Function Definition and 1000 Threads

(a) ##f(x)## is continuous only at ##x=3##: 1- If ##x\in\mathbb Q##, ##f(x)=9## at ##x=3##; around, there is ##\mathbb Q## 2- If ##x\in \mathbb R\setminus \mathbb Q##, this is the set of irrational numbers. Intuitively, if ##x## was in ##\mathbb R##, ##x^2## and ##6(x-3)+9## would meet at...

Given f(x) = [sqrt{2x^2 - x + 10}]/(2x - 3), find the horizontal asymptote. Top degree does not = bottom degree. Top degree is not less than bottom degree. If top degree > bottom degree, the horizontal asymptote DNE. The problem for me is that 2x^2 lies within the radical. I can rewrite...

this seems to come up frequently in undergrad math classes so it is worth asking, what is the simplest and most efficient way to show ##f(x)\in C^2(\mathbb{R})## given $$f(x)=\begin{cases} (x+1)^4 & x<-\frac{1}{2} \\ 2x^4-\frac{3}{2}x^2+\frac{5}{16} & -\frac{1}{2}\leq x \end{cases}$$ And what...

Hey there! I have two questions regarding the Double Slit Experiment and the Wave Function Collapse. How effective does a measuring device have to be to cause a collapse? As in, say that every second the device has a 50% chance to turn off or on for one second, does the collapse still occur...

I can solve (i), I got x = -1.6 For (ii), I did like this: $$(f^{-1} o ~g)(x)<1$$ $$g(x)<f(1)$$ But it is wrong, the correct one should be ##g(x) > f(1)##. Why? Thanks

Got B = 9. My school doesn't provide answers, hence could someone double check if my answers is right? Thanks!

Non-homegenous first order ODE so start with an integrating factor ##\mu## $$\mu=\textrm{exp}\left(\int a dt\right)=e^t.$$ Then rewrite the original equation as $$\frac{d}{dt}\mu y = \mu g(t).$$ Using definite integrals and splitting the integration across the two cases, $$\begin{align}...

I simplified the given function into a single minterm and a single maxterm F(A,B,C,D) = ABC + (A + B + C) + AB F(A,B,C,D) = AB(C+1) + (A+B+C) F(A,B,C,D) = AB(1) + (A+B+C) F(A,B,C,D) = AB + (A+B+C) The only terms that involve a logical AND operation are AB as (A+B+C) is a maxterm of the...

Solve the following equation: $[x]=2x+1$,where [x] is the floor function

I am having some trouble find the domain with this function: ##f(x)=\frac{1}{\sqrt{x^2-4x\cos(\theta)+4}}## and ##\theta\in[0,\pi]##.I know that the denominator needs to be greater than 0. So ##\sqrt{x^2-4x\cos(\theta)+4}>0##. I squared both side of the inequality, ##x^2-4x\cos(\theta)+4>0##...

Dear Everyone, I am having some trouble find the domain with this function: $f(x)=\frac{1}{\sqrt{x^2-2x\cos(\theta)+4}}$ and $\theta\in[0,\pi]$. My attempt: I know that the denominator needs to be greater than 0. So $\sqrt{x^2-2x\cos(\theta)+2}>0$. I squared both side of the inequality...

I'm solving this exercise, first I did a force diagram for the transformer nucleus and I got this: ∑Fx = ma P(t) - Fk - Fb = ma P(t) = mx''+ bx' + kx So I got that dynamic equation, my question is, after transform that dynamic equation to Laplace Domain how can I relate it with the Output...

My questions: 1) What about if t = 2? Is there a certain meaning to ##G_X (2)## ? 2) PGF for uniform distribution is ##G_X (t)=\frac{t(1-t^n)}{n(1-t)}## and for t = 1 ##G_X (1)## is undefined so ##G_X (1) =1## is not true for all cases? Thanks

Verify the given function has a zero in the indicated interval. Then use the Intermediate Value Theorem to approximate the zero correct to three decimal places by repeatedly subdividing the interval containing the zero into 10 subintervals. f (x) = x3 − 4x + 2; interval: (1, 2) I don't...

Hi, PF, I think I've found a typo in my textbook. It says: "In the case of a multiplication by a constant, we've got $$(Cf)'(x)=\displaystyle\lim_{h \to{0}}{\dfrac{Cf(x+h)-Cf(x)}{h}}=\displaystyle\lim_{h \to{0}}{\dfrac{f(x+h)-f(x)}{h}}=Cf'(x)$$" My opinion: it should be...

The function h is defined as h : x x 2 – 8x – 9 where x ≥4 (a) h(x) can be written in the form (x+a) 2 +b, find the value of ‘a’ and ‘b’ (b) Express the inverse function h –1 in the form h –1 : x ... (c) Find the range of function ‘h’.

