Chapter 8

Graphical Reasoning In Exercises 1-4, use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. \(r=\frac{2 e}{1+e \cos \theta}\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=3 t-1, \quad y=2 t+1 $$

Find \(d y / d x\). $$ x=t^{2}, y=5-4 t $$

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$ (4,3 \pi / 6) $$

Graphical Reasoning In Exercises 1-4, use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. \(r=\frac{2 e}{1-e \cos \theta}\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=2 t^{2}, \quad y=t^{4}+1 $$

Find \(d y / d x\). $$ x=\sqrt[3]{t}, y=4-t $$

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$ (-2,7 \pi / 4) $$

Graphical Reasoning In Exercises 1-4, use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. \(r=\frac{2 e}{1-e \sin \theta}\)

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

In Exercises 3 and 4, find the area of the region bounded by the graph of the polar equation using (a) a geometric formula and (b) integration. $$ r=8 \sin \theta $$

Find \(d y / d x\). $$ x=\sin ^{2} \theta, y=\cos ^{2} \theta $$

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$ (-4,-\pi / 3) $$

Graphical Reasoning In Exercises 1-4, use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. \(r=\frac{2 e}{1+e \sin \theta}\)

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

Find the area of the region bounded by the graph of the polar equation using (a) a geometric formula and (b) integration. $$ r=3 \cos \theta $$

Find \(d y / d x\). $$ x=2 e^{\theta}, y=e^{-\theta / 2} $$

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$ (-3,-1.57) $$

Writing Consider the polar equation \(r=\frac{4}{1+e \sin \theta} .\) (a) Use a graphing utility to graph the equation for \(e=0.1\), \(e=0.25, e=0.5, e=0.75,\) and \(e=0.9 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{-}\) and \(e \rightarrow 0^{+}\) (b) Use a graphing utility to graph the equation for \(e=1\). Identify the conic. (c) Use a graphing utility to graph the equation for \(e=1.1\), \(e=1.5,\) and \(e=2 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{+}\) and \(e \rightarrow \infty\).

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=\sqrt{t}, \quad y=t-2 $$

In Exercises 5-10, find the area of the region. One petal of \(r=2 \cos 3 \theta\)

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. $$ x=2 t, y=3 t-1 \quad t=3 $$

Use the angle feature of a graphing utility to find the rectangular coordinates for the point given in polar coordinates. Plot the point. $$ (5,3 \pi / 4) $$

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=\sqrt[4]{t}, \quad y=3-t $$

Find the area of the region. One petal of \(r=6 \sin 2 \theta\)

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. $$ x=\sqrt{t}, y=3 t-1 \quad t=1 $$

Use the angle feature of a graphing utility to find the rectangular coordinates for the point given in polar coordinates. Plot the point. $$ (-2,11 \pi / 6) $$

In Exercises \(7-16,\) 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. \(r=\frac{-1}{1-\sin \theta}\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=t-1, \quad y=\frac{t}{t-1} $$

Find the area of the region. One petal of \(r=\cos 2 \theta\)

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. $$ x=t+1, y=t^{2}+3 t \quad t=-1 $$

In Exercises \(7-16,\) 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. \(r=\frac{6}{1+\cos \theta}\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=1+\frac{1}{t}, \quad y=t-1 $$

Find the area of the region. One petal of \(r=\cos 5 \theta\)

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. $$ x=t^{2}+3 t+2, y=2 t \quad t=0 $$

Use the angle feature of a graphing utility to find the rectangular coordinates for the point given in polar coordinates. Plot the point. $$ (8.25,1.3) $$

In Exercises \(7-16,\) 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. \(r=\frac{6}{2+\cos \theta}\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=2 t, \quad y=|t-2| $$

Find the area of the region. Interior of \(r=1-\sin \theta\)

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. $$ x=2 \cos \theta, y=2 \sin \theta \quad \theta=\frac{\pi}{4} $$

The rectangular coordinates of a point are given. Plot the point and find \(t w o\) sets of polar coordinates for the point for \(0 \leq \theta<2 \pi\). $$ (1,1) $$

In Exercises \(7-16,\) 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. \(r=\frac{5}{5+3 \sin \theta}\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=|t-1|, \quad y=t+2 $$

Find the area of the region. Interior of \(r=1-\sin \theta\) (above the polar axis)

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. $$ x=\cos \theta, y=3 \sin \theta \quad \theta=0 $$

The rectangular coordinates of a point are given. Plot the point and find \(t w o\) sets of polar coordinates for the point for \(0 \leq \theta<2 \pi\). $$ (0,-5) $$

In Exercises \(7-16,\) 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. \(r(2+\sin \theta)=4\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=e^{t}, \quad y=e^{3 t}+1 $$

In Exercises 11 and 12, use a graphing utility to graph the polar equation and find the area of the given region. Inner loop of \(r=1+2 \cos \theta\)

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. $$ x=2+\sec \theta, y=1+2 \tan \theta \quad \theta=\frac{\pi}{6} $$