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).

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24 views

Maxima of $f(x)/e^x$ where $f(x)$ is an approximation of $e^x$ using Stirling's

Let $$f(x)=1+\sum_{n=1}^\infty\frac{x^n}{\sqrt{2\pi n}(n/e)^n}\tag1$$ and let $$g(x)=\frac{f(x)}{e^x}\tag2$$ If we plot $g(x)$ we get a graph that looks like this: Clearly there is a maximum at ...
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0answers
14 views

(Conceptual_Calculus) Differential Conditions v. Derivative Conditions

I have few questions regarding the reason we learn about the differential conditions in higher dimensions and dealing with multivariable calculus. In the context of optimization (e.g. finding ...
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0answers
15 views

How can we use the Lindley's method to approximate the following expression?

The Lindley's(1980) approximation is one of the most popular methods that is used to obtain Bayes estimates. In this method we need to maximum likelihood estimators(MLEs) of the unknown parameters. ...
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1answer
20 views

Iteration: Approximation and Errors, finding all possible iterative arrangements

I am looking at a relatively simple problem to reiterate: $x^4=e^x$ I've found 5 different possible forms 1: $x_{r+1}=\frac{e^x}{x^3}$ 2: $x_{r+1}=(\frac{e^x}{x^2})^{0.5}$ 3: $x_{r+1}=(\frac{e^x}...
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14 views

Approximating $\chi_0$ pointwise with holomorphic functions

Define $\chi_0:\mathbb{C}\to \mathbb{C}$ as $$ \chi_0(z)=\left\{ \begin{gathered} 1 \quad z=0\hfill \\ 0 \quad z\ne 0 \hfill \\ \end{gathered} \right.$$ Does there exist a sequence of ...
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1answer
38 views

(Terminology_Taylor Series) “expand at $x_0$, evaluate at x, affine approximation”

I am reading one-variable calculus book where it explains Taylor series and little confused with the following terms: (1) Expand $f(x)$ at $x_0$ (2) Evaluate $f(x)$ at x (3) Best Affine, ...
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6answers
3k views

How to show this formula to get a square root of a number in “just few seconds” is true?

I don't remember in which topic I found it but I know it was there. And I still have not find a proof of this nice approximation. Let $x$ be a non perfect square number. If $y$ is the closer ...
2
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1answer
58 views

Taylor Series for a Function of $3$ Variables

The Taylor expansion of the function $f(x,y)$ is: \begin{equation} f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)...
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2answers
41 views

Partial Derivatives Approximation

By definition we know the following: \begin{equation} \frac{\partial f(x,y)}{\partial x} \approx \frac {f(x+ \delta x,y)-f(x,y)}{\delta x} \end{equation} \begin{equation} \frac{\partial f(x,y)}{\...
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0answers
26 views

Chebyshev's inequality and quadratic function

I am trying to use Chebyshev's inequality in order to find sample sizes $n$ such that some condition is met with probability $p$ or larger. That is, find $n\in \mathbb{N}$ such that $\mathbb{P}(X_n &...
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1answer
22 views

finding a close enough point with implicit function theorem

Let $a_1, ...,a_n ,B\in \mathbb{R}^n$ , not all on the same plane. Prove that for a small enough neighborhood of zero $U$ and $\forall u_1,...,u_n \in U $ there is a point $C \in \mathbb{R}^n$ that $...
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0answers
24 views

Formal approximation for second-order ODE with varying coefficients

I have a differential equation of the form $$0=a+by(x)+cf(x)+z(x)f''(x)$$where the functions $y$ and $z$ are known and we want to find $f$. If $z$ is constant, i.e. $z(x)=Z$, it is straightforward to ...
2
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3answers
73 views

Finding an approximation of a function's root

I have the polynomial function $f (x) = x^5+2x^2+1$. I am trying to find an approximation to its root in $[-2,-1]$, with the precision of $0.1$, and with a minimal number of steps. The answer I was ...
0
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1answer
32 views

Behaviour of the Spectral Weight Function $\frac{\sin^2{(\pi f t)}}{(\pi f)^2}$

I'm looking into the properties of the so called spectral weight function $W_0 = \frac{\sin^2{(\pi f t)}}{(\pi f)^2}$. While not important for the question, this function is is encountered in the ...
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0answers
35 views

Maximum of Contour Line

Consider the potential $U(x,y)=ay^{2}+b(e^{x-y}-1)^{2}+c(e^{x+y}-1)^{2}$ where $a$, $b$ and $c$ are known constants. I want to move through a contour line of this potential $U(x,y)=k$, say $y=g(x)$. ...
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0answers
11 views

