Tag Archives: rate of change

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 k_{phot}(z) that gives the rate of photolysis for naproxen at a depth z , 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.

Rates of rates of change

Some instructors (like me!) like to foreshadow the ideas of concavity early in the semester. When I talk about rate of change, average and instantaneous, I like to throw out some discussion of the rate of change of the rate of change. This is a language puzzle for many students -- they may see that a function is increasing but need to think harder about whether it is increasingly increasing or decreasingly increasing. What does that all mean, anyway?! It's a great time to discuss precise mathematical language, communication skills, and the usefulness of equations. It is easy to be precise when symbolically indicating that a function is concave up, but our English language can obscure meaning here. Politicians certainly take advantage of this when discussing the decreased rate of growth in the budget or slowing the rate of budget cuts for social programs!

(Any examples a reader would like to publicize here? I know I've heard some great political lines like this but I cannot find a citation...)

This worksheet goes back to the air pressure activity introduced earlier. It is a fairly straightforward exercise in

  • computing average rates of change,
  • plotting secant lines, and
  • taking a first pass at the concept of concavity. 

Because it's straightforwardly computational rather than deeply conceptual, use this for a moment in class when you want students to work through the ideas but also want to give them a little mental break. It's a good time for getting a drink of water or chatting a bit about how things are going. Sometimes students need some computation and a stretch, as the ability to concentrate on mathematics for more than twenty minutes at a time takes development through repeated practice.

Rates Of Change: Mountains

I've been working on a post about interpretation of story problems and graphs, so that will probably make an appearance next week. It's also time to go toward derivative rules and derivative graphing. Good old-fashioned non-applied explanations of the derivative at a point and the derivative as a function are up to you, as I find students need a purely mathematical or formal explanation before applications. We'll revisit lynx and naproxen and hopefully add another story to the mix!

Aquifers and Rate of Change

Since childhood I'd had a mental image of an aquifer as a big underground lake, but it turns out that's not so accurate. Aquifers are layers of permeable rock or ground-up rock (dirt, silt, etc.) below the earth's surface that contain groundwater. When we sink a well down to water, we're trying to extract water from an aquifer. Any time you hear about groundwater usage (as opposed to surface water usage) you should think "aquifers!"

Why care about aquifers? Well, I like to drink water. When I visited Charleston, South Carolina and Savannah, Georgia this spring I learned that many wells in the region have been rendered useless because of saltwater intrusion -- if you pump out fresh water and you're near the coast, salty water comes in! Also, I would like it if my house did not collapse into a sinkhole. Apparently in 2010 about 130 sinkholes appeared in Florida, because of rapid removal of water from aquifers. That water in the spaces in the rock is pretty important. Last of all, many people enjoy lakefront property and recreation. The site just linked is for the White Bear Lake Restoration Association. Why does White Bear Lake in Minnesota need a restoration association? Because it's been shrinking dramatically, and now the docks are on dry land and lakefront property isn't on the lakefront any more. The US Geological Survey (USGS) and Minnesota DNR have concluded it's because of the draining of an aquifer that has contact with the lake.

Alright. We like drinking water, not falling into sinkholes, and waterskiing rather than trudging through muck. How, then, do scientists look at aquifer health? One way to do this is through keeping track of well levels across a region and coupling that data with geological information about aquifer locations. The graph of water level for a well is called a hydrograph, and this one is shared directly from the Minnesota DNR page with their permission:


The USGS maintains a groundwater watch page from which you can find all sorts of data for your local wells, and many states maintain similar pages. I used the Minnesota Water Level Monitoring Page to find the raw data for the above well, and then used the R program to create graphs for this worksheet on rate of change.

You'll notice a huge seasonal variation in well depth. Groundwater is commonly used for industrial applications, which may be year-round, but also for agriculture and lawn care, which are seasonal. According to one source, the city of Woodbury in Minnesota pumps around 5 million gallons a day in winter and 20 million gallons a day in summer.

This worksheet looks at real, messy data. Students are asked to estimate a lot of numbers and discuss their estimates with group members and the instructor. It's a good time for discussion of estimates and how we deal with real, messy data -- shill for your local statistics class here! The worksheet covers:

  • graphical approximation of average rate of change,
  • graphical approximation of instantaneous rate of change,
  • creating a linear model using approximations of rate of change,
  • and analyzing the model.

The graphs take up a lot of room but the questions are pretty straightforward, so print it double sided and it won't take that long to do in class. Have students work in groups and ask different groups to report their results by writing them on the board: they will have different numbers and can discuss the validity of each approximation. The last question, in particular, is open for a lot of debate: what does it mean for a well to "run dry" if there is seasonal variation in water level?

 Rate Of Change: Aquifers worksheet

Oh, if you've forgotten, remember we're still under federal budget sequestration: the USGS is going to have to turn off a number of streamgages used to monitor stream health and warn of flood events because it can't afford to keep them going...