Welcome to Maple

Copyright 2001
Department of Mathematics
University of Georgia
Athens, Georgia

Carol W. Penney

This introduction will help you begin your work with Maple.  Work through the examples and do the exercises to gain practical experience with this powerful and useful computer algebra program.  Do not try to memorize commands now; you will learn more about particular Maple commands as you work on projects. The last section, Project 1:  Investigations of Functions   and Graphs , outlines a short project that will give you practice working with Maple.  You will investigate two functions and write a report on the results of your investigations.  Begin your work by clicking on the plus sign below to open the first section of this introduction.

The Maple Worksheet

You are reading a Maple worksheet.  It consists of a combination of text, commands, and output.  Your projects are presented in the form of worksheets and you work in a worksheet, combining your exposition, input, and output to produce a report on each of your investigations.

Text:  As you see, Maple acts like a word processor to produce text.  It has the usual word processing capabilities, some of which you can see in the Edit menu above.

• Exercise:   Look at the menus in the tool bar at the top of the page to get an idea of what choices are readily available. Click on each topic to reveal the submenus.  Do not change settings now; this worksheet assumes, for example, that your Options  are set to  Input Display:  Maple Notation .

Commands:   Command lines appear in red. Here is one; to enter it click your mouse in the command line and press  [Enter].

>	factor(x^4+13*x^3+45*x^2-25*x-250);

As you see, output is in blue.  Notice that your cursor automatically jumps to the next command line, ready to execute the next command.

Maple consists of two parts:  the front end , which is the content of the display, and the kernel , the computational engine of Maple.  Even though you see a command in the worksheet, Maple's kernel will not know it exists until you enter that command.  In order to enter a command, you must send it to the kernel by placing your cursor on the command line and pressing  [Enter].  If you enter a command now, save your program, close it, and open it later, you may see blue output displayed by the front end, but Maple's kernel will not remember having calculated it.  You must re-enter any command you want the kernel to know about because Maple begins each session with an empty kernel.

• Exercise:  Enter each of these commands.

Addition:

>

12+56;

Subtraction:

>

768-12.4;

The asterisk indicates multiplication, and the  ^  symbol indicates exponentiation.  Here we calculate ( )( ):

>

5*2^4;

Compare these two calculations of the square root of 200---first the exact value then a decimal approximation, which is obtained by inserting a decimal point in the number.

>	sqrt(200);

>	sqrt(200.0);

The standard rules of parentheses apply.  Compare these two calculations:

>

2/(3+5);

>

2/3+5;

A plot of the graph of  on the domain [-2,3]:

>	plot(x^2,x=-2..3);

>

Notice that your cursor moved to the blank execution group below the plot command.  Each time you enter a command, the cursor will move to the next command line.  The empty execution group above this paragraph has a function; it stops the cursor so that you can look at the graph.   If you want to see what happens if this empty execution cell is not present, do the following exercise:

• Exercise :  Highlight the empty execution group above the preceding paragraph  (the line under the plot command) by double-clicking its bracket, then delete that group by pressing your delete  button.  Go back to the plot command line and re-enter the plot command.  Notice what happens.  What do you have to do in order to view the graph?  

Entering Commands:  In order to create an execution group so you can type a new command, place your cursor where you want to type the command and click on the prompt  icon, which looks like  [>  and lies on the line of icons at the top of the page immediately to the right of the T .  Clicking the prompt icon creates an Execution Group .  Type your command followed with a semicolon.  Then enter the command.

• Exercise :  Place your cursor on this line and click the prompt  icon.  Type sqrt(10.0)  and press [Enter], to compute the square root of 10.  Your cursor will move to the next execution group, so you will have to scroll back up to see your result.  
• Exercise:   Remove the semicolon from your command line of the previous exercise and enter the command again.  What happens?
• Exercise:   Put the semicolon back and insert an empty execution group below the command cell.  Then again enter the command to find the square root of 10.  Do you like having this empty execution cell present?
• Exercise :  Go back to the graph of the parabola.  Click on the cell bracket to the left of the graph, then click the prompt icon to insert a new execution group below the graph.  Then scroll up to the plot command and re-enter it.  Notice what happens.

