Questions tagged [derivation-of-formulae]
For questions about derivations of formulas and questions about how to derive a formula.
124 questions
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Can someone explain to me how a formula for the sine function is derived?
I'm 14 years old, and I'm taking a geometry course over the summer to get ahead in school. We've reached a unit on right triangles and trigonometry. I have knowledge of mathematics up to Algebra 1, ...
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1
answer
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Taylors inequality open vs closed interval
Taylors theorem states that if $f$ and all it's derivatives up to $f^{(n)}$ exist and are continuous on the interval $[a,b]$ and $f^{(n+1)}$ derivative exists on the interval $(a,b)$, then there ...
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Is this formula valid for polynomial function extrema [closed]
solve for a such that the remainder of the expression
(f(x) - f(a))/(x-a)^2
is 0 for a is any constant and f(x) is any polynomial function. That a is candidate(s) for possible extrema for f(x).
Of ...
1
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3
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Fraction of numbers with a number of divisors divisible by $3$
I think that the fraction of numbers with a number of divisors divisible by $3$ is $1-\frac{6}{\pi^2}\zeta(3)$.
To formally define what I mean by the fraction of numbers with a certain property, if $f(...
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2d rotation matrix
I was watching a YouTube video on the 2D rotation matrix, and the person in the video derived the rotation matrix as follows:
$$R(\psi) = \begin{bmatrix} \cos(\psi) & -\sin(\psi) \\ \sin(\psi) &...
0
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0
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Why does $\pi$ equal $4 \left(\frac{1}{2}!\right)^2$? [duplicate]
I was messing around with factorials and found out that you can do factorial's of fractions using the gamma function $ \Gamma(z+1) $
which I thought was cool, and I was plugging in a couple of numbers,...
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1
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Should there be sign reversals in this formula?
I am working on rotating point P(x,y) to point P$^\prime$(x$^\prime$,y$^\prime)$ around any point C with horizontal coordinate h and vertical coordinate v. Someone posted the formula:
$$x^\prime=h+(...
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Floor-related: $f(x) = 1 + x + \left\lfloor\frac{ x - 1}{n}\right\rfloor + \left\lfloor\frac{x-2}{n}\right\rfloor$ is 1-1, so how can we invert it?
Desmos Calculator, Exhibit A
Similar Question, Exhibit B
Therefore, after having examined the evidence, and that a function is one-to-one if and only if it has a left inverse, I'm on the lookout for ...
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Deriving Merton's share correctly
I am trying to derive Merton's share for two assets and I would like to, in the first place, double check if I am deriving correctly, and secondly, two specific questions regarding the rule for ...
1
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2
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Is there any alternative to the integral $\int_\pi^{\frac{3 \pi}{2}} \arctan^n \left(\frac{\cos x}{1-\sin x}\right) d x ?$
In the youtube by Prime Newtons, the integral
$$\int_\pi^{\frac{3 \pi}{2}} \arctan \left(\frac{\cos x}{1-\sin x}\right) d x$$
is evaluated by integration by parts as:
$$
\begin{aligned}
\int_\pi^{\...
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2
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2
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What is the closed form expression of $\int_0^{2\pi}(1 + a \cos\theta)^bd\theta$
I am wondering whether it is possible to give some kind of closed form expression for the following integral
\begin{equation}
\int_0^{2 \pi}{(1 + a \cos\theta)^b d\theta}, -1 < a < 1?
\end{...
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A question on "Prove Ramanujan's formula for nested cubic roots"
Someone asked a question about a proof:
Ramanujan found that
$$\begin{align*} & \sqrt[3]{(m^2+mn+n^2)\sqrt[3]{(m-n)(m+2n)(2m+n)}+3mn^2+n^3-m^3}\\ =&\sqrt[3]{\tfrac {(m-n)(m+2n)^2}9}-\sqrt[3]{\...
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Seeking Feedback on the Clarity and Presentation of a Formula for Decimal Expansions of Fractions with Residual Terms
I have been working on a formula describing decimal expansions of fractions with a residual term, particularly how repeating sequences behave when written in a structured way. My notation defines ...
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3
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3
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The derivation of the approximation $2\pi \sqrt{(a^2+b^2)/2}$ of the perimeter of an ellipse
I am writing about the approximations of the perimeter of an ellipse and their proofs, but for the life of me, I cannot produce, nor find the derivation of $$P \approx 2\pi \sqrt{(a^2+b^2)/2}$$
This ...
1
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0
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Recursive formula with derivatives
$$\frac{d^n}{dx^n}\left( \frac{1}{1+f(x)}\right)$$
I'm trying to figure what expression we get doing n derivatives of the reciprocal of a function $f(x)+1$.
First, I thought that it would be a good ...
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