Category Archives: Derivatives

Changing tides (the prodigal returns)

There's a reason I started this project over the summer... now that I'm back teaching, there are so many urgent tasks to complete that blogging can fall to the side! I'm used to making a point of carving out time for research: on a personal level, I don't feel like a vital mathematician or teacher if I'm not creating mathematics. I have less practice, though, making time for math communication outside the classroom.

How are your semesters going? It is that time when you might feel that the semester's been going on forever, and yet the end is not near. We all know the end is rushing toward us, though, and it's the time when we try to remind students who might have been slacking or who have started getting discouraged that we have almost two months to wrestle a class into success or let it sink into failure.

In my calculus class right now we're dealing with derivatives of arcsine and arccosine, but I am not going to talk about those today. Instead I'll put up the natural follow-up to the last post, a spreadsheet worksheet on modeling ocean water level over 24 hours.
Continue reading

Our leaky atmosphere

Backing away from the sensational headlines of last week, here's something a bit more relaxing. Our atmosphere is leaking away into space and maybe someday eventually we won't have one anymore. How's that?

Alright. I exaggerate a bit here -- the force of gravity is enough to keep nitrogen and oxygen around, as well as lovely greenhouse gases like carbon dioxide and methane. These are heavy molecules! Hydrogen, on the other hand, is really light. It reaches escape velocity fairly easily, especially at the outer reaches of the earth's atmosphere.

For a more nuanced look at atmospheric escape, check out the Scientific American article. I'm linking to an author's website because I can't get this article at the Scientific American site even via Cornell's library.

Today's worksheet is another longish one. I might consider splitting it up. The first two pages guide the student through a derivation of escape velocity using the chain rule, integration, and limits. It's one of those activities that dredges up everything a student had to learn all semester, yet in the end none of the integration or limits are that hard. The last page (the third) asks some very concrete questions about the escape velocity for hydrogen molecules, relating this velocity to temperature. "Is it hot enough for hydrogen to escape?" is the main question.

I have to confess that I spent all day traveling to a math conference and I'm a bit underslept. As I went through my calculations I realized that hydrogen could escape from anywhere in the earth's atmosphere, and this really alarmed me! We are all going to perish! until I realized that indeed that's the case, and that's why we basically have no hydrogen in our atmosphere. It's ok. Can't breathe that stuff anyway.

Let me know if you use this worksheet and whether you think it would be better broken up into parts. I like the surprising results, and I like the fact that it brings together the chain rule, integration, and limits. It would be fun to extend it and talk about the moon's lack of atmosphere and Venus's crushing anvil of an atmosphere. We sometimes take for granted our Goldilocks world...

Calc Roundup: Atmospheric Escape

Derivation of equation for atmospheric pressure

Flashbacks to the past: one of the first worksheets I published in this project was on using linear approximation to estimate the atmospheric pressure at various altitudes, and a later one was about a power function for atmospheric pressure. The derivation of the formula for atmospheric pressure is actually pretty straightforward. I'll assume that your students have not yet encountered integrals per se, but this worksheet pushes them to use their knowledge of differentiation to deduce an antiderivative.

This is a worksheet that puts together a few disparate concepts:

  • dimensional analysis, using units to understand equations
  • antiderivatives,
  • and baby differential equations.

It's certainly an activity for the end of the section on differentiation. The very last question asks students to think about a more accurate equation, and I wouldn't expect most students to be able to solve it alone -- but sometimes a good challenge is important as it points to concepts you'll be dealing with later on. Knowing how to integrate would really help in solving that last problem 🙂

Derivation: Atmospheric Pressure

As mentioned in the earlier posts, a great resource on atmospheric pressure and rocketry and all sorts of fun things is found at Portland State Aerospace Society's rocketry pages.

Related Rates: Compost oh compost

I'm on a nitrogen kick, I guess. While researching the previous post on beets, which ought to have a follow-up when we get to integration (did you do the worksheet?!), I learned a bit about carbon-nitrogen ratios in compost. The ratio of carbon to nitrogen is important because compost gets hottest when this ratio is around 30:1. Empirical evidence supporting this is not too hard to find, but a mathematical model is hard to find! This is why you need to mix "browns" and "greens" in your home compost.

Again, nitrogen is a very important nutrient for vegetable and grain growth -- if you use up the nitrogen from your soil you'll have small flowers and no tomatoes. Nitrogen conservation in your compost is thus pretty important. But again, I had a hard time finding math equations for this -- unless you count all the cool papers that solved twelve nonlinear systems simultaneously.

