Approximation Definition and 766 Threads

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 "At the very low temperature, freeze-out occurs; the Boltzmann approximation is no longer valid."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...

Please verify my answer. Homework Statement Use Newton’s method with the specified initial approximation to find , the third approximation to the root of the given equation. (Give your answer to four decimal places.) x^5+2=0, x_1=-1 Homework Equations The Attempt at a...

Approximate 1.58800 - (1.5402 sqrt(-1))

Can anyone tell me about how to use the local density approximation in density functional theory analytically if it possible?

I would like to know the basic experimental observations or the logic which prove that the 3-d space which we inhabit is a close approximation of Euclidean Geometry. is it because parallel lines don't appear to converge or diverge? But how is this established, as we can't draw perfect straight...

Any ideas appreciated!

Is there a way ( a theorem ) to find a rational number for a given irrational number such that it is an approximation to it to the required decimal places of accuracy. For example 22/7 is an approximate for pi for 2 decimal places.

Hey everyone, in doi:10.1016/0375-9601(82)90182-7 I found the following claim: "Any vector of L^2(\mathbb{R}^3) can be arbitrarily well approximated by a finite sum of gaussian vectors." Is this actually true? I lack the insight on how to prove this, but it would be a useful argument I could...

Homework Statement \int^{ \pi}_{0} sin(x)dx \;\;\;\;\;\;\;\; dx=\frac{ \pi}{2} Homework Equations Trapezoidal Approximation: |f''(x)| \leq M \;\;\;\;\; for \;\;\;\;\; a \leq x \leq b \frac {b-a}{12}(M)(dx)^{2} = Error Simpson's Rule: |f^{(4)}(x)| \leq M \;\;\;\;\; for...

Homework Statement \int^{5}_{1} \frac{x}{x+1} dx using dx = .5 Homework Equations \sum^{a}_{b} \frac {f(x)+f(x+dx)}{2} dx = [\frac{1}{2}f(x_{0})+f(x_{1})+\cdots+f(x_{n-1})+\frac{1}{2}f(x_{n})]dx The Attempt at a Solution TAI = Trapazoidal Approximation Input value x_{0}=1 \rightarrow...

problem with integration for WKB approximation in MATLAB hi all, i have been having troubles with getting MATLAB to solve the following problem (the language is not the MATLAB one, the functions are not solvable by the symbolic integration and i was trying to get one of the quad functions to...

Could somebody explain what exactly a "piecewise quadratic approximation" is? Problem Statement Find a piecewise quadratic approximation P(x) of f(x), where f(x)=\sin{4x}\; on \; [0,\pi] Plot f(x) and P(x) on [0,\pi]. What is the maximum value of the following: |f(x)-P(x)| \...

Could somebody explain what exactly a "piecewise quadratic approximation" is? Problem Statement Find a piecewise quadratic approximation P(x) of f(x), where f(x)=\sin{4x}\; on \; [0,\pi] Plot f(x) and P(x) on [0,\pi]. What is the maximum value of the following: |f(x)-P(x)| \; on \;[0,\pi]...

Homework Statement I'm trying to find a root-finding method for a function [tex] f: \mathbb R^n \to \mathbb R [/itex] Homework Equations x is a root of f(x) if f(x) = 0 The Attempt at a Solution There is lots of work done for this problem when n=1, and also lots of work done when...

Sir, Suppose we are asked to find the basic feasible solution for maximizing transportation cost using Vogel approximation method (VAM). We then write the row penalty and column penalty. Suppose there is tie between 2 penalty values, which should be taken first? I have this doubt because I get...

Homework Statement Given f(x)=sqrt(2x+2) Question : Find the linear approximation of f(x) at a=7 AND use it to approximate sqrt(18). Homework Equations L(x)=f(a)+f'(a)(x-a) The Attempt at a Solution Using the linear approximation formula I am getting the value 6.75, but when...

Homework Statement Obtain the Taylor polynomials Tnf(x) as indicated. In each case, it is understood that f(x) is defined for a11 x for which f(x) is meaningful. Problem one Tn = (a^x) = sigma from k = 0 to n of ((log a)^k)/k! x^k Problem two Tn = (1/(1+x)) = sigma from k = o...

I once wanted to find the nth order Fourier approximation for f(x)=x. Since this function is odd, the projections on all cosines will be zero, hence it will be expressed through the sines only. So I just needed to find the sine coefficients. The problem is that I checked the answer to this...

Homework Statement A sample survey interviews an SRS of 267 college women. Suppose (as is roughly true) that 70% of all college women have been on a diet within the past 12 months. Use a Normal approximation to find the probability that 75% or more of the women in the sample have been on a...

So I've been wrestling with something I was reading in a stat mech text. It's the derivation of the partition function for an ideal gas but I imagine the technique is used again. The author starts with the partition function for a single particle but then approximates the sum as an integral...

Homework Statement \Sigma(-1)^{n+1}\frac{1}{n!} How many terms will suffice to get an approximation within 0.0005 of the actual sum? Find that approximation. Homework Equations No idea.The Attempt at a Solution What I tried doing is setting my absolute value of the series less than 0.005, but...

Hello, I am trying to find an interpolating curve between a few points that has minimal curvature. That means, as close to a straight line as possible. Reading a document about cubic splines, they say that \kappa \left ( x \right )=\frac{|f''\left ( x \right )|}{\left ( 1+\left [ f'\left...

Homework Statement Let D={x in the set of real numbers: -3<x<3, x does not equal 0,1,2} and define g(x)=(cosx-1)/x + (x3-2x2-x+2)/(x2-3x+2) on D. Find G:R→R such that G is continuous everywhere and G(x)=g(x) when x is in set D. Homework Equations The Attempt at a Solution From a...

Homework Statement Determine the order two Taylor polynomial, p2(x, y), for f(x, y) = log e (1 + x2 + y4) about point (0, 1) ANSWER: loge (2) + 2y - 2 + \frac{1}{2} [ x2 - 2y2 + 4y - 2 ] Managed that question and should be correct. If not, do let me know =) Part 2: Using...

I was reading Chpater 13 in A&M solid state physics, which is about relaxation time approximation. But there is one fundamental expression I'm trying to understand. It's the formulation of the relaxation-time approximation (Eq. 13.3). dg = dt/\taug0 But most of other textbooks including...

Homework Statement \int_1^\infty \! \frac{(sin(x)+5)}{x^3} \, dx "Find two simpler integrals, one larger and one smaller." Homework Equations The Attempt at a Solution How could I make this a simpler (ie, solvable) integral? It's been straight forward with other integrals like...

In a QM textbook (Newton), I found the below expression for large x: coth(x)\cong 1+2e^{-2x} I tried coth(x)=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=\frac{1+e^{-2x}}{1-e^{-2x}}

Homework Statement find a value of h such that for |x|<h implies sin(x)=x - x^3/6 + x^5/120 + R where |R|<10^(-4) Homework Equations The Attempt at a Solution it's tedious to type out my working but I've got h= (6!/10^4)^1/6 but I'm not sure about this...

Homework Statement A hot fluid is flowing through a thick-walled cylindrical metal tube at a constant temperature of 450C. The cylinder wall has an inner radius of 1 cm and an outer radius of 2 cm and the surrounding temperature is 20C. The temperature distribution u(r) in the metal is defined...

Hi, I have three questions about the application of quadratic approximation, what it is & when to use it. It ties in with a question about linear approximation also, I'll give an example first of what I'm talking about, just for you to evaluate if I'm wrong in the way I see the whole process, I...

I'm looking to evaluate the magnetic field using Jefimenko's equations. There is two parts to it but I'm just looking at the first. The approximation is r>>r' where r' is localized about the origin. The Jefimenko's equation for the magnetic field (the first term that I'm having trouble with)...

u''(x)=f(x), boundary conditions u(a)=0,u(b)=0. (u(x+h)-2u(x)+u(x-h))/h^2=f(x); maltab code: clear all a=0; b=1; n=10; h=(b-a)/(n+1); x_with_boundary=linspace(a,b,n+2)'; x=x_with_boundary(2:n+1); A=h^(-2).*(diag(ones(1,n-1),-1)+diag(-2.*ones(1,n),0)+diag(ones(1,n-1),1))...

