What is Approximation: Definition and 761 Discussions

"Differential Approximation" A person of height 1.9m is walking away from a lamp-post at 1m/s. The light on the lamp-post is 5.1m above the ground. 1. At what rate (in m/s to the nearest cm/s) is the length of the person's shadow changing? 2. At what speed (in m/s to the nearest cm/s)...

Homework Statement Let f(x) = cos(pi*x), 0 < x < 1. Find a quartic approximation to f(x): By solving the continuous least squares problem using Chebyshev polynomials. By solving the continuous least squares problem using Legendre polynomials.Homework Equations The Attempt at a Solution For...

Homework Statement I have this equation: T=(1+\frac{U_{0}^{2}}{4E(U_{0}-E)}sinh^{2}(2 \alpha L))^{-1} Where α is given by: \alpha = \sqrt{ \frac{2m(U_{0}-E)}{\hbar^{2}}} I have to show that in the limit αL>>1 my equation is approximately given by...

Homework Statement storage of heat, T at time, t (measured in days) at a depth x (measured in metres) T(x,t)=T0 + T1 e^{-\lambda} x sin (wt - \lambdax) where w = 2pi/365 and \lambda is a positive constant show that \deltaT/\deltat = k \delta^2 T / \deltax^2Derive the second order Taylor...

Pls. answer in the simplest and the most intuitive way. 1. What is the reason our quantum field theory needs perturbative approach. Is it because in the concept of fields, there is an infinite number of freedom in the oscillations of the virtual particles, or is it because the field is...

Electric Dipole and Electric Potential.. and binomial approximation! Homework Statement An electric dipole at the origin consists of two charges +q and -q spaced distance s apart along the y-axis. a.)Find an expression for the potential V(x,y) at an arbitrary point in the xy-plane...

Homework Statement A player located 18.1 ft from a basket launches a successful jump shot from a height of 10 ft (level with the rim of the basket), at an angle Θ = 34 degrees and initial velocity of v = 25 ft/s.A. Show that the distance s of the shot changes by approximately 0.255∆Θ ft if the...

When we study something with our physics theory, we may always ignore some "unimportant" factors to simplify the culculation.And then, we get a approximation. But if we don't ingore any factors, we will get the absolutely accurate result. Is it possible? I think the physics theory isn't...

Homework Statement I trying to figure this out, its part of a bigger question. When ka \ll 1, what happens to, \frac{1}{\left ( 2-k^{2}a^{2} \right )\textup{sin}\, ka\, - 2\, ka \textup{cos}\, ka} Homework Equations Its something to do with the lowest order approximation. The...

"The Probing of, Approximation to and Idealization of Structure" for Foundations Hey, Over the past six years I have worked with Lucien Hardy at Perimeter in Waterloo and Prakash Panangaden at McGill. This paper is the culmination of thoughts on physics gleaned from that work...

Two lasers are incident on a photodetector, one has a wavelength of 780.56160 nm the other has a slightly shorter wavelength. They produce a beat frequency of 460 Hz. Attempt df = f2 - f1 = c ((1/ lambda1) - (1/ lambda2)) = c ((lambda1 - lambda2) / (lambda1 * lambda2)) Is it ok to...

"Random Phase Approximation" (Why they call it so?) Hi, I'm wondering that why the scientists call this phenomena in many body physics "Random Phase Approximation". (Why Random? Why Phase?) It seems that when we want calculate a correlation function in quantum field theory, it leads to...

Im trying to calculate the ground state energy of Helium using a density functional theory approach combined with the local density approximation. So far I have set up universal functionals and I mainly need help with the actual algorithm the evaluation of the Hartree energy functional.

Homework Statement find the 2nd, 3rd, and 6th degree taylor approximation of: f(x) = 10(x/2 -0.25)5 + (x-0.5)3 + 9(x-0.75)2-8(x-0.25)-1 for h = 0.1 to h = 1, with \Deltah = 0.05 and where xo=0; and x = h Homework Equations N.A The Attempt at a Solution I just need to...

Prove \hbar\omega << k_{B}T \Rightarrow \frac{\hbar\omega}{e^{\frac{\hbar\omega}{k_{B}T} - 1}.

In A First Course in General Relativity, the use of the weak field approximation is confusing to me. I constantly get confused when the term "f(x) is only valid to first order in f..." for the Newtonian potential in the metric comes up. At a certain point the book states: ...1/2(-1/(1 +...

