Calculo 2 De dos variables_9na Edición - Ron Larson

SECCIÓN 10.4 Coordenadas polares y gráficas polares 739
10.4 Polar Coordinates and Polar Graphs 739
10.4 Polar Coordinates and Polar Graphs 739
58. Fórmula para la distancia
58. Distance Formula
58. a) Distance Verificar Formula que la fórmula para la distancia entre dos puntos
(a)
r
Verify
y dados en coordenadas polares es
(a) Verify 1 , 1
that
r
that 2 , 2
the Distance Formula for the distance between
the two points the Distance r 1 , Formula for the distance between
1 and r 2 ,
the d two rpoints 12 r 22
r 1 , 2r 1 1
and r 2
r 2 , in polar coordinates is
cos1 2
2 in polar coordinates is
2.
d r2 1 r
2
2 2r 1 r 2 cos 1 2
b) Describir las posiciones de los puntos, en .
d r2 1 r
2
2 2r 1 r 2 cos relación uno con
1 2
(b) otro, Describe .
si 1 Simplificar la fórmula de distancia para
(b) este Describe caso. ¿Es the the 2. positions of the points relative to each other
for positions la simplificación of the points lo que relative se esperaba? to each Explicar other
1
por for qué. 1 2 .
2 . Simplify the Distance Formula for this case. Is
the simplification Simplify what the you Distance expected? Formula Explain. for this case. Is
the simplification what you expected? Explain.
c) (c) Simplificar Simplify the la fórmula Distance de Formula distancia for si 11 ¿Es
(c) la Simplify simplificación the Distance lo que se Formula esperaba? for
22 Explicar 1 2 por 90 90 90. . Is the
simplification what you expected? Explain. qué. . Is the
simplification what you expected? Explain.
d) (d) Elegir Choose dos two puntos points en on el the sistema polar coordinate de coordenadas system polares and find y
(d) encontrar Choose the distance two la distancia points between on entre the them. polar ellos. Then coordinate Luego choose elegir system different representaciones
the representations distance polares between diferentes of the them. same para Then los two choose mismos points different and dos apply puntos polar the y
and find polar
aplicar representations Distance la fórmula Formula of para again. the la same distancia. Discuss two the Analizar points result. and el resultado. apply the
Distance Formula again. Discuss the result.
En In los Exercises ejercicios 59– 59 62, a use 62, the usar result el resultado of Exercise del 58 ejercicio to approximate 58 para
aproximar In the Exercises distance la 59– distancia between 62, use the entre the two result los points dos of Exercise puntos in polar descritos 58 coordinates.
approximate en coordenadas
the distance polares. between the two points in polar coordinates.
59. 1, 5 59. 1, 5 60. 8, 7
60. 8, 7 5,
6 , 4, 5,
6 , 4, 3
4 ,
3
4 ,
61. 2, 0.5 , 7, 1.2
62. 4, 2.5 , 12, 1
61. 2, 0.5,
, 7, 1.2
62. 4, 2.5,
, 12, 11
In Exercises 63 and 64, find dy/dx and the slopes of the tangent
En In lines Exercises los
shown
ejercicios 63 on and the
63 y 64, graph
64, find hallar
of dy/dx the
dy/dx
polar and y the equation.
las pendientes slopes of the de tangent las rectas
tangentes shown on que the se graph muestran of the en polar las gráficas equation. de las ecuaciones
lines
polares. 63. r 2 3 sin
64. r 21 sin
63. r 2 3 sin
64. r 21 sin
63. sen sin π
64.
