Approximation Definition and 705 Threads

Often you use taylor series to approximate differential equations for easier solving. An example is the small angle approximation to the pendulum. My question is: Is there mathematical tool for calculating the error you make as time goes with such an approximation? Because I could Imagine it...

The Fresnel diffraction integral is: A(x_0 , y_0 ) = \frac{i e^{-ikz}}{λz} \int \int dx dy A( x , y ) e^{\frac{-ik}{2z} [(x - x_0)^2 + (y - y_0)^2]} When we want to obtain the Fraunhofer diffraction integral from here, we need to somehow convert it to: A(x_0 , y_0 ) = \frac{i...

Most General Form of the "Rate-Equation Approximation" In quantum optics or laser physics, while solving an ordinary differential equation (ODE) using the integrating factor, the so-called Rate-Equation Approximation is used. I have come across different sources implementing it differently. For...

Hey, guys. Having problems with this question because I don't exactly know how to begin it. Homework Statement The problem states to: "Find the Taylor polynomial of smallest degree of an appropriate function about a suitable point to approximate the given number to within the indicated...

Homework Statement Consider ##n## independent trials, each of which results in one of the outcomes ##1,...k## with respective probabilities ##p_1,...p_k, \sum_{i=1}^{k} p_i = 1##. (I interpret this summation as just saying mathematically that definitely one of the outcomes has to occur on each...

Homework Statement One uses the approximation sin(x) = x to calculate the oscillation period of a simple gravity pendulum. Which is the maximal angle of deflection (in degree) such that this approximation is accurate to a) 10%, b) 1%, c) 0.1%. You can estimate the accuracy by using the next...

I am asking for simple guidance on this problem. f(x) = 3x^2-1, (2,11)I do believe I need to obtain an equation for tan line so first step I think is to use point slope or slope intercept (a friendly reminder to the name of formula would be very nice :)) y - ysub1 = m(x-xsub1) = y -...

Hi everyone, I have an equation that contains the derivative of the Bessel Function of the first kind. I need to evaluate Jn'(x) for small values of x (x<<1). I know that Jn(x) is (x)n/(2n*n!). What is it for the derivative?

Homework Statement Use a graphing calculator or computer to verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. (Round the answers to two decimal places.) tan(x) ≈ x Homework Equations derivative...

1)What exactly is meant by the 'static limit' where the frequency is taken to zero, but the wavenumber is finite? I am getting confused because if the frequency is zero, then surely the probing electrons/photons/whatever have no wavelength, so how can the wavenumber be finite and non-zero? 2)...

I am modelling the atmosphere as a perfect, static gas subject to uniform gravity, assuming ideal gas equation, the density is found to follow: p=A*exp(-z/H) where A is a const, z is the heigh, and L is the scale height. I want to know when this approximation breaks down! at what density? i am...

Hi! How do I approximate the integral \begin{equation} \int_0^{\infty} dt \:e^{-iA(t-B)^2} \end{equation} with A, B real, A > 0, and B=b \cos\theta where 0 \leq \theta < 2\pi? I guess for B\ll 0 the lower limit may be extended to - \infty to yield a full complex gaussian integral, but what...

Homework Statement Derive the Derive the two variable second order Taylor series approximation, below, to f(x,y) = x^3 + y^3 – 7xy centred at (a,b) = (6,‐4) f(x,y) ≈ Q(x,y) = f(a,b) + \frac{∂f}{∂x}| (x-a) + \frac{∂f}{∂x}|(y-b) + \frac{1}{2!}[\frac{∂^2f}{∂x^2}| (x-a)^2 + 2\frac{∂^2f}{∂x∂y}\...

Hey everyone - just a bit of a conceptual question regarding the sudden approximation for a particle in an infinite square well. In theory, if we were to suddenly decrease the width of the potential from say L, to L' << L, in a very quick period of time - wouldn't this in some sense constitute a...

Homework Statement f(x) = sqrt(1 - x) a = 0 Approximate sqrt(0.9). Homework Equations L(x) = f'(a)(x - a) + f(a) The Attempt at a Solution I understand that linear approximation is finding the equation of a line of a point tangent to a function. But now this question is asking me...

Homework Statement Hey guys I'm having a hard time understanding how the book obtained the solution. Here is the question A function f is given along with a local linear approximation of L to f at a point P. Use the information given to determine point P. f(x,y)= x2+y2; L(x,y)=2y-2x-2...

