Question

True or False A cardioid passes through the pole.

Question

Multiple Choice The equations for cardioids and limaçons are very similar. They all have the form \begin{equation*} r=a\pm b \ \cos \theta \qquad\hbox{or}\qquad r=a\pm b \ \sin \theta ,\;a>0, b>0 \end{equation*} The equations represent a limaçon with an inner loop if [(a) \(a\lt b\), (b) \(a>b\), (c) \(a= b]\); a cardioid if [(a) \(a\lt b\), (b) \(a>b\), (c) \(a= b];\) and a limaçon without an inner loop if [(a) \(a\lt b\), (b) \(a>b\), (c) \(a= b].\)

Question

True or False The graph of \(r=\sin ( 4\theta )\) is a rose.

Question

The rose \(r= \ \cos ( 3\theta )\) has __________ petals.

Question

\(r=2+2 \ \cos \theta\)

Question

\(r=3-3\sin \theta\)

Question

\(r=4-2 \ \cos \theta\)

Question

\(r=2+\sin \theta\)

Question

\(r=1+2\sin \theta\)

Question

\(r=2-3 \ \cos \theta\)

Question

\(r=\sin ( 3\theta )\)

Question

\(r=4 \ \cos ( 4\theta )\)

Question

\(r=8\cos \theta, \,\;\, r=2\sec \theta\)

Question

\(r=8\sin \theta,\,\;\, r=4\csc \theta\)

Question

\(r=\sin \theta, \,\;\, r=1+ \ \cos \theta\)

Question

\(r=3, \,\;\, r=2+2 \ \cos \theta\)

Question

\(r=1+\sin \theta, \,\;\, r=1+ \ \cos \theta\)

Question

\(r=1+ \ \cos \theta, \,\;\, r=3 \ \cos \theta\)

Question

\(r=f(\theta )=e^{\theta /2}\, \text{from} \,\theta = 0\, \text{to} \,\theta =2\)

Question

\(r=f(\theta )=e^{2\theta }\, \text{from} \,\theta = 0\, \text{to} \,\theta =2\)

Question

\(r=f(\theta )= \ \cos ^{2}\dfrac{\theta }{2}\, \text{from} \,\theta = 0\, \text{to} \,\theta =\pi\)

Question

\(r=f(\theta )=\sin ^{2}\dfrac{\theta }{2}\, \text{from} \,\theta = 0\, \text{to} \,\theta =\pi\)

Question

Question

Question

Question

Question

\(r=2 \ \cos ( 3\theta )\, \text{at} \,\theta =\dfrac{\pi }{6}\)

Question

\(r=3\sin ( 3\theta )\, \text{at} \,\theta =\dfrac{\pi }{3}\)

Question

\(r=2+ \ \cos \theta\, \text{at} \,\theta =\dfrac{\pi }{4}\)

Question

\(r=3-\sin \theta\, \text{at} \,\theta =\dfrac{\pi }{6}\)

677

Question

\(r=4+5\sin \theta\, \text{at} \,\theta =\dfrac{\pi }{4}\)

Question

\(r=1-2 \ \cos \theta\, \text{at} \,\theta =\dfrac{\pi }{4}\)

Question

\(r^{2}=4 \ \sin ( 2\theta )\)

Question

\(r^{2}=9 \ \cos ( 2\theta )\)

Question

\(r^{2}= \ \cos ( 2\theta )\)

Question

\(r^{2}=16 \ \sin ( 2\theta )\)

Question

\(r=\dfrac{2}{1- \ \cos \theta }\) (parabola)

Question

\(r=\dfrac{1}{1- \ \cos \theta }\) (parabola)

Question

\(r=\dfrac{1}{3-2 \ \cos \theta }\) (ellipse)

Question

\(r=\dfrac{2}{1-2 \ \cos \theta }\) (hyperbola)

Question

\(r=\theta\); \(\theta \geq 0\) (spiral of Archimedes)

Question

\(r=\dfrac{3}{\theta }\); \(\theta >0\) (reciprocal spiral)

Question

\(r=\csc \theta -2; 0\lt\theta \lt\pi\) (conchoid)

Question

\(r=3-\dfrac{1}{2}\csc \theta\) (conchoid)

Question

\(r=\sin \theta \tan \theta\) (cissoid)

Question

\(r= \ \cos \dfrac{\theta }{2}\)

Question

\(r=\tan \theta\) (kappa curve)

Question

\(r=\cot \theta\) (kappa curve)

Question

Show that \(r=4(\cos \theta +1)\) and \(r=4(\cos \theta -1)\) have the same graph.

