Approximation Definition and 766 Threads

Homework Statement There is a fluid flowing over a hot plate. We non-dimensionalized the problem from three partial diff eq's to two ode's. I am modeling I have two coupled differential equations that are a system of initial value problems. I am supposed to numerically integrate the two...

Homework Statement The tangent line through given points (pi/4,pi/4) m=1 y= cos(y)cos(x)/sin(x)sin(y) The Attempt at a Solution dy/dx= d/dx[cos(y)cos(x)/sin(x)sin(y)] First use quotient rule ? vu'-uv'/v^2 v= sin(x)sin(y) v'= do i need to use product rule? for product rule...

Let f(x) = x^(1/3). The equation of the tangent line to f(x) at x = 8 can be written in the form y = mx+b where m is: and where b is: Using this, we find our approximation for 8.1^(1.3) is: I found the slope to be 1/12 I found b to be 1.3333333333333333333 I still can't get the answer...

Homework Statement I am given a table of data derived from experiment. A force (F) is applied to a spring and the extension (x) is measured and recorded. An additional column of data for the derivative (dF/dx) is also provided. Here is the data: x(m) F(kN) df/dx (kN/m) 0.0...

Find an asymptotic approximation as p goes to infinity: f_{\lambda}(p)=\oint_{C}exp(-ipsinz+i\lambda z)dz where C is a square contour and p, lambda are real. Taking C to be of side length pi and centered at the origin, I applied the method of steepest descent at the point z=-pi/2...

Use the binomial expansion of (1+x)^(-1/2) to find an approximation for 1/(rt4.2). I've got the expansion of (1+x)^(-1/2) as 1-(1/2)x+(3/8)x^2... but the obvious idea of substituting x=3.2 gives me the wrong answer. I think it's something to do with the expansion being valid but can't...

Homework Statement Show that the integrand of \Gamma(s+1)=\int_{0}^{\infty} t^se^{-t}dt may be written as e^{f(t)} where f(t)=s\ln{t}-t. Show that f(t) is maximum at t=t_0 and find t_0. If the integrand is sharply peaked, expand the integrand about this point (ie Taylor expansion) and...

Can the method of steepest descent (saddle point method) be used if an integral has the following form: \int exp\left[M f(x) + g(x)\right]dx where M goes to infinity? I ask because all the examples I've seen of this method involve a function which is multiplied by a very large number...

I'm having problems understanding how \frac{e^{-\hbar \omega / 2k_BT}}{1-e^{-\hbar \omega / k_BT}} approximates to k_BT/ \hbar\omega when T >> \hbar\omega/k_B Seems like it should be simple but don't quite see how to arrive at this result. *update* I have tried using taylor...

Hi, I need your help with a very standard proof, I'll be happy if you give me some detailed outline - the necessary steps I must follow with some extra clues so that I'm not lost the moment I start - and I'll hopefully finish it myself. I am disappointed that I can't proof this all by myself...

On my last test I got four problems wrong. I'd like to know what I did wrong on these for my final. 1. Given f(x) = x^(3/2) ; x=4; and delta x = dx = 0.1; calculate delta y 2. Use differentials to approximate the change in f(x) if x changes from 3 to 3.01 and f(x) = (3x^2-26)^10 3. f(x) =...

In quantum optics and bose-einstein condensates, this is a well known technique however, i still cannot grasp its essense. in bec, what is its advantage over the gross-pitaevskii equation?

Can anyone give an approximation as to when CERN starts testing?

Hi, In my textbook they derive that a solution to the law of Faraday and the law of Ampère-Maxwell is an electromagnetic wave. In one of the steps they have to calculate E(x+dx,t) where E is the magnitude of the electric field of the wave. They say E(x+dx,t) \approx E(x,t)+\frac{dE}{dx}...

Hi, Homework Statement The electric field of a uniform charged disk at a point on its axis at a distance x from the disk is given by E = 2k_e\pi\sigma(1-\frac{x}{\sqrt{x^2+R^2}}) where R the radius of the disk and \sigma the surface charge density. In my notes it says that when x\gg R, that is...

Hello, This question is more conceptional - I think I can do the algebra (mostly approximations) in this problem ok. Homework Statement I am wondering why it is for a radiation zone, that a linear electric quadrupole can be approximated as a dipole. I am wondering if this is...

Homework Statement An oscillator is in the ground state of H = H^0 + H^1 , where the time-independent perturbation H^1 is the linear potential (-fx). It at t = 0, H^1 is abruptly turned off, determine the probability that the system is in the nth state of H^0 . Homework...

