Approximation Definition and 768 Threads

  1. Leo Liu

    How does this approximation work?

    My physics textbook does the approximation in the homework statement. Here, x and y are variables and are much smaller than h. I attempted to figure out why it is valid with ##(1+x)^-1\sim 1-x##. However, after trying to convert the initial equation into 1+x form, I obtained ##h(1-(h+x-y-1))##...
  2. Leo Liu

    B Approximation of the equation of ellipse

    My physics textbook does the approximation that $$r=\frac{r_0}{1-\frac{A}{r_0}\sin\theta}\approx r_0\left( 1+\frac A r_0\sin\theta\right)$$ when ##A/r_0 \ll 1##. Can someone please explain how it is done?
  3. L

    Thermodynamics: gas expansion formula or approximation error?

    FIRST TYPE: REVERSIBLE PROCESS At the temperature of 127 ° C, 1 L of CO2 is reversibly compressed from the pressure of 380 mmHg to that of 1 atm. Calculate the heat and labor exchanged assuming the gas is ideal. Q = L = - 34.95 J CONDUCT 380 mmHg = 0.5 atm L = P1 * V1 * ln (P1 / P2) = 0.5 * 1...
  4. S

    Integral as approximation to summation

    Writing down several terms of the summation and then doing some simplifying, I get: $$\sum_{r=1}^n \frac{1}{n} \left(1+\frac{r}{n} \right)^{-1}= \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...\frac{1}{2n}$$ How to change this into integral form? Thanks
  5. M

    MHB Approximation of the integral using Gauss-Legendre quadrature formula

    Hey! :giggle: Let $\displaystyle{I_n(f)=\sum_{i=0}^na_if(x_i)}$ be a quadrature formula for the approximate calculation of the integral $I(f)=\int_a^bf(x)\, dx$. Show that a polynomial $p$ of degree $2n+2$ exists such that $I_n(p)\neq I(p)$. Calculate the approximation of the integral...
  6. VexCarido

    Runge-Kutta Projectile Approximation From Initial Conditions

    Hi everyone. I'm a new member, great to be here:) I have a few questions that I wanted to ask you guys regarding the method by which we implement the Runge-Kutta approximation of Projectile Motion if we should do it using a numerical iterative method with a Spreadsheet like Excel. I have...
  7. J

    Debye Approximation of Heat Capacity in 1D

    So really i am just unsure how to answer the last part of the question. I am unsure how to apply the low and high temperature limits the way i have done it. Do i set upper/lower limits on the integral and solve? If so i am not sure what to put Here is what he book has for 3d
  8. U

    I What is the approximation used here?

    I'm reading a paper (Beamwidth and directivity of large scanning arrays, R. S. Elliott, Appendix A) in which the author starts from this expression: ##\frac{\sin\left [ \left ( 2N+1 \right ) u_0\right ]}{\sin(u_0)}\sum_{p=-P}^Pa_p\cos(p\pi)\left [\frac{\sin(u_0)}{\sin(u_p)} -1+1 \right ]##...
  9. U

    I Approximation of a function with another function

    Hi, I am wondering if it is possible to demonstrate that: tends to in the limit of both x and y going to infinity. In this case, it is needed to introduce a measure of the error of the approximation, as the integral of the difference between the two functions? Can this be viewed as a norm...
  10. SamRoss

    I How is the weak force related to a change in velocity?

    Hi everyone, The four fundamental forces are gravity (I understand that G.R. does not look upon gravity as a force but I'm not worried about that here), the Lorentz force, the weak force, and the strong force. I'm familiar with the inverse square law for gravitation and the Lorentz force...
  11. L

    I Simplifying limit with Stirling approximation

    I'm trying to determine why $$ \lim_{N \rightarrow +\infty} ln( \frac {N!} {(N-n)! N^n}) = 0$$ N and n are both positive integers, and n is smaller than N. I want to use Stirling's, which becomes exact as N->inf: $$ ln(N!) \approx Nln(N)-N $$ And take it term by term: $$ \lim_{N...
  12. D

    Numerical approximation of the 2nd derivative across a diffuse interface

    Imagine you create a diffuse interface in space and determine which side of the interface you are on by a local scalar value that can be between 0 and 1. We could create a circle, centered in a rectangular ynum-by-xnum grid, with such a diffuse interface with the following MATLAB code: xnum =...
  13. F

