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Topic: Exponential functions
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Posts: 447
Registered: 12/3/04
Exponential functions
Posted: Jun 4, 2001 12:49 PM

The following is from the NY Times. It's a nice example of how
exponential function grow, and, of course, it is timely.

June 4, 2001
The Mirage of a Growing Fuel Supply

SCOTTSDALE, Ariz. When I discussed the exponential function in
the first-semester calculus classes that I taught, I
invariably used consumption of a nonrenewable natural resource
as an example. Since we are now engaged in a national debate
about energy policy, it may be useful to talk about the
mathematics involved in making a rational decision about
resource use.

In my classes, I described the following hypothetical
situation. We have a 100-year supply of a resource, say oil --
that is, the oil would last 100 years if it were consumed at
its current rate. But the oil is consumed at a rate that grows
by 5 percent each year. How long would it last under these
circumstances? This is an easy calculation; the answer is
about 36 years.

Oh, but let's say we underestimated the supply, and we
actually have a 1,000-year supply. At the same annual 5
percent growth rate in use, how long will this last? The
answer is about 79 years.

Then let us say we make a striking discovery of more oil yet
-- a bonanza -- and we now have a 10,000-year supply. At our
same rate of growing use, how long would it last? Answer: 125

Estimates vary for how long currently known oil reserves will
last, though they are usually considerably less than 100
years. But the point of this analysis is that it really
doesn't matter what the estimates are. There is no way that a
supply-side attack on America's energy problem can work.

The exponential function describes the behavior of any
quantity whose rate of change is proportional to its size.
Compound interest is the most commonly encountered example --
it would produce exponential growth if the interest were
calculated at a continuing rate. I have heard public
statements that use "exponential" as though it describes a
large or sudden increase. But exponential growth does not have
to be large, and it is never sudden. Rather, it is inexorable.

Calculations also show that if consumption of an energy
resource is allowed to grow at a steady 5 percent annual rate,
a full doubling of the available supply will not be as
effective as reducing that growth rate by half -- to 2.5
percent. Doubling the size of the oil reserve will add at most
14 years to the life expectancy of the resource if we continue
to use it at the currently increasing rate, no matter how
large it is currently. On the other hand, halving the growth
of consumption will almost double the life expectancy of the
supply, no matter what it is.

This mathematical reality seems to have escaped the
politicians pushing to solve our energy problem by simply
increasing supply. Building more power plants and drilling for
more oil is exactly the wrong thing to do, because it will
encourage more use. If we want to avoid dire consequences, we
need to find the political will to reduce the growth in energy
consumption to zero ÃÂ or even begin to consume less.

I must emphasize that reducing the growth rate is not what
most people are talking about now when they advocate
conservation; the steps they recommend are just Band-Aids. If
we increase the gas mileage of our automobiles and then drive
more miles, for example, that will not reduce the growth rate.

Reducing the growth of consumption means living closer to
where we work or play. It means telecommuting. It means
controlling population growth. It means shifting to renewable
energy sources.

It is not, perhaps, necessary to cut our use of oil, but it is
essential that we cut the rate of increase at which we consume
it. To do otherwise is to leave our descendants in an
impoverished world.

Evar D. Nering is professor emeritus of mathematics at Arizona
State University.

Copyright 2001 The New York Times Company



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