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I have a prolate cycloid: $$\begin{align*} x &= 2 - \pi\cos(t)\\ y &= 2t - \pi\sin(t) \end{align*}$$ over the interval $-\pi \leq t \leq \pi$, crossed itself at point $P$ on the $x$-axis

a) Find the equations of the 2 tangent lines at $P$

b) find the points on curve where tangent line is horizontal.

c) find the point on curve where tangent line is vertical.

So I know for part $b$ and $c$, you just need to use the derivatives $dx$ and $dy$ and then solve for when $dx=0$ and $dy=0$... however for part a), do I just simply take the derivative of both $x$ and $y$ and that is my solution?

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  • $\begingroup$ Why is the answer not accepted? $\endgroup$
    – ViktorStein
    Feb 8, 2019 at 15:47
  • $\begingroup$ @ViktorGlombik: Because the person asking the question never accepted it; and he has not been on the site since August 2012. $\endgroup$
    – Arturo Magidin
    Feb 12, 2019 at 20:26
  • $\begingroup$ Something should be done in this case, right? So people are not mislead thinking this question is unanswered, even though it has a good answer. $\endgroup$
    – ViktorStein
    Feb 12, 2019 at 20:39
  • $\begingroup$ @ViktorGlombik: No, "something" need not be done. This question is not "unanswered", as it has an answer with upvotes. It does not have an accepted answer, but that is different from being "unanswered". That's not how the site works. Only the person who asks the question can "accept" an answer. Acceptance only reflects that the person who posed the question is satisfied with the given answer. $\endgroup$
    – Arturo Magidin
    Feb 14, 2019 at 23:41
  • $\begingroup$ @Arturo I'm making the case for us not being able to determine if the person is satisfied with the answer since he/she hasn't been of the site since roughly seven years and isn't likely to return and tell us. Maybe he just forgot to accept... $\endgroup$
    – ViktorStein
    Feb 15, 2019 at 10:54

1 Answer 1

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Note that $dx$ is not a derivative, it's a differential. Same with $dy$.

What you mean, presumably, is that you will take $\frac{dx}{dt}$ and $\frac{dy}{dt}$ for parts (b) and (c). Note that $$\frac{dy}{dx} = \frac{\quad\frac{dy}{dt}\quad}{\frac{dx}{dt}}$$ so you can use this for (a), (b), and (c). For (a), this can be used to get the slope of the tangent, but to find the equation of the tangent you'll have to do a bit more work. For (b), you want $\frac{dy}{dx}$ to be $0$, so you want $\frac{dy}{dt}=0$ and $\frac{dx}{dt}\neq 0$. For (c), you want $\frac{dy}{dt}\neq 0$ and $\frac{dx}{dt}=0$.

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  • $\begingroup$ Alright, so I think I got part b and c down alright.. but for part a, i solved for dy/dx and I get (2-picos(t))/(pisin(t))... but not sure what the next step is to find the EQN of the tangent? $\endgroup$
    – Nick
    Jun 10, 2012 at 19:59
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    $\begingroup$ @Nick: If you know the point and you know the slope(s), you should be able to get the equation. $\endgroup$
    – Arturo Magidin
    Jun 10, 2012 at 20:02

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