Mathematics 133 -- Intensive Calculus for Science
Lab 1: Getting Started with Maple, Plotting Functions
September 25 and 26, 2000
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 another graphical way to
understand the "zooming in" operation involved in finding
the instantaneous rate of change of a function.
Lab Activity 1
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.
The above plot command is a template. To do an actual plot, you
will need to fill in the expression and range of x values
for the plot you want and add the appropriate options.
For example, suppose we wanted to plot y = x4 - 2x3
+ x - 5 for x between -2 and 1. We could use the Maple
plot command with no options:
plot(x^4-2*x^3+x-5,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 > 1 (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 display several plots together by putting the formulas
for the functions together inside the plot command, in a set of
curly brackets ( { } ), separated by commas. For example, try entering
the command
plot({x^4-2*x^3+x-5,cos(x)},x=-2..1);
to plot our polynomial function from before together with y = cos(x).
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.
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 functions we studied in Chapter 1 of
our text. 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.
Lab Activity 2
Recall, last Monday in class, we introduced
rates of change of functions. The average rate of change of
a function f(x) over the interval [a,b]
in x is (f(b) - f(a))/(b - a). We can attempt to
measure the "instantaneous" rate of change
of a function f at x = a by studying
(f(a+h) - f(a))/((a + h) - a)} and letting h
approach zero. Today, we will use Maple graphing to understand
the meaning of this process in another, geometrical way.
Questions
In these questions, we will study the graph
y = f(x) = sin(x) - x cos(x).
To plot this function in Maple, you will need to translate the
mathematical formula into a Maple expression.
- Plot the portion of the graph y = f(x),
for -5 <= x <= 5.
- Our function f(x) satisfies
f(-1) = -.3012
(approximate value). We want to use the graph to
find an instantaneous rate of change at
x = -1. The process of computing average
rates of change over shorter and shorter intervals
can be seen visually if
we "zoom in" on x = -1. Here "zooming in" will mean
plotting smaller and smaller pieces of the graph
taking x in smaller and smaller intervals containing
x = -1. For instance, try plotting the function
on the new intervals -1.5 <= x <= -.5, then
-1.05 <= x <= -.95, then -1.005 <= x <= -.995.
What are you noticing?
To be sure this is a real pattern, try adding the option
scaling = CONSTRAINED in the plot command from now
on. (This plots with equal scales on the x and y axes.)
What do you see as we ``zoom in'' this way? Keep only your
final graph in the worksheet (the others are not
necessary -- and long worksheets take longer to print out).
- A "zoomed" graph like the one
on the interval [-1.005, -.995] can
be used to compute an approximation to
the instantaneous rate of change at x = -1.
If you place the cursor over the graph in your Maple worksheet
and click the left mouse button, the approximate coordinates
of the point at the head of the cursor arrow will be displayed
in a box under the toolbar in the Maple V Release 5 window.
Clicking on two different points will give you the information
you need to compute a slope. Maple can also be used as a
numerical calculator if you want to compute the slope value.
For example, try entering a Maple command like
(6.2-5.3)/(.443-.323); When you press ENTER the value
will be computed and displayed.
Use these ideas to find an approximate value for the instantaneous
rate of change of f at x = -1.
- Also find the equation of the line through (-1,f(-1))
whose slope is your instantaneous rate of change, and plot it
together with f(x) for x on the interval
-5..5.
- Repeat the ``zooming process'' near the point
(4,f(4)),
describe what happens, include your final "zoomed" graph in your
worksheet, and use it to approximate instantaneous rate of change
- Find the
(approximate) equation of the line through (4,f(4)) with
slope equal to your instantaneous rate of change value, and plot
it together with f(x) for x on the interval
x=-5..5 and take y=-5..5 to control the "plotting
window".
Assignment
The lab write-up (a paper print-out of your worksheet, including all
graphs and explanations in text regions) is due on
Tuesday, October 3.