91Ó°ÊÓ

Find all points at which \(|r|\) is a maximum and show that the tangent line is perpendicular to the radius connecting the point to the origin. $$r=2-4 \sin 2 \theta$$

In exercises identify all points at which the curve has (a) a horizontal tangent and (b) a vertical tangent. In exercises identify all points at which the curve has (a) a horizontal tangent and (b) a vertical tangent. $$\left\\{\begin{array}{l} x=2 \cos t+\sin 2 t \\ y=2 \sin t+\cos 2 t \end{array}\right.$$

Find rectangular coordinates for the given polar point. $$\left(2,-\frac{\pi}{3}\right)$$

Show that the curve starts at the origin at \(t=0\) and reaches the point \((\pi, 2)\) at \(t=1 .\) Then use the time formula ( 3.2 ) to determine how long it would take a skier to take the given path. $$\left\\{\begin{array}{l} x=\pi t \\ y=2 \sqrt{t} \end{array}\right.$$

Use your CAS or graphing calculator to sketch the plane curves defined by the given parametric equations. $$\left\\{\begin{array}{l}x=t^{2}-1 \\\y=t^{4}-4 t\end{array}\right.$$

Identify the conic section and find each vertex, focus and directrix. $$y=2(x+1)^{2}-1$$

Find polar equations for and graph the conic section with focus (0,0) and the given directrix and eccentricity. Directrix \(y=-2, e=0.4\)

Find rectangular coordinates for the given polar point. $$\left(-1, \frac{\pi}{3}\right)$$

In exercises identify all points at which the curve has (a) a horizontal tangent and (b) a vertical tangent. $$\left\\{\begin{array}{l} x=2 \cos 2 t+\sin t \\ y=2 \sin 2 t+\cos t \end{array}\right.$$

Identify the conic section and find each vertex, focus and directrix. $$y=-2(x+2)^{2}-1$$