In the context of control systems, if I have a vibratory second-order system, (ω_n)^2 / [s^2 + 2ζ(ω_n) + (ω_n)^2], I know how to get the natural frequency ω_n. So, if I have something like 2 / (3s^2+5s+2), I know how to get the natural frequency ω_n. However, if I instead have something like...

That's my attempt: $$\int (\frac{1}{cos^2x\cdot tan^3x})dx = \int (\frac{1}{cos^2x}\cdot tan^{-3}x) dx$$ Now, being ##\frac{1}{cos^2x}## the derivative of ##tanx##, the integral gets: $$-\frac{1}{2tan^2x}+c$$ But there is something wrong... what?

Step 1: I first started by reducing the inside of the block diagram of picture "bd" (the portion with G1 and the negative feedback G2) I obtained G1/[1 + G1G2] I'll call this term "F" Step 2: Then I'm left with two terms feeding into a summing point: F - G3 I'll call this term "K" I can...

Very early in the development of thermodynamics, it was realized that the 2nd Law of Thermodynamics is not a law fundamental to the fabric of our cosmos, but only becomes true in the limit of the number of particles. It was none other than Boltzmann himself who realized and articulated this...

Ok, first I tried to show that ##A = \left \{a^{r}|r\in\mathbb{Q},r<x \right \}## does not have a maximum value. Assume ##\left\{ a^{r}\right\}## has a maximum, ##a^{r_m}##. By this hypothesis, ##r_{m}<x## and ##r_{m}>r\forall r\neq r_{m}\in\mathbb{Q}##. Consider now that ## q\in\mathbb{Q}|q>0##...

##y=x^2ln x-x## ##\frac {dy}{dx}=2x ln x+x-1## ##\int [2xln x+x-1]\,dx##=##x^2ln x-x##, since ##\int -1 dx= -x## it follows that, ##\int [2x ln x +x]\,dx##=##x^2 ln x## ##\int 2x ln x \,dx = x^2ln x##+##\int x\,dx## ##\int_1^2 xln x\,dx =\frac {x^2ln x}{2}##+##\frac{x^2}{4}##=##2ln2+1-0.25##

The book is asking me to write my own unique_ptr template (after just covering a bit about templates). I called my template single_ptr, and I gave it two template parameters, T and D. T is supposed to be the type that the raw pointer points to. D is supposed to represent a function type so that...

Hi guys, I've attempted to integrate this function by parts, which seemed to be the most appropriate method... but apparently, I'm getting something wrong since the result doesn't match the right one. Everything looks good to me, but there must be something silly missing :) My attempt:

Is it possible to do the integration? That is the full question I don't know where to start, try to use ##u=\cos x## and also ##\cos^2 (x) = \frac{1}{2} + \frac{1}{2} \cos (2x)## but failed. Thanks

This is an issue I've been stuck on for about two weeks. No matter how many times I take this derivative, I keep getting the same answer. However, this answer is inevitably wrong. Please help me to understand why it incorrect. To start, I will define an input matrix ##X##, where ##n## is the...

(a) Let the center of the concentric spheres be the origin at ##r=0##, where r is the radius defined in spherical coordinates. The potential is given by the piece-wise function $$V(r)=\infty, r<a$$ $$V(r)=0, a<r<R$$ $$V(r)=\infty, r<a$$ (b) we solve the Schrodinger equation and obtain...

The proof is given in two steps 1. Prove the lemma. 2. Use lemma to prove result. %%1-Lemma%% Assume ##a\neq0##. Define ##g:(-(|a|+1),|a|+1)\longrightarrow \mathbb{R}## by ##g(x)=\sqrt[3]{x^2}+\sqrt[3]{xa}+\sqrt[3]{a^2}##. Then ##g## is bounded from below by some positive number ##m##...

I’m coming at this question with a physics application in mind so apologies if my language is a bit sloppy in places but I think the answer to my question is grounded in math so I’ll post it here. Say I have a function F(z) defined in the complex z plane which has branch points at z=0 and z =...

Solution attempt: $$F(F(x,y))=(z,w)$$ is the map given by $$x=z$$ $$y=w$$

My attempt at this: From the general result $$\int \frac{d^Dl}{(2\pi)^D} \frac{1}{(l^2+m^2)^n} = \frac{im^{D-2n}}{(4\pi)^{D/2}} \frac{\Gamma(n-D/2)}{\Gamma(n)},$$ we get by setting ##D=4##, ##n=1##, ##m^2=-\sigma^2## $$-\frac{\lambda^4}{M^4}U_S \int\frac{d^4k}{(2\pi)^4} \frac{1}{k^2-\sigma^2} =...

Hi PF! Do you know of any examples of the Ritz method which use Bessel functions as trial functions? I’ve seen examples with polynomials, Legendre polynomials, Fourier modes. However, all of these are orthogonal with weight 1. Bessel functions are different in this way. Any advice on an...

