Stewart Section 10.3: Polar Coordinates

78 questions

--:--
0 / 78
Stewart Section 10.3: Polar Coordinates 0/78
1 Polar Coordinates - Plotting · Level 1
Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with \(r > 0\) and one with \(r < 0\).
(a) \(\left(1, \dfrac{\pi}{4}\right)\)
(b) \(\left(-2, 3 \dfrac{\pi}{2}\right)\)
(c) \(\left(3, -\dfrac{\pi}{3}\right)\)

Enter your answer directly below each part above.

2 Polar Coordinates - Plotting · Level 1
Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with \(r > 0\) and one with \(r < 0\).
(a) \(\left(2, 5 \dfrac{\pi}{6}\right)\)
(b) \(\left(1, -2 \dfrac{\pi}{3}\right)\)
(c) \(\left(-1, 5 \dfrac{\pi}{4}\right)\)

Enter your answer directly below each part above.

3 Polar Coordinates - Cartesian Conversion · Level 2
Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point.
(a) \(\left(2, 3 \dfrac{\pi}{2}\right)\)
(b) \(\left(\sqrt{2}, \dfrac{\pi}{4}\right)\)
(c) \(\left(-1, -\dfrac{\pi}{6}\right)\)

Enter your answer directly below each part above.

4 Polar Coordinates - Cartesian Conversion · Level 2
Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point.
(a) \(\left(4, 4 \dfrac{\pi}{3}\right)\)
(b) \(\left(-2, 3 \dfrac{\pi}{4}\right)\)
(c) \(\left(-3, -\dfrac{\pi}{3}\right)\)

Enter your answer directly below each part above.

5 Polar Coordinates - Polar Conversion · Level 2
The Cartesian coordinates of a point are given. (i) Find polar coordinates \((r, \theta)\) of the point, where \(r > 0\) and \(0 \leq \theta < 2 \pi\). (ii) Find polar coordinates \((r, \theta)\) of the point, where \(r < 0\) and \(0 \leq \theta < 2 \pi\).
(a) \((-4, 4)\)
(b) \((3, 3 \sqrt{3})\)

Enter your answer directly below each part above.

6 Polar Coordinates - Polar Conversion · Level 2
The Cartesian coordinates of a point are given. (i) Find polar coordinates \((r, \theta)\) of the point, where \(r > 0\) and \(0 \leq \theta < 2 \pi\). (ii) Find polar coordinates \((r, \theta)\) of the point, where \(r < 0\) and \(0 \leq \theta < 2 \pi\).
(a) \((\sqrt{3}, -1)\)
(b) \((-6, 0)\)

Enter your answer directly below each part above.

