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1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1 " -1 260 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 4 1 0 1 0 1 2 0 1 }{PSTYLE "Heading 1" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 6 6 1 0 1 0 1 2 0 1 }{PSTYLE "Heading 1" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 6 6 1 0 1 0 1 2 0 1 }{PSTYLE "Normal " -1 263 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 1 "I" }{TEXT 256 20 "ntroduc tion to Maple" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Maple is a compu ter algebra system. It can be used to" }}{PARA 257 "" 0 "" {TEXT -1 41 "help you with manipulations of equations" }}{PARA 257 "" 0 "" {TEXT -1 24 "find numerical solutions" }}{PARA 257 "" 0 "" {TEXT -1 31 "display the results graphically" }}{PARA 257 "" 0 "" {TEXT -1 52 " develop, implement and debug mathematical algorithms" }}{PARA 257 "" 0 "" {TEXT -1 51 "write documents containing mathematical expressions " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 204 "The Maple software package i s available for several operating systems, among them UNIX platforms, \+ including Linux, and Microsoft Windows (TM). There are student version s available at an affordable price. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Maple fulfills the tasks of a" }}{PARA 257 "" 0 "" {TEXT -1 46 "symbolic calculator with graphing capabilities" }}{PARA 257 "" 0 "" {TEXT -1 29 "flexible programming language" }}{PARA 257 "" 0 "" {TEXT -1 34 "word processor with formula editor" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "All these features are integrated into Maples " }{TEXT 257 10 "worksheet " }{TEXT -1 200 "concept. A worksheet is a Maple doc ument, usually stored as a file with the \".mws\" extension. It may co ntain passive text, hyperlinks, plots and formulas as well as \"execut ion groups\" with Maple code." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 271 "It is easiest to get to know Maple by experimenting with a worksheet. This document itself is available as a Maple worksheet. If you have n ot done so, yet, we reccomend you install Maple on your computer and o pen this document from within Maples graphical user interface." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 270 "It is possible to structure Maple worksheets by dividing them into sections. The content of a section c an be hidden or shown by clicking on the grey [+] or [-] button at the top left corner of a section. Try it out by clicking on the [+] butto n of the following section:" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "W orking with execution groups" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Yo u can recognize execution groups by the [> prompt on the left and the \+ (usually) red font." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Move the \+ cursor somewhere into the execution group below (using mouse or cursor keys) and press enter." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "4 +5;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "The result of the calculat ion should appear below in blue." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Try inserting a new execution group by clicking the [> button or c hoosing " }{TEXT 260 6 "Insert" }{TEXT -1 1 "-" }{TEXT 261 15 "Executi on Group" }{TEXT -1 1 "-" }{TEXT 262 12 "After Cursor" }{TEXT -1 22 " \+ from the menu. Type \"" }{TEXT 259 9 "25^2*4^4;" }{TEXT -1 18 "\" and \+ press enter." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Right click on t he appearing number. A context menu will offer you various functions t hat can be applied to the result. Try " }{TEXT 258 15 "integer factors " }{TEXT -1 13 " for example." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "Of course you can copy, cut and paste and drag/drop almost everything visible on the maple worksheet using your mouse or keyboard in the us ual manner." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 5 "Note:" }}{PARA 257 "" 0 "" {TEXT -1 84 "Within execution groups, manual line breaks c an be created by pressing shift-enter. " }}{PARA 257 "" 0 "" {TEXT -1 45 "Execution can be interrupted by pressing the " }{TEXT 264 5 "stop \+ " }{TEXT -1 6 "button" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "The Maple help system" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 296 "Executing the following execution group will directly lead you into Maples help system. On the top sect ion of the help window, you can navigate through the hierarchy of Mapl es help topics. Use the hyperlinks and the button bar of the main wind ow to browse through it like in the world wide web. