Approximation Definition and 768 Threads

Homework Statement Homework Equations The Attempt at a Solution I think this problem is probably a lot simpler than I am making it out to be. However, when I apply sterling's approx., I get a very nasty equation that does not simplify easily. One of the biggest problems I have though is...

Homework Statement A point particle of mass m slides without friction within a hoop of radius R and mass M. The hoop is free to roll without slipping along a horizontal surface. What is the frequency of small oscillations of the point mass, when it is close to the bottom of the hoop...

Obviously ##\mathbb{R^2}## is convex, that is, any points ##a,b\in\mathbb{R^2}## can be connected with a line segment. In addition, ##f## is differentiable as a composition of two differentiable functions. Thus, the conditions of mean value theorem for vector functions are satisfied. By applying...

I am solving the simple 2nd-order wave equation: $$ \frac {\partial ^2 E}{\partial t^2} = c^2 \frac {\partial ^2 E}{\partial z^2} $$ Over a domain of (in SI units): ## z = [0,L=10]##m, ##t = [0,t_{max} = 10]##s and boundary/initial conditions: $$ E(z=0) = E(z=L) = 0 $$ $$ E(t=0) =...

Homework Statement An object of mass m is initially at rest and is subject to a time-dependent force given by F = kte^(-λt), where k and λ are constants. a) Find v(t) and x(t). b) Show for small t that v = 1/2 *k/m t^2 and x = 1/6 *k/m t^3. c) Find the object’s terminal velocity. Homework...

Hey there, I have two questions - the first is about an approximation of a central gravitational force on a particle (of small mass) based on special relativity, and the second is about the legitimacy of a Lagrangian I'm using to calculate the motion of a particle in the Schwarzschild metric...

Homework Statement .[/B] The full range input of a12-bit, successive-approximation type ADC is 1 volt. Determine: a) the maximum input change required to give a one bit change in output of the ADCb) The number of approximations made to complete the conversion of an input signal of 0.8125...

You have an infinitesimally small mass in the center of octahedron. Mass is connected with 6 different springs (k_1, k_2, ... k_6) to corners of octahedron. Equilibrium position is in the center, you don't take into account gravity, only springs. Find normal modes and frequencies. Relevant...

Is Differentiation exact or just an approximation? I am wonder whether this question is meaningful or not. Slope is expressed as "it is approaching to a value as x is approaching 0" so it is inappropriate to ask such question. But when I deal with uniform circular motion, it is very confusing...

I studied Taylor series but I would like to have an answer to a doubt that I have. Suppose I have ##f(x)=e^{-x}##. Sometimes I've heard things like: "the exponential curve can be locally approximated by a line, furthermore in this particular region it is not very sharp so the approximation is...

I am examining the polynomial approximation for $e^x$ near $x = 2$. From Taylor's theorem: $$e^x = \sum_{n = 0}^{\infty} \frac{e^2}{n!} (x - 2)^n + \frac{e^z}{(N + 1)! } (x - 2)^{N - 1}$$ Now, I don't get the next part: We need to keep $\left| (x - 2)^{N + 1} \right|$ in check so we can...

Hello, I would like to ask, if somebody knows anything about comparison post-Newtonian approximation of gravitational waves and these which were detected. Or generally post-Newtonian predictions vs. facts found in detection. I tried find some article but I didn't find. Please let me know what...

Homework Statement Suppose that the number of asbestos particles in a sam- ple of 1 squared centimeter of dust is a Poisson random variable with a mean of 1000. What is the probability that 10 squared cen- timeters of dust contains more than 10,000 particles? Homework Equations E(aX+b) =...

Hello, Can someone explain this to me? In the above case ct=yt-gt I tried to solve it as a three variable taylor approximation but got a few extra terms that weren't included in the above. So I am a little confused now. I only need to understand how the first line was derived because I get...

I am calculating the strain in a cantilever beam with a point load for a given deflection. The deflection is around .5mm for a beam that is just over 5mm long (width is 3.8mm and height is 0.15mm). I was told for the assumption of small deflections to be valid, deflection should be 2% of the...

Hello, could anyone please explain me some logic or derivation behind the approximation: Found it in the Hull Derivatives book without further explanation. Thanks for help

To solve the full many-body electron problem one often uses the approximation that the dynamics of the electron system, is that of N interacting electrons living in a periodic lattice of positive ion cores. What justifies this approximation and does it have a particular name?

