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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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How would you determine the measurement error in the following example?

I'm at a bit of a loss as to how to determine the error in measurement in a project I'm working on. The project involves taking a picture of an object, and then using the image to determine the width of the object. So, a sample image would be like…
Ryan
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Does it matter which term do I abandon during approximation? And if it is, is there any consistency occurs when doing approximation?

I am trying to solve this problem: Give that $p=(1 + \lambda) e^{-\lambda}$, and $q=1-p$. Show that, when $\lambda$ is sufficiently small: $q=\frac{1}{2}\lambda^2$. So, what I did is $(1 + \lambda) \approx 1$, and $e^{-\lambda} \approx 1 - \lambda +…
Henry Cai
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Why does the approximation of $(1+x^2)^n$ work this way?

I was trying to solve this problem: Show that, for large value of $x$, $\frac{1}{y} = \frac{2 + 3x^2}{3(1+x^2)^\frac{3}{2}} + \frac{1}{3} \approx 3 -\frac{9}{x}$. So, I looked that the mark scheme, it indicated that $(1+x^2)^\frac{3}{2} \approx…
Henry Cai
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What is 0.25801227546?

Due to specific knowledge, I know that this is a constant which has a very nice closed form expression (probably short and in terms of square roots, rationals, and $\pi$). However, the Inverse Symbolic Calculator and I can't find it. What is…
Display name
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show this $\{m\sqrt[n]{a}\}>\frac{c}{m^{n-2}}$

Prove or disprove let $n>4$ be give postive integers,if $a$ is also be give postive integers,such $\sqrt[n]{a}$ be not postive integers,show that: for any postive $m$,there exist constant $c$,such $$\{m\sqrt[n]{a}\}>\dfrac{c}{m^{n-2}}$$ …
wightahtl
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Using approximation to find a value of theta

So I have one vector of alpha, one vector of beta, and I am trying to find a theta for when the sum of all the estimates (for alpha's i to n and beta's i to n) equals 60. Basically what I did is start from theta = 0.0001, and iterate through,…
PascalVKooten
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Approximation of $2^n$ for large $n$

Is there some sort of approximation for $2^n$? I'm specifically interested in how the ratio $\frac{2n}{2^n}$ scales for large $n$ (apart from decreasing to zero in the limit)
user3037237
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How get a good numerical approximation for exponential quantile function?

I want to calculate the "scaled" quantiles for an exponential distribution and I have a function for the inverse CDF, $iCDFExp(p,a,b)$ (e.g. $a=0, b=1$). The argument $p$ is in my application close to $1.0$, like $0.999\dots$, and in this situation,…
user32038
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Approximations with differentials

My book states that for large n and small a, $$\dfrac{1}{(n+a)^2} - \dfrac{1}{(n)^2} \approx -\dfrac{2a}{n^3}$$ Let $f(x) = \dfrac{1}{x^2}$, $$ df = -\dfrac{2}{n^3}dx$$ with $$dx = a$$ and so the above result follows. But the only restriction with…
Ron
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In search for a method to approximate a curve into horizontal segments

In order to model a 24-hour load profile I need to approximate it into smaller number of pieces. Is there any method to approximate such a curve? Would anybody help me to figure out this?
rgholizadehr
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How does WolframAlpha or other software get such precise values for zeta(3), etc.?

I've been looking with a friend at the values of zeta at the odd integers. WolframAlpha can give us over 100 digits in a second or two, but it seems that if you take the sum out to n, say, then you are off by an order of 1/n^2, so getting 100 digits…
user387394
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Inconsistencies when approximating to the nearest whole number

Does 22.449 approximate to 22 or 23? If we see it one way $22.449≈22$ But on the other hand $22.449≈22.45≈22.5≈23$ Which one is correct?
Decepti Brine
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Error term for a rational function approximation to $x^y=y^x$

Define a function $\mathscr E(x)$ on the interval $(e,\infty)$ implicitly via $$y=\mathscr E(x)~\Leftrightarrow~x^y=y^x,\quad x\neq y.$$ Show that $\frac{x+2}{x-1}$ approximates $\mathscr E(x)$ to an accuracy of $<0.05~$ for all $~x>e$. The…
Supware
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Approximation of tan(f(n))

I have the function $f(n)= \pi (n+\frac{1}{2})+\epsilon \frac{2\pi(n+1/2)-a}{2\pi(n+1/2)}$ when $\epsilon<<1$, I can't understand what identity I need to use to prove that: $\tan(f(n))\approx \frac{2\pi(n+1/2)}{\epsilon(2\pi(n+1/2)-a)}$ I tried with…
Daniel Vainshtein
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Find an example to show Greedy TSP algorithm is not a good approximation

In the greedy Traveling Salesman algorithm, the algorithm starts from a starting vertex $v_1 = s$, and in $i$th stage, it goes to the closest vertex to $v_i$ that was not visited yet. I want to find a counter example that the greedy traveling…
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