It's that time of year where derivatives and inverses both start creeping into calculus classes, and all that notation at the upper right of a function name starts to get confusing! Continuing on a theme, I'm using pressure and altitude this week to do a quick conceptual worksheet on derivatives and inverses and derivatives of inverses. It's good to write out these relationships.
Not a ton of new science in this one, but perhaps useful if you're "following the story" of atmospheric pressure 🙂 I may add some similar worksheets with other settings for interpretation, just because they're fun!
Derivatives and Inverses: Altitude
This is why I started the EarthCalc blog over the summer -- teaching is always like an oncoming train! With the other things I'm still working on, the weeks have been crazy!! Hopefully things will calm down as the semester progresses.
Well, my running in the mountains went pretty well. I'm a slow runner and the 11,000 foot altitude made me slower. This issue of getting enough oxygen is pretty important.
My previous worksheets on altitude and pressure talked about linear functions, power functions versus exponentials, and deriving the equations themselves, which was pretty sophisticated. This one has two pages (if printed duplex) asking basic questions about one-to-one functions and inverse functions. Interestingly, while atmospheric pressure decreases with altitude (monotonically if not constantly), temperature decreases and then increases due to the sun's warmth. Funky stuff happens at the very outer edges of the atmosphere, which we don't discuss. (But check out the leaky atmosphere worksheet if you're interested!)
I'll put up answers to this soon but wanted to get the worksheet out first. Here it is:
This week's post is
- the first this semester (fall semester! I am teaching again! I wasn't quite sure if I would or not!)
- curiously appropriate, since I am up in the mountains this weekend.
I'm at Brian Head Resort briefly, running down a mountain. This running race starts at around 10,000 feet of elevation and I'm not used to it. There's a short initial uphill run, and then a chairlift ride, and then a lot of downhill. I am thinking keenly about the amount of oxygen available to my brain and muscles.
(I'm hearing from my running-mates that you can't drive a rental car up Mauna Kea -- they all die because the air is thin enough that combustion is affected. You've got to get a good SUV. So throw this out to your students as a real-life application!)
Regular readers might remember the first set of worksheets I did on altitude. As I come back to class this September I'm reminded that there's always room for working through the most basic concepts in a clear and straightforward way. The worksheet I'll post in a day or two (next internet access!) looks at the concepts around inverse functions: the horizontal and vertical line tests. It uses some data taken from empirical measurements of temperature at different altitudes. Remember that we're hot down here, the temperature drops as we climb mountains and fly up in planes, but then if we're on the sunny side of the planet the temperature rises again in the upper atmosphere. There are some wiggles in the middle, too. (Check out the graph at this site.) Clearly, me telling you what temperature I am does not tell you uniquely what altitude I'm at! On the other hand, pressure is monotonically decreasing as altitude increases, and so me telling you an atmospheric pressure would allow you to estimate my altitude.
More later: time to adjust to 10,000 feet!
No new math today, but some updates:
- I'd initially planned to run this blog from June through August. I did it, and it was fun. I took two weeks out to travel and think about what's next and I've decided to do it again. Here comes EarthCalculus Fall Semester edition!
- I'm almost done with a decent draft of an e-book, Conceptual Climate Modeling for All. It cleans up and expands the notes I took at the MAA-NCS Summer Course on conceptual climate modeling. If you'd like to get a free copy in exchange for giving feedback, sign up for the email list on the right-hand side of the page. The email list is updated each night, rather than instantaneously, so you should get email the next day.
- Speaking of the e-book, would you rather have a pdf file, an iBook, a Kindle version...? Why? Feel free to email me or leave a comment.
- Since it's the beginning of the semester, I'll be putting up a more elementary worksheet on inverse functions in the next day or two. We'll go from there.