Bounded Definition and 514 Threads

Dear friends, I just joined the forums, and I'm looking forward to being a part of this online community. This semester, I signed up for Analysis II. I'm a math major, so I should be able to understand pretty much everything you say (hopefully). However, I'd really appreciate it if you try...

Homework Statement how do i determine whether a function is bounded using differentiation eg: f(x)=x/(2^x) Homework Equations The Attempt at a Solution i know it has something to do with maximums and minimums but i can't figure out how to do it. any help would be appreciated...

Homework Statement how do show whether the following sequences are bounded? 1) {an}=sqrt(n)/1000 2) {an}=(-2n^2)/(4n^2 -1) 3) {an}=n/(2^n) 4) {an}=(ncos(npi))/2^n Homework Equations i have to show whether the sequences are bounded by a number but i don't know how to find that number...

Determine whether the sequence {an} defined below is (a) monotonic (b) bounded (c) convergent and if so determine the limit. (1) {an}=(sqrt(n))/1000 a) it is monotonic as the sequence increase as n increases. b) it's not bounded (but I'm not sure why) c) divergent since limit doesn't...

Homework Statement This is not a homework question. I am solving this from the lecture notes that one of my friend's has got from last year. If C(X) denotes a set of continuous bounded functions with domain X, then if X= [0,1] and fn(x) = x^n. Does the sequence of functions {fn} closed ...

Homework Statement Prove that every sequence of bounded functions that is uniformly convergent is uniformly bounded. Homework Equations Let {fn} be the sequence of functions and it converges to f. Then for all n >= N, and all x, we have |fn -f| <= e (for all e >0). ---------- (1)...

This problem was suggested by Gokul43201, based on this year's Putnam A2. Suppose that K is a convex set in \mathbb{R}^2 which is contained in the region bounded by the graphs of the hyperbolas xy=1, xy=-1 (so the set is in the inner + shaped region which contains the origin also). What is...

Homework Statement Find the area bounded by y = 3x^2 + 1, x = 0, x = 2, y = 0 Homework Equations The Attempt at a Solution Not sure how to do this? Is this like finding the upper and lower sums?

[SOLVED] Inverse Image of a Compact Set -- Bounded? Problem: Let f : X → Y be a continuous function, K ⊂ Y - compact set. Is it true that f^{-1}(K)– the inverse image of a compact set– is bounded? Prove or provide counterexample. Questions Generated: 1. Why does compactness matter? (I...

[SOLVED] Variational calculus with bounded derivative constraints After learning about the calculus of variations and optimal control for a bit this semester, I've decided to tackle a "simple" (in the words of my professor) problem meant to illustrate a simplified example of highway...

Homework Statement prove that a monotone sequence which has a bounded subsequence is bounded Homework Equations The Attempt at a Solution

Definitions: Let {x[n]} be a bounded sequence in Reals. We define {y[k]} and {z[k]} by y[k]=sup{x[n]: n \geq k}, z[k]=inf{x[n]: n \geq k} Claim: (i) Both y[k] and z[k] are bounded sequences (ii){y[k]} is a decreasing sequence (iii){z[k]} is an increasing sequence Proof: (i)...

Evaluate the volume of the solid bounded by the plane z=x and the paraboloid z = x^2 + y^2 I have tried to graph this, and they don't bound anything? have i graphed it wrong. and is there a way to do these problems where you don't need to draw the graph.

How can you deduce that nad bounded sequence in R has a convergent subsequence?

Homework Statement Prove that if {a_{n}} is a sequence of rational numbers such that {a_{n+1}} > {a_{n}} for all n \in \textbf{N} and there exists an M\in \textbf{Q} such that {a_{n}} \leq M for all n \in \textbf{N}, then {a_{n}} is a Cauchy sequence of rational numbers.Homework Equations Do...

Evaluate \int\int_{Q}\left(1 - x^{3}\right)y^{2} dA where Q is the region bounded by y=x^2 and x = y^2 So I have drew the graphs of y=x^2 and x=y^2 and found that they intersect at (0,0) and (1,1). Now I am confused what to replace Q with, but I think it should be this: please tell me if I am...

Homework Statement I won't post the entire problem since I'm only stuck on one part of it. I need to find where y=cos x and y=sin 2x intersect. Homework Equations sin(x)=cos(x +- pi/2) The Attempt at a Solution cos x = sin 2x since sin(x)=cos(x +- pi/2), sin 2x =...

The way they use the terms:"closed set" and "bounded set" make me thinking that a closed set is different from a bounded set but i can not figure out how to prove that. So can some body show me clearly the difference between those two terms?

So the theorem states if a sequence is monotonic and bounded, it converges. WEll, it's easy enough to prove is a sequence is monotonic, but how would one go about proving that a sequence is bounded?

Homework Statement True or false: 1)If f is bounded in R and is uniformly continues in every finate segment of R then it's uniformly continues for all R. 2)If f is continues and bounded in R then it's uniformly continues in R. Homework Equations The Attempt at a Solution 1) If...

Homework Statement Ok I just wanted to make sure of this one. Write inequalities to describe the region: The solid rectangular box in the first octant bounded by the pane x=1 y=2 z=3. The Attempt at a Solution I thought of it as the volume of the box bounded boy the planes so: 0\leq...

Homework Statement Show that if f: S -> Rn is uniformly continuous and S is bounded, then f(S) is bounded. Homework Equations Uniformly continuous on S: for every e>0 there exists d>0 s.t. for every x,y in S, |x-y| < d implies |f(x) - f(y)| < e bounded: a set S in Rn is bounded if it is...

Homework Statement Why doesn't a set being totally bounded imply the set has the Heine Borel property? Another related question is what happens if a cover consist of open balls that cover the set and more of it? i.e. A=(-1,2) U (1,3) covers (0,1) but really covers more than (0,1). Is A...

Homework Statement Thm: If a sequence is Cauchy than that sequence is bounded. However Take the partial sums of the series (sigma,n->infinity)(1/n). The partial sums form a series which is Cauchy. But the series diverges so the sequence of partial sums is unbounded. Sequence of partial sums...

Suppose T: X -> Y and S: Y -> Z , X,Y,Z normed spaces , are bounded linear operators. Is there an example where T and S are not the zero operators but SoT (composition) is the zero operator?

Hi everyone. I'm a math student still learning to do proofs. Here is a problem I encountered that seems easy but has me stuck. 1. The problem statement, all variables and given/known data Let a be a positive rational number. Let A = {x e Q (that is, e is an element of the rationals) |...

Homework Statement Find the exact total of the areas bounded by the following functions: f(x) = sinx g(x) = cosx x = 0 x = 2pi Homework Equations the integral of (top equation - bottom equation) The Attempt at a Solution Change the window on the graphing calculator to...

Rotate the area bounded by y = 5,\,y = x + (4/x) about x=-1 Verify the limits of integration x + \left( {4/x} \right) = 5,\,\,x = 1\& 4 solve \begin{array}{l} \int\limits_1^4 {2\pi r\,h\,dx} \\ h = 5 - \left( {x + \left( {4/x} \right)} \right),\,\,r = x + 1 \\ 2\pi...

Homework Statement The integral of f on [a,b] exists and is positive. Prove there is a subinterval J of [a,b] and a constant c such that f(x) >= c > 0 for all x in J. Hint: Consider the lower integral of f on [a,b] Homework Equations The Attempt at a Solution I don't see how...

Homework Statement Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis: x+3=4y-y^2 x=0 About x-axis Homework Equations The Attempt at a Solution I tried solving x+3=4y-y^2 for y, but I can't get it even though it seems simple...

Hi, I would like to know if the function f(x) = (x+2)^-1 is bounded on the open interval (-2,2)? The interval doesn't include the point x = -2 but I'm not sure if I can say that there is a K>=0 such that |f(x)| < K for all x in (-2,2). The function is defined everywhere in that interval but...

Consider the region bounded by y = x^2, y = 5x+6, and the negative x-axis Compute the area of this region. Im somewhat confused by what they mean by the negative x-axis? The points of intersection between the two functions are [-1,1] and [6,36] A = Integral -1 to 6 (5x-6-x^2)? I'd...

Lets say we have a solution u, to the cauchy problem of the heat PDE: u_t-laplacian(u) = 0 u(x, 0) = f(x) u is a bounded solution, meaning: u<=C*e^(a*|x|^2) Where C and a are constant. Then, does u is necesseraly the following solution: u = integral of (K(x, y, t)*f(y)) Where K...

Equation of a hypocycloid is: x^{3/2} + y^{3/2} = a^{3/2}. Find the area of the figure bounded by this hypocycloid. My work: I can use the plane polar coordinates here taking x = a\cos t & y = a\sin t with t = [0, 2\pi]. But I don't know how to obtain the surface integral for evaluating the...

Find the area of a figure bounded by the equilateral hyperbola xy = a^2, the x-axis, and the lines x = a, b = 2a. My work: The equations of the lines and curves involved here are: xy = a^2 y = 0 x = a I don't know how b=2a is treated as an equation of a line here & hence I am puzzled as how to...

Let f be a real uniformly continuous function on the bounded set A in \mathbb{R}^1. Prove that f is bounded on A. Since f is uniformly continuous, take \epsilon = m, \exists \delta > 0 such that |f(x)-f(p)| < \epsilon whenever |x-p|<\delta and x,p \in A Now we have |f(x)| < m +...

I'm having trouble with this for some reason. If A:\mathcal{H}\to \mathcal{H} is a bounded operator between Hilbert spaces, the norm of A is ||A|| = \inf\limits_{\psi \neq 0} \frac{||A\psi||}{||\psi||}. My trouble is in verifying that ||A|| is in fact a bound for A in the sense that...

Could someone explain me how these three concepts hang together? (When is a set bounded but not closed, closed but not bounded, closed but not compact and so one?)

Suppose the universe is bounded and not infinite, would it require more dimensions for the universe to curve back upon itself? Like most, I find it impossible to picture 3 dimensional space having a boundary.

Hi. I'm new here. :) I was wondering if anyone could help me out with this problem... i'm supposed to find the region bounded by: y=x+1 y=e^-x x=1 i think i should find the other point of intersection but i forgot to do that (i haven't taken a math course for about 4 years). please help!

Some rule says that not all bounded sequence must be convergent sequence , one example is the sequence with general bound: Xn=(-1)^n could anyone help?! thanks in advance!

Prove that if f is uniformly continuous on a bounded set S then f is bounded on S. Our book says uniform continuity on an interval implies regular continuity on the interval, and in the previous chapter we proved that if a function is continuous on some closed interval then it is bounded...

hey Can someone please give me an example of an essentially bounded function?? I'm a bit lost.

If f is defined on [a,b] and for every x in [a,b] there is a d_x such that if is bounded on [x-d_x, x+d_x]. Prove that f is bounded on [a,b]. This question seems very odd. If every point, and indeed the neighbourhood of every point is bounded, then of course the function itself must be bounded...

I have absolutely no clue how to start here. Let F be the set of expressions of the form a = sum from i in Z of a-sub-i*x*i, where each a-sub-i is an element of R and {i < 0 : a-sub-i does not equal 0) is finite. (X is a formal symbol, not a number). An element a belonging to F is positive if...

"If I and J are bounded, then I\capJ is also bounded." Now, I was able to do this using the definition of suprema and infima and so fourth, but it is one godawful mess. I could sumbit it as is, but I was wondering if there's an easier way.

Find the volume of the solid region R bounded above by the paraboloid z=1-x^2-y^2 and below by the plane z=1-y The solution to this problem is: V=\int_{0}^{1} \int_{-\sqrt{y-y^2}}^{\sqrt{y-y^2}} (1-x^2-y^2)dxdy -\int_{0}^{1} \int_{-\sqrt{y-y^2}}^{\sqrt{y-y^2}}(1-y)dxdy I thought that...

I am having trouble finding the area between 2 polar curves... I have the procedure down, but the bounds are throwing me off. Any help with understanding how to bound would be great appreciated! I have attatched one problem that I am having hard time with and the work I have done. I know...

Q. Find the volume of the solid which is bounded by the cylinders x^2 + y^2 = r^2 and y^2 + z^2 = r^2. To me they don't really look like equations of cylinders, more like circles. Would the term "r" be constant in this case? Or would it be a variable? Even if r is a variable, I don't understand...

Can anybody help me find the volume y=sinx, bounded by the y axis, and the line y=1. The axis of revolution is the line y=1. I tried so many times and I can't find the correct answer, in the sheet it says (pie^2)/2 - 2pie