Summer of Math Exposition

The motivation from the start is clear: we use the second fundamental theorem of calculus often as a computational tool, so knowing why it works is helpful. For the target audience of someone in an introductory calculus course, everything is explained in a comprehensible way. On the more negative side, regarding the video’s novelty, explanations of the FTC have been done to death, and this video doesn’t bring much new to the table. This video will probably be memorable to some of its target audience, but with my experience, it likely won’t be for me.

More minor notes, given by timestamp:

0:00 So the description says that part 1 of the FTC is the “derivative of the integral” part, and part 2 is the “integral equals antiderivative at upper bound minus it at lower bound” part. But at the start of the video itself, that seems to have been swapped. I know there are different conventions (I follow the one in the description), but can the video at least stay consistent with itself?

1:03 It’s always the signed area. When the signed area happens to be positive, that’s the same as ordinary area. Also, even if f is positive, we still have to worry about a minus sign if the lower bound of the integral is greater than the upper bound.

7:37 Well, why not write it like f(x star) Δx? First of all, that maintains the order that l ⋅ w was written in. Secondly, that sets you up to establish the connection to the f(x) dx in the integral notation, and it’s a huge missed opportunity not to explain that. It’s one of the typical clichés not incorporated in this video, which works to its detriment.

7:58 Similarly missed opportunity to write that f(x star) = ΔF(x) / Δx. It’s just like f(x) = dF(x) / dx. Calculus is all about these tiny changes in value.

11:27 Again, this ignores the other assumption that a is less than b. If a were greater than b, then Δx would be negative (so that you go in the negative direction from a to b), and the sign of Δx ⋅ f(x star) would account for that.

Conclusion: this video is pretty good, with no real major flaws to speak of. I have quite a liking for calculus and for the fundamental theorem of calculus in particular. Unfortunately, the main thing keeping down the score is that it’s based on “how valuable [this entry is] to the space of online math exposition”, and considering how readily one can find other explanations of the FTC that touch on the same points with at least the same quality, I’ve decided not to score it too high in that regard. Therefore, my rating is 6.50.

I like that it’s giving me a Khan academy vibe with an easy to follow instruction style. This video seems to satisfy most of the scoring criteria, although the “innovative” part can still use some work. i would say overall it’s a great addon to the class room materials.

Exposition/explanation was adequate for task but not especially earthshattering - perfect fodder for a college calculus lecture. I think I would have preferred if a numeric example were used in conjunction with the drawn one - hand-drawn not-to-scale figures sometimes feel misleading or muddying to me.

I have never seen the FTC being visualised in that way and I really enjoyed it. Your presentation was simple, didn’t require much. Just beautiful and it gave the viewer a nice intuition.

Good video! Th e presentation was well thought out, although rather simplistic. There was a satisfying aha moment and I’m optimistic I’ll be able to remember it for a while. My main reasons to deduce points are the novelty and pacing: The proved theorem is already widely discussed — nothing you can do about that — and I think you could have gone a bit faster.

Nice explanation, easy to follow. I was a bit confused by the (lack of) dimensional analysis - you’re stating that an area is equal to a length - but I suspect that that would make perfect sense to someone with more calculus knowledge than I.

Very efficient, clear explanation.

The only thing that could trouble the viewer is that on the drawing the length of the line and the area do not look the same (naive geometric intuition says that are is lot bigger then a thin line). Maybe a couple of computer drawn exact example at the end would help this issue.