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Simplifying calculus by assuming linearity

January 11, 2010, 7:30 am

Last semester I stumbled upon an approach for teaching the concept of the derivative, and later the integral, that worked surprisingly well with my students. It stems from a realization I had that much of what students see when they first learn about derivatives has very little to do with understanding what a derivative is. The typical approach to introducing the derivative throws students directly into the trickiest possible case: a smooth nonlinear curve, and we want to calculate the slope of a tangent line to this curve at a point. To do this, we have to bring in a lot of “stuff”: average rates of change, tables of sequences of average rates of change, and in a vague and non-rigorous sort of way the notion of a limit. It’s this “stuff” that confuses students — not because it’s hard, but because maybe it’s not suited for their first contact with the idea of the derivative. Maybe we need to build their intuition first.

In a nutshell, the approach is: Assume linearity. All too frequently, students do assume linearity, but in the algebraic sense; they tend to want to think that \(\ln(x+y) = \ln(x) + \ln(y)\) and so on. But I mean, assume linearity in the graphical sense. More specifically, the pedagogical idea is use only piecewise-linear functions until students have a sufficiently solid grasp on the concept of the derivative. No smooth curves, no tangent lines, no average rates or limits, until students can explain what a derivative is and what it has to do with slopes.

Here’s how this approach might play out in a classroom.

Consider Alex, a student at our college. Let’s suppose Alex is leaving his dorm room for the cafeteria, which is 100 meters away. His distance \(y\) from his dorm room is a function of time \(t\) (measure distance in meters, time in seconds). Suppose the graph of this function looks like this:

Question: How fast was Alex going? It’s crucial for students to understand that his speed was the same at all points. If his speed changed, we’d see a difference in shape in the graph; going faster means a steeper graph since he covers more distance in the same amount of time, similarly for going slower. This is the essence of his distance being a linear function of time — his distance changes at the same rate all the time. That rate, or speed, is the slope. So the question is trivial to answer. Alex covered 100 meters in 120 seconds, so that’s a speed of \(100/120 \approx 0.833\) meters per second. (That’s about 3/4 of a normal human walking pace.) Students learn at this point that the rate of change in a function has something to do with slope; for linear functions, the rate of change is equal to the slope of the line.

Now suppose Bob, Alex’s roommate, also leaves from the dorm room for the cafeteria, but his distance function looks like this:

Question: How fast was Bob going? It is extremely important for students at this stage to recognize that the answer is: It depends. Bob, unlike Alex, has two different speeds, one prior to the 60-second mark and another afterwards. So the question “How fast was Bob going?” is ambiguous. We have to ask instead: How fast was Bob going at a particular point in time? The answer to this question is what in calculus we call an instantaneous velocity and unless we have a function that is changing at the same rate at all times, any time we talk about a velocity we must be talking about an instantaneous velocity.

OK, so: How fast was Bob going at, say, 30 seconds? Well, at this point on the graph the function is linear, so we can calculate speed by calculating a slope. He is on pace to cover 20 meters in 60 seconds, so his speed at t = 30 is \(20/60 \approx 0.33\) meters per second. And of course this is the same speed throughout the first minute. (The 60-second mark will need a little separate treatment.) And what about the second half of the trip? Well, Charlie covered 80 meters in 60 seconds, so the slope/speed is \(80/60 \approx 1.33\) meters per second.

Now suppose Charlie, who lives next door to Alex and Bob, is also leaving the dorm room for the cafeteria. (Must be feeding time.), but his distance function looks like this:

First of all, what’s his story? How would you give a play-by-play announcement for Charlie’s short trip to the cafeteria? In particular, what’s different about his trip versus the other two? Students tend to be good at reconstructing stories like this, and they’d say that Charlie headed out the door and made it most of the way to the cafeteria, then had to turn around and go most of the way back, and then moved really quickly back to the cafeteria.

How fast was Charlie going? Again, it depends; but it’s easy to calculate. From 0 to 60 seconds he was going \(80/60 \approx 1.33\) meters per second. From 60 to 90 seconds, the slope of the line is negative: \(\frac{80 – 20}{60 – 90} = -60/30 = -2\) meters per second. (Implication: Negative velocities indicate an opposing direction.) Then in the final phase, he was going  \(\frac{100-20}{120-90} = 80/30 \approx 2.67\) meters per second.

So now students have learned the following important concepts/facts about calculus:

  1. Rates of change are calculated with slopes;
  2. Functions that aren’t linear have different slopes in different places, so we must talk about the slope at a point; and
  3. Rates of change can have different signs (positive or negative) and these signs indicate some notion of direction. (Essentially, we learn that rates of change are vector quantities, not scalar.)

Note well that we have developed all these fundamental concepts without introducing formulas (except the well-known slope formula), limits, epsilons, deltas, Δx’s, or any other technical jargon. This is because we are building students’ intuition and conceptual understanding first, using the simplest possible functions — piecewise-linear functions — before introducing the general case of a smooth curve. Once the students’ intuition and conceptual understanding is built up, then they’re ready to tackle the much trickier case of a smooth nonlinear curve and all the notational “stuff” that this important problem requires.

I have at least 2-3 more posts about this planned. The next one will discuss the crucial step of dealing with functions that are not piecewise linear; how do we use the piecewise-linear function approach to ramp up into the general case of differentiable functions? Then, I’ll talk about how this approach works for developing the idea of the integral. And in a later post, I’ll try to go in to some of the devils in the details of this approach, such as how to deal (pedagogically) with the junction points between the linear pieces and to what extent this assumption of piecewise linearity actually works in general — although some of you who are more knowledgeable in analysis than I am might beat me to it in the comments.

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