Lab Instructions for the MAGNIFY Computer Activity
Two of the most fundamental concepts in calculus are the limit and the derivative. Each
of these concepts can be visualized graphically, in terms of zooming in on a point on the
graph of a function. Here, zooming in refers to repeatedly magnifying a part of the
graph, so that it can be examined on finer and finer scales. Think of looking at the graph
with ever more powerful microscopes. Both the idea of limit and the idea of derivative
concern the properties of the graph under indefinitely high magnifications.
For a limit, we imagine that the graph is missing a point for one particular x. By zooming
in, it may become clear that there is a single obvious position for the missing point. If so,
at that position the y coordinate is what we call the limit. It allows us to infer a natural y
value for a missing point, based on nearby points.
For a derivative, we again imagine zooming in at a single point of the graph. This time,
we want to know whether the graph will appear to become a straight line under
sufficiently powerful magnifications. If so, the slope of the straight line is what we call
the derivative. It gives us a sense of the slope of a curve at a single point.
This computer activity will allow you to experience this zooming in process for each
concept. At the bottom of the OVERVIEW screen, click on Limits or Derivatives to
begin an activity. The succeeding sections of this handout provide detailed instructions
for each page of the computer activity. You can also look at the instructions within the
computer activity if you wish (for example, in order to use cut and paste to define
functions). The Show Instructions button on each screen is for this purpose.
LIMITS PAGE
Anything printed on the screen in blue can be changed. Click after the part you wish to
change, delete it, and then type in the new information. When you click Compute!, the
computer will show f(x) for several values of x near the point at which the limit is to be
computed. That will be referred to as the x target point. It is the blue number indicated
on screen by
.
The main point of this activity is to tabulate and graph the result of the input formula for
several x values near the target point. You can control how near these x values will be by
changing the number labeled step. The From Left and From Right options indicate
whether the x's are to the left, or right, or on both sides of the x target point. Turn these
options on or off by clicking in the little box for each option. If a check mark is showing,
the option is on.
The Auto Size option, if checked, will automatically set the size of the graph. To change
graph settings manually, un-check the Auto Size option, then right-click on the graph, and
click on Settings ... .
Sample Functions
Enter the formula for each function below, and the corresponding x target point. Try
several different values for the step size. Based on the data that you see, do you observe
a clear position for the missing point on the graph? Can you formulate a clear value
where **** appears in the table? As you work, keep a record of what you do and what
conclusions you form.
Explanation
Suppose you are given a graph for y = f(x) without a point for x = 4. Based on x values
near 4, can you tell what y should be for x = 4? If so, that guessed value is what is called
the limit of f(x) as x approaches 4. Note that this guessed value need not equal f (4). In
fact, f (4) might not even exist. That is the situation for f (x ) 
x2
. For that
x4
function, you cannot compute f (4) . On the graph there is no point for x = 4. But if you
look at the graph, it sure seems clear that the missing point should have a y value of .25.
That value, .25, is NOT f (4) . We use the limit terminology to emphasis the difference
between substituting 4 into the function, and postulating a presumed value based on f (x)
for nearby values of x .
There is a graphical interpretation of this idea. Imagine repeatedly magnifying a section
of the curve centered at x = 4., zooming in closer and closer. Do the two parts of the
curve on either side of the missing point seem to match up? Is it obvious where that
missing point belongs? If so, then the y value at that location is what we mean by the
limit. On the other hand, if zooming in reveals a jump, or if for any other reason there is
no one obvious place for the missing point, then no limit exists.
Although the preceding discussion refers to the missing point as x = 4, there is nothing
special about 4. Any other number could be used as well. As mentioned above, the
missing value of x is called the x target point. It is the x coordinate we focus on as we
zoom in to ever higher magnifications.
For this part of the computer activity, you can easily experience the zooming in process
by making the step size smaller and smaller. Each time you click Compute!, a graph is
shown using 7 points on either side of the x target point. The smaller the step size, the
closer these 7 points will be to the x target. The results are shown numerically and
graphically, and you can see whether or not the there seems to be an obvious value for the
missing point.
Sometimes one is concerned only with the data points to the left of the x target point, or
only with the data points to the right of the x target point. The checkoff boxes at the top
of the screen can be used to illustrate this. Experiment with the sample functions again,
this time using data either from the left or from the right. For which samples can you
clearly tell where the missing point belongs using only data from the left? From the
right?
