Our Second Limit Part 2
The code to create the second point on
g(x) close to E in the previous activity is (x(E)+h,g(x(E)+h)).
Slide h to see it in action. Can you detect a trend in the slope of the secant lines as h tends to 0?What is the approximate value of the limit of the slope of the secant lines as h tends to 0?
Note that I've also added a tracking point near the origin keeping track of the slope of the secant lines.
Before we move on, the units of the slope of this secant line are very important to consider: The slope of a line is a "rise" divided by a "run", so the units of the slope of any line will be the units of the rise divided by the units of the run. In the case of the secant line above the units of the run are the units of
x, and as we saw when we first created this function, those are seconds. The units of the rise are the units of g(x), and those are meters. Thus the units of slope are meters-divided-by-seconds, or you might say "meters per second." Thus the units of the secant line are telling us something about the rate of change or "growth rate" of g(x). We'll look at this more in the next chapter when we study derivatives.
For now, let's keep looking at this one limit, and not worry much about "rates of change" or "growth rates."