Chapter 10

Give the property that defines all parabolas.

Plot the points with polar coordinates \(\left(2, \frac{\pi}{6}\right)\) and \(\left(-3,-\frac{\pi}{2}\right) .\) Give two alternative sets of coordinate pairs for both points.

Express the polar equation \(r=f(\theta)\) in parametric form in Cartesian coordinates, where \(\theta\) is the parameter.

Explain how a pair of parametric equations generates a curve in the \(x y\) -plane.

Give the property that defines all ellipses.

Write the equations that are used to express a point with polar coordinates \((r, \theta)\) in Cartesian coordinates.

How do you find the slope of the line tangent to the polar graph of \(r=f(\theta)\) at a point?

Give two pairs of parametric equations that generate a circle centered at the origin with radius 6

Give the property that defines all hyperbolas.

Write the equations that are used to express a point with Cartesian coordinates \((x, y)\) in polar coordinates.

Explain why the slope of the line tangent to the polar graph of \(r=f(\theta)\) is not \(\frac{d r}{d \theta}\).

Give parametric equations that describe a full circle of radius \(R,\) centered at the origin with clockwise orientation, where the parameter varies over the interval \([0,10].\)

Sketch the three basic conic sections in standard position with vertices and foci on the \(x\) -axis.

What is the polar equation of a circle of radius \(|a|\) centered at the origin?

Give parametric equations that generate the line with slope \(-2\) passing through \((1,3).\)

What is the polar equation of the vertical line \(x=5 ?\)

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. $$r=1-\sin \theta ;\left(\frac{1}{2}, \frac{\pi}{6}\right)$$

Find parametric equations for the parabola \(y=x^{2}.\)

What is the equation of the standard parabola with its vertex at the origin that opens downward?

What is the polar equation of the horizontal line \(y=5 ?\)

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

Describe the similarities and differences between the parametric equations \(x=t, y=t^{2}\) and \(x=-t, y=t^{2},\) where \(t \geq 0\) in each case.

What is the equation of the standard ellipse with vertices at \((\pm a, 0)\) and foci at \((\pm c, 0) ?\)

Explain three symmetries in polar graphs and how they are detected in equations.

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. $$r=8 \sin \theta ;\left(4, \frac{5 \pi}{6}\right)$$

In which direction is the curve \(x=-2 \sin t, y=2 \cos t,\) for \(0

What is the equation of the standard hyperbola with vertices at \((0, \pm a)\) and foci at \((0, \pm c) ?\)

Explain the Cartesian-to-polar method for graphing polar curves.

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. $$r=4+\sin \theta ;(4,0) \text { and }\left(3, \frac{3 \pi}{2}\right)$$

In which direction is the curve \(x=-2 \sin t, y=2 \cos t,\) for \(0

Given vertices \((\pm a, 0)\) and eccentricity \(e,\) what are the coordinates of the foci of an ellipse and a hyperbola?

Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates. \(\left(2, \frac{\pi}{4}\right)\)

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. $$r=6+3 \cos \theta ;(3, \pi) \text { and }(9,0)$$

Explain how to find the slope of the line tangent to the curve \(x=f(t), y=g(t)\) at the point \((f(a), g(a)).\)

Give the equation in polar coordinates of a conic section with a focus at the origin, eccentricity \(e,\) and a directrix \(x=d,\) where \(d>0\)

Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates. \(\left(3, \frac{2 \pi}{3}\right)\)

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. \(r=2 \sin 3 \theta ;\) at the tips of the leaves

Explain how to find points on the curve \(x=f(t), y=g(t)\) at which there is a horizontal tangent line.

What are the equations of the asymptotes of a standard hyperbola with vertices on the \(x\) -axis?

Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates. \(\left(-1,-\frac{\pi}{3}\right)\)

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. \(r=4 \cos 2 \theta ;\) at the tips of the leaves

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. $$x=2 t, y=3 t-4 ;-10 \leq t \leq 10$$

How does the eccentricity determine the type of conic section?

Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates. \(\left(2, \frac{7 \pi}{4}\right)\)

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. $$r=1+2 \sin 2 \theta ;\left(3, \frac{\pi}{4}\right)$$

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. $$x=t^{2}+2, y=4 t ;-4 \leq t \leq 4$$

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

Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates. \(\left(-4, \frac{3 \pi}{2}\right)\)

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. $$r^{2}=4 \cos 2 \theta ;\left(0, \pm \frac{\pi}{4}\right)$$

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. $$x=-t+6, y=3 t-3 ;-5 \leq t \leq 5$$