Estimate the drift and diffusion function numerically

I have a 1D problem as following $$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x} \Big[ \frac{1}{2} \frac{\partial (g(x) f)}{\partial x} -\mu(x)f \Big]$$ I have a time-series of ...
0
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0answers
34 views

Fitting a continuous curve over a piecewise constant data

I have some measurements that are piecewise constant over a certain variable. For example, in the following image, the vertical axis represents the measurement data and the variable is on the ...
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1answer
38 views

Normal Approximation - how many bookings so probability for “overbooking” stays under certain value

I need some help with the following: A hotel has $r$ rooms. The probability that a guest who booked a room also appears (which means: no cancellation) is $p = 0.9$. I'd like to know how many rooms ...
0
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1answer
92 views

What is the value of $e^{-10000}$?

What is the value of $e^{-10000}$? We know that the function $e$ does not attain value $0$ anymore. But in R and Matlab the value of $e^{-10000}$ is given as $0$ which is not correct anymore. I ...
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0answers
72 views

What is the field of mathematics that describes the transition into statistical mechanics?

There are interesting changes that occur in a sample of interacting objects, such as gas particles, as you approach a statistically significant sample. The position or velocity of any given particle ...
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1answer
24 views

Central Limit Theorem and Normal Approximation

having started 'learning' all that is related to the Central Limit Theorem just one day ago, I am already a bit confused - maybe you can help me seeing through the cloud of misunderstanding. Let's ...
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1answer
50 views

When is $1-(1-p)^n \sim pn$

Let $0<p=p(n)<1$ with $p=o(1)$. For which $p$ is it true that $1-(1-p)^n \sim pn$? With $\sim$ I mean that they are asymptotically the same, so $\frac{1-(1-p)^n}{pn}\rightarrow 1$, or at least ...
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0answers
19 views

Spherical Wave Approximation

Lets suppose i have $ K $ data points $(r_i,\phi_i,\theta_i,p_i)$ and i want to approximate my data points with the following function: $$p(r,\theta,\phi) = \sum_{n=0}^{N} \sum_{m=-n}^{n} c_{n,m}h_n^{...
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0answers
33 views

The approximation formula $\left|\alpha -\frac{p}{q}\right| \le \frac{1}{\sqrt{5}q^2}$

I have seen a result about the approximation of irrational numbers and want to find its proof. Suppose $\alpha$ is an irrational number, then there are infinitely many integers $p,q$ with $(p,q)=...
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0answers
33 views

A geometric proof for the “small angle approximation” for the sine, cosine and tangent

How can be deduced the so called "small angle approximations" for sine, cosine and tangent, namely $\sin \theta \approx \theta$ $\tan\theta \approx \theta$ $\cos\theta \approx 1-\frac{\theta^2}{2}$ ...
2
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1answer
97 views

Calculating large exponential probabilities

Earlier today there was Youtube video attempting to solve a problem for a certain game. In it he tries to calculate the probability of certain events happening which narrows down to this equation: $P(...
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2answers
44 views

Perturbation: compute an approximation to the solution of the equation $y+\epsilon\sin y=x^2$

Compute approximation to the solution of the equation $y+\epsilon \sin y=x^2$ using perturbation method. Assume that terms involving powers of $\epsilon$ of order 3 or more can be ignored. So far I ...
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2answers
67 views

Least Squares Alternates- approximating functions

I was given this least squares problem to solve: Find a linear function $\ell(x)$ such that $\displaystyle\int_0^1(e^x-\ell(x))^2{\rm d}x$ is minimized. As an answer, I got $\ell(x)=0.5876+0....
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1answer
37 views

Optimal approximation of spline curve using linear interpolation

I have a parametric cubic spline which I need to draw in graphics. I am restricted to using a set of lines to draw this, and for performance reasons I need as few lines as possible. So I need an ...
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2answers
43 views

Approximation of $\ln(x+1)$ with $\Psi$ function

I found the following approximation for the function $$f=\ln(x+1)$$ $$f\simeq\Psi\left(x+\dfrac{3}{2}\right)-2+\gamma+\ln(2)$$ where $\Psi(x)$ is the 'Digamma' function: $\Psi(x)=\dfrac{\dfrac{d}{dx}\...
2
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1answer
63 views

A curious approximation to $\cos (\alpha/3)$

The following curious approximation $\cos\left ( \frac{\alpha}{3} \right ) \approx \frac{1}{2}\sqrt{\frac{2\cos\alpha}{\sqrt{\cos\alpha+3}}+3}$ is accurate for an angle $\alpha$ between $0^\circ$ ...
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0answers
20 views

Reduced Chebyshev approximation?

Few days ago my teacher mentioned a method of approximation for a function. I think it was called "reduced chebyshev approximation", where you find the taylor series of degree $N$ then subtract from ...
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1answer
35 views

What is an accurate approximation and asymptotic for this function?

$$f(n)=\Bigg(1-\Big(1-\frac{1}{2^{n/2}}\Big)^n\Bigg)^{n^7}$$ I am interested in large $n$. The number $7$ can be replaced by any fixed integer. I have $$f(n)\rightarrow\Bigg(1-e^{-n2^{-n/2}}\Bigg)^{...
2
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1answer
38 views

Upper bound for ratio of modified Bessel functions of second kind

I was wondering if someone has an idea if for $0 < x < y$, and $0< \nu \leq \frac{1}{2}$, one can obtain an upper bound for the ratio $$ \frac{K_{\nu}(x)}{K_{\nu}(y)} $$ Thanks.
4
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1answer
88 views

Alternate proof of the integral: $\int_0^1 x^x(1-x)^{2x}\,dx\neq3/8$

I am looking into the integral: $$I=\int_0^1 x^x(1-x)^{2x}\,dx\neq\frac{3}{8}$$ How might you prove this to be true? What's tough is that the integral $$3/8\lt I<0.37503$$ numerically. I managed ...
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1answer
53 views

Proof that Newton's Method gives better and better approximations with each iteration?

I've seen this question and answer: Why does Newton's method work? It gives some geometric intuition as to what is going on when applying Newton's method, but what I really need to know is why it ...
0
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1answer
56 views

How to prove that your approximation using Newton's Method is correct to $x$ decimal places?

I just watched a video about a problem stated as: "Find where $f(x)= x^7-1000$ intersects the $x$ axis. Find solution correct to 8 decimal places." The author uses Newton's Method, repeating the ...
2
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0answers
53 views

Ideas on how to simplify or approximate this nasty sum

I have a sum (let's call it $p$): $$p:= \frac{1}{n!}\sum_{i=2}^l (i-1)\frac{(n-k)!}{(n-k-i+2)!}(n-i)!$$ where $l, n, k$ are fixed positive integers, and $k \leq n$. I'd like to either simplify or ...
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1answer
27 views

Optimization algorithms for Distribution and Logistics scenario

I am looking for a way to express the following logistics/distribution problem as an equation that can be run thru a solver to find an approximate solution. The problems is described as follows: ...
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2answers
45 views

Binomial theorem estimate for very large samples

I have around $2^{105}$ balls, of which 1 in 20 is white. I expect that when I draw a random sampling of them, roughly 5% of all balls drawn would be white. What is the probability that, if I draw $...
3
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2answers
65 views

When is $\frac{dx}{dt}=\frac{\Delta x}{\Delta t}$ a valid approximation?

It is often said that when the change in e.g. $\Delta x$ is small than we can make the approximation: $$\frac{dx}{dt}=\frac{\Delta x}{\Delta t}$$ But it is not enough to say $\Delta x$ is small ...
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1answer
11 views

Approximately equal too symbol operations permitted

I have to doubt to clear if A is approximately to 2/3 B can I write it as A approximately equal to 4/6 B? Since it is approximate relation is it permitted to multiply both numerator and denominator as ...
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1answer
33 views

Why is $\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$?

I came across this approximation in the book Principles of Population Genetics by Hartl and Clark (page 130). $$\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$$ ...
2
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0answers
18 views

What is the name of this approximation?

I remember studying a while back about an approximation method where the error is calculated using $$ E_{n}=M_{n+1}-a_{n+1} \widetilde{T}_{n+1} $$ Where $\widetilde{T}_{n}=\frac{{T}_{n}}{2^{n-1}}$, ...
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1answer
29 views

Solve using succesive approximation methods

Consider the equation $f(x) = xe^x-1=0$. We wanna solve it using successive approx method, solving the fix point problem $x=e^{-x}$ I don't understand Banach's theorem, how can I solve this? Any idea,...
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2answers
43 views

Separation of integral by approximation

I'm working with the following integral $\displaystyle\int_0^y \frac{dx}{x \sqrt{1-ax-bx^2}}$ and would like to split it in something like $$\int_0^y\frac{dx}{x \sqrt{1-ax}}+\int_0^y\frac{dx}{x \sqrt{...
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0answers
17 views

methods function approximation

My goal is to modelize the temperature of a motor: I want to be able to calculate the temperature of a motor at a given moment, with a given torque and a given speed, in a constant environment. For ...
2
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3answers
49 views

Why transform degrees into radians when computing linear approximation to find $\tan{44^\circ}$?

I am asked to find the linear approximation of $\tan{44^\circ}$. Why should I transform degrees into radians to do that? I understand that using degrees would give me a wrong solution (which would be ...