21 sen sin π
π
π 2 5, π 2
π π
π
2 5,
2
5,
π
( )
)
2
2
(2, 0)
2
(2, 0) 0
1 (2, 2 0) 3
7π
0
3, 1 2 3
3π
7π
3,
6
3,
7π
( )
)
−1,
3π
6
−1,
2
−1,
3π
( )
)
2
0
(2, π)
2 3
3π
0
4,
(2, π)
2 3
3π
4,
2
(2, 4,
3π
( )
)
2
In Exercises 65– 68, use a graphing utility to (a) graph the polar
En In equation, Exercises los ejercicios (b) 65– draw 68, 65 a use the 68, a tangent usar graphing una line herramienta utility at the to given (a) graph de value graficación the of polar , and y
a) equation, (c) trazar find dy/dx la (b) gráfica draw at the the de given la tangent ecuación value line of polar, at . the Hint: b) given dibujar Let value the la of recta increment , and tangente
(c) between find en dy/dx el the valor values at the dado of given de equal , value y c) hallar of
/24..
dy/dx Hint: en Let el the valor increment dado de
. between Sugerencia: the values Tomar of incrementos equal /24. de iguales a /24.
65. r 31 cos , 66. r 3 2 cos , 0
65. r 31 31 cos , , 2
66.
r 3 2 cos , , 0
2
67. r 3 sin ,
68. r 4,
67. r 3 sen
sin , , 3
68.
r 4, 4
3
4
En
In
los
Exercises
ejercicios
69 and
69 y
70,
70,
find
hallar
the
los
points
puntos
of horizontal
de tangencia
and
horizontal
vertical
In Exercises 69 and 70, find the points of horizontal and vertical
tangency
y vertical
(if
(si
any)
los
to
hay)
the
a
polar
la curva
curve.
polar.
tangency (if any) to the polar curve.
69.
69. r
1
sin sen
sin
70.
70.
r
a
sin sen
sin
69. r 1 sin
70. r a sin
En In los Exercises ejercicios 71 71 and y 72, 72, hallar find the los points puntos of de horizontal tangencia tangency horizontal
In (if Exercises
(si any) los to hay) the 71
a polar and curva
72, curve. find
polar.
the points of horizontal tangency
(if any) to the polar curve.
71. 71. r 2 csc csc 3
72. 72.
r a sin sen sin cos cos 2
2
71. r 2 csc 3
72. r a sin cos 2
En In los Exercises ejercicios 73–76, a 76, use usar a graphing una herramienta utility to de graficación graph the polar para
representar In equation Exercises and la 73–76, ecuación find all use points polar a graphing y of hallar horizontal utility todos tangency. los to puntos graph de the tangencia
equation horizontal. and find all points of horizontal tangency.
polar
73. r 4 sin cos 2 73. r 4 sin cos 2 2
74. r 3 cos 2 sec
sen
74. r
3 cos 22 sec
)
)
)
)
)
)
)
)
75.
75. r
2
csc
csc
5
76.
76.
r
2 cos
cos3
3
2
2
En
75.
los
r
ejercicios
2 csc
77
5
a 84, dibujar
76.
la
r
gráfica
2 cos
de
3
la ecuación
2
polar
y
In
hallar
Exercises
las tangentes
77–84, sketch
en el polo.
a graph of the polar equation and
In find Exercises the tangents 77–84, at sketch the pole. a graph of the polar equation and
find 77. the r tangents 3 sen sin at the pole. 78. r 5 cos
77. r 5 sin
78. r 5 cos
77. 79. rr 5 21 sin sen sin 78. 80. rr 5 31 cos cos
79. r 21 sin
80. r 31 cos
79. 81. rr 21 2 cos sin 3
80. 82. rr sin 31sen
cos 5
81. r 4 cos 3
82. r sin 5
81. 83. rr 4 3 cos sen sin 32
82. 84. rr 3 sin cos 52
83. r 3 sin 2
84. r 3 cos 2
En 83. los r ejercicios 3 sin 2 85 a 96, trazar 84. la gráfica r 3 cos de la 2 ecuación polar.
In Exercises 85–96, sketch a graph of the polar equation.
In 85. Exercises r 8 85–96, sketch a graph 86. of rthe 1 polar equation.
87.
85. r
41
8
cos 88.
86.
r
1
sen sin
85. r 8
86. r 1
89.
87. r
3
41
2 cos
cos
90.
88.
r
5
1
4
sin
sin sen
87. r 41 cos
88. r 1 sin
89. r 3 2 cos
90. r 5 4 sin
6
89. 91. rr 3 csc 2 cos
90. 92. rr 5 4 sin
2 sen sin 6
91. r 3 csc
92. r
3 cos
91. 92.
