Working through Calculus with my older son

This year my older son wanted to learn calculus. I was hesitant, but we’ve finished the Art of Problem Solving books and also have done (as of September 2018) about 850 stand alone math projects, so I couldn’t find a good reason to object.

The collection of calculus books I had from 20 years ago seems to have dwindled down to just 2 -> a 3rd edition of Stewart’s Calculus, and the book by Spivak. Spivak feels too hard for an introductory course, so I chose Stewart.

Last week we worked through the chapter on limits and continuity, and this week we are starting the section about derivatives. I’ve also been using Grant Sanderson’s fantastic video series “Essence of Calculus” and used the 3rd video in that series to introduce some ideas about derivatives last night. Here’s that video:

Before diving into Stewart’s chapter on derivatives, I reviewed the ideas and challenges in Sanderson’s video this morning. Here’s the conversation about the derivative of $f(x) = x^2$

The main idea I want to be sure that he understands is that small changes in functions as the input changes can be represented as:

f(x + \Delta x) \approx f(x) + f^{'}(x) \Delta x

Next we worked through 2 of the challenges that Sanderson gave in his video.  The first is to use a geometric argument to find the derivative of f(x) = \sqrt{x}.

The second challenge in Sanderson’s video is a bit more challenging -> Use a geometric argument to find the derivative of f(x) = 1/x.  I helped out here a bit more because the arguments are bit more subtle, but he was able to find the right argument.

I’m hoping to be able to write about our calculus conversations once or twice a week starting with today’s conversation. Hopefully this will be an exciting year studying calculus.