Approximation Definition and 705 Threads

Homework Statement A string of length a is stretched to a height of y when it is attached to the origin so making a triangle with length L=\sqrt{a^{2}+\frac{y^{2}}{a^{2}}} and therefore a length extension ΔL= \sqrt{a^{2}+\frac{y^{2}}{a^{2}}}-a which simplifies to...

Hi group, I'm a theoretical ecologist with fairly adequate training in applied math (ODE, linear algebra, applied probability, some PDEs). In my current work, I've encountered the use of adiabatic approximation to a joint probability distribution of two ever-fluctuating spatial variables. A...

Wiki says: Isn't this exactly what every A/D converter does? For a graph of Vin to digital output it basically approximates the nearest digital value to the continuous signal -> So I don't see the difference between them.

Homework Statement I've attached the questionHomework Equations x(n+1) = x(n) - f(x(n)) / f '(x(n))The Attempt at a Solution okay so x2= 1.3517323300 and I've already calculated x3 to be 1.3483949227 then how do i estimate the error in x2? do i subtract or something?

Homework Statement Prove that any function f(x) can be approximated to any accuracy by a linear combination of sign functions as: f(x) ≈f(x_{0})+ \sum{[f(x_{i+1})-f(x_{i})]} \frac{1+ sgn(x -x_i)}{2} Homework Equations The Attempt at a Solution Looks like taylors theorem...

i have found the area of sphere in two ways using the same approximation.but i get two different answers ;one the correct value 4∏R^2?how does this happen? i'm attaching the solution below?please refer to the attachments and give a solution?

Homework Statement The Type Ia supernova SN 1963p in the galaxy NGC 1084 had an apparent blue magnitude of B = 14.0 at peak brilliance. Then, with an extinction of 0.49 mag to that galaxy, the distance to the supernova is approximately d = 10(m - M - A + 5)/5 = 41.9 Mpc Homework...

Hi, I have to approximate an irrational number x by rationals r = p/q. Let ε>0 in ℝ, then, for almost all x exist α and r in (x-ε,x+ε) such that q ≈ c(x) ε^-α, c(x) in ℝ? I know, from Hurwitz theorem (and a conseguence) that α>2, if exists.

I got quite confused with the math in Thomas-Fermi's approximation. I thought it was supposed to approximate a length but the math from a textbook gives energy instead. I don't understand what is it trying to approximate. My professor told me that normal conductors screen electric field...

Homework Statement This problem will guide you through the steps to obtain a numerical approximation of the eigenvalues, and eigenvectors of A using an example. We will define two sequences of vectors{vk} and {uk} (a) Choose any vector u \in R2 as u0 (b) Once uk has been determined, the...

Diamond has a Debye temperature of Dt = 2000 K and a density of 3500 kg/m3. The distance between nearest neighbors is 0.15 nm. Determine the velocity of sound using the Debye approximation. I have no idea where to even start with this question. Most books don't even mention the Debye...

How do you come up with this approximation? [1+H(t-t_0)- \frac{1}{2}qH^2(t- t_0)^2]^{-1}\approx1+ H(t_0-t)+ \frac{1}{2}qH^2(t-t_0)^2+ H^2(t-t_0)^2 Is there a rule that leads to this approximation?

Homework Statement Approximate the function f(x)=sin(\pi x) on the interval [0,1] with the polynomial ax^{2}+bx+c with finding a, b and c. Homework Equations f(x)=a_{0}+\sum^{\infty}_{n=1}(a_{n}cos(nx)+b_{n}sin(nx)) a_0=\frac{1}{2\pi}\int^{\pi}_{-\pi}f(x)dx...

Homework Statement In a manufacturing process for electrical components, the probability of a finished component being defective is 0.012, independently of all others. Finished components are packed in boxes of 100. A box is acceptable if it contains not more than 1 defective component...

Moderator's note: This thread is a perfect example of what not to do in the homework help forums. It is unacceptable for the opening poster not to work through the problem and to demand answers. It is inappropriate for the helper to give out those answers, or tell the poster exactly what to do...

Hello! I was wondering if the following statement is true for large n: \sum_{i=1}^{n} \ \left( 1 \ - \ \frac{i}{(n+1)} \right) \ \approx \ \lim_{n \rightarrow \inf}\ \sum_{i=1}^{n} \ \left( 1 \ - \ \frac{i}{(n+1)} \right) \left( \frac{1}{(n+1)} \right) Firstly, the RHS is an integral...

I've been reading the book "why chemical reactions happen", and according to my understanding, it seems as though orbital hybridization is just an "approximation" and not real, as in there is no such orbital, while MO are (real). Is my understanding correct? Or are MOs also just approximations...

