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Aug
22
answered Sources of problems for teaching/tutoring young mathematicians
Aug
22
comment Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?
At least yours has color, Don. :)
Aug
22
comment Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?
In the world of complex variables, for instance, yes. The exponential is defined as the infinite series, and all its properties are derived from there.
Aug
22
comment Is there an analytic solution for the density function of this complex random variable?
I should probably also add: if the system supports regularized versions of both incomplete gamma functions (the versions where the factorial denominators are incorporated), it's better to use those than the vanilla incomplete gammas.
Aug
22
comment Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?
Thanks for showing the picture of the Euler polygon! In the Hairer-Norsett-Wanner book, the picture of the Euler polygons slowly converging to the true solution is referred to as "Lady Windermere's fan", after the character by Oscar Wilde.
Aug
22
answered Is there an analytic solution for the density function of this complex random variable?
Aug
22
comment Which one result in mathematics has surprised you the most?
I actually wasted some time back in the day trying to look for an analytic function that looked like a circular paraboloid. :)
Aug
22
comment Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?
This is the rigorous way of stating what the animation in my answer was showing.
Aug
22
comment Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?
Now that i think about how it was taught back then, it was quite a while in between deducing the properties of $\ln$ and $\exp$ ab initio, and then revealing that they are in fact the logarithm and exponential that was taught in previous courses!
Aug
22
comment Which one result in mathematics has surprised you the most?
And for the people who'd rather see pictures, here's how bad a tiny perturbation can get: books.google.com/books?id=YHXU4W3Ez2MC&pg=PA202 . Here's Wilkinson's prize-winning paper: mathdl.maa.org/images/upload_library/22/Chauvenet/Wilkinson.pdf . Perfidious indeed!
Aug
22
answered Which one result in mathematics has surprised you the most?
Aug
22
comment Which one result in mathematics has surprised you the most?
All the more galling is that Wilkinson demonstrated that computing the eigenvalues of the corresponding symmetric tridiagonal matrix is in fact more stable than computing the roots of the characteristic polynomial. That certainly outmoded the old-school methods (e.g. Danilewski's)!
Aug
22
comment Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?
On the other hand, way back when I was still in school, the way it was done was to define the natural logarithm as an integral, the exponential as its inverse, and then one deduces the properties of $\exp(x)$ from those definitions.
Aug
22
comment Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?
Well, this is the formulation assuming that the power series is the definition of $\exp(x)$. One now shows that this power series satisfies the differential equation in the title.
Aug
22
revised Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?
added 27 characters in body
Aug
22
comment Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?
The approach I would have taken though, was to express $\exp(x)$ first as 1+(other terms), take the derivative so that you now have the general term $\frac{n x^{n-1}}{n!}=\frac{x^{n-1}}{(n-1)!}$, and shift indices accordingly.
Aug
22
comment Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?
Thanks Agusti. :)
Aug
22
answered Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?
Aug
22
comment Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?
Would showing the power series count as "intuitive"?
Aug
22
comment logic operations on proposals
Ah, so it is. I forgot to use the expansion to (not a or b). Thanks for correcting Niel! :)