Typing Text:  In order to type text following an execution group, click on that execution group bracket and select  Insert Paragraph  from the Insert Menu.  Begin to type.  You notice that text is black. The Maple kernel ignores text, which is present for the benefit of the reader---you do not  [Enter]  text.  An alternative method of creating a place to type text is to click on the prompt  icon then the T  icon ( T  stands for text)   on the left of the prompt  icon.  This will create an execution group in which you can type text.

• Exercise :  Insert a new paragraph below this one, using Insert Paragraph .  Type a sentence or paragraph.  Then insert another paragraph below that one by clicking on the prompt  icon then the T  icon.  Type a sentence in this paragraph.  Do you see the difference between these two methods of inserting a paragraph?
• Exercise:  Create a command cell and enter the command to plot the graph of y=sin(x) on the interval [0,5 ].  In Maple,  is typed Pi, with a capital P;  is typed 5*Pi.  Your plot command will be:  plot(sin(x),x=0..5*Pi).   You can either type this command in a new execution cell, or highlight it here, copy it, and paste it into your execution cell.  Try both methods.

Opening two worksheets at once:   You can work with two or more worksheets open at once.  You can choose to toggle back and forth from one to the other or view two worksheets simultaneously.  Choose your arrangement by selecting, from the Window menu,   Cascade , Tile , Horizontal , or Vertical .

• Exercise:  Open a new file by clicking the icon that looks like a blank page.  Practice arranging the two worksheets.

Saving your work:   Save frequently as you work.  If there is a power outage or if you crash you will not lose anything except the work you have done since the previous Save , so from time to time you will appreciate having developed the habit of saving frequently.  In order to save this file, choose Save As  from the  File menu.  A dialog box will appear, asking you to give the file a name and choose a place to save it.  Name your file something you can remember (such as Welcome followed by your initials), then type  a: filenameyourinitials    and click O.K.  This will save the file to your floppy disk (in drive a).

• Exercise :  Save this file now, using Save As .

Once you have saved your work, you can save updates as you work by simply clicking on Save  in the File menu; you only have to tell Maple where to save and what to name the file the first time you save.  Before making your final Save of each session, select Remove Output from Worksheet  from the Edit  menu.  This will save room on your floppy disk, may prevent difficulty opening your file on a different computer, and will ensure that when you next open your file output that is unknown to the computer will not appear to be known.

Functions

Before we begin more investigations of Maple, we clear the kernel's memory with the restart command.  This is good way to ensure that Maple will not be remembering any values that you assigned earlier in this session and have forgotten about.

>

restart:

Maple knows the usual functions, such as sin(x), ln(x), arctan(x), and so forth.  Most are entered in the obvious way:

>

sin(x);

>

ln(x);

The notation for the natural exponential function is this:

>

exp(x);

Defining Functions:  You can also define your own functions.  For example:  to define the function f(x)=x^3-x^2+1, think of the function f as a rule that operates on the number x to transform it to x^3-x^2+1.  Define f as follows:

>	f:=x->x^3-x^2+1;

• Note:  You must use this notation to define a function.  If you attempt to define a function by typing f(x):=x^2, for example, Maple will not recognize that f(x) is a function; you must type f:=x->x^2.

Now that Maple knows this function, you can refer to it by name throughout your work.  Maple will be able to evaluate f(x), plot its graph, solve equations involving f(x), and so forth.

Evaluation of a Function:  You can evaluate a built-in function or a function you have defined.  By hand, you would denote the value of f at x=2 by f(2); Maple notation is the same.  Calculate an exact value by entering the value of x without a decimal point, a floating point (decimal) approximation by entering x with a decimal point.

>	sin(Pi/4);

>

f(2);

>

f(2.0);

>	f(sqrt(2));

>	f(sqrt(2.0));

>

f(Pi);

Since  has no place to put a decimal point, use the command evalf ("evalf" is short for "evaluate as a floating point number")  to evaluate f(Pi) as a decimal.

