livingcalculus.com

Category: Pedagogy and Teaching Calculus

  • Is the MVT the Integral of the IVT?

    In a word, no.

    But it’s a great question with a really interesting answer. If you ever have a student in your calculus class ask this question, then you are doing something very right.

    But let’s take a step back. What does the title of this post even mean? As discussed in my last post, the MVT states that f’ (if it exists on a closed interval) must equal its average value somewhere in that interval. And it is clearly true (thanks to the IVT) that any continuous function equals its average value at some point in a closed interval. This suggests that we can easily prove the MVT by invoking the IVT. The “proof” goes like this:

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  • Visualizing the Mean Value Theorem

    If your students have seen that (for a differentiable function on a closed interval) the average rate of change of f always equals the average value of f’, then you are ready to show them a nice way of visualizing the Mean Value Theorem. I think almost every calculus teacher illustrates the MVT with a picture something like this one:

    Here, we can see the point(s) at which the average rate of change matches the instantaneous rate of change. If we view the same situation from the perspective of f’, then we get this picture:

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  • Average Value and Average Rate of Change

    One connection I like to make in my class is that, on any interval, the average rate of change of a differentiable function is equal the the average value of its derivative. This follows directly from the Fundamental Theorem, since the average value of \(\scriptsize f’\) on \(\scriptsize[a,b]\) is $$\frac{\int_a^bf'(x){\rm d}x}{b-a}=\frac{f(b)-f(a)}{b-a} .$$From my point of view, this connection is what justifies the use of the word “average” in describing a rate of change. What are we averaging? The derivative, which is to say the rate of change, of f. What are we averaging over? The entire interval \(\scriptsize[a,b]\).

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  • Enduring Understandings For Derivatives

    As I’ve been preparing to give a talk at NCTM in October, I’ve been thinking about my much-beloved growth rate activity and why I like it so much. Thinking about this helped me name three enduring understandings for derivatives: I want students to

    1. Understand the difference between change and rate (or ratio) of change.
    2. Understand slope (or rate of change) as quantity which is dynamic, and is therefore worth viewing as a function in its own right.
    3. Understand the distinction between slope over an interval and slope at a single point.

    Understandings 1 and 2 are, perhaps, the most important to the widest range of students: these are concepts that everyone should understand in order to be an intelligent consumer of quantitative information. And yet understanding 3, which is of interest to a relatively select audience, is what often gets the most emphasis in calculus classes. Moreover, I believe that students have a hard time engaging with number 3 if they don’t first understand numbers 1 and 2. So, I start my class by teaching understandings 1 and 2, then get to understanding 3 later in the year (See here for more thoughts on delaying limits).

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  • Introducing the Chain Rule Part 3: Teaching Related Rates First

    In my last post, I described the importance of drawing a clear connection between the idea of related rates and the concept of a composed function. In this post, I will describe a way to introduce the chain rule using related rates. I like to think that this is the “this actually makes sense” intuition for the chain rule that many people wish they had and many calculus teachers wish they could convey. Sam J Shah has another take on how to make the chain rule approachable, which is a bit more mathematical and a bit less physical.

    Here is the image at the top of the packet I use to introduce the chain rule:

    We have someone walking up a hill, so that their elevation is a function of position which, in turn, is a function of time. How fast are they gaining elevation?

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  • Introducing the Chain Rule, Part 2: Connecting Composition and Related Rates

    One reason that so many students struggle with the chain rule is that they never really got the point of composition in the first place. Sure they can “put one function inside another,” but they are missing this essential point:

    Composite functions appear in situations where the value of one quantity is determined by another, which in turn is determined by another.

    Functions appear in situations where the value of one quantity is determined by the value of another.

    Understanding composition this way builds a natural bridge between related rates and the idea of composition. For example, consider the following scenarios:

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  • What is Calculus (Part 2)?

    In a previous post, I wrote that Calculus is the branch of math that deals with understanding the relationship between rate of change and accumulation of change.

    The website Better Explained says that “calculus finds patterns between equations,” and specifically mentions the connection between the area of a circle \(\scriptsize (\pi r^2)\) and the perimeter of a circle \(\scriptsize( 2\pi r)\). Calculus relates these two: the perimeter is the derivative of the area.

    But what does the area of a circle have to do with rate of change and accumulation of change? How is this situation similar to the COVID cases used as an example in my previous post?

    Suppose we have a circle, and we imagine growing it slowly: starting with a radius of 0 and getting bigger. In the circle below, we have the accumulated area (the full circle) in white.  The gray strip around this circle shows the change in the area resulting from increasing the circle’s radius just a bit.

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  • What is Calculus?

    One of my all-time favorite complements was from a student who told me that he loved calculus because the class is one topic all year long: we just keep learning about the same idea in more detail. Indeed, part of what I love about teaching calculus is that I see it as being a unified class, a single multifaceted concept.

    So what is the topic that I claim is the subject of calculus?

    I claim that calculus is the branch of math dedicated to understanding the relationship between accumulated change and rate of change. This is a fairly radical claim: many would say that limits (or continuous change) are the heart and soul of calculus.1

    The graphs below give a specific example of accumulation and rate of change. On the left, we have a graph showing the new daily COVID cases in the US in the first couple years of the pandemic. On the right, we have the cumulative number of cases: total cases ever recorded in the US, up to a given day.

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  • Introducing the Chain Rule, part 1: Comparing Leibniz and Prime Notation

    I have so many thoughts about teaching the chain rule. If you just want some problems to help you introduce the chain rule in a problem-based way, I can cut to the chase: start with these problems and then move on to these ones. If you look at those problems and they make sense, you can probably use them without reading more. But if you want some extensive rumination on teaching the chain rule, I have a whole series of posts coming up for you.

    To my mind, half the battle of teaching the chain rule is helping students understand how it is that the Leibniz form and the prime form mean the same thing. On the one hand, $$\frac{{\rm d}y}{{\rm d}x}=\frac{{\rm d}y}{{\rm d}u}\cdot\frac{{\rm d}u}{{\rm d}x}$$ feels so obvious that students can’t quite see why one would bother commenting on it. But this is deceptive: derivatives aren’t just fractions and the importance of the fact that you can treat them like they are is deep and vast. On the other hand, $$(f(u))'(t)=f'(u(t))\cdot u'(t)$$ feels impenetrable: it’s hard to even make sense of this unless you are thinking very closely about where the primes are. And it takes some thought to see why one derivative is evaluated at u(t), while the other is evaluated just at t.

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  • Some Thoughts on Introducing Limits

    Almost all calculus classes start with limits. The consensus seems to be that it is impossible to discuss derivatives or integrals without first discussing limits. But of course, calculus without limits is possible: it is just numerical calculus.

    Instead of asking “how can you talk about calculus without limits,” this post will argue that we should instead ask “how can we expect students to engage thoughtfully with limits unless they first see a problem where they are needed?”

    Sam Shah’s blog has many interesting posts, and this post is related to Sam’s post on getting rid* of limits. In brief, I don’t think that we should get rid of limits, but I do think that they are a terrible place to start a calculus class, and I also think that they are much less central to calculus than many people seem to think they are.

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