Your whole "equivalent to but not equal" thing, for instance, is a matter of much discussion - the = sign does indeed mean different things to different people at different times It's valid, you can say that if you want
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Yeah - does that dx in every derivative actually exist? It's not an easy question.
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It was a great big dilemma and multiple people (Newton, Cauchy, etc) worked really hard over a century to try to find a way to say what calculus was about without using words like "infinite", to "banish the infinitesimal" It's fun to read about
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I mean it's this kind of thing that makes the human process of doing math fun to learn about and that demystifies this idea that mathematical truths are holy tablets received from God
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This is why I go into a bit of this history, and *why* people hated it so much, when I teach intro calc. Because students seeing it for the first time hate it for the exact same reasons, and they're not sophisticated enough w/proofs yet for epsilon-delta to satisfy them.
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It's also really useful for explaining why we have both f'(x) and dy/dx notation.
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