Awhile back it occurred to me that with gas prices as high as they are, it doesn’t seem to do any good to shop around for the cheapest price. I was taking a business calculus class at the time, so I set about to prove it mathematically.
First, I assigned the price of gas per gallon to the variable x. Then, for simplicity, I decided to study the amount of gas you can buy for $1.00 — because it’s usually very easy to calculate things involving the number 1. Expressed algebraically, the amount of gas you can buy for $1.00 is $1.00 over x, or f(x)=1/x. (Proof: if gas is $1/gallon, f(x)=1/1=1 gallon; if gas is $2/gallon, f(x)-1/2=half a gallon.)
Enter calculus; the first derivative of that simple function would tell us the instantaneous rate of change of how much gas you could buy for a dollar. In English, that would tell us at any given price how quickly the amount of gas you can buy is dropping as 
the price goes up. That first derivative is negative one over x squared, or f’(x)=-1/x^2. (Proof: The first derivative tells you the slope of a curve at a given point, so at x=1.4, 1/x=.714 and x’=-1/1.96=-0.51. We should be able to approximate 1/x at 1.5 by adding x’ times .1 to 1/x: -0.51*.1+.714 = .66. 1/1.5 = .66.)
The second derivative will tell us what the slope of the tangent line of the instantaneous rate of change of how much gas you can buy for a dollar is. Or, how big the rate of quantity loss at any given price. The second derivative is two over x cubed, or f”(x)=2/x^3. This is what I would call the significance of gas price differences.
That is what this graph is showing you (click on it for the full, clear version). As you can see, from $1.00 to $2.00 it was fairly important to choose the gas station with the lowest price. From $2.00 and especially $2.50 on up, it hardly matters any more which gas station you go to. Gas is too expensive for it to matter whether you buy gas for $2.799 or $2.699.
In conclusion, don’t waste your time going out of your way to find a cheap gas station any more. Your driving habits are now more significant to your gas budget than which gas station you stop at: you can save more money monitoring the way you move your feet.
[I wrote this January 6th, 2006 and posted it on another blog. It still seems relevant today.]