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "If you already know what you are looking for (e.g. for help on Maple worksheets), you may use the following syntax" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "? worksheet" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Maple expre ssions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "For simple algebraic ex pressions, all you have to remember is to always write down the operat or (e.g. " }{TEXT 0 3 "5*x" }{TEXT -1 12 " instead of " }{TEXT 0 2 "5x " }{TEXT -1 34 " ). Terminate each statement with " }{TEXT 0 1 ";" } {TEXT -1 67 " if you want Maple to display the result, otherwise termi nate with " }{TEXT 0 1 ":" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 114 "Maple will automatically simplify your input to a certain degr ee and display the result in standard math notation." }{TEXT 397 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 398 9 "Examples:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "2/15+1/15;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "3*(12*x - 6)^(2+3) - 4*x = 9*x - 4*x + 2; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "2+3:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "(output suppressed)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "2*I^3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "(" } {TEXT 0 1 "I" }{TEXT -1 33 " denotes the complex unit number)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Variables and Assignments" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "We ha ve already used the variable " }{TEXT 0 1 "x" }{TEXT -1 77 " in the ex amples above. Variable names may contain the following characters: " } {TEXT 0 16 "a..z,A..Z,_,0..9" }{TEXT -1 35 " and must not begin with a number. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The following statem ent will assign the equation " }{TEXT 0 13 "2*x = 5*x - 9" }{TEXT -1 17 " to the variable " }{TEXT 0 6 "my_var" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "my_var := 2*x = 5*x - 9;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "While the " }{TEXT 0 1 "=" }{TEXT -1 58 " operator symbolizes an equality, the assignment operator " } {TEXT 0 2 ":=" }{TEXT -1 161 " actively stores something in a variable . Once you assigned something to a variable, you can retrieve the vari able's contents from anywhere within the worksheet:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "my_var;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "We are working with two variables now: " }{TEXT 0 6 "my_var" } {TEXT -1 38 ", to which we assigned something, and " }{TEXT 0 1 "x" } {TEXT -1 63 ", which we use as a mere symbol. If we now assign somethi ng to " }{TEXT 0 1 "x" }{TEXT -1 27 ", we will get a new result:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x := 4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "my_var;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "It may sometimes become necessary to prevent Maple from evaluatin g an expression. This can be done by enclosing it in a pair of '...'" }{TEXT 0 0 "" }{TEXT -1 6 ", e.g." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "3*(12*x - 6)^(2+3) - 4*x = 9*x - 4*x + 2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "'3*(12*x - 6)^(2+3) - 4*x = 9*x - 4*x + 2';" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Clearing Variables" }}{PARA 0 "" 0 "" {TEXT -1 194 "Once \+ you have defined a variable, Maple will remember its value during your entire working session. If you want to overwrite the variable with a new value, you can simply make a new assignment." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x:=3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "my_var;" }}}{PARA 0 "" 0 "" {TEXT -1 100 "Sometimes you will want to \"clear\" a variable in memory so t hat you can use it in a new situation. " }}{PARA 0 "" 0 "" {TEXT -1 277 "In order to get x to be a general variable again we must first \" clear\" (i.e. erase from Maple's memory) our earlier value for x. This is accomplished by entering x:='x'; The single quotes prevent the e xpression x on the right hand side form evaluation, resulting in the n ame " }{TEXT 0 1 "x" }{TEXT -1 30 " instead of the assignd vlaue " } {TEXT 0 1 "3" }{TEXT -1 1 ":" }{TEXT 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Execute the next two lines to see how this works." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x:='x';" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "my_var;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "You may also unassign all variables of the worksheet by t yping" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Note: Maple is case sensitive. So for example Maple consi ders k and K to be different variables.. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "k := 1/4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "k;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "K;" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Numerical Calculations" }}{PARA 0 "" 0 "" {TEXT -1 119 "In this section you will learn how to use Mapl e to do some standard numerical calculations. Maple's ability to produ ce " }{TEXT 276 13 "exact answers" }{TEXT -1 1 " " }{TEXT 277 39 "in a ddition to numerical approximations" }{TEXT -1 44 " gives you more opt ions in solving problems." