91Ó°ÊÓ

15-16 Graph the curve and find the area that it encloses. $$ r=1+2 \sin 6 \theta $$

\(15-20\) Identify the curve by finding a Cartesian equation for the curve. $$r=2$$

Find the vertices and foci of the ellipse and sketch its graph. $$x^{2}+3 y^{2}+2 x-12 y+10=0$$

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. $$r=\frac{10}{5-6 \sin \theta}$$

\(15-20\) Identify the curve by finding a Cartesian equation for the curve. $$ r \cos \theta=1 $$

Find \(d y / d x\) and \(d^{2} y / d x^{2} .\) For which values of \(t\) is the curve concave upward? \(x=\cos 2 t\), \(y=\cos t\), \(0<\)t\(<\pi\)

(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. \(x=\ln t, \quad y=\sqrt{t}, \quad t \geqslant 1\)

Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. $$x=10-t^{2}, \quad y=t^{3}-12 t$$

\(15-20\) Identify the curve by finding a Cartesian equation for the curve. $$r=3 \sin \theta$$

(a) Find the eccentricity and directrix of the conic \(r=1 /(1-2 \sin \theta)\) and graph the conic and its directrix. (b) If this conic is rotated counterclockwise about the origin through an angle 3\(\pi / 4\) . write the resulting equation and graph its curve.