Tag Archives: trig functions

Lynx: the chain rule and a better model

As promised, a return to lynx. In my previous post about lynx I posted a worksheet modeling lynx populations with a cosine function, and mentioned that this is not the best model. Look at the derivative to see how bad it is -- the green and red lines ought to be matching up:


Graphing the log of the lynx data gives a transformed graph that is much more sinusoidal! The better model for the lynx data, then, is exp(something sinusoidal). Look at the graph below to compare Model 2 and its derivative to the data. The green and yellow curves are much more alike:


This worksheet guides students to developing this model after having them evaluate the previous sinusoidal model via technology.

The worksheet I'll include below is meant for a day when you have computer lab time with students. I know that this does not include everyone... but if you can head down to the lab for such an activity, there is a lot students can learn!

This worksheet applies knowledge of:

  • the chain rule, on compositions of trigonometric and exponential functions
  • numerical approximation of the derivative
  • shapes of graphs.

Along the way students must evaluate models and create one of their own.

As the instructor, you'll have to decide what software you want to use for this activity. I have had success using Excel, asking every student to email me their work on the way out of the lab, and these days you can use Google Drive if your institution uses Gmail. If you and your students are already quite familiar with R you could also use that. Beware of differences between Mac Excel and Windows Excel, especially in graphing -- work through the activity yourself on whatever platform students will use.

Chain Rule: Lynx

Lynx Pelt Data Spreadsheet


Trigonometry is often a terrifying topic for students taking a first calculus class in college. America's trig teaching seems a bit haphazard, and we can't assume students have a good grasp of periodic functions.

This summer project is clearly not offering a full calc curriculum, but instead offering supplemental material. A good lesson on the unit circle is a first step in any college calc discussion of trig. After that, though, students may find practice with a variety of trig problems useful. You can look at particle motion and springs and all sorts of mechanical applications, or you can use a cute and fuzzy animal that also has sharp teeth and claws.... the lynx!

A lot of arctic populations including lynx, snowshoe hare, and lemmings have cyclic population fluctuations and there is a lot of research about why this is. If you've got students in ecology you can ask them to tell you about this in more detail: this is a great topic for a small project or extra credit assignment.

I love this family of examples: there is so much room for discussion of the real world, from foxes and rabbits in my own neighborhood to the wolves and moose of Isle Royale. The examples work so well from precalculus through multivariable, linear algebra, and differential equations. Students can easily experiment with changing parameters in their models, using Excel or more sophisticated programs. And it's all about the circle of life, one of the most compelling stories we as humans know!

Alright then. To the worksheet. I may be a bit dissatisfied with this one yet for reasons I'll outline below. The instructor should display a graph of the actual data, included below. Students then work through 

  • constructing a trigonometric function from the amplitude and period, 
  • considering the shape of the actual data as they construct the function, and
  • critiquing their model at the end.

As so often is true in modelling, our first instinct -- the sinusoidal function -- is not actually most accurate. If you've got access to the lynx data (you can also find it as a dataset in R, the stats program) you can check that the logarithm of the data is actually more sinusoidal! Look at the long troughs in the the non-log graph, along with the sharper peaks. With students you can use the idea of symmetry to discuss why a sinusoidal approximation is not the best for the non-log-transformed data: the graph does not have reflect-and-glide symmetry, unlike cosine and sine. I, though, honestly don't yet know a good mathematical explanation for the fact that the logarithm of the population data has a more sinusoidal shape. I'd love to hear one.

Worksheet constructing periodic function for lynx trapping

Plot of lynx trapping over time

Plot of logarithm of lynx trapping over time

I know that someone other than me has looked at this blog in the past week! Question of the day: do you use cute fuzzy animals to increase interest and engagement in class?