91Ó°ÊÓ

In Exercises 1–18, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=t^{3}, \quad y=\frac{t^{2}}{2} $$

Finding Slope and Concavity In Exercises \(5-14\) , 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. $$ \text{Parametric Equations} \quad \text{Parameter} $$ $$ x=\sqrt{t}, y=3 t-1 \quad t=1 $$

Polar-to-Rectangular Conversion In Exercises \(1-10,\) plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. $$ \left(-2, \frac{11 \pi}{6}\right) $$

In Exercises 1–18, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=t^{2}+t, \quad y=t^{2}-t $$

Finding the Area of a Polar Region In Exercises \(5-16\) , find the area of the region. Interior of \(r=3 \cos \theta\)

Finding the Area of a Polar Region In Exercises \(5-16\) , find the area of the region. One petal of \(r=2 \cos 3 \theta\)

Finding Slope and Concavity In Exercises \(5-14\) , 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. $$ \text{Parametric Equations} \quad \text{Parameter} $$ $$ x=t+1, y=t^{2}+3 t \quad t=-1 $$

Sketching a Parabola In Exercises \(7-14\) , find the vertex, focus, and directrix of the parabola, and sketch its graph. $$ y^{2}=-8 x $$

Polar-to-Rectangular Conversion In Exercises \(1-10,\) plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. $$ (\sqrt{2}, 2.36) $$

Finding Slope and Concavity In Exercises \(5-14\) , 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. $$ \text{Parametric Equations} \quad \text{Parameter} $$ $$ x=t^{2}+5 t+4, y=4 t \quad t=0 $$