Hi I'm trying to figure out how to get the electric dipole selection rules for an atom with many electrons. In all textbooks that I've seen it's shown for Hydrogen, or in the central field approximation (which is, in some sense, equivalent to Hydrogen). Obviously the central field...

Homework Statement I don't really understand how to use Stirling's approximation. here's an example you flip 1000 coins, whts the probability of getting exactly 500 head and 500tailsHomework Equations N!=NNe-N(2pieN)1/2The Attempt at a Solution wht they did was 21000 total number outcome...

y' = 3 + t - y, y(0) = 1 A) Find the approximate values of the solution of the given initial value problem at t = 0.1, 0.2, 0.3, 0.4 using the Euler method with h = 0.1. B) Repeat part A with h = 0.05. Compare the results found in A. I did part A correctly, but cannot get the right...

Homework Statement For a single large two-state paramagnet, the multiplicity function is very sharply peaked about N_{\uparrow} = N/2. (a) Use Stirling's approximation to estimate the height of the peak in the multiplicity function. (I am fairly confident in my answer here) (b) Use the...

Homework Statement Here is a drawing with all the needed variables: http://i.imgur.com/192GI.jpgHomework Equations The Attempt at a Solution I have been trying to figure out how this approximation is derived for some time now and have no progress to show for it. Any help in figuring out the...

Homework Statement Suppose a set of N data points {(xk,yk)}Nk=1 appears to satisfy the relationship for some constants a and b. Find the least squares approximations for a and b. Homework Equations The Attempt at a Solution I really have no idea about this problem.

Homework Statement Let f: R-->R be continuous. For δ>0, define g: R-->R by: g(x) = (1/2δ) ∫ (from x-δ to x+δ) f Show: a) g is continuously differentiable b) If f is uniformly continuous, then, for every ε>0, there exists a δ1>0 such that sup{∣f(x) - g(x)∣; x∈R} < ε for 0<δ≤δ1The Attempt at...

1. The problem statement, all variables and givennown data 1)FInd the nth left endpoint approximation Ln for f(x) = 3x^2-2 on [0,2]. What is the limit as n approaches infinity Ln in this case? 2)Evaluate: \sum45i=5 (2i-5) Homework Equations Ln = \sumNj=1 f(cj)(xj-xj-1) The...

Can you tell me something more about Tyablikov approximation? \langle\langle \hat{S}_i^z\hat{S}_j^{\pm}|\hat{B}_l\rangle\rangle \approx \langle\hat{S}_i^z\rangle \langle\langle \hat{S}_j^{\pm}|\hat{B}_l\rangle\rangle \qquad i\neq j I'm confused here? Is that approximation work in real...

Hi Homework Statement I'm trying to follow and work through a derivation in my textbook, making sure I can replicate the steps myself and understand what's happening. However, I came across this approximation and can't seem to figure out where it comes from or why and the book gives no...

Hi, Where can I find the expression of the central difference approximation of the first and second derrivative (spatial) for a NON uniform grid?

Can someone explain the attached image for me please? I do not understand how 2\delta_{k, k'}a_{k'}^{\dagger}a_{k} becomes a_{k}^{\dagger}a_{k} + a_{-k}^{\dagger}a_{-k} to me it should just be 2a_{k}^{\dagger}a_{k} and also I do not understand how e^{-ik}a_{-k}a_{k} +...

I am reading the H. A. Bethe's book ---quantum mechanics of one and two electron atoms. In section 39, it is shown how to go from general form to pauli approximation form through momentum space. who knows how to go directly in position space?

Homework Statement Show that if f is continuously differentiable on [a, b], then there is a sequence of polynomials pn converging uniformly to f such that p'n converge uniformly to f' as well.Homework Equations The Attempt at a Solution Let pn(t) = cn t^n Use uniform convergence and integrate...

Homework Statement The IPA potential-energy function U(r) is the potential energy "felt" by an atomic electron in the average field of the other Z - 1 electrons plus the nucleus. If one knew the average charge distribution rho(r) of the Z - 1 other electrons, it would be a fairly simple matter...

