quiz1.mws
What is Maple?
Much of your mathematical background has probably been focused on developing the ability to solve equations and explore functions. The sophisticated manipulation of symbols and expressions that you use to solve equations and investigate functions can also be performed by software packages called computer algebra systems (CAS). A CAS can be use to generate the exact symbolic solutions you obtained by hand, the numerical approximations you found using a calculator, and the graphs you have drawn. Maple is one of several CAS's; other major examples are Mathematica, Macsyma, Derive, and MathCad.
Documentation and Online Help
The on-line help for Maple is very good. The help pages describe the syntax of each command, a brief description of the algorithm that has been implemented, and a few examples illustrating the use of the command. To obtain help on the command
func
, type:
?func
. To close a help window use the small button in the top left corner; to simply turn it into a small icon for later reference use the dot in the top right corner (this works for any window actually, and the box next to it opens a window up to full size). The Help Browser, an interactive interface to the entire Maple help system, can be brought up by clicking on
Help
, found at the top right of the Maple window. To get started trying activating
Balloon Help
under the Help Browser (this will be useful when you start to experiment with the Menu Bar).
Getting Started
Information about these commands can be found in the on-line help:
?plots
,
?restart
.
Basic Facts
In the following lines, use paper and pencil to first predict what you think Maple will do. Then execute the command and see what actually happens!
| > |
number:= 4 * 6 + 12 / 6 - 1 ;
|
| > |
3 / ( 5 - sqrt( number ) ) ;
|
Now enter some commands of your own! To produce the input prompt
>
use the
[>
button in the Menu Bar. Be generous with the space bar to make your commands easy to read, and to edit (modify after the fact). Go back to the tangent calculation above (remember how to reposition the cursor!) and change it to compute the tangent of
.
Next we define some functions for later use.
| > |
h:= 0.25 * cos ( 8 * x ) ;
|
Plotting
Graphs of functions are produced by the plot command. In its simplest form, plot needs to know the function to be plotted and the range of values for the independent variable. Note that a..b is Maple's way of describing the interval [a,b]. Observe that I have placed signed titles on some of these graphs. When you actually print plots to turn in with your homework, they MUST have a title and signature [by your initials or last name]. Again, try to predict the output before tapping the return key!
| > |
plot( 3 * t - 2 , t = -3 .. 10 ) ; # WAIT FOR THE GRAPH TO APPEAR!!
|
| > |
plot( f , x = -Pi .. 2 * Pi ) ;
|
| > |
plot( p , x = -6 .. 8 , title = ` f * g [by DM] ` );
|
These plots are nice, but what information do they convey? Let's concentrate on the plot of p, that is f*g. Position the cursor on a point on the graph and click the left button. The numbers that appear on the upper-left corner are the coordinates of the current location of the cursor. Use this technique to identify the global maximum and minimum values of f*g on the interval [-6, 8], and the x-values at which these are found. Where do other (local) minima and maxima occur? Can you guess the exact values?
| > |
plot(2 * u^3 + 4 * u - 5 , u = -10 .. 7 , title = ` cubic function [by DM] ` ) ;
|
Where does the cubic function cross the u-axis? Maybe it would be helpful to cut down the plotting interval from [-10, 7] to [-1, 3], or even narrower, say to [0, 1.5]. Try it. Can you find the u-intercept to 2-decimal point accuracy by this process? It is a powerful method, which we call ZOOMING IN.
| > |
plot( s , x = 0 .. 3 * Pi );
|
| > |
plot( q , x = -1.2 .. 1.2 ) ;
|
Can you explain why the graphs of s and q look the way they do?
Housecleaning
As you scroll up and down you may find that old plots take time to be redrawn and aren't really needed anyhow. If you click on a plot you will box it (and for printing purposes this box can be resized by dragging on the sides or corners). To remove it hold down the shift key and press delete.
Graphing Several Functions in One Plot
| > |
plot( { p , g , -g } , x= -6 .. 8 );
|
| > |
plot( { f , h , s }, x=0 .. 3*Pi );
|
Can you explain the little bumps that appear inside the large dips in the graph of s?
Plotting A Discontinuous Function
The tangent function is well-known to all of us. In particular, we know that the graph of
has vertical asymptotes at all odd multiples of
. Let's see how Maple handles this.
| > |
plot( f , x = -Pi .. Pi );
|
The key is to restrict the vertical range of the plot and tell Maple that the function is discontinuous.
| > |
plot( f, x = -Pi .. Pi , -10 .. 10 );
|
| > |
plot( f, x = -Pi .. Pi , -10 .. 10 , discont=true );
|
| > |
plot ( ( 1 - 0.4*t^3) / ( 1 - t^2 ) , t = -3 .. 5 , -10 .. 10 ,
discont = true, title = `vertical asymptotes [by DM]` ) ;
|
This technique is very useful for any function that has vertical asymptotes.
Congratulations!
You have just completed your first Maple worksheet. If you wish to save it select File in the menu bar, and then either Save (which will destroy the original version of day1.mws) or Save As... (which will leave day1.mws untouched, and will request a new name for this modified version). Worksheet names should always end with .mws; also unless you really need the output it is usually a good idea to remove it before saving -- to do this, select Edit, then Remove Output.
Quiz 1 (due midnight, Thursday, August 22, 2002)
Instructions
This week's quiz is to answer the following five (5) questions. Prepare your answers in a Microsoft Word document and submit this file in the Drop Box on the Blackboard page for the appropriate section of the course. (Be sure to use a filename that follows the conventions described in the syllabus.)
Consider the function
.
1.
What is the name of the Maple command used to factor a polynomial?
2.
Write the function in factored form.
3.
What command(s) did you use to obtain this factorization?
4.
Create a graph of this function on the interval [-2,3]. Be sure the graph clearly shows all of the oscillations in this interval.
5.
Do you think there is more area between the curve and the x-axis that is below the x-axis or above the x-axis? (Explain how you determined your answer.)
Extra Credit
What is the URL for the Maple Applications Center? What has your professor contributed to the Maple Applications Center?