In this section we give you examples of
how you can use Mathematica for some familiar and elementary
operations.
Opening a workspace and executing a Mathematica
command
In the first example, we illustrate some of the algebraic and
graphical capabilities of Mathematica. First first you must "open" a
line to type in as follows:
- If not at the top of a "clean"
notebook window, move the cursor slowly down the window until the cursor
becomes a horizontal I-beam.
- Click the mouse; a grey line will
appear across the window.
You can now enter new text in the
notebook window. The first example defines y, as a function of x. Type the
following exactly as it is written:
y = x^3 - x^2 - 9 x + 9
The line
you've just typed is called a command, or "input". When you "enter" such an
input, Mathematica processes it and returns "output" (or a result).
To obtain the result you must first "execute" the command by following
these steps:
- If not already there, move the
cursor to the end of the command; in this case, after the second 9.
(Alternatively, put the cursor on the ] to the far right of the command),
- Click the mouse to select the command,
- Hit the Enter key,
located on the extreme lower right of the keyboard. (Note: this is NOT the
same as the (carriage) Return key that you have been using.)
Once
the command is completed, you will see an input line labeled In[1] that
contains the original command, and an output line labeled Out[1]. Notice
that the result in the output line is written differently than the way we
typed it; Mathematica prefers to write polynomials in increasing
powers of x.
Writing a Mathematica command
The input line above
is an example of writing the definition of y as a cubic polynomial in x.
There are two items you should notice in the above command that are
peculiar to the Mathematica program:
- "x cube"and "x square" are typed using the
carat (^ ) .
- The product "9 x" does not need a multiplication
symbol. The two possible forms for writing a product are "9 x" (employing a
space as above) and "9*x" (employing an asterisk).
Plotting a
function
The command below is a very common plotting format that will
soon become familiar to you. (Note: this plot command assumes the above
definition of y; if you have not executed a command defining y, scroll back
and do it now.) Open a new line in the notebook window and type the
following command exactly as written.
Plot[y, {x,-7,7}, PlotRange ->
All]
There are some items to note in this plot command that are
peculiar to Mathematica:
- Notice the square brackets, [
and ], around the arguments of the Plot command. Also note {x,-7,7}
specifies a domain interval for x. It is important to remember when to use
the different types of brackets. Mathematica is very sensitive to
different bracket styles.
- "PlotRange -> All" is a plotting option,
telling Mathematica that---"yes, we do want to see the entire
graph". (Other plotting options will be introduced as we need them.) The
arrow (->) is created with the "hyphen" followed by the "greater than" keys
on the keyboard.
- Notice the upper case letters in Plot and
PlotRange. This is typical of all Mathematica commands and options,
so you will need to get used to this! In contrast, if YOU define a new
function, like y above, you are free in the use of upper and lower case
letters.
Execute the plot command below to see what the graph of y
looks like. (Select the command by locating the cursor after it, and hit
Enter). You can see that the graph depicts the essential character of
function y.
Modifying a command
You can now practice modifying
a Mathematica command and, at the same time, get a closer look at
the graph.
Follow these steps to change the domain interval for x in
the plot command below:
- Carefully place the cursor in front of
the -7;
- Press and drag the mouse across the -7,7 (it will be
shaded grey);
- Type -4,4 (the -7,7 will be replaced).
Now
execute the modified plot command. (If the command does not execute, it's
always a good idea to check for typographical errors. If there are no
errors and the graph still does not plot, ask for help.)
You will now
see a more detailed graph of y near a point where it crosses the x-axis.
Factoring polynomials
In the above graph it looks to the eye
as if the function y is equal to 0 somewhere around x = -3, x = 1, and x =
?. That is, it appears that y has roots at x = -3, x = 1, and x = 3.
- Locating roots is an important problem that we will pursue from
several directions this semester. But in this case, the question is easily
answered because Mathematica is good at factoring polynomials.
To see this, open a new command line and execute the following
command.
Factor[y]
It is now clear that y = 0 precisely at x=-3,
x=1, and x=3.
Zooming in on a graph
Let's return to the graph
of y in order to discuss an issue that will be very important throughout
our study of calculus.
Again, execute:
Plot[y, {x,-4,4}, PlotRange
-> All]
A recurring theme in calculus can be paraphrased: most
functions are "almost linear" if you look closely enough. To illustrate
this point, you can "zoom in" on the graph at x near 2:
- Modify
the plot command below so that the domain interval is 1.9 to 2.1. (If
necessary, you may scroll back to review the directions Modifying a
command.)
- Execute the plot command.
You should see a
smaller portion of the graph and note that it is starting to straighten out
a bit. To further illustrate the point, modify the domain interval in the
plot command to 1.99 to 2.01 and execute the command. Notice that the graph
is almost linear on this restricted domain interval.
Is there anything
special about the point (2, -5)? No. You may want to experiment on your own
by modifying the command with any value for the domain of x that you
choose, and "zoom in" as we have done. Remember the fundamental point being
made:
most functions are "almost linear" if you look closely
enough.
We will revisit this fundamental notion often in Calculus.
http://math.colgate.edu/mathlab/basicelts.html
Revised: March 1, 1996.
Questions to:
valente@colgate.edu
Copyright 1996 © Colgate University. All rights reserved.