7 Polar Coordinates - Regions · Level 2
Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. \(r \geq 1\)
8 Polar Coordinates - Regions · Level 2
Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. \(0 \leq r < 2\), \(\pi \leq \theta \leq 3 \dfrac{\pi}{2}\)
9 Polar Coordinates - Regions · Level 2
Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. \(r \geq 0\), \(\dfrac{\pi}{4} \leq \theta \leq 3 \dfrac{\pi}{4}\)
10 Polar Coordinates - Regions · Level 2
Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. \(1 \leq r \leq 3\), \(\dfrac{\pi}{6} < \theta < 5 \dfrac{\pi}{6}\)
11 Polar Coordinates - Regions · Level 2
Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. \(2 < r < 3\), \(5 \dfrac{\pi}{3} \leq \theta \leq 7 \dfrac{\pi}{3}\)
12 Polar Coordinates - Regions · Level 2
Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. \(r \geq 1\), \(\pi \leq \theta \leq 2 \pi\)
13 Polar Coordinates - Distance · Level 2
Find the distance between the points with polar coordinates \(\left(4, 4 \dfrac{\pi}{3}\right)\) and \(\left(6, 5 \dfrac{\pi}{3}\right)\).
14 Polar Coordinates - Distance · Level 3
Find a formula for the distance between the points with polar coordinates \((r_1, \theta_1)\) and \((r_2, \theta_2)\).
15 Polar Coordinates - Identify Curve · Level 2
Identify the curve by finding a Cartesian equation for the curve. \(r^2 = 5\)
16 Polar Coordinates - Identify Curve · Level 2
Identify the curve by finding a Cartesian equation for the curve. \(r = 4 \sec \theta\)
17 Polar Coordinates - Identify Curve · Level 2
Identify the curve by finding a Cartesian equation for the curve. \(r = 5 \cos \theta\)
18 Polar Coordinates - Identify Curve · Level 2
Identify the curve by finding a Cartesian equation for the curve. \(\theta = \dfrac{\pi}{3}\)
19 Polar Coordinates - Identify Curve · Level 3
Identify the curve by finding a Cartesian equation for the curve. \(r^2 \cos 2 \theta = 1\)
20 Polar Coordinates - Identify Curve · Level 3
Identify the curve by finding a Cartesian equation for the curve. \(r^2 \sin 2 \theta = 1\)
21 Polar Coordinates - Polar Equation · Level 2
Find a polar equation for the curve represented by the given Cartesian equation. \(y = 2\)
22 Polar Coordinates - Polar Equation · Level 2
Find a polar equation for the curve represented by the given Cartesian equation. \(y = x\)
23 Polar Coordinates - Polar Equation · Level 2
Find a polar equation for the curve represented by the given Cartesian equation. \(y = 1 + 3x\)
24 Polar Coordinates - Polar Equation · Level 2
Find a polar equation for the curve represented by the given Cartesian equation. \(4y^2 = x\)
25 Polar Coordinates - Polar Equation · Level 3
Find a polar equation for the curve represented by the given Cartesian equation. \(x^2 + y^2 = 2c x\)
26 Polar Coordinates - Polar Equation · Level 3
Find a polar equation for the curve represented by the given Cartesian equation. \(x^2 - y^2 = 4\)
27 Polar Coordinates - Conceptual · Level 2
For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve.
(a) A line through the origin that makes an angle of \(\dfrac{\pi}{6}\) with the positive \(x\)-axis
(b) A vertical line through the point \((3, 3)\)

Enter your answer directly below each part above.

28 Polar Coordinates - Conceptual · Level 2
For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve.
(a) A circle with radius 5 and center \((2, 3)\)
(b) A circle centered at the origin with radius 4

Enter your answer directly below each part above.

