TABLE OF CONTENTS The idea of rate of change occurs all the time in our everyday experience. We are all familiar with expressions such as miles per hour, miles per gallon, pressure per square inch, price per pound, and so on. They all represent rates of change. An analogous idea in mathematics is that of the rate of change of a function.
Given some function y = f( x), we can change x by some ? x. This, of course, causes a change in the dependent variable y, say ? y. If ? y is proportional to ? x, then the function changes uniformly, or at a uniform rate. This means that the change in y, corresponding to a given change in x, is constant, independent of the actual value of x. The rate is the change in the function divided by the change in the argument:
where m is a constant. The graph of this function is a straight line whose slope is m:
When a function does not vary uniformly, the ratio ? y/ ? x is merely the average rate of change over the interval ? x. If we let ? x approach zero, this ratio approaches a definite limiting value, or the instantaneous rate of change:
which is the rate of change of y with respect to x.
We need nothing more than our...