Finally, after days of trolling through the library files and becoming much more aware of what a HUGE BUSINESS waste processing is, I found a paper with some polynomials. It's also got some great 3D graphs and some visualizations, in Figure 4, of how these functions depend on their independent variables. Might be food for another post. The authors of "Optimizing composting parameters for nitrogen conservation in composting" took an approach similar to the beet-research people: they did a bunch of experiments, measuring values for aeration of the compost heap, moisture content, particle size, and time after start, and ran a big backward computation to come up with polynomials in those variables (A, M, P, and t) that predicted the carbon/nitrogen ratio (among other things) pretty well. That is the polynomial that the worksheet below focuses on.

This worksheet covers

  • related rates, applying the product and chain rules a few times
  • physical reasoning: I ask students to examine the assumptions of the model, which I then violate for mathematical purposes!
  • and writing English sentences explaining a math result.

Teaching tips: students often freak out about all the symbols in here. Reassure them that many are constants in the problems they're asked to work out. Remind them that the derivative of a constant is zero, even if the constant is one they don't know!

I would like to have had some other "real-life scenarios" or more interpretation, so I'll think about what sorts of related-rates problems could be added to this.

Related Rates: Nitrogen

If you've got refinements or modifications, let me know!

Give me some optimal sugar... calculus-style

I was on a math trip this week, so have been a bit delayed in posting. After four days of intensive pure math thought, I've returned to my little farm in the city, the minuscule plot that is my back yard. Today was spent doing math and picking cherries. The beans and peas are doing well, too; we've got peas planted where we used to grow tomatoes in an effort to increase the nitrogen content of the soil without applying fertilizer.

I have the luxury of not depending on my little garden for my primary food source. Instead, I buy food from farmers either at the farmer's market or at the grocery store. It is nice to live in the city and be able to take the bus to the opera, but it means I depend on others for agriculture. They use fertilizer and irrigate their land because farmers must do everything they can to control growing conditions for their crops.

Today is about sugar beet production. (I also looked into optimization of conditions for composting, but there are no equations I can find!) Sugar beets are a major crop across the US, particularly in North Dakota, Minnesota, and Idaho. We love sugar and want it in many foods (until we find it's killing us) and of course farmers want to optimize their yields. Sugar beets are interesting because simply adding more nutrients to the soil can be counterproductive: you don't want the biggest sugar beets, you want the sweetest ones! Too little nitrogen means yellow leaves and poor growth. Too much nitrogen means impurities in your beets and reduced sucrose, or at worst killing your seedlings (source). It's the Goldilocks question.

An older report on how nitrogen levels affect recoverable sugar yields has some very nice equations. G.L. Malzer and Greg Buzicky looked at many variables and came up with several equations that predicted recoverable sugar yield pretty well, all in terms of the soil's nitrate-nitrogen content. And they're quadratic! This is a nice way to do a pretty easy optimization exercise with applications to something... sweet!

The first page is all about finding the optimum recoverable sugar yields given different levels of nitrate-nitrogen in the soil. The second page mixes in some experimentation and treats a two-variable function, foreshadowing multivariable calculus techniques. Including discussion of multivariable functions in a first-semester calculus course is a really cool idea that deserves more attention -- it does not disturb students, but only people who have a set idea of what one "should" learn in first-semester calculus. The third page asks students to use the first and second derivative tests to prove the results they've already produced, and asks them to think about the applicability of the Extreme Value Theorem. As in many situation, physical constraints could lead to a closed-interval phrasing of the problem, although it's not necessary mathematically. Provoke an argument!

Optimization: Sugar Beets

Agriculture is often ignored in calculus and STEM classes, as it's not so sexy these days. Universities like the University of Minnesota and Cornell have big ag programs, though, and they're hugely important. You sure can't be a vegan or vegetarian in the north without the products of modern agriculture!

I've learned a lot from my ag students and they need to deal with optimization often: they need to optimize nutrient composition in animal feed, optimize nutrient composition of fertilizers for soybean growth, optimize temperature for dry-matter intake of chickens, and of course look at the economics of all the above. Their decisions impact the diets, waterways, air quality, climate, and fuel prices of city-dwellers. Don't forget the ag, even if you live in LA.

Drugs in waterways: derivative mix

We return to naproxen (sold under the brand name Aleve). Naproxen is my "drug of choice" for these worksheets because it apparently occurs in a lot of our waterways and its decay is pretty well understood. Last time we discussed naproxen in particular, we looked at a function that gives the rate of photolysis for naproxen at a depth , the rate at which the substance breaks down in the presence of sunlight. There are a few different ways that substances like naproxen, ciprofloxacin (an antibiotic), cocaine, or bisphenol-A get taken out of waterways: breakdown in sunlight, breakdown by organic processes, or sedimentation. Naproxen breaks down easily in sunlight but it doesn't like to be filtered by sand or settle out into sediment even when the water is treated with ferric sulfate to make coagulation happen.

