Objective: This
interactive demo illustrates the generation of
a logistic curve.
Level: This demo is appropriate
for a precalculus course, but
is quite effective in a calculus class immediately after a discussion of
inflection points.
Prerequisites: For a precalculus class, students need
to
understand the notions of increasing and
decreasing along a curve. In calculus, students
need concepts of increasing, decreasing, maxmin, and inflection
points. Platform: This demo can be
used with calculators
which can randomly generate an integer in a specified interval. (See
Using Calculators below.) We
include a MATLAB routine to simulate the classroom demo and a
utility to
generate the random numbers in
a classroom setting in which individual
students do not possess a calculator with a random number generation
feature. Mathematica and Maple worksheets are also
provided. In addition, a Java applet
provides a platformindependent routine for the generation of random
integers.
Instructor's
Notes:
The logistic curve is used to model a variety of
physical situations in
which a quantity's growth is "selflimited," that is, the
growth rate of the quantity depends on the size of the quantity in such
a way that if the quantity grows beyond a certain level, the growth rate
decreases. The
logistic model nicely describes the
behavior of certain types of growth in business, economics, populations, the spread of disease, and sales forecasts. In situations of
"controlled growth" the logistic curve frequently provides an easily
constructed graph that is readily understood. For example:
 Currently, our economy is continuing to grow, but
at a slower rate.
Often this and similar statements are misunderstood,
but can be clarified with pictures and the discussion of the logistic
curve.
The story goes that Sam Walton of WalMart fame made
his money by having inventory reported every night. As soon as products
were being bought at a decreasing rate he would stop stocking that
product. Thus there was never a surplus of inventory that needed to be
sold at a low price. This short tale has made many business students
stop asking why they take calculus.
A Simple Classroom Demo.
Here we describe a simple demo of the spread of a
virus that leads to a logistic curve and involves every student in the
room.
Suppose you have 25 students in your class. Assign
each student a number from 1 to 25. This is easily done by having them
countoff. You might have each student
write his or her number on a Postit. (You
bring the Postits.) Let's assume that each student has a calculator
which can randomly generate an integer between 1 and 25. ( Alternatively,
appropriate random integers can be generated
by some other means, as needed
in the classroom if calculators are not available.)
On the board in the class draw a set of axes.
Horizontally we represent time in days and vertically the number of
students infected with the virus. In addition draw a table with 5 rows
and 5 columns which are numbered 1  25. As the instructor, use your
calculator to generate a random
integer between 1 and 25. Take this
event to mean that on DAY 1 the student whose number you generated
becomes the first to be infected with the virus. To indicate this ask
this student to stand and put his or her Postit in the corresponding
numbered box in the table on the board. (You might want to recruit
the first infected student as the recorder of information on the graph
that is on the board.) Record the point (1,1) on the graph on the board.
(This indicates on DAY 1 there is 1 infected student.)
To simulate a second day, have the first infected
student use his/her calculator to generate a random integer between 1 and
25. If the number generated
is different from the number assigned to the infected student, then
there is a second infected student. Have that student stand and place his/her
Postit in the corresponding numbered box
in the table drawn on the board. Plot the point (2,2) on the board; otherwise only one student is
infected so plot point (2,1).
For Day 3 have each previously
infected student generates a random
integer between 1 and 25 using
their calculator. Have newly infected students stand and then place their
Postits in the corresponding numbered box in
the table on the board. The pool of infected students may grow or, by chance, remain the
same. Plot the point (3, total number of infected students).
The preceding actions of the infected
students are repeated on each successive day. In addition, the point
with coordinates
(number of the DAY, total number of infected
students)
is plotted. This continues until the entire
class is standing (infected by the virus). The graph recorded on the
board will have the shape of a logistic curve as shown in Figure
1.
Figure 1.
The curve displayed in Figure 1 shows
that the entire class of 25 students was infected in just 8 days.
Results can vary as shown by the graphs in Figures 2  4 which used the
same procedure for a group of 25 students. The demo depicted on Figure 2
took 9 days to infect all 25 students, while those in Figures 3 and 4
took 12 days and 8 days respectively.
