Approximation Definition and 768 Threads

Homework Statement Find a value for n for which the nth partial sum is ensured to approximate the sum of the alternating harmonic infinite series to three decimal places. Homework Equations Sn = Ʃ(-1)^k+1*1/k = 1 - 1/2 + 1/3 - 1/4 + 1/5 - . . . S1 = 1 S2 = 1 - 1/2 S3 = 1 - 1/2 + 1/3 S4...

I've been working on a Theremin Excel simulation for the past couple of months. For those who don't know what a Theremin is, it was one of the very first electronic instruments to be invented, and has two "antennas" that independently change the pitch and amplitude of a tone via hand...

The Simple Approximation Lemma Let f be a measurable real-valued function on E. Assume f is bounded on E, that is, there is an M \geq 0 for which |f|\leq M on E. Then for each \epsilon > 0, there are simple functions \phi_{\epsilon} and \psi_{\epsilon} defined on E which have the following...

So are Newton's laws also an approximation to quantum phenomena. Can it be derived from quantum laws?

My questions are from lecture 9, MIT OCW SV Calculus, Jerison, 2009; At 27:50 he is deriving the linear approximation for the function e^(-3x)(1+x)^(-1/2)≈(1-3x)(1-1/2x)≈1-3x-1/2x+3/2x^2≈1-7/2x, for x near 0. In the last step he drops the x squared term since it is negligible(no questions so...

Homework Statement \mu = \frac{mM}{m+M} a. Show that \mu = m b. Express \mu as m times a series in \frac{m}{M} Homework Equations \mu = \frac{mM}{m+M} The Attempt at a Solution I am having trouble seeing how to turn this into a series. How can I look at the given function...

I'm currently teaching myself some QFT trough Peskin and Schroeders Introduction to QFT and I've noticed that in several arguments they rely on appealing to the Born approximation of non-relativistic QM scattering theory. For example on page 121 equation (4.125) they appeal to the scattering...

Homework Statement What is the probability that an electron in the ground state of hydrogen will be found inside the nucleus? a) First calculate the exact answer, assuming the wave function \psi(r,\theta,\phi) = \frac{1}{\sqrt{\pi a^3}} e^{-r/a} is correct all the way down to r=0. Let b be the...

Homework Statement obtain the number r = √15 -3 as an approximation to the nonzero root of the equation x^2 = sinx by using the cubic Taylor polynomial approximation to sinxHomework Equations cubic taylor polynomial of sinx = x- x^3/3!The Attempt at a Solution Sinx = x-x^3/3! + E(x) x^2 =...

Hello! In order to prepare for an exam I have started solving exercies problems and have gotten most of them right but have quetion a regarding a solution. In this probelm I used the first order Born approximation in order to calculate the differential and total cross section for the...

Homework Statement consider the function f(x) = aln(x+2). Given that f'(1) = a/3, what is the approximate value of f(0.98)?Homework Equations f(x1) = f(x0) + f'(x0)x(x1-x0)The Attempt at a Solution I solved it and get f(.98) = aln(1+2) + (.098-1) = aln(3) - (.02)(a/3) <= not an answer the...

I am confused with a couple of terms usually used in the context of non-radiative transitions. I believe that I understand the concept of diabatic and adiabatic states described in http://en.wikipedia.org/wiki/Adiabatic_theorem. The basic finding is that the coupling terms in the Hamiltonian...

Here is the question: Here is a link to the question: CALCULUS HELP PLEASE!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement I just stumbled upon an approximation I don't get where comes from Homework Equations F(x+dx) -F(x) = dF/dx dx The Attempt at a Solution My textbook just stated it out of nothing, so I have no idea where to start.

Hi I'm trying to understand how the sum over spin polarizations is treated when using the Narrow Width Approximation(NWA) with a spin 1 resonance. For a spin zero resonance there is no such problem and the factorization is quite straightforward. I'll go through some details to explain where...

Hi, I'm reading about the Born-Oppenheimer approximation for a solid and they're doing the formalism of it. They say that we can basically consider the ions stationary with respect to the electrons because they move so little and so slowly in comparison to them. They say that ##R_i## are the...

Homework Statement Find, by comparison with exact trigonometry, the angle,  (provide a numerical value in degrees), above which the small angle approximation departs from the exact result by more than 1 percent. Homework Equations Approx.: d = s = rθ Exact: d = 2*r*Sin(θ/2) The...

Homework Statement f(x,y) = y' = \frac{y+x^2-2}{x+1} , y(0) = 2 Write the formula for the 2nd order Taylor approximation I just want to ask a question Homework Equations Taylor seriesThe Attempt at a Solution Taylor: y(x) = y(x_0) + y'(x_0)(x-x_0) + \frac{y''(x_0)(x-x_0)^2 }{2} = \\...

Is there a way to "crudely" approximate PDEs with Fourier series? By saying crudely, I meant this way: Assuming I want a crude value for a differential equation using Taylor series; y' = x + y, y(0) = 1 i'd take a = 0 (since initially x = 0), y(a) = 1, y'(x) = x + y; y'(a)...

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...

Hello! I have tried for a whole afternoon to solve this problem but I didn't succeed. Let \cos(2 \pi (f_0 + i/T_N) t + \phi_i) and \cos(2 \pi (f_0 + j/T_N) t + \phi_j) be two quasi-orthogonal functions: \int_{0}^{T_N} \cos(2 \pi (f_0 + i/T_N) t + \phi_i) \cos(2 \pi (f_0 + j/T_N) t + \phi_j) dt...

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...

Homework Statement A friend of mine asked me on the Fresnel approximations earlier, and I couldn't really remember many of the details other than it was an approximation for spherical waves based on the Taylor series. So basically I had to look it up in a textbook. One of the exercises...

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 a,b\in R, a<b, n\in N\\ h=\frac{b-a}{n} , x_i = a+ih , i=0..n \\ f\in C^1[a,b] we approximate the integral of f in a,b with Q_n(f) = h\left[f(x_1) + f(x_1) + ... + f(x_n)\right] Find the error R_n(f) = \int_a^bf(x)dx - Q_n(f), as function of the first derivative of f...

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...