Originally posted by Benarius
[f(x + h) - f(x)]/h^2 as the limit of h--> 0
Wonderfull. I like your explanation of the limit.
No, I mean, the limit of h as it goes to 0. As h gets infinitely close to, but not exactly, zero. That formula is for a derivative.
f(x) means the function with x plugged in. So if f(x) = 2x + 4, f(a) = 2a + 4, and f(5) = 2(5) + 4 .
The limit is just when something gets infinitely close to but doesn't touch something.
What that formula is doing is taking the distance between two points (x and x + h) where h is the distance between these points. If h is 2, and x is
6, you're taking the distance between the points where x=6 and x=6 + 2=8.
You can draw a line between two points and find the average slope between them, easy. Change in Y over the change in X. But what if you want to find
the instantaneous slope, that is, the slope at one point, instead of between two points? You can't use change in y over change in x because for
change, you need two points!
By defining h as the limit as h gets infinitely close to zero, you're making the distance between these two points infinitely small. By doing this,
you can use change in y over change in x, but since the two points are infinitely close to one another, it's effectively one point. And therefore,
the slope at a point! This is much easier to understand if you see it graphically. You can probably find explanations with graphing pretty easily,
and I don't want to seem to patronizing since you really do have more wisdom by virtue of age.
