James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 1.4 Exponential Functions 49
It seems from this pattern that, in general,
pstd − 2 t 3 1000 − s1000d2 t
Table 1
t
(years since 1900)
Population
(millions)
0 1650
10 1750
20 1860
30 2070
40 2300
50 2560
60 3040
70 3710
80 4450
90 5280
100 6080
110 6870
This population function is a constant multiple of the exponential function y − 2 t , so it
exhibits the rapid growth that we observed in Figures 2 and 7. Under ideal conditions
(unlimited space and nutrition and absence of disease) this exponential growth is typical
of what actually occurs in nature.
What about the human population? Table 1 shows data for the population of the world
in the 20th century and Figure 8 shows the corresponding scatter plot.
P
5x10'
figure 8
Scatter plot for world
population growth
0
20 40 60 80 100 120
Years since 1900
t
The pattern of the data points in Figure 8 suggests exponential growth, so we use a graphing
calculator with exponential regression capability to apply the method of least squares
and obtain the exponential model
P − s1436.53d ? s1.01395d t
where t − 0 corresponds to 1900. Figure 9 shows the graph of this exponential function
together with the original data points. We see that the exponential curve fits the data reasonably
well. The period of relatively slow population growth is explained by the two
world wars and the Great Depression of the 1930s.
P
5x10'
FIGURE 9
Exponential model for
population growth
0
20 40 60 80 100 120
Years since 1900
t
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