So… I just re-read your brain dump post and realized that you described an issue that I not only encountered but the exact example for which it happened!
so i might remember the intuition behind newton’s approximation, but i won’t know how to apply it or won’t remember that it’s useful in proving the chain rule.
I indeed have a card for Newton’s approximation but didn’t remember this fact! That said, I don’t know whether I would have noticed the connection had I tried to re-prove the chain rule, but I suspect not. The one other caveat is that I created cards very sparsely when I reviewed calculus so I’d like to think I might have avoided this with a bit more card-making.
I want to highlight a potential ambiguity, which is that “Newton’s approximation” is sometimes used to mean Newton’s method for finding roots, but the “Newton’s approximation” I had in mind is the one given in Tao’s Analysis I, Proposition 10.1.7, which is a way of restating the definition of the derivative. (Here is the statement in Tao’s notes in case you don’t have access to the book.)
So… I just re-read your brain dump post and realized that you described an issue that I not only encountered but the exact example for which it happened!
I indeed have a card for Newton’s approximation but didn’t remember this fact! That said, I don’t know whether I would have noticed the connection had I tried to re-prove the chain rule, but I suspect not. The one other caveat is that I created cards very sparsely when I reviewed calculus so I’d like to think I might have avoided this with a bit more card-making.
I want to highlight a potential ambiguity, which is that “Newton’s approximation” is sometimes used to mean Newton’s method for finding roots, but the “Newton’s approximation” I had in mind is the one given in Tao’s Analysis I, Proposition 10.1.7, which is a way of restating the definition of the derivative. (Here is the statement in Tao’s notes in case you don’t have access to the book.)
Ah that makes sense, thanks. I was in fact thinking of Newton’s method (which is why I didn’t see the connection).