Approximating the Instantaneous Rate of Change in Calculus

I’ve been trying something new this year in calculus… really having students grapple with the concepts of what they can definitively know, what they can definitively not know, and what they can know with some certainty (but not total certainty) when they are given some information about a car trip. I’m hammering home the conceptual underpinnings of average and instantaneous rates of change. And I’ll blog about that soon I hope. But today’s post comes from where we went with this…

This week, we got to the point where we were estimating the instantaneous rate of change of a function at a point by using the average rate of change for a small interval near the point. And we’re used to seeing things like this in a textbook:

We’re getting our interval smaller and smaller and seeing the average rate of change get closer to some value. This value it is getting closer and closer to is the instantaneous rate of change.

That’s a deep and important thing. And we all know that.

But when we were generating a table like this, one of my students asked “Why do we have to do this? Why can’t we just pick two points really really close together instead of doing this horrible calculation like 4 times? Like super close together. Then we only have to do it once if we’re just trying to estimate the instantaneous rate of change.”


Because who wants to do that horrible calculation like 4 times? It’s tedious, even with a calculator. I wasn’t ready to talk about the derivative but I did want to answer his question. Why do we have to do so many calculations instead of just one?

Unfortunately, I fumbled through it.

And as always is the case, a genius idea strikes me right after class ends. So I decided to use it for my other section.

In that section, I have them think about what the use is of doing this calculation for smaller and smaller intervals, instead of just one interval… one student came up with the idea that “it gives us more certainty… more data to work with…” but that was ambiguously stated. More certainty about what?

So here’s where the idea came in. I had each student individually use only one small interval of their choice (instead of four) to estimate the instantaneous rate of change of y=sin(921,364x) at x=0.

What was great is that some students picked intervals like [0,0.0001] and others [0,0.00001]. Were they similar? Different? WHOA they were very different. Students got VERY VERY different estimates even though everyone used really small intervals. So what’s going on?

When we looked at the average rate of changes for various intervals, we saw this:

So yeah, if you happened to choose two numbers really close to each other, they might not be close enough! You just don’t know. Even if they’re really close. So doing a series of smaller and smaller intervals indeed gives us more certainty that we have a good estimation.

This was just sort of thrown into my lesson, so I don’t know exactly how much they got out of it. But I hope that next year either I use it as a do now, a new conceptual skill that I add to my calculus Standards Based Grading skill list, and make it a little more formalized [1]. Maybe after doing this next year, have a sheet with a few different functions, some which are wildly erratic and fluctuate a lot and some which are nice — and have students pick out merely from the graph and the point I want to estimate the average rate of change, if they can make do with two points “pretty close together” to estimate the instantaneous rate of change, or if they truly do need two points “very very close together.” That would be a good check to see if they understood the conceptual underpinnings of what’s going on.

[1] Idea. Have a sheet with two columns. On the left column, the function y=x^2. On the right column y=\sin(921,364x). Have them use the interval [0,0.001] to estimate the instantaneous rate of change at x=0. Then say: “You have $5 to bet on which one is closest to the true instantaneous rate of change. What are you going to bet on, and why?” Have groups whiteboard their ideas/thoughts for 5 minutes and present. Then show the graphs of the functions. Have then talk for 2 minutes to see if the graphs change their thoughts. Finish up student discussion.



  1. My intro about the car trip went like this: You use an awesome camera (like this one in your cop car and use it to video a speeding car. Unfortunately, the monitor on your computer is broken. So, to see the car, you have to tell the computer to print out certain frames. Which frames do you want to print out to figure out how fast it was going?

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