91Ó°ÊÓ

Find \(d y / d x\) and \(d^{2} y / d x^{2}\), and find the slope and concavity (if possible) at the given value of the parameter. $$ \begin{array}{ll} \underline{\text { Parametric Equations }} & \underline{\text { Point }} \\ x=2+\sec \theta, y=1+2 \tan \theta &\quad \theta=\frac{\pi}{6} \end{array} $$

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=t-1, \quad y=\frac{t}{t-1}\)

The rectangular coordinates of a point are given. Plot the point and find two sets of polar coordinates for the point for \(0 \leq \theta<2 \pi\). $$ (1,1) $$

Find the area of the region.Interior of \(r=1-\sin \theta\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=1+\frac{1}{t}, \quad y=t-1\)

The rectangular coordinates of a point are given. Plot the point and find two sets of polar coordinates for the point for \(0 \leq \theta<2 \pi\). $$ (0,-5) $$

Find \(d y / d x\) and \(d^{2} y / d x^{2}\), and find the slope and concavity (if possible) at the given value of the parameter. $$ \begin{array}{ll} \underline{\text { Parametric Equations }} & \underline{\text { Point }} \\ x=\sqrt{t}, y=\sqrt{t-1}& \quad t=2 \end{array} $$

Find the area of the region.Interior of \(r=1-\sin \theta\) (above the polar axis)

Use a graphing utility to graph the polar equation and find the area of the given region.Inner loop of \(r=1+2 \cos \theta\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=2 t, \quad y=|t-2|\)