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Doin g Exact Arithmetic with Maple" }}{PARA 0 "" 0 "" {TEXT -1 126 "Using M aple to do numerical computations is very straightforward. Just enter \+ the numerical expression and end the line with a " }{TEXT 266 9 "semic olon" }{TEXT -1 11 ". Pressing " }{TEXT 267 7 "[Enter]" }{TEXT -1 97 " will then execute the line and the result will be displayed in blue i n the center of the screen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 0 "" }}{PARA 0 "" 0 "" {TEXT 287 10 "Example 1:" }{TEXT -1 73 "\nF irst some simple calculation. Click anywhere in the red line and press " }{TEXT 268 7 "[Enter]" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "2+4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "12* 34567890;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "134^39;" } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 0 "" }} {PARA 0 "" 0 "" {TEXT 288 10 "Example 2:" }{TEXT -1 67 "\nMaple can ca lculate with fractions without converting to decimals:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "3/5 + 5/9 + 7/12;" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 271 10 "Example 3:" }}{PARA 0 "" 0 "" {TEXT -1 51 "To enter the square root of a number use sqrt( ) :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "sqrt(24);" }}}{PARA 0 "" 0 "" {TEXT -1 34 "Notice that Maple has simplified " }{XPPEDIT 18 0 "sqrt(24)" " 6#-%%sqrtG6#\"#C" }{TEXT -1 126 " but has left the answer in exact for m. In the next section you will learn how to get a decimal approximati on for this number." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 289 10 " Example 4:" }}{PARA 0 "" 0 "" {TEXT 273 0 "" }{TEXT -1 95 "Unlike your calculator, Maple gives you the exact answer when applying trigonomet ric functions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sin(5*Pi/3 );" }{TEXT -1 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }{TEXT 272 0 "" } }{PARA 0 "" 0 "" {TEXT 290 11 "Example 5:\n" }{TEXT -1 147 "Maple has \+ many special purpose commands for working with numbers. e.g. if we hav e an integer and want to factor it into primes we can use Maple's " } {TEXT 265 11 "ifactor( )" }{TEXT -1 12 " command. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ifactor(31722722304);" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 10 "Example \+ 6:" }}{PARA 0 "" 0 "" {TEXT -1 55 "To calculate and display a sequence of numbers use the " }{TEXT 275 7 "seq(..)" }{TEXT -1 73 " command. H ere we calculate the squares of the first 100 natural numbers." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(k^2,k=1..100);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Numerical App roximations using the " }{TEXT 286 8 "evalf( )" }{TEXT -1 8 " command " }}{PARA 0 "" 0 "" {TEXT -1 248 "Recall that in the previous section \+ we asked Maple to add three fractions and the result was also displaye d as a fraction. This sort of exact arithmetic is very useful but ther e are times when we prefer an answer in decimal form. The Maple comman d " }{TEXT 278 11 " evalf( )" }{TEXT -1 29 " performs this task for us. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 10 "Example 1:" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(3/5+5/9+7/12);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 285 10 "Example 2:" }}{PARA 0 "" 0 "" {TEXT -1 73 "We can also do this with the item 'Approximate...' from the context menu:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "sqrt(3/5+5/9+7/12);" }{TEXT -1 0 " " }}}{PARA 263 "" 0 "" {TEXT -1 10 "Example 3:" }}{PARA 0 "" 0 "" {TEXT -1 123 "If we want fewer or more digits of accuracy than the def ault number which is 10 digits we can add an extra argument to the " } {TEXT 284 9 "evalf( )" }{TEXT -1 25 " command as shown below. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "w:=4*(3+Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(w);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(w,4);" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 291 10 "Example 4:" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 129 "If you enter numbers with a de cimal point Maple automatically gives decimal results. Compare the res ults of the two lines below. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "sqrt(34);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "sqrt(34.0 );" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 283 0 "" }{TEXT -1 0 "" }} }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "Note: Quick reference to last o utput" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 0 " " }{TEXT -1 209 "There will be many times when using Maple that you wi ll string together a sequence of computations. Rather than giving a na me to each result as you go along, you can use the percent sign ( % ) \+ to refer to the " }{TEXT 280 34 "last expression computed by Maple." } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "Here is an example of ho w it works. