Mathematics 36, Section 1 -- AP Analysis
Lab 1: Getting Started with Maple, Plotting Functions
September 5, 1997
Goals
Today, we will begin using a program called Maple to draw graphs
of functions of one variable (given by symbolic formulas). You will be
using the commands covered here for most of the other lab days this semester.
On the mathematical side, we will study the graphical effects of some algebraic
transformations on formulas defining functions.
A Sample Graphing Session
Let's get right down to work and walk through a sample graphing session!
First, you will need to get into Windows and Maple as described in the
General Information handout. The basic Maple command for 2D plotting graphs
of the form y = f(x) is called plot. The basic format is
plot(function,range,options)
where
- function is the function to be plotted -- the simplest way to
specify one is via a formula (an expression in Maple)
- range is the range of x-values you want to see plotted, and
- options can be used to control the form of the plot if desired.
No options need be specified however if you don't want to. More on this
later.
For example, suppose we wanted to plot y = x4 - 2x3
+ x - 5 sin(x2) for x between -2 and 1. We could use the Maple
plot command with no options:
plot(x^4-2*x^3+x-5*sin(x^2),x=-2..1);
(Note: The close parenthesis comes directly after the range of x-values
if there are no options.) Type in this command line (exactly as here) and
press ENTER. If you make a typing mistake, Maple will let you know about
it (!) Fortunately, if this happens, the whole command does NOT need to
be re-entered. Just move the cursor arrow to the place on the input line
you want to change, press the left mouse button, and edit the input as
needed. Typing from the keyboard will insert new stuff at the cursor
location; the DELETE and BACKSPACE keys will remove stuff (DELETE removes
the character in front of the "insert point"; BACKSPACE removes
the character in back). You can also move around on the input line with
the arrow keys if more than one thing needs to be changed. When you think
it's OK, press ENTER again to have Maple execute the command again.
When all goes well you will see the graphics output displayed under
the input command in the worksheet.
From the formula, you might guess that there is at least one other x-intercept
for this graph for x > 2 (why?). To see that part of the graph as well,
edit your previous command line to change the right hand endpoint of the
interval of x values (do not retype the whole command). Press ENTER on
that input line to have Maple execute the command again. Experiment until
you are sure that your plot shows all the x-intercepts of this graph. (You
can repeat this process of editing a command and re-running it as often
as you want; the previous output is replaced by the new output each time.)
You can also manipulate the graphics output in place within the
worksheet in several ways. For instance:
- If you click the left mouse button once over the graphics output, you
will see a black box with eight ``tabs'' displayed at the corners and the
midpoints of the edges of the box. If you place the cursor on one of the
tabs, hold down the left mouse button, drag the cursor and release, you
can resize the graphics. Try it!
- Maple 2D graphics output regions have another nice feature: If you
place the cursor arrow at a point in a 2D graphics region and click the
LEFT mouse button once, approximate coordinates of the point at
the head of the arrow are printed out in a box at the upper left of the
window. How good approximations can you get that way to the x-intercepts
of this graph?
- Other features of the graph can be changed from the tool bar. Experiment
and see if you can figure out what the different tool bar icons do.
When you get a graphics window with all the intercepts shown,
let's add one more thing. Sometimes, an informative title makes a graph
much more understandable. To add a text title to a plot, you can insert
a comma after the range of x-values, and include an option in the plot
command of the form
title=`whatever you want`
NOTE: Those are ``backquotes,'' not ordinary apostrophes. Add an appropriate
title to your plot.
Frequently, it's the relationship between two or more different graphs
that you want to understand by looking at a plot. You can also use the
plot command to generate plots showing several graphs with the same
coordinate axes. You do this by plotting an unordered list or set
of functions. You do this by putting the formulas for the functions you
want, separated by commas, between curly braces -- { ,} -- where
the single formula went before. (In fact this is the way sets or unordered
lists are always entered in Maple.) For example, to plot y = sin(x)
together with y = sin(x + 2) for x between 0 and 4 Pi, you could use:
plot({sin(x),sin(x+2)},x=0..4*Pi);
Try it.
