UniCalc: Lab 3
(Adapted from NCSSM’s Contemporary Calculus Through Applications, Jansen, 1996). The purpose of this
activity is to explore how a function (or at least some values of the function) can be found based on
knowledge of derivatives. We will essentially explore graphical and numerical solution techniques to
differential equations. We will learn a little later about analytic solution techniques.
Instructions:
Thomas Malthus in 1798 proposed a model for population growth that could be represented by the
differential equation
dP
 kP , where P(t) is the population at time t and k is a (positive) constant [think of
dt
k as the difference between birth and death rates].
1. Use the U.S. Census data on the first page of the spreadsheet to estimate population growth rates for
each year; then, using data from 1790 – 1860 only, come up with a value for k in the model. Some
P
suggestions: 1) Look at the ratios of
t for each year, and let k be the average of those ratios, or
P
2) Graph P ’ vs. P; let k be the slope of the line of best fit (Excel will give you a linear “Trendline,”
and even let you force a y-intercept of 0).
2. Go to the “Malthus A” page. Use Euler’s method to predict population with a step size of 1 year and
the k value you found in part 1, for the years 1790-1860. Examine the difference between actual and
predicted populations at the decade points; if necessary, revise your value of k to minimize the
differences. [You’ll want to graph actual and predicted populations as functions of the year—
highlight all 3 columns, and make a scatter plot].
3. Once you are satisfied with the value for k that “best” fits the data for 1790 – 1860, use it with Euler’s
method (t = 1 yr) and construct a table which compares predicted and actual populations for each
year, 1790-2000 (set up on “Malthus B” page). What factors in the growth of the U.S. population
might account for discrepancies between the predicted values and the actual values for these later
years?
4. About 40 years after Malthus’ model was published, P.F. Verhulst proposed a modification to account
for the capacity of the Earth to sustain human population. Using M to represent the maximum
capacity, the model he proposed was
dP
P

 kP1   . Note that if P is small relative to M, the
dt
 M
term (1 – P/M) is close to 1, so this model is similar to Malthus’. On page “Verhulst A” set up
Euler’s method to predict population for the years 1790-1910. Start with the same k as in part 3, and
start with M about 2-3 times the current population of the U.S. Adjust k and M to make your
prediction fit the data as closely as possible.
5. On page “Verhulst B,” use the k and M you found in part 4 to extend the prediction through 2000.
What factors in the growth of the population in this time span might account for any discrepancies?
What are the implications for U.S. population of the value for M that you used?
6. In the 1960’s, an electrical engineer named Heinz von Foerster tinkered with Malthus’ model a bit,
claiming that technological improvements speed up growth. His model is
dP
 kP1 r , where r is
dt
some positive constant. Notice that in this model P  is a power function of P. Use the world
population data on page “von Foerster” to estimate growth rates; then plot P  vs. P (scatter plot).
Have Excel give you a power Trendline for this plot, then set up Euler’s method using von Foerster’s
model to predict world population in each decade (so that t = 10) for the next 400 years (the second
set of data columns is set up by decade for you). This model is also known as the “doomsday
model.” Why do you think so? From a graph of the predicted world population, about when does
“Doomsday” occur?
Calculus 1 Lab 3 (continued)
7. Close the Excel file; start the program “MDEP” by double-clicking on its icon. Press “Enter” and
change the Function Type to “Nth Order ODE”/Euler. Press F1 to “edit” the formula for y’; enter the
differential equation you used in part 6 above. Press Ctrl-g to get a graph. From the graph screen, set
up the Window to have x from 0 - 600 and y from 0 up to a “fairly large population” (how many
people do you think the Earth can hold?). Press I to enter Initial conditions with x0 = 0 (setting year
1750 to be zero, so that year 2350 will be correspond to x = 600) and y0 to be the population in 1750.
Press L to cLear the graph, D to obtain a Direction field, then G to re-Graph. You now have a slope
field, with the prediction curve corresponding to the particular solution to the differential equation!
Lab Report:
Although this is a Lab, your “report” will be different than before. You should submit your Excel files,
complete with graphs from parts 3, 5, 6, and 7 above, clearly showing the values for k, M, and r that
you used with each, and answer the questions in parts 3, 5, and 6 (there are blue-bordered text boxes on
appropriate pages of the spreadsheet for your convenience).
The creative/narrative part is: alone or with your partner, come up with a television news-style narrative
describing something you learned or examined as part of the lab. You could be “responsible” journalists
(reporting the news as it happened), sports journalists, meteorologists, or tabloid journalists (using
hyperbole and imagination to sensationalize the news). Your report does not need to be limited to the
mathematics that you did. In fact, some of the answers to some of the questions don’t have anything to
do with math, so your report might not either! If you have a partner, you must have graphics to go along
with your news story (mathematical content irrelevant).
Scheduling issues:
I expect that you will be able to complete all computer portions of this lab during class periods. You
should prepare your oral presentation as homework. You do NOT have to be finished with the
spreadsheet work in order to begin planning your news report!
In use: I gave five in-class days for work on the lab, then had Monday “news” (which I videotape, and
play for subsequent years).
BOTTOM LINE: HAVE A LITTLE FUN WITH THIS LAB REPORT.