91Ó°ÊÓ

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=t^{3}, \quad y=3 \ln t\)

Find the center, foci, and vertices of the ellipse. Use a graphing utility to graph the ellipse. $$ 12 x^{2}+20 y^{2}-12 x+40 y-37=0 $$

Convert the rectangular equation to polar form and sketch its graph. $$ y=4 $$

Use a graphing utility to approximate the points of intersection of the graphs of the polar equations. Confirm your results analytically.\(r=4 \sin \theta\) \(r=2(1+\sin \theta)\)

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=\ln 2 t, \quad y=t^{2}\)

Find all points (if any) of horizontal and vertical tangency to the curve. Use a graphing utility to confirm your results. $$ x=t^{2}-t+2, \quad y=t^{3}-3 t $$

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=e^{-t}, \quad y=e^{3 t}\)

Write the equation for the ellipse rotated \(\pi / 4\) radian clockwise from the ellipse \(r=\frac{5}{5+3 \cos \theta}.\)

Find all points (if any) of horizontal and vertical tangency to the curve. Use a graphing utility to confirm your results. $$ x=3 \cos \theta, \quad y=3 \sin \theta $$

Convert the rectangular equation to polar form and sketch its graph. $$ 3 x-y+2=0 $$