For questions about linear approximations, $f(x) \approx f(a)+f'(a)(x-a)$ for $x$ around $a$.
Questions tagged [linear-approximation]
271 questions
7
votes
1 answer
What is the difference between linear approximation and a differential?
From my understanding, linear approximations and differentials both use the tangent line to a function to estimate the value of the function at a point. I understand that their respective equations are different - but conceptually how are they…
maddie
- 441
7
votes
3 answers
What is the meaning of a differential in terms of an exact differential?
As I understand it a differential is an outdated concept from the time of Liebniz which was used to define derivatives and integrals before limits came along. As such $dy$ or $dx$ don't really have any meaning on their own. I have seen in multiple…
RonGiant
- 113
6
votes
1 answer
How can I formulate the 3-SAT problem as a 0-1 Linear integer program?
I understand the 3-Sat problem but I do not understand 0-1 Linear Integer Program. I know in a linear integer program I would have an indicator variable $X_i$ that indicates whether a clause is true or not, and I want to maximize this? Can someone…
theGuy05
- 197
5
votes
2 answers
Computer Algebra Systems for Experimental Mathematics (especially Integer Relations with PSLQ)
I would like to use a computer algebra system to do some experimental mathematics, particularly Integer Relation problems using the PSLQ algorithm.
I know that Maple has a PSLQ implementation, but I'd rather not pay for the entire Maple suite just…
hatch22
- 1,116
5
votes
0 answers
Reason for the name "Ring of dual numbers"
The ring of dual numbers over a field $k$ is defined as the quotient
$$k[\varepsilon]/\varepsilon^2.$$
I was reading this question with an interesting answer about some of their basic properties and, if I understand correctly, this ring offers a way…
Abramo
- 7,155
5
votes
1 answer
$\zeta(1 + 2/x)$ 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 regression gives me $\zeta\left(1 +…
NoOneIsHere
- 334
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- 10
5
votes
1 answer
Intuition of error in Taylor appoximation and finding error in approximation of a function by a constant function
I am reading up on Taylor approximation of a function and I'm trying to develop the intuition for the remainder, when approximating a function with $n^{th}$ degree polynomial which has a continuous $(n+1)^{th}$ derivate, given by
…
Srini Vas
- 160
5
votes
0 answers
Linear Approximations from Differential Algebras
Suppose we have a differential ring $(R,+,\cdot)$ with derivation $\partial: R\to R$ which is linear
$$\partial(f+g)=\partial f+\partial g$$
and obeys the Leibniz rule:
$$\partial(f\cdot g)=(\partial f) \cdot g + f\cdot (\partial g)$$
and we…
Sean D
- 109
- 9
4
votes
1 answer
Particular Integral, with sequence
Subject: Seeking Help for a Computer Science Contest - Integral Estimation
Hello everyone,
I hope this message finds you well. I am currently preparing for an ongoing computer science contest, and I have encountered a challenging problem related to…
Henry D
- 145
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4
votes
3 answers
Contradiction in derivatives as linear approximations
From the definition of a derivative, we have that
$$f'(a) = \lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}$$
or
$$\lim\limits_{x\to a}f'(x) = \lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}$$
This leads me to believe we can write
$$\lim\limits_{x\to…
Ark1409
- 123
4
votes
1 answer
The approximation function of $\frac{x}{y}$
Is there a approximation function of $$\frac{x}{y},$$ and the approximation function is in the form of $f(x) + f(y)$ or $f(x) - f(y)$. That's to say the approximation function can split $x$ and $y$.
UniMilky
- 71
4
votes
1 answer
Estimating error of linear approximation
I have this question on my assignment which I just cannot seem to wrap my brain around. I've been reluctant to post it to any forum because I don't like the answer handed to me, but I seriously do need some help with this:
I'm not sure how I am…
Skylineblue
- 155
3
votes
2 answers
When can I plug in value of the parts of limits?
As a part of a limit that I was working on, I had to rewrite $$\frac{2x\tan\left(a\right)+2x}{1-x\tan\left(a\right)}$$ for $x$ near $0$. My understanding is that I can apply limit to the parts of the limit, so I did the…
3
votes
1 answer
Comment regarding the error made approximating $f$ with its differential
In general, it holds that: $$f(x_0+\Delta x)\approx f(x_0)+f'(x_0)\cdot \Delta x$$
Clearly, an $e(\Delta x)$ error is made in the approximation.
Let us consider the equation $$f(x_0+\Delta x)-f(x_0)=f'(x_0)\cdot \Delta x + e(\Delta x)\cdot \Delta…
Sigma Algebra
- 1,855
3
votes
1 answer
Response surface and variables importance
Disclaimer: I'm not expert in math; the solution to this problem may be trivial.
I have a process based on a deterministic computer simulation where two continuous input variables $x_1$ and $x_2$ (under my control) produce a continuous output…
omega
- 129