91Ó°ÊÓ

Finding the Area of a Polar Region In Exercises \(5-16\) , find the area of the region. One petal of \(r=4 \sin 3 \theta\)

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. $$ x=\sqrt[4]{t}, \quad y=8-t $$

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. $$ \text{Parametric Equations} \quad \text{Parameter} $$ $$ x=t^{2}+5 t+4, y=4 t \quad t=0 $$

Sketching a Parabola In Exercises \(7-14\) , find the vertex, focus, and directrix of the parabola, and sketch its graph. $$ x^{2}+6 y=0 $$

Polar-to-Rectangular Conversion In Exercises \(1-10,\) plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. $$ (-4.5,3.5) $$

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. $$ x=t-3, \quad y=\frac{t}{t-3} $$

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. $$ \text{Parametric Equations} \quad \text{Parameter} $$ $$ x=4 \cos \theta, y=4 \sin \theta \quad \theta=\frac{\pi}{4} $$

Sketching a Parabola In Exercises \(7-14\) , find the vertex, focus, and directrix of the parabola, and sketch its graph. $$ (x+5)+(y-3)^{2}=0 $$

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

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. $$ \text{Parametric Equations} \quad \text{Parameter} $$ $$ x=\cos \theta, y=3 \sin \theta \quad \theta=0 $$