 While we can consider the rate of change between any two quantities, the most common type of rate of change is that which occurs over an interval of time, so we can define many average rates of change as the amount of change divided by how long the change took. It will be useful to remember that an average rate of change is always associated with an interval with a definite start and end. For example, suppose we have a table giving a town's population and we want to know how rapidly the town's population changed between 1990 and 2000. We know this is asking for an average rate of change because we're asking about a change in population and we're given an interval, in this case between 1990 and 2000, and it is useful to keep in mind that an interval has a beginning and an end. Now, we have two changes, so we could find the rate of change of population with respect to time or the rate of change of time with respect to population. And in general, unless otherwise specified, we'll assume the rate of changes with respect to the input or independent variable, which in this case is time, and so we'll find the first. So the change in population is end, population in 2000, minus beginning, population in 1990, which will be the change took 10 years, and so the average rate of change will be the change in population divided by how long the change took, and again we can simplify the numerical part while leaving the units as a quotient. And remember we should answer the question in the same language it was asked. The question was in a natural language, so our answer should be in a natural language. And in English and other natural languages we should interpret using direction words, so this negative amount should be read as a decrease, and so we might say the town's population decreased by an average of 2000 persons every year between 1990 and 2000. As another example, suppose we have the altitude of a ball and we want to know how rapidly the ball was moving during the first three seconds. So again we know this is asking for an average rate of change because how rapidly is asking about a rate of change, and during the first three seconds is an interval with a definite beginning and end. Since we have altitude as a function of time, we'll find the average rate of change of altitude with respect to time. So we need to find the altitude at the beginning that's at 0 seconds, and at the end that's at 3 seconds, and so we find, we'll find the change in altitude which is the difference between the end and the start. This was during the first three seconds, so the change took three seconds, and note we can still compute this as a difference, the change in time, time at end, minus time at start, that's 3 minus 0. We find the quotient, change in altitude divided by how long it took, which will be, and so the average rate of change is 5 meters per second, and again giving this in a natural language we might say during the first three seconds, the ball's altitude increased by an average of 5 meters every second.