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I was trying to solve some calculus problems and I came across with some doubts related to two of them.

  1. Given the parametric equation $$ (x,y)=(3-3\cos^2 (t),3-3\cos(t)\sin(t)) \quad 4 \pi /3 < t < 3 \pi /2 $$ prove if there exists a value for $t$ so that the tangent line to the point of the curve determined by that value is parallel to the straight line $f(x)=x/3$

  2. Given the parametric equation $$ (x,y)=(4\cos (2t)\cos(t),4\cos(2t)\sin(t)) \quad 0 < t < \pi /4 $$ prove if there exists a value for $t$ so that the tangent line to the point of the curve determined by that value is parallel to the $x$ axis.


Well, I know that for 1) I have to use Lagrange's theorem and for 2) Rolle's theorem. But the thing is that I know how to use them when solving "simple" functions, but these are parametric functions. So, for example, in 1) I know that the function is continuous and differentiable in that interval of t, and using Rolle's theorem "formula" I have $f(b)$ and $f(a)$ but $a$ and $b$ are $x$ values and now I'm being asked to use $t$ values, not $x$ or $y$. Moreover I can't write that function as a cartesian one. To sum up, I don't how to apply these theorems in parametric functions.

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For example, with the first curve

$$r(t):=\left(\,3-3\cos^2t\,,\,\,3-3\cos t\sin t\,\right)=\left(\,3\sin^2t\,,\,\,3-\frac32\sin2t\,\right)\implies$$

$$r'(t)=\left(\,3\sin2t\,,\,\,3-3\cos2t\,\right)$$

Thus we get:

$$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{3-3\cos2t}{3\sin2t}=\frac{1-\cos2t}{\sin 2t}=\frac{2\sin^2t}{\sin2t}=\tan t$$

and we must assume $\;t\neq k\pi\,,\,\,k\in\Bbb Z\;$ , which is fine as in the given interval there aren't any integer multiples of $\;\pi\;$ .

so we want to check whether

$$\;\frac{dy}{dx}=\frac13 \iff\tan t=\frac13$$

Try now to take it from here...and I'm not sure where is Lagrange or Rolle needed.

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