When we talk about average rate of change, what we're really talking about is the total
change in the dependent variable over the total change in the independent variable.
A lot of discussions about average rate of change relate to speed.
And in that case, the average rate of change represents the total distance covered over
a certain amount of time.
But really, we can talk about any change in y over any change in x, regardless of what
x and y represent.
And this brings us to the formula for the average rate of change between two points
on a function.
The average rate of change of a function is given by f(x_2) minus f(x_1) over x_2 minus
x_1, where (x_1, f(x_1)) is the starting point of the interval we are interested in and (x_2,
f(x_2)) is the endpoint.
If you've spent any time studying linear equations, you'll probably recognize that this formula
for average rate of change is exactly the same as the equation to find the slope of
the line!
More specifically, the formula for the average rate of change calculates the slope of the
secant line between two points on the graph of a function.
A secant line is simply a straight line connecting two points on a function, which is exactly
what the average rate of change calculates.
The average rate of change of a function can be negative, positive, or even zero, just
like the slope of the secant line.
The average rate of change of a function will be negative when f(x_2) is less than f(x_1),
which makes sense both when we draw the secant line on the graph and when we realize what
subtracting f(x_1) from f(x_2) will do to our equation.
Likewise, the average rate of change will be positive whenever f(x_2) is greater than
f(x_1).
In the same way, if f(x_2) and f(x_1) are equal to each other, the secant line will
be horizontal and the numerator of the formula will be zero, so the average rate of change
will also be zero.
This can be tricky, because even if the average rate of change is zero, it doesn't necessarily
mean that no change occurred.
After all, if I threw a ball into the air, and it came right back into my hand, the graph
of its change in height over time would look like this.
The ball definitely moved, but if I asked you for the average rate of change from this
point to this point, you would get zero.
So clearly, finding the average rate of change is not the same thing as knowing the rate
of change at any particular moment.
Rate of change is represented on a graph as the slope of a curve, and looking at this
function and the secant line found using the average rate of change formula, it's clear
that the average rate of change doesn't tell us anything at all about the actual rate of
change at this point or the slope of the curve at this point.
To find out more about these, we would need to calculate the instantaneous rate of change
by using the derivative of the function.
In a lot of cases though, we only need to know the average rate of change of a function,
like if we were writing a report on marathon runners.
It would be pretty boring to report the speed of the runner at every given second, but it
is pretty cool to know a runner's average speed over the course of the race.
Let's look at how we could figure that out.
Say we're given this table of this runner's time for each mile of the marathon.
We could easily find her average speed at each mile by plugging into the formula the
change in distance, one mile, over the amount of time elapsed.
But if we wanted to find her average speed over the course of the race, we would need
the total change in distance, 26.2 miles, over the total time elapsed.
In a table like this, where the times are not cumulative, we would need to sum up the
increments of time in order to plug the right amount into the average rate of change formula.
So to summarize, we've talked all about average rate of change and its relationship to the
secant line.
We now know how to calculate the average rate of change when given two points on a curve
to use as an interval, and we know how to use information from a chart to do the same
thing.
We definitely know that the average rate of change doesn't tell us everything there is
to know about a function, but it does give us a general idea about its behavior!