91Ó°ÊÓ

Find the slope of a tangent line to a polar curve \(r=f(\theta)\). Let \(x=r \cos \theta=f(\theta) \cos \theta\) and \(y=r \sin \theta=f(\theta) \sin \theta, \quad\) so the polar equation \(r=f(\theta)\) is now written in parametric form. $$ r=2 \sin (3 \theta) ; \text { tips of the leaves } $$

Find the points on the interval \(-\pi \leq \theta \leq \pi\) at which the cardioid \(r=1-\cos \theta\) has a vertical or horizontal tangent line.

For the cardioid \(r=1+\sin \theta,\) find the slope of the tangent line when \(\theta=\frac{\pi}{3}\).

Find the slope of the tangent line to the given polar curve at the point given by the value of \(\theta\). $$ r=3 \cos \theta, \theta=\frac{\pi}{3} $$

Find the slope of the tangent line to the given polar curve at the point given by the value of \(\theta\). $$ r=\theta, \quad \theta=\frac{\pi}{2} $$

Find the slope of the tangent line to the given polar curve at the point given by the value of \(\theta\). Use technology: \(r=2+4 \cos \theta\) at \(\theta=\frac{\pi}{6}\)

Find the points at which the following polar curves have a horizontal or vertical tangent line. $$ r=4 \cos \theta $$

Find the points at which the following polar curves have a horizontal or vertical tangent line. $$ r^{2}=4 \cos (2 \theta) $$

Find the points at which the following polar curves have a horizontal or vertical tangent line. $$ r=2 \sin (2 \theta) $$

Find the points at which the following polar curves have a horizontal or vertical tangent line. $$ \text { The cardioid } r=1+\sin \theta $$