11.06 Average rates of change
Recall that the slope of a linear function is also its rate of change. For a linear function, the slope is always the same, therefore we say that the rate of change is constant.
For a non-linear function, the rate of change varies along the graph. Therefore, we say a non-linear function has a variable rate of change.
To get an idea of how the graph of the function changes, we can take the average rate of change over a specific interval of the domain.
If the variable $x$x represents the independent variable of a function, and $y$y represents the dependent variable of a function, then
$\text{Average rate of change}=\frac{\text{change in }y\text{ values}}{\text{change in }x\text{ values}}$Average rate of change=change in y valueschange in x values
Exploration
Suppose that the relationship between the population ($n$n) of a colony of bacteria and the time ($t$t, in days) is described by the relationship $n=2^t$n=2t. We want to find the growth rate, which is the rate of change of the population. We can plot this relationship:
To find the average rate of change, we can draw a secant line. A secant is a line which touches a curve at two specific points. We can choose two points on this curve, $\left(1,2\right)$(1,2) and $\left(5,32\right)$(5,32), and draw the line connecting them, $y=7.5x-5.5$y=7.5x−5.5:
This secant has a slope of $7.5$7.5 so we say that the average rate of change from day $1$1 to day $5$5 is $7.5$7.5 per day.
Worked example
The volume of a lake over five weeks has been recorded below:
| Week | $0$0 | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 |
|---|---|---|---|---|---|---|
| Volume (m3) | $123000$123000 | $142000$142000 | $135000$135000 | $111000$111000 | $104000$104000 | $123000$123000 |
(a) Find the average rate of change of volume in the first week
(b) Find the average rate of change of volume over the whole five weeks
(c) Find the average rate of change of volume in the last three weeks
Think: We don't know the function mapping the week to the volume. However, the average rate of change only requires two points on the graph. So we can find the average rate of change from just the data points.
Do: For each period, the average rate of change will be the change in volume divided by the number of weeks:
| (a) | average rate of change of volume in the first week | $=$= | $\frac{\text{change in volume}}{\text{number of weeks}}$change in volumenumber of weeks |
| $=$= | $\frac{142000-123000}{1-0}$142000−1230001−0 | ||
| $=$= | $\frac{19000}{1}$190001 | ||
| $=$= | $19000$19000 | ||
| (b) | average rate of change of volume over the whole five weeks | $=$= | $\frac{\text{change in volume}}{\text{number of weeks}}$change in volumenumber of weeks |
| $=$= | $\frac{123000-123000}{5-0}$123000−1230005−0 | ||
| $=$= | $\frac{0}{1}$01 | ||
| $=$= | $0$0 | ||
| (c) | average rate of change of volume in the last three weeks | $=$= | $\frac{\text{change in volume}}{\text{number of weeks}}$change in volumenumber of weeks |
| $=$= | $\frac{135000-123000}{5-2}$135000−1230005−2 | ||
| $=$= | $\frac{-12000}{3}$−120003 | ||
| $=$= | $-4000$−4000 |
Reflect: We can tell that the function is non-linear because the average rate of change is variable. Also notice that the average rate of change can be positive, negative or zero depending on the interval we choose. This is a significant limitation of average rates of change.
A secant is a line which intersects with a curve at two points
The average rate of change of a function over an interval is the slope of the secant on the function between the endpoints of the interval
Practice questions
Question 1
Does the graphed function have a constant or a variable rate of change?
Constant
AVariable
B
Question 2
Consider a function which takes certain values, as shown in the table below.
| $x$x | $3$3 | $6$6 | $8$8 | $13$13 |
|---|---|---|---|---|
| $y$y | $-12$−12 | $-15$−15 | $-17$−17 | $-22$−22 |
Find the average rate of change between $x=3$x=3 and $x=6$x=6.
Find the average rate of change between $x=6$x=6 and $x=8$x=8.
Find the average rate of change between $x=8$x=8 and $x=13$x=13.
Do the set of points satisfy a linear or non-linear function?
Linear
ANon-linear
B
Question 3
The graph shows the height of a cricket ball in feet after it is thrown.
What is the rate of change of the height of the ball in the interval between when it is thrown and $t=1$t=1?
What is the rate of change of the height of the ball in the interval between $t=1$t=1 and when it is at its greatest point?
What is the rate of change of the height of the ball in the interval between when it is at its greatest point and $t=3$t=3?
What is the rate of change of the height of the ball in the interval between $t=3$t=3 and when it returns to the ground?
In which interval(s) is the ball traveling at its fastest speed?
Between $t=0$t=0 and $t=1$t=1
ABetween $t=1$t=1 and $t=2$t=2
BBetween $t=2$t=2 and $t=3$t=3
CBetween $t=3$t=3 and $t=4$t=4
DWhat do the negative rates of change in the interval between $t=2$t=2 and $t=3$t=3 and in the interval between $t=3$t=3 and $t=4$t=4 indicate?
The speed of the ball is decreasing
AThe ball is decelerating
BThe ball is below ground level
CThe ball is falling down
D