This question arose while studying Cosmology (section 38.2 in Lecture Notes in GR) but it is purely mathematical, that is why I ask it here. I do not see why the equation $$H^2 = H_0^2 \left[\left( \frac{a_0}{a}\right)^3 (\Omega_M)_0 + (\Omega_{\Lambda})_0 \right] \tag{1}$$ Has the following...

Hi I want to see whether there is a way to make this program work without much complication. I read from Ivor book that you can work with two different type of variables for example mixing int x = 1 and double y = 2.5 and use template to swap them using declaration <double> in the program as...

I have a integral with unknown h. My integral looks like this where C, a, b are constants F(x) and G(x) are two functions. So the only unknows in the integral is h. How can I solve it ? I guess I need to use scipy but I don't know how to implement or use which functions. Thanks

For the sake of this question, I am primarily concerned with the position wave function. So, from my understanding, the wave function seems to 'collapse' to a few states apon measurement. We know this because, if the same particle is measured again shortly after this, it will generally remain in...

Attached is the problem and my work through the problem. I got the problem correct, but my teacher said this could be done quicker on a calculator. Any idea how it could be done quicker.

This is from Evans page 37. I seem to be missing a basic but perhaps subtle point. Definition. Green's function for the half-space ##\mathbb{R}^n_+,## is \begin{gather*} G(x,y) = \Phi(y-x) - \Phi(y-\tilde{x}) \qquad x,y \in \mathbb{R}^n_+, \quad x \neq y. \end{gather*} What's the proper way to...

I am studying the 'toy' Lagrangian (Quantum Field Theory In a Nutshell by A.Zee). $$\mathcal{L} = - \frac{1}{4} F_{\mu \nu}F^{\mu \nu} + \frac{m^2}{2}A_{\mu}A^{\mu}$$ Which assumes a massive photon (which is of course not what it is experimentally observed; photons are massless). The...

Hello, I’m trying to better my understanding of how the total emissivity changes with temperature for ceramic materials. Currently it is my understanding that non-metals typically have a high emissivity. A sanded surface will result in a higher emissivity, and that spectral emissivity varies...

The wave function ψ(x) of a particle confined to 0 ≤ x ≤ L is given by ψ(x) = Ax, ψ(x) = 0 for x < 0 and x > L. When the wave function is normalized, the probability density at coordinate x has the value? (A) 2x/L^2. (B) 2x^2 / L^2. (C) 2x^2 /L^3. (D) 3x^2 / L^3. (E) 3x^3 / L^3 Ans : D

I started out by rewriting the function as (f(x^2))^(1/2). I then did chain rule to get 1/2(f(x^2))^(-1/2) *(f'(x^2). - I think I need to go further because it is an x^2 in the function, but not sure.

Hallo at all! I'm learning statistic in python and I have a problem to show you. I have this parametric function: $$P(S|t, \gamma, \beta)=\langle s(t) \rangle \left( \frac{\gamma-\beta}{\gamma\langle s(t) \rangle -\beta}\right)^2\left( 1- \frac{\gamma-\beta}{\gamma\langle s(t) \rangle...

I have a problem where I am given the density of states for a Fermion gas in terms of momentum: ##D(p)dp##. I need to express it in terms of the energy of the energy levels, ##D(\varepsilon)d\varepsilon##, knowing that the gas is relativistic and thus ##\varepsilon=cp##. Replacing ##p## by...

In QFT by peskin scroeder page 30 the action of Klein Gordon Operator on propagator (∂2+m2)DR(x-y)=∂2θ(x0-y0)... how to compute this ∂2θ(x0-y0)?

Integral \int^{\pi}_0\sin^3xdx=\int^{\pi}_0\sin x \sin^2xdx=\int^{\pi}_0\sin x (1-\cos^2 x)dx=\frac{4 \pi}{3} Is it possible to write integral ##\int^{\pi}_0\sin^3xdx## in form of Beta function, or even Bessel function?

I say the answer is A.

This is more of a conceptual question. To find the horizontal velocity as a function of time for the above wave function, you take its partial t derivative and insert x=4. In other words the function would be -2.4sin(1-12t). Im wondering why you take the partial t derivative and not to partial...

Hey! :giggle: Let $f:\mathbb{Q}\rightarrow \mathbb{Q}$, $f(x):=x^2$. Show that : (i) $f$ is well-defined. (ii) $f(1)=1$, $f(3)=9$ (iii) $f$ does not satisfy the intermediate value theorem (e.g. not on $[1,3]\cap \mathbb{Q}$) For (i) do we just say that $f$ is well-defined from $\mathbb{Q}$...

First off let me say I am a bit confused by this question. Searching for some references I found the following related to the KG propagator, given by (P&S, chapter 2 pages 29, 30) Then they Fourier-transformed the KG propagator Is this what is aimed with this exercise? If yes, could you...