Use the following values, where needed: radius of the Earth = 4000 mi = 6440 km, 1 year (Earth year) = 365 days (Earth days), 1 AU =

$$92.9 \times 10 ^ { 6 }$$

mi=

$$150 \times 10 ^ { 6 } \mathrm { km }$$

. The dwarf planet Pluto has eccentricity e = 0.249 and semimajor axis a = 39.5 AU. Find the perihelion and aphelion distances.

Use the formula below to calculate the arc length of the polar curve.

Formula:

$$L=\int_\alpha^\beta\sqrt{[f(\theta)]^2+[f'(\theta)]^2}d\theta=\int_\alpha^\beta\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$$

r= r raised to (3 theta) from theta = 0 to theta = 2.

Find an equation for the ellipse that satisfies the given conditions. Ends of major axis $(0,\pm 6)$; passes through (−3, 2).

Use the following values, where needed: radius of the Earth = 4000 mi = 6440 km, 1 year (Earth year) = 365 days (Earth days), 1 AU =

$$92.9 \times 10 ^ { 6 }$$

mi=

$$150 \times 10 ^ { 6 } \mathrm { km }$$

. The dwarf planet Pluto has eccentricity e = 0.249 and semimajor axis a = 39.5 AU. Find the period T in years.

Show that the graph of the given equation is an ellipse. Find its foci, vertices, and the ends of its minor axis.

$$288 x ^ { 2 } - 168 x y + 337 y ^ { 2 } - 3600 = 0$$

Show that the curve $x = t ^ { 3 } - 4 t , y = t ^ { 2 }$ intersects itself at the point $( 0,4 ) ,$ and find equations for the two tangent lines to the curve at the point of intersection.

Find the area of the region that is enclosed by the cardioid

$$r = 2 + 2 \cos \theta$$

.

Graph the equation using a graphing utility.

$x=\cot y, 0<y<\pi$

Graph the equation using a graphing utility.

$$x=\tan y, - \pi / 2 < y < \pi / 2$$

What can you say about the arc length of the portion of the curve

$$r=1 / \theta$$

for

$$\pi / 4 \leq \theta \leq \pi / 2$$

that lies inside the circle r=1?

Sketch the curve in polar coordinates.

$$\theta = - \frac { 3 \pi } { 4 }$$

Use the formula below to calculate the arc length of the polar curve.

Formula:

$$L=\int_\alpha^\beta\sqrt{[f(\theta)]^2+[f'(\theta)]^2}d\theta=\int_\alpha^\beta\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$$

The entire cardioid r = a(1 − cos theta)

Graph the equation using a graphing utility.

$$x=y-3 y^3+y^5$$

Find the arc length of the polar curve

$$r=1 / \theta$$

for

$$\pi / 4 \leq \theta \leq \pi / 2$$

.