What is Approximation: Definition and 761 Discussions

Homework Statement "Use the binomial expansion to derive the following results for values of v << c. a) γ ~= 1 + 1/2 v2/c2 b) γ ~= 1 - 1/2 v2/c2 c) γ - 1 ~= 1 - 1/γ =1/2 v2/c2" (where ~= is approximately equal to) Homework Equations As far as I can tell, just γ = (1-v2/c2)-1/2The Attempt at a...

Approximation of the FE feat. "loose notation" I'm looking for a (professional) relativist to help me clarify something. I refer to the article General Relativity Resolves Galactic Rotation Without Exotic Dark Matter by Cooperstock and Tieu, available here...

I've found a new way for finding the circumference of a circle by using a visual perspective ,an angle of 18 degrees, and law of sines, its formula is: R is the radius of the circle r is the new radius C is the Circumference h=17.7062683767 t=3.23606808139 (R/h)= r (r/t)*360=C with...

I've came up to a problem, where I would like to prove that a differentiable function f(x) can be approximated by f(x) = f(x_0) \left(\frac{x}{x_0}\right)^{\alpha} where \alpha = \frac{d \ln f(x)}{d \ln x} \Big |_{x=x_0} But I'm not sure this is true. The problem and solution can be...

Homework Statement OK, I'm doing this linear approximation problem: Approximate \sqrt{4.1} - \sqrt{3.9} Homework Equations f(a + h) ~ f(a) + hf`(a) The Attempt at a Solution This is what I have done so far: I approximated each square root separately... 4.1 = 4 + h h = .1 f(x) =...

Homework Statement I need to use Taylor's thm to get an approximation to sqrt(5) with an error of no more than 2^(-9) and am totally lost. Homework Equations Taylor's theorem: Rn(x) = f(n)(y)/n! *x^n -- where f(n) is the nth derivative of f and Rn is R sub n. The Attempt at a...

Is there a possibility that general covariance be an approximation or violated ?

For small x, it seems sqrt(1+x) can be approximated by 1+x/2. Why exactly is this? Is there a theorem that I can refer to? Some kind of infinite series where the x^4 power term dies out? Thanks!

How come in the weak field approximation, where the metric is equal to, ds^2=-(1+2phi)dt^2 + dr^2(1-2phi). where of course dr is the three distance. why is phi multiplied by 2? I have two more stupid question regarding a different approach. please just explain it to me as i want to to see...

Hi all I have a question about WKB approximation Why is it that WKB method can be applied only to problems that are one dimensional or those which can be reduced to forms that are one dimensional ones? any help is deeply appreciated

Homework Statement Approximate f by a Taylor polynomial with degree n at the number a. f(x) = x^(1/2) a=4 n=2 4<x<4.2 (This information may not be needed for this, there are two parts but I only need help on the first) Homework Equations Summation f^(i) (a) * (x-a)^i / i! The Attempt at...

Does this hold in general ?? (as an approximation only) for every real or pure complex number 'a' can we use as an approximation: \sum _{n} exp(-aE_{n}) \sim \int_{-\infty}^{\infty} dx \int_{-\infty}^{\infty} dp exp(-ap^{2}-aV(x)) So for every x V(x) > 0 in case of real and positive a...

Hey! In deriving the WKB approximation the wave function is written as \psi \left( x \right) = exp\left[ i S\left( x \right) \right ] Now, in some of the deriviations I've seen, the function S(x) is expanded as a power series in \hbar as S(x) = S_0(x) + \hbar S_1(x) +...

Homework Statement Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.100000 cm thick to a hemispherical dome with a diameter of 45.000 meters. Homework Equations Surface Area of sphere=4\pi(r^2) Since it is hemipshereical, the...

In the context of a Lagrangian mechanics problem (a rigid pendulum of length l attached to a mass sliding w/o friction on the x axis), I found the following equations of motion and now I must solve them in the small oscillation limit. (I know the equations are correct)...

Hi, In his notes, our teacher makes this approximation: \log(1 + 3e^{-2\frac{E_o}{\tau}}) \approx \log(3e^{-2\frac{E_o}{\tau}}) For \tau << E_o Also, and I don't think this matters, the logs are assumed to be natural logs. I was wondering what the justification for this was...

hey can som1 please help, i know how to find the quadratic approximation for a given function but i don't know how the quadratic approximation determines a local max/min :confused: This is with regard to multivariable functions. thanks

The description in P.252in liboff's quantum mechanics, I cannot not figure out the continuity and continue in first order derivative of the wave function \varphi_I = \frac{1}{\sqrt{\kappa}} \exp {(\int_{x_1}^{x} \kappa dx)}...

