Playing around with Newton’s method

The next section in my son’s Calculus book is Newton’s method. I think it is a neat topic, but I chose to do a high level overview today because I wanted my younger son to join in. He’s learning algebra this year and I think (obviously) that the calculus details would be both over his head and not interesting to him.

We have looked at Newton’s method before in this project:

Exploring Newton’s method with kids

and I used the Mathematica code from A. Peter Young at UC Santa Cruz in this project, too. That code can be found here:

The page from A. Peter Young at U.C. Santa Cruz that gave me the Newton’s method code for Mathematica

So, here’s the high level overview I gave for Newton’s method and, more generally, the problem of finding roots of equations.

One fun thing that came up at the end of this video is that my older son noticed that Newton’s method might not always find the root you were hoping to find.

In the next part of the project we explored the idea my older son brought up at the end of the last video -> Are there cases where Newton’s method might not work as expected?

Next we looked at the function f(x) = x^2 - 4. We used Newton’s method with an initial guess of x = 3 to try to find approximations to the root x = 2.

Finally, we explored Newton’s method for complex numbers. This part was just for fun and to explore a few pretty pictures.