DERIVATIVES PAGE
On this page you can zoom in on one point on a graph. The main idea is to see whether
zooming in eventually makes the graph appear to be a straight line, and in case it does, to
find the slope of that line. As you work with this page, record your activities and results
in a lab report as, as described in the instructions below.
GENERAL INFORMATION
You can change any text that appears in blue type by selecting it with the mouse and
editing it.
The first line of the page is where the function f(x) is defined. Here are some points to
keep in mind when you change the function:
 Some functions must be spelled out: sqrt(x) for x , exp(x) for ex etc.
 You must use a * symbol each place a multiplication appears (e.g., 3*x, NOT 3x)
 The ^ symbol indicates an exponent (e.g. x^2 means x2 )
The magnify factor controls zooming. When the setting says magnify 2, each time you
zoom in or zoom out the scale will expand or shrink by a factor of 2. When you zoom in,
a red square will appear briefly to show the part of the graph that will be expanded to fill
the entire graph window. If you change any of the blue inputs, you must click on the
Restart button.
There is a slider bar near the top of the screen
(as shown at right). It allows you to specify the
slope of the green line on the graph. Adjust this
as you zoom in to make the green line and the
blue graph come as close as possible.
Activity 1.
Familiarize yourself with how the page works. As you proceed, keep notes in your lab
report about what you do and what you observe. Zoom in and out on the graph shown,
and see if the graph becomes a straight line. Experiment with different settings for the
magnification amount, zooming both in and out. Pay attention to the max and min values
of x and y on the graph as you do this. Change the setting so you can zoom in at a
different x. Zoom in until the blue curve becomes straight. Adjust the slope so the green
line matches the blue one. What is the slope when they match?
When the green line and the blue curve seem to be the same, we say that the
slope of the green line is also the slope of the blue curve at the point where
you are zooming in. Finding the slope of a curve at a point means zooming
in until the curve looks like a line and then finding the slope of that line.
When your blue curve looks like a straight line, and the green line matches it as nearly as
possible, zoom back out until you can again see the original curve. How would you
describe the appearance of the green line in relation to the blue curve?
Activity 2.
For this activity, use the function f(x) = x^2. For 5 different values of x, zoom in until
the graph becomes a straight line. In your lab report, record your results in a table,
something like this:
x value
slope
1
2.01
2
4.00
3
5.99
(your results may differ). When your table is complete, make a graph with the x values
on the x axis, and the slope values on the y axis. What is its shape?
Do the same activity for the function f(x) = .5*x^3 - 4*x. For best results, zoom in for
values of x near 0, such at -2, -1, 1, 2, etc. Record your results in a table, and again make
a graph for the results.
Activity 3.
For this activity consider these functions: 2^x , 3^x , and 1.5^x . For each function,
zoom in at x = 0 until the graph appears to be a straight line. Find the slope of that line
by manipulating the slider bar until the green line matches the blue one, just as you did
above. Record in your lab report what the slope is (at x = 0) for each of these functions.
These functions are called exponential functions. In each one, x is an exponent attached
to a constant number, referred to as the base. Try several more exponential functions
with bases between 2 and 3. For example, you might consider 2.2^x or 2.5^x . Can
you find a base for which the slope of the graph at x=0 is exactly 1? Record in your lab
report the bases you tried and what you found for the slopes at 0 when you zoomed in.
Activity 4.
Zooming in can sometimes be misleading. Set f(x) equal to sin(x) + sin(50000*x)/50000
and zoom in x = .5 until the curve becomes straight. Find the slope using the green line.
Record your results in your lab report. Now continue zooming in, and report your
observations in your lab report. How far must you zoom in on a graph if you want to be
sure you have found the correct slope? How can you be sure? Write your answer in your
lab report.
Activity 5.
Although most functions that you see in algebra and calculus courses become straight
lines if you zoom in far enough, there are some odd ball examples where this does not
occur. In the examples below, there are some that never appear to become straight lines
no matter how far you zoom in. Explore each of these functions, recording in your lab
report for each function what happens when you zoom in. In each example, zoom in at
x = 0.
f(x) = (x^2)^(1/3)+.3*x
f(x) = (x^2+.0001)^(1/3)+.3*x
f(x) = .8*x-sqrt(exp(x)-x-1)
f(x) = .8*x-sqrt(exp(x)-x-.9999)