My guess is Samuel simply forgot to include the attachment that was meant to accompany his post.
The Stew Boy wrote:
lopsidation wrote:
Here's a great one using calculus:
The derivative of x^2 is 2x.
But x^2 can also be written x+x+x+x+x... x times.
Taking the derivative of x+x+x+x+x... x times gives 1+1+1+1+1... x times.
1+1+1+1+1... x times is, of course, equal to x.
So the derivative of x^2 is x.
Thus, x=2x.
Letting x=1, this becomes 1=2. QED.
There's a simple explanation for why this doesn't work, but the real reason is much, much deeper.
I'm not quite sure why this doesn't work.
It's relatively simple, actually. Remember that the derivative of f(x) is the rate of change, i.e. the ratio of how much f(x) is changing compared to how much x is changing. Now, it's true that x^2 = x + x ... + x, but when we look at how x + x ... + x is changing, we have to remember that the value of x
and the number of x's in the sum are both changing. The third step ignores this; in effect it rewrites x^2 = x + x ... + x as xy = x + x ... + x where x and y just happen to be the same.
Another way of putting it -- the derivative of (x + x ... + x) (x times) simply is not (1 + 1 ... + 1) (x times), and there's no reason why it should be, since this cannot be derived from the standard laws of calculus; you can't just ignore the fact that the x in "
x times"
is the variable you're differentiating with respect to.
I'm not sure what was meant by the "
real reason"
that goes deeper than this.
____________________________
50th Skywatcher
[Last edited by Nuntar at 06-07-2009 05:09 PM]
/dy
