Mathematics Stack Exchange
Mathematics Stack Exchange

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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Approximation in $L^2$ by piecewise constant functions

I would like to know if there is any general result on the approximation of $L^2$ functions by piecewise constant functions. More specifically, I'd like to know if the following approximability property is correct for all $w\in V$,…
Sebastien
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Root calculation by hand (division-like algorithm)

I remember from my highschool days a division-like algorithm for calculating square, even cubic roots. I know the continued fraction method, some series and Newton's method. I have checked similar questions in this site with no success. Do anybody…
dot dot
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Approximation by Taylor polynomial

Let $f(x) = (1 − x)^{-1}$ and $x_{0} = 0.$ (a) Find the nth Taylor polynomial $P_{n}(x)$ for $f(x)$ about $x_{0}$. (b) Find the smallest value of $n$ necessary for $P_{n}(x)$ to approximate $f(x)$ to within $10^{-6}$ on $[0; 0.5]$. For part (a) I…
drawar
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Bilinear Transformation

I'm studying about digital control systems and I was given this transformation $$s=\frac{2}{T}\frac{z-1}{z+1}$$ But I don't like just accepting things so I looked it up to find the proof and I found this on wikipedia I'm aware of the taylor and…
John Katsantas
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Atomic weights: rational or irrational?

I teach a Theory of Knowledge course in the I.B. curriculum when I'm not teaching Algebra. I presented this thought experiment to a class: If God, or some supernatural entity, could tell you the exact weight of your school mathematics textbook,…
Samuel Williams
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trying to approximately compute $0.93^{2.98}$ without using a calculator

I tried to do that using this method : let $f(x,y) = x^y$ so now I'm looking for $f(0.93,2.98)$ $\Delta f = f(1,3)-f(0.93,2.98) \approx df = f_xdx+f_ydy$ here $x=1,y=3,dx=-0.07,dy=-0.02$ $f_x=yx^{y-1},f_y=x^y\ln x,f(1,3)=1$ so $$df =…
the_firehawk
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Rounding to fraction

If I write the number 0.333333, you will immediately think to $1/3$. Somehow 0.333333 has been "rounded" to $1/3$. I wonder if there is a general way to find such fraction. So let's formulate the problem in a precise way. I suppose here that all…
Olivier Esser
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Approximation for $2^r\ln \frac{2^r}{2^r-r}$

I know the function $$2^r\ln \frac{2^r}{2^r-r}$$ is about linear in $r$, but I need an argument that an undergraduate could follow. Is there a simple way to explain this? I'd be happy with a simple upper-bound.
Fixee
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Sequence of smooth functions approximating a 2d cylinder step function

Let $f(x,y)=1$ if $(x,y)$ is in the unit disk and $f=0$ otherwise. I would like to approximate $f$ by a sequence of smooth functions. The functions need to be evaluated quickly so the results I'm getting from mollification are to awkward to deal…
alext87
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Finding the cubic near minimax approximation for sin(x) on (0,pi/2)

I am really stuck here. Here is the question that I have. Find the cubic near minimax approximation for $f(x)=\sin(x)$ on $(0,\pi/2)$. So I defined $h(x)=ax^3 + bx^2 + cx + d - sin(x)$ The max of this occurs at $dh/dx=0$, so that…
kingneptune117
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Approximating $\ln{\frac{x-y}{x+y}}$

For $x>>y$, $$\ln{\frac{x-y}{x+y}} = \ln{\left[ x\left( 1-y/x \right) \right]} -\ln{\left[ x\left( 1+y/x \right) \right]} \approx -2\frac{y}{x}$$ However, the following does not work: $$\ln{\frac{x-y}{x+y}} \approx \ln{\frac{x-y}{x}} = \ln{\left(…
ChickenGod
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How to fit a 3-D parametric equation to datapoints

Consider that I have $3$ parametric equations as function of time and describe the motion of a body in space: $x = f(t)$ $y = g(t)$ $z = h(t)$ These curves are pretty simple and can be modeled within a certain interval by a 2nd order polynomial. Is…
Padu Merloti
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How to do this approximation?

Question: Of the following, which is the best approximation of $$\sqrt{1.5}(266)^{3/2}$$ $$(A)~1,000~~~~(B)~2,700~~~~(C)~3,200~~~~(D)~4,100~~~~(E)~5,300$$ I used $1.5\approx1.44=1.2^2$ and $266\approx256=16^2$. Therefore the approximation by me is…
xzhu
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How to approximate unknown two-dimensional function?

I have a surface defined by values on a two-dimensional grid. I would like to find an approximate function which would give me a value for any arbitrary point within certain range of xs and ys. My general idea is to construct some sort of…
Vilius Normantas
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How do I calculate ln(12) approximate value to two decimals?

(edit) The problem I've been assigned literally says just to "Calculate ln(12) without a calculator to two decimals using ln(1+x) series." I've been trying to figure out what series I'm dealing with here and don't seem to find an answer anywhere.
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