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "3/5+5/9+7/12 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 22 "Algebraic Calculations" }}{PARA 0 "" 0 "" {TEXT 292 30 "M aple is a \"C.A.S\" , i.e. a " }{TEXT 306 1 "C" }{TEXT 307 8 "ompute r " }{TEXT 308 1 "A" }{TEXT 309 7 "lgebra " }{TEXT 310 1 "S" }{TEXT 311 294 "ystem. This means that Maple knows almost every rule of algeb ra that you know (and probably some more!). As you progress through Ca lculus, Differential Equations and Linear Algebra you will find that M aple also has the essential operations from those subjects built into \+ its large command set. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 204 "In this section you will learn how to enter an alg ebraic expression and substitute values in for the variables. Then you will learn the commands that allow you to expand, factor and simplify expressions. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{SECT 1 {PARA 4 "" 0 "subs( )" {TEXT -1 4 "The " }{TEXT 312 7 "subs ( )" }{TEXT -1 9 " command\n" }{TEXT 300 10 "Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 55 "For our first example let's start with the expressio n " }{XPPEDIT 18 0 "3*x^2+8" "6#,&*&\"\"$\"\"\"*$%\"xG\"\"#F&F&\"\")F &" }{TEXT -1 28 " and assign it the name W. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "W:=3*x^2+8;" }}}{PARA 0 "" 0 "" {TEXT -1 77 "Sup pose now that you want to substitute the value 4 for x in the expressi on " }{XPPEDIT 18 0 "3x^2+8" "6#,&*&\"\"$\"\"\"*$%\"xG\"\"#F&F&\"\")F &" }{TEXT -1 48 ". The quickest way to do this is to use Maple's " } {TEXT 299 7 "subs( )" }{TEXT -1 36 " command. Here's what it looks lik e:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(x=4,W);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 10 "Example \+ 2:" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 302 7 "subs( )" }{TEXT -1 49 " command works equally well with symbolic values:" }}{PARA 0 " " 0 "" {TEXT -1 17 "To replace x by " }{XPPEDIT 18 0 "5+2*u" "6#,&\" \"&\"\"\"*&\"\"#F%%\"uGF%F%" }{TEXT -1 19 " in the expression " } {XPPEDIT 18 0 "3*x^2+8" "6#,&*&\"\"$\"\"\"*$%\"xG\"\"#F&F&\"\")F&" } {TEXT -1 64 " execute the following line. In this case we label the re sult M." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "W:=3*x^2+8;" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "M:=subs(x=5 +2*u,W);" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 303 10 "Example 3:" }}{PARA 0 "" 0 "" {TEXT -1 21 "You ca n also use the " }{TEXT 304 7 "subs( )" }{TEXT -1 229 " command to sub stitute a value into an equation. This is the sort of thing you might \+ want to do to test whether a particular value \"satisfies\" the equati on. In the next few lines we substitute different values into the equa tion " }{XPPEDIT 18 0 "x^3-5*x^2+7*x-12=0" "6#/,**$%\"xG\"\"$\"\"\"*& \"\"&F(*$F&\"\"#F(!\"\"*&\"\"(F(F&F(F(\"#7F-\"\"!" }{TEXT -1 55 " . A re any of these values a solution to the equation?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Note we use \" := \" to \+ assign the name and just \"=\" for the equation itself." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn:=x^3-5*x^2+7*x-12=0;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(x=3,eqn);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "The " }{TEXT 318 9 "expand( )" } {TEXT -1 9 " command " }}{PARA 0 "" 0 "" {TEXT -1 8 "Use the " }{TEXT 296 10 "expand( ) " }{TEXT -1 124 "command is to distribute products o ver sums. It can also be used to expand trigonometric and other more g eneral functions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 315 10 "Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 11 "Apply the \+ " }{TEXT 316 9 "expand( )" }{TEXT -1 28 " command to the expression \+ " }{XPPEDIT 18 0 "(x+2)^2*(3x-3)*(x+5)" "6#*(,&%\"xG\"\"\"\"\"#F&F',&* &\"\"$F&F%F&F&F*!\"\"F&,&F%F&\"\"&F&F&" }{TEXT -1 2 " ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "k:=(x+2)^2*(3*x-3)*(x+5);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(k);" } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 317 10 "Exampl e 2:" }}{PARA 0 "" 0 "" {TEXT -1 63 "Maple applies some familiar trigo nometric identities to expand " }{XPPEDIT 18 0 "sin(2*x)" "6#-%$sinG6# *&\"\"#\"\"\"%\"xGF(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(2*x)" "6 #-%$cosG6#*&\"\"#\"\"\"%\"xGF(" }{TEXT -1 2 " ." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "expand(sin(2*x));" }}}}{SECT 1 {PARA 4 "" 0 "f actor( )" {TEXT -1 4 "The " }{TEXT 313 9 "factor( )" }{TEXT -1 8 " co mmand" }}{PARA 0 "" 0 "" {TEXT -1 69 "Conversely, Maple can compute th e factors of a polynomial expression:" }}{PARA 0 "" 0 "" {TEXT 294 10 "Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 23 "Factor the expression: " } {XPPEDIT 18 0 "3*x^2-10*x-8" "6#,(*&\"\"$\"\"\"*$%\"xG\"\"#F&F&*&\"#5F &F(F&!\"\"\"\")F," }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "factor( 3*x^2-10*x-8);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 295 10 "Ex ample 2:" }}{PARA 0 "" 0 "" {TEXT -1 7 "Factor " }{XPPEDIT 18 0 "sin^2 x -cos^2x:" "6#,&*&%$sinG\"\"#%\"xG\"\"\"F(*&%$cosGF&F'F(!\"\"" } {TEXT -1 2 " ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "factor((si n(x))^2-(cos(x)^2));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "simplify( )" {TEXT -1 4 "The " }{TEXT 314 11 "simplify( )" }{TEXT -1 8 " command" }}{PARA 0 "" 0 "" {TEXT 297 10 "Example 1: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 293 24 "Consider the expressi on " }{XPPEDIT 18 0 "cos(x)^5 + sin(x)^4 + 2*cos(x)^2 - 2*sin(x)^2 - c os(2*x)" "6#,,*$-%$cosG6#%\"xG\"\"&\"\"\"*$-%$sinG6#F(\"\"%F**&\"\"#F* *$-F&6#F(F1F*F**&F1F**$-F-6#F(F1F*!