Chapter 1: Parametric Equations and Polar CoordinatesContent check — Calculus Vol. 3, Chapter 1: Parametric Equations and Polar Coordinates

Sketch the curve described by the parametric equations

Eliminate the parameter for the plane curve defined by the following parametric equations and describe the resulting graph.

For the following exercises, sketch the curves below by eliminating the parameter t. Give the orientation of the curve.

$x=t^2+2t,$ $y=t+1$

For the following exercises, sketch the curves below by eliminating the parameter t. Give the orientation of the curve.

$x=$ $2t+$ $4,$ $y=$ $t-$ $1$

For the following exercises, eliminate the parameter and sketch the graphs.

$x=$ $2t^2,$ $y=$ $t^4+$ $1$

For the following exercises, use technology (CAS or calculator) to sketch the parametric equations.

[T] $$\begin{array}{cc}x=e^{\text{−}t},\hfill & y=e^{2t}-1\hfill \end{array}$$

For the following exercises, use technology (CAS or calculator) to sketch the parametric equations.

[T] $$\begin{array}{cc}x=\text{sec}t,\hfill & y=\text{cos}t\hfill \end{array}$$

For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph.

$x=$ $6\text{sin}(2\theta ),$ $y=$ $4\text{cos}(2\theta )$

For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph.

$$\begin{array}{cc}x=3-2\text{cos}\theta ,\hfill & y=\mathrm{-5}+3\text{sin}\theta \hfill \end{array}$$

For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph.

$$\begin{array}{cc}x=\text{sec}t,\hfill & y=\text{tan}t\hfill \end{array}$$

For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph.

$$\begin{array}{cc}x=e^t,\hfill & y=e^{2t}\hfill \end{array}$$

For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph.

$$\begin{array}{cc}x=t^3,\hfill & y=3\text{ln}t\hfill \end{array}$$

For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form.

$$\begin{array}{cc}x=t^2-1,\hfill & y=\frac{t}{2}\hfill \end{array}$$

For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form.

$x=$ $4\text{cos}\theta ,$ $y=$ $3\text{sin}\theta ,$ $\theta \in (0,2\pi ]$

For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form.

$$\begin{array}{cc}x=2t-3,\hfill & y=6t-7\hfill \end{array}$$

For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form.

$$\begin{array}{cc}x=1+\text{cos}t,\hfill & y=3-\text{sin}t\hfill \end{array}$$

For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form.

$$\begin{array}{cc}x=\text{sec}t,\hfill & y=\text{tan}t,\pi \le t<\frac{3\pi }{2}\hfill \end{array}$$

For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form.

$$\begin{array}{cc}x=\text{cos}(2t),\hfill & y=\text{sin}t\hfill \end{array}$$

For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form.

$$\begin{array}{cc}x=t^2,\hfill & y=2\text{ln}t,t\ge 1\hfill \end{array}$$

For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form.

$$\begin{array}{cc}x=t^n,\hfill & y=n\text{ln}t,t\ge 1,\hfill \end{array}$$ where n is a natural number

For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form.

$$\begin{array}{c}x=2\text{sin}(8t)\hfill \\ y=2\text{cos}(8t)\hfill \end{array}$$

For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents.

$$\begin{array}{c}x=3t+4\hfill \\ y=5t-2\hfill \end{array}$$

For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents.

$$\begin{array}{c}x=2t+1\hfill \\ y=t^2-3\hfill \end{array}$$

For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents.

$$\begin{array}{c}x=2\text{cos}(3t)\hfill \\ y=2\text{sin}(3t)\hfill \end{array}$$

For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents.

$$\begin{array}{c}x=3\text{cos}t\hfill \\ y=4\text{sin}t\hfill \end{array}$$

For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents.

$$\begin{array}{c}x=3\text{cosh}(4t)\hfill \\ y=4\text{sinh}(4t)\hfill \end{array}$$

For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation.

[T] $$\begin{array}{c}x=\theta +\text{sin}\theta \hfill \\ y=1-\text{cos}\theta \hfill \end{array}$$

For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation.

[T] $$\begin{array}{c}x=t-0.5\text{sin}t\hfill \\ y=1-1.5\text{cos}t\hfill \end{array}$$

For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation.

The trajectory of a bullet is given by $x=$ $v_0\left(\text{cos}\alpha \right)t,$ $y=$ $v_0\left(\text{sin}\alpha \right)t-$ $\frac{1}{2}g{t}^2$ where $v_0=500\text{m/s,}$ $g=$ $9.8=$ $9.8{\text{m/s}}^2,$ and $\alpha =$ $30\text{degrees}.$ When will the bullet hit the ground? How far from the gun will the bullet hit the ground?

For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation.

[T] Use technology to sketch $x=$ $2\text{tan}(t),$ $y=$ $3\text{sec}(t),$ $\text{−}\pi <$ $t<$ $\pi .$

For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation.

[T] Use technology to sketch the spiral curve given by $x=$ $t\text{cos}(t),$ $y=$ $t\text{sin}(t)$ from $\mathrm{-2}\pi \le t\le 2\pi .$

For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation.

[T] Sketch the curve given by parametric equations $$\begin{array}{c}x=\text{cosh}(t)\hfill \\ y=\text{sinh}(t),\hfill \end{array}$$ where $\mathrm{-2}\le t\le 2.$

Calculate the derivative $dy\text{/}dx$ for the plane curve defined by the equations

and locate any critical points on its graph.

Find an equation of the tangent line to the curve defined by the equations

Calculate the second derivative $d^{2}y\text{/}d{x}^2$ for the plane curve defined by the equations

and locate any critical points on its graph.

Find the area under the curve of the hypocycloid defined by the equations

Find the arc length of the curve defined by the equations

Find the surface area generated when the plane curve defined by the equations

is revolved around the x-axis.

For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.

$$\begin{array}{cc}x=8+2t,\hfill & y=1\hfill \end{array}$$

For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.

$$\begin{array}{cc}x=\mathrm{-5}t+7,\hfill & y=3t-1\hfill \end{array}$$

For the following exercises, determine the slope of the tangent line, then find an equation of the tangent line at the given value of the parameter.

$$\begin{array}{cc}x=\text{cos}t,\hfill & y=8\text{sin}t,\hfill \end{array}t=\frac{\pi }{2}$$

For the following exercises, determine the slope of the tangent line, then find an equation of the tangent line at the given value of the parameter.

$$\begin{array}{cc}x=t+\frac{1}{t},\hfill & y=t-\frac{1}{t},t=1\hfill \end{array}$$

For the following exercises, find all points on the curve that have the given slope.

$$\begin{array}{cc}x=4\text{cos}t,\hfill & y=4\text{sin}t,\hfill \end{array}$$ slope = 0.5

For the following exercises, find all points on the curve that have the given slope.

$$\begin{array}{cc}x=t+\frac{1}{t},\hfill & y=t-\frac{1}{t},\text{slope}=1\hfill \end{array}$$

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t.

$$\begin{array}{cc}x=e^{\sqrt{t}},\hfill & y=1-\text{ln}{t}^2,t=1\hfill \end{array}$$

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t.

$$\begin{array}{cc}x=e^t,\hfill & y={(t-1)}^2,\text{at}(1,1)\hfill \end{array}$$

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t.

For $x=$ $\text{sin}(2t),$ $y=$ $2\text{sin}t$ where $0\le t<2\pi .$ Find all values of t at which a vertical tangent line exists.

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t.

Find $\frac{dy}{dx}$ for $x=$ $\text{sin}(t),$ $y=$ $\text{cos}(t).$

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t.

For the curve $x=$ $4t,$ $y=$ $3t-$ $2,$ find the slope and concavity of the curve at $t=3.$

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t.

Find the slope and concavity for the curve whose equation is $x=$ $2+$ $\text{sec}\theta ,$ $y=$ $1+$ $2\text{tan}\theta$ at $\theta =\frac{\pi }{6}.$

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t.

Find all points on the curve $x=$ $\text{sec}\theta ,$ $y=$ $\text{tan}\theta$ at which horizontal and vertical tangents exist.

For the following exercises, find $d^{2}y\text{/}d{x}^2.$

$$\begin{array}{cc}x=\text{sin}\left(\pi t\right),\hfill & y=\text{cos}\left(\pi t\right)\hfill \end{array}$$

For the following exercises, find points on the curve at which tangent line is horizontal or vertical.

$$\begin{array}{cc}x=t(t^2-3),\hfill & y=3(t^2-3)\hfill \end{array}$$

For the following exercises, find $dy\text{/}dx$ at the value of the parameter.

$$\begin{array}{cc}x=\text{cos}t,\hfill & y=\text{sin}t,t=\frac{3\pi }{4}\hfill \end{array}$$

For the following exercises, find $dy\text{/}dx$ at the value of the parameter.

$$\begin{array}{cc}x=4\text{cos}\left(2\pi s\right),\hfill & y=3\text{sin}\hfill \end{array}\left(2\pi s\right),s=-\frac{1}{4}$$

For the following exercises, find $d^{2}y\text{/}d{x}^2$ at the given point without eliminating the parameter.

$x=$ $\sqrt{t},$ $y=$ $2t+$ $4,$ $t=$ $1$

For the following exercises, find $d^{2}y\text{/}d{x}^2$ at the given point without eliminating the parameter.

Determine the concavity of the curve $x=$ $2t+$ $\text{ln}t,$ $y=$ $2t-$ $\text{ln}t.$

For the following exercises, find $d^{2}y\text{/}d{x}^2$ at the given point without eliminating the parameter.

Find the area bounded by the curve $x=$ $\text{cos}t,$ $y=$ $e^t,$ $0\le t\le \frac{\pi }{2}$ and the lines $y=1$ and $x=0.$

For the following exercises, find $d^{2}y\text{/}d{x}^2$ at the given point without eliminating the parameter.

Find the area of the region bounded by $x=$ $2{\text{sin}}^2\theta ,$ $y=$ $2{\text{sin}}^2\theta \text{tan}\theta ,$ for $0\le \theta \le \frac{\pi }{2}.$

For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.

[T] $x=$ $2a\text{cos}t-$ $a\text{cos}(2t),$ $y=$ $2a\text{sin}t-$ $a\text{sin}(2t),$ $0\le t<$ $2\pi$

For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.

[T] $x=$ $2a\text{cos}t-$ $a\text{sin}(2t),$ $y=$ $b\text{sin}t,$ $0\le t<$ $2\pi$ (the “teardrop”)

For the following exercises, find the arc length of the curve on the indicated interval of the parameter.

$$\begin{array}{ccc}x=\frac{1}{3}{t}^3,\hfill & y=\frac{1}{2}{t}^2,\hfill & 0\le t\le 1\hfill \end{array}$$

For the following exercises, find the arc length of the curve on the indicated interval of the parameter.

$$\begin{array}{ccc}x=1+t^2,\hfill & y={\left(1+t\right)}^3,\hfill & 0\le t\le 1\hfill \end{array}$$

For the following exercises, find the arc length of the curve on the indicated interval of the parameter.

$x=$ $a{\text{cos}}^3\theta ,$ $y=$ $a{\text{sin}}^3\theta$ on the interval $[0,2\pi )$ (the hypocycloid)

For the following exercises, find the arc length of the curve on the indicated interval of the parameter.

Find the distance traveled by a particle with position $\left(x,y\right)$ as t varies in the given time interval: $$\begin{array}{ccc}x={\text{sin}}^{2}t,\hfill & y={\text{cos}}^{2}t,\hfill & 0\le t\le 3\pi \hfill \end{array}.$$

For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis.

$$\begin{array}{ccc}x=t^3,\hfill & y=t^2,\hfill & 0\le t\le 1\hfill \end{array}$$

For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis.

[T] Use a CAS to find the area of the surface generated by rotating $x=$ $t+$ $t^3,$ $y=$ $t-$ $\frac{1}{t^2},$ $1\le t\le 2$ about the x-axis. (Answer to three decimal places.)

For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis.

Find the area of the surface generated by revolving $x=$ $t^2,$ $y=$ $2t,$ $0\le t\le 4$ about the x-axis.

In the following exercises, plot the point whose polar coordinates are given by first constructing the angle $\theta$ and then marking off the distance r along the ray.

$\left(3,\frac{\pi }{6}\right)$

In the following exercises, plot the point whose polar coordinates are given by first constructing the angle $\theta$ and then marking off the distance r along the ray.

$\left(0,\frac{7\pi }{6}\right)$

In the following exercises, plot the point whose polar coordinates are given by first constructing the angle $\theta$ and then marking off the distance r along the ray.

$\left(1,\frac{\pi }{4}\right)$

In the following exercises, plot the point whose polar coordinates are given by first constructing the angle $\theta$ and then marking off the distance r along the ray.

$\left(1,\frac{\pi }{2}\right)$

For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point.

Coordinates of point B.

For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point.

Coordinates of point D.

For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in $\left(0,2\pi \right].$ Round to three decimal places.

$\left(3,\mathrm{-4}\right)$

For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in $\left(0,2\pi \right].$ Round to three decimal places.

$\left(\mathrm{-6},8\right)$

For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in $\left(0,2\pi \right].$ Round to three decimal places.

$\left(3,\text{−}\sqrt{3}\right)$

For the following exercises, find rectangular coordinates for the given point in polar coordinates.

$\left(\mathrm{-2},\frac{\pi }{6}\right)$

For the following exercises, find rectangular coordinates for the given point in polar coordinates.

$\left(1,\frac{7\pi }{6}\right)$

For the following exercises, find rectangular coordinates for the given point in polar coordinates.

$\left(0,\frac{\pi }{2}\right)$

For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the $x$-axis, the $y$-axis, or the origin.

$r=3\text{sin}(2\theta )$

For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the $x$-axis, the $y$-axis, or the origin.

$r=\text{cos}\left(\frac{\theta }{5}\right)$

For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the $x$-axis, the $y$-axis, or the origin.

$r=1+\text{cos}\theta$

For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.

$\theta =\frac{\pi }{4}$

For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.

$r=\text{csc}\theta$

For the following exercises, convert the rectangular equation to polar form and sketch its graph.

$x^2-y^2=16$

For the following exercises, convert the rectangular equation to polar form and sketch its graph.

$3x-y=2$

For the following exercises, convert the polar equation to rectangular form and sketch its graph.

$r=4\text{sin}\theta$

For the following exercises, convert the polar equation to rectangular form and sketch its graph.

$r=\theta$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

$r=1+\text{sin}\theta$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

$r=2-2\text{sin}\theta$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

$r=3\text{cos}\left(2\theta \right)$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

$r=2\text{cos}\left(3\theta \right)$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

$r^2=4\text{cos}\left(2\theta \right)$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

$r=2\theta$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

[T] Use a graphing utility and sketch the graph of $r=$ $\frac{6}{2\text{sin}\theta -3\text{cos}\theta }.$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

[T] Use technology to graph $r=$ $e^{\text{sin}(\theta )}-$ $2\text{cos}\left(4\theta \right).$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

Without using technology, sketch the polar curve $\theta =\frac{2\pi }{3}.$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

[T] Use technology to plot $r=e^{\mathrm{-0.1}\theta }$ for $\mathrm{-10}\le \theta \le 10.$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

[T] Use the results of the preceding two problems to explore the graphs of $r=e^{\mathrm{-0.001}\theta }$ and $r=e^{\mathrm{-0.0001}\theta }$ for $\left|\theta \right|>100.$

For the following exercises, determine a definite integral that represents the area.

Region enclosed by $r=3\text{sin}\theta$

For the following exercises, determine a definite integral that represents the area.

Region enclosed by one petal of $r=8\text{sin}(2\theta )$

For the following exercises, determine a definite integral that represents the area.

Region below the polar axis and enclosed by $r=1-\text{sin}\theta$

For the following exercises, determine a definite integral that represents the area.

Region enclosed by the inner loop of $r=2-3\text{sin}\theta$

For the following exercises, determine a definite integral that represents the area.

Region enclosed by $r=1-2\text{cos}\theta$ and outside the inner loop

For the following exercises, determine a definite integral that represents the area.

Region common to $r=$ $2\text{and}r=$ $4\text{cos}\theta$

For the following exercises, find the area of the described region.

Enclosed by $r=6\text{sin}\theta$

For the following exercises, find the area of the described region.

Below the polar axis and enclosed by $r=2-\text{cos}\theta$

For the following exercises, find the area of the described region.

Enclosed by one petal of $r=3\text{cos}\left(2\theta \right)$

For the following exercises, find the area of the described region.

