Chapter 11

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find andgraph the ellipse's foci and directrices. $$16 x^{2}+25 y^{2}=400$$

Which polar coordinate pairs label the same point? $$\begin{array}{llll}{\text { a. }(3,0)} & {\text { b. }(-3,0)} & {\text { c. }} & {(2,2 \pi / 3)}\end{array}$$ $$d. (2,7 \pi / 3) \quad e. -3, \pi) \quad f. (2, \pi / 3)$$ $$g. (-3,2 \pi) \quad h. (-2,-\pi / 3)$$

Identify the symmetries of the curves. Then sketch the curves in the \(x y\) -plane. \(r=1+\cos \theta\)

Find the areas of the regions in Exercises \(1-8\) Bounded by the spiral \(r=\theta\) for \(0 \leq \theta \leq \pi\)

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. $$x=3 t, \quad y=9 t^{2}, \quad-\infty< t <\infty$$

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. $$ x=2 \cos t, \quad y=2 \sin t, \quad t=\pi / 4 $$

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find andgraph the ellipse's foci and directrices. $$7 x^{2}+16 y^{2}=112$$

Match the parabolas in Exercises \(1-4\) with the following equations: $$ x^{2}=2 y, \quad x^{2}=-6 y, \quad y^{2}=8 x, \quad y^{2}=-4 x $$ Then find each parabola's focus and directrix.

Which polar coordinate pairs label the same point? $$\begin{array}{lll}{\text { a. }(-2, \pi / 3)} & {\text { b. }(2,-\pi / 3)} & {\text { c. }(r, \theta)} \\ {\text { d. }(r, \theta+\pi)} & {\text { e. }(-r, \theta)} & {\text { f. }(2,-2 \pi / 3)}\end{array}$$ $$\text { g. }(-r, \theta+\pi) \quad \text { h. }(-2,2 \pi / 3)$$

Find the areas of the regions in Exercises \(1-8\) Bounded by the circle \(r=2 \sin \theta\) for \(\pi / 4 \leq \theta \leq \pi / 2\)

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. $$x=-\sqrt{t}, \quad y=t, \quad t \geqslant 0$$

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. $$ x=\sin 2 \pi t, \quad y=\cos 2 \pi t, \quad t=-1 / 6 $$

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

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find andgraph the ellipse's foci and directrices. $$2 x^{2}+y^{2}=2$$

Plot the following points (given in polar coordinates). Then find all the polar coordinates of each point. $$\begin{array}{ll}{\text { a. }(2, \pi / 2)} & {\text { b. }(2,0)} \\ {\text { c. }} {(-2, \pi / 2)} & {\text { d. }(-2,0)}\end{array}$$

Identify the symmetries of the curves. Then sketch the curves in the \(x y\) -plane. \(r=1-\sin \theta\)

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. $$ x=4 \sin t, \quad y=2 \cos t, \quad t=\pi / 4 $$

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. $$x=2 t-5, \quad y=4 t-7, \quad-\infty< t <\infty$$

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find andgraph the ellipse's foci and directrices. $$2 x^{2}+y^{2}=4$$

Plot the following points (given in polar coordinates). Then find all the polar coordinates of each point. $$\begin{array}{ll}{\text { a. }(3, \pi / 4)} & {\text { b. }(-3, \pi / 4)} \\\ {\text { c. }(3,-\pi / 4)} & {\text { d. }(-3,-\pi / 4)}\end{array}$$

Identify the symmetries of the curves. Then sketch the curves in the \(x y\) -plane. \(r=1+\sin \theta\)

Find the areas of the regions in Exercises \(1-8\) Inside the cardioid \(r=a(1+\cos \theta), \quad a>0\)

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. $$ x=\cos t, \quad y=\sqrt{3} \cos t, \quad t=2 \pi / 3 $$

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. $$x=3-3 t, \quad y=2 t, \quad 0 \leq t \leq 1$$

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find andgraph the ellipse's foci and directrices. $$3 x^{2}+2 y^{2}=6$$

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

Find the areas of the regions in Exercises \(1-8\) Inside one leaf of the four-leaved rose \(r=\cos 2 \theta\)

