0 1 1
1 10 2
2 100 4
3 1,000 8
4 10,000 16
5 100,000 32(^) However, when the term “exponential function” is invoked, the meaning frequently
is the sequence of values for y = ex, where “e” is the irrational number 2.71828. ...
This particular exponential function has the special property that the slope of the
function (the change in y divided by the change in x, or dy/dx) at x = 0 is 1, and that
the [slope of ex] = dex/dx = ex at all x. This function turns out to be (yes, lots of
mathematics is hidden in that phrase) the exact relationship for any compound-interest
problem when bank interest is compounded continually. The important thing isn’t the
compounding interval, but the interest rate (however much banks may try to convince
you otherwise). Let’s try an example. Let the interest rate, r, be 8% per year. If
interest is compounded once per year, then the principal at T years PT = Po(1 + r)T. If
it is compounded n times per year, then it is PT = Po(1 + r/n)nT. If it is compounded
continuously, then the principal at T is PT = PoerT.
(^) Let Po be $1000 (or yen, or rubles, or euros):
Even continuous compounding doesn’t get you much. In fact, a change from 8% to
8.3%, i.e. a small change in the annual rate, is all it takes to cover all the possible
effect of more frequent compounding. This continuous compounding is an excellent
model for many processes such as the decline in concentration of phytoplankton when
animals are filtering parts of the water and returning the water to the suspending
volume.
(^) Let us expand slightly on the mathematics for downward light extinction; the results
are so important that immediate review is in order. The continuous-interest formula
(or exponential function) turns out to be the solution (or integral) for an equation for
the slope at any point of the curve of light intensity vs. depth. Such equations are