Expert: Josh - 12/10/2003

Question
Josh,

If I graph,

1 2 4 8 16 24 48 96 192 ...

then I thought this was an exponential curve... is it? If not what function does it describe?  Secondly in normal speak what is an exponential curve? A decent link would do!

Pleeease help


Adam

PS thanx for existing  

Answer
Hi Adam,

Thanks for the question. It was a good question by the way:)
The data suggests that this function grows at a rate which doubles every time unit, except, it contains a break point in the middle.
If we represent this sequence as r(n),
when n=0, r(0)=1,
when n=1, r(1)=2 =2*r(0),
when n=2, r(2)=4 =2*r(1),
when n=3, r(3)=8 =2*r(2),
when n=4, r(4)=16=2*r(3),
-/\/\/\-
when n=5, the intermediate condition changes to
r(5)=24, that is, r(5)=2*r(4) no longer holds.
So, strictly speaking, it is not a homogenous exponential curve.

Although you have every reason to model the data as an exponential curve which best fits the description (however, it won't be exact, if you do).
...
But then the trend continues (as far as we can tell)
r(n)=2*r(n-1) for n>=5

To describe this discrete function properly, we need to specify the initial and intermediate value.

Initial condition: r(0)=1.
r(n)= { 2*r(n-1), for 1<=n<5
     { 24,       for   n=5
     { 2*r(n-1), for   n>5
------------------------------------------------
EXPONENTIAL BEHAVIOR IN THE GENERIC SENSE:
All functions of the form f(t)=a^t + C, exhibits exponential behavior. Here, "a" is call the exponential base, "C" is just some arbitrary offset.

Example 1:
If a=2 and C=0, then we have
f(0)=2^0=1, f(1)=2^1=2, f(2)=2^2=4, etc. Evaluating at integer "t" values, we obtain a corresponding sequence {1,2,4,8,16,32,64,128,256,512,1024,...}

Example 2:
If a=3 and C=1, then we have
f(0)=3^0+1=2, f(1)=3^1+1=4, f(2)=3^2+1=10, etc. Evaluating at integer "t" values, we obtain a corresponding sequence
{3,4,10,28,82,244,730,...} which equals
{0,3,9,27,81,243,729,...}+1.

INTERPRETATION OF EXPONENTIALS AND THE EULER CONSTANT "e":
Formally, exponential is synonymous with the Euler constant "e" which has an approximate value of 2.71828182845905. This is a magic number. When used in place of "a" in f(t)=a^t+C, it turns out that many natural phenomena agree with this model. It is also the basis for the general solution to problems described by differential equations. So, exponential is often understood to be f(t)=e^(bt)+C, where e=2.7182...
By changing the exponential decay constant b, we can model various growth rates. If b is negative (b<0), it decays away. If b is positive (b>0), it grows exponentially.

EXPONENTIALS CONTAIN NO ESSENTIAL DISCONTINUITY BY DEFINITION
We normally think of exponentials as a continuous function, which is everywhere differentiable. Specifically, we cannot allow there to be any jump (known as essential discontinuity). A more thorough explanation will be given if you study "limits" in calculus.

For the time being, you may think of a discrete sequence (such as the one you have given) as a sampled version of some underlying function along the continuous curve. If you conduct an experiment, your measurements will generally be discrete.

I'm not sure if you've done "differentiation" and "inverse functions" yet. Amongst all functions, it is the only one whose derivative is a scaled version of itself. In other words, the slope (aka, gradient or rate of change) on an exponential curve is proportional to the value at which you evaluate the slope.

if you let f(x)=exp(ax), f'(x)=df(x)/dx=a*exp(ax).
Exponential also has a counter part, call the natural logarithm, often written as log_e(x) or ln(x).

RELEVANCE OF EXPONENTIALS IN THE REAL WORLD.
Anyway, exponentials describe many physical phenomenon, such as the behavior of electronic circuits with inductors and capacitors, radioactivity in biomedical applications (ever heard of the Urea test for a stomach ulcer causing bacteria), natural population growth and decay, it also serves as economic models.

Cheers,
Josh