 The amount of change per quantity is the amount at the end minus the amount at the start. The change usually occurs while or because something else is changing, so we're often interested in the ratio of the amounts of change, and so the quotient of the two amounts of change give us the average rate of change. We could give a formula for finding the average rate of change, but we won't. Remember, don't memorize formulas, understand concepts. It's convenient to think of the average rate of change as a fraction, the change of glurb divided by the change of quazit, where glurb and quazit are the quantities whose changes we're interested in. We call this the average rate of change of glurb with respect to quazit. For example, suppose the volume of water in a conical tank can be computed when the water fills the bottom h meters. Let's find the average rate of change of volume with respect to the water level as the water level goes from 2 meters to 6 meters. So remember the average rate of change of glurb with respect to quazit is the ratio expressed as a fraction, the change in glurb divided by the change in quazit. And remember the change is the amount at the end minus the amount at the beginning. So we need to find the change in volume and the change in water level. So the water has gone from 2 meters to 6 meters, and our function will tell us what the volume is once we know the water level. So at the end, when the water level is 6 meters, the volume is. And when the water level is 2 meters, the volume is. So the change is end minus beginning. And remember always include units, in this case cubic meters. The change in level is the difference between 2 meters and 6 meters. So again, end minus beginning is 4 meters. And the average rate of change will be the quotient, change in volume, divided by change in level. And we can reduce the numerical part. And remember, units act like algebraic variables. So it's as if we have an x cubed in the numerator as an x in the denominator. And we could simplify. But when dealing with units, it's often best if we don't always simplify. So why don't we always simplify our units? The reason is that we've gone through all these computations for a purpose to find something. And the useful idea, answer questions in the same language they were asked. So for average rate of change is the change in glurb, divided by the change of quazit. We can interpret it as a change in glurb, as quazit increases by 1 unit. Equivalently, we could say it's the change of glurb per unit of quazit. And note that we assume the denominator is increasing. So if we go back to that rate of change, since the question was asked in English, or more generally in a natural language, we should give an answer in a natural language as well. So let's see if we can try to make sense out of what that rate of change actually means. Again, always look to the units. Our numerator units are cubic meters, which measure volume. Our denominator units are meters, which measure the water level. So we can read this as the volume increases by 208-3rds pi cubic meters for every additional meter of water level. Or we could say the volume is increasing by 208-3rds pi cubic meters per meter. Or we could say that every additional meter of water level corresponds to a volume increase of 208-3rds pi cubic meters.