Chapter 10

The points of intersection of the cardioid \(r=1+\sin \theta\) and the spiral loop \(r=2 \theta,-\pi / 2 \leqslant \theta \leqslant \pi / 2,\) can't be found exactly. Use a graphing device to find the approximate values of \(\theta\) at which they intersect. Then use these values to estimate the area that lies inside both curves.

\(29-48\) Sketch the curve with the given polar equation. $$r^{2}=9 \sin 2 \theta$$

Find an equation for the conic that satisfies the given conditions. Hyperbola, vertices \((0, \pm 2), \quad\) foci \((0, \pm 5)\)

When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it sup- presses noise from the audience. Suppose the microphone is placed 4 \(\mathrm{m}\) from the front of the stage (as in the figure) and the boundary of the optimal pickup region is given by the car.dioid \(r=8+8 \sin \theta,\) where \(r\) is measured in meters and the microphone is at the pole. The musicians want to know the area they will have on stage within the optimal pickup range of the microphone. Answer their question.

\(29-48\) Sketch the curve with the given polar equation. $$r^{2}=\cos 4 \theta$$

Find an equation for the conic that satisfies the given conditions. $$\begin{array}{l}{\text { Hyperbola, vertices }(-3,-4),(-3,6)} \\ {\text { foci }(-3,-7),(-3,9)}\end{array}$$

Graph the curve and find its length. $$x=e^{t} \cos t, \quad y=e^{t} \sin t, \quad 0 \leqslant t \leqslant \pi$$

\(45-48\) Find the exact length of the polar curve. $$ r=3 \sin \theta, \quad 0 \leqslant \theta \leqslant \pi / 3 $$

\(29-48\) Sketch the curve with the given polar equation. $$r=2 \cos (3 \theta / 2)$$

Suppose that the position of one particle at time \(t\) is given by $$x_{1}=3 \sin t \quad y_{1}=2 \cos t \quad 0 \leqslant t \leqslant 2 \pi$$ and the position of a second particle is given by $$x_{2}=-3+\cos t \quad y_{2}=1+\sin t \quad 0 \leqslant t \leqslant 2 \pi$$ (a) Graph the paths of both particles. How many points of intersection are there? (b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place at the same time? If so, find the collision points. (c) Describe what happens if the path of the second particle is given by $$x_{2}=3+\cos t \quad y_{2}=1+\sin t \quad 0 \leqslant t \leqslant 2 \pi$$

Find an equation for the conic that satisfies the given conditions. $$ \begin{array}{l}{\text { Hyperbola, vertices }(-1,2),(7,2),} \\ {\text { foci }(-2,2),(8,2)}\end{array}$$

Graph the curve and find its length. $$x=\cos t+\ln \left(\tan \frac{1}{2} t\right), \quad y=\sin t, \quad \pi / 4 \leqslant t \leqslant 3 \pi$$

\(45-48\) Find the exact length of the polar curve. $$ r=e^{2 \theta}, \quad 0 \leqslant \theta \leqslant 2 \pi $$

\(29-48\) Sketch the curve with the given polar equation. $$r^{2} \theta=1$$

If a projectile is fired with an initial velocity of \(v_{0}\) meters per second at an angle \(\alpha\) above the horizontal and air resistance is assumed to be negligible, then its position after \(t\) seconds is given by the parametric equations $$x=\left(v_{0} \cos \alpha\right) t \quad y=\left(v_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2}$$ where \(g\) is the acceleration due to gravity \(\left(9.8 \mathrm{m} / \mathrm{s}^{2}\right).\) (a) If a gun is fired with \(\alpha=30^{\circ}\) and \(v_{0}=500 \mathrm{m} / \mathrm{s},\) when will the bullet hit the ground? How far from the gun will it hit the ground? What is the maximum height reached by the bullet? (b) Use a graphing device to check your answers to part (a). Then graph the path of the projectile for several other values of the angle \(\alpha\) to see where it hits the ground. Summarize your findings. (c) Show that the path is parabolic by eliminating the parameter.

Find an equation for the conic that satisfies the given conditions. Hyperbola, vertices \((\pm 3,0),\) asymptotes \(y=\pm 2 x\)

Graph the curve and find its length. $$x=e^{t}-t, \quad y=4 e^{t / 2}, \quad-8 \leq t \leqslant 3$$

\(45-48\) Find the exact length of the polar curve. $$ r=\theta^{2}, \quad 0 \leqslant \theta \leqslant 2 \pi $$

\(29-48\) Sketch the curve with the given polar equation. $$r=1+2 \cos 2 \theta$$

Investigate the family of curves defined by the parametric equations \(x=t^{2}, y=t^{3}-c t .\) How does the shape change as \(c\) increases? Illustrate by graphing several members of the family.

Find an equation for the conic that satisfies the given conditions.$$\begin{array}{l}{\text { Hyperbola, foci }(2,0),(2,8),} \\\ {\text { asymptotes } y=3+\frac{1}{2} x \text { and } y=5-\frac{1}{2} x}\end{array}$$

\(45-48\) Find the exact length of the polar curve. $$ r=\theta, \quad 0 \leqslant \theta \leqslant 2 \pi $$

\(29-48\) Sketch the curve with the given polar equation. $$r=1+2 \cos (\theta / 2)$$

The swallowtail catastrophe curves are defined by the parametric equations \(x=2 c t-4 t^{3}, y=-c t^{2}+3 t^{4} .\) Graph several of these curves. What features do the curves have in common? How do they change when \(c\) increases?

The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude 110 \(\mathrm{km}\) and apolune altitude 314 \(\mathrm{km}(\) above the moon). Find an equation of this ellipse if the radius of the moon is 1728 \(\mathrm{km}\) and the center of the moon is at one focus.

Use Simpson's Rule with \(n=6\) to estimate the length of the curve \(x=t-e^{t}, y=t+e^{t},-6 \leq t \leq 6\).

The curves with equations \(x=a \sin n t, y=b \cos t\) are called Lissajous figures. Investigate how these curves vary when \(a, b,\) and \(n\) vary. (Take \(n\) to be a positive integer.)

Investigate the family of curves defined by the parametric equations \(x=\cos t, y=\sin t-\sin c t,\) where \(c>0 .\) Start by letting \(c\) be a positive integer and see what happens to the shape as \(c\) increases. Then explore some of the possibilities that occur when \(c\) is a fraction.

Find the distance traveled by a particle with position \((x, y)\) as \(t\) varies in the given time interval. Compare with the length of the curve. \(x=\sin ^{2} t, \quad y=\cos ^{2} t, \quad 0 \leq t \leqslant 3 \pi\)

In the LORAN (LOng RAnge Navigation) radio navigation system, two radio stations located at \(A\) and \(B\) transmit simul- taneous signals to a ship or an aircraft located at \(P\) .The onboard computer converts the time difference in receiving these signals into a distance difference \(|P A|-|P B|,\) and this, according to the definition of a hyperbola, locates the ship or aircraft on one branch of a hyperbola (see the figure). Suppose that station \(B\) is located 400 mi due east of station \(A\) on a coastline. A ship received the signal from B 1200 microseconds (\mus) before it received the signal from A. (a) Assuming that radio signals travel at a speed of 980 ft/\mus, find an equation of the hyperbola on which the ship lies. (b) If the ship is due north of \(B\) , how far off the coastline is the ship?

Show that the polar curve \(r=4+2 \sec \theta\) (called a conchoid) has the line \(x=2\) as a vertical asymptote by showing that $$\lim _{r \rightarrow \pm \infty} x=2 .$$ Use this fact to help sketch the conchoid.

Find the distance traveled by a particle with position \((x, y)\) as \(t\) varies in the given time interval. Compare with the length of the curve. \(x=\cos ^{2} t, \quad y=\cos t, \quad 0 \leq t \leq 4 \pi\)

Show that the function defined by the upper branch of the hyperbola \(y^{2} / a^{2}-x^{2} / b^{2}=1\) is concave upward.

Show that the total length of the ellipse \(x=a \sin \theta\) \(y=b \cos \theta, a>b>0,\) is $$L=4 a \int_{0}^{\pi / 2} \sqrt{1-e^{2} \sin ^{2} \theta} d \theta$$ where \(e\) is the eccentricity of the ellipse \((e=c / a,\) where \(c=\sqrt{a^{2}-b^{2}} )\)

Show that the curve \(r=\sin \theta \tan \theta\) (called a cissoid of Diocles) has the line \(x=1\) as a vertical asymptote. Show also that the curve lies entirely within the vertical strip \(0 \leqslant x<1\) Use these facts to help sketch the cissoid.

Find an equation for the ellipse with foci \((1,1)\) and \((-1,-1)\) and major axis of length \(4 .\)

Find the total length of the astroid \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta\) where \(a>0\)

\(53-54\) Graph the curve and find its length. $$r=\cos ^{2}(\theta / 2)$$

Sketch the curve \(\left(x^{2}+y^{2}\right)^{3}=4 x^{2} y^{2}\)

Determine the type of curve represented by the equation $$\frac{x^{2}}{k}+\frac{y^{2}}{k-16}=1$$A in each of the following cases: (A) $$ k>16,(\mathrm{b}) 0

(a) Graph the epitrochoid with equations $$\begin{array}{l}{x=11 \cos t-4 \cos (11 t / 2)} \\ {y=11 \sin t-4 \sin (11 t / 2)}\end{array}$$ What parameter interval gives the complete curve? (b) Use your CAS to find the approximate length of this curve.

(a) Show that the equation of the tangent line to the parabola \(y^{2}=4 p x\) at the point \(\left(x_{0}, y_{0}\right)\) can be written as $$y_{0} y=2 p\left(x+x_{0}\right)$$ (b) What is the \(x\) -intercept of this tangent line? Use this fact to draw the tangent line.

(a) Find a formula for the area of the surface generated by rotating the polar curve \(r=f(\theta), a \leqslant \theta \leqslant b\) (where \(f^{\prime}\) is continuous and \(0 \leqslant a