For Tom Wiederman Only

Notice that


2 squared = 2 + 2,


3 squared = 3 + 3 + 3,


4 squared = 4 + 4 + 4 + 4,


and therefore


X squared = X + X + X ... + X (X amount of times)


If we take the derivative of each side we get


2X = 1 + 1 + 1 ... + 1 (X amount of times) = X


Now divide by X and we get


2 = 1 QED

Wow, memories from high school come flooding back, seeing this again. I'm posting this response with the understanding that you will delete this thread because it's so far off topic.


A derivative is a measure of how fast something is changing with respect to something else's change. X squared is indeed equal to X + X + X... (X times). Suppose we want to know how fast this is changing with respect to X. We do this by adding a small amount to X (call it h) in the function (X squared), subtract what we had before the change (this difference gives the change in the function) and then divide this difference by h. That is,


change in X squared function = [(X+h)+(X+h)+... (X+h times)] - [X + X +... (X times)]


Now what do we do? All of the X's in the second bracket are cancelled by X's in the first bracket, but there are some EXTRA in the first bracket, because there were more of them (X+h of them is more than X of them). How many more? That's easy to answer... h more of them. Therefore the change in the X-squared function is:


change in X squared function = [h+h+... (X times)] + [(X+h)+(X+h)+... (h times)]


Rewriting this will make the next step clearer...


change in X squared function = [h+h+... (X times)] + [X+X+... (h times)] + [h+h+... (h times)]


Okay, now divide this by the small change h:


rate of change = [1+1+...(X times)] + [ X ] + [ h ]


Now let's make the small change (h) VERY small (so that we get the rate of change of X squared at a single instant). that is, we let h->0. This instantaneous rate is what a derivative measures. Then we have:


derivative = X+X = 2X


un-QED


The "trick" in your version comes when you take a derivative of all those X's in the sum, but did not take a derivative of the "X amount of times" (which was accounted for above when I said that a small increase in X created a small increase in the number of X's summed).


Okay, so now I have three requests:


1. Please delete this thread.


2. Please spell my name correctly in the future.


3. And PLEASE, no math puzzles that involve envelopes containing unknown amounts of money or that end with the phrase, "So where did the other dollar go?"


Tom Weideman

2 + 2 does not equal 4 and there has to be a new name for the company....gl...after all that work to build a corporate identity and then lose that marketing advantage...gl

I just finished grading a test so I've seen much worse. Tell you what. You can teach my class, and I'll go play with your bankroll!

Tom:


1.Please accept my apologies for not spelling your name correctly.


2. Don't worry, I don't remember but else from my education so there will be no more problems (unless David wants to get in the act).


Best wishes,

Mason