Page 186 - Calculus Demystified

Transcendental Functions
CHAPTER 6
Notice that when the number of bacteria is large, then different generations of 173
bacteria will be reproducing at different times. So, averaging out, it makes sense
to hypothesize that the growth of the bacteria population varies continuously as
in Fig. 6.13. Here we are using a standard device of mathematical analysis: even
though the number of bacteria is always an integer, we represent the graph of the
population of bacteria by a smooth curve. This enables us to apply the tools of
calculus to the problem.

Fig. 6.13

6.5.1 A DIFFERENTIAL EQUATION

If B(t) represents the number of bacteria present in a given population at time t,
then the preceding discussion suggests that
dB
= K · B(t),
dt
where K is a constant of proportionality. This equation expresses quantitatively
the assertion that the rate of change of B(t) (that is to say, the quantity dB/dt)is
proportional to B(t). To solve this equation, we rewrite it as

1 dB
· = K.
B(t) dt
We integrate both sides with respect to the variable t:

1 dB
· dt = Kdt.
B(t) dt
The left side is
ln |B(t)|+ C

and the right side is
Kt + C,