91Ó°ÊÓ

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

Use a graphing utility to graph the polar equation and find the area of the given region.Inner loop of \(r=4-6 \sin \theta\)

Find the points of intersection of the graphs of the equations.\(r=3(1+\sin \theta)\) \(r=3(1-\sin \theta)\)

Find all points (if any) of horizontal and vertical tangency to the curve. Use a graphing utility to confirm your results. $$ x=t+1, \quad y=t^{2}+3 t $$

Find all points (if any) of horizontal and vertical tangency to the curve. Use a graphing utility to confirm your results. $$ x=1-t, \quad y=t^{3}-3 t $$

Antenna Radiation The radiation from a transmitting antenna is not uniform in all directions. The intensity from a particular antenna is modeled by \(r=a \cos ^{2} \theta\) (a) Convert the polar equation to rectangular form. (b) Use a graphing utility to graph the model for \(a=4\) and \(a=6\) (c) Find the area of the geographical region between the two curves in part (b).

Describe what happens to the distance between the directrix and the center of an ellipse if the foci remain fixed and \(e\) approaches \(0 .\)

The path of a projectile is modeled by the parametric equations \(x=\left(90 \cos 30^{\circ}\right) t \quad\) and \(\quad y=\left(90 \sin 30^{\circ}\right) t-16 t^{2}\) where \(x\) and \(y\) are measured in feet. (a) Use a graphing utility to graph the path of the projectile. (b) Use a graphing utility to approximate the range of the projectile. (c) Use the integration capabilities of a graphing utility to approximate the arc length of the path. Compare this result with the range of the projectile.

The planets travel in elliptical orbits with the sun as a focus, as shown in the figure. (a) Show that the polar equation of the orbit is given by \(r=\frac{\left(1-e^{2}\right) a}{1-e \cos \theta}\) where \(e\) is the eccentricity. (b) Show that the minimum distance (perihelion) from the sun to the planet is \(r=a(1-e)\) and the maximum distance \((\) aphelion \()\) is \(r=a(1+e)\).

Consider the parametric equations \(x=4 \cot \theta\) and \(y=4 \sin ^{2} \theta, \quad-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\). (a) Use a graphing utility to graph the curve represented by the parametric equations. (b) Use a graphing utility to find the points of horizontal tangency to the curve. (c) Use the integration capabilities of a graphing utility to approximate the arc length over the interval \(\pi / 4 \leq \theta \leq \pi / 2\)