Question

Show that \(r=5(\sin \theta +1)\) and \(r=5(\sin \theta -1)\) have the same graph.

Question

Arc Length Find the arc length of the spiral \(r=\theta\) from \(\theta =0\) to \(\theta =2\pi\).

Question

Arc Length Find the arc length of the spiral \(r=3\theta\) from \(\theta =0\) to \(\theta =2\pi .\)

Question

Perimeter Find the perimeter of the cardioid \(r=f(\theta )=1- \ \cos \theta ,\) \(-\pi \leq \theta \leq \pi \).

Question

Exploring Using Graphing Technology

1. Graph \(r_{1}=2 \ \cos ( 4\theta ) .\) Clear the screen and graph \(r_{2}= 2 \ \cos ( 6\theta ) .\) How many petals does each of the graphs have?
2. Clear the screen and graph, in order, each on a clear screen, \(r_{1}=2 \ \cos ( 3\theta ),\;r_{2}=2 \ \cos ( 5\theta )\), and \(r_{3}=2 \ \cos ( 7\theta )\). What do you notice about the number of petals? Do the results support the definition of a rose?

Question

Exploring Using Graphing Technology Graph \(r_{1}=3-2 \ \cos \theta\). Clear the screen and graph \(r_{2}=3+2 \ \cos \theta\). Clear the screen and graph \(r_{3}=3+2\sin \theta .\) Clear the screen and graph \(r_{4}=3-2\sin \theta .\) Describe the pattern.

Question

Horizontal and Vertical Tangent Lines Find the horizontal and vertical tangent lines of the cardioid \(r=1-\sin \theta\) discussed in Example 1.

Question

Horizontal and Vertical Tangent Lines Find the horizontal and vertical tangent lines of the cardioid \(r=3+3 \ \cos \theta\).

Question

Horizontal and Vertical Tangent Lines Find the horizontal and vertical tangent lines of the limaçon with an inner loop \(r=1+2 \ \cos \theta\) discussed in Example 3.

Question

Horizontal and Vertical Tangent Lines Find the horizontal and vertical tangent lines of the rose with four petals \(r=2 \ \cos ( 2\theta ),\;0\leq \theta \leq 2\pi\), discussed in Example 4.

Question

Horizontal and Vertical Tangent Lines Find the horizontal and vertical tangent lines of the spiral \(r=e^{\theta /5}\) discussed in Example 5.

Question

Horizontal and Vertical Tangent Lines Find the horizontal and vertical tangent lines of the lemniscate \(r^{2}=4\,\sin\,( 2\theta )\).

Question

Test for Symmetry Symmetry with respect to the polar axis can be tested by replacing \(\theta\) with \(-\theta .\) If an equivalent equation results, the graph is symmetric with respect to the polar axis.

1. Explain why this test is valid.
2. Use the test to show that \(r=3+2 \ \cos \theta\) is symmetric with respect to the polar axis.

Question

Test for Symmetry Symmetry with respect to the pole can be tested by replacing \(r\) by \(-r\) or by replacing \(\theta\) by \(\theta +\pi .\) If either substitution produces an equivalent equation, the graph is symmetric with respect to the pole.

1. Explain why these tests are valid.
2. Show that \(r^{2}=4 \ \sin\, ( 2\theta )\) is symmetric with respect to the pole.

Question

Test for Symmetry Symmetry with respect to the line \(\theta =\dfrac{\pi }{2}\) can be tested by replacing \(\theta\) by \(\pi -\theta .\) If an equivalent equation results, the graph is symmetric with respect to the line \(\theta =\dfrac{\pi }{2}\).

1. Explain why this test is valid.
2. Use the test to show that \(r=2 \ \cos ( 2\theta )\) is symmetric with respect to the line \(\theta =\dfrac{\pi }{2}\).

Question

Testing for Symmetry The graph of \(r=\sin ( 2\theta )\) (a rose with four petals) is symmetric with respect to the polar axis, the line \(\theta =\dfrac{\pi }{2}\), and the pole. Show that the test for symmetry with respect to the pole (see Problem 63) works, but the test for symmetry with respect to the polar axis fails (see Problem 62).

678

Question

Arc Length Find the entire arc length of the curve \(r=a \ \sin ^{3}\dfrac{\theta }{3},\) \(a>0\). (Hint: Use parametric equations.)

Question

Arc Length of a Rose Petal

1. Graph the rose \(r=4\sin ( 5\theta )\).
2. The petal in the first quadrant begins when \(\theta =0\) and ends when \(\theta =\alpha .\) Find \(\alpha .\)
3. Find the length \(s\) of the petal described in (b).