\"\"-F&6#*&F1F*F(F*F9" }{TEXT -1 116 " . Maple can apply identities to simplify many lengthy mathematic al expressions, such as trigonometric expressions. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "V:=cos(x)^5 + sin(x)^4 + 2*cos(x)^2 - 2*sin (x)^2 - cos(2*x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(V);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 258 "" 0 "" {TEXT 298 10 "Example 2:" }}{PARA 0 "" 0 "" {TEXT -1 65 "The simplify( ) command can be used to \+ add rational expressions. " }}{PARA 0 "" 0 "" {TEXT -1 17 "Rewrite the sum " }{XPPEDIT 18 0 "1/(x+1)+x/(x-1)" "6#,&*&\"\"\"F%,&%\"xGF%F%F%! \"\"F%*&F'F%,&F'F%F%F(F(F%" }{TEXT -1 23 " as a single fraction." }} {PARA 259 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "M:=(1/(x+1))+(x/(x-1));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(M);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 261 "" 0 "" {TEXT 338 8 "Graphing" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 266 "In this sect ion you will learn how to plot the graph of a function defined by an e xpression. Other topics covered include: combining the graphs of sever al expressions into a single plot, plotting points, and combining diff erent plot structures into a single picture." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 1 {PARA 4 "" 0 "plot( )" {TEXT -1 28 "Plotting an Expression: the " }{TEXT 335 7 "plot( )" }{TEXT -1 8 " command" }}{PARA 0 "" 0 "" {TEXT 322 10 "Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 11 "We use the " }{TEXT 337 7 "plot( )" }{TEXT -1 31 " c ommand to plot the graph of " }{XPPEDIT 18 0 "3*x^2-8" "6#,&*&\"\"$\" \"\"*$%\"xG\"\"#F&F&\"\")!\"\"" }{TEXT -1 5 " for" }{TEXT 319 2 " x" }{TEXT -1 20 " between - 5 and 5 ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(3*x^2-8,x=-5..5);" }}}{PARA 0 "" 0 "" {TEXT -1 29 "Notice that Maple scales the " }{TEXT 320 1 "y" }{TEXT -1 32 "-axi s automatically, choosing a " }{TEXT 321 1 "y" }{TEXT -1 74 "-scale th at shows the entire graph corresponding to the specified domain. " }} {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 324 10 "Exampl e 2:" }}{PARA 0 "" 0 "" {TEXT -1 87 "Automatic scaling is a useful fea ture but there are times when you may want to set the " }{TEXT 323 1 " y" }{TEXT -1 102 " range manually. For example automatic scaling isn't appropriate for graphs with vertical asymptotes. " }}{PARA 0 "" 0 "" {TEXT -1 122 "Compare the next two graphs. Notice how we have set the \+ limits for y to the interval [-20,20] in the second plot command. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(x/(x-2),x=-5..5);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(x/(x-2 ),x=-5..5,y=-20..20);" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 325 10 "Example 3:" }}{PARA 0 "" 0 "" {TEXT -1 201 "To show more than one graph in the same pictur e, list them in square brackets [ ] separated by commas. You can spec ify the colors for each function by adding a color option at the end o f the command. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot([cos (x),x^2],x=-1..5,y=-4..4,color=[blue,black]);" }{TEXT -1 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Plotting points" }}{PARA 0 "" 0 " " {TEXT 326 10 "Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 171 "To graph a set of points list them in the plot command. Note the commas and reme mber square brackets for each point and an extra pair of square bracke ts surround the list." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plo t([ [2,3],[-2,5],[1,-4] ],x=-7..7,y=-7..7,style=point);" }{TEXT -1 0 " " }}}{PARA 0 "" 0 "" {TEXT 327 10 "Example 2:" }}{PARA 0 "" 0 "" {TEXT -1 66 "Changing style to \"line\" connects the points in the ord er listed. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot([ [2,3], [-2,5],[1,-4] ],x=-7..7,y=-7..7,style=line);" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 328 10 "Example 3:" }}{PARA 0 "" 0 "" {TEXT -1 131 "Op tional extensions can be used to specify point color and symbol (e.g. \+ diamond, circle, cross is default) to indicate the points. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot([[3,2],[-2,3],[2,-1]],style=po int,color=blue,symbol=circle);" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "with(plots)" {TEXT -1 48 "Combi ning Graphs of Expressions and Points: the " }{TEXT 336 10 "display( ) " }{TEXT -1 8 " command" }}{PARA 0 "" 0 "" {TEXT -1 34 "A special plot ting package called " }{TEXT 330 5 "plots" }{TEXT -1 122 " contains ma ny additional graphing features. To use these commands, you need to \+ execute the following line which loads " }{TEXT 331 5 "plots" }{TEXT -1 148 ". Recall, the colon at the end of the statement allows this l ine to be executed without displaying any distracting output. To see \+ the contents of " }{TEXT 332 5 "plots" }{TEXT -1 41 " you can change t he colon to a semicolon." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " with(plots):" }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 329 11 "disp lay( ) " }{TEXT -1 173 "command allows you to combine graphs of expres sions and points in the same picture. The first step is to name the in dividual picture components. IMPORTANT: Be sure to use a " }{TEXT 334 5 "colon" }{TEXT -1 79 " at the end of the line to suppress output (se e first three lines below). The " }{TEXT 333 11 "display( ) " }{TEXT -1 74 "command is then used to do the actual plot (this ends with a se micolon). " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "pict1:=plot([ -3*x+5,9-x^2],x=-3..5,color=[green,red]):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 66 "pict2:=plot([[-1,8],[4,-7]],style=point,color=blue, symbol=circle):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display( [pict1,pict2]);" }{TEXT -1 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Parametric Equations" }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " } {TEXT 394 7 "plot( )" }{TEXT -1 76 " command can also be used to graph curves described by parametric equations." }}{PARA 0 "" 0 "" {TEXT 393 0 "" }{TEXT -1 41 "Plot the parametric curve determined by " } {XPPEDIT 18 0 "x=t^2-t" "6#/%\"xG,&*$%\"tG\"\"#\"\"\"F'!\"\"" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "y=2*t-t^3" "6#/%\"yG,&*&\"\"#\"\"\"%\" tGF(F(*$F)\"\"$!\"\"" }{TEXT -1 30 " over the t interval [-2,2] . " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([t^2-t,2*t-t^3,t=-2..2] ,x=-2..5,y=-5..5);" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 1 {PARA 4 "" 0 "implicitplot( )" {TEXT -1 14 "Implicit Plots" } }{PARA 0 "" 0 "" {TEXT -1 90 "Maple can plot curves that are implicitl y defined by an equation in the variables x and y." }}{PARA 0 "" 0 "" {TEXT 395 0 "" }{TEXT -1 58 "To plot the graph of the hyperbola given \+ by the equation: " }{XPPEDIT 18 0 "x^2/4 - y^2/9 = 1" "6#/,&*&%\"xG\" \"#\"\"%!\"\"\"\"\"*&%\"yGF'\"\"*F)F)F*" }{TEXT -1 9 " use the " } {TEXT 396 15 "implicitplot( )" }{TEXT -1 94 " command. To use this com mand we must first load the \"plots\" library using the \"with\" comma nd." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {PARA 0 "" 0 "" {TEXT -1 51 "\nNote the syntax for this command on the next line." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "implicitplot( x^2/4-y^2/4=1,x=-5..5,y=-5..5);" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 339 17 "Solving Equations" }}{PARA 0 "" 0 "" {TEXT -1 52 "In this section you will learn how to apply Maple's " }{TEXT 347 8 "solve( )" }{TEXT -1 21 " command to find the " }{TEXT 346 5 "exact" }{TEXT -1 79 " solutio ns of equations (when this is possible). Moreover you will use Maple's " }{TEXT 348 10 "fsolve( ) " }{TEXT -1 121 "command to find decimal a pproximations for solutions. The solution to linear systems of equatio ns will also be discussed." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Entering and Manipulating Equa tions: The " }{TEXT 381 6 "lhs( )" }{TEXT -1 5 " and " }{TEXT 382 6 "r hs( )" }{TEXT -1 9 " commands" }}{PARA 0 "" 0 "" {TEXT 383 10 "Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 40 "On the next line we enter the equa tion " }{XPPEDIT 18 0 "x^3-5*x^2+23=2*x^2+4*x-8 " "6#/,(*$%\"xG\"\"$ \"\"\"*&\"\"&F(*$F&\"\"#F(!\"\"\"#BF(,(*&F,F(*$F&F,F(F(*&\"\"%F(F&F(F( \"\")F-" }{TEXT -1 31 " and give it the name \"eqn1\" ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqn1:=x^3-5*x^2+23=2*x^2+4*x-8;" }} }{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 389 10 "Example 2:" }}{PARA 0 "" 0 "" {TEXT -1 79 "We can isolate the left-hand and right-hand sides of the equation by using the " }{TEXT 384 6 "lhs( )" }{TEXT -1 5 " an d " }{TEXT 385 6 "rhs( )" }{TEXT -1 11 " commands. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lhs(eqn1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "rhs(eqn1);" }}}}{SECT 1 {PARA 4 "" 0 "solve( )" {TEXT -1 29 "Finding Exact Solutions: The " }{TEXT 377 8 "solve( )" } {TEXT -1 8 " command" }}{PARA 0 "" 0 "" {TEXT -1 18 "We first consider " }{TEXT 349 10 "polynomial" }{TEXT -1 49 " equations. Algorithms exi st for calculating the " }{TEXT 353 5 "exact" }{TEXT -1 15 " solutions for " }{TEXT 354 10 "polynomial" }{TEXT -1 17 " equations up to " } {TEXT 355 8 "degree 4" }{TEXT -1 10 ". Maple's " }{TEXT 358 8 "solve( \+ )" }{TEXT -1 38 " command implements these algorithms. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 352 10 "Example 1:" }} {PARA 0 "" 0 "" {TEXT -1 64 "We want to find the exact solutions to th e polynomial equation " }{XPPEDIT 18 0 "3*x^3-4*x^2-43*x+84 = 0.;" "6 #/,**&\"\"$\"\"\"*$%\"xGF&F'F'*&\"\"%F'*$F)\"\"#F'!\"\"*&\"#VF'F)F'F. \"#%)F'$\"\"!F3" }}{PARA 0 "" 0 "" {TEXT -1 114 "Note that the second \+ argument of the command tells Maple that x is the unknown variable tha t we are solving for. