Instantaneous Rate of Change
When driving, the speedometer measures the instantaneous speed at any moment. It's how fast the car is actually going at that moment.
If our speedometer broke, we could estimate the instantaneous speed by looking at the average speed over shorter and shorter time intervals. Or maybe it's time to go the the shop.
Consider the table of values below, which shows how far Sadako was from home at various times:
Sadako's average speed during the whole trip was
We can estimate how fast Sadako was going at 2 pm by finding her average speed over smaller and smaller time intervals that have 2 pm as an endpoint. Between 2pm and 2:30 pm, her average speed was
Between 2 pm and 2:10 pm her average speed was
If we had measurements at, say, 2:05 pm and 2:01 pm, we would be able to have a better idea of how fast Sadako was going at exactly 2 pm.
Here's another table of values showing Sadako's distance from home at various times:
Use the table to estimate Sadako's instantaneous speed at 2 pm.
We find the average speed over intervals getting narrower and narrower around 2 pm.
To find speeds in miles per hour, we need to think of the time intervals in hours rather than minutes. One minute is one sixtieth of an hour.
From 2 pm to 2:05 pm, Sadako's average speed is
From 2 pm to 2:04 pm, Sadako's average speed is
From 2 pm to 2:03 pm, Sadako's average speed is
From 2 pm to 2:02 pm, Sadako's average speed is
From 2 pm to 2:01 pm, Sadako's average speed is
We can't quite tell what the numbers 24, 22.5, 22, 21, 18 are approaching, but 18 mph is a reasonable guess.
Similarly, for any independent variable x and corresponding dependent variable y, the instantaneous rate of change of y with respect to x at a particular moment is:
lim(average r.o.c. of y with respect to x)
where the average r.o.c. is found on an interval around the moment in question, and we take the limit as the size of the interval gets smaller.
Since the average rate of change of y = f(x) with respect to x on the interval [a, a + h] is
the instantaneous rate of change of y = f(x) with respect to x at a is
This quantity is also known as the derivative of f at a, written f ' (a). The derivative f ' (a)is the slope of f at the single point x = a.