Saw “Up” yesterday. How Pixar manages to be so consistent in their astonishing quality is entirely beyond me. In a bit of a tribute, this Sunday Function is not about any dramatically important special function, but instead it’s about filling a balloon. Air or water, as your preference.
You’ll have noticed that when you start to fill a balloon, its radius expands very rapidly at first before slowing dramatically. A water balloon will go from 1 to 2 inches much faster than it will go from 6 to 7 inches. Why? Because if you fill the balloon at a constant rate, you’re increasing the volume at a constant rate. The volume and radius of a balloon don’t change at the same rate. Pretend the balloon is a sphere for simplicity. The volume is related to the radius in the usual way:
Now we can use calculus to find how that static relationship turns into a relationship between the rates of change of the volume and radius. First, remember that the chain rule will ensure that the rate of change of volume with respect to time is equal to the rate of change of volume with respect to radius times the rate of change of radius with respect to time. A little cumbersome to say in words, but easy in the language of calculus after differentiating both sides:
But the term on the left isn’t some complicated function, it’s just our constant fill rate since the change in volume is just the rate at which we add volume via whatever we’re filling. This is true for water balloons, and a reasonably good approximation for air (which is a little compressible). But since it’s a constant, why not drop the derivative notation and just give it a constant name like the Greek letter rho? Do that and rearrange a little bit and you get our Sunday function.
What it means is that the rate of change of the radius is inversely proportional to the square of the radius at that moment. When the sphere has grown by a factor of five, the rate of growth of the radius has slowed by a factor of 25.
The equation we derived is implicitly a differential equation in the time variable, but we don’t need to go through the effort of solving it to see that we’ve already figured out mathematically why the rate of growth slows. As in making a film, simplicity is often the best choice.