    A Cross correlations with 2 probes: Approximation of a 2D + 3D synthesis

    I am interested, in the context of my work, in the cross correlations between a spectroscopic probe (which gives a 3D distribution of galaxies with redshifts, which is also called spectroscopic Galaxy clustering, GCsp) and a photometric probe (which gives an angular distribution, that is to say...
  14. S

    I Understanding different Monte Carlo approximation notations

    Currently working on a project involving Monte Carlo integrals. I haven't had any prior studies of this method, so hence the following question. Consider the following expectation: $$E[f(X)]=\int_A f(x)g(x)dx.$$ Let ##X## be a random variable taking values in ##A\subseteq\mathbb{R}^n##. Let...
  15. K

    How Does Refraction Affect Perceived Fish Size Underwater?

    From This picture, I think the fish will be smaller but the problem is how small will it be? (Fish "L" is the image of fish "K") Let ##H## be the depth of fish "K", ##\theta## be the angle of eyes to y-axis and ##n## is the index of refraction of water.
  16. J

    MHB Linear Approximation: Check Your Answer

    Have I solved this linear approximation question correctly?
  17. A

    I Approximation of a function of two variables

    In a manner analogues to the linearization of functions of a single variable to approximate the value of a function of two variables in the neighbourhood of a given point (x0,y0,z0) where z=f(x,y) using a tangent plane. The tangent plane must pass through the point we wish to approximate z...
  18. J

    Understanding the second approximation of diodes

    My understanding of barrier potential is as follows : Due to the holes and electrons combining in the diode a depletion region is formed. Essentially this region has no mobile charge carriers (as the holes and electrons have combined) and thus the ions (whose holes/ electrons have combined)...
  19. T

    I Dispersive approximation (limit) in the Jaynes-Cummings Model

    I wanted to know what is understood as the dispersive approximation (or limit) in the context of the Jaynes-Cummings model for one mode of the field.
  20. J

    Why does the Euler approximation fail for the Airy or Stokes equation?

    I had thought it would be failure of structural stability since in structural stability qualitative behavior of the trajectories is unaffected by small perturbations, and here, even tiny deviations using ##h## values resulted in huge effects. However, apparently that's not the case, and I'm not...
  21. T

    Adiabatic approximation in the derivation of the speed of sound

    The speed of sound in a gas at temperature T is given to be ## v=\sqrt{\frac{\gamma RT}{M}}##, where ##\gamma## is the adiabatic exponent, R is the gas constant and M is the molar mass of the gas. In deriving this expression, we assumed that the compression and expansion processes were so fast...
  22. H

    I Polynomial approximation of a more complicated function

    There is an arbitrarily complicated function F(x,y,z). I want to find a simpler surface function G(x,y,z) which approximates F(x,y,z) within a region close to the point (x0,y0,z0). Can I write a second-order accurate equation for G if I know F(x0,y0,z0) and can compute the derivatives at the...
  23. Diracobama2181

    Virial Expansion Approximation of of Lennard Jones Potential

    I get $$B_2(T)=2\pi N\int_{0}^{\infty} (1-e^{-\beta E_0((\frac{r_0}{r})^{12}-2(\frac{r_0}{r})^6)})r^2dr$$ as the coefficient. I was just unsure how to evaluate it numerically from here. Any suggestions would be appreciated. Thank you.
  24. Biochemgirl2002

    Approximation to the binomial distrubution

    a) since np has to be greater than 5, n*p= 50*.5 =25 so yes, we can use this since it is much larger than 5. now, for mean, i believe the equation is saying that the mean is np, which is 25 but in this equation we do not have a q value, so this is where my issue begins... what should i use...
  25. V

    Linear approximation and percentage error

    I found the linearization, L(x) = -0.0001x+0.2 and I found L(1/99) = 0.0199989899. Then I tried to put that value into my percentage error formula along with 1/99 and got: the absolute value of (1/99)-L(1/99) and then we divide that by our actual value which is 1/99, then I multiply everything...
  26. dRic2

    When can I use the prompt jump approximation?