Looks like I really don't have a feel for it. So I was working on this the other day. (arranged in order) http://img218.imageshack.us/img218/9613/1gx9.jpg http://img68.imageshack.us/img68/4677/2te4.jpg http://img68.imageshack.us/img68/7853/3jg9.jpg...

Let phi(x) and phi_dagger(x) be field operators which satisfy the appropriate commutation relations. Then is there any analytic approximation for the two particle density matrix given by <phi_dagger(x)phi_dagger(x')phi(x')phi(x)> Thanks!

Does anyone know any relatively simple approximation methods for approximating complex zeros of an analytic function, like say, cos(1/z), for example? Inquisitively, Edwin G. Schasteen

Hello ~ I be in dire need of help with this problem because I fell asleep in math class. Could anyone be so kind as to thoroughly guide me through the following problem? "A school has enrolled the same number of boys and girls. Two hundred students are selected at random to participate...

Pade Approximation states that a power series can be written as a rational function. Which is a series divided by another series. (An easy example of this will be the geometric series with mod'r' < 1) I've read books about the abstract bit of this. But I am completely stuck when it goes onto...

So I have this as the last thing I don't understand before tommorrow's test.. I have tried reading in the book and online, but it's just not clicking for me! There are so many numbers, and it they seem to just plug them in from nowhere. Like some example problems would be like sqrt(25.1)...

I'm in AP Calculuc and was given a homework package, which is an old university introductory calculus exam. There is one particular question with which I'm having a terrible time. It is known that f(0)=5 and the tangent line to the graph of f(x) at (0,5) is y=5+3x. It is also known that...

I need to use Taylor's Theorem to obtain the upper bound for the error of the approximation on the following e^{\frac{1}{2}} \approx 1 + \frac{1}{2} + \frac{{\left( {\frac{1}{2}} \right)^2 }}{{2!}} + \frac{{\left( {\frac{1}{2}} \right)^3 }}{{3!}} Here is an example problem in the textbook I...

I have the the follow 3rd order polynomial approximation for e^3x f(x) = e^{3x} \approx 1 + 3x + \frac{9}{2}x^2 + \frac{9}{2}x^3 In an earlier part of the problem, I found f\left( {\frac{1}{3}} \right) = e^{3\left( {\frac{1}{3}} \right)} \approx 1 + 3\left( {\frac{1}{3}} \right) +...

Hello everyone, I got stuck on a probability question and would be very thankful if someone could give me a hint: An Opaque bag contains 10 green counters and 20 red. One couner is selected at random and then replaced: green scores 1 and red scores zero. 1) Calculate the probability of...

We all know the least squares method to find the best fit line for a collection of random data. But I wonder if it is the best method. Suppose we have two random variables y and x that appear to have a linear relation of the type y = ax+b. What we want is, given the next type x signal to...

Dear PF, Would you please be so kind and help me with one question? Ive put my question in attached word file since my LATEX does no show me the formulas I type. Would you pls have a glance on my question and give any feedback? Thanks a lot Georeg

My book gives a formula for linear approximation of two independent variables, but I needed one for three. So I modified the formula given in the book, but I need someone to please just quickly see if it looks okay. Given: f(x,y)=z=f(x_0,y_0)+(\frac{\partial f}{\partial x} (x_0,y_0)) (x-x_0) +...

Hi everybody. Some students have asked me about problem 2.13 in Mallat's book "A wavelet Tour of Signal Proccessing". After some work on it, I think is not completely correct. I think some hypostesis on modulus of continuity are needed. I attach the statement. Esentially, what it says...

I have this algebraic term: (1+i)^-1 where i is very very small i<<<1 if we used second approximation what would it equal to? thanks in advance!

basically, i don't get it at all. i understand that x0 P0 P01 x1 P1 P012 P12 x2 P2 let's approximate f(x) where x is some number. i have some Pi given and a Pi(i+1) and Pi(i+1)(i+2) i also have the xi i don't know what f(x) is, some...

Hello, The effect of a 2pi periodic function f is defined as P(f) = 1/(2\pi) \int_{-\pi}^\pi |f(t)|^2 \ dt and Parsevals Theorem tells us that P(f) = \sum_{n=\infty}^\infty |c_n|^2 . Now, it seems rather intuituve that the effect of the N'te partial sum is P(Sn) = \sum_{n=-N}^N |c_n|^2 But...

