Q&A: Why is Newton’s Method Useful?

Cross-posted from here.

Question

If you teach Newton’s method for finding roots of real functions on the high school (or freshmen) level, I think some students may reason like a variant of the following:

Why do I need learn such a “complicated” method if I simply can use the following?

• plot it, for example, via Geogebra, and zoom in and just read off the root to the desired precision
• or just use the table value function of my calculator and do some kind of interval nesting

What would be good examples or activities to make it clear that the Newton method is useful and in some ways better than the approaches above? How can I explain where it makes sense to use Newton’s method over the more simple methods above?

Answer

I only taught this to students who had some baseline level of interest in math/CS and were amenable to “here’s why this is cool/powerful” examples (not just “here’s why you might need to know this in the future” examples), so YMMV, but what worked for me was to start off with the following question:

How can you estimate $\sqrt[3]{2}$ without directly using fractional exponents?

The most straightforward option is to use bisection search, but Newton’s method (finding the root of $x^3-2$ is faster). Here is the problem set I had them do.

(In the broader scope of that course, Newton’s method helped instill some intuition that the slope of a function is a key property that can be leveraged to computationally search the graph of the function for places of interest. That way, by the time we got to gradient descent, the students already had some intuition about how the gradient can be useful.)