Math Is Fun Forum

Abstract algebra is the one I enjoy most.

If you differentiate (−1/2exp(−r[sup]2[/sup])) with respect to r, you get rexp(−r[sup]2[/sup]), don’t you? Therefore, if you integrate rexp(−r[sup]2[/sup]) with respect to r, you should get (−1/2exp(−r[sup]2[/sup])). (Plus an arbitrary constant – but as you’re working with a definite integral, you don’t need to bother with arbitrary constants.) It’s that simple.

Also you’re wondering why

Why? Because it’s a definite integral, so the x and the y are just dummy variables. Indeed:

Ah. So it’s not a puzzle at all. It’s just doing arithmetic in illogical ways.

I think what bayogang means is this:

There are three crates. One contains oranges only, one contains apples only, and one contains both oranges and apples. However, the labels “Apples”, “Oranges” and “Oranges & Apples” are all mixed up on the crates. The problem: Take one fruit from one of the crates, see what it is, and then work out which label should go with which crate.

Well, the 2D case works because

holds for all non-negative integers n and r with r ≤ n – and the

For the 3D pyramid, I suppose you just need to find a corresponding formula for the coefficients of the terms in the trinomial expansion of (x+y+z)[sup]n[/sup].

One number, though, brought everything alive:
3.1415926535…

That would just mean that the object isn't accelerating.

The formula for converting from (x,y) to polar co-ordinates for double integration is given here:
http://ltcconline.net/greenl/courses/20 … ration.htm

So if you substitute r and θ into your double integral, the integrand becomes rexp(−r[sup]2[/sup]), which can be easily integrated with respect to r.

Only in subordinate clauses (otherwise it’s pretty much the same as in English). Examples:

Der Apfel ist schlecht.
The apple is bad.

Er sagt, dass der Apfel schlecht ist.
He says that the apple is bad.

Ich esse die Banane, weil der Apfel schlecht ist.
I eat the banana because the apple is bad.

1. I think the magnitude of the force needs to be given in order to get the answer (unless you want the answer to be in terms of the force). But yes, once the external force is removed the object will no longer accelerate.

2. I am not an expert on physics but I’ll take a swipe at it here.

1. No.
2. The friction has to be 2500 N. This is equal to μR, where μ is the coefficient of kinetic friction of the road surface and R is the reaction of the road on the car, which is equal to the weight of the car, which is 1200g N, where g is the acceleration due to gravity. Hence 2500 = μ(1200g) ⇒ μ ≈ 0.21 (taking g to be 9.81 ms[sup]−2[/sup]).

3. Looks fine to me. (But then, as I said, I’m no physics expert.)

means

That’s how it’s defined in standard real analysis. In other words, given any real positive number epsilon, you can always find a big enough positive number delta such that if you plug in any value of x greater than delta into ƒ(x), ƒ(x) will be within epsilon of 0. This avoids any mention of infinity, see?