Working through a challenging calculus problem

I stumbled on a neat, but challenging, calculus problem in my son’s book yesterday afternoon. We talked through the problem and then I wanted to revisit it today by having my son solve it from scratch.

Here’s the problem and his solution – It was too bad that the whiteboard didn’t leave enough room for the picture:

Following his presentation, we talked through a few of the math / calculus ideas in the problem:

Finally, I wanted to show him a different solution involving a u-subsitition. I was just doing this on the fly and didn’t realize how confusing a u-substitution would be. But we got through that part (and the upside is that I’ll remember that this topic is a lot more tricky than I think when we eventually get to it!):

Definitely a fun problem, and one that really forces you to think pretty hard about tangent lines and derivatives.