# Questions tagged [linear-approximation]

For questions about linear approximations, $f(x) \approx f(a)+f'(a)(x-a)$ for $x$ around $a$.

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### How do we determine what small angle and small $x$ are for a simple pendulum to justify linear approximation?

Consider a simple pendulum consisting of a point-like mass $m$ attached to a massless string of length $L$ from a fixed support and constrained to move in a vertical plane. Here is a picture of this ...
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1 vote
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### Best piecewise with $n$ points for a portion of a log function.

Let $f(x) = |\log_2(x)|$ for $x$ belonging to the domain $(0,1]$. I would like to know if there is an algorithm to fit $f(x)$ using a piecewise linear function g(x) in an optimal way?. That is, on ...
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### Approximation of a function using n intervals

Consider the function 𝑓 𝑥 = 𝑠𝑖𝑛 𝑛𝜋𝑥 and divide the domain 0 ≤ 𝑥 ≤ 1 into m intervals. For the “exact” approximation of the function, we will use 𝑚 = 100 intervals. Plot the “exact” function ...
1 vote
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### $\zeta(1 + 2/x)$ has a strange, nearly linear behaviour

I was messing around with some infinite sums in $\ell^p$ spaces and I encountered a strange result: $\zeta\left(1 + \frac{2}{x}\right)$ looks like it is linear in $x$ for $x > 1$! A simple linear ...
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### Why is the linear approximation of my function, after I remove negligible terms, more accurate than the linear approximation not removing the terms?

I have the function $F(x)$ where $x >> a$ and I have derived two linear approximations of $F(x)$: $L_{1}(x)$, where I take that $\frac{\left(a^{2}-2xa\right)}{x^{2}}$ is way way less ...
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### Taylor approximation of $\phi_1(\lambda) = \frac{1}{\sqrt{\psi(\lambda)}} - \frac{\sigma}{\lambda}$

I am reading this paper. In some point of the analysis the non linear equation $$\phi_1(\lambda) = \frac{1}{\sqrt{\psi(\lambda)}} - \frac{\sigma}{\lambda} \tag{A}$$ is studied, i.e. $(6.7)$ in the ...
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### Number of fixed points in a fixed point iteration function

I'm trying to understand if a fixed point iteration function can have more than 1 fixed points. In theory I tried to find an f(x) such that I could generate an iteration function g(x) in a way that g(...
1 vote
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### How exactly is the value of $\frac{1}{y\sqrt{y^2+\frac{l^2}{4}}}$ as $y \gg l$ calculated?

I am having trouble understanding if a particular calculation is or is not a limit calculation. I suspect it is not, but a particular set of notes from an MIT OCW physics course (problem starts on ...
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### linear approximation of a function at a number

question: Find linear approximation of the function f(x) = √x at x=9. Use it to approximate √9.1 linearisation formula L(x) = f(a) + f'(a)(x-a) In this question, a=9 right? and when we substitute ...
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