Biology Reference
In-Depth Information
Year
U.S. Population (millions)
1800
5.3
1810
7.2
1820
9.6
1830
12.9
1840
17.1
1850
23.2
1860
31.4
TABLE 1-1.
Population of the United States from 1800 to 1860.
U.S. Population
35
30
25
20
15
10
5
0
0
1
2
3
4
5
6
7
Time [decades] after year 1800
FIGURE 1-1.
Plot of U.S. population versus time. A graph of the data shown in Table 1-1.
time t, the data points in Figure 1-1 may have come from sampling the
function P(t)
at
2
or P(t)
at
3
, where a
0 is a constant, or some other
power law. It may also be that the data follow an exponential law of
increase with the general form P(t)
¼
¼
>
ae
bt
where a
0 are
constants. To determine the specific nonlinear function that provides the
best fit for the data, we examine the change in U.S. population per decade;
that is, the rate of change. In our example, they appear to be growing with
time—the population change is 1.9 million from 1800 to 1810 but 8.2
million from 1850 to 1860 (more than four times as large). Thus, the rate of
population growth increases as the U.S. population increases.
¼
>
0 and b
>
These observations lead to two different ways of plotting the data: (1) The
change in population size per decade versus time, and (2) the change in
population size per decade versus population size at the beginning of
decade. While the graph in Figure 1-2(A) is still not very telling,
the one in Figure 1-2(B) is strikingly linear. Is this amere coincidence, or are
