Chapter 11

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=\sin t, \quad y=\cos 2 t, \quad-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}$$

Exercises \(9-12\) give the foci or vertices and the eccentricities of ellipses centered at the origin of the \(x y\) -plane. In each case, find the ellipse's standard-form equation in Cartesian coordinates. $$\begin{array}{l}{\text { Foci: }( \pm 8,0)} \\ {\text { Eccentricity: } 0.2}\end{array}$$

Exercises \(9-16\) give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch. $$ x^{2}=6 y $$

Find the polar coordinates, \(-\pi \leq \theta<2 \pi\) and \(r \leq 0,\) of the following points given in Cartesian coordinates. $$\begin{array}{ll}{\text { a. }(-2,0)} & {\text { b. }(1,0)} \\ {\text { c. }(0,-3)} & {\text { d. }\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)}\end{array}$$

Find an equation for the line tangent to the curve at the point defined by the given value of \(t\) . Also, find the value of \(d^{2} y / d x^{2}\) at this point. $$ x=1 / t, \quad y=-2+\ln t, \quad t=1 $$

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=1+\sin t, \quad y=\cos t-2, \quad 0 \leq t \leq \pi$$

Exercises \(9-12\) give the foci or vertices and the eccentricities of ellipses centered at the origin of the \(x y\) -plane. In each case, find the ellipse's standard-form equation in Cartesian coordinates. $$\begin{array}{l}{\text { Vertices: }(0, \pm 70)} \\ {\text { Eccentricity: } 0.1}\end{array}$$

Exercises \(9-16\) give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch. $$ x^{2}=-8 y $$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$r=2$$

Find an equation for the line tangent to the curve at the point defined by the given value of \(t\) . Also, find the value of \(d^{2} y / d x^{2}\) at this point. $$ x=t-\sin t, \quad y=1-\cos t, \quad t=\pi / 3 $$

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=t^{2}, \quad y=t^{6}-2 t^{4}, \quad-\infty< t <\infty$$

Identify the symmetries of the curves. Then sketch the curves in the \(x y\) -plane. \(r^{2}=-\sin \theta\)

Exercises \(9-12\) give the foci or vertices and the eccentricities of ellipses centered at the origin of the \(x y\) -plane. In each case, find the ellipse's standard-form equation in Cartesian coordinates. $$\begin{array}{l}{\text { Vertices: }( \pm 10,0)} \\ {\text { Eccentricity: } 0.24}\end{array}$$

Exercises \(9-16\) give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch. $$ y^{2}=-2 x $$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$0 \leq r \leq 2$$

Find an equation for the line tangent to the curve at the point defined by the given value of \(t\) . Also, find the value of \(d^{2} y / d x^{2}\) at this point. $$ x=\cos t, \quad y=1+\sin t, \quad t=\pi / 2 $$

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=\frac{t}{t-1}, \quad y=\frac{t-2}{t+1}, \quad-1< t <1$$

Identify the symmetries of the curves. Then sketch the curves in the \(x y\) -plane. \(r^{2}=-\cos \theta\)

Exercises \(9-16\) give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch. $$ y=4 x^{2} $$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$r \geq 1$$

Find an equation for the line tangent to the curve at the point defined by the given value of \(t\) . Also, find the value of \(d^{2} y / d x^{2}\) at this point. $$ x=\frac{1}{t+1}, \quad y=\frac{t}{t-1}, \quad t=2 $$

Graph the lemniscates. What symmetries do these curves have? \(r^{2}=4 \cos 2 \theta\)

Find the areas of the regions in Exercises \(9-18\) Inside the lemniscate \(r^{2}=6 \cos 2 \theta\) and outside the circle \(r=\sqrt{3}\)

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=t, \quad y=\sqrt{1-t^{2}}, \quad-1 \leq t \leq 0$$

Exercises \(9-16\) give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch. $$ y=-8 x^{2} $$