Homework Statement Homework Equations ΔE = hf The Attempt at a Solutionf = ΔE/h = ##k\left [ \frac{1}{(n-1)^2} - \frac{1}{n^2} \right ]## = ## \frac{k(2n-1)}{n^2(n-1)^2}## = ## \frac{k}{n^4}(2n-1)(1-\frac{1}{n})^{-2}## = ## \frac{2kn}{n^4}(1-\frac{1}{2n})(1+\frac{2}{n} +## higher powers...

Homework Statement For the magnetic circuit: Derive the circuit approximation. Compute all magnetic fluxes if the total solenoid current is I. Homework Equations Rm = L / μS The Attempt at a Solution [/B] Mostly, right now, I'm just trying to determine the magnetic circuit equivalent. From...

using binomial theorem can I write sqrt(1+(d^2)/(x^2)) = 1+ .5(d^2)/(x^2)? d is a variable. X known constant.

Homework Statement In regards to linearization of a nonlinear system in differential equations. What does it mean for a linear approximation to be reliable to describe the long term behavior of the non-linear system around the equilibrium point? Homework Equations jacobian matrix The Attempt...

Ever since the discovery of Pi, Mathematicians have been obsessed with finding methods of approximating Pi. I think I've a unique way of doing so via the Newton-Raphson. Newton-Raphson Formula: Let ## ƒ(x)=Sin(x) ⇒ ƒ'(x)=Cos(x) ⇒ X_n= X_{n-1} - tan(X_{n-1})## For example: Let ##X_0=X ⇒ X_3=...

X = # of cars that pass in one hour E(X) = λ = n * p λ cars/1hour = 60min/hour * (λ/60) cars/min In this old video (5:09) on poisson process Sal asks: "What if more than one car passes in a minute?" "We call it a success if one car passes in one minute, but even if 5 cars pass, it counts as 1...

Homework Statement Good day all! I'm studying for finals and i'd like to know how to do this problem (its not homework): "Using the WKB method, find the bound state energies E_n of a particle of mass m in a V-shaped potential well: V(x)= \begin{Bmatrix} -V_0 (1- \begin{vmatrix}...

I'm struggling about finding a way to find the upper bound of the error of Taylor polynomial approximation. I will explain better using a solved example I found... > $f: ]-3;+\infty[ \rightarrow \mathbb{R} $ $f(x)=ln(x+3) +1 $ >Find the upper bound of the error approximating the function in...

I have noticed that in a lot of theoretical modelling of semiconductors you assume that the electrons living in the bottom of the conduction band obey a free particle Hamiltonian: H = p^2/2m* , where m* is the effective mass in the conduction band and p^2 is the usual differential operator. I...

Homework Statement Consider a D dimensional Ising model with N sites, defined by the Hamiltonian $$\mathcal H = -J \sum_{\langle i j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i$$ where the sum extends over nearest neighbours and each spin variable ##\sigma_i = \pm 1##. For a given spin...

Hi. I can't for the life of me understand the math behind the SVEA. I graphically/intuitively understand what it means that the envelope varies slowly, but I can't connect that with the mathematical expression: $$ \left \vert \frac{\partial ^2 E_0}{\partial t ^2} \right \vert << \left \vert...

Homework Statement Find the least squares approximation of cos^3(x) by a combination of sin(x) and cos(x) over the interval (0, 2pi) Homework EquationsThe Attempt at a Solution I know how to find a least squares approximation with vectors, but I don't even know how to start with a function...

Any good book for starting wkb approximation except Griffth... Please suggest some...

I am across this equation with no known analytical solutions: x^5 - x + 1 = 0. I asked this before and the answer was that you can get as good an approximation you want by approximation methods. It is possible that when applying approximation methods, you will get singularities. These...

Homework Statement Suppose that T4 is used to approximate the ∫ from 0 to 3 of f(x)dx, where -2 ≤ f ''(x) ≤ 1 for all x. What is the maximum error in the approximation? Homework Equations |ET|≤ (K(b-a)^3)/(12n^2) The Attempt at a Solution So I know how to find the error of the trapezoidal...

1. The question is. Show that if |nx| <1, the following is exact up to (and including) the x^2 order. The hint giving says to use the Taylor Expansion for both sides of the equation2. (1+x)^n = e^n(x-(1/2)x^2) ; the n(x-(1/2)x^2) is all an exponent3. My first attempt was to take the taylor...

I am working with the general Hamiltonian for an electron gas of density ρ(r): H = -ħ2/2m∑∂2/∂xi2 + 1/4πε ∫dr∫dr' ρ(r)ρ(r')/lx-x'l I wonna do a Hartree Fock approximation on this Hamiltonian. How does that work in general?