29 Polar Curves - Sketching · Level 2
\( r = -2 \sin \theta \)
30 Polar Curves - Sketching · Level 2
\( r = 1 - \cos \theta \)
31 Polar Curves - Sketching · Level 2
\( r = 2(1 + \cos \theta) \)
32 Polar Curves - Sketching · Level 2
\( r = 1 + 2 \cos \theta \)
33 Polar Curves - Sketching · Level 2
\(r = \theta\), \(\theta \geq 0\)
34 Polar Curves - Sketching · Level 2
\(r = \theta^2\), \(-2 \pi \leq \theta \leq 2 \pi\)
35 Polar Curves - Rose Curves · Level 2
\( r = 3 \cos 3 \theta \)
36 Polar Curves - Rose Curves · Level 2
\( r = -\sin 5 \theta \)
37 Polar Curves - Rose Curves · Level 2
\( r = 2 \cos 4 \theta \)
38 Polar Curves - Rose Curves · Level 2
\( r = 2 \sin 6 \theta \)
39 Polar Curves - Limacons · Level 2
\( r = 1 + 3 \cos \theta \)
40 Polar Curves - Limacons · Level 2
\( r = 1 + 5 \sin \theta \)
41 Polar Curves - Lemniscate · Level 3
\( r^2 = 9 \sin 2 \theta \)
42 Polar Curves - Lemniscate · Level 3
\( r^2 = \cos 4 \theta \)
43 Polar Curves - Sketching · Level 2
\( r = 2 + \sin 3 \theta \)
44 Polar Curves - Sketching · Level 3
\( r^2 \theta = 1 \)
45 Polar Curves - Sketching · Level 2
\( r = \sin\left(\dfrac{\theta}{2}\right) \)
46 Polar Curves - Sketching · Level 2
\( r = \cos\left(\dfrac{\theta}{3}\right) \)
47 Polar Curves - Graph to Curve · Level 3
The figure shows a graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. Use it to sketch the corresponding polar curve.
question image
48 Polar Curves - Graph to Curve · Level 3
The figure shows a graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. Use it to sketch the corresponding polar curve.
question image
49 Polar Curves - Asymptotes · Level 4
Show that the polar curve \(r = 4 + 2 \sec \theta\) (called a conchoid) has the line \(x = 2\) as a vertical asymptote by showing that \(\operatorname*{lim}\limits_{r \rightarrow \infty} x = 2\). Use this fact to help sketch the conchoid.
50 Polar Curves - Asymptotes · Level 4
Show that the curve \(r = 2 - \csc \theta\) (also a conchoid) has the line \(y = -1\) as a horizontal asymptote by showing that \(\operatorname*{lim}\limits_{r \rightarrow \infty} y = -1\). Use this fact to help sketch the conchoid.
51 Polar Curves - Asymptotes · Level 4
Show that the curve \(r = \sin \theta \tan \theta\) (called a cissoid of Diocles) has the line \(x = 1\) as a vertical asymptote. Show also that the curve lies entirely within the vertical strip \(0 < x < 1\). Use these facts to help sketch the cissoid.
52 Polar Curves - Sketching · Level 3
Sketch the curve \((x^2 + y^2)^3 = 4 x^2 y^2\).
53 Polar Curves - Limacons · Level 4
(a) In Example 11 the graphs suggest that the limacon \(r = 1 + c \sin \theta\) has an inner loop when \(|c| > 1\). Prove that this is true, and find the values of \(\theta\) that correspond to the inner loop.
(b) From Figure 19 it appears that the limacon loses its dimple when \(c = \dfrac{1}{2}\). Prove this.

Enter your answer directly below each part above.

54 Polar Curves - Graph Matching · Level 3
Match the polar equations with the graphs labeled I-VI. Give reasons for your choices. (Don't use a graphing device.)
question image
(a) \(r = \ln \theta\), \(1 \leq \theta \leq 6 \pi\)
(b) \(r = \theta^2\), \(0 \leq \theta \leq 8 \pi\)
(c) \(r = \cos 3 \theta\)
(d) \(r = 2 + \cos 3 \theta\)
(e) \(r = \cos\left(\dfrac{\theta}{2}\right)\)
(f) \(r = 2 + \cos\left(3 \dfrac{\theta}{2}\right)\)

Enter your answer directly below each part above.

