One of the most useful mathematical insights I gained from working as an astronomer is that the bounds in an integral are inextricably linked to the variable of integration, and that this plays an absolutely critical role in integration by substitution. Changing the bounds when you change the variable isn’t just a shortcut or a convenience: it is an essential part of the process of substitution. And it makes total sense if we look at substitution in a physical context in which we change between two variables that actually mean something to us. The problems linked here introduce substitution by placing it in a physical context and clarifying what it really means to change the variable of integration.
(more…)Category: Pedagogy and Teaching Calculus
-
Visualizations of the Fundamental Theorem
Suppose you have done this activity, which has students use a numerical derivative to produce a growth-rate graph based on CDC growth charts. And suppose you have also done this sequence of activities, which introduces students to integration using the energy released by a supernova and culminates in the computation of an accumulation function, showing cumulative energy over time. And perhaps you’ve also had students do this problem, which gets them thinking about the connection between accumulation and rate in the context of COVID. If so, then your students have seen all three pairs of graphs shown below.
(more…) -
Signed Area and the Definition of the Definite Integral
There is a long-standing tradition of defining integrals as being area and introducing accumulation as an application of this core mathematical idea. What happens if, instead, we define integrals as being accumulation and then link this idea to area? For example, I use this activity to introduce Riemann sums by first having students estimate the total energy released by a supernova over a period of time, and then leading them represent their work in terms of area.
This approach has a clear benefit to student buy-in: students are likely to find accumulation problems more intrinsically interesting than areas. It also has a benefit in terms of giving students some intuition for integrals. For example, in the graphs above, students find it intuitive that the volume graph is steep when the flow rate graph has a large value. There is also a strong historical case for foregrounding accumulation: Newton was certainly more interested in motion than in area per se, and Babylonian astronomers tracking the motion of Jupiter were among the very earliest to use Riemann sums.1
But you may be wondering: is this rigorous? Shouldn’t integrals be defined in terms of a mathematically clear idea like area, rather than a more ill-defined one like accumulation? This post focuses on the theoretical benefits of foregrounding accumulation. I would argue that foregrounding accumulation actually leads to a more theoretically rich (and more rigorous) framework for understanding Riemann integrals.
(more…) -
Numerical Derivatives and the Mean Value Theorem
As I outlined in an earlier post, it is challenging to give first-year calculus students a compelling reason to care about the MVT. Its theoretical importance is often lost on this audience: using the MVT to prove the Increasing Function Theorem often creates confusion rather than providing clarity. And the sorts of “applications” often introduced to a first-year class feel contrived: the classic “cop giving a speeding ticket” example is amusing, but it hardly conveys the importance of this theorem.
However, if you’ve spent some time (even just a couple classes) with numerical derivatives, then the MVT has a clear and important meaning. Indeed, the MVT gives the answer to important questions students have likely been asking since they first encountered numerical derivatives. Before posing these questions, let’s take a brief detour to review numerical derivatives.
(more…) -
On Teaching the Mean Value Theorem (part 1)
The Mean Value Theorem (MVT) is a central piece of a first class in calculus. However, the role it plays in the class is often not clearly defined. In many classes and books, students are told that the MVT is the “backbone of calculus,” or some such thing: it is the foundational result on which many important things are built. From the standpoint of analysis, this is a reasonable claim: the MVT and extensions of it are essential to proving everything from the Increasing Function Theorem (IFT) to some of the more general versions of L’hospital’s rule. But how does this come across to a typical calculus student? Such a student has not yet taken (and may never take) a course in analysis and lacks the context to see why the MVT is so manifestly important.
(more…)