# Tagged Questions

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### Second difference $\to 0$ everywhere $\implies f$ linear

Exercise 20-27 in Spivak's Calculus, 4th ed., asks us to show that if $f$ is a continuous function on $[a,b]$ that has $$\lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}=0\,\,\,\text{for all }x,$$ then ...
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### How to evaluate the integral $e^{-(c\ln(\frac{1}{x}))^s} dx$?

Can anyone help me evaluate $$\int_{\alpha}^1 \exp{\left\{-\left(c\ln\left(\frac{1}{x}\right)\right)^s\right\}} dx$$, Where $0 \leq \alpha \leq 1$ and $s \in \mathbb{R}$. I tried changing ...
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### Simplify function with polynomial via least-squares

I want to "adjust" (simplify) $f(x)$, a function, by $g(x)$, a polynomial, via least-squares. I want to write code for that. Apperently my code is issuing wrong results, so I was wondering if my ...
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### Product of Gamma functions II

What is the value of the product of Gamma functions \begin{align} \prod_{k=1}^{10} \Gamma\left(\frac{k}{10}\right) \end{align} and can it be shown that \begin{align} \prod_{k=1}^{20} ...
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### Moving average as ODE

Is it possible to represent or approximate the moving average $m(t) = \frac{1}{w}\int_{t-w}^t x(\tau) d\tau$ of a function $x(t)$ as a set of ordinary differential equations $\dot{y} = \ldots$? I am ...
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### Taking limits on integration limits.

For some function $f$ and $g$ lets say that I have an integral which looks like, $\int ^{f(\epsilon)}_0 g(t,\epsilon) dt$. So if I want to compute this to zeroth order in $\epsilon$ can I just ...
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### Integration and differentiation of an approximation to a function - order of approximation

For my research I am working with approximations to functions which I then integrate or differentiate and I am wondering how this affects the order of approximation. Consider as a minimal example the ...
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### Using newthon method to find nth root - did wolframalpha get it wrong?

I'm trying to implement the n-th root algorithm as outlined here: http://en.wikipedia.org/wiki/Nth_root My code however, takes a lot of iterations (more than 100) to converge. I tried to check with ...
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### Finding/approximating 2 unknowns using one equation

I’m doing experimental data in a chemistry lab and I have faced this mathematical problem at a point of my work. Hope you guys can help me with that. What would be the best way to find two constants m ...
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### Where did the linear approximation/linearization formula come from?

Where did the linear approximation/linearization formula come from? I understand that it takes root in the point-slope form and slope intercept form of a linear equation, but I don't understand where ...
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### Approximate $|x|$ with a smooth function

I am trying to get the derivative of $|x|$, and I want that derivative function, say $g(x)$, to be a function of x. So it really needs the |x| to be smooth (ex. $x^2$); I am wondering what is the ...
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### Approximating an interval [duplicate]

Corollary C is derived from $E_(x) = \frac{f''(x)}{2} \cdot (x-a)^2$ I'm having serious issues understanding this problem and I'm just not getting it right. I'm preparing for a test so this isn't ...
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### Error formula for linearization

Can anyone shed some light on this formula? I can't find any information on it. It has three corollaries that I also need to understand:
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### How to evaluate the derivate of a hypergeometric function w.r.t. one of its parameters?

I have to numerically evaluate the derivative of the hypergeometric function w.r.t. its first and second parameters $\large\frac{\partial}{\partial a}{_2F_1}\left(a , b ,c;z\right)$ and ...
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### Why is Simpson's rule exact for cubics?

I can't understand: Why is Simpson's rule exact for cubic polynomials?
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### When to consider an approximation as a Good approximation?

What is the criteria for the Good approximation ? e.g. we can approxiamte $\sin(x)$ to $x$ for $x<0.16 rad$ why 0.16 ? why not 0.23 enother e.g. $\tanh(x/2s)=x$ for $x<s$ and so on .. how ...
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### Midpoint approximation over/under estimation

So left handed approximation underestimates the area under a increasing curve and over estimates for decreasing curves. And right handed approximation overestimates for increasing curves and ...
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### Evaluating the precision in the calculation of $\mathrm{e}$

I'm calculating $\mathrm{e}$ using a computer like this: $$\mathrm{e} \approx \sum\limits_{i=0}^n {1\over i!}$$ I'm storing it as a rational number. I was wondering, if I write down my rational ...
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### I want to show that $\left | f(3)-P(3) \right |\leq \frac1{1000}$ without a calculator.

Let $f(x)=\ln x$. Then the Taylor polynomial of degree 2 at $x=e$ is $P(x)=1+\frac1e(x-e)-\frac1{2e^2}(x-e)^2$ I want to show that $\left | f(3)-P(3) \right |\leq \frac1{1000}$ without a calculator. ...
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### Using $(1+x)^k \approx 1+kx$ to approximate?

Use the approximation $(1+x)^k \approx 1+kx$ to estimate $(1.0003)^{26}$ and $\sqrt[4]{1.006}$. I know how to solve this step-by-step, but I don't understand what I'm doing exactly: why does ...
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### Approximating $\tanh(B\sqrt{A} )$ for small $A$ and arbitrary $B$ by correcting for asymptotic behaviour of $\tanh$

I would like to approximate a function containing terms of the form $\tanh( B\sqrt{A})$ for small $A$. I have tried doing a Taylor series, but I consistently find that it is not only $A$ that has to ...
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### How to find a sequence by its limit?

Is there any way to construct non-trivial sequence by its limit? Something like $\begin{cases} a_1=2 & \\ a_{n+1}=\dfrac1{2}\left(a_n+\dfrac2{a_n}\right) \end{cases}$for $\sqrt2$. I'm especially ...
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### Variational calculus applied to the strain energy functional in solid mechanics

The question is basically about when to apply the variational operator... Given the general functional representing the strain energy of a solid under a given stress state $\sigma$ and strain state ...
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### On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?)

The Stone-Weierstrass Theorem tells us that we can approximate any continuous $f:\mathbb{R}^n\to\mathbb{R}$ arbitrary well on a compact subset of $\mathbb{R}^n$ by some polynomial. Suppose that $f$ is ...
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