Approximation Definition and 705 Threads

Homework Statement Show that if f is continuously differentiable on [a, b], then there is a sequence of polynomials pn converging uniformly to f such that p'n converge uniformly to f' as well.Homework Equations The Attempt at a Solution Let pn(t) = cn t^n Use uniform convergence and integrate...

Homework Statement stimate Δf using the Linear Approximation and use a calculator to compute both the error and the percentage error. f(x) =1/(1+x^2) , a = 3, Δx = 0.5 Homework Equations f'(a)(x) percentage error= abs(error) divided by actual value The Attempt at a Solution So...

Homework Statement If the price of a bus pass from Albuquerque to Los Alamos is set at x dollars, a bus company takes in a monthly revenue of R(x) = 1.5x − 0.01x2 (in thousands of dollars). Suppose that x = 80. How will revenue be affected by a small increase in price? Explain using the...

On a production line, only 45% of items produced meet quality standards. A random sample of 500 items will be taken. Using the normal approximation to the binomial distribution, approximate the probability that less than half of the sampled items meet quality standards. 500*.5 = 250...

Can you take the sin terms out of Snell's Law when dealing with angles below 10 degrees? so Snell's Law would be become n1\theta1=n2\theta2 Thanks

Homework Statement There is a derivation in the text that I'm having problems replicating. The text gives the formula for tidal potential as: U_{tid}=-GM_{m}m(\frac{1}{d}-\frac{x}{d^{2}_{0}}) Where M_{m} is the mass of the moon, d is the distance from the CM of the moon to the point of...

Homework Statement I have to program a three component decay chain using finite difference approximation. I understand finite difference and have written my code, but I have an error I can not find which is giving me an erroneous answer. The curve is correct, but the magnitude of the...

For certain computations I need a quick approximation of the left singular vector of a matrix G( nxk ; n>k ). Also, the corresponding singular value would be needed. Perhaps after approximating the singular value I could use the Conjugate Gradient method to obtain the approximation of the left...

So my book merely mentions that this holds as a result of the central limit theorem for values of lambda greater than 10, but ideally greater than 32. Anyway I was wondering if anyone knew this actual proof as I am interested in seeing it step by step and I could not have found it anywhere...

Homework Statement I'm reffering to http://hyperphysics.phy-astr.gsu.edu/hbase/electric/dipole.html and the approximation r_1 - r_2 \approx = d \cos \theta. I see that it is correct if I draw it up, but I wondered if there were any "more mathematical" ways to see this? Where does these...

Hi! I have a question about approximation of functions with small angles. I was looking through some notes from my teacher and didnt understand why the following approximation is valid. We have a system which is at equlibrium at an angle, say a. Now we wanted to se what happens with the...

Homework Statement In an inertial navigation system on switching ON an intial error of 0.0001m/s^-2 exists in one direction. After 30 minutes an additional error of -0.00005m/s^-2 in acceleration builds up. The system navigates for 2hrs 30 minutes after switching ON. compute the error in...

Homework Statement Well, this problem is quiet similar to the one I've posted before. It asks me to approximate to the e number using taylors polynomial, but in this case tells me that the error must be shorter than 0.0005 Homework Equations...

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...

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]...

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...

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...

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...

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...