Wrap Up
In this chapter we have learned how limits are a way of investigating functions. Specifically, we've seen that limits can give us insight into the growth rate of a function at a point.
We looked at two functions,
f(x)=x^2+2*x, and g(x), our model of the height of a missile over time from earlier in the book. In both cases, we investigated how the slope of secant lines limits towards the growth rate of the function at a point.
The most important outcome of our study is that we came up with a process (albeit a clunky one) for calculating the growth rate of functions. To recap, the process is:
- Plot a point on the function, usually called
A - create a "dummy" variable
hset to a small number - plot a second point on the function close to
Aby "nudging" the x-coordinate ofAwithh - create the so-called "secant line" between the two points
- study the limit of the slopes of the secant lines when
htrends towards 0
A.
This growth rate of the function at a point is extremely important in calculus, and it has two other names: the derivative of f at A; the instantaneous rate of change of f at A. We'll study this concept at length, and refine our clunky process in the next chapter.
Side-note: If you're curious in some of the other uses for limits aside from derivatives, check out this Geogebra activity in the Miscellany chapter at the end of this book