Census figures
for the U. S. population (in millions) are listed in the table below.
Let f
be the function such that P = f(t) is the population (in millions)
in the year t.
| Year | Population | Year | Population | Year | Population |
| 1790 | 3.9 | 1860 | 31.4 | 1930 | 122.8 |
| 1800 | 5.3 | 1870 | 38.6 | 1940 | 131.7 |
| 1810 | 7.2 | 1880 | 50.2 | 1950 | 150.7 |
| 1820 | 9.6 | 1890 | 62.9 | 1960 | 179.0 |
| 1830 | 12.9 | 1900 | 76.0 | 1970 | 205.0 |
| 1840 | 17.1 | 1910 | 92.0 | 1980 | 226.5 |
| 1850 | 23.1 | 1920 | 105.7 | 1990 | 248.7 |
1.
a. Estimate the rate of change of the population for the years 1900, 1945,
and 1990.
Make sure your answer includes the correct units.
b. When, approximately, was the rate of change of the population greatest?
c. Estimate the U. S. population in 1956.
d.
Based on the data in the table, what would you predict for the population
in 2000?
2.
Assume that f is an increasing function (as the values in the table
suggest).
Then f is one-to-one and so it has an inverse function; call the
inverse function g, so t = g(P).
a.
What is the meaning of g(100)?
What are the units here?
b.
What does the slope of the graph of g at P=100 represent?
What are its units?
c. Estimate g(100).
d.
Estimate the slope of the graph of g at P=100.
3.
a. Usually we think of the U. S. population P = f(t) as a smooth
function of time.
To what extent is this justified: what happens if we zoom in at a point
of the graph?
What about events such as the Louisiana Purchase?
Or the moment of your birth?
b. What do we in fact mean by the rate of change of the population at a particular time t?
c. Give another example of a real-world function which is not smooth but is usually treated as if it were smooth.
(From
Hughes-Hallet, Gleason, et al., Calculus, 2nd edition,
Wiley.)
(To plot the data in the table, you need the Data/Matrix Editor. You'll need to enter the data from the keyboard.)