91Ó°ÊÓ

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. \(x=4 \cos \theta, \quad y=5 \sin \theta, \quad-\pi / 2 \leqslant \theta \leqslant \pi / 2\)

Find \(d y / d x\) and \(d^{2} y / d x^{2} .\) For which values of \(t\) is the curve concave upward? $$x=t^{3}-12 t, \quad y=t^{2}-1$$

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. $$r=\frac{3}{2+2 \cos \theta}$$

\(7-12\) Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $$r \geqslant 1, \quad \pi \leqslant \theta \leqslant 2 \pi$$

Find \(d y / d x\) and \(d^{2} y / d x^{2} .\) For which values of \(t\) is the curve concave upward? $$x=t-e^{t}, \quad y=t+e^{-t}$$

Find the vertices and foci of the ellipse and sketch its graph. $$4 x^{2}+y^{2}=16$$

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. \(x=\sin t, \quad y=\csc t, \quad 0< t<\pi / 2\)

\(9-14\) Sketch the curve and find the area that it encloses. $$ r=2 \cos 3 \theta $$

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. $$r=\frac{9}{6+2 \cos \theta}$$

Find the distance between the points with polar coordinates \((2, \pi / 3)\) and \((4,2 \pi / 3) .\)