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$\noindent It'll be the pair of parallel lines parallel to the given line and shifted $4$ units away.$ $\noindent To get the equations of this pair of lines, let $P=(x,y)$. The condition for $P$ to be distance $4$ from the given line is that $\frac{\left|3x+4y+5\right|}{\sqrt{3^2 + 4^2}}=4$...

Re: MATH1231/1241/1251 SOS Thread You essentially have two conditions: x -(1/4)y + 0z = 0 and 0x -(3/4)y + z = 0 (assuming your answer was right, I didn't check it). So the vector (x,y,z) is in the column space of A iff it satisfies those two conditions. You can turn those conditions...

Re: MATH1231/1241/1251 SOS Thread Essentially, here is a sketch. $\noindent Let the product in question equal $\rho$. Recall from Paradoxica's earlier remarks that $PA_k ^2 = \left(x -\omega_k \right)\left(x -\overline{\omega}_k\right)$. Take the product of both sides from $k=1$ to $n$...

Re: First Year Uni Calculus Marathon Solve the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}t} = 2\sqrt{y}$$.

A car is resting on the ground; a ball thrown in the air isn't.

I said ignoring things like air resistance. If you don't ignore them, there's things like wind etc. What normal force are you expecting?

$\noindent Call the integral $I$. Complete the square in the denominator. We have$ $$\begin{align*}3-12t -18t^2 &= -18\left(t^2 + \frac{2}{3}t -\frac{1}{6}\right) \\ &= -18\left[ \left(t+\frac{1}{3}\right)^2 -\frac{1}{9}-\frac{1}{6}\right] \\ &= -18\left(t+\frac{1}{3}\right)^2 + 5.\end{align*}$...

Once the projectile is in flight, the only force acting on it is gravity (ignoring things like air resistance).

$\noindent Dummy variables would only get used by TB's to say something like $\int _0 ^3 \left(3-u\right)^3\, \mathrm{d}u =\int _0 ^3 \left(3-x\right)^3\, \mathrm{d}x$. (In other words, the functions are the \emph{same} and we \emph{only} changed a letter.) Dummy variables wouldn't justify...

$\noindent Well as said above (which I know was posted later than your comment), $2-x^3$ is irrelevant to the discussion. What $2-x^3$ is is $2-f(x)$, rather than $f(2-x)$.$ $\noindent Whenever we use dummy variables in these integrals, we're just saying that if the function is the \emph{same}...

$\noindent Well first of all, in that example, $f(2-x)$ isn't $2-x^3$, it's $\left(2-x\right)^3$. This indeed has the same area from $0$ to $2$. $\noindent Let $I = \int _0 ^a f(x)\, \mathrm{d}x$. Let $u=a-x$, then we get $I = \int _a ^ 0 f(a-u) \, \cdot (-1) \, \mathrm{d}u = \int _0 ^a...

Yes that dummy variables idea is used there too. $\noindenf In general, $\int _a ^b f(t)\, \mathrm{d}t$ means \emph{exactly the same thing} as $\int _a ^b f(x) \, \mathrm{d}x$. It's the actual function and bounds that matter, not the symbol used inside the function. For this reason, we could...

It is definitely possible to have a successful surgery career if you go to Newcastle, or other places. Basically, the university you go to isn't that important for Medicine, just getting in is. So you should probably apply everywhere you can and accept any place you get if you get an offer, if...

One way to think of it is that integral from x = 0 to x = a of f(x) dx is the (signed) area under the graph of y = f(x). The integral from u = 0 to u = a of f(u) du is the signed area under the graph of y = f(u). These two graphs are the same, since they have the same equation and bounds; the...

The second part of the question intends us to also meet the condition of the first part, i.e. the writers should still all be separated.

The method would be similar. Use the given substitution and you should end up with a linear ODE, which you can solve via an integrating factor for example. Make sure to find the value of the constant C by using the initial condition. Once you have found the solution, you should be able to find...

$\noindent Note if $X$ is log-normal with parameters $\mu, \sigma^2$, ($\sigma >0$) then the PDF is $f(x) = \frac{1}{\sqrt{2\pi}\sigma x} \exp \left(-\frac{1}{2} \left(\frac{\ln x -\mu }{\sigma}\right)^2\right)$, for $x >0$. Then $\mathbb{E}\left[X\right] =\int _0 ^\infty xf(x)\, \mathrm{d}x$...

(For interest's sake, this is a Bernoulli differential equation: https://en.wikipedia.org/wiki/Bernoulli_differential_equation . The method for solving it is typically via a substitution like the one suggested.) $\noindent Let $z= y^{-1}$, $y\neq 0$, so $z^{\prime} = -y^\prime y^{-2}$. If we...

Answer to your first Q: it's not Poisson(2). For the dice one (note Y is the sample average): 1) The expectation of the sample average is just the expectation of each Xi, i.e. 3.5 2) The variance of the sample average is (sigma^2)/n, where sigma^2 is the variance of each Xi, which you can find...

$\noindent You could try finding the central angle using geometry and then using $\ell = r\theta$.$