Figure
2. 
Figure 3. 
Figure
4. 
Curves generated while using this demo
procedure will be increasing (but not necessarily strictly
increasing).
To help explain phrases like "our economy
is continuing to grow, but at a slower rate", we can use the data
generated in the demo as follows. Make a table of the coordinates of the
ordered pairs that were graphed. (See Figure 5a.) Construct a third row
(label it ydiff) which is the difference of successive ycoordinates;
make the first entry in this row 0. (See Figure 5b.) Next plot x vs.
ydiff on another set of axes. Figure 6a shows the graph of x vs. y,
while Figure 6b displays x vs. ydiff.
x 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
y 
1 
2 
3 
6 
10 
15 
21 
24 
24 
25 
Figure 5a.
ydiff 
0 
1 
1 
3 
4 
5 
6 
3 
0 
1 
Figure 5b.
Figure 6b shows the rate of change
of the number of students infected daybyday. From this graph we can
infer that the rate of change of the number of students infected "slowed
down" starting at Day 7. That is, the number of students infected is still
increasing but at a decreasing rate. The assimilation of this idea is an
important step in the learning process for students in a variety of
majors.
There are other features of Figure
6b that can be discussed; for example the horizontal segment from Day 2 to
Day 3 and the behavior from Day 9 to Day 10. Each time we repeat this demo
we expect to see variations in both the graphs of x vs. y and x vs.
ydiff. Pairs of such graphs can be used as takehome exercises where
students are asked to write a description of the behavior of the spread of
the virus.
In a calculus class,
figures like that in 6b provide a nice foundation for the discussion of
inflection points. For the demo shown in Figures 6a and 6b, there is the
behavior of an inflection point at DAY 7.
This demo has been used in a variety
of classes, precalculus, business calculus, and calculus classes with
great success. Students are involved in the process to generate the
information used to explain a behavior that appears in many applications.
They seem to personalize the experience provided by this demo and are
better able to explain such behavior when they meet it in the
future.
Using Calculators
If you have a TI89
graphing
calculator and have
downloaded the 2.0 operating system and Statistical APPS you have the
randInt command available for the generation of the random
integers. If you go to Catalog, then F3 Flash Apps, then
R, you will see randInt. For a class of 25 students, the
number of the student first infected is obtained from the command
randInt(1,25,1). Subsequently the command randInt(1,25,k), generates k random
integers between 1
and 25.
For additional information on using
the TI89 and randInt click here.
You will see a PDF file developed by Dr. Roseanne Hofmann for introducing
the use of random integers and the Virus Spread demo. Included are
calculator screen pictures.
If you do not
have the flash update for the TI89 that contains the randInt program,
a program, randmint, is available by clicking
here.
The program prompts the user to input the number of students (n) and the
number of random integers (m) that are needed at each stage in the
progression of the disease. A new random number is generated each
time the user presses the Enter key, until m random integers have been
produced. If you have the TI Graph Link hardware and program, you can
download the file to your computer and then directly into your calculator.
The TI83
calculator has a builtin randInt program, so the demo can be
effectively performed using TI83 calculators. The command,
randInt(1,25,k), generates k random integers between 1 and 25. For additional information on using
the TI83 click here.
You will see a PDF file developed by Dr. Roseanne Hofmann for introducing
the use of random integers and the Virus Spread demo. Included are
calculator screen pictures.
Software available
If you have a large class the physical
aspects of this demo may not be practical. However you can simulate the
demo using software.
If you have MATLAB available in the
classroom, a utility to generate
the random numbers is available by clicking here.
Credits: This demo was submitted
by
Dr.
Roseanne Hofmann Department of Mathematics Montgomery
County Community College
and is included in Demos with Positive Impact with her permission.
The MATLAB routines were written by David R. Hill for this project. The
randmint program for the TI89 and the Mathematica notebook were written
by Lila F. Roberts. The Java applet was developed by Saritha
Somasundaram, a student at Temple University.
