Math 170
Lab 2
Lab Goals
1. Use logistic regression to model the world population to model world population growth.
2. Calculate the average growth rate.
3. Calculate the instantaneous growth rate
4. Compare average growth rate and instantaneous growth rates.
5. Predict the population and marginal population, based on logistic growth.
Procedure
Name___________
Look at Wikipedia page for population estimates by “experts.” Below are population averages from that page, for selected years with the highest
number of scholarly estimates per row (at least three estimates).
0
1000
1200
1300
1500
1700
1750
1800
1850
1900
1950
1960
1970
1980
1990
2000
2010
2.294E+08 2.753E+08 3.985E+08 3.730E+08 4.501E+08 6.248E+08 7.616E+08 9.290E+08 1.235E+09 1.625E+09 2.498E+09 3.035E+09 3.669E+09 4.442E+09 5.278E+09 6.021E+09 6.861E+09
1. Load this data into your calculator and generate a scatter plot.
a) On which intervals is the graph most linear?
b) Approximately when does the population begin increasing rapidly? Conjecture several reasons for this increase.
c) One study on world population growth claims that
Two-thirds of carrying capacity estimates place the number between 4 billion and 16 billion people. World
Population Monitoring 2001.)
Add a single point to your data, set far enough in the future to induce the asymptotic behavior of carrying capacity. Do a bit of research and
pick your carrying capacity wisely. You may have to experiment with this, as you perform logistic regression on the data set. You must have
this point placed far enough in the future, the logistic regression fails!
2. Use your regression graphing utility to fit a logistic function to the data, and record the function below, using proper functional notation.
Visually, how well does the mathematical model fit the data? i) State the logistic function given by your calculator, and ii) Comment about the
accuracy of your logistic model. iii) Use your model to predict the population of the world when you are 50 years old.
Math 170
Lab 2
Name___________
3. Calculate the average rates of change of population with respect to time for the following intervals, using raw data. i) Specifically, what does
“average rate of change mean in this case? ii) Which average rate of change most accurately fits the rate of growth on its interval? Explain.
a) [0, 1000]
b) [1,000, 1800]
c) [1800, 2010]
4. For each of the intervals calculate the derivative halfway between the endpoints using the numerical derivative tool on your calculator. Then
compare your average rate of change with the numerical derivative, centered on each interval. Compare the derivatives to the average rate of change.
That is, calculate the absolute value of the difference between average rate of change and derivative in all three cases.
a) [0, 1000]
b) [1,000, 1800]
c) [1800, 2010]
5. Explain the units and the physical significance of the derivative as a rate of change in exercise 4. Specifically, what do the words, “instantaneous
rate of change of population at a point,” mean here? Also, how does the interpretation of the derivative differ physically from that of “average rate of
change”?
6. Economists use the term “marginal population” to predict the “number of people added to a population 1 year in the future, provided linear growth
occurs. (Think derivative here!) Calculate the marginal population growth following the time points below.
a) 1000
b) 1850
c) 2010