the tree

Hi there, To get started with the first few, you'll just need to recall a few basic identities - knowing ax =eln a x, the product rule and the quotient rule should get you through those three. Once you've done that, think about the rules you've used and the question with s(t) should look easy. The next one is a little messy so I'd suggest that first you try to work out da/dx for a=a2+ln(x). For the graphing problem, think about range, value at x=0 and limits for both ends of x. For the last bit on that page - just work through it real slow, so long as you remember the basic properties of logarithms then you should be fine. Good luck tho!

From a pure maths perspective you could say that it's a consequence of looking for general rules - the type of relationships that we look for are ones that are symmetrical, invertible etc - which limits the amount of operations that could possibly be used to describe the ones we find.

Taking the limit as h -> 0 is different to just substituting in h=0. You'll need a better idea of limits before trying to work your way through that proof.

You just made a tiny mistake, you should be looking at: [math]\lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}[/math] you'll find it works out easily.

That would be what logarithms are for. If y=2x then log2(y)=x. In the case of your example, log2(1080)=10.077... So we know that it's not 1 doubled a whole number of times. The best we can say at that point is 210<1080<211. A little further investigation will tell you that 1080=210+56. If your calculator cannot do base 2 logarithms then you'll have to use the natural logarithm ( ln or loge ) and to do that you'll need to know that logb(x)=ln(x)/ln(b). Well I think since 4=22 you might be able to work that out.

Because evolution's solution is a lot better than anything we've ever come up with. The problem is that things will always fall apart eventually, entropy increases, nothing should last forever and that's sort of built into the universe. We can patch things up or whatever. But things that make new things, with adaptation to a varying environment, we haven't found a better solution.

That isn't how it works in the slightest. 0 is a number.It doesn't fall under the range of the domain of the division function. But it's still a number. [*]You're correct in that there are no problems or paradoxes.[*]I really doubt any mathematicians are confused by this.

I'd sort of given up the non geometric view of pi, since connector seemed so obsessed with the idea that circles were a requirement in some way. Also he's saying that sqrt(2) doesn't exist. What the hell.

Nah that's fine I'll take it. I guess we can throw out the numbers 5, 17, 130.2 and 9, while we're at it.

0 is defined as the number with these two properties:x+0=x x*0=0 So in fact, it is.

Are you seriously suggesting that just because a number doesn't have a finite decimal expansion, that means it doesn't exist?

Okay we all know that you can't draw a perfect circle in the physical realm, no-one is contending that. But we're talking about pi as a mathematical object, not a physical one. A mathematical object that has plenty definitions that don't mention a single geometric concept, let alone circles. If you're trying calculate pi then you're doing a calculation which is different than a measurement in that it's not the same thing The proof that pi is irrational is actually a little bit complicated It's not just "rah rah smooth surfaces approximate unsmooth ones" That's not what a proof is Or an argument Or really anything that has any place in mathematics As an aside, the square root of two is also irrational and can fairly easily be depicted with a finite amount of straight lines, how does that contend?

How do these follow?For that matter it'd be trivial to contrive a smooth curve with an integer length, even though it's length actually would be regarded as the sum of infinite straight lines. We don't expect them to: they do.

-0 = -1*0 by definition of "-" -1*0 = 0 by definition of "0" -0 = 0 by transitive property of "=" Q.E.D.

The period of e^ix? A consequence of the axioms of euclidean geometry? Hmm? So? Then you wouldn't be doing mathematics would you? Prey tell, for what? Seriously. Most real numbers have an infinite decimal expansion. It's not a big deal and you don't need to make up ridiculous explanations for it.