT 298 16 "anti-derivative " }{TEXT -1 36 "and assign names to the \+ expressions," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "g := 3*x^2 \+ + 6*x + 3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "G := int(g,x) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "For single points, you can evaluate the anti-derivative expression using " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "eval(G,x=1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "as well a s plotting it over a range of values:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot( [g, G], x=0..10, colour=[blue, red]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Be careful that the variable " }{TEXT 334 1 "x" }{TEXT -1 62 " in the co mmands matches the one used in your definitions of " }{TEXT 335 1 "g " }{TEXT -1 5 " and " }{TEXT 336 2 "G." }{TEXT -1 3 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 337 15 "Differen tiation" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Differe ntiation is straightforward using the " }{TEXT 320 4 "diff" }{TEXT -1 70 " command with the variable of differentiation supplied as an argum ent:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(sin(x),x);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "A gain, you can assign names to the expressions. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "F:= (x+1)^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "F1:=diff(F,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "F2:=diff(F,x,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "F3:=diff(F,x,x,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot ([F,F1,F2,F3],x=-1..2);" }}}{EXCHG {PARA 0 "" 0 "Multivariate calculus " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 " " 0 "" {HYPERLNK 17 "Return to contents." 1 "" "Contents" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "The SOLVE command" {TEXT -1 17 "The SOLVE command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Here \"solving\" can \+ describe a number of different operations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve(y = m*x + b , x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "You can also use " }{TEXT 279 5 "solve" }{TEXT -1 38 " to determine the roots of an equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "quintic := x^5+2*x^4-6*x^3-4*x^2+13*x-6;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(quintic=0,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Which is \+ easily verified since this quintic equation is just" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "expand( (x+2)*(x+3)*(x-1)^3 );" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "In deed, you can picture this using the " }{TEXT 295 4 "plot" }{TEXT -1 8 " command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot( quinti c, x=-4..4, y=-20..25);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Observe the triple root at " }{TEXT 296 1 "x" }{TEXT -1 3 "=1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 10 "Data types" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "You are already familiar wi th some data types: scalar values and arrays such as vectors and matri ces. Maple also has " }{TEXT 338 5 "lists" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "l1:=[5,1,2,2,3,3,0,0];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "and " }{TEXT 339 4 "sets" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "s1:=\{5,1,2,2,3,3,0,0\}; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 340 5 "Note " }{TEXT -1 64 "the use of different brackets \+ and the difference between them. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 40 "You can extract an entry from either one " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "l1[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "s1[4];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 341 7 "Remark." }{TEXT -1 25 " Vector s and arrays use " }{TEXT 342 5 "lists" }{TEXT -1 57 " in their defini tions, since the order is very important." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 281 5 "solve" }{TEXT -1 19 " command expects a " } {TEXT 282 3 "set" }{TEXT -1 39 " of equations to solve for a specified " }{TEXT 283 3 "set" }{TEXT -1 185 " of unknowns, so you can use it t o solve systems of equations in several unknowns. Since it expects se ts, you must use the curly brackets, \{\}, to collect the equations an d the unknowns" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "By default, " }{TEXT 285 5 "solve" }{TEXT -1 94 " wi ll solve for all three unknowns, so you do not need to specify the unk nowns in the command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "eq us1 := \{u+v+w=1 , u+2*v-w=-2, u+3*v=4\};" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "solve(equs1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "The following non-linear equations have more than one solution. \+ These are given as two separate sets," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "equs2 := \{x+2*y=3, y+1/x=1\};" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "sols2:=solve(equs2);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 17 "We thus have two " }{TEXT 301 4 "sets" }{TEXT -1 14 " o f solutions," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "sols2[1];" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "sols2[2];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "You can verify these results using the " } {TEXT 284 4 "eval" }{TEXT -1 10 " command, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eval(equs2, sols2[1]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "and" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eval(equ s2, sols2[2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "**Insert a new Maple input line after this paragrap h and use Maple to find the stationary points of the function " } {XPPEDIT 18 0 "p(q) = q^3-3*q^2+24*q-65;" "6#/-%\"pG6#%\"qG,**$F'\"\"$ \"\"\"*&F*F+*$F'\"\"#F+!\"\"*&\"#CF+F'F+F+\"#lF/" }{TEXT -1 17 ". You will need " }{TEXT 351 5 "diff " }{TEXT -1 5 " and " }{TEXT 352 6 "so lve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {HYPERLNK 17 "Return to contents." 1 "" "C ontents" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "Manip ulating expressions" {TEXT -1 24 "Manipulating expressions" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Try some commands which allow you to mani pulate expressions and think about what they do." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Expanding brackets " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expand((u+v)^4);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "expand((x+1)*(x+2)*(x-9));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "expand((x+1)/(x-1));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "expand(tan(2*x));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Co nversely" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "factor(x^3+7*x^ 2-21*x-27);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "factor(x/(x+ 2) - 3/(x+2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "factor((x ^3-y^3)/(x^4-y^4));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {HYPERLNK 17 "Return to contents." 1 "" "Contents" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "Defining functio ns" {TEXT -1 18 "Defining functions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 10 " In Maple " }{TEXT 358 9 "functions" } {TEXT -1 26 " are defined as follows. " }{TEXT 370 4 "Note" }{TEXT -1 79 " the arrow notation which is a dash \"-\" followed by the great er than sign, \">\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f:= x->x^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "g:= x->x-1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Maple then allows simple " }{TEXT 321 11 "composition" }{TEXT -1 41 " of these functions using the `@' symbo l:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(f@g)(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "which, as we would expect, is different from:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(g@f)(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 80 "The same arrow notation is used to define functions of more than one variable. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "paraboloid1:= (x,y) -> x^2 + y^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 329 7 "NOTE: " }{TEXT -1 29 "Maple differentiates between " }{TEXT 326 10 "functions," } {TEXT -1 23 " as defined above, and " }{TEXT 322 11 "expressions" } {TEXT -1 9 " such as " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "F: = x^3-1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "This assigns the " } {TEXT 328 6 "name, " }{TEXT -1 1 "\"" }{TEXT 331 1 "F" }{TEXT -1 1 "\" " }{TEXT 330 2 ", " }{TEXT -1 20 " to the expression " }{XPPEDIT 18 0 "x^3-1;" "6#,&*$%\"xG\"\"$\"\"\"F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "You can evaluate t he " }{TEXT 348 10 "function f" }{TEXT -1 4 " at " }{TEXT 323 1 "x" } {TEXT -1 5 "=2 by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "or obtain the " }{TEXT 349 11 "expression " }{TEXT -1 15 "of the func tion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "However, if you try to do the same for th e " }{TEXT 366 10 "expression" }{TEXT -1 2 ", " }{TEXT 367 1 "F" } {TEXT -1 1 "," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "yo u get something very confusing, because the expression " }{TEXT 324 1 "F" }{TEXT -1 34 " is only defined for the symbol \"" }{TEXT 332 1 "x" }{TEXT -1 55 "\" and is not a function. You can get around this u sing" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "eval(F,x=2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "In contrast to this, " }{TEXT 325 1 "f" }{TEXT -1 18 " is defined as a " }{TEXT 327 5 "rule " }{TEXT -1 8 "so that " }{TEXT 343 5 "f(x) " } {TEXT -1 45 "produces an expression in the input variable " }{TEXT 344 1 "x" }{TEXT -1 19 ". Thus evaluating " }{TEXT 345 4 "f(2)" } {TEXT -1 27 " produces 4, and evaluating" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "produces the correct expression in the variable \"" }{TEXT 333 1 "y" }{TEXT -1 4 "\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 368 15 "It is important" }{TEXT -1 100 " to recognise this difference \+ when you use Maple commands. The Maple commands in this tutorial use \+ " }{TEXT 346 12 "expressions " }{TEXT -1 28 "to represent real functio ns." }}{PARA 0 "" 0 "" {TEXT -1 53 "In particular, to integrate, diffe rentiate or plot a " }{TEXT 359 5 "Maple" }{TEXT -1 1 " " }{TEXT 350 9 "function," }{TEXT -1 18 " you must use the " }{TEXT 347 18 "express ion f(x) " }{TEXT -1 84 "in the command. This has the advantage tha t you will see that your arguments match." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Note the difference in the command s below for the " }{TEXT 360 13 "function, f, " }{TEXT -1 8 "and the \+ " }{TEXT 361 15 "expression, F, " }{TEXT -1 14 "above: Using " } {TEXT 362 9 "functions" }{TEXT -1 67 " has the advantage of seeing and being able to change the variable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f(x),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(F,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(F,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Yo u should understand exactly why the last result was 0 and " }{TEXT 363 3 "not" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3*t^2;" "6#*&\"\"$\"\"\"*$ %\"tG\"\"#F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "int(f(x),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(F,x);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "One more example using \+ " }{TEXT 371 9 "functions" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "h:= t -> (t-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "H:=t->int(h(t),t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Since both of these are " } {TEXT 364 9 "functions" }{TEXT -1 34 ", you can use any variable in th e " }{TEXT 365 13 "expressions, " }{TEXT -1 30 "as long as you are con sistent." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "h(x); H(x);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot([h(s), H(s)], s=0..4, \+ colour=[plum,cyan]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {HYPERLNK 17 "Retur n to contents." 1 "" "Contents" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 180 "This is the end of the basic tutoria l. You should now be able to use Maple in your mathematics courses. Y ou can use Help to find information on topics not covered, and experim ent!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "If you have any comments or questions on this tutorial, please send an email to maple@maths.lse.ac.uk." }}}}{MARK "1 0 2" 43 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }