Question

consider the parametric curve given by

( x=cos (2 t) ) and ( y = 3 cos ( t ) ) with ( 0 < t < pi ).

a. find ( \frac { d y } { d x } ) and ( \frac { d ^ { 2 } y } { d x ^ { 2 } } ) in terms of ( t ).

( \frac { d y } { d x } = )

( \frac { d ^ { 2 } y } { d x ^ { 2 } } = )

b. determine the ( t )-values where the curve is concave upward. use interval notation, or enter dne if no such intervals exist. include the union symbol when entering multiple intervals.

concave up on:

Explanation:

Step1: Find \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\)

We know that if \(x = \cos(2t)\), then by the chain rule \(\frac{dx}{dt}=- 2\sin(2t)=-4\sin t\cos t\) (using \(\sin(2t) = 2\sin t\cos t\)).
If \(y = 3\cos t\), then \(\frac{dy}{dt}=-3\sin t\).

Step2: Calculate \(\frac{dy}{dx}\)

By the formula \(\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\), we substitute the values of \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\):
\(\frac{dy}{dx}=\frac{-3\sin t}{-4\sin t\cos t}=\frac{3}{4\cos t}=\frac{3}{4}\sec t\) (since \(\sin t
eq0\) for \(0 < t<\pi\) and \(t
eq\frac{\pi}{2}\)).

Step3: Calculate \(\frac{d}{dt}(\frac{dy}{dx})\)

We have \(\frac{dy}{dx}=\frac{3}{4}\sec t\), so \(\frac{d}{dt}(\frac{dy}{dx})=\frac{3}{4}\sec t\tan t\).
Then, by the formula \(\frac{d^{2}y}{dx^{2}}=\frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}\), substituting \(\frac{d}{dt}(\frac{dy}{dx})=\frac{3}{4}\sec t\tan t\) and \(\frac{dx}{dt}=-4\sin t\cos t\):
\(\frac{d^{2}y}{dx^{2}}=\frac{\frac{3}{4}\sec t\tan t}{-4\sin t\cos t}=\frac{3\tan t}{-16\sin t\cos^{2}t}=\frac{3}{-16\cos^{3}t}\) (since \(\tan t=\frac{\sin t}{\cos t}\)).

Step4: Determine where the curve is concave - upward

The curve is concave upward when \(\frac{d^{2}y}{dx^{2}}>0\).
We have \(\frac{d^{2}y}{dx^{2}}=\frac{3}{-16\cos^{3}t}>0\).
Since \(3>0\) and \(- 16<0\), we need \(\cos^{3}t<0\), which implies \(\cos t<0\).
For \(0 < t<\pi\), \(\cos t<0\) when \(\frac{\pi}{2}

Answer:

a. \(\frac{dy}{dx}=\frac{3}{4\cos t}\), \(\frac{d^{2}y}{dx^{2}}=-\frac{3}{16\cos^{3}t}\)
b. Concave up on: \((\frac{\pi}{2},\pi)\)