Homework Statement In a beta decay H3 -> He3+, use the sudden perturbation approximation to determine the probability of that an electron initially in the 1s state of H3 will end up in the |n=16,l=3,m=0> state of He3+ Homework Equations |<n'l'm'|nlm>|^2 The Attempt at a Solution...

I'd be greatful for a bit of help on this question, can't seem to get the answer to pop out: A particle moving in a potential V is described by the Klein-Gordon equation: \left[-(E-V)^2 -\nabla^2 + m^2 \right] \psi = 0 Consider the limit where the potential is weak and the energy is...

Can anyone tell me what is the "Brus Approximation" in case of the bang gap of semiconductors?:rolleyes:

For a system with minimum cut set (C1,C2,C3),(C4,C5,C6) and (C4,C5,C7), Using the F-V algorithm, calculate the first and second order approximations given P[C1] =P[C1] = P[C2] =0.107, P[C3] =0.168,P[C4] =P[C5] =0.0478,P[C6] =0.0921,P[C7] =0.102 Answers, Ist order= 0.002366919, 2nd...

My understanding of the Hartree approximation is that the product wavefunction is symmetric rather than antisymmetric, therefore the Hartree approximation effectively ignores the Pauli exclusion principle. So how does the Pauli-exclusion principle get taken account of in the Hartree model...

Use Taylor's theorem to determine the degree of the Maclaurin polynomial required for the error in the approximation of the function to be less than .001. e^.3 I really, really don't know what to do for this one, and I have a quiz tomorrow. I have read through the section in the book, but...

Homework Statement Use Taylor's theorem to obtain an upper bound of the error of the approximation. Then calculate the exact value of the error. cos(.3) is approximately equal to 1 - (.3)^2/2! + (.3)^4/4! Homework Equations The Attempt at a Solution I came up with upper...

1. Homework Statement Determine the number of terms required to approximate the sum of the series with an error of less than .001 Sum ((-1)^(n+1))/(n^3) from n=1 to infinity 2. Homework Equations 3. The Attempt at a Solution I guess this is what you do: 1/(n+1)^3 <...

Homework Statement Determine the number of terms required to approximate the sum of the series with an error of less than .001 Sum ((-1)^(n+1))/(n^3) from n=1 to infinity Homework Equations The Attempt at a Solution I guess this is what you do 1/(n+1)^3 < 1/1000 and...

Function f(t) specified on [t_0;t_1] has a necessary number of derivatives. Find algorithm which can build uniform approximations of this function with help of partial sums: \sum_{i=1}^{N}\alpha_i e^{-\beta_i t}. That is, find such \alpha_i, Re(\beta_i)\geq 0 satisfying the expression...

I am trying to write a program that estimates the derivative of a polynominal and determines the error. So far my code is % The code for Problem 3. a=5-4*x^2+3*x^3-2*x^4+x^5; % ask for a function to be differentiated x=input('Enter the value x at which to find the derivative '); % ask...

[SOLVED] Taylor approximation Homework Statement I have an exact funktion given as: P(r)=1-e^{\frac{-2r}{a}}(1+\frac{2r}{a}+\frac{2r^2}{a^2}) I need to prove, by making a tayler series expansion, that: P(r)\approx \frac{3r^3}{4a^4} When r \prec \prec a The Attempt at a Solution...

Homework Statement Given some dispersion relation for the tight binding approximation in 2D: e(k_x,k_y) = -2t_1[cos(k_x*a)+cos(k_y*a)]-4t_2[cos(k_x*a)cos(k_y*a)] Show that the density of states has a logarithmic singularity for some choice of parameters t_i. Homework Equations g(e)de=g...

Homework Statement Write an algorithm and Matlab function JN, which uses the Composite Trapez- ium Rule (CTR) to compute an approximation of the integral f(x) dx for an arbitrary function f of one variable. The inputs should be a, b and N (the number of subintervals), and f (the name of a...

The parallel transport equation is \frac{d\lambda^{\mu}}{d\tau} = -\Gamma^{\mu}_{\sigma\rho} \frac{dx^{\sigma}}{d\tau} \lambda^{\rho} If I take the derivative of this with respect to tau, and get \frac{d^2\lambda^{\mu}}{d\tau^2} =...

