Mathematics 162                                  Laboratory 2                            Week of Feb. 1, 1993
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

© Lafayette College, 1994.

Exponential Growth

In this lab, you will examine an exponential growth model for the population of the United States.  First, we will make a brief comparison of exponential and polynomial growth rates.

Part 1:  Exponential and Polynomial Growth

Use

Plot[{E^x, x^2},{x, 0, 5},PlotStyle->{Red,Blue}]

to display the graphs of y = ex and y = x2 on the same set of axes.  Write one or two sentences describing the growth rates of these functions over the interval [0, 5].

Now modify the command to compare the functions ex and x4 on the same interval.  How do the growth rates of these functions compare over [0, 5]?

Now compare the second pair of functions on the interval [0, 12].  How do their growth rates compare on this interval?

Formulate a hypothesis relating the growth rate of ex  to the growth rates of power functions, xr where r is any positive number.  Ideally, your hypothesis should be consistent with all the graphs you have created above, and still be concise—a single sentence.

Part 2:  An Exponential Model for U.S. Population

The following table gives information on the population of the United States in the years 1790 through 1890.

The information in this table has been stored in the file dataset1.dat.  To retrieve the data and give it the name popdata1, enter

popdata1 = <<i:\course\root\math162\dataset1.dat

Each ordered pair {t,p} gives the time t in years from 1790 and the population that year p in millions.  Notice that time is being measured from 1790 (the year of the 1st Census).  So, t = 0 corresponds to 1790, t = 10 corresponds to 1800, and so forth.

Use

plot1 = ListPlot[popdata1, PlotStyle->Red,
                 PlotRange->{0,65},AxesOrigin->{0,0}]

to plot the data points and give the graph the name plot1.  Do these points appear to lie near the graph of an exponential function or a power function?  Try to explain what made you choose the function you did, but don't spend a lot of time.

If we assume that the population of the U.S. grows at a rate proportional to the current population, then the population in year t, p(t), should satisfy the differential equation

where k is a constant.  Any function satisfying this equation can be written in the form

where a is another constant.  Define the "template" function p(t) by

p[t_] := a E^(k t)

and use the populations given for 1790 and 1840 to determine appropriate values for a and k, using the command below:

Solve[{p[0] == 3.929, p[50] == 17.07},{a,k}] //N

You will need to choose the only real-valued solution pair from the many pairs Mathematica offers.  (Mathematica uses I to represent the square root of –1.)  Assign the solution pair you want the name soln1; for instance, if the second pair is the one with only real values, enter soln1 = %[[2]].

The command

model1[t_] = p[t] /. soln1

uses the solutions you have found to define a particular function that models the population growth of the United States.  We use p[t] /. soln1 to cause a and k in the formula for the template function p to be replaced by the values given in our chosen solution pair.  Write the definition for model1[t] below.

Use

plot2 = Plot[model1[t], {t,0,100}, PlotStyle->Blue]

to plot your population function and give the graph the name plot2.  Then you can use

Show[plot1, plot2]

to show your plot of the data points and your plot of model1[t] on the same set of axes.  Does your model fit the data well?  Where is the fit the best, and where is it the worst?  Offer reasons (mathematical and/or historical) why these are reasonable places for the best and worst fits.

There was no special reason to use the populations in 1790 and 1840 to determine the values of a and k.  Repeat the steps above using the data points p(0) = 3.929 and p(100) = 62.95 to obtain a new model, model2, of the population.  Make a plot, plot3, of that model and then use

Show[plot1, plot2, plot3]

to show your plot of the data points and your plots of the two models on the same set of axes.  Compare the fits of the two models, indicating where each is best and worst.  Which is the best model overall?  Justify your answer based on a visual analysis of the plot.

Part 3:  A Test of the Exponential Model

Is an exponential model really appropriate in this situation?  Here is way to test the fit of an exponential model to the data you were given.  Notice that if we take the natural logarithm of both sides of our equation  p(t) = aekt, we get

Now suppose we were to plot time on the horizontal axis and the logarithm of the population on the vertical axis.  If an exponential model fits our data, what will the graph of ln(p(t)) vs. t be?  (Hint: In equation above, replace ln(p(t)) by y on the left, and t by x on the right.  What does the graph of the new equation look like?)

Use the following steps to make a plot of ln(p(t)) vs. t for the data you have been given.

time = Table[popdata1[[n]][[1]], {n, 1, 11}]

pop = Table[popdata1[[n]][[2]], {n, 1, 11}]

ListPlot[Table[{time[[n]],Log[pop[[n]]]}, {n, 1, 11}]]

What conclusion do you draw from this plot about the fit of an exponential model?  Explain your conclusion; you may find a printed copy of the plot useful.

Part 4:  Predictions of Future Populations

If the population of the United States is really growing at an exponential rate, then we should be able to use model2 to predict future populations of the U.S.  What population would it predict for the year 2000?  Does this prediction seem reasonable to you?

U.S. population data for the years 1790 - 1980 is stored in the file dataset2.dat.  Retrieve this file and give the data the name popdata2 by entering

popdata2 = <<dataset2.dat

Use ListPlot to make a plot of this data (like plot1 above, but change the PlotRange option to {0,250}).  From your plot, does it appear that the population of the U.S. has continued to grow at an exponential rate?