Using the “Higher Order Reflections” post from the Calculus VII blog with my son

I’ve been looking for fun ways to review calculus topics with my older son and found a great post from the Calculus VII blog:

Higher order reflections

Today I had my son read through the post and then we discussed the ideas. His initial thoughts are in the video below – he understood most of the post and also had a couple of good questions:

After we talked through the post we went to Mathematica to take a look at some of the example “reflections” which preserve the 1st and 2nd derivatives:

One of my son’s questions in the first video was why the blog post was using functions like f(x/2) and f(x/4) to make reflections. I’d mentioned that these were essentially arbitrary choices. Below we saw what would happen if we used f(x/3) instead:

We finished up by going back to Mathematica to see what these new “reflections” would look like: