Chapter 10

Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. $$y^{2}=20 x$$

Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates. $$r=2 \theta ;\left(\frac{\pi}{2}, \frac{\pi}{4}\right)$$

Consider the following parametric equations. a. Make a brief table of values of \(t, x,\) and \(y.\) b. Plot the \((x, y)\) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing \(t\)). c. Eliminate the parameter to obtain an equation in \(x\) and \(y.\) d. Describe the curve. $$x=-t+6, y=3 t-3 ;-5 \leq t \leq 5$$

Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. $$x=-y^{2} / 16$$

Express the following polar coordinates in Cartesian coordinates. \(\left(3, \frac{\pi}{4}\right)\)

Find the points at which the following polar curves have a horizontal or a vertical tangent line. $$r=4 \cos \theta$$

Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in \(x\) and \(y.\) b. Describe the curve and indicate the positive orientation. $$x=\sqrt{t}+4, y=3 \sqrt{t} ; 0 \leq t \leq 16$$

Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. $$4 x=-y^{2}$$

Express the following polar coordinates in Cartesian coordinates. \(\left(1, \frac{2 \pi}{3}\right)\)

Find the points at which the following polar curves have a horizontal or a vertical tangent line. $$r=2+2 \sin \theta$$

Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in \(x\) and \(y.\) b. Describe the curve and indicate the positive orientation. $$x=(t+1)^{2}, y=t+2 ;-10 \leq t \leq 10$$

Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. $$8 y=-3 x^{2}$$

Express the following polar coordinates in Cartesian coordinates. \(\left(1,-\frac{\pi}{3}\right)\)

Find the points at which the following polar curves have a horizontal or a vertical tangent line. $$r=\sin 2 \theta$$

Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in \(x\) and \(y.\) b. Describe the curve and indicate the positive orientation. $$x=\cos t, y=\sin ^{2} t ; 0 \leq t \leq \pi$$

Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. $$12 x=5 y^{2}$$

Express the following polar coordinates in Cartesian coordinates. \(\left(2, \frac{7 \pi}{4}\right)\)

Find the points at which the following polar curves have a horizontal or a vertical tangent line. $$r=3+6 \sin \theta$$

Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in \(x\) and \(y.\) b. Describe the curve and indicate the positive orientation. $$x=1-\sin ^{2} s, y=\cos s ; \pi \leq s \leq 2 \pi$$

Express the following polar coordinates in Cartesian coordinates. \(\left(-4, \frac{3 \pi}{4}\right)\)

Find the points at which the following polar curves have a horizontal or a vertical tangent line. $$r=1-\sin \theta$$

Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in \(x\) and \(y.\) b. Describe the curve and indicate the positive orientation. $$x=r-1, y=r^{3} ;-4 \leq r \leq 4$$

Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola that opens downward with directrix \(y=6\)

Find the points at which the following polar curves have a horizontal or a vertical tangent line. $$r=\sec \theta$$

Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola with focus at (3,0)

Express the following Cartesian coordinates in polar coordinates in at least two different ways. (2,2)

Eliminate the parameter to find a description of the following circles or circular arcs in terms of \(x\) and \(y .\) Give the center and radius, and indicate the positive orientation. $$x=3 \cos t, y=3 \sin t ; \pi \leq t \leq 2 \pi$$

Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola with focus at (-4,0)

Express the following Cartesian coordinates in polar coordinates in at least two different ways. (-1,0)

Make a sketch of the region and its bounding curves. Find the area of the region. The region inside the right lobe of \(r=\sqrt{\cos 2 \theta}\)

Eliminate the parameter to find a description of the following circles or circular arcs in terms of \(x\) and \(y .\) Give the center and radius, and indicate the positive orientation. $$x=3 \cos t, y=3 \sin t ; 0 \leq t \leq \pi / 2$$

Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola symmetric about the \(y\) -axis that passes through the point (2,-6)

Express the following Cartesian coordinates in polar coordinates in at least two different ways. \((1, \sqrt{3})\)

Make a sketch of the region and its bounding curves. Find the area of the region. The region inside the circle \(r=8 \sin \theta\)

Eliminate the parameter to find a description of the following circles or circular arcs in terms of \(x\) and \(y .\) Give the center and radius, and indicate the positive orientation. $$x=\cos t, y=1+\sin t ; 0 \leq t \leq 2 \pi$$

Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola symmetric about the \(x\) -axis that passes through the point (1,-4)

Express the following Cartesian coordinates in polar coordinates in at least two different ways. (-9,0)

Make a sketch of the region and its bounding curves. Find the area of the region. The region inside the cardioid \(r=4+4 \sin \theta\)

Eliminate the parameter to find a description of the following circles or circular arcs in terms of \(x\) and \(y .\) Give the center and radius, and indicate the positive orientation. $$x=2 \sin t-3, y=2 \cos t+5 ; 0 \leq t \leq 2 \pi$$

Express the following Cartesian coordinates in polar coordinates in at least two different ways. \((-4,4 \sqrt{3})\)

Make a sketch of the region and its bounding curves. Find the area of the region. The region inside the limaçon \(r=2+\cos \theta\)

Eliminate the parameter to find a description of the following circles or circular arcs in terms of \(x\) and \(y .\) Give the center and radius, and indicate the positive orientation. $$x=-7 \cos 2 t, y=-7 \sin 2 t ; 0 \leq t \leq \pi$$

Express the following Cartesian coordinates in polar coordinates in at least two different ways. \((4,4 \sqrt{3})\)

Make a sketch of the region and its bounding curves. Find the area of the region. The region inside all the leaves of the rose \(r=3 \sin 2 \theta\)

Eliminate the parameter to find a description of the following circles or circular arcs in terms of \(x\) and \(y .\) Give the center and radius, and indicate the positive orientation. $$x=1-3 \sin 4 \pi t, y=2+3 \cos 4 \pi t ; 0 \leq t \leq \frac{1}{2}$$

Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$\frac{x^{2}}{4}+y^{2}=1$$

Convert the following equations to Cartesian coordinates. Describe the resulting curve. \(r \cos \theta=-4\)

Make a sketch of the region and its bounding curves. Find the area of the region. The region inside one leaf of \(r=\cos 3 \theta\)

Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of \(x\) and \(y .\) Answers are not unique. A circle centered at the origin with radius \(4,\) generated counterclockwise

Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$$