.
A second model is power growth. The model for this type of
growth is given by .
| t | y(t) | dy/dt | d2y/dt2 | d3y/dt3 | d4y/dt4 |
| 0 | 0 | ||||
| 1 | 2 | 2 | |||
| 2 | 16 | 14 | 6 | ||
| 3 | 54 | 38 | 12 | 2 | |
| 4 | 128 | 74 | 18 | 2 | 0 |
| 5 | 250 | 122 | 24 | 2 | 0 |
| 6 | 432 | 182 | 30 | 2 | 0 |
| 7 | 686 | 254 | 36 | 2 | 0 |
| 8 | 1024 | 338 | 42 | 2 | 0 |
| 9 | 1458 | 434 | 48 | 2 | 0 |
| 10 | 2000 | 542 | 54 | 2 | 0 |
An example of power growth is given in the table and graph above with a=2 and b=3. Notice that this function increases faster as you move from left to right.
In the table above the first column is the t values, the
second is the function values, the third is the approximate slope
with the first entry given by 
.Notice that the values in this column
get larger as you go down the column. That is this function grows
faster as you move from left to right. The fourth column is
related to the second derivative on each of the intervals and the
first entry in it is given by 
. The fifth column is related to the
third derivative on each of the intervals and the first entry in
it is given by 
. Since this column is constant the model is a third
degree polynomial or power function which is, of course, what we
started with. The values in the first position of each column of
the table can be used to write the formula for the model (beyond
the scope of this appendix).
Lets look at this model from the calculus standpoint. The
derivative is 
. The second and third derivatives are given by 
and 
respectively. Observe that the third derivative like the third
difference is constant.
Another tool to use with power functions is the log-log plot.
Observe that by taking the logarithm of each of the variables, t
and y(t), a straight line is attained. That is in 
we have a
linear function relating log(y(t)) and log(t). So if a plot y
versus t on log-log paper reveals a straight line then a good
model for the data is a power function. The graph on the left is
a log-log plot of our made-up model. In this example a scatter
plot of the data is constructed and the axes are each changed to a log scale by double clicking on the axis and changing it to a log scale in the dialog box that comes up.