Hello, I am reading a paper, and the author claimed that in asymptotic sense as M goes to infinite: \sum_{i=1}^M\sum_{l=0}^L|h_i(l)|^2=M where: \sum_{l=0}^L\mathbb{E}\left\{|h_i(l)|^2\right\}=1. How is that asymptotic follows? Thanks in advance

Find the linear approximation to the equation f(x,y) = 3 sqrt((x y)/4) at the point (2,8,6), and use it to approximate f(2.28,8.22). I know you take the derivative of fx(x,y) and fy(x,y), I think I'm taking the derivative wrong. Then after that you put x and y in the equation and solve for...

Homework Statement So in my biology class, my professor wants us to use the Nernst equation without using calculators. I personally think this is stupid. However, I have no choice, so today, I tried coming up with approximations of the log function. Homework Equations We start with loga(b) =...

I've asked this question before. But still I got some unanswered ones. I am really tired, but I cannot sleep if I got something laying there, tingling me. http://pokit.org/get/cfb750b79f49cfc12dc51a74a37f576e.jpg This is digital ramp. In attachments I added a full circuit, from...

Homework Statement Hi! I have a couple of problems on Taylor Series Approximation. For the following equations, write out the second-order Taylor‐series approximation. Let x*=1 and, for x=2, calculate the true value of the function and the approximate value given by the Taylor series...

y(t) = y0 + \int^{ t}_{t_0} f(s, y(s)) ds. Picard’s method starts with the definition of what it means to be a solution: if you guess that a function φ(t) is a solution, then you can check your guess by substituting it into the right-hand side of equation (2) and comparing it to the...

Homework Statement A particle is in a region with the potential V(x) = κ(x2-l2)2 What is the approximate ground state energy approximation for small oscillations about the location of the potential's stable equilibrium? Homework Equations ground state harmonic oscillator ~ AeC*x2...

This isn't a coursework problem. I'm on winter break. Homework Statement A common approximation used in physics is: (1+x)n ≈ 1+nx for small x This implies that lim(x→0) (1+x)n = lim(x→0) 1+nx which is a true statement. However, lim(x→0) (1+x)n = lim(x→0) [(1+x)1/x]xn = lim(x→0) exn This...

Homework Statement The interatomic force in the chlorine molecule Cl2 may be represented by the Lennard-Jones potential: with e = 1.79.10-19 J and r0 = 0.2 nm. (i) Find the interatomic distance at which V(r) is minimized. What is the interatomic force at this separation? (ii) Calculate the...

Homework Statement Suppose the point (pi/3, pi/4) is on the curve sinx/x + siny/y = C, where C is a constant. Use the tangent line approximation to find the y-coordinate of the point on the curve with x-coordinate pi/3 + pi/180. Homework Equations TLA: f(a) + f'(a)(x-a) Where a is...

Homework Statement I need to find the approximation to: X = m_N\>\bigg[\frac{m^2+\mu^2}{m_N^2 - (m^2+\mu^2)}\>\mathrm{ln} \bigg(\frac{m_N^2}{m^2+\mu^2} \bigg) - \frac{m^2-\mu^2}{m_N^2 - (m^2-\mu^2)}\>\mathrm{ln} \bigg(\frac{m_N^2}{m^2-\mu^2} \bigg) \bigg] for m^2 \gg \mu^2 ...

Hello, In my QM class we're using power series which don't converge but apparently still give a good approximation if one only takes the lower-order terms. Is there any way to understand such a phenomenon? Is it a genuine area of mathematics? Or is it impossible to say something general on...

When we setup the Schrodinger equation for the hydrogen atom we use the classical electric potential. It seems that we would need a new potential instead of mixing QM and classical. Is this just a very good approximation or is there something subtle.

Homework Statement Use Stirling's Approximation formula to evaluate the following: 1) Γ(12.3) where Γ(N) is the gamma function 2) (π^2)! This is suppose to be pi squared 3) (1/2)! Homework Equations So here is Stirling's approximation formula: N! ~= sqrt(2πN)(N/e)^N (where...

Homework Statement This was a test question I just had, and I'm fairly certain I got it wrong. I'm confused as to what I did wrong, though. We were told that our potential was infinite when x<0, and Cx where x>0. We were asked to approximate the ground state potential using the...

I have been studying for the GRE and taking note of various approximations to use on the exam, but I am having a difficult time finding a way to evaluate the following without the aid of a calculator e^{-x}. The GRE practice book has a problem to which the answer is e^{-10} = 4.5 \times...

