Recent content by Harambe1

Thanks for the reply. I'm not really sure how I would go about drawing a direction field for this particular function so have opted to use the "If $f'(q)>0$ at equilibrium point $q$, then it's unstable. If $f'(q)=0$, then $q$ is neither." Although having obtained $f'(q)=0$ using the...

A one-dimensional dynamical system is given by $x′ = f(x), t \in [0,+\infty)$, where $f : \mathbb{R} \to \mathbb{R}$ is the smooth function defined as follows: $$f(x) = \begin{cases} x^4 \sin \left(\frac{1}{x}\right) & x \neq 0\\ 0 & x = 0. \end{cases}.$$ Find all the equilibrium points and...

Hi, I'm struggling to understand probability coupling. I have the following problem: Let X and Y each be uniformly distributed on the discrete set {1,...6} (i.e. the distribution of the roll of 1 fair die). (a) If X and Y are independent, what is Pr[X = Y]? (b) Couple X and Y so that Pr[X = Y]...

Hi, I am struggling with the following problem: "Let $V=P_3(\Bbb{R})$ and let $t_1=3x^3-x-2$ and $t_2=x^3-3x+2$ with $T=\left\{ t\in V \:|\: t(1)=0 \right\}$. Find ${t_3}\in\left\{T\right\}$ such that $\left\{t_1, t_2, t_2\right\}$ is a basis of T. Not sure where to go as each column matrix...

Hi, I'm struggling to prove that a limit ceases to exist as x tends to 0 for the following function: $$f(x)=\begin{cases}\sin(\frac{1}{x}), & \text{if $x \notin \mathbb{Q}$} \\[3pt] 1, & \text{if $x \in \mathbb{Q}$} \\ \end{cases}$$ I've attempted a proof by contradiction, assuming the...