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Questions tagged [approximation]

For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

Approximations are representations of numbers that aren't exact, for example $\sqrt{2}\approx 1.41$. Such representations may be obtained using differentials (more generally, Taylor's formula), linear interpolation, etc.

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Finding the second-degree polynomial that is the best approximation for cos(x)

So, I need to find the second-degree polynomial that is the best approximation for $f(x) = cos(x)$ in $L^2_w[a, b]$, where $w(x) = e^{-x}$, $a=0$, $b=\infty$. "Best approximation" for f is a function $\hat{\varphi} \in \Phi$ such that: $||f -…
scribu
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Selecting $x_0$ in approximations using Taylor polynomials?

Are there any general rules for how to pick $x_0$ (relative to $x$) in approximations using Taylor polynomials? What if $x=x_0$?
mavavilj
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How is the approximation justified and how to improve it?

In an attempt to find the solution to the equation $Mx=e^x$ with $M$ being a large real number and the solution $w \gt 1$ I was asked to justify why $\ln M$ is a reasonable approximation to $w$. I was also asked to improve the approximation by…
Rescy_
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How exacting must we be for these powers of roots-of-unity?

If, given a root-of-unity approximation $\omega \approx e^{2 \pi i/n}$, and we want to take this approximation to the power $m$, how exacting must we be with the approximation so that: $$\left|\left( e^{2 \pi i/n} \right)^m - \omega^m \right| <…
Matt Groff
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Estimation of $x$ if $x! = N^{\log N}$

If $x! = N^{\log N}\;,$ How can I estimate $x$ in terms of $N$?
alexander
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Physics Approximations Quadratic Equation

I'm having a hard time following one of the solutions to this physics problem. In particular, the math. Consider, $$a\Omega ^2 + b\Omega + c = 0$$ The solutions to this quadratic equation are, $$\Omega = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Consider…
DWade64
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Deriving the approximation formula

$f'(x) \thickapprox$ $\frac{1}{2h} [ 4f(x+h) - 3 f(x) + f(x + 2h)]$ I need to derive the approximation formula for the function above. And I need to show that it's error term is of the form $\frac{1}{3}h^2 f'''(\xi)$ How do I go around doing this?…
user906153
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Approximate the largest and the smallest values of the integral

How do I solve this:approximate the largest and the smallest values of the integral for
Ravishay Prasad
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Determining when the approximation fails

Context Suppose we have a grid-based game where a unit has a range parameter that serves as the upperbound of the sum of the costs of his movements in a single turn. Moving orthogonally to an adjacent tile costs 1, and moving diagonally to an…
Sylin
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Can someone explain what destructive cancellation is as well as how to answer the question?

Let $f(x)= \sqrt{x^2 + 1} - 1$ (taking the positive real square root, as usual). When $a = 10^{−3}$, compute $f(a)$, working to $5$ significant figures at every stage of the calculation. Also it can be shown (algebraically) that $$f(x) =…
Jed
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Please, Let me know this approximation

$$ N_v = 0.5(t^{2}+2t^{6/7})\ln(1+2t^{-8/7})-t^{6/7} \tag{1} $$ $$ N_v =(0.871+0.125\ln t)^2 \tag{2}$$ Eq(2) is the approximated version of Eq(1). Does anyone know how to derive (2) from (1)? I'd appreciate your intuiton.
angbong
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Understanding an approximation equation

In this biology textbook I found the following approximation: $$\frac{1}{2N}\left( 1-\frac{1}{2N} \right)^t ≈ \frac{1}{2N}e^{\frac{-t}{2N}}$$ Can you help me to understand this approximation and help me to understand what assumption are needed for…
Remi.b
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Matching First and Second Derivatives: Taylor Series

I have $$ f(x) = \sqrt{\ln\left(a\cosh^{2}(mx)(1+bx^{2})\right)} $$ If I expand this as a series I should get something of the form $$ \sqrt{\ln a}+gx^{2}+\mathcal{O}(x^{4}) $$ but I'm having real difficulty getting Mathematica to give me the…
apg
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Divide $n$ seats by a list of $\mathbf{w}$ weights proportionally.

I have $n$ number of seats and I have list of weights $\mathbf{w} \in \mathbb{R}^{k}$ which is a probability distribution with $k$ possible values, $\sum_{i=1}^k{w_i}=1$. I want to divide the $n$ seats proportional to the weights as much as…
siamii
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