Homework Statement stimate Δf using the Linear Approximation and use a calculator to compute both the error and the percentage error. f(x) =1/(1+x^2) , a = 3, Δx = 0.5 Homework Equations f'(a)(x) percentage error= abs(error) divided by actual value The Attempt at a Solution So...

Homework Statement If the price of a bus pass from Albuquerque to Los Alamos is set at x dollars, a bus company takes in a monthly revenue of R(x) = 1.5x − 0.01x2 (in thousands of dollars). Suppose that x = 80. How will revenue be affected by a small increase in price? Explain using the...

On a production line, only 45% of items produced meet quality standards. A random sample of 500 items will be taken. Using the normal approximation to the binomial distribution, approximate the probability that less than half of the sampled items meet quality standards. 500*.5 = 250...

Can you take the sin terms out of Snell's Law when dealing with angles below 10 degrees? so Snell's Law would be become n1\theta1=n2\theta2 Thanks

Homework Statement There is a derivation in the text that I'm having problems replicating. The text gives the formula for tidal potential as: U_{tid}=-GM_{m}m(\frac{1}{d}-\frac{x}{d^{2}_{0}}) Where M_{m} is the mass of the moon, d is the distance from the CM of the moon to the point of...

Homework Statement I have to program a three component decay chain using finite difference approximation. I understand finite difference and have written my code, but I have an error I can not find which is giving me an erroneous answer. The curve is correct, but the magnitude of the...

Homework Statement I'm reading about fluid mechanics and in one of the examples they have approximated the velocity field. The field is two dimensional u = (u,v) I have never seen this before so cold someone tell me what it is called so I can look it up? The notes I am reading are hand...

Homework Statement Approximation of e^(x^2+3x+1) from 0 to 3 within .2 of the actual integral. Homework Equations Riemann Sum or Trapezoidal. (We haven't learned Taylor yet). The Attempt at a Solution Last n value i found was 148068 which gave me a delta x of 3/148068...im very...

For certain computations I need a quick approximation of the left singular vector of a matrix G( nxk ; n>k ). Also, the corresponding singular value would be needed. Perhaps after approximating the singular value I could use the Conjugate Gradient method to obtain the approximation of the left...

So my book merely mentions that this holds as a result of the central limit theorem for values of lambda greater than 10, but ideally greater than 32. Anyway I was wondering if anyone knew this actual proof as I am interested in seeing it step by step and I could not have found it anywhere...

Homework Statement I'm reffering to http://hyperphysics.phy-astr.gsu.edu/hbase/electric/dipole.html and the approximation r_1 - r_2 \approx = d \cos \theta. I see that it is correct if I draw it up, but I wondered if there were any "more mathematical" ways to see this? Where does these...

Hi! I have a question about approximation of functions with small angles. I was looking through some notes from my teacher and didnt understand why the following approximation is valid. We have a system which is at equlibrium at an angle, say a. Now we wanted to se what happens with the...

Homework Statement In an inertial navigation system on switching ON an intial error of 0.0001m/s^-2 exists in one direction. After 30 minutes an additional error of -0.00005m/s^-2 in acceleration builds up. The system navigates for 2hrs 30 minutes after switching ON. compute the error in...

Homework Statement Well, this problem is quiet similar to the one I've posted before. It asks me to approximate to the e number using taylors polynomial, but in this case tells me that the error must be shorter than 0.0005 Homework Equations...

Homework Statement Use the Simpson method to estimate \displaystyle\int_{0}^{1}\cos(x^2)dx with an approximation error less than 0.001. Well, I have a problem. Actually I'm looking for a bound for the error of approximate method integration by using Simpson's method. I have to bring...

Studying the semiconductor in equilibrium, i found a sentence which i don't understand. "At the very low temperature, freeze-out occurs; the Boltzmann approximation is no longer valid." I know that freeze-out occurs at the very low temperature, but why is it that the Boltzmann...

How would one know when to find the least squares approximation?

Hi For small angles or points near the vertex of a parabola we can approximate a parabolic surface with a circle. The focus of the parabola is a unique point specifically for optics (Parallel light will converge at the focus), and vice versa. Has anyone come across an derivation that...

I have the following problem: Assume g is a (smooth enough) function, X a random variable and \varepsilon^h a sequence of random variables, whose moments converge to 0 as h goes to zero. I would then like to prove that \mathbb{E}\left|g(X+\varepsilon^h)-g(X)\right| converges to zero as well...