Tag Archives: mountain climbing

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!

Group work and a power function for atmospheric pressure

We instructors of calculus know that linear models aren't everything, even though linearization is in some sense the point of the differential calculus. Since the first week of many calculus courses begins with a precalc review including power functions, I'll just move smoothly along to a power function model for atmospheric pressure! (Don't worry: we'll get to lynx trapping in the Yukon for a trig review activity in a day or two -- not everything is about physics or the atmosphere.)

It's always important to remind students about the difference between power functions and exponential functions, not least because they've got different differentiation rules. One nice way to look at power functions and exponential functions is by looking at growth -- we know that  x^2 and  2^x grow at very different rates. But everyone does that... and I was having fun with atmospheric pressure! This worksheet has a very funky power -- 1/0.19... -- and might be a good way to acquaint students with the messiness of real-life models. I will return to this topic when we get to derivative and integrals, too, because this equation is actually fairly easy to derive.

The worksheet below tries to foreshadow the idea of the derivative fairly heavily. It asks about 

  • composition of functions,
  • intervals of increase and decrease, and
  • slope of the tangent line.

As usual, I try to incorporate a bit of writing and thinking about the meaning of a model as well. There's definitely room for discussion around these worksheets.

In addition, you might notice that there's a bit of tedious calculation at the beginning. Why would an enlightened modern instructor do that? I like to give these worksheets to students in groups. At the beginning of the semester I always give a student survey asking about past math experience, major, problems or gifts I should know about, outside interests, favorite dessert, favorite color.  In the first few weeks of class I use this to arrange student groups and ask them to figure out how I've grouped them (by major, dessert, color, last name...!). Giving them just a bit more tedious calculation than most people would enjoy gives me a chance to encourage and incentivize conversation within groups even more.

This week's worksheet: atmospheric pressure as a power function!

A better model: power function

There's also a natural place to discuss solving for the inverse function here, and I might add a worksheet about that soon too.

Do you use group work or worksheets with students? Why or why not? What kinds of constraints do you have to deal with in considering group work?

A linear beginning

I've taught calculus now at several different college and universities. Calc is a funny class these days: students who have done quite well in high school math now often enter a linear algebra or multivariable calculus class directly, so students in calculus come from a variety of backgrounds but often did not have a good high school math experience. Often students in college calculus classes took the AP calculus offered in high school with varying results. A surprisingly high number of students I've seen in college calc did not even take precalculus in high school. This is generally a recipe for disaster.

During the fall and spring semesters I start every calculus section with a conversation about what calculus is good for and what other classes a student could take to fulfill requirements. I like to push statistics and math for liberal arts majors courses: statistics is becoming crucial to survival and innovation in a number of fields, from finance to medicine, and math for liberal arts majors classes often expose a student to graph theory, voting theory, and other useful techniques for looking at problems we all encounter in life. There is so much beautiful mathematics! Don't get stuck on calculus as the only way forward!

After that, it's time to start with review. I phrase it as a warm-up: here are things you will need to dredge out of your brain and reacquaint yourself with to be successful in calculus. First is the point-slope equation for a line.

The worksheet attached explores the change in atmospheric pressure as we increase in altitude from San Francisco to Denver to the peak of Mount Everest. In it, students develop a linear approximation for pressure in kilopascals from two data points (no use of derivatives), and then examine the validity of their approximation. Use this to explore:

  • how to work through a word problem
  • the point-slope equation for a line, introducing the idea of slope as rate of change
  • critical thinking about models: comparing theoretical and actual results can point out weaknesses in a model!

When discussing this worksheet, remember that temperature, humidity, and weather patterns affect the pressure of our atmosphere as well. Some of the lowest pressure readings ever taken near the Earth's surface have been in the centers of hurricanes, for instance. In calculus we study "baby problems" so that we can eventually build up the techniques to model situations more accurately -- like scales before playing Beethoven.

Linear Functions -- Head in the Clouds Worksheet

How do you start out your calculus classes in the first few days? How much review do you do? Do you think all of your students are best served by calculus?