Recent content by mscbuck

Homework Statement Prove that if \int_{-\infty}^{+\infty} f exists, then \lim_{N\rightarrow \infty of {\int_{-N}^{N} f} exists and is equal to the first equation. Show moreover, that \lim_{N\rightarrow \infty of {\int_{-N}^{N+1} f} and \lim_{N\rightarrow \infty of {\int_{-N^2}^{N} f}...

Well I decided to take those outside the sum aftewards, figuring that the sum of ti-1, and ti all together would be the interval that it is on. So that should probably edited (my bad, I'm learning Latex and trying to get good at it but I'm forgetting some other things trying to get syntax down)...

For the first: U(f,P) = \sum Mi(a-1) and U(f,P') = \sum Mi(ab-b) In both I replaced the change in "t" with the respective intervals. where Mi = sup{f(t): ti-1 <= x <= ti} in general

Unfortunately I am not seeing this result. When I am writing my summations, the change in "t" can be replaced by (a-1) and for the second the change in "t" can be replaced by (ab - b)? What would my Mi's be though (i know they are the supremum of f(x) on the interval ti-1 <= t <= ti+1)?

Sorry, I must've just evaluated them wrong. In terms of seeing if their upper and lower sums are equal, how would I go about doing this? I understand that being integrable would imply that we have inf{U(f,P)} = sup{L(f,P)}, but does this become an exercise in notation of various partitions...

Homework Statement For a,b > 1 prove that: \int_{1}^{a} (1/t) dt + \int_{1}^{b} (1/t) dt = \int_{1}^{ab} (1/t) dt Homework Equations Hint: This can be written \int_{1}^{a} (1/t) dt = \int_{b}^{ab} (1/t) dt "Every partition P = {t0, ..., tn} of [1,a] gives rise to a partition P' = {bt0...

Would a proof by contradiction work better? Could we suppose that f' is indeed continuous at a hopefully to find something? I'm trying right now but not really getting anywhere :/

Homework Statement Suppose that f is differentiable in some interval containing "a", but that f' is discontinuous at a. a.) The one-sided limits lim f'(x) as x\rightarrow a+ and lim f'(x) as x\rightarrowa- cannot both exist b.)These one-sided limits cannot both exist even in the sense of...

It appears from that that I have received x = (a/b) + 1 as my final answer? Does this appear to be correct?

Here is what I have: f(x)=\sqrt{a^2+x^2}+\sqrt{(x-1)^2+b^2} f'(x) = x/(\sqrt{a^2+x^2}) + (x-1)/(\sqrt{(x-1)^2+b^2} = 0 Then I squared both sides to get rid of any square roots if need be, but from there I'm kind of stuck deciding what algebra to use. I found my mistake with x=0, so I'm...

Hi micromass, I actually had gotten up to that point, I probably should've written some of it down, but I ignored it because I kept getting led nowhere. But perhaps it's because I was using the information wrong. I would assume since we are finding a minimum that I'd like to find any critical...

Homework Statement A straight line is drawn from the point (0,a) to horizontal axis, and then back to (1,b). Prove that the total length is shortest when the angles \alpha and \beta are the same. 2. Homework Equations /graphs [PLAIN]http://dl.dropbox.com/u/23215/Graph.jpg The...

Thanks a lot Mark44. I find my problem so far in this intro to analysis class is simply that I often approach problems the wrong way, or drastically over think them (like in this case!). I was able to figure out 3 and 4 from your hints, thanks a lot!

Homework Statement Show that there is a polynomial function f of degree n such that: 1. f('x) = 0 for precisely n-1 numbers x 2. f'(x) = 0 for no x, if n is odd 3. f'(x) = 0 for exactly one x, if n is even 4. f'(x) = 0 for exactly k numbers, if n-k is odd Homework Equations The...

Homework Statement -F is a continuous function on [0,1], so let ||f|| be the maximum value of |f| on [0,1] a. Prove that for any number c we have ||cf|| = |c|\ast||f|| b. Prove that ||f + g|| \leq ||f|| + ||g||. c. Prove that ||h - f|| \leq ||h - g|| + ||g - f|| Homework Equations Based...