You may not have noticed it, but in the previous activity when you used the Monkey Rules to calculate the derivative of [url=https://www.geogebra.org/m/x39ys4d7#material/ntenkuu6]our model of the length of the day[/url], you actually did a remarkably powerful calculation. Just like we [url=https://www.geogebra.org/m/x39ys4d7#material/ybp9bfdt]estimated the maximum height of the missile with the derivative[/url] a few activities ago, we can similarly use the derivative to identify what the model thinks is the longest day of the year at Northern Vermont University Johnson. [br][br]To get started, let's use Geogebra to calculate the derivative of the model of the length of the day below by typing [code]derivative(g)[/code] into the input bar.

Don't worry if you get ugly fractions when you used [code]derivative[/code]. This is normal and due to some of the code Geogebra uses to calculate derivatives. If you use a calculator to get a decimal value for these fractions, you'll find they are the same as you obtained in the previous activity. Also, don't worry that you can barely see [code]g'[/code]. This is also normal; [code]g'[/code] is a lot smaller than [code]g[/code], and is simply located down by the x-axis. [br][br]Now, go back up to the applet and type [code]intersect(g', y=0, 1, 365) [/code]which will intersect [code]g'[/code] (which we can barely see) with the line [code]y=0 [/code](which is just x-axis) to find all days where the derivative is 0 between day 1 and day 365. You could do this algebraically by solving the equation [code]g'(x)=0 [/code]which would be a challenging exercise in trigonometry that's not at all needed to do calculus. [br][br]Regardless of how you find them, you should obtain the points [code](171.95281,0)[/code] and [code](358.73002,0)[/code]. This of course simply means that g' is 0 at 171.95281 and 358.73002, which in turn means that the model g predicts these days are days when the rate of change of the length of the day will be 0. In other words, on these days, the length of the day is neither getting longer or shorter. This essentially means that these are days where the length of the day is either at a maximum or a minimum. [br][br]Before we continue unpacking this idea, let's round these to the more convenient days 172 and 359 (more on why this is appropriate to do when working with models later in the Miscellany Chapter at the end of the book). [br][br]Using [url=https://www.esrl.noaa.gov/gmd/grad/neubrew/Calendar.jsp]this page on NOAA's website[/url] we see that these days of the year correspond to June 21 and December 25, which are very close to the actual longest and shortest days of the year at Northern Vermont University -- Johnson (June 22 and December 22). [br][br]If you step back and reflect for a moment on what you've done, you should find it truly remarkable. By using [i]just 7 data points[/i], we not only created a model of the length of the day, but with that model plus the power of calculus, we were able to make quite accurate predictions about the longest and shortest days of the year! Don't short sell your achievement here! This is really cool.[br][br]And before you worry too much about it: we'll talk about why this process didn't exactly find the longest and shortest days later in the Miscellany Chapter. The reason is pretty simple though in summary, so I'll tip you off now: g is just a model, or if you prefer, a "mathematical sketch" of the length of the day based on a limited set of observations; g isn't reality, and so it is simply doing the best it can with limited resources. [br][br]Before we move forward, let's take a moment to discuss why using g' like this worked: The reason the derivative, [code]g'[/code], is able to help us identify these days is because places where the derivative is 0 are places where the model thinks the length of the day stops growing or shrinking and pauses for a moment. These are precisely the days of the year where the length of the day is longest and shortest. Put another way: the rate of change of the length of the day must be 0 when we are at either the longest or the shortest day of the year. Pause for a moment to reflect on this! It's quite a powerful way of thinking about maximums and minimums.[br][br]Move forward to the next activity where we will begin a formal discussion of this important application!