Homework Statement A wave with an initial profile ##U(z)=Ae^{ik_0z}e^{-\frac{z^2}{2\sigma^2}}## is traveling in the z direction (yes, the Gaussian profile and the optical axis are not perpendicular). It then passes through a prism with apex angle ##\alpha## and refractive index...

For some small parameter ##\epsilon##, how would one go about making an approximation such as ##\sqrt{k^2-\epsilon^2}\approx k-\frac{\epsilon^2}{2k}##? I was thinking that these types of approximations came from truncating Taylor series expansions, but I can't see how it would be obvious which...

Hello! I'm just trying to make sure I understand Maxwell, Biot-Savart, Faraday, etc well enough. What I'm wondering about is when the quasistatic treatment is indeed approximate, and when it is, assuming an equal if time-varying current, correct. Suppose, first, that we have a steadily...

I am studying in IGCSE and I learned simple techniques to find, say, approximate change in Area of a circle for a small change in its radius, making use of : δy/ δx ≈ dy/dx . or δA/ δr ≈ dA/dr . or δA ≈ dA/dr x δr so what I basically have to do is find the derivative of A ( πr​2 ) , which...

Hi all, I am reading a book on spacecraft engineering in the section about trajectory dynamics. They define linear and angular momentum as: ##I = \int_{0}^{\tau}{F}dt## (Linear Momentum) ##L = \int_{0}^{\tau}{T}dt## (Angular Momentum) But they (and so many other sources) always mention the...

Hi all, hopefully this is in the correct section here. Any help is really gratefully received. 1. Homework Statement I have a coursework, one question asks us to use a 2nd order approximation of the transfer function to..."estimate the settling time (5% of the settling value of output, peak...

The following comes from Griffiths Intro. to QM (2nd Ed) page 53. We want to solve the Schrödinger Equation for the harmonic oscillator case using a power series method. The details aren't important but you want to solve ##h''(y)-2yh'(y)+(K-1)h=0## whose recursion formula is...

Consider the Dirac equation for bounded electron in hydrogen atom. I am trying to get a clear physical explanation for all mathematical terms that appear in the Hamiltonian and energy spectrum. Kinetic and Coulombic potential and rest energies are the first terms and easy to identify. Then we...

Homework Statement There are ##N = 10000## clients of an insurance company. One-half of them will file claims with probability ##p_1 = .05##, another half of them will file claims with probability ##p_2 = .03##. Each claim is worth ##$1000##. Find the Value-at-Risk at the level α = 0.99, that...

Good evening. Is there a way to take a decimal approximation and see if there is a relatively simple expression? I'm guessing there might be software for this, but I'm not sure I'm even asking the appropriate question. If it matters, the number I'm after is...

Hi I'm having trouble visualizing why in a function such as 1/(1-x2) linear approximation of 1/(1-u) where u = x2 is the same as quadratic approximation of 1/(1-x2) The linear approximation is 1+u or 1+x2 Quadratic approximation is the same, 1+x2 Can someone explain to me why this happens...

Consider E>V(x). WKB states the wavefunction will remain sinusoidal with a slow variation of wavelength $ \lambda $ and amplitude given that V(x) varies slowly. From the equation \begin{equation} k(x)=\frac{\sqrt{2m(E-V(x))}}{\hbar} \end{equation}, I can see that the k(x) is directly...

Hello, I don't understand the following. I have this function: V(x,y,z)=\frac{(A+B+C)r^2-3(Ax^2+By^2+Cz^2)}{r^5} with r=\sqrt{x^2+y^2+z^2} and on the textbook they say that if x,y,z are approximately equal or comparable as order of magnitude to r, and if they are all "big" enough (they are...

Homework Statement Find linear approximation of the surface z2 = xy + y + 3 at the point (0,6, -3) and use it to approximate f(-0.01, 6.01, -2.98) Homework EquationsThe Attempt at a Solution [/B] so this means the total surface has decrease by -0.17?

Hi there, in my notes for Heun's method for solving an ODE, I have y(new) = y(old) + 0.5(k1 + k2)Δh And k1 is supposed to be f(y(old)) while k2 is f(y(old) + q11k1Δh) and q11 is 1 So if for example I have a simple differential equation like du/dt = au It would be du/dt = 0.5(k1 + k2) du/dt...

Hi, I've attached an image of an equation I came across, and the text describes this as an approximation to the second derivative. Everything seems to be exact to me (i.e. not an approximation) if the limit of h was taken to 0. Is that the only reason why it's said to be an approximation or is...