. For example let a=2 and b=1, then the
graph is like that on the right. Again notice that the graph
rises rapidly as you move from left to right and that the rate of
change increases as you move from left to right. A third type of growth and possibly the most common growth
function in biology is exponential growth. The formula for
exponential growth is given by . For example let a=2 and b=1, then the
graph is like that on the right. Again notice that the graph
rises rapidly as you move from left to right and that the rate of
change increases as you move from left to right.
The tabular form of the function is shown below. The first three columns are like the first three columns of the previous functions. The fourth column, on the other hand, is different than any that we have calculated so far. Here the difference quotient is divided by y. In this case the second item in column 4 is 4.670774 divided by 2.718282 (check it on your calculator). The fact that this column is a constant tells me that the function is exponential.
| t | y(t) | dy/dt | (dy/dt)/y |
| 0 | 1 | ||
| 1 | 2.718282 | 1.718282 | 1.718282 |
| 2 | 7.389056 | 4.670774 | 1.718282 |
| 3 | 20.08554 | 12.69648 | 1.718282 |
| 4 | 54.59815 | 34.51261 | 1.718282 |
| 5 | 148.4132 | 93.81501 | 1.718282 |
| 6 | 403.4288 | 255.0156 | 1.718282 |
| 7 | 1096.633 | 693.2044 | 1.718282 |
| 8 | 2980.958 | 1884.325 | 1.718282 |
| 9 | 8103.084 | 5122.126 | 1.718282 |
| 10 | 22026.47 | 13923.38 | 1.718282 |
The logarithm of the function: 
is a linear
function of t. Thus a straight line plot on semi-log paper would
indicate an exponential model.