A odd 2pi periodic function, for which x \in [0;\pi] is given by f(x)=\frac \pi{96}(x^4-2\pi x^3+\pi^3x) was found to have the Fourier series f(x) = \sum_{n=1}^\infty \frac{\sin(2n-1)x}{(2n-1)^5}, \ x \in \mathbb{R} The problem is now: prove that |f(x) - \sin x| \leq 0.01, \forall x \in...

It's a happy day :biggrin: Today is pi approximation day. The date, written in day/month format, is 22/7. I doubt the celebration will be quite as extravagant as the pi day of 1592 - March 14, 1592 at 6:53:58 (3/14/1592 6:53:58) touched off one the great celebrations in man's history. I...

I have a HSC physics assessment task (Yr 12 Australia) due in a few days where we had to take measurements of the photoelectric effect (VStop, Wave no/length, F) etc with different filters and find an approximation for H, by manipulating different equations etc. I already found an fairly...

Last semester in my course QM 2 we discussed the Rayleigh-Schrödinger perturbation theory. A very elegant theory, based on a general principle: when terms are depending on a certain factor to the nth order, where the factor is very small if not infinitesimally small, you can collect the terms...

How do i show that B_{x}(x+dx,y,z)-B_{x}(x,y,z)\approx \frac{\partial B_{x}(x,y,z)}{\partial x} dx using a Taylor series to the first term. Using a Taylor series does B(x) = B(a) + B'(a)(x-a)? In that case what would B(x+dx) be and how can i obtain the desired result from this? Thanks in...

Every day, Jack and Jill agree to meet at a certain time at the nearby bus interchange, where buses depart at equal periods of time. Once, Jill came 15 minutes later and Jack saw 6 buses depart. On a second occasion, Jill came 26 minutes later, and Jack saw 8 buses depart. On another occasion...

For a function f(x,y): The error is: \Delta f(x,y) = \sqrt{(\frac{df}{dx})^2+(\frac{df}{dy})^2} Is this a form of the approximation in algebraic error determination: \Delta f(x,y) = \sqrt{f(x+\Delta x,y) + f(x,y+\Delta y)} ? Now I was trying to do this in my bio lab for genetics...

Hi, I'm having trouble doing my work where I have to find the Taylor Approximation of function. My real work is the program this thing when the function, x, a, and ErrorBound is given. I don't knwo what to do with the ErrorBound to get n, where n is the number of terms. do i make any sense...

For a binomial distribution with n=10 and p=0.5 ,we should not use the poisson approximation because both of the conditions n>=100 and np<=10 are not satisfied. SUppose we go way out on a limb and use the Poisson aproximation anyway. Are the resulting probabilities unacceptable...

i'm stuck trying to figure out this probabilities problem for my thermodynamics class. the question is: consider an idealized drunk, restricted to walk in one dimension (eg. back and forward only). the drunk takes a step every second, and each pace is the same length. let us observe the...

It seems to me that I've got part (a) right, but I'm not so sure about what I have in part (b). I just need to know whether or not I am on the right direction. Any help is highly appreciated. :smile: Problem The force due to gravity on an object with mass m at a height h above the surface...

Hi, I'm trying to evaluate the following integral to calculate the scattering cross section for a spherically symmetrical potential e^{\frac{-r^2}{a^2}}? f(\theta)=\int r e^{\frac{-r^2}{a^2}} sin(kr) dr where a is a constant. What is the easiest way to evaluate this? I was able to get...

I'm using the laplace approximation (also known as MacKay's evidence framework) to the posterior volume of a baysian model. The standard procedure is as follows: 1) Find the (local) maximum point of the posterior pdf i.e optimise the parameter values. 2) Evaluate the hessian matrix(H) by a...

Derive \frac{n_1}{s_o} + \frac{n_2}{s_i} = \frac{n_2-n_1}{R} for Gaussian optics from the following equation \frac{n_1}{l_o} + \frac{n_2}{l_i} = \frac{1}{R} \left( \frac{n_2s_i}{l_i} - \frac{n_1 s_o}{l_o} \right) by approximating l_o = \sqrt{R^2 + \left( s_o + R \right) ^2 -...

I need to find thelocal approximation of f(x)=x^(1/3) at x=26.6, knowing f(27)=3. Here's what I did, don't know if I did it right: f '(x)=(1/3)x^(-2/3)=[x^(-2/3)]/3 slope of the tanget = slope of the secant [x^(-2/3)]/3=(y-y1)/(x-x1) [26.6^(-2/3)]/3=(y-3)/(x-27) now I sub in X and...