general
How do I find x as a function of x?
Sorry for the title, I don't know how else to put this into words. Basically I wanted to know how to get the result below: I have no idea about why t...
Apr 13, 2026
Archive: General
general
Finding basis for orthogonal subspace
Find a basis for $S^\perp$ for the subspace $$ S = span\left\{\left[\begin{matrix}1\\1\\-2\end{matrix}\right]\right\} $$ How do I start this question?
Apr 13, 2026
general
The meaning of 'integral powers' and how $\{2,3\}$ generates $\mathbb Z_6$
There is a line in my text This shows that all such products of integral powers of $a$ and $b$ form a subgroup of $G$... What does 'integral powers&#...
Apr 13, 2026
general
Derivative of 1/sin x
I want to find out the derivative of 1/sin(x) without using the reciprocal rule. Let f(x) = 1/sin(x) Df/dx = (f(x+h) - f(x))/ h I keep getting 0 as the an...
Apr 13, 2026
general
Precise definition of the support of a random variable
$\newcommand{\F}{\mathcal{F}} \newcommand{\powset}[1]{\mathcal{P}(#1)}$ I am reading lecture notes which contradict my understanding of random variables. ...
Apr 13, 2026
general
Integral of $1/z$ over the unit circle
Let $ f(z) = \dfrac{1}{z}, \gamma$ the unit circle in $\mathbb{C}$. So \begin{align} \int_{\gamma} f(z) dz = \int _0 ^{2 \pi} \dfrac{i e^{i z}}{e^{i z}} d...
Apr 13, 2026
general
How come $\pi$ is usually approximated as 3.14 or 22/7?
I've heard that $\pi$ is usually approximated as 3.14, but it can also be approximated as 22/7, which is equal to 3.142857142857142857.... Guess what...
Apr 13, 2026
general
In what situations should I use and not use a pooled estimator for $\hat{p}$
In a question, it says that a true-false exam is used to discriminate between well-prepared students and poorly prepared students. There are $\frac{205}{2...
Apr 13, 2026
general
Prove formula $\operatorname{arctanh} x = \frac12\,\ln \left(\frac{1+x}{1-x}\right)$
Problem Prove formula $\operatorname{arctanh} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$ Attempt to solve To start off with definition of functions...
Apr 13, 2026
general
The interconnection between Hyperbolic functions and Euler's Formula
From Euler's identity one may obtain that, $$\sin x=\dfrac{e^{ix}-e^{-ix}}{2i}$$ $$\cos x=\dfrac{e^{ix}+e^{-ix}}{2}$$ However, it looks quite same to...
Apr 13, 2026
