Summary

How fast is a function changing at a? At what rate is it changing? We tell how fast a linear function is changing by the slope. So how do we define a slope for a wiggly function?

We've discussed the problem of finding the slope of a curve, rather than a straight line. We should make use of the straight line wherever possible, however, because we understand that case.

What does it mean to talk of the ``slope'' of a curve at a particular point? We can approximate the slope to a curve at a point by a well-chosen line. If we don't choose well, then the slope won't be a good ``fit'':

 

But even if it's not a good fit, a line can still do a decent job. As an example of this, consider the calculation of average speed.

Average speed is calculated as total distance divided by total time. While the instantaneous speed will vary, over a long trip the average speed may do a good job of estimating it. The definition of average speed is our motivation for the general definition of average rate of change:

Average rate of change: the average rate of change of f(x) with respect to x for a function f as x changes from a to b is

You can think of it simply as the slope of the line joining the locations of a particle at two different times.

Example: #32, p. 176

Now in order to improve the approximation, we can use a line that fits the curve well. We let the point b approach the point a - i.e., we use a limit:

Instantaneous rate of change: the instantaneous rate of change of f(x) with respect to x for a function f when x=a is

or, if we define h=b-a (representing the diminishing distance between b and a), then b=h+a, b-a=h, and

provided the limit exists.

Note: this limit is of the unusual sort, where both the numerator and denominator are tending to zero. If we can't factor out a term of h in the numerator, then we're going to get into trouble....

Example: Consider a linear function. We ought to be able to use the instantaneous rate of change limit to ``recover'' the slope. So consider the function .

and now we can cancel factors to obtain

So we recover the slope! Let's use it using the other formula:

Whew! Good thing!

Example: Let's try it with a quadratic (things get hairier!). Consider the function , and let's find out the rate of increase of the function f as x approaches the value 1. Then we write

so, expanding out,