Example 1.  Use the method of "data linearization" to find the logistic curve that fits the data for the population of the U.S. for the years 1900-1990.  Fit the curve    to the census data for the population of the U.S.

Solution 1.

Enter the data points into a two dimensional array using millions.  Be careful with your typing !

Next, a limiting population L, or "carrying capacity" must be estimated.  For this data the number L is not too sensitive, but must be larger than the largest ordinate so that the values    are not complex numbers. For illustration, we choose  L = 800  million.

Do a series of intermediate computations.

Now glue together the transformed parts to form the pairs  .  

Now use the Mathematica procedure  Fit  to get the least squares line in XY-space.  Then we shall graph this line in the transformed XY-plane.  

Plot the least squares line in XY-space.

So the coefficients  A  and  B  are located at nodes  (2,1)  and  (1), respectively:

Use    and  a = A  to get the coefficients of     back in the original  xy-plane.

When we form the function, we must adjust "x" because we shifted the abscissas to the left.  The actual form of the answer is a little different than what we original planned.

Now graph the function  .  

(c) John H. Mathews 2003