91Ó°ÊÓ

Eliminate the parameter and obtain the standard form of the rectangular equation. Hyperbola: \(x=h+a \sec \theta, \quad y=k+b \tan \theta\)

Find an equation of the ellipse. Foci: \((0, \pm 5)\) Major axis length: 14

Convert the polar equation to rectangular form and sketch its graph. $$ r=2 \csc \theta $$

Determine the \(t\) intervals on which the curve is concave downward or concave upward. $$ x=2 \cos t, \quad y=\sin t, \quad 0

Find an equation of the ellipse. Center: \((0,0)\) Major axis: horizontal Points on the ellipse: \((3,1),(4,0)\)

Use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. $$ r=3-4 \cos \theta $$

Arc Length write an integral that represents the arc length of the curve on the given interval. Do not evaluate the integral. $$ \begin{array}{ll} \underline{\text { Parametric Equations }} & \underline{\text { Interval }} \\\ x=2 t-t^{2}, \quad y=2 t^{3 / 2} &\quad 1 \leq t \leq 2 \end{array} $$

Conjecture Find the area of the region enclosed by \(r=a \cos (n \theta)\) for \(n=1,2,3, \ldots \ldots\) Use the results to make a conjecture about the area enclosed by the function if \(n\) is even and if \(n\) is odd.

Use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. $$ r=5(1-2 \sin \theta) $$

Arc Length write an integral that represents the arc length of the curve on the given interval. Do not evaluate the integral. $$ \begin{array}{ll} \underline{\text { Parametric Equations }} & \underline{\text { Interval }} \\\ x=\ln t, \quad y=t+1& \quad 1 \leq t \leq 6 \end{array} $$