Geology Reference
In-Depth Information
Appendix A
Exponential growth and decay
At a number of places in this topic, a situation has given rise to exponential growth or
decay. Although these situations have been analysed without using calculus, as promised,
it may be useful to those who know some basic calculus to present the more rigorous
solution that calculus allows. Exponential behaviour arises in a standard way, so we can
start with a general situation. The resulting solution can then be adapted to particular
situations by appropriately identifying the variables. Some of the particular situations
will be covered in this appendix, whereas others will be covered in later appendices.
A.1 Exponential solution
Suppose something is growing larger, and the rate at which it grows is proportional to its
present size. Let's call the something
y
and its rate of growth
v
. Then our situation is
described by the relationship
v
=
ay.
(A.1)
But if
v
is the rate of change of
y
, we can also write
d
y
d
t
,
v
=
so Eq. (A.1) becomes
d
y
d
t
=
ay.
(A.2)
To solve this equation, we can treat the differentials as infinitesimal increments and
rearrange the equation as
δ
y
y
=
a
δ
t.
(A.3)
Each side is now in a form whose integral is known:
δ
y
y
=
ln(
y
)
,
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