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.
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.