Homework Statement use an appropriate local quadratic approximation to approximate the square root of 36.03 Homework Equations not sure The Attempt at a Solution missed a day of class

Homework Statement How is, [(N+Q)!Q!]/[(Q+1)!(N+Q-1)!] equal to (N+Q)/(Q+1) when N,Q>>1 ?? It looks like the Q!/(N+Q-1)! cancels but i don't see how, I am going from my lecturers notes here. Homework Equations The Attempt at a Solution

Homework Statement Hi, I've read from Wikipedia that in the Hartree-Fock approximation, "Each energy eigenfunction is assumed to be describable by a single Slater determinant". My question is... if the approximation fails and the system has to be described by linear combinations of more than...

Homework Statement Find the linear harmonic oscillator approximation for potential energy function: \ V=\frac{a}{x^2}+\ b \ x^2 Homework Equations The Attempt at a Solution The 2nd term will be present in the expression of V(approx).But what about the first term. Should we make...

Homework Statement Suppose the exact value of a particular definite integral is 6. The following questions refer to estimates of this integral using the left, trapezoid, and Simpson's rules. Use what you know about approximate errors to answer the following questions. Give your answer to 4...

Homework Statement Use Sterling's approximation to show that the multiplicity of an Einstein solid, for any large values of N and q is approximately \Omega(N,q) = \frac{(\frac{q+N}{q})^q(\frac{q+N}{N})^N}{\sqrt{2\pi q(q+N)/N}}Homework Equations \Omega(N,q) = \frac{(N+q-1)!}{q!(N-1)!} \ln(x!)...

Homework Statement Okay, so this is a three-part question, and I need some help with it. 1. I need to show that the function f(x) = e^{-1/x^{2}}, x > 0 and 0 otherwise is infinitely differentiable at x = 0. 2. I need to find a function from R to [0,1] that's 0 for x \leq 0 and 1 for x...

I'm curious if the question posed my Smolin Could quantum mechanics be an approximation to another theory? "We consider the hypothesis that quantum mechanics is an approximation to another, cosmological theory, accurate only for the description of subsystems of the universe. Quantum theory...

How to find a polynomial P(x) of the smallest degree such that sin(x-x^2)=P(x)+o(x) as x->0? Do I have to calculate the first six derivatives of f(x)=sin(x-x^2) to get Taylor polynomial approximation?

Homework Statement A particle is charge +e is incident on an electric dipole of charge +e and a charge of -e separated by a vector d (which runs from -e to +e). Use the Born approximation to calculate the differential scattering cross section as a function of the initial wave vector, the...

I was wondering what series approximation I can use to approximate y: y=(1-(dx/dy)^2)^1/2 when dx/dy is not trigonometric, and contains the derivative of a floor function

Homework Statement in the first part of the problem we were told to estimate the lowest energy of a electron in a hydrogen atom for a certain orbital angular momentum l by evaluating the equation for effective potential at it's minimum. That is to say that for any given l the minimum energy...

Some interesting calculus... Which of the following expressions yields the best approximation to df/dx (h<<1)? A. \frac{f(x+h)-f(x)}{h} B. \frac{f(x+\frac{h}{2})-f(x-\frac{h}{2})}{h} C. \frac{f(x)-f(x-h)}{h} D. \frac{f(x+h)-f(x-h)}{h} From school days I have been taught A...

Ok, we are asked to determined the degree of the the taylor polynomial about c =1 that should be used to approximate ln (1.2) so the error is less than .001 the book goes throught the steps and arrives at: |Rn(1.2)| = (.02)^(n+1)/(z^(n+1)*(n+1) but then, it states that...

Hi, I am doing a phys experiment, and I find myself trying to obtain some pattern of quantization of some measurements, i.e., I'm trying to find a number (double) that divides at least a significant portion of my data, with an arbitrary remainder. Does anyone know of any algorithm that does this...

Homework Statement To the right is the graph of 5x^3y-3xy^2+y^3=6. Verify that (1,2) is a point on the curve. There's a nearby point on the curve whose point is (1.07,u). What is the approx. value for u? There's a nearby point on the curve whose coordinates are (.98,v). What is the approx...

Homework Statement Use the "Three Term" Taylor's approximation to find approximate values y_1 through y_20 with h=.1 for this Initial Value Problem: y'= cosh(4x^2-2y^2) y(0)=14 And write a computer program to do the grunt work approximation Homework Equations The Attempt...

[SOLVED] Approximation to simple harmonic motion. Homework Statement A small mass m, which carries a charge q, is constrained to move vertically inside a narrow, frictionless cylinder. At the bottom of the cylinder is a point mass of charge Q having the same sign as q. Show that if the mass m...