I'm almost done. In fact the best advice I have for you is to get the online version of Strang's

*Calculus*, (or better yet, buy your own copy), and read Chapter 6.

In the last post we came to:

and flipping things around to look at x as a function of y:

It turns out that the constant c = log

_{b}e, and then of course, c = 1 when b = e. We can get an expression for that. Since I get a little freaked out looking at dx/dy, let's express the log function in the normal way, where y = log

_{b}x. So now we want dy/dx. As before, call Δx, the small change in x, h, then:

In this case, y(x) = log

_{b}x so we want

At the particular value x = 1:

If we substitute n = 1/h

The part in the brackets is e:

There are other definitions in the book, but this one about e as the limit of an infinite series is certainly fundamental. In exactly the same way we can develop:

and this will give us the famous series for e

^{x}(although to be honest I am still confused about how that works):

## No comments:

Post a Comment