Appendix A: Mathematics of Exponential Growth

Exponential growth is characterized by the speed of growth being proportional to the size of the thing that is growing the bigger the thing gets, the faster it grows. This is another way of saying that the rate of growth is constant when the rate of growth is measured as the percentage increase over a fixed unit of time. If a church grows exponentially, its size at any point in time t is given by  (4)  where A is the size of the church at  and δ is the continual growth rate, or the force of growth. If delta is a positive constant then the church is growing exponentially, and if it is a negative constant then the size of the church is decaying exponentially.  If we know the size of the church at the beginning and ending of a time period and we assume it grew exponentially over that time period, we can calculate that period's  by    (5)  Over a 1-year period, the change in the size of the church can be broken down into 3 components: the number of converts baptized, denoted b; the number of children of record baptized, denoted c; and the number of decrements due to death, excommunication, or resignation, denoted d. Thus the number of members at time  is equal to    (6)  Substituting (6) into (5) results in the formula  (7)  Which is equivalent to  (8)  It follows that for a church to grow at a constant exponential rate, the following must be constant for all years:  (9)  That is nothing other than the rate of growth expressed on an annual basis, broken down into 3 components.

But what if the magnitude of  changes over time? Let  be the value of  at any given point in time. Assuming that the function  is continuous, formula (4) can be generalized as          (10)    