Bounded Definition and 514 Threads

Hi, I want to answer the following question: x=x(t) is continuous on [0,T) and satisfies 1 ≤ x(t) ≤ C_{1} + C_{2}∫^{t}_{0} x(s)(1+logx(s)) ds for 0 ≤ t < T. Prove x(t) is bounded on [0,T].Using Gronwall's inequality I get to x(t) ≤ C_{1}exp( C_{2} ∫^{t}_{0} (1+logx(s)) ds ) ≤ C_{1}exp(...

Let $$(T_{n}) $$be a sequence in $${B(l_2}$$ given by $$T_{n}(x)=(2^{-1}x_{1},...,2^{-n}x_{n},0,0,...). $$Show that $$T_{n}->T$$ given by $$T(x)==(2^{-1}x_{1},2^{-2}x_{2},0,0,...). $$ I get a sequence of geometric series as my answer for the norm, but not sure whether that's correct.

was looking at a proof of this here: http://gyazo.com/8e35dc1a651cec5948db1ab14df491f8 I have two questions, why do you set K = max of all the terms of the sequence plus the 1 + |A| term? Why do you need the absolute value of all the terms? i.e. why |a_1| instead of |a_1|?

Homework Statement Suppose ##\phi(x)## is a function with a continuous derivative on ##0\leq x<\infty## such that ##\phi'(x)+2\phi(x)\leq 1## for all such ##x## and ##\phi(0)=0##. Show that ##\phi(x)<\frac{1}{2}## for ##x\geq 0##. The Attempt at a Solution I tried to solve this like I...

Let C_b^\infty(\mathbb{R}^n) be the space of infinitely differentiable functions f, such that f and all its partial derivatives are bounded. Is C_b^\infty(\mathbb{R}^n) dense in L^2(\mathbb{R}^n)? I think the answer is yes, because C_b^\infty(\mathbb{R}^n) contains C_0^\infty(\mathbb{R}^n), the...

Homework Statement 2.11. Determine (explicitly) a convergent subsequence of the sequence in R2 given for n = 1; 2; : : : by xn =(e^{n}sin(n\pi/7),((4n+3/3n+4)cos(n\pi/3)) I know that the Bolzano-weierstrass theorem says that every bounded sequence has a convergent subsequence. I...

Here is the question: I have posted a link there to this topic so the OP can see my work.

Problem: Use an appropraite parametrization x=f(r,\theta), y=g(r,\theta) and the corresponding Jacobian such that dx \ dy \ =|J| dr \ d\theta to find the area bounded by the curve x^{2/5}+y^{2/5}=a^{2/5} Attempt at a Solution: I'm not really sure how to find the parametrization. Once I...

Assume that a function f:[a,b]\to\mathbb{R} is differentiable in all points of its domain, and that the derivative f':[a,b]\to\mathbb{R} is bounded. Is the derivative necessarily Riemann integrable? This what I know: Fact 1: Assume that a function is differentiable at all points of its domain...

Here is the question: I have posted a link there to this topic so the OP can see my work.

Homework Statement Set up the integral (but do not solve) for the volume of the object created by rotating the region bounded by y = arctan(x) and y = arcsin(x) in the first quadrant. Homework Equations I = ∏∫(f(x)^2 - g(x)^2) dx The Attempt at a Solution a.) rotate about he x...

The following is a problem statement. locally bounded (or locally (weakly) compact) differential operators of the Schwartz space of smooth functions on a sigma-compact manifold I realize this is very abstract. I expect the solution to be just as abstract. Thanks in advance.

How exactly might one go about showing that \left| \frac{1 - e^{-iy}}{-iy} \right| is bounded by 1 for y\in \mathbb R? I thought this would be easy to show using the series expansion of e^{-iy} in some way: \left| \frac{1 - e^{-iy}}{-iy} \right| = \left| 1 - \frac{iy}{2} -...

Proof on Linear 1st Order IVP solution being "bounded" A function h(t) is called "bounded" for t≥t0 if there is a constant M>0 such that |h(t)|≤M for all t≥0 The constant M is called a bound for h(t). Consider the IVP x'=-x+q(t), x(0)=x0 where the nonhomogeneous term q(t) is bounded...

Homework Statement The question is "Use double integration to find the volume of the solid bounded by the cylinder x2+y2=9 and the planes z=1 and x+z=5" Homework Equations The Attempt at a Solution I tried to draw the curves and the solid that i formed is a cylinder with a...

I need to solve an integral equation of the form $$\forall \omega \in [0,1], ~ \int_{\mathbb{R}} K(\omega,y)f(y)dy = \omega$$ where - f is known and positive with $$\int_{\mathbb{R}} f(y)dy = 1$$ - K: [0,1] x R -> [0,1] is the unknown kernel I am looking for a solution other than...

I will do my best to describe the problem I am working on. The problem is not from a textbook or anything but something I am working on independently to strengthen my first year calculus knowledge. What I did is I took sin(x) and -sin(x) and graphed them together. Sin(x) and -sin(x)...

Homework Statement Find the centroid of the region bounded by the graphs of y = sqrt (x) and y = (1/2) * x Homework Equations A = [f(x)-g(x)]dx from point a -> b The Attempt at a Solution x = [0,4] ; p(0,0) and p(4,2) I am just checking on if I did the integral correctly. A...

Homework Statement a<b<c and, f is bounded on [a,b] and f is bounded on [b,c] prove that f is bounded on [a,c] The Attempt at a Solution there exist M1≥0 s.t. for all x ε [a,b] |f(x)|≤M1 there exist M2≥0 s.t. for all x ε [b,c] |f(x)|≤M2 for x ε [a,b] and x ε [b,c] Let M>0...

1. Homework Statement Let A and B be nonempty bounded subsets of \mathbb{R}, and let A + B be the set of all sums a + b where a ∈ A and b ∈ B. (a) Prove sup(A+B) = supA+supB .Homework Equations The Attempt at a Solution Let Set A=(a_1,...,a_t: a_1<...a_i<a_t) and let set B=(b_1,...,b_s...

Homework Statement Sketch the region R bounded by the curves y = x, x = 2 - y^2 and y = 0. This is the initial part of an integral problem and I'm just curious about the method here. Homework Equations The Attempt at a Solution So, would it be proper to take the x = 2 - y^2...

Homework Statement Find the area bounded by the curves, y= √x, y= (5-x)/4, and y= (3x-8)/2 Homework Equations The Attempt at a Solution I found the intersection between each of the three curves to each other. Not sure what exactly the area bounded is. Is it the small triangular area...

Hello MHB, I got stuck on an old exam determine the area of the finite region bounded by the curves $$y^2=1-x$$ and $$y=x+1$$ the integration becomes more easy if we change it to x so let's do it $$x=1-y^2$$ and $$x=y-1$$ to calculate the limits we equal them $$y-1=1-y^2 <=> x_1=-2 \ x_2=1$$ so...

Homework Statement Find the error in this proof and give an example in (ℝ,de) to illustrate where this proof breaks down. Proof that every totally bounded set in a metric space is bounded. The set S is totally bounded and can therefore be covered by finitely many balls of radius 1, say N...

A little explanation here. My professor assigned a homework question without attempting the problem herself. When we were assigned this problem, we were forbidden to use the notion of a Taylor series in our proof (at least not without proving Taylor's Theorem on our own) as we had not covered...

Homework Statement The question : http://gyazo.com/7eb4b86c61150e4af092b9f8afeaf169 Homework Equations Sup/Inf axioms Methods of constructing sequences ##ε-N## ##lim(a_n) ≤ sup_n a_n## from question 5 right before it. I'll split the question into two parts. The Attempt at a...

Homework Statement The problem : http://gyazo.com/aa487398b3658600b98deabca8086334 Homework Equations The Attempt at a Solution Let A be a nonempty subset of reals which is bounded above. ##("\Rightarrow")## Assume ##sup(A)## exists, call it s. Since s exists, we know ##a...

Here is the question: Here is a link to the question: Quick Calculus 1 question!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Consider the functional Tf = f(5) - i f(7). If we take the domain T to be C_0(ℝ) with supremum norm, is T a bounded linear functional? What if we take the domain to be C_c(ℝ) with L^2 norm || . ||_2?I know I should post what I have so far but this time I have no idea because I had to missed 2...

Homework Statement Consider the region bounded by the curves y= lnx and y=( x-3)^2 Find the volume of the solid obtained by rotating the region about the y-axis Homework Equations The Attempt at a Solution For this I solve for the x so i got x= e^y and x= (y)^(1/2) +3...

Find the area bound by the curve y = x^3 - 2x^2 - 5x + 6, the x-axis and the lines x = -1 and x = 2. The answer is 157/12. The curve cuts the x-axis at x = -2, 1 and 3. I've shown my general idea on the attachment. I didn't end up with the correct answer so could somebody explain to me where...

Homework Statement Determine if the sequence is Monotonic and Bounded. Homework Equations an = 2 - (3/n) The Attempt at a Solution Depending on the domain: Ex: a1, a2, a3 ... n=1 ; n=2 it would be bounded by [1,2] however, if we have negative n values and values as fractions we...

Homework Statement Prove that if f is uniformly continuous on a bounded set S, then f is a bounded function on S.Homework Equations Uniform continuity: For all e>0, there exist d>0 s.t for all x,y in S |x-y| implies |f(x)-f(y)| The Attempt at a Solution Every time my book has covered a...

Here is the question: Here is a link to the question: Find the area between those curves: x=y^2, and x=4-y^2? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

So I ran into a question; Show that there is no metric on S^2 having curvature bounded above by 0 and no metric on surface of genus g which is bounded below by 0. honestly I have no idea what is going on here. I know that a Genus is the number of holes in some manifold or the number of...

(Hey guys and gals!) Homework Statement Given a bounded set x_n and for any y_n the following condition holds: \limsup_{n \rightarrow ∞}(x_n+y_n) = \limsup(x_n)+\limsup(y_n) Show that x_n converges. Homework Equations Definition of limsup(x_n) = L: \forall \epsilon > 0 \mid...

Homework Statement Find the values of m for y = mx that enclose a region with y = \frac{x}{x^2 + 1} and find the area of this bounded region. Homework Equations The Attempt at a Solution So I set the two functions equal to each other to solve for x in terms of m: mx = \frac{x}{x^{2}...

Homework Statement find the center of mass of a thin plate with constant density in the given region. region bounded by y-axis, x=y-y^3 ; 0<=y<=1Homework Equations x(bar) = (integral)(a to b) α(x) * x * (f(x) - g(x)) ---------------------------------- (integral)(a to...

Homework Statement Suppose c_{n} > 0 for each n\geq 0. Prove that if \sum ^{\infty}_{n=0} c_{n} is Cesaro summable, then the partial sums S_{N} are bounded. Homework Equations -- The Attempt at a Solution I tried contraposition; that was getting me nowhere. I have a few...

Homework Statement Find the volume of the solid generated by rotating the region bounded by the x-axis, the curve y=3x^4, and the lines x=1 and x= -1. The axis of rotation is the y-axis. Homework Equations Washers method: V=∏∫ [(R)^2 - (r)^2]dr x = (y/3)^(1/4) The Attempt at a...

Homework Statement Let X be a metric space and let E be a subset of X. Show that E is bounded if and only if there exists M>0 s.t. for all p,q in E, we have d(p,q)<M. Homework Equations Use the definition of bounded which states that a subset E of a metric space X is bounded if there exists...

Is something wrong in my assertions below? Suppose we have two quantum systems N and X. Let N is described by discrete observable \hat{n} (bounded s.a. operator with discrete infinite spectrum) with eigenvectors |n\rangle. Let X is described by continuous observable \hat{x} (unbounded s.a...

Homework Statement Find the area bounded by the cardioid x^2 + y^2 = (x^2+y^2)^{1/2} - y Homework Equations Area of R = \int \int_R dxdy = \int \int_{R'} |J| dudv J Is the Jacobian. The Attempt at a Solution Switching to polars, x=rcosθ and y=rsinθ our region becomes r^2 = r(1-sinθ) → r =...

a) find the volume of the region enclosed by z = 1 - y^2 and z = y^2 -1 for x greater or equal to 0 and less than or equal to 2. b) would i split up the volume into two integrals, each integral for each z function and then add them together? I also don't know how to find the bounds...

Homework Statement I'm having some difficulty deriving the equation for a concentration of CO2 as a function of length and time. Ultimately I end up with an equation that includes the summation of two error function terms that appear to have incorrect signs. Given: A cylinder of...

Homework Statement Prove that if S is a bounded subset of ℝ^n, then the closure of S is bounded. Homework Equations Definitions of bounded, closure, open balls, etc. The Attempt at a Solution See attached pdf.

Homework Statement Prove that if a is a real number, a > 1, then the set {a, a^2, a^3, ...} is not bounded from above. Hint: First find a positive integer n such that a > 1 + 1/n and prove that a^n > (1 + 1/n)^n >/= 2. Homework Equations The Attempt at a Solution Showing that...

Homework Statement Find the volume bounded by the following surfaces: z = 0 (plane) x = 0 (plane) y = 2x (plane) y = 14 (plane) z = 10x^2 + 4y^2 (paraboloid) Homework Equations The above.The Attempt at a Solution I think it has something to do with triple integrals? But...

Pretty much what the title says. Suppose we have a topological vector space $(X,\tau)$ and $U\subseteq X$ is topologically bounded. Is it possible for there to be some $x\in X$ such that $cx\in U$ for arbitrarily large $c$? I'm thinking of a real vector space here. If we try to prove...

Hi all, I am working with some beta functions. I want to show that the following is positive and bounded between 0 and 1. Is it possible to show this? $$ \frac{ \frac{B( a + b , \frac{2}{ c} )}{B(a, \frac{2}{c}) } - \big\{\frac{B( a + b , \frac{1}{ c} )}{B(a, \frac{1}{c}) }\big\}^{2} }{...