A Closer Look at US Population Growth

  Since undergoing open heart surgery last month I've been trying to analyze the US census data in hopes of being able to make a better prediction of future growth. If we assume the rate of change in the number of people, X, is a linear function of X and integrate we find X is a simple exponential function of the time.  

Separating factors involving the variable X from constant factors we can simplify the function that we need to fit.

If one tries to fit blocks of five census population observations one finds the exponential rate of growth is not constant but slowly decreases. 

After 1920 the rate drops to zero for the exponential term. The reason for this is X is replaced by a linear function of time.

So instead of the rate of change of X being a quadratic function of X which involved a carrying capacity K, the census fits suggest the constants for the fits changed with time and the current rate of change in the US population is now a linear function of time. 

Supplemental (Oct 9): The census fits were 3 point fits centered on the year indicated and using the census values +/- 20 years.