Let P(t) be the population of the town t years after the initial population of 245.
The growth rate is 6%, or 0.06, so the population after one year is 1.06P(0), the population after two years is
1.06(1.06 P(0)) = 1.06^tP(0), and in
general, the population after t years is 1.06^t P(0).
Therefore, the function that describes how fast the population grows each year is the derivative of this function, which is:
d/dt[1.06^tP(0)]= 1.06^t * 0.06 * P(0)
~ NOTE THE ‘ * ‘ are the times sign
This function tells us how much the population grows each year as a percentage of the initial population P(0).
For example, if P(0) = 245, then the
population grows by approximately 6.3% each year.