My colleague is currently struggling a bit with how to introduce and motivate the Fundamental Theorem of Calculus. So I looked in the essay I wrote for my teaching diploma and found a quote by Leibniz, which is useful, and some less than useful “intuitive” arguments based on distance and speed.

While it’s worth remembering that I had done virtually no teaching at the time of writing, the essay does still manage to provide some insights into teaching concepts in calculus.

Maybe someone else will find it useful – it’s available here.

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I thought what you wrote about the geometrical argument was pretty good. I always found the fundamental theorem sort of magical even though I got it. I mean the fact that the area depends only on the value of the primitive function on the edges of the interval. It's so cool!Which Leibniz quote did you mean? The one with sum and difference?

I think the geometrical argument is quite confusing and requires careful handling. I guess if one is well-versed in physics that makes things easier, however. The primitive function carries a lot of information about the original graph – so I don't see anything that "only" about it. To be honest, I never found the FTC that fascinating. Useful, but not awesome or that surprising.Yes, the "integral is a sum and derivative is a difference" quote. It's neat.

Neat? That's the part that's obvious! And sure, the primitive function carries lots of information, but does that make this theorem less profound? I've never had trouble motivating the fundamental theorem for my students. But I get very enthusiastic when I teach it. That helps.