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. $$ x=t, \quad y=\sqrt{t}, \quad t=1 / 4 $$

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. $$x=\cos 2 t, \quad y=\sin 2 t, \quad 0 \leq t \leq \pi$$

Find the Cartesian coordinates of the following points (given in polar coordinates). $$\begin{array}{ll}{\text { a. }(\sqrt{2}, \pi / 4)} & {\text { b. }(1,0)} \\\ {\text { c. }(0, \pi / 2)} & {\text { d. }(-\sqrt{2}, \pi / 4)}\end{array}$$ $$\begin{array}{ll}{\text { e. }(-3,5 \pi / 6)} & {\text { f. }\left(5, \tan ^{-1}(4 / 3)\right)} \\ {\text { g. }(-1,7 \pi)} & {\text { h. }(2 \sqrt{3}, 2 \pi / 3)}\end{array}$$

Find the areas of the regions in Exercises \(1-8\) Inside one leaf of the three-leaved rose \(r=\cos 3 \theta\)

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. $$ x=\sec ^{2} t-1, \quad y=\tan t, \quad t=-\pi / 4 $$

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. $$x=\cos (\pi-t), \quad y=\sin (\pi-t), \quad 0 \leq t \leq \pi$$

Match each conic section in Exercises \(5-8\) with one of these equations: $$\begin{array}{ll}{\frac{x^{2}}{4}+\frac{y^{2}}{9}=1,} & {\frac{x^{2}}{2}+y^{2}=1} \\ {\frac{y^{2}}{4}-x^{2}=1,} & {\frac{x^{2}}{4}-\frac{y^{2}}{9}=1}\end{array}$$ Then find the conic section's foci and vertices. If the conic section is a hyperbola, find its asymptotes as well.

Identify the symmetries of the curves. Then sketch the curves in the \(x y\) -plane. \(r=1+2 \sin \theta\)

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find andgraph the ellipse's foci and directrices. $$6 x^{2}+9 y^{2}=54$$

Find the polar coordinates, \(0 \leq \theta<2 \pi\) and \(r \geq 0,\) of the following points given in Cartesian coordinates. $$\begin{array}{ll}{\text { a. }(1,1)} & {\text { b. }(-3,0)} \\ {\text { c. }(\sqrt{3},-1)} & {\text { d. }(-3,4)}\end{array}$$

Find the areas of the regions in Exercises \(1-8\) Inside one loop of the lemniscate \(r^{2}=4 \sin 2 \theta\)

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

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. $$ x=\sec t, \quad y=\tan t, \quad t=\pi / 6 $$

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. $$x=4 \cos t, \quad y=2 \sin t, \quad 0 \leq t \leq 2 \pi$$

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find andgraph the ellipse's foci and directrices. $$169 x^{2}+25 y^{2}=4225$$

Find the polar coordinates, \(-\pi \leq \theta<\pi\) and \(r \geq 0,\) of the following points given in Cartesian coordinates. $$\begin{array}{ll}{\text { a. }(-2,-2)} & {\text { b. }(0,3)} \\ {\text { c. }(-\sqrt{3}, 1)} & {\text { d. }(5,-12)}\end{array}$$

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

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. $$x=4 \sin t, \quad y=5 \cos t, \quad 0 \leq t \leq 2 \pi$$

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. $$ x=-\sqrt{t+1}, \quad y=\sqrt{3 t}, \quad t=3 $$

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. $$\begin{array}{l}{\text { Foci: }(0, \pm 3)} \\ {\text { Eccentricity: } 0.5}\end{array}$$

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. $$ y^{2}=12 x $$

Find the polar coordinates, \(0 \leq \theta<2 \pi\) and \(r \leq 0,\) of the following points given in Cartesian coordinates. $$\begin{array}{ll}{\text { a. }(3,3)} & {\text { b. }(-1,0)} \\ {\text { c. }} {(-1, \sqrt{3})} & {\text { d. }(4,-3)}\end{array}$$

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. $$ x=2 t^{2}+3, \quad y=t^{4}, \quad t=-1 $$