Average Rate of Change
A rate is a value that expresses how one quantity changes with respect to another quantity. For example, a rate in "miles per hour" expresses the increase in distance with respect to the number of hours we've been driving.
If we drive at a constant rate, the distance we travel is equal to the rate at which we travel multiplied by time:
Dividing both sides by time, we have
If we drive at 50 mph for two hours, the distance we'll travel is 50 mph × 2 hrs = 100 miles.
If we drive at a constant speed for 3 hours and travel 180 miles, we must have been driving
In real life, though, we don't drive at a constant rate. When we start our trip through Shmoopville, we first climb into the car, traveling at a whopping 0 miles per hour. We speed up gradually (hopefully), maybe need to slow down and speed up again for traffic lights, and finally slow down back to a speed of 0 when reaching our destination, The Candy Stand. We can still divide the distance we travel by the time it takes for the trip, but now we'll find our average rate:
To calculate the average rate of change of a dependent variable y with respect to the independent variable x on a particular interval, we need to know
- the size of the interval for the independent variable, and
- the change in the dependent variable from the beginning to the end of the interval.
Depending on the problem, we may also need to know
- the units of the independent and dependent variables.
Then we can find
The average rate of change of y with respect to x is the slope of the secant line between the starting and ending points of the interval:
Relating this to the more math-y approach, think of the dependent variable as a function f of the independent variable x. Let h be the size of the interval for x:
and let a be one endpoint of the interval, so the endpoints are a and a + h, with corresponding y-values f(a) and f(a + h):
Then the slope of the secant line is
We also write this as
This is the definition of the slope of the secant line from (a, f(a)) to (a + h, f(a + h)).