Homework Statement Let f(x) = sin x a) find p_6 (taylor polynomial 6th degree) for f at x = 0 b) How accurate is this on the interval [-1,1] Homework Equations The Attempt at a Solution I got p_6 = x + (x^3)/6 + (x^5)/120, which was correct as per the solution manual. My...

clear all; nx=50; ny=30; hx=pi/nx; x=linspace(0,pi,nx+1); y=linspace(0,pi,ny+1); x_plus_h=x+hx.*ones(1,nx+1); x_minus_h=x-hx.*ones(1,nx+1); for i=1:nx+1 for j=1:ny+1 f_xx(j,i)=(f9(x_plus_h(i),y(j))-2*f9+f9(x_minus_h(i),y(j)))./(hx.^2); end; end; [xx,yy]=meshgrid(x,y)...

1. Find the length of the curvature of a concave mirror of 20cm that comply with paraxial approximation for all incident rays 2.conventional geometry formula , sinθ≈θ or tanθ≈θ for paraxial rays 3. I had try drafting out the diagram , labeling all the unknown angle with symbol and...

Is there any physical reason why second order approximation to ground state in time independent perturbation theory is always negative. I know how to prove it mathematicly but I wonder whether one may justify it using only physical arguments.

Hi, I get an a^4 whilst the answer has an a^2. Where am i going wrong? Is the delta function throwing a spanner in my work? see attachments for question/equations and my attempt. Thanks

Homework Statement [U][t]=-U+k[U][xx] u(x,0)=U(L,0)=0 u(x,0)sin(pix/L) Write down difference equations for the approximate solution of this problem using the following methods: 1)forward difference 2)backward difference 3)crank nicholson Homework Equations I can do...

Homework Statement Using the WKB approximation, estimate the lifetime of an electron in the ground state of a 1D quantum well with 10 nm width, surrounded from both sides by 0.3 eV high and 8nm wide barriers. Homework Equations Hint: Estimate the tunneling probability and find the...

I have been looking for a proof of the fact that for a large parameter lambda, the Poisson distribution tends to a Normal distribution. I know the classic proof using the Central Limit Theorem, but I need a simpler one using just limits and the corresponding probability density functions. I was...

I read this in a book (it was stats and about poisson approx to normal) Given was this: n(n-1)(n-2) \cdots (n-r+1) = \frac{n!}{(n-r)!} \approx n^r Stating that "Stirling's approximation" had been used. So I looked the up and found: \ln n! \approx n\ln n - n\ In the poisson...

Effective mass and bands in semiconductors In the study of the basic semiconductor physics devices we usually draw flat bands without taking into account the spatial dependence of them. Now why is it correct? I suppose that the "real band diagram" informations are included into the effective...

I'm trying to figure out what the Hamiltonian for a simple molecule is using the Born-Oppenheimer approximation. 1) My textbook gives the Hamiltonian for a simple system like H2 when you hold the internuclear distance constant. The only terms that drop out are the ones where you take the...

For a class project, I need to make an application that, among other things, can sort students in a class into project teams that takes students' schedules into consideration and groups students with overlapping schedules together. The way I'm storing schedule data is a collection of times...

Homework Statement Given u_tt = F(x,t,u,u_x, u_xx), give the finite difference approximation of the pde (ie using u_x = (u(x + dx; t) - u(x - dx; t))/(2dx) etc.)Homework Equations Well, clearly, u_x = (u(x + dx; t) - u(x - dx; t))/(2dx)The Attempt at a Solution I really have no idea how that...

intermediate-value theorem (approximation) ? Homework Statement Use the Intermediate-Value Theorem to show that there is a right circular cylinder of height h and radius less than r, whose volume is equal to that of a right circular cone of height h and radius r. Homework Equations...