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On Sat, 19 Jul 2025 11:00:53 -0700, The Horny Goat wrote:

On Mon, 19 Aug 2024 20:54:58 -0400, Joy Beeson
<jbeeson@invalid.net.invalid> wrote:

Many years later, I learned that this was because my teachers not only >>didn't explain the fundamental thereom to me, they didn't even tell me
that calculus *had* a fundamental thereom.

You scared me for a moment since 40 years ago I was a math major so
obviously would have learned that. So I Googled and realized I hadn't forgotten the theorem, just the name of it...

Yes, it's a pretty obvious calculus fact - that the derivative of the
integral of a function is the original function, so integration is the
inverse of differentiation. That lets one work out the first few integrals
to get started.

Even though you could never approach teaching first-year calculus to first graders, I thought that you could probably explain the fundamental theorem
of calculus to them with a felt board. Although the usefulness of that
would be limited.

John Savard
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John Savard <quadibloc@invalid.invalid> wrote or quoted:

Yes, it's a pretty obvious calculus fact - that the derivative of the >integral of a function is the original function, so integration is the >inverse of differentiation. That lets one work out the first few integrals >to get started.

Even though you could never approach teaching first-year calculus to first >graders, I thought that you could probably explain the fundamental theorem >of calculus to them with a felt board. Although the usefulness of that
would be limited.

Everybody gets that a bucket fills up faster if the stream of water
coming in is stronger. What ends up in the bucket is the integral,
basically the sum. The force of the stream, though, that's the
derivative, the rate of change. If we watch how the water level
rises, we can figure out how strong the stream is, and the other
way around too. That's the fundamental theorem of calculus.

Like I mentioned earlier, if you want legit hard science
fiction, check out that Springer book with stories written by
actual scientists. The quality's all over the place, but a few
of them have been pretty solid so far (I have not yet read all).

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ram@zedat.fu-berlin.de (Stefan Ram) wrote or quoted:

Everybody gets that a bucket fills up faster if the stream of water
coming in is stronger. What ends up in the bucket is the integral,
basically the sum. The force of the stream, though, that's the
derivative, the rate of change. If we watch how the water level
rises, we can figure out how strong the stream is, and the other
way around too. That's the fundamental theorem of calculus.

And if there's already some water in the bucket at the start,
that's the integration constant!

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John Savard <quadibloc@invalid.invalid> wrote:

Even though you could never approach teaching first-year calculus to first >graders, I thought that you could probably explain the fundamental theorem >of calculus to them with a felt board. Although the usefulness of that
would be limited.

I could see it being kind of useful in special cases... like how if you
know your speed and time you can find your distance and so that means if
you know your distance and time you can figure our your speed. That is
a thing that a first grader might find valuable.

I took calculus for engineers and we got a mention of how the integral
and the derivative were opposites, but it was never described as a
theorem and never proven. It was just something we were expected to
take for granted as part of the definition of what the operations were.

Engineering calculus was all about how to do operations as quickly as
possible rather than what they actually meant.
--scott
--
"C'est un Nagra. C'est suisse, et tres, tres precis."
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