Hello there if anyone could help me out with this question I would very much appreciate it. (Note this is my first post) For my optics lab Fraunhofer(single slit) diffraction patterns were observed(measured distance to screen(x) and had readings for intensity in the y direction.) I...

For #4, I'm mostly confident I did it correctly. In determining the error, we're supposed to find the maximum absolute value on an interval I. I set I = (0,2pi). Is that right? http://i111.photobucket.com/albums/n149/camarolt4z28/4-1.png For #5...

Homework Statement 68% of students favor a new policy, we are interested in the proportion of 45 students in a math class who favor the same policy. What is the probability that the proportion of the class that favors the policy is no more than two-thirds? Homework Equations The...

I'm attempting to perform interpolation in 3 dimensions and have a question that hopefully someone can answer. The derivative approximation is simple in a single direction: df/dx(i,j,k)= [f(i+1,j,k) - f(i-1,j,k)] / 2 And I know that in the second order: d2f/dxdy(i,j,k)= [f(i+1,j+1,k)...

Problem - Find backward finite difference approximations to first, second and third order derivatives to error of order h^3 Attempt By Tailor’s series expansion f(x-h) = f(x) - h f’(x) + h^2/2! f’’(x) - h^3/3! f’’’(x) + … Therefore, f’(x) with error of order h^3 is given by f(x-h) = f(x)...

Homework Statement (a) The quadratic Chebyshev approximation of a function on [-1, 1] can be obtained by finding the coefficients of an arbitrary quadratic y = ax^2 + bx + c which fit the function exactly at the points (-sqrt(3)/2), 0, (sqrt(3)/2). Find the quadratic Chebyshev approximation of...

Homework Statement Let f(x,y) = (xe^y)^8 i) Find \frac{∂f}{∂x} \frac{∂f}{∂y} \frac{∂^2f}{∂x^2} ii) Using a tangent plane of f(x,y) find an approximate value of (0.98e^0.01)^8 Homework Equations The Attempt at a Solution i) \frac{∂f}{∂x} = 8e^{8y}x^{7}...

Homework Statement A H2 molecule can be approximated by a simple harmonic oscillator having spring constant k = 1.1*10^3 N/m. Find a() the energy levels, and (b) the possible wavelengths of photons emitted when the H2 molecule decays from the third excited state eventually to the ground...

Homework Statement "Derive a more accurate approximation for the heat capacity at high temperatures, by keeping terms through x^{3} in the expansions of the exponentials and then carefully expanding the denominator and multiplying everything out. Throw away terms that will be smaller than...

Hello everyone! I have been on this website for quite a while, and found some interesting answers to many questions, and I decided to create an account to seek you help with a particular issue I encountered in my assignment. Please have a read, and thank you for any input! Homework...

Homework Statement This is a problem form my calculus book, which states: According to the ideal gas law, the pressure, temperature, and volume of a confined gas are related by P=kT/V, where K is a constant. Use differentials to approximate the percentage change in pressure if the temperature...

Is there a name for these types of derivative approximation? I've encountered several different derivative approximation formulas, all of which work. For this series of formulas, the variable "a" denotes the nesting depth (a refers only to the nested part of the equation, not the initial x -...

While discussing Taylor's theorem, my professor pointed out that for n=2, Taylor's Theorem says: f(x) = f(x_{0}) + f'(x_{0})(x - x_{0}) + O(|x - x_{0}|^{2}) He then emphasized that O(|x - x_{0}|^{2}) is a much better approximation than o(|x - x_{0}|). But how is O(|x - x_{0}|^{2}) a...

I have a formula to be solved for x that is y=x^3-x I have a solution given x=((27y^2-4)^.5/23^2/3+y/2)^1/3 + 1/3((27y^2-4)^.5/23^2/3+y/2)^1/3 which seems to work Is this solution an approxiamation?

Homework Statement I just have a question regarding "estimating" the value of a function at a given point. Say we have a function f(x, y, z, w) and we want to know the value of that function at f(1.99, 1.001, 8.02, 2.01) Can we simply do the following: let f=f(2, 1, 8, 2) and...

MFA \hat{A}\hat{B}\approx \hat{A}\langle\hat{B}\rangle+\hat{B}\langle\hat{A}\rangle-\langle\hat{A}\rangle\langle\hat{B}\rangle What this mean physically? What we neglect here? If I calculate Neel temperature using this method T_N^{MFA} and using RPA method T_N^{RPA} is there some relation...

Homework Statement Hi every body I am triyng to find a polynolial approximation to the function: f(x)= (x+2)ln(x+2) using the chebyshev polynomials, the idea is to use MATLAB to find the coeefficients of the approximation poly. using the comand double(int(...)) but this command...