    I know that it is only an approximation to get an idea, but at times it works quite well (in class we solved the kinetics equation for a PWR reactor (point-reactor model) with MATLAB and then we plotted the solution along with the prompt jump approximation... It was very good). But I did not...
  27. Wrichik Basu

    B "Slowly varying" potential for WKB approximation

    In order to use WKB approximation, the potential has to be "slowly varying". I learned the method from this video: But the Professor hasn't mentioned in detail what the measure of "slowly varying" is. What is the limit beyond which we cannot use the WKB method accurately?
  28. bob45

    I Why do we use the sin x = x approximation for calculating waves on a rope?

    hi, when we try to find the speed of a wave on a rope v = (F/u)^1/2, we use the fact that if the angles are small then sin x = x. I understand the approximation but not WHY we use the approximation. We say delta(Theta) is small (and then amplitude is small) then ... . So the proof is only...
  29. S

    I Running through a complex math derivation of plasma frequency

    Background of problem comes from Drude model of a metal (not necessary to answer my problem but for the curious): Consider a uniform, time-dependent electric field acting on a metal. It can be shown that the conductivity is $$\sigma = \frac{\sigma_0}{1-i\omega t}$$ where $$\sigma_0 =...
  30. mertcan

    I The Cochran and Cox Approximation Pair T test Unequal Variance

    Hi everyone I hope you are well. Maybe as you know according to Behners-Fisher problem (unequal variance case of samples) there are some kind of approximations. I have recently covered the Satterthweiths Approximations and comprehended the logic of it. But I got stuck with the Cochran-Cox...
  31. QuarkDecay

    Born-Oppenheimer approximation & distance

    According to the Born-Oppenheimer approximation, what does the internuclear distance Req depend on? Atomic number Z? Rotational Energy of the nuclei? Electrons' kinetic energy? Coulomb interaction between the two nuclei? Coulomb interaction between the electrons? Vibrational energy of the...
  32. W

    I 1D scattering: Taylor expansion

    Hi all, I'm having a problem understanding a step in an arxiv paper (https://arxiv.org/pdf/0808.3566.pdf) and would like a bit of help. In equation (29) the authors have $$R = \frac{\sigma}{\sqrt{\pi}} \int dk \ e^{-(k - k_0)^2 \sigma^2} \ \Big( \frac{ k - \kappa}{ k+ \kappa} \Big)^2$$ where...
  33. Boltzman Oscillation

    I Vector math (small angle approximation)

    Given the following vectors: how can i determine that Θ = Δp/p ? I can understand that p + Δp = p' but nothing arrives from this. Any help is welcome!
  34. M

    MHB Least squares method : approximation of a cubic polynomial

    Hey! :o I want to determine an approximation of a cubic polynomial that has at the points $$x_0=-2, \ x_1=-1, \ x_2=0 , \ x_3=3, \ x_4=3.5$$ the values $$y_0=-33, \ y_1=-20, \ y_2=-20.1, \ y_3=-4.3 , \ y_4=32.5$$ using the least squares method. So we are looking for a cubic polynomial $p(x)$...
  35. J

    MHB Approx. in $L^p$: Proving Existence of Subseq & $g$

    Problem: Assume $E$ is a measurable set, $1 \leq p < \infty$, and $f_n \rightarrow f$ in $L^p(E)$. Show that there is a subsequence $(f_{n_k})$ and a function $g \in L^p(E)$ for which $\left| f_{n_k} \right| \leq g$ a.e. on $E$ for all $k$. Proof: Maybe use?: $f_n \rightarrow f$ in $L^p(E)$...
  36. Abhishek11235

    Deriving the small-x approximation for an equation of motion

    Homework Statement The problem is taken from Morin's book on classical mechanics. I found out Lagrangian of motion. Now to solve, we need small angle and small x approximation. The small angle approximation is easy to treat. But how to solve small x approximation i.e how do I apply it...
  37. A

    MHB How to proof stirling approximation

    i want to know about stirling approximation. why lnx! = xlnx - x
  38. opus

    Trapezoidal Approximation Help

    Homework Statement Approximate each integral using the trapezoidal rule using the given number for ##n##. ##\int_1^2 \frac{1}{x}dx## where ##n=4## Homework Equations Trapezoidal Approximation "Rule": Let ##[a,b]## be divided into ##n## subintervals, each of length ##Δx##, with endpoints at...
  39. hilbert2

    I Is there a name for this approximation?

