Class things
May. 13th, 2014 10:46 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
davvI have been very busy lately with lots of assignments to finish and suchlike. But there was one thing I partly discovered on my own in the last mathematics lecture, that I'd like to mention:
We were on the subject of inverse functions, and the professor showed us how to find the derivative of an inverse function without knowing what that inverse function actually is. (This was in the multivariate case, but the logic holds both there and if you have only one variable.)
But then I started thinking: couldn't this be used to find the inverse function itself? Just repeat finding the derivative for the second, third, fourth, etc, derivative and then construct a Taylor series. And I asked about this, and he said that, indeed, that was possible.
One of the nice things about mathematics is that if you're not violating any of the premises, any trick you can think of will work. There's no god of mathematics that simply says "now you're just being a smartass" :)
(Later, he dealt with the theory of finding a graph curve (y = f(a, b, c, ...)) for a more general function (that inputs and outputs a vector), in the vicinity of a point. He proved that the curve will always exist given certain preconditions, but not how to find it. So I asked how one may find it, and he said that "in general, you'll have to do it numerically. There may be closed-form tricks in certain special cases, but I don't know of them". I later thought that sounded a lot like integration: there are tricks but if you want something fully general, numerical is the way to go :) )
We were on the subject of inverse functions, and the professor showed us how to find the derivative of an inverse function without knowing what that inverse function actually is. (This was in the multivariate case, but the logic holds both there and if you have only one variable.)
But then I started thinking: couldn't this be used to find the inverse function itself? Just repeat finding the derivative for the second, third, fourth, etc, derivative and then construct a Taylor series. And I asked about this, and he said that, indeed, that was possible.
One of the nice things about mathematics is that if you're not violating any of the premises, any trick you can think of will work. There's no god of mathematics that simply says "now you're just being a smartass" :)
(Later, he dealt with the theory of finding a graph curve (y = f(a, b, c, ...)) for a more general function (that inputs and outputs a vector), in the vicinity of a point. He proved that the curve will always exist given certain preconditions, but not how to find it. So I asked how one may find it, and he said that "in general, you'll have to do it numerically. There may be closed-form tricks in certain special cases, but I don't know of them". I later thought that sounded a lot like integration: there are tricks but if you want something fully general, numerical is the way to go :) )