In an introductory calculus course I am doing I have just come across the following problem: "Given that $\sin(x)=e^{-x}$ has a solution near x=1, use Newton's method to find the solution to 4 decimal places." My question will strike you as very basic, however, I *am* a beginner and I *have*...

Homework Statement For x near 0, local linearization gives the following equation. e^x ≈ 1 + x Estimate to one decimal place the magnitude of the error for −1 ≤ x ≤ 1. Homework Equations The Attempt at a Solution I'm no exactly sure what to do here to be honest, but what I thought I'd...

Homework Statement The Attempt at a Solution Thus my answer is N = 20. I wasn't sure if I should use ≤ or just <. Also to get 2 decimal place accuracy would using 0.005 be correct?

Everyone knows the general thrust equation: T = {{\dot m}_i}\left[ {(1 + f){V_e} - {V_\infty }} \right] + ({P_e} - {P_\infty }){A_e} Where mdot_i is the incoming mass flow rate, f is the fuel flow rate, and the subscripts ∞ and e represent free-stream and exit conditions, respectively...

Hi, I have a question regarding finite difference approximation: Consider the finite difference approximation u'(xj-1/2) + λu(xj−1/2) ≈ 1/h*[u(xj ) − u(xj−1)] + λ(θu(xj ) + (1 − θ)u(xj−1)) how can I Find the order of approximation as a function of θ? I am really new in this field, so...

Homework Statement I was recently given this question and very little explanation of the concept. I've struggled with this for a week and read absolutely everything I can find and I'm still not any closer to understanding it. Can anyone please point me in the right direction or explain how...

Homework Statement Pade approximation [N/D]=\frac{a_0+a_1x+...+a_Nx^N}{1+b_1x+...+b_Dx^D} With this approximation we approximate Maclaurin series f(x)=\sum^{\infty}_{i=0}c_ix^i=[N/D]+O(x^{N+D+1}) How to calculate [1/1] for f(x)=1-\frac{1}{2}x+\frac{1}{3}x^2-... ? Homework Equations...

According to the link below, fractal dimension is an exponent of some sort: http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html The Hausdorff Dimension (aka fractal dimension) is denoted as D in the website above. And r is the base number. If we were to look at...

Homework Statement I have a long EM question in which there is a Hertzian dipole at a point (0,0,-100), (unknown orientation) and I am told the equation of the physical magnetic field detected 100m away at the origin of Cartesian coordinates. $$(B_0 \sin (2 \pi f t)\mathbf{e}_x$$, and $$B_0 =...

Hey guys, I am looking for a textbook that I can cite as a source for a project, for which I am doing the math on. I know that for a 22° approximation sinθ=θ and cosθ=1-\frac{θ^{2}}{2} but for a 5° approximation sinθ=θ but now cosθ=1 and that's all fine and dandy, but I am looking...

f(x)=x^3/2 with x=4 and deltax=dx=0.1 calculate deltay and dy I got 0.4 for dy but its wrong, and i can't find deltay. please help! thxs

Hi all, I'm having a bit of trouble getting my head round approximations to a function in the limit of small and large values of the x parameter. The function is: y = x\left\{ {\left[ {1 + \left( {{1 \over x}} \right)^2 } \right]^{{1 \over 2}} - 1} \right\} The paper I'm reading says...

Homework Statement Firstly sorry if this is in the wrong place. I have never submitted a question on this forum about a comp sci question. I got an assignment that asked me to solve for a variable using Bisection of successive approximations. This however is not why I am here as I know you...

Homework Statement It's attached. The problem and solution are given. Homework Equations The Attempt at a Solution I circled a part of the image in red. Is this substitution supposed to be an approximation? I was thinking it was because one is referring to angular motion, so...

Hello! So I was reading a paper in which I came across the following: k = \pi + \pi\ell \tanh{(k)} \approx \pi\ell where "l" is very small. What on Earth is the origin of this approximation? I'm sure it's very simple, but I can't seem to derive it from the angle-sum and small angle...

Hi, fellow physicists (to be). This is my first post on the forum, so I hope I get it right. If not so, please let me know :) introduction to the problem At the moment I am working on my physics bachelor's thesis at the theoretical department of my university (Amsterdam). My thesis focusses...

I am aware that for a function of two variables f(x,y) a linear approximation of a point f(x,y) close to f(x_0,y_0) can be approximated by the tangent plane approximation f(x_0+\Delta x,y_0+\Delta y)\approx f(x_0,y_0)+f_x(x_0,y_0)\Delta x+f_y(x_0,y_0)\Delta y where \Delta x=x-x_0 and \Delta...