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "s olve(3*x^3-4*x^2-43*x+84=0,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 350 10 "Example 2:" }}{PARA 0 "" 0 "" {TEXT -1 172 "Sometimes you will want to select one solution from the list of s olutions and use it in another computation. You can do this by first \+ assigning a name (we use the letter N" }{TEXT 371 1 " " }{TEXT -1 35 " in this case) to the output of the " }{TEXT 343 8 "solve( )" }{TEXT -1 15 " command. Then " }{TEXT 341 4 "N[1]" }{TEXT -1 34 " is the firs t number in the list, " }{TEXT 342 4 "N[2]" }{TEXT -1 59 " is the seco nd number and so on. Note the square brackets." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "N:=solve(x^2-5*x+3=0,x); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "N[1];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 351 10 "Example 3:" }}{PARA 0 "" 0 "" {TEXT -1 114 "Sometimes the \"exact\" solutions are too cumbersome to be of muc h use. In the next two lines we solve the equation " }{XPPEDIT 18 0 "x ^3-34*x^2+4=0" "6#/,(*$%\"xG\"\"$\"\"\"*&\"#MF(*$F&\"\"#F(!\"\"\"\"%F( \"\"!" }{TEXT -1 3 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "e qn1:=x^3-34*x^2+4=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "H:=solve(eq n1,x);" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 83 "As you can see, reading these exact solutions is quite a challenge! Note that the " }{TEXT 388 1 "I" }{TEXT -1 13 " stands for " }{XPPEDIT 18 0 "sqrt(-1) " "6#-%%sqrtG6#,$\"\"\"!\"\"" }{TEXT -1 99 ". When a solution is this \+ complicated it is more useful to look at the approximate solutions usi ng " }{TEXT 387 8 "evalf( )" }{TEXT -1 4 ": . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(H);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 26 "A good alternative to the " } {TEXT 356 8 "solve( )" }{TEXT -1 41 " command in a situation like this is the " }{TEXT 357 9 "fsolve( )" }{TEXT -1 23 " command (look below) .\n" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 361 8 "solve( )" } {TEXT -1 58 " command can also be used to find the exact solutions for " }{TEXT 340 14 "non-polynomial" }{TEXT -1 120 " equations. However i f the equations are complicated, then an exact solution will typically not be available. Again the " }{TEXT 360 9 "fsolve( )" }{TEXT -1 28 " command is an alternative. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 359 10 "Example 4:" }}{PARA 0 "" 0 "" {TEXT -1 23 "Sol ve the equation: " }{XPPEDIT 18 0 "5*exp(x/4)=43 " "6#/*&\"\"&\"\" \"-%$expG6#*&%\"xGF&\"\"%!\"\"F&\"#V" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(5*exp(x/4)=43,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "fsolve( )" {TEXT -1 35 "Finding Approxi mate Solutions: The " }{TEXT 378 9 "fsolve( )" }{TEXT -1 8 " command" }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 364 9 "fsolve( )" } {TEXT -1 73 " command can be used to find approximate solutions for an y equation. For " }{TEXT 366 10 "polynomial" }{TEXT -1 11 " equations \+ " }{TEXT 365 8 "fsolve()" }{TEXT -1 12 " produces a " }{TEXT 372 13 "c omplete list" }{TEXT -1 79 " of all of the real solutions in one step \+ (see Example 1). For other equations " }{TEXT 367 9 "fsolve( )" } {TEXT -1 20 " can be used to get " }{TEXT 373 22 "one solution at a ti me" }{TEXT -1 25 " (see Examples 2 and 3). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 362 10 "Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 50 "Approximate all real solutions for the equation: " } {XPPEDIT 18 0 "x^4-x^3-17*x^2-6*x+2=0" "6#/,,*$%\"xG\"\"%\"\"\"*$F&\" \"$!\"\"*&\"# " 0 "" {MPLTEXT 1 0 28 "eqn:=x^4-x^3-17*x^2-6*x+2=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsolve(eqn,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 363 10 "Example 2:" }}{PARA 0 "" 0 "" {TEXT -1 33 "Find a solution of the equation " }{XPPEDIT 18 0 "x^3+1-exp(x) =0" "6#/,(*$%\"xG\"\"$\"\"\"F(F(-%$expG6#F&!\"\"\"\"!" }{TEXT -1 12 " \+ using the " }{TEXT 368 9 "fsolve( )" }{TEXT 369 10 " command. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eqn:=x^3+1-exp(x)=0;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsolve(eqn,x);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Maple returns " } {TEXT 375 3 "one" }{TEXT -1 116 " solution. This time Maple has not gi ven us the whole story. Are there any other solutions? How do we fin d them? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 370 10 "Example 3:" }}{PARA 0 "" 0 "" {TEXT -1 49 "Find the other real solutions for the equation " }{XPPEDIT 18 0 "x^3+1-exp(x)=0" "6#/,( *$%\"xG\"\"$\"\"\"F(F(-%$expG6#F&!\"\"\"\"!" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "The first step in finding the other solutions is to plot a graph of the left-hand si de of the equation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot( x^3+1-exp(x),x=-3..5,y=-5..15);" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We can extend the " } {TEXT 376 9 "fsolve( )" }{TEXT -1 139 " command to look for a solution in a particular interval. For example to find the negative solution w e ask Maple to search on the interval " }{XPPEDIT 18 0 "[ -1 , -0.2]" "6#7$,$\"\"\"!\"\",$$\"\"#F&F&" }{TEXT -1 104 " since we can see from \+ the graph that there definitely is one (and only one) solution on that interval. