91Ó°ÊÓ

State the definition of a smooth curve.

Use a graphing utility to graph the polar equation and find all points of horizontal tangency. $$ r=3 \cos 2 \theta \sec \theta $$

Consider a projectile launched at a height \(h\) feet above the ground and at an angle \(\theta\) with the horizontal. If the initial velocity is \(v_{0}\) feet per second, the path of the projectile is modeled by the parametric equations \(x=\left(v_{0} \cos \theta\right) t\) and \(y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}\). The center field fence in a ballpark is 10 feet high and 400 feet from home plate. The ball is hit 3 feet above the ground. It leaves the bat at an angle of \(\theta\) degrees with the horizontal at a speed of 100 miles per hour (see figure). (a) Write a set of parametric equations for the path of the ball. (b) Use a graphing utility to graph the path of the ball when \(\theta=15^{\circ} .\) Is the hit a home run? (c) Use a graphing utility to graph the path of the ball when \(\theta=23^{\circ} .\) Is the hit a home run? (d) Find the minimum angle at which the ball must leave the bat in order for the hit to be a home run.

Folium of Descartes A curve called the folium of Descartes can be represented by the parametric equations \(x=\frac{3 t}{1+t^{3}} \quad\) and \(\quad y=\frac{3 t^{2}}{1+t^{3}} .\) (a) Convert the parametric equations to polar form. (b) Sketch the graph of the polar equation from part (a). (c) Use a graphing utility to approximate the area enclosed by the loop of the curve.

Use the formula for the arc length of a curve in parametric form to derive the formula for the arc length of a polar curve.

Define the eccentricity of an ellipse. In your own words, describe how changes in the eccentricity affect the ellipse.

Describe the differences between the rectangular coordinate system and the polar coordinate system.

Earth moves in an elliptical orbit with the sun at one of the foci. The length of half of the major axis is \(149,598,000\) kilometers, and the eccentricity is \(0.0167\). Find the minimum distance (perihelion) and the maximum distance (aphelion) of Earth from the sun.

The apogee (the point in orbit farthest from Earth) and the perigee (the point in orbit closest to Earth) of an elliptical orbit of an Earth satellite are given by \(A\) and \(P\). Show that the eccentricity of the orbit is \(e=\frac{A-P}{A+P}\)

How are the slopes of tangent lines determined in polar coordinates? What are tangent lines at the pole and how are they determined?