We are given
, defining the percent
of a population
with a given characteristic with t
= 0 corresponding to 1960 and t
measured in decades.
and its second derivative can be entered into the Y= screen as shown at below.
For this type of function, called a “logistic” model, we know that
is always positive. We set a WINDOW for Y values from 0 to 100 and X values from 0 to 4. The graph
of this function
and the WINDOW are shown below.
It appears that concavity changes from up to down somewhere near t = 1.
To see if the second derivative is 0 at a point
= 1, we graph
the second derivative which is Y5 in the first screen above. A choice of different values for Y makes the second derivative graph
easier to see.
To find the zero of this second derivative, use
and choose item number 2. You should get about 1.
02 which represents the passage of one decade or the beginning of the year 1970 in our problem.