Another thread in this section (https://www.physicsforums.com/showthread.php?t=619945) raises some interesting questions. I can't figure what the original poster there is asking, so I think it best to put what I'm asking in a new thread. A simple "field test" of a system that detects a...

I read that when v≈c, sqrt(1-β2) = sqrt(2*(1-β)). How do you show this mathematically? I have no idea. Thanks! :)

Dear all, in what sense the tangent space is the best approximation of a manifold? The idea is clear to me when we think about a surface in Rn and its tangent plane at a point. But what does this mean when we are referring to very general manifolds? In what sense "approximation" and in what...

Homework Statement Firstly, I'd just like to point out that this is not actually a course related question. I have been trying to teach myself mathematics, and have been grappling with this for a couple of days. The book has no answer at the back for this particular question. Variables...

I took a LA course in the spring, and was interested by the least squares method for building models. I decided to practice this concept by attempting to build a model that would predict ticket sales for the Mega Millions lottery given the jackpot amount. I have 249 data pairs of jackpot and...

This problem arises in a paper on population genetics (Kimura 1962). 1. The problem statement Let f(p) = \int_0^p ((1 - x)/x)^k dx. For a small value of p, we have approximately f(p) = (p ^ (1-k)) / (1-k) How is this obtained? 2. My attempt at a solution I tried to expand the f(p) around p =...

Find an approximate value of the number e-0.1 with an error less than 10-3 ı know that ex = Ʃ(from zero to ınfinity) xn / n!=1+x/1!+x 2/2!+... ı don't know how to use e-0.1 in this question.Do ı write -0.1 instead of x in ex series?

I am trying to understand the elctric dipole approximation when an atom interacts with an electromagnetic wave. I know that if the size of the atom is much much smaller than the wavelength of the radiation, then the dot product od the wavevector and the position vector becomes constant. I...

By chance I stumbled on this "almost" equality: \frac{1}{5}(1/2+2/3+3/4+4/5+5/6+6/7+7/8+8/9+9/10) ≈ √2 - 7.2×10^{-6} I'm just wondering, are these funny coincidences simply, well, coincidences :biggrin: or is there some kind of explanation? I've see a ton of other funny stuff like...

I created a method for both approximating a function and extending a it's domain from a Natural to a Real Domain. Does this have a name already or any interesting application? Basically. Add polynomial of degree 0, 1, 2, 3, etc. Making at the same time the approximation function equal to f(0)...

\frac{Ns-L}{L+Ns} What does that reduce to if L << Ns ? Obviously setting L to zero leads me nowhere since that argument above is actually inside a logarithm. I don't know how to perform the approximation. And the answer can't be zero by the way. Is there something I can do here? Usually the...

I was having this debate with a friend, and I wanted to know if I was correct. My friend was saying that SR is an approximation for GR (albeit a very good one) with the specific conditions of only inertial reference frames, and I was saying that SR is exactly accurate with GR, and so is directly...

y(4C) ≈ 7.3 + C + \frac{C}{3^{10C}} + \frac{C}{3^{20C}} + \frac{C}{3^{30C}} Would: y(nC) ≈ 7.3 + C\sum_{n = 0}^{\infty}{\frac{1}{3^{10Cn}}} Be an acceptable answer? If not, what am I doing wrong here?

Homework Statement I've attached the questionHomework Equations Pr(X<=x)= (x + 0.5 - n*p) / sqrt(n*p*(1-p))The Attempt at a Solution okay so n=1150, p=0.02 , Pr(X<23) =23 + 0.5 - 1150(0.02) / sqrt(1150*0.02*0.98) =0.105316 is that bit right so far. Because it is less than i thought x...

The following component is connected to a reference voltage of Vref = 6V. The component is given the value (2C)16 to convert. Calculate Vout. http://img69.imageshack.us/img69/7396/figuringout.jpg Basically I treated this supposedly successive approximation converter like any other...

Hi all, I am reading now Zee's book "Quantum Field Theory in a Nutshell", there in Apendix 2 of Chapter I.2 the method of steepest descent is briefly described. The part where I have a question is almost self contained and half a page long, so I attached the screen shot of it (formula 19)...

Homework Statement Hi Whenever I read about the dipole approximation in QM, then the Hamiltonian is given as \hat {V}_{\text{dipole}} = -\mathbf{d}\cdot \mathbf{E} where E is the electric field and d the dipole operator. What I am wondering about is that d is an operator. Is it wrong to...