Question

the table below gives the population in a small coastal community for the period 1990 - 1999. figures shown are for january 1 in each year.

year	1990	1991	1992	1993	1994	1995	1996	1997	1998	1999
population	650	878	1302	1553	1618	1488	975	838	778	733

(a). what was the average rate of change of population between 1991 and 1994?
(b). what was the average rate of change of population between 1995 and 1997?
(c). for what period of time was the population increasing? from to
(d). for what period of time was the population decreasing? from to
note: for (c) and (d), give the starting and ending years for the periods where the population was increasing/decreasing.

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[x_1,x_2]$ is $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$. Here, $x$ is the year and $y$ is the population.

Step2: Calculate average rate of change for (a)

For the years 1991 and 1994, $x_1 = 1991$, $x_2=1994$, $f(1991)=878$, $f(1994)=1618$.
$\frac{f(1994)-f(1991)}{1994 - 1991}=\frac{1618 - 878}{3}=\frac{740}{3}\approx246.67$

Step3: Calculate average rate of change for (b)

For the years 1995 and 1997, $x_1 = 1995$, $x_2=1997$, $f(1995)=1488$, $f(1997)=838$.
$\frac{f(1997)-f(1995)}{1997 - 1995}=\frac{838 - 1488}{2}=\frac{- 650}{2}=-325$

Step4: Determine increasing period for (c)

We check when the population values are getting larger. The population is increasing from 1990 to 1994.

Step5: Determine decreasing period for (d)

We check when the population values are getting smaller. The population is decreasing from 1994 to 1999.

Answer:

(a) $\frac{740}{3}\approx246.67$
(b) $- 325$
(c) From 1990 to 1994
(d) From 1994 to 1999