    Because it holds that ##\displaystyle\int_{1}^{x}\frac{dt}{t} = \log x##, and ##\displaystyle\int_{1}^{x}\frac{dt}{t^a} = \frac{1}{a-1}\left(1-\frac{1}{x^{a-1}}\right)\hspace{20pt}##when ##a>1## it could be expected that ##\displaystyle\frac{1}{a-1}\left(1-\frac{1}{x^{a-1}}\right)...
  40. Hiero

    I Can anyone justify this derivation of Stirling’s approximation?

    The famous Stirling’s approximation is ##N! \approx \sqrt{2\pi N}(N/e)^N## which becomes more accurate for larger N. (Although it’s surprisingly accurate for small values!) I have found a nice derivation of the formula, but there is one detail which bothers me. The derivation can be found...
  41. J

    I Legendre polynomials in boosted temperature approximation

    Hi all, In S. Weinberg's book "Cosmology", there is a derivation of the slightly modified temperature of the cosmic microwave background as seen from the Earth moving w.r.t. a frame at rest in the CMB. On Page 131 (1st printing), an approximation (Formula 2.4.7) is given in terms of Legendre...
  42. F

    I A steady-state solution for the flow in the Boussineq approximation in a star

    Hello, I am trying desperately to find the solution indicated in this question : If I compute the equations on the 3 axis, I can't get the flow to be directed along ##\vec{e_y}##. I have only : ##\dfrac{\partial v_{z}}{\partial t} = -\dfrac{1}{\rho_0}\dfrac{\partial \delta P}{\partial...
  43. D

    Approximations with the Finite Square Well

    Homework Statement Consider the standard square well potential $$V(x) = \begin{cases} -V_0 & |x| \leq a \\ 0 & |x| > a \end{cases} $$ With ##V_0 > 0##, and the wavefunctions for an even state $$\psi(x) = \begin{cases} \frac{1}{\sqrt{a}}cos(kx) & |x| \leq a \\...
  44. J

    MHB Estimate Paint for Hemispherical Dome - Linear Approximation

    Hello I have tried to resolve the problem below Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.040000 cm thick to a hemispherical dome with a diameter of 45.000 meters. My procedure was: the volume of the sphere is V=4/3 pi r^3...
  45. L

    Geometric Optics Approximation - validity

    How is the "geometric optics approximation" exactly defined? Given all the source of visible radiation's parameters, all the apparatus, instruments, screen, etc, specifications, how can I know if, e. g. there will be diffraction, interference or other wave properties or if I'll be able to...
  46. T

    MATLAB Numerical approximation of the area under curve

    I am very new too Matlab and how it all works but I am having trouble understanding at what axis the numerical integration is occurring from on the graph that I plotted. So I am currently doing an experiment in gamma ray spectroscopy and due to issue with the software we found it hard to...
  47. P

    MHB How Does Simpson's Rule Estimate the Volume of an Artificial Lake?

    An artificial lake is made up of 5m width and 100m length in dimension. The depth of the lake varies every 20m length as recorded in the following table. Use Simpson's rule approximation to estimate the volume of the water in the lake. Distance (m) 0 20 40 60 80 100 120 Depth (m) 2.0...
  48. A

    I Accuracy of the Normal Approximation to Binomial

    What is the preferred method of measuring how accurate the normal approximation to the binomial distribution is? I know that the rule of thumb is that the expected number of successes and failures should both be >5 for the approximation to be adequate. But what is a useful definition of...
  49. G

    Equation for the resolving power of a microscope?

    Hi I'm reading through a Quantum Mechanics textbook called Quantum Mechanics by Book by Alastair I. M. Rae and in the opening chapter it talks about the Heisenberg uncertainty principle and talks about how a measurement of position of a particle causes an uncertainty from the momentum due to the...
  50. W

    I Understanding Confusing Expansion: Taylor Series Expansion

    I came across the following working in my notes and would like some help understanding how the step was done. Many thanks in advance! The following is the working, and we assume that ##\beta## is small $$\frac{1}{1+ \beta \hbar \omega /2 + (7/12)(\beta \hbar \omega)^2 +...} \approx 1 - (\beta...
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