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fsolve(eqn,x= -1..-.2);" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "To find the \+ other two solutions we use " }{TEXT 374 9 "fsolve( )" }{TEXT -1 75 " a gain, this time with search interval [1,2] and then with interval [4, 5]." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "fsolve(eqn,x=1..2);\n fsolve(eqn,x=4..5);" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Solving a Linear System of Equa tions using the " }{TEXT 379 8 "solve( )" }{TEXT -1 9 " command " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 344 8 "solve( )" }{TEXT -1 47 " command can also be used to solve a system of " }{TEXT 399 1 " m" }{TEXT -1 21 " linear equations in " }{TEXT 345 1 "n" }{TEXT -1 13 " variables. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 386 8 "Example:" }}{PARA 0 "" 0 "" {TEXT -1 28 "Solve the 2 by 2 system: " }{XPPEDIT 18 0 "3*x+2*y=3" "6#/,&*&\"\"$\"\"\"%\"xGF'F'* &\"\"#F'%\"yGF'F'F&" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "x-y=-4" "6#/ ,&%\"xG\"\"\"%\"yG!\"\",$\"\"%F(" }{TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(\{3*x+2*y=3,x-y=-4\});" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 247 "A graph of the two underlying functio ns shows the solution corresponds to the point of intersection at (-1, 3). But we first need to find the explicit form for each of the linea r functions before we can graph them. So we solve each equation for y. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "y1:=solve(3*x+2*y=3,y); " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "y2:=solv e(x-y=-4,y);" }}}{PARA 0 "" 0 "" {TEXT -1 147 "Now we construct a pict ure made up of two parts: \"part1\" contains the graphs the two equati ons and \"part2\" plots the solution point that we found. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "part1:=plot([y1,y2],x=-5..5): " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "part2:=plot([[-1,3]],style =point,color=blue,symbol=circle):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display([part1,part2]);" }{TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 262 "" 0 "" {TEXT 391 44 "Funct ions: Defining, Evaluating and Graphing" }}{PARA 0 "" 0 "" {TEXT -1 186 "In this section you will learn how to define a function f(x) in M aple. The remainder of the section covers evaluating functions, solvin g equations with functions, and graphing functions." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Defining and Clearing a Functio n in Maple" }}{PARA 0 "" 0 "" {TEXT -1 138 "To distinguish a function \+ from an expression, Maple requires special notation when defining a fu nction. For example, the function f(x) = " }{XPPEDIT 18 0 "cos (Pi*x) + 3" "6#,&-%$cosG6#*&%#PiG\"\"\"%\"xGF)F)\"\"$F)" }{TEXT -1 15 " is d efined as:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:=x->cos(Pi*x )+3; " }}}{PARA 0 "" 0 "" {TEXT -1 132 "Take note of the syntax here . It is necessary to type the \"arrow\" - > made by typing a \"minu s sign\" and a \"greater than\" symbol. " }{TEXT 380 44 "Maple will no t define a function if you type" }{TEXT -1 1 " " }{TEXT 0 18 "f(x):=co s(Pi*x)+3;" }{TEXT 392 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "Below is a comparison of an expression and a f unction. Note the difference in syntax and how Maple returns the outp ut for each." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y:=(x + 2)/( x^3 + 5*x + 2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f:=x->(x + 2)/(x^3 + 5*x + 2);" }{TEXT -1 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Evaluating a Function" }}{PARA 0 "" 0 "" {TEXT -1 208 "Once a function has been defined, you can evaluate it at various values or literal expressions using function notation. It's \+ often a good idea to clear the function name first before entering a n ew function." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f:='f';" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f:=x->3*x+x ^2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(-1); " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f(2+sqrt (5));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(f(2+sqrt(5)) );" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(x+4); " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify (%);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "(f(x +h)-f(x))/h;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 136 "Notice that if you define a function, there is no need t o evaluate the function using the \"subs\" command like you do with ex pressions. " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Graphing a Funct ion" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 390 4 "plot" }{TEXT -1 39 " function works the same for functions:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "h:='h'; y:='y'; x:='x';" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "h:=x->x*exp(-x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(h(x),x=-1..4,y= -2..1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{MARK "10" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }