Page 185 - Calculus Demystified

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hence CHAPTER 6 Transcendental Functions

2 cos x
2 x
F (x) = + + ln 5 ·[x · (sin x) · 5 ].
x sin x
You Try It: Calculate (d/dx)[(ln x) ln x ].

6.5 Exponential Growth andDecay

Many processes of nature and many mathematical applications involve logarithmic
and exponential functions. For example, if we examine a population of bacteria,
we notice that the rate at which the population grows is proportional to the number
of bacteria present. To see that this makes good sense, suppose that a bacterium
reproduces itself every 4 hours. If we begin with 5 thousand bacteria then
after 4 hours there are 10 thousand bacteria
after 8 hours there are 20 thousand bacteria
after 12 hours there are 40 thousand bacteria
after 16 hours there are 80 thousand bacteria ...
etc.
The point is that each new generation of bacteria also reproduces, and the older
generations reproduce as well.Asketch (Fig. 6.12) of the bacteria population against
time shows that the growth is certainly not linear—indeed the shape of the curve
appears to be of exponential form.

Fig. 6.12