The linked abstract is for a paper about a pilot-scale drinking water purification plant, looking at how water from the River Vantaa could be used for drinking water if the groundwater source for Helsinki, Finland, were to be rendered unusable. Remember that groundwater usage is increasing enormously across the world, and so our nice clean aquifers are overtaxed in many locations. We should not waste so much water (agriculture and lawns, folks!) but will also need to learn a lot about how surface water can be purified so that we can drink it again.

The worksheet below has a mix of derivative and rate of change questions. It asks about some derivatives that require the chain rule (quotient rule and exponential function rule combined) and it also asks students, at the end, to switch variables and look at how the rate of photolysis changes as turbidity changes. After every heavy rain a lot of sediment enters a river and then settles out over time. Development and construction can also change turbidity substantially: digging up a lot of trees and plants to expose dirt allows a lot of that dirt to run off. Agriculture also has its role, as during the planting season fields can be vulnerable to erosion and run-off.

Chain Rule: Photolysis of Naproxen

If you're in a position to work with a science teacher or run experiments yourself, I found a fun page on experiments with turbidity appropriate to junior to senior high school students (and what college student wouldn't mind playing with mud, really?). This could make a cool big brother/big sister activity: high school seniors do the math and the freshmen or junior high students do some experiments on turbidity. In addition, there's a World Water Monitoring project and day (September 18) that you could join.

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

Stories and Graphing

I've been spending the last few days trying to edit some math papers I'm working on and learn more about polygon spaces. (What's the space of quadrilaterals in with side lengths (1,2,1.3,1.8) and one vertex at the origin, for instance? If you guessed that it looks like , you'd be right!)  It's always interesting to talk with students about what mathematicians do. They expect we do more with numbers, and when I talk about what I am working on they're surprised at all the funny little pictures I draw and all the writing we all do.

This activity is a good one for discussion. You could either print this or use a projector to put it up in front of the class for a group discussion. Group discussions can be a great way to get a class comfortable with each other and facilitate the verbal discussion of mathematics, which is another skill that must be learned and practiced. I had a hard time learning to "talk math" despite my good reading and essay-writing skills, and even now I need to gather my thoughts to produce coherent mathematical sentences in a way that I don't need to when talking about politics, for instance. I've concluded that I think about mathematics nonverbally and that the translation process simply takes some time, even now that I've had a lot more practice than your average freshman.

When I run a class discussion like this, I try to do a few things:

  • Give students a minute or two to think in silence and write down a tentative answer.
  • Start cold-calling students -- "Pahoua, what do you think about the first one? Jordan, what about you?" This requires a level of trust that is built over time.
  • If students are right, I don't necessarily say so and move on -- I keep asking other students, and ask other students if they think the first student is right or not. I don't want students practicing the Clever Hans strategy. (Clever Hans is a horse who counted by tapping his hoof, but he just watched his trainer until the trainer looked happy and then stopped.)
  • If students are wrong and I can identify the assumption that led to the mistake, pointing out that assumption can often help. "Danielle, are you assuming that the graph must be increasing?" Sometimes I ask other students for their thoughts. "It looks like Jason disagrees. Why?"
  • If a student is visibly hesitant or reluctant to answer, ask for a question or the beginning of the thought instead. "Alexi, what do you think about part c? ...   What strategy are you thinking of trying?" or "What do you notice first about the graph?" or "What question are you asking yourself right now?" These often prompt good discussions.
  • I do try to use students names consistently as I call on them, and
  • I try to keep track of who I call on and rotate through the whole class. There's plenty of research that shows that even the most enlightened and conscious instructors call more on boys than girls, etc., so bringing a class list and discreetly checking names off is an easy way of making sure you involve everyone in discussions. Telling students that you track who you ask is sometimes useful, too, because then they know that everyone will be asked and they should be prepared! Students do think that it's fair to do this, especially if you offer the outs suggested above if they are stuck.

So, with no further ado, here is a worksheet on matching graphs and stories, using the language of derivatives:

Graphing Stories and Stories of Graphs

Copyright details: The graphs on page one are:

(A) Water levels in Lake Superior measured at Marquette, MI, using data from as suggested by commenter TAO. 

(B) Well water depth data from observation well 27010, with data from

(C) Data on lynx populations using the dataset included in the R programming package. The data is from the Hudson Bay company's logs.

(D) A graph from, since I can't seem to find raw data for lemmings. Used under the fair use policy for copyrighted documents and documents produced by the federal government.