91Ó°ÊÓ

(a) Graph the epitrochoid with equations $$\begin{array}{l}{x=11 \cos t-4 \cos (11 t / 2)} \\ {y=11 \sin t-4 \sin (11 t / 2)}\end{array}$$ What parameter interval gives the complete curve? (b) Use your CAS to find the approximate length of this curve.

(a) Find a formula for the area of the surface generated by rotating the polar curve \(r=f(\theta), a \leqslant \theta \leqslant b\) (where \(f^{\prime}\) is continuous and \(0 \leqslant a

(a) Show that the equation of the tangent line to the parabola \(y^{2}=4 p x\) at the point \(\left(x_{0}, y_{0}\right)\) can be written as $$y_{0} y=2 p\left(x+x_{0}\right)$$ (b) What is the \(x\) -intercept of this tangent line? Use this fact to draw the tangent line.

\(57-62\) Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta .\) $$r=2 \sin \theta, \quad \theta=\pi / 6$$

Set up an integral that represents the area of the surface obtained by rotating the given curve about the \(x\) -axis. Then use your calculator to find the surface area correct to four decimal places. $$x=1+t e^{t}, \quad y=\left(t^{2}+1\right) e^{t}, \quad 0 \leqslant t \leqslant 1$$

Show that the tangent lines to the parabola \(x^{2}=4 p y\) drawn from any point on the directrix are perpendicular.

Show that if an ellipse and a hyperbola have the same foci, then their tangent lines at each point of intersection are perpendicular.

\(57-62\) Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\) . $$r=2-\sin \theta, \quad \theta=\pi / 3$$

Find the exact area of the surface obtained by rotating the given curve about the \(x\) -axis. $$x=t^{3}, \quad y=t^{2}, \quad 0 \leq t \leq 1$$

\(57-62\) Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\) . $$r=1 / \theta, \quad \theta=\pi$$