 UBC Calculus Online Course Notes 
Doublings: The exponential function with base 2On the last page, we studied how the process of doubling can lead to a very sharp increase after a few "cycles" (or doubling time periods) of growth. But what would we see if we made observations at more frequent intervals? How big would the population be? For example, we could think of the following experiment:
An E. coli culture is initiated with a drop of fluid containing 100 cells. The cells are allowed to grow and multiply for 2.5 hrs. The experiment is stopped, and a cell count is made. How many cells would be found in the culture at the conclusion of the experiment?
You might wonder what happens to the bacteria between the doublings. Do they all divide at once exactly every twenty minutes? In that case, we would think of this graph as a sequence of steps, with the heigths of each of the steps getting bigger and bigger. Or do some cells divide sooner and others later so that the growth looks smooth? Although the first situation can occur (in specially designed cultures of cells in which the cell cycles are synchronized) it is rather rare in nature. It is more likely that the cells will divide at intervals which are not exactly synchronized. Even though each division will make the number of cells increase by an integer, the effect of each additional cell on the whole population will be very tiny. (Remember that after a few divisions, we will be dealing with thousands and then millions of cells, so that the axes we use to follow this graphically will hardly detect change by one or two cells.) This leads us to think that perhaps the process of growth need not be described as discrete jumps from one level to the next, but rather as a smooth transition. We might think about defining a new function that represents the growth more smoothly.
