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

Section 3.7 Rates of Change in the Natural and Social Sciences 229
Biology
ExamplE 6 Let n − f std be the number of individuals in an animal or plant popu lation
at time t. The change in the population size between the times t − t 1 and t − t 2
is Dn − f st 2 d 2 f st 1 d, and so the average rate of growth during the time period
t 1 < t < t 2 is
average rate of growth − Dn
Dt
− f st 2d 2 f st 1 d
t 2 2 t 1
The instantaneous rate of growth is obtained from this average rate of growth by letting
the time period Dt approach 0:
Dn
growth rate − lim − dn
Dt l 0 Dt dt
Strictly speaking, this is not quite accurate because the actual graph of a population
function n − f std would be a step function that is discontinuous whenever a birth or
death occurs and therefore not differentiable. However, for a large animal or plant
population, we can replace the graph by a smooth approximating curve as in Figure 7.
n
FIGURE 7
A smooth curve approximating
a growth function
0
t
To be more specific, consider a population of bacteria in a homogeneous nutrient
medium. Suppose that by sampling the population at certain intervals it is determined
that the population doubles every hour. If the initial population is n 0 and the time t is
measured in hours, then
Eye of Science / Science Source
E. coli bacteria are about 2 micrometers
(mm) long and 0.75 mm wide. The
image was produced with a scanning
electron microscope.
f s1d − 2f s0d − 2n 0
f s2d − 2f s1d − 2 2 n 0
f s3d − 2f s2d − 2 3 n 0
and, in general,
f std − 2 t n 0
The population function is n − n 0 2 t .
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