As I spent a lot of time driving around this past weekend, it gave me ample time to think about much that was going on around. We can see this evidenced as we look at my past two posts All that (air) pressure and Early morning ride. One other thing I thought about while I was driving was the connection between speed and distance traveled. In particular, as I am teaching calculus right now, the fact that speed is the derivative of distance traveled and, therefore, distance traveled is the integral of speed, it a routinely used to give applications within class.
As I thought about this, I decided that while I was driving around would be a great time to create an exercise for my students. Often, outside of mathematics I find that students will actually use data much more frequently than they will be given an equation. While I try to use data for examples in class, they tend to get bored if they know the data was just compiled for the sake of an exercise. I decided to keep track of my speed at five second intervals, so that we could try to determine how far I traveled. I also set the trip odometer so that I could compare to the actual distance traveled.
Therefore, I attached my GoPro to my steering column (making sure it didn’t interfere with the steering wheel) and set it for a time-elapse and took photos every 5 seconds. When I got home, I went through the tedious process of looking at the photos and entering the data into a spreadsheet. From there I drew up directions for my students on what I would want them to do with it.
The best use for such data I saw was in explaining the utility of Riemann Sums other than right Riemann Sums. That is, when giving the definition of a definite integral, we describe it as finding the area under the curve. In order to do this, we can approximate the area using rectangles. In order to get the exact area, we then take a limit as the width of the rectangles gets infinitesimally small. This causes the error in the approximation to go to zero, resulting in being able to find the exact area under the curve. Notationally we would have
Here Δx is the width of the rectangles and xi is some point within the width of the rectangle. If we take the limit, the choice of xi won’t matter, and the limits will always be the actual area as long as the function is integrable. Since we focus on integrable functions, whether we use right endpoints, left endpoints, midpoints, or even different shapes than rectangles, we will end up with the same answer, so the use of other techniques seems to have no point. I already have made a lab for my students to explore this topic, Motorcycle Stopping Distance, which is available at Calculus Labs.
In that lab, however, we are given an explicit formula to work with. If we are instead given data, there is no way to take the above limit because we won’t have enough information to be able to tell what happens at every instant. Therefore the best we can do is approximate the integral by finding the areas with a set number of subintervals. Hence, I decided to work with the above data and look at what happens if we only measure the speed every minute. We can then approximate the area using rectangles, trapezoids, or a shape where the top is determined by a quadratic function. In each case, we will get a slightly different result, so this can give the students the opportunity to reflect on which technique was easiest and which technique was most accurate.
I then have them repeat this process with speed measured every 30 seconds and then every 5 seconds. We can then talk about how having more data points should lead to a better accuracy, but that it also requires more work. Again we can compare the shapes to see which give the best approximation. In all, this really gives the students the opportunity to look at the use of topics of Calculus in a physics setting, so that they don’t just see it as a course that is independent of the rest of their studies. Hopefully this will lead to better retention of the material as they move onto other classes.
If you’d like to see the results I obtained or the data collected, the spreadsheet with calculations is available here. If you would like to use this lab in your own calculus class I have a preview available, or you can purchase the full lab instructions on the Calculus Labs page, or using the link below. It is also now part of the collection of all of my calculus labs as well. Even if you don’t want to use my lab, I hope that this will inspire you to find a way to collect actual data for use in your class, since it will help to get your students more engaged.