93. r 2
94. r 1 6
r 3 csc
r 2 sin 3 cos
2 sin 3 cos
1
93. r 2
94. r 1
93. 95. rr 2 2
4 cos 2
94. 96. rr 2 4 sin sen
95. r 2 4 cos 2
96. r 2 4 sin
En 95. los r 2 ejercicios 4 cos 2 97 a 100, usar 96. una rherramienta 2 4 sin de graficación
para In Exercises representar 97–100, la ecuación use a graphing y mostrar utility que la to recta graph dada the es equation
una
Exercises and de show la 97–100, gráfica. that the use given a graphing line is an utility asymptote graph of the graph. equa-
In asíntota
tion and show that the given line is an asymptote of the graph.
Nombre Name of de Graph la gráfica Polar Ecuación Equation polar Asymptote Asíntota
Name Concoide of Graph
97. Conchoid
Polar r Equation
r 2 2 sec sec
Asymptote
x x 1 1
97. Conchoid
98. Concoide Conchoid
98. Conchoid
99. Espiral Hyperbolic hiperbólica spiral
99. Hyperbolic Estrofoide spiral
100. Strophoid
r
r
r
r
r
r
r2 2sec
2 csc csc
r2 csc
2 2
r2
2 cos 2 cos 2 sec 2 sec
x y 1
y 1 1
y y 1
y 2
2
y x 2
x 2 2
100. Strophoid
r 2 cos 2 sec x 2
Desarrollo WRITING ABOUT de conceptos
CONCEPTS
WRITING ABOUT CONCEPTS
101. Describir Describe the las differences diferencias between entre el the sistema rectangular de coordenadas coordinate
101. Describe the differences between the rectangular coordinate
rectangulares system and the y el polar sistema coordinate de coordenadas system. polares.
system and the polar coordinate system.
102. Dar Give las the ecuaciones equations para for the pasar coordinate de coordenadas conversion rectangulares
rectangular a coordenadas to polar polares coordinates y viceversa. and vice versa.
from
102. Give the equations for the coordinate conversion from
rectangular to polar coordinates and vice versa.
103. ¿Cómo How are se the determinan slopes of las tangent pendientes lines de determined rectas tangentes in polar en
103. How are the slopes of determined in polar
coordenadas coordinates? polares? What are ¿Qué tangent son lines las rectas at the tangentes pole and en howel
coordinates? What are tangent lines at the pole and how
polo are they y cómo determined? se determinan?
are they determined?
Para discusión
CAPSTONE
CAPSTONE
104. Describir Describe las the gráficas graphs of de the las following siguientes polar ecuaciones equations. polares.
104. Describe the graphs of the following polar equations.
a) r 7
b) r 2 7
a) r 7
b) r 2 7
7
7
c) r 7
d) r 7
c) r cos
d) r sen
e) r cos 7 cos f) r sen 7 sen
e) r 7 cos f) r 7 sen
105. Trazar la gráfica de r sin en el intervalo dado.
105. Sketch the graph of r 4
sen
sin over each interval.
105. Sketch the graph of r 4
a) b) c)
2 ≤ ≤
2 ≤ sin over each interval.
0 ≤ ≤
≤
(a) 0 2 (b) (c) 2
(a) 0 2 (b) 2
(c) 2 2
106. Para pensar 2 Utilizar una 2 herramienta graficadora 2 para representar
About la ecuación It Use polar a graphing r 61 utility cos to graph the para polara)
equation r 61 cos for (a) 0, (b) 4,
2
106. Think About It Use a graphing utility to graph the polar
106. Think
equation b) r 61 cos y c) for (a) Usar 0, las (b) gráficas para
and (c) 2. Use the graphs to describe the effect of 4, the
and describir (c) el efecto 2. Use del ángulo the graphs . Escribir to describe la ecuación the effect como of the función
de . sen Write para the equation el inciso as c). a function of sin for part
angle . Write the equation as a function of sin for part (c).
angle (c).
0,
4,
2.