de 30 de noviembre de 1833)
1.7. El origen moderno de la seguridad privada: La Guardería Rural
1.7.5. Fracasos y reformas de la Guardería Rural
SOLUTION We can see from Figure 6.54 that the curve is a limaçon with an inner
loop and maximum r-value 5. The graph is symmetric only about the x-axis. Domain: All reals.
Range:1, 5 Continuous
Symmetric about the x-axis. Bounded
Maximum r-value: 5
No asymptotes. Now try Exercise 39.
Graphs of Limaçon Curves
The graphs of ra b sin and ra b cos , where a0 and b 0, have the following characteristics:
Domain: All reals Range:a b,a b Continuous
Symmetry: ra b sin , symmetric about y-axis ra b cos , symmetric about x-axis Bounded
Maximum r-value:a b No asymptotes
EXPLORATION 1 Limaçon Curves
Try several values for aand bto convince yourself of the characteristics of limaçon curves listed above.
FIGURE 6.54 The graph of a limaçon with an inner loop. (Example 6)
[–3, 8] by [–4, 4]
=0
θ
R=51
FIGURE 6.55 The graph of rfor (a)0 (set min 0,max 45, step 0.1) and (b) 0 (set min 45, max 0,step 0.1). (Example 7)
[–30, 30] by [–20, 20] (b) [–30, 30] by [–20, 20]
(a)
Other Polar Curves
All the polar curves we have graphed so far have been bounded. The spiral in Example 7 is unbounded.
EXAMPLE 7
Analyzing the Spiral of Archimedes
Analyze the graph of r.SOLUTION We can see from Figure 6.55 that the curve has no maximum r-value
and is symmetric about the y-axis. Domain: All reals.
Range: All reals. Continuous
Symmetric about the y-axis. Unbounded
No maximum r-value.
The are graphs of polar equations of the form r2a2sin 2 and r2a2cos 2. EXAMPLE 8
Analyzing a Lemniscate Curve
Analyze the graph of r24 cos 2for 0, 2.SOLUTION It turns out that you can get the complete graph using r2cos 2.
You also need to choose a very small step to produce the graph in Figure 6.56. Domain:0,
434, 5474, 2Range:2, 2
Symmetric about the x-axis, the y-axis, and the origin. Continuous (on its domain)
Bounded
Maximum r-value: 2
No asymptotes. Now try Exercise 43.
lemniscate curves
FOLLOW-UP
Have students try to confirm the x-axis symmetry of Example 4 by using the replacement (r,) instead of using (r,). Discuss the results.
ASSIGNMENT GUIDE
Day 1: Ex. 1–8 all, 15–30, multiples of 3 Day 2: Ex. 9–12, 33–48, multiples of 3, 58, 61–65
COOPERATIVE LEARNING
Group Activity: Ex. 57
NOTES ON EXERCISES
Ex. 45–48 require students to find the dis- tance from the pole to the furthest point on the petal, not the arc length. Ex. 58–60 provide additional discussion of examples in the text.
Ex. 61–66 provide practice with standard- ized tests.
ONGOING ASSESSMENT
Self-Assessment: Ex. 13, 21, 23, 29, 33, 39, 41, 43
Embedded Assessment: Ex. 59, 60, 73
FIGURE 6.56 The graph of the lemniscate r24 cos 2. (Example 8)
[–4.7, 4.7] by [–3.1, 3.1]
EXPLORATION 2 Revisiting Example 8
1. Prove that -values in the intervals
4, 34and 54, 74are not in the domain of the polar equation r24 cos 2.2. Explain why r 2cos2produces the same graph as r2cos2 in the interval 0, 2.
3. Use the symmetry tests to show that the graph of r24 cos 2is sym-
metric about the x-axis.
4. Use the symmetry tests to show that the graph of r24 cos 2is sym-
metric about the y-axis.
5. Use the symmetry tests to show that the graph of r24 cos 2is sym-
metric about the origin.
EXPLORATION EXTENSIONS
Graph r24 sin 2. How is this graph
related to the graph of r24 cos 2?
QUICK REVIEW 6.5
(For help, go to Sections 1.2 and 5.3.) In Exercises 1 – 4, find the absolute maximum value and absoluteminimum value in 0, 2and where they occur.
1.y3 cos 2x 2.y23 cos x
3.y2cos2x 4.y33 sin x
In Exercises 5 and 6, determine if the graph of the function is sym- metric about the (a)x-axis,(b)y-axis, and (c)origin.
5.ysin 2x no; no; yes 6.ycos 4x no; yes; no
In Exercises 7–10, use trig identities to simplify the expression.
7.sin sin 8.cos cos
SECTION 6.5 EXERCISES
In Exercises 1 and 2,(a)complete the table for the polar equation, and
(b)plot the corresponding points.
1.r3 cos 2
2.r2 sin 3
In Exercises 3 – 6, draw a graph of the rose curve. State the smallest -interval 0 kthat will produce a complete graph.
3.r3 sin 3 4.r 3 cos 2
5.r3 cos 2 6.r3 sin 5 Exercises 7 and 8 refer to the curves in the given figure.
7.The graphs of which equations are shown? r3is graph (b).
r13 cos 6 r23 sin 8 r33cos 3
8.Use trigonometric identities to explain which of these curves is the graph of r6 cos 2sin 2. (a)
In Exercises 9 – 12, match the equation with its graph without using your graphing calculator.
