Planning Calculus: What Is My 2nd Unit?

A few weeks ago, just before the semester started, I had unit 2 as calculating the derivative, and unit 3 as using the derivative, much like Math Boelkins does. It seems like a logical division. But now I'm thinking it's not a good way to frame things pedagogically. We would spend way too much time coming up with 'rules' and learning to use them. That would get the students so stuck on the procedural that I'd never be able to get them to go back to trying to make sense out of these new ideas.

I want to introduce the 'rules' in the context of real problems. I don't have the facility with equipment that Shawn Cornally does. If I could spend a year as an intern in his classroom, I think I'd take lots of his ideas back home with me. But every time I think about using lab equipment, I get nervous. (We used a lab thermometer in pre-calc to measure the temperature of a hot cup of coffee during our murder mystery. And we built inclinometers with a protractor, tape, a straw, thread and a weight. That's about the extent of 'lab equipment' for me so far.)

To rethink this, I started with the sections of our textbook (Briggs Calculus) that I'm supposed to 'cover':

3.2: rules (constant, sum, power, constant multiple, e^x!)
3.3 product rule
3.4 trig function derivatives

3.5: derivative as rate of change

3.6: chain rule
3.7 implicit
3.8 derivative of log and exp fcns
3.9 derivative of inverse trig fcns

3.10 related rates

4.1 maxima and minima

4.2 what derivatives tell us

4.3 graphing

4.4 optimization

4.5 linear approximation and differentials
4.6 mean value theorem

[3.1 was part of the first unit, and the last two sections of chapter 4 will be part of the last unit.]

I played around with separating the 'rules' and the 'applications', and it seemed like there were only 4 applications (highlighted, with 4.1 to 4.3 as one). I'm not sure why the text splits things up the way it does. Graphing could come earlier, and seems like a simple way to begin seeing what derivatives can do for you.

In the first week of class I had the students read some history from Morris Kline's Calculus: An Intuitive and Physical Approach (pages 1 to 6). He gives 4 major problems that needed calculus (no wonder two people invented it at once):

1. Motion:  planetary & projectile,
2. Tangents to curves: for projectiles (will it hit head on?) & for lenses (telescopes and microscopes),
3. Optimizing (best angle to shoot a cannon, when will a planet be closest and farthest from sun?), and
4. Lengths of curves, areas, volumes, also center of gravity (for example, the volume of the earth, which is an oblate spheroid; these topics are mostly delayed to calc II).

I'm wondering if any of these historical applications are useful to bring into the mix.

What I have for now is:

App1: Rate of change & Graphs
Rate of change:
3.5 derivative as rate of change, velocity, rate of growth, cost (we mostly did this and this can be review)

In pre-calc, we factored to find x-intercepts of polynomials. What we couldn’t find was the maxes and mins, which are often pretty important.
3.2: rules (constant, sum, power, constant multiple, e^x! skip e^x for now)

4.1: maxima and minima
4.2: what derivatives tell us
4.3: graphing


App2: Sound!
3.4 trig function derivatives
3.3 product and quotient rule  (Shawn C uses 1/x * sin x to model guitar string)


App3: Differential Equations & Growth, Related Rates & Optimization
Infection (logistic application by Bowman D)
e^x
3.6: chain rule
3.7: implicit
3.10: related rates
3.8 derivative of log and exp fcns,
3.9 derivative of inverse trig
4.4: optimization

I still need a home for:

4.5 linear approximation and differentials
4.6 mean value theorem

This would make 3 units instead of two. That might be fine. I have to decide today, and get a unit sheet made by this evening. I keep wishing I could teach for a week and prep for a week. (Dream on!) But as hectic as this is, I'm having fun working on getting it right. And I know it will go more smoothly next semester.