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5
Perhaps equally interesting are theorems that can only be proved
"by exhaustion"---by a long case analysis---as far as we know.
For example, that there are only a finite number of sporadic simple groups
seems to be such a theorem.
According to Jack Schmidt, Gerhard Michler studied the issue
and thinks that
"it should not be taken for granted ...
5
Again, the more standard label would be "by brute force" or "case by case", rather than "by exhaustion", since the latter refers to a proto-calculus method dating back at least to Archimedes.
My own reaction to lengthy case-by-case treatment is that such a discussion seems to give no reason for the phenomenon. That is, a "...
4
A classical example of proof by exhaustion is to establish an integer $N$ is prime by trial divisions. One uses trial division by candidates for divisors up to $\sqrt{N}$. This can be made slightly less exhausting by using the method of the Sieve of Eratosthenes.
If $N$ is large, say 250 digits, then this approach to establishing primality is not practical, ...
0
Here are several real-world problems that are mostly accessible to first-year students. Some of them require a little bit of differential equation manipulation, but a minimum amount:
Geometry (where do equations of volumes of cones / spheres / etc come from)
Deriving the Interest Rate Equation
Deriving the Logistic Curve
Deriving the Kelly Criterion
Lotka-...
1
More of a comment than an answer but I don't have the rep to post comments. When my AP calc professor was introducing derivatives, we dropped Barbie dolls out of a window and timed their descent lol. We found average speeds that way, and then he asked us if we could determine the speed it was falling at exactly 1 second. There's not a good way to do this ...
3
I worry about throwing a bunch of application/modeling that relies on diffyQs beyond the AP curriculum. You are combining tougher aspect (applications are harder than straight math) along with new math content. This is sort of what I hate about how PDEs are taught (relatively unfamiliar application, like string tension, plus the new math needed to solve it)...
2
While you're talking about physics, you might want to mention generally the way physics is computed in video games. The positions and velocities of various objects are updated every fraction of a second, with the new velocity being the old velocity plus acceleration (due to forces) times time and the new position being the old position plus velocity times ...
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