More Detailed Information on Maple Commands
Now that we have seen some first examples of Maple commands, here is
some more information about the syntax rules that Maple uses to
decide if what you have typed in is a well-formed command it can execute.
For a function described by a formula, the formula is entered in something
like usual mathematical notation:
- The symbols for addition, subtraction, multiplication, and division
are +,-,*,/ respectively.
- The caret (^) is the Maple symbol for raising to a power.
- The asterisk symbol for multiplication MUST be included whenever you
are performing a product in a formula. Moreover, everything must be entered
in one string of characters, so you will need to use parentheses
to group terms to get the expressions you want. The rule to keep in mind
is: Maple always evaluates expressions by doing powers first, then products
and quotients, then sums and products, left to right, unless parentheses
are used to override these built-in rules. For example, the Maple expression
a + b^2/c + d is the same as the mathematical formula: a + b2/c
+ d.$$ If you really wanted a + b2 in the numerator and c +
d in the denominator of a fraction, you will need to enter the expression
(a + b^2)/(c + d). What if you really wanted (a+b)2 in the numerator?
- Maple ``knows'' all the usual elementary functions from calculus. The
names of the most common ones are sin, cos, tan, exp, ln. To use
one of these functions in a Maple formula, you put the name, followed by
the "argument" (that is the expression you are applying the function
to) in parentheses.
- The range of x-values to plot for a graph y = f(x) MUST be specified,
in the format x = low..high. A range of y-values to plot can also
be specified, in the same format.
- Every Maple command must be terminated either with a semicolon or a
colon. This is the signal that the command is finished and that Maple should
try to execute what you asked it to do. The difference between a semicolon
and a colon is that for commands terminated with a semicolon, any output
produced will be displayed. If you terminate the command with a colon,
the calculation will be performed but not displayed (this is useful sometimes
for intermediate steps in a big computation where you don't need to see
the output). Until you enter either a semicolon or a colon and press ENTER,
Maple will not do anything with your command. Also, if a command you want
to enter doesn't all fit on one line, just keep typing, but don't press
ENTER until you are finished. Maple automatically wraps around to a new
line if you need it.
Assignment
Your first lab assignment is to use Maple plots to help you develop
answers to each of the questions below. For your lab write-up, create a
worksheet containing graphics input commands, output, and text regions.
For each question, generate a plot that illustrate your answer. (One plot,
combining several graphs on one set of axes, will be sufficient for each
of the questions 1,2,3,4.) In complete English sentences, answer the "describe"
and "explain" questions. One write-up per lab pair is sufficient.
- What is the general relationship between the graph y = f(x) and the
graph y = f(x) + c, where c is a constant value? (Describe how the graph
y = f(x) could be transformed to yield the graph y = f(x) + c.) How does
the exact value of c used -- for instance whether c is positive or negative
-- affect the result? Explain why the relationship you describe is valid.
(Suggestion: Use f(x) = sin(x).)
- What is the general relationship between the graph y = f(x) and the
graph y = f(x + c), where c is a constant value. (Describe how the graph
y = f(x) could be transformed to yield the graph y = f(x+c).) How does
the exact value of c used -- for instance whether c is positive or negative
-- affect the result? Explain why the relationship you describe is valid.
- What is the general relationship between the graph y = f(x) and the
graph y = cf(x), where c is a constant value. How does the exact value
of c used -- for instance whether c is positive or negative -- affect the
result? Explain why the relationship you describe is valid.
- What is the general relationship between the graph y = f(x) and the
graph y = f(cx), where c is a constant value. Describe in full English
sentences. How does the exact value of c used -- for instance whether c
is positive or negative, greater or less than 1 in absolute value -- affect
the result? Explain why the relationship you describe is valid.
Due
The lab write-up (as a paper print-out of your worksheet) is due on
Friday, September 12.