Enclosed by the inner loop of $r=3+6\text{cos}\theta$

For the following exercises, find the area of the described region.

Common interior of $r=$ $4\text{sin}\left(2\theta \right)\text{and}r=$ $2$

For the following exercises, find the area of the described region.

Common interior of $r=$ $6\text{sin}\theta \text{and}r=$ $3$

For the following exercises, find the area of the described region.

Common interior of $r=$ $2+$ $2\text{cos}\theta \text{and}r=$ $2\text{sin}\theta$

For the following exercises, find a definite integral that represents the arc length.

$r=1+\text{sin}\theta$ on the interval $0\le \theta \le 2\pi$

For the following exercises, find a definite integral that represents the arc length.

$r=$ $e^{\theta }\text{on the interval}0\le \theta \le 1$

For the following exercises, find the length of the curve over the given interval.

$r=$ $e^{3\theta }\text{on the interval}0\le \theta \le 2$

For the following exercises, find the length of the curve over the given interval.

$r=$ $8+$ $8\text{cos}\theta \text{on the interval}0\le \theta \le \pi$

For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve.

[T] $r=$ $3\theta \text{on the interval}0\le \theta \le \frac{\pi }{2}$

For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve.

[T] $r=$ ${\text{sin}}^2\left(\frac{\theta }{2}\right)\text{on the interval}0\le \theta \le \pi$

For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve.

[T] $r=$ $\text{sin}\left(3\text{cos}\theta \right)\text{on the interval}0\le \theta \le \pi$

For the following exercises, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral.

$r=$ $\text{sin}\theta +$ $\text{cos}\theta \text{on the interval}0\le \theta \le \pi$

For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral.

$r=$ $3\text{sin}\theta \text{on the interval}0\le \theta \le \pi$

For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral.

$r=$ $6\text{sin}\theta +$ $8\text{cos}\theta \text{on the interval}0\le \theta \le \pi$

For the following exercises, find the slope of a tangent line to a polar curve $r=f\left(\theta \right).$ Let $x=$ $r\text{cos}\theta =$ $f(\theta )\text{cos}\theta$ and $y=$ $r\text{sin}\theta =$ $f(\theta )\text{sin}\theta ,$ so the polar equation $r=f\left(\theta \right)$ is now written in parametric form.

Use the definition of the derivative $\frac{dy}{dx}=$ $\frac{dy\text{/}d\theta }{dx\text{/}d\theta }$ and the product rule to derive the derivative of a polar equation.

For the following exercises, find the slope of a tangent line to a polar curve $r=f\left(\theta \right).$ Let $x=$ $r\text{cos}\theta =$ $f(\theta )\text{cos}\theta$ and $y=$ $r\text{sin}\theta =$ $f(\theta )\text{sin}\theta ,$ so the polar equation $r=f\left(\theta \right)$ is now written in parametric form.

$r=4\text{cos}\theta ;$ $\left(2,\frac{\pi }{3}\right)$

For the following exercises, find the slope of a tangent line to a polar curve $r=f\left(\theta \right).$ Let $x=$ $r\text{cos}\theta =$ $f(\theta )\text{cos}\theta$ and $y=$ $r\text{sin}\theta =$ $f(\theta )\text{sin}\theta ,$ so the polar equation $r=f\left(\theta \right)$ is now written in parametric form.

$r=4+\text{sin}\theta ;$ $\left(3,\frac{3\pi }{2}\right)$

For the following exercises, find the slope of a tangent line to a polar curve $r=f\left(\theta \right).$ Let $x=$ $r\text{cos}\theta =$ $f(\theta )\text{cos}\theta$ and $y=$ $r\text{sin}\theta =$ $f(\theta )\text{sin}\theta ,$ so the polar equation $r=f\left(\theta \right)$ is now written in parametric form.

$r=4\text{cos}\left(2\theta \right);$ tips of the leaves

For the following exercises, find the slope of a tangent line to a polar curve $r=f\left(\theta \right).$ Let $x=$ $r\text{cos}\theta =$ $f(\theta )\text{cos}\theta$ and $y=$ $r\text{sin}\theta =$ $f(\theta )\text{sin}\theta ,$ so the polar equation $r=f\left(\theta \right)$ is now written in parametric form.