Volumes of Revolution

We just started volumes in Calculus today – it is one of my favorite topics. Learning this material is one of my favorite memories from high school.

My son seemed to enjoy it, too. We worked through the 6 examples at the start of the section together this morning. When he got home from school I had him try a few examples on his own.

He picked two problems from the book. The first problem was finding the volume when the area between and is rotated around the x-axis:

The second problem he chose asked him to find the volume of a “cap” of a sphere – this is both a really neat problem and a pretty challenging one. I was surprised that he chose this one, but it was fun to talk through:

For the final problem we worked in Mathematica and made a 3d picture of the volume created when you rotated the curve between and around the x-axis:

I’m excited to pick a few more of these shapes to print as we work through this topic.

Working through a differential equation problem from the 2015 BC calculus exam

I’m using old problems from the BC Calculus exams to make sure I’m pitching the course at the right level. A few of the old questions have surprised me, but many are really nice problems.

This differential equation problem from the 2015 exam was a nice way to explore some basic ideas about derivatives with my son:

We started by reading the question and then drawing in the slope fields:

The next question asked you to find the 2nd derivative in terms of x and y only, and then asked you to talk about the concavity of solutions to this differential equation in the 2nd quadrant:

The third part of the question asked about a solution to the differential equation at a specific point – in particular if that specific point was a maximum or minimum:

Finally, a pretty neat question about a linear solution to the differential equation. Unfortunately I forgot to zoom out after reading the question – hopefully my son’s words explain what he’s doing:

I thought this was a really nice introductory differential equation problem – it was nice to see that we could talk through it even though we haven’t really talked about differential equations formally, yet.