91Ó°ÊÓ

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. $$ \text{Parametric Equations} \quad \text{Parameter} $$ $$ x=\sqrt{t}, y=\sqrt{t-1} \quad t=2 $$

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. $$ x=|t-1|, \quad y=t+2 $$

Finding the Area of a Polar Region In Exercises \(5-16\) , find the area of the region. Interior of \(r=1-\sin \theta\) (above the polar axis)

Rectangular-to-Polar Conversion In Exercises \(11-20\) , the rectangular coordinates of a point are given. Plot the point and find two sets of polar coordinates for the point for \(\mathbf{0} \leq \boldsymbol{\theta}<2 \pi .\) $$ (0,-6) $$

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. $$ x=e^{t}, \quad y=e^{3 t}+1 $$

Sketching a Parabola In Exercises \(7-14\) , find the vertex, focus, and directrix of the parabola, and sketch its graph. $$ x^{2}+4 x+4 y-4=0 $$

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. $$ \text{Parametric Equations} \quad \text{Parameter} $$ $$ x=\cos ^{3} \theta, y=\sin ^{3} \theta \quad \theta=\frac{\pi}{4} $$

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

Rectangular-to-Polar Conversion In Exercises \(11-20\) , the rectangular coordinates of a point are given. Plot the point and find two sets of polar coordinates for the point for \(\mathbf{0} \leq \boldsymbol{\theta}<2 \pi .\) $$ (-3,4) $$

Sketching and Identifying a Conic In Exercises \(13-22\) , find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. $$ r=\frac{1}{1-\cos \theta} $$