[–4.7, 4.7] by [–4.1, 2.1] (d) [–3.7, 5.7] by [–3.1, 3.1] (c) [–4.7, 4.7] by [–3.1, 3.1] (b) [–4.7, 4.7] by [–4.1, 2.1] (a) [–4.7, 4.7] by [–3.1, 3.1] (b) [–4.7, 4.7] by [–3.1, 3.1] (a) 0
6 3 2 23 56 r 0 2 0 2 0 2 0 0 4 2 34 54 32 74 r 3 0 3 0 3 0 3 09.Does the graph of r22 sin or r22 cos appear in the figure? Explain. Graph (b) isr 2 2 cos .
10.Does the graph of r23 cos or r23 cos appear in the figure? Explain. Graph (c) is r23 cos .
11.Is the graph in (a) the graph of r22 sin or r22 cos ? Explain. Graph (a) is r22 sin .
12.Is the graph in (d) the graph of r21.5 cos or r21.5 sin ? Explain. Graph (d) is r21.5 sin .
In Exercises 13– 20, use the polar symmetry tests to determine if the graph is symmetric about the x-axis, the y-axis, or the origin.
13.r33 sin 14. r12 cos 15.r43 cos 16. r13 sin 17.r5 cos 2 18. r7 sin 3 19.r 1 3 sin 20. r 1 2 cos
In Exercises 21– 24, identify the points for 0 2 where maxi- mum r-values occur on the graph of the polar equation.
21.r23 cos 22. r 32 sin
23.r3 cos 3 24. r4 sin 2 In Exercises 25 – 44, analyze the graph of the polar curve.
25.r3 26. r 2 27.3 28. 4 29.r2 sin 3 30. r 3 cos 4 31.r54 sin 32. r65 cos 33.r44 cos 34. r55 sin 35.r52 cos 36. r3sin 37.r25 cos 38. r34 sin 39.r1cos 40. r2sin 41.r2 42. r4 43.r2sin 2, 0 2 44. r29 cos 2, 0 2 In Exercises 45–48, find the length of each petal of the polar curve.
45.r24 sin 2 46. r35 cos 2
47.r14 cos 5 48. r34 sin 5
In Exercises 49–52, select the two equations whose graphs are the same curve. Then, even though the graphs of the equations are identical, describe how the two paths are different as increases from 0 to 2.
49.r113 sin , r2 13 sin , r313 sin
50.r112 cos , r2 12 cos , r3 12 cos
51.r112 cos , r212 cos , r3 12 cos
In Exercises 53 – 56,(a)describe the graph of the polar equation,
(b)state any symmetry that the graph possesses, and (c)state its maximum r-value if it exists.
53.r2 sin22sin 2 54. r3 cos 2sin 3
55.r13 cos 3 56. r13 sin 3
57.Group Activity Analyze the graphs of the polar equations racos nand rasin nwhen nis an even integer.
58.Revisiting Example 4 Use the polar symmetry tests to prove that the graph of the curve r3 sin 4is symmetric about the y-axis and the origin.
59.Writing to Learn Revisiting Example 5 Confirm the range stated for the polar function r33 sin of Example 5 by graphing y33 sin xfor 0 x 2. Explain why this works.
60.Writing to Learn Revisiting Example 6 Confirm the range stated for the polar function r23 cos of Example 6 by graphing y23 cos xfor 0 x 2. Explain why this works.
Standardized Test Questions
61. True or False A polar curve is always bounded. Justify your answer. False. The spiral ris unbounded.
62. True or False The graph of r2 cos is symmetric about the x-axis. Justify your answer.
In Exercises 63–66, solve the problem without using a calculator.
63. Multiple Choice Which of the following gives the number of petals of the rose curve r3 cos 2? D
(A)1 (B) 2 (C) 3 (D) 4 (E) 6
64. Multiple Choice Which of the following describes the symme- try of the rose graph of r3 cos 2? D
(A) only the x-axis
(B) only the y-axis
(C) only the origin
(D) the x-axis, the y-axis, the origin
(E) Not symmetric about the x-axis, the y-axis, or the origin
65. Multiple Choice Which of the following is a maximum r-value for r23 cos ? B
(A)6 (B) 5 (C) 3 (D) 2 (E) 1
66. Multiple Choice Which of the following is the number of petals of the rose curve r 5 sin 3? B
(A)1 (B) 3 (C) 6 (D) 10 (E) 15
Explorations
67.Analyzing Rose Curves Consider the polar equation racos nfor n,an odd integer.
(a)Prove that the graph is symmetric about the x-axis.
(b)Prove that the graph is not symmetric about the y-axis.
(c)Prove that the graph is not symmetric about the origin.
(d)Prove that the maximum r-value is a.
(e)Analyze the graph of this curve.
68.Analyzing Rose Curves Consider the polar equation rasin nfor nan odd integer.
(a)Prove that the graph is symmetric about the y-axis.
(b)Prove that the graph is not symmetric about the x-axis.
(c)Prove that the graph is not symmetric about the origin.
(d)Prove that the maximum r-value is a.
(e)Analyze the graph of this curve.
69.Extended Rose Curves The graphs of r13 sin52and r23 sin 72may be called rose curves.
(a)Determine the smallest -interval that will produce a complete graph of r1; of r2.
(b)How many petals does each graph have?
Extending the Ideas
In Exercises 70– 72, graph each polar equation. Describe how they are related to each other.
70. (a)r13 sin 3 (b)r23 sin 3
(
1 2
)
(c)r33 sin 3(
4)
71. (a)r12 sec (b)r22 sec
(
4
)
(c)r32 sec(
3)
72. (a)r122 cos (b)r2r1(
4)
(c)r3r1(
3)
73.Writing to Learn Describe how the graphs of rf, rf, and rf are related. Explain why you think this generalization is true.
6.6
De Moivre’s Theorem and nth Roots
What you’ll learn about