5 ms·

> I've had lots of classes that worked like this, particularly my Calculus I/II classes where there was a textbook but homework from it was just suggested, not

> I've had lots of classes that worked like this, particularly my Calculus I/II classes where there was a textbook but homework from it was just suggested, not collected, and the lectures were entirely sufficient to understand the material and do well on the exams. I'm guessing the exams were easy but you didn't learn much calculus.

jimmaswell 10y ago
I definitely learned calculus from the Calculus classes. I took calculus 3 recently for credits with no difficulty remembering anything relevant from 1/2. Some things I could give right now off the top of my head are the definition of an integral/derivative, squeeze theorem, mean value theorem, integration by parts, epsilon delta definition of a limit, second/third derivative tests, and a lot more. Test me with something if you want.
- thanatropism 10y ago
  So what's the basic idea behind Taylor series? (A "proof sketch", or even a sequence of theorems relying on each other building up to the lightbulb step.)
  - Chris2048 10y ago
    If OP doesn't reply, can you? I'd like any insight I can :-)
    - thanatropism 10y ago
      (1) Rolle's theorem says that if f(x) if differentiable in the interval (a,b) and f(a)=f(b), there must be a point c such that f'(c) = 0. Proving this one uses maths that, although simple, more analysis-like than calculus-like, but it should be intuitive if you try to draw a function with f(a)=f(b). It has to come up and down so it has a point where it has a horizontal tangent. (2) The Mean Value Theorem is like Rolle's theorem, but "tilting" or "rotating" your function, so that if f(b) = r(f(b)-f(a)) then there's a point c with f'(c) = r. And this is already an order 1 Taylor series! It gives f(b) = f(a) + (b-a) f'(c) for some c (3) Now what remains is just induction. For an order-2 series, write the order-1 series for f'(c) as a function of f'(b). You'll start to see the pattern. If you really want to put a bow on it, you can assume the order (n-1) series exists with the known formula and prove the order (n) series works out.
  - jimmaswell 10y ago
    One of the things that was used less and less time was spent on, but they're convergent sum of a series that eventually approximates some function's value with increasing accuracy, from what I remember. You can get a sine value from one. Actually proving them, I can't remember how, but the class treated it pretty much as a short bonus topic.

jimmaswell 10y ago

I definitely learned calculus from the Calculus classes. I took calculus 3 recently for credits with no difficulty remembering anything relevant from 1/2. Some things I could give right now off the top of my head are the definition of an integral/derivative, squeeze theorem, mean value theorem, integration by parts, epsilon delta definition of a limit, second/third derivative tests, and a lot more. Test me with something if you want.

thanatropism 10y ago
So what's the basic idea behind Taylor series? (A "proof sketch", or even a sequence of theorems relying on each other building up to the lightbulb step.)
- Chris2048 10y ago
  If OP doesn't reply, can you? I'd like any insight I can :-)
  - thanatropism 10y ago
    (1) Rolle's theorem says that if f(x) if differentiable in the interval (a,b) and f(a)=f(b), there must be a point c such that f'(c) = 0. Proving this one uses maths that, although simple, more analysis-like than calculus-like, but it should be intuitive if you try to draw a function with f(a)=f(b). It has to come up and down so it has a point where it has a horizontal tangent. (2) The Mean Value Theorem is like Rolle's theorem, but "tilting" or "rotating" your function, so that if f(b) = r(f(b)-f(a)) then there's a point c with f'(c) = r. And this is already an order 1 Taylor series! It gives f(b) = f(a) + (b-a) f'(c) for some c (3) Now what remains is just induction. For an order-2 series, write the order-1 series for f'(c) as a function of f'(b). You'll start to see the pattern. If you really want to put a bow on it, you can assume the order (n-1) series exists with the known formula and prove the order (n) series works out.
- jimmaswell 10y ago
  One of the things that was used less and less time was spent on, but they're convergent sum of a series that eventually approximates some function's value with increasing accuracy, from what I remember. You can get a sine value from one. Actually proving them, I can't remember how, but the class treated it pretty much as a short bonus topic.