55 Polar Curves - Tangent Lines · Level 3
Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\). \(r = 2 \cos \theta\), \(\theta = \dfrac{\pi}{3}\)
56 Polar Curves - Tangent Lines · Level 3
Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\). \(r = 2 + \sin 3 \theta\), \(\theta = \dfrac{\pi}{4}\)
57 Polar Curves - Tangent Lines · Level 3
Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\). \(r = \dfrac{1}{\theta}\), \(\theta = \pi\)
58 Polar Curves - Tangent Lines · Level 3
Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\). \(r = \cos\left(\dfrac{\theta}{3}\right)\), \(\theta = \pi\)
59 Polar Curves - Tangent Lines · Level 3
Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\). \(r = \cos 2 \theta\), \(\theta = \dfrac{\pi}{4}\)
60 Polar Curves - Tangent Lines · Level 3
Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\). \(r = 1 + 2 \cos \theta\), \(\theta = \dfrac{\pi}{3}\)
61 Polar Curves - Horizontal/Vertical Tangents · Level 3
Find the points on the given curve where the tangent line is horizontal or vertical. \(r = 3 \cos \theta\)
62 Polar Curves - Horizontal/Vertical Tangents · Level 3
Find the points on the given curve where the tangent line is horizontal or vertical. \(r = 1 - \sin \theta\)
63 Polar Curves - Horizontal/Vertical Tangents · Level 3
Find the points on the given curve where the tangent line is horizontal or vertical. \(r = 1 + \cos \theta\)
64 Polar Curves - Horizontal/Vertical Tangents · Level 3
Find the points on the given curve where the tangent line is horizontal or vertical. \(r = e^\theta\)
65 Polar Curves - Proof · Level 4
Show that the polar equation \(r = a \sin \theta + b \cos \theta\), where \(a b \neq 0\), represents a circle, and find its center and radius.
66 Polar Curves - Proof · Level 4
Show that the curves \(r = a \sin \theta\) and \(r = a \cos \theta\) intersect at right angles.
67 Polar Curves - Graphing · Level 3
Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. \(r = 1 + 2 \sin\left(\dfrac{\theta}{2}\right)\) (nephroid of Freeth)
68 Polar Curves - Graphing · Level 3
Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. \(r = \sqrt{1 - 0.8 \sin^2 \theta}\) (hippopede)
69 Polar Curves - Graphing · Level 3
Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. \(r = e^{\sin \theta} - 2 \cos(4 \theta)\) (butterfly curve)
70 Polar Curves - Graphing · Level 3
Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. \(r = |\tan \theta|^{|\cot \theta|}\) (valentine curve)
71 Polar Curves - Graphing · Level 3
Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. \(r = 1 + \cos^{999} \theta\) (Pac-Man curve)
72 Polar Curves - Graphing · Level 3
Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. \(r = 2 + \cos\left(9 \dfrac{\theta}{4}\right)\)
73 Polar Curves - Transformations · Level 3
How are the graphs of \(r = 1 + \sin\left(\theta - \dfrac{\pi}{6}\right)\) and \(r = 1 + \sin\left(\theta - \dfrac{\pi}{3}\right)\) related to the graph of \(r = 1 + \sin \theta\)? In general, how is the graph of \(r = f(\theta - \alpha)\) related to the graph of \(r = f(\theta)\)?
74 Polar Curves - Optimization · Level 3
Use a graph to estimate the \(y\)-coordinate of the highest points on the curve \(r = \sin 2 \theta\). Then use calculus to find the exact value.
75 Polar Curves - Families · Level 3
Investigate the family of curves with polar equations \(r = 1 + c \cos \theta\), where \(c\) is a real number. How does the shape change as \(c\) changes?
76 Polar Curves - Families · Level 3
Investigate the family of polar curves \(r = 1 + \cos^n \theta\) where \(n\) is a positive integer. How does the shape change as \(n\) increases? What happens as \(n\) becomes large? Explain the shape for large \(n\) by considering the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates.
77 Polar Curves - Tangent Angle · Level 4
Let \(P\) be any point (except the origin) on the curve \(r = f(\theta)\). If \(\psi\) is the angle between the tangent line at \(P\) and the radial line \(O P\), show that \(\tan \psi = \dfrac{r}{d \dfrac{r}{d} \theta}\) [Hint: Observe that \(\psi = \phi - \theta\) in the figure.]
question image
78 Polar Curves - Tangent Angle · Level 4
(a) Use Exercise 77 to show that the angle between the tangent line and the radial line is \(\psi = \dfrac{\pi}{4}\) at every point on the curve \(r = e^\theta\).
(b) Illustrate part (a) by graphing the curve and the tangent lines at the points where \(\theta = 0\) and \(\dfrac{\pi}{2}\).
(c) Prove that any polar curve \(r = f(\theta)\) with the property that the angle \(\psi\) between the radial line and the tangent line is a constant must be of the form \(r = C e^{k \theta}\), where \(C\) and \(k\) are constants.

Enter your answer directly below each part above.

Answered: 0 / 78