>	evalf(f(Pi));

>	evalf(sin(Pi/4));

• Example:  Define the function g(x)=1/(1+x^2) and the function h(x)=sqrt(x).  Evaluate each of these functions at x=2, x=Pi, and x=sqrt(3), to obtain both exact values and decimal approximations.  Begin by completing this:

>

g:=x->

>

h:=x->

>

Graphs of Functions:  To plot the graph of f(x)=x^3-x^2+1 on the interval [-0.5,1]:

>	plot(f(x),x=-0.5..1);

>

You can insert various options in the plot command to change the way the graph looks.  You may want to zoom in on the graph, change the color of the graph, display or suppress the display of asymptotes, change the relative proportions of units in the x- and y- directions, and so forth.

If you want to set the range of the plot, insert bounds on y.  For example, to display values of y between 0.9 and 1.1:

>	plot(f(x),x=-0.5..1,y=0.9..1,title="Nessie");

If you want to select a different color for your graph, insert a color command.  Readily available colors are:

aquamarine black  blue      navy   coral cyan  
brown      gold   green     gray   grey  khaki
magenta    maroon orange    pink   plum  red   
sienna     tan    turquoise violet wheat white
yellow                                          

>	plot(tan(x),x=-10..10,y=-5..5,color=navy);

>

Notice that Maple drew lines that look like vertical asymptotes to the graph.  In fact, Maple is sketching the graph by connecting points on the graph; because it does not recognize the discontinuities of the tangent function, it is simply attempting to draw a continuous function by drawing "almost vertical" lines between points on separate branches of the graph.  If you do not want to see these "asymptotes", warn Maple to look for discontinuities by inserting discont=true  in your plot command:

>	plot(tan(x),x=-10..10,y=-5..5,discont=true,color=navy);

>

Sometimes the graph is best displayed if the scale of the horizontal units is different from those in the vertical direction:

>	plot(x^2,x=-5..5);

But sometimes you want the units to be equal in the horizontal and vertical directions in order to show the scale of the graph accurately.  In this case use scaling=constrained in your plot command:

>	plot(x^2,x=-5..5,scaling=constrained);

>

To plot the graphs of two functions on the same coordinate system, enclose the pair of functions in brackets.

>	plot([x^2,1-3*x],x=0.1..0.5);

>

If you have defined a function, you may refer to it either by typing its formula or, more simply, by name.  The following two commands produce identical graphs.

>	plot(x^3-x^2+1,x=-0.5..2);

>	plot(f(x),x=-0.5..2);

• Exercise :  Plot the graphs of the functions g(x)=1/(1+x^2) and h(x)=sqrt(x) that you defined in the previous exercise, first on separate coordinate systems, then together on the same coordinate system.
• Exercise:  Define the two functions m(x)=sin(5*x) and n(x)=5*sin(x).  Plot the graphs of m, n, and m +n on one coordinate system.  Experiment with the choice of domain and range in order to get a nice display.

>

Summary of Important Commands:

You can obtain many results with only a few commands.  The four essential tools that you will need to use in every project are these:

• Define a function.  For example, to define the function   :

>	f:=x->x^3-3*x+4;

• Evaluate a function.  For example, to evaluate f(x) at x=5:

>

f(5);

• Plot the graph of a function or several functions.  For example:

>	plot(f(x),x=-3..5);

>	plot([f(x),x^2],x=-3..5);

• Solve an equation.  For example, to solve     for , to get both an exact solution and a decimal approximation to the exact solution:

>	solve(4*x^3+6*x+7=10,x);

>	fsolve(4*x^3+6*x+7=10,x);

>

Exercises:  
Use these exercises to practice using the commands you have learned.  Try to remember how to enter the appropriate commands before you look them up.  At the end of this project you will find a section called Tools for this Project; it contains all of the commands you will need to work these exercises.  State each problem and present your answer to problem 4 clearly in a complete sentence.

• 1.  Define the functions     and   .
• 2.   Evaluate the function     at  x=1.25.
• 3.   Plot, on the same coordinate systerm, graphs of the functions     and     that you defined in problem 1.
• 4.   Find x-coordinates of all points at which these graphs intersect.  Give these coordinates both in exact form and as decimal approximations.  

Now you are ready to begin your project.