1. J

    Electric Field of a Cylinderical Shell

    Electric Field of a uniformly charged thin-walled right cylinderical shell. (a) A uniformly charged thin-walled right cylinderical shell has total charged Q, radius R, and height h. Determine the electric field at a point a distance d from the right side of the cylinder as shown below. Hint...
  2. H

    Uniformly Distributed Probablity Question

    The speed of cars passing through the intersection of Blossom Hill Road and the Almaden Expressway varies from 15 to 35 mph and is uniformly distributed. None of the cars travel over 35 mph through the intersection. Question: Find the probability that the speed is more than 27 mph given (or...
  3. M

    Pointwise and uniformly convergence of a series.

    See the uploaded picture. Here are my attempts, a) If I am using the (big) $\mathcal{O}$ notation (Bachmann-Landau notation) as a remainder of a Taylor polynomial of degree $1$ around the point $0$, then I have...
  4. A

    Determine whether the function is uniformly continuous

    I have the question determine whether f(x)=(4x-3)/(x-2) is uniformly continuous on the open interval (1,2). I have only answered these types of questions with closed intervals so I'm not sure how to proceed? Thanks, Alex
  5. A

    Does this infinite sum of functions converge uniformly? I appreciate any help.

    Hi guys, I am curious whether these infinite sums converge uniformly. To me, it seems they are not so I do not need to prove it? Here goes the question. Suppose u^\prime(c_t) >0, u^{\prime \prime}(c_t)<0, \lim_{c_t \to 0}u^\prime(c_t)=\infty, \lim_{c_t \to \infty} u^\prime(c_t)=0 and for...
  6. G

    Math help--- prove that the composition gof:A-> C is uniformly continuous

    prove that the composition gof:A-> C is uniformly continuousLet A,B,C be nonempty sets of real numbers, and let f:A->B and g:B->C be uniformly continuous functions. Prove that the composition gof:A->C is also unfiormly continuous,(Bow)
  7. J

    Proving that a sequence converges uniformly or pointwise

    Hi, so I have the solutions to the following 2 problems, but I don't quite understand them. It would be great if someone could explain it to me... I took a year out so my math is a bit rusty now. Thanks a lot! Problem 1: Prove that x/n -> 0 uniformly, as n -> Infinity, on any closed interval...
  8. O

    proving uniformly continousity

    i need to prove that f(x)=ln(x^{2}+cos^{2}x) is uniformly continous. now, i thought of a way. i know that f'(x)=\frac{2x-sin2x}{x^{2}+cos^{2}x} so i acctually need to prove that f'(x) is bounded ( N\leq f'(x)\leq M ). but how do i prove it?
  9. O

    proving uniformly continuous function

    i need to show that f(x)=\frac{x^{2}}{1+x} is uniformly continuous at [0,\infty) . i was trying to show that \forall(\varepsilon>0)\exists(\delta>0)\forall|x-y|<\delta maintains |\frac{x^{2}}{1+x}-\frac{y^{2}}{1+y}|\leq\varepsilon . \:what i was getting is that in the given range -...
  10. W

    Find all values of k for which f is uniformly continuous

    Find all k \in R such that f= x^{k} is uniformly continuous, x \in (0, + \infty ) I can check if a certain function is uniformly continuous, but I don't know how to check for what values it is uniformly continuous. Please, help.
  11. X

    converge uniformly?

    On [a,b], the function sequence \{f_n\},\{g_n\} converge uniformly to f,g respectively. Suppose there exists positive sequence M_n such that f_n(x)\leq M_n, g_n(x)\leq M_n,\ \forall\ x\in [a,b]. Prove that f_ng_n converge unformly to fg on [a,b] PS: If M_n=M, I know how to prove. But...
  12. C

    uniformly convergent for function

    define function fn:[0,1]->R by fn(x)=(n^p)x*exp(-(n^q)x) where p , q>0 and fn->0 pointwise on[0,1] as n->infinite.,with the pointwise limit f(x)=0 ,and sup|fn(x)|=(n^(p-q))/e assume that ε is in (0,1) does fn converges uniformly on [1,1-ε]? how about on[0.1-ε]???? my idea is checking whether...
  13. C

    converges uniformly

    define function fn:[0,1]->R by fn(x)=(n^p)x*exp(-(n^q)x) where p , q>0 and fn->0 pintwise on[0,1] as n->infinite. find ||fn||(infinite) and deduce that if p<q then fn converges uniformly on[0.1] whereas if p>=q then fun does not converge uniformly on[0,1] how about [0,1-esillope] and...
  14. R

    PLease prove uniformly continuous

    Please prove whether the function f(x)= 1/ sinx is uniformly continuous or not on (0,1)
  15. S

    Uniformly continuous functions

    Hi! Help me please to solve this. Prove or disprove that: 1. If f,g are uniformly continuous functions in [0,\infty ) then also fg (product) is unif.continuous in [0,\infty ). 2. Same question, but about the composition of f and g. Thanks beforehand.
  16. I-Think

    Conditional distribution of a uniformly distributed random variable

    Let U denote a random variable uniformly distributed over (0,1). Compute the conditional distribution of U given that U>a where 0<a<1 So if I understand this correctly, we desire f_{U|U>a}(U|U>a)=\frac{f(U, U>a)}{f_{U>a}(U>a)} How do I proceed from here, assuming I'm correct?
  17. T

    Extension of Uniformly Continuous Function into a Complete Metric Space

    Let E$ be a subset of X , and f : E -> Y be uniformly continuous, where Y is complete. Show that there exists a continuous function which maps the closure of E to Y such that F(x) = f(x). So basically, we are trying to prove that a function can be extended to its domains closure, and...
  18. M

    Sequence converging uniformly.

    Hey, I have come across these two questions and I think I have to use an example for both of them to justify them. Is anyone able to quickly tell me an example I could use for both of them? Thanks so much http://i56.tinypic.com/9r4npd.jpg
  19. C

    uniformly convergent of sequence of integrals

    Let f \in C(\mathbb{R}) and M>0. Prove that the sequence of functions: f_n(z)=\frac{n}{2} \int^{z+\frac{1}{n}}_{z-\frac{1}{n}} f(y) dy is uniformly convergent to f in [-M,M] .
  20. C

    uniformly convergent

    Let f_n : \mathbb{R} \rightarrow \mathbb{R} be a sequence of functions that is unifromly convergent in \mathbb{R} to funcion f: \mathbb{R} \rightarrow \mathbb{R}. For n \in \mathbb{N} we define: g_n(x)=\exp (-(f_n(x))^2), \ \ \ g(x)=\exp(-(f(x))^2) h_n(x)=(f_n(x))^2, \ \ \ h(x)=(f(x))^2...