drink some water and go home

My high school calculus teacher was fabulous. Fabulous because of her cute outfits and fun sense of humor, but mostly because of her incredible gift for teaching math. Mrs. Weiss made custom notes, homework, quizzes, and tests that set us up for success. She taught us the ideas behind calculus, not just how to solve calculus problems. She taught us the why. By the end of the year, I could see functions and graphs forming in my head after just glancing at numbers or problems. Don’t tell my mom, but my trick in college was to do integration problems in my head at the bar. When I couldn’t integrate anymore, it was time to drink some water and go home.

On the very first day of class, Mrs. Weiss told us “calculus is the study of how things change.” That didn’t sink in immediately, but I understood it over time. And last night, as I struggled to find my keys in the dark, it popped into my head again.

Here in Alaska, summer is sadly at an end. Fall is in full swing, and the day lengths are changing. I got home last night a few minutes after 9:00, as I usually do, and for the first time since summer had to use a flashlight to find my key. It’s that dark now. How does something so gradual and metered sneak up on you all at once? It seems like such a noticeable change all of the sudden. (In my earlier blog post shut up you stupid little birds, I touched on how the whole season changy, day length changy thing works and how it’s exaggerated at extreme latitudes.)

As I turned the key to my dark apartment, a graph popped into my head. An infinite sine curve. The graph of how the day length changes must be a beautiful, perfect sine curve.

In calculus, the derivative of a function is the rate at which something changes at any given point along that graph. If we want to know at which time during the year the changes in day length are greatest from one day to the next, we look at the derivative.

On our graph to the right, the path of the graph levels out to almost not changing at all around the solstices (June 21 and Dec 21). This means the derivative is 0 or very close to it right around the solstices, and that the sun may only set or rise a matter of seconds later or earlier than it did the previous day.

The derivative is greatest on September 23 and March 20 (the equinoxes). You can visually see here that the graph is taking a steep nose dive/climb.  Here in Haines, there may be more than five minute or so difference in how much daylight we get from day to day around the equinoxes.

So maybe I’m not crazy for thinking the days seem to be getting really short all the sudden. The autumnal equinox is in a matter of days, and the change is especially noticeable because we’re losing daylight at a faster rate than we have been all year. The graph is plunging.

It’s worth noting that yet another function can be tied into this little story. Depending on what latitude you live in, you see this day length change at varying magnitudes. At the poles, 24 hour light/night is experienced, which would mean the peaks of the graph are higher and the valleys are lower. It also means the derivatives are greater. Conversely, at the equator, they experience 12 hour days and 12 hour nights year round, so there are no peaks and no valleys. Their graph is completely flat.

So what is the derivative of the graph for the folks that live on the equator where there is no change in day length?

If you can’t tell me, you need to drink some water and go home.