Find an equation for the line tangent to the curve at the point defined by the given value of \(t\) . Also, find the value of \(d^{2} y / d x^{2}\) at this point. $$ x=t+e^{t}, \quad y=1-e^{t}, \quad t=0 $$

Graph the lemniscates. What symmetries do these curves have? \(r^{2}=4 \sin 2 \theta\)

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=\sqrt{t+1}, \quad y=\sqrt{t}, \quad t \geqq 0$$

Exercises \(13-16\) give foci and corresponding directrices of ellipses centered at the origin of the \(x y\) -plane. In each case, use the dimensions in Figure 11.47 to find the eccentricity of the ellipse. Then find the ellipse's standard-form equation in Cartesian coordinates. $$\begin{array}{l}{\text { Focus: }(-4,0)} \\ {\text { Directrix: } x=-16}\end{array}$$

Exercises \(9-16\) give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch. $$ x=-3 y^{2} $$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$0 \leq \theta \leq \pi / 6, \quad r \geq 0$$

Assuming that the equations define \(x\) and \(y\) implicitly as differentiable functions \(x=f(t), y=g(t)\) , find the slope of the curve \(x=f(t), y=g(t)\) at the given value of \(t\) . $$ x^{3}+2 t^{2}=9, \quad 2 y^{3}-3 t^{2}=4, \quad t=2 $$

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=\sec ^{2} t-1, \quad y=\tan t, \quad-\pi / 2< t <\pi / 2$$

Exercises \(9-16\) give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch. $$ x=2 y^{2} $$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$\theta=2 \pi / 3, \quad r \leq-2$$

Assuming that the equations define \(x\) and \(y\) implicitly as differentiable functions \(x=f(t), y=g(t)\) , find the slope of the curve \(x=f(t), y=g(t)\) at the given value of \(t\) . $$ x=\sqrt{5-\sqrt{t}}, \quad y(t-1)=\sqrt{t}, \quad t=4 $$

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=-\sec l, \quad y=\tan t, \quad-\pi / 2< t <\pi / 2$$

Graph the lemniscates. What symmetries do these curves have? \(r^{2}=-\cos 2 \theta\)

In Exercises \(17-24\) , find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices. $$x^{2}-y^{2}=1$$

Exercises \(17-24\) give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch. $$ 16 x^{2}+25 y^{2}=400 $$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$\theta=\pi / 3, \quad-1 \leq r \leq 3$$

Assuming that the equations define \(x\) and \(y\) implicitly as differentiable functions \(x=f(t), y=g(t)\) , find the slope of the curve \(x=f(t), y=g(t)\) at the given value of \(t\) . $$ x+2 x^{3 / 2}=t^{2}+t, \quad y \sqrt{t+1}+2 t \sqrt{y}=4, \quad t=0 $$

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=-\cosh t, \quad y=\sinh t, \quad-\infty< t <\infty$$

Find the slopes of the curves at the given points. Sketch the curves along with their tangents at these points. Cardioid \(r=-1+\cos \theta ; \quad \theta=\pm \pi / 2\)

In Exercises \(17-24\) , find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices. $$9 x^{2}-16 y^{2}=144$$

Exercises \(17-24\) give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch. $$ 7 x^{2}+16 y^{2}=112 $$

Assuming that the equations define \(x\) and \(y\) implicitly as differentiable functions \(x=f(t), y=g(t)\) , find the slope of the curve \(x=f(t), y=g(t)\) at the given value of \(t\) . $$ x \sin t+2 x=t, \quad t \sin t-2 t=y, \quad t=\pi $$

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=2 \sinh t, \quad y=2 \cosh t, \quad-\infty< t <\infty$$

Find the slopes of the curves at the given points. Sketch the curves along with their tangents at these points. Cardioid \(\quad r=-1+\sin \theta ; \quad \theta=0, \pi\)

In Exercises \(17-24\) , find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices. $$y^{2}-x^{2}=8$$