Recent content by steelphantom

For reflexive, show that (m, n) ~ (m, n) is true. For symmetric, show that if (m, n) ~ (p, q) then (p, q) ~ (m, n). Just check these explicitly to see if they work out.

Is it basically by definition? Since f is continuous, f-1(N) is a neighborhood of x for every neighborhood N of f(x). Maybe I don't understand it, heh.

Ok, if I use that fact, then if f(x) != g(x), there exist subsets U, V disjoint and open with f(x) in U and g(x) in V. Since f is continuous, f-1(U) is open in X (because U is open in Y). Similarly, g-1(V) is open in X (because V is open in Y). Thus {x in X | f(x) != g(x)} is open, and it's...

Homework Statement Let X be a topological space, Y a Hausdorff space, and let f:X -> Y and g:X -> Y be continuous. Show that {x \in X : f(x) = g(x)} is closed. Hence if f(x) = g(x) for all x in a dense subset of X, then f = g. Homework Equations Y is Hausdorff => for every x, y in Y with...

Homework Statement Let f be of class C1 on [a, b], with f(a) = f(b) = 0. Show that \int_a^b xf(x)f'(x)dx = -1/2 \int_a^b [f(x)]^2 dx.Homework Equations If F is an antiderivative of f, then \int_a^b f(t)dt = F(b) - F(a) The Attempt at a Solution I'm just really not sure how to begin this one. I...

Got it. Thanks!

Thanks statdad, but I've already figured that one out with Dick's help. See the post right before yours for the one I'm now having trouble with.

Ok, I have another Riemann sum, but this time, it actually says what the integral evaluates to. The question is the following: Show that the limit as n -> infinity \sum_{k=1}^nn/(n2+k2) = pi/4. So I'm thinking that f(x) = sqrt(1-x2) on [0, 1]. But I'm having a hard time figuring out what to...

Cool. Thanks!

Ok, so if I choose \xi = k/n = xk, then I get \sum(k/n)3(k/n - (k-1)/n) = \sum(k3/n3)(1/n) = \sumk3/n4. So the limit of this sum is \int_{0}^1x^3 ?

As usual, I'm just not seeing it. :redface: I guess the k^3 should be a dead giveaway, but I'm not sure what to do about the n^4. Is the limit = \intx^3 from a to b? By using the definition of a Riemann sum, I get: \sum_{k=1}^n \xi3(xk-xk-1). But what are the xks?

You should be calculating \int2sqrt(x) - x from 0 to 4.

Homework Statement Find the limit, as n -> infinity, of \sum_{k=1}^nk3/n4 Homework Equations Riemann sum: S(f, \pi, \sigma) = \sum_{k=1}^nf(\xi)(xk - xk-1) The Attempt at a Solution My guess is that I should try to put this sum in terms of a Riemann sum, and then taking n -> infinity will...

Use the product rule with -2x and sin(x^2). You'll have to use the chain rule again when you calculate the derivative of sin(x^2).

Homework Statement Let p > 1, and put q = p/(p-1), so 1/p + 1/q = 1. Show that for any x > 0, y > 0, we have xy <= xp/p + yq/q, and find the case where equality holds. Homework Equations The Attempt at a Solution This is in the differentiation chapter of my analysis book (Browder)...