Recent content by analysis001

Homework Statement I'm trying to understand what compact sets are but I am having some trouble because I am having trouble understanding what open covers are. If someone could reword the following definitions to make them more understandable that would be great. Homework Equations...

Homework Statement Suppose f: (a,b)→R where (a,b)\subsetR is an open interval and f is a differentiable function. Assume that f'(x)≠0 for all x\in(a,b). Show that there is an open interval (c,d)\subsetR such that f[(a,b)]=(c,d), i.e. f is surjective on (c,d). Homework Equations f is...

Homework Statement I am trying to model an equilibrium for the attached picture. The picture is of a square with a small circle in the center of the square. There is a force from the sides of the square that attracts particles to the edges. There is also a force from the circle that attracts...

I would like to model rain hitting a surface and then running off of the surface. The surface would be a square, such as a slab of concrete, and the concrete would not be at any angle, so the rain would run off of all sides equally. This is a starting point for a small research project I am...

Yeah, I've gotten to that point, so as of now I have: \sum_{n=1}^{\infty}\frac{1}{n(\sqrt{n+1}+\sqrt{n})} but I'm still not sure what to compare it to.

Yes, it's talking about the inverse image, not the function inverse. I don't really see how f(D)=sin(D) would work though. If the question was to find a f(D)\subseteqR where f(D) is compact but f-1(D) is not then I see how f(D)=sin(D) would work, because f(D)=[-1,1] is compact but f-1(D)=R is...

Ok, I might have summarized the problem wrong. I'll write it word for word here: Consider a function f:R→R which is continuous on all of R. Find an example satisfying the following: D\subseteqR is compact but f-1(D) is not.

Homework Statement Prove that the series \sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n} converges. The Attempt at a Solution I think I'm going to use the comparison test but I'm having trouble coming up with a series to compare it to. Any clues would be great. Thanks!

Homework Statement I need to find an example of a set D\subseteqR is compact but f-1(D) is not. Homework Equations f-1(D) is the pre-image of f(D), not the inverse. The Attempt at a Solution I'm having trouble visualizing a function that would work for this scenario. Any clues...

Thanks everyone!

I don't get why sin((-\pi,\pi)=(-1,1). I thought because sin(\pi/2)=1 and since \pi/2\in (-\pi,\pi) and also because sin(-\pi/2)=-1 and since -\pi/2\in (-\pi,\pi) then sin((-\pi,\pi)=[-1,1]. If sin((0,\pi) then it equals (0,1] which is neither open nor closed. I'm not sure where my reasoning...

If I take A to be the interval (-\pi,\pi) which is open, then sin(A) would be on the interval [-1,1], which is closed. Am I understanding this correctly?

Oh ok would f(A)=sin(A) work?

Homework Statement I'm trying to do a problem, and in order to do it I need to find a function f:R→R which is continuous on all of R, where A\subseteqR is open but f(A) is not. Can anyone give an example of a function that satisfies these properties? I think once I have an example I'll...

Homework Statement Prove that the series diverges: \sum_{i=1}^{\infty}\sqrt{n+1}-\sqrt{n} The Attempt at a Solution I'm trying to use the comparison test, but I don't know what to compare it to. All I have done so far is change the terms to be in fraction form...