Calculo 2 De dos variables_9na Edición - Ron Larson

1059997_1006.qxp 9/8/08 3:40 PM Page 756
1059997_1006.qxp 9/8/08 3:40 PM Page 756
756 CAPÍTULO 10 Cónicas, ecuaciones paramétricas y coordenadas polares
En los ejercicios 31 a 34, usar una graficadora para representar
la cónica. Describir en qué difiere la gráfica de la del ejercicio
Desarrollo de conceptos
756 Chapter 10 Conics, Parametric Equations, and Polar Coordinates
indicado. 756 Chapter 10 Conics, Parametric Equations, and Polar Coordinates 51. Clasificar las cónicas de acuerdo con su excentricidad.
14
52. Identificar cada cónica.
31. r
(Ver ejercicio 15.)
In 1 Exercises sen sin 31–34, 4 use a graphing utility to graph the conic. WRITING 5 ABOUT CONCEPTS5
In Exercises
Describe
31–34,
how the
use
graph
a graphing
differs
utility
from
to
the
graph
graph
the
in the
conic.
indicated WRITING a) r
b) r
6
32. (Ver ejercicio 16.)
51. Classify
1ABOUT 2 cos
the
CONCEPTS
conics by their
10
eccentricities.
sin
Describe r exercise.
how the 4
sen
graph differs from the graph in the indicated
1 cos 3
51. Classify the conics
52. Identify
5 by their eccentricities.
exercise.
each conic.
5
c) r
d) r
6 4
52. Identify 3each 3 conic. cos
1 3 sen sin 4
33. r 31. r 4
(Ver ejercicio (See Exercise 17.) 15.)
5
5
31. r 2 cos 1 sin 6 4 (See Exercise 15.)
53. Describir (a) qué r pasa 5
(b) r
1
con
2 cos
la distancia entre la 5directriz 10 sin
y el centro
de una 1 elipse 2 cos si los focos permanecen 10 fijos sin y e se apro-
1 sin 4
(a) r
(b) r
6 4
34. r 32. r 4
(Ver ejercicio (See Exercise 22.) 16.)
5
5
32. r 3 7 sen sin 1 cos23
3(See Exercise 16.)
xima a (c) 0. r 5
(d) r 5
1 cos 3
(c) r 3 3 cos (d) r 1 3 sin 4
6
3 3 cos
1 3 sin 4
35. Dar 33. la ecuación r 6 de la elipse que se obtiene (See Exercise al girar 6 17.) radianes
53. Describe what happens to the distance between the directrix
33. r 2 cos 6(See Exercise 17.)
53. Describe
en sentido de las manecillas del reloj la elipse
and
what
the
happens
center of
to
an
the
ellipse
distance
if
between
the foci remain
the directrix
2 cos 6
fixed and e
6
and the
34. r
(See Exercise 22.)
approaches
center of an
0.
ellipse if the foci remain fixed and e
34. rr
85
6
3
.
7 sin 2 (See 3 Exercise 22.)
Para approaches discusión 0.
85 3 53 7 cos sin
2 3
35. Write the equation for the ellipse rotated 6 radian clockwise
54. Explicar en qué difiere la gráfica de cada cónica de la grá-
36. 35. Dar Write la ecuación the
from
equation
the
de
ellipse
la for parábola the ellipse que rotated se obtiene 6al radian girar 4 clockwise radianes
from en the sentido ellipse contrario a las manecillas del reloj la parábola
fica 54. de Explain r how the . graph of each conic differs from the graph
CAPSTONE
CAPSTONE 4
8
54. Explain how 1the graph sen sin
4
of each conic differs from the graph
r92
8 .
of r 4
8 5 cos
.
1 sin
4
rr
. .
of a) r
. b) r
18 sen sin 5 cos
1 sin
cos
1 sen sin
36. Write the equation for the parabola rotated 4 radian
4
4
(a) r
(b) r
36. Write
counterclockwise
the equation for
from
the
the
parabola
parabola
rotated 4 radian
4
4 4
(a) c) rr
1 cos d) (b) r r
1 sin
En los counterclockwise ejercicios 37 a 48, from hallar the parabola una ecuación polar de la cónica
1 cos
1 1 sen sin sin 4
4
4
con foco en el polo. (Por 9 conveniencia, la ecuación de la directriz
(c) r 4
(d) r 4
r 9 .
(c) r 1 cos (d) r 1 sin 4
está dada r en forma 1 . rectangular.)
sin
1 cos
1 sin 4
1 sin
x
55. Demostrar que la ecuación polar de 2
es
Cónica Excentricidad Directriz
a y2
In Exercises 37–48, find a polar equation for the conic with its
2 b 1
x 2 In Exercises
focus at
37–48,
the pole.
find
(For
a polar
convenience,
equation for
the
the
equation
conic with
for the
its
directrix
is given in rectangular form.)
r 2
2 that the polar equation for 2 y 2
37. Parábola
55. Show
b e 1
x 1
x 1 is
focus at the pole. (For convenience, the equation for the directrix
Parábola is given in rectangular e 1form.)
1 e 2 cos 2 a
55. Show that the polar . equation for 2 y 2
a 2 1bis
2 Elipse.
38.
2 b 2
y 1
b
Conic
Eccentricity Directrix
r 2 2
. Ellipse
39. Elipse
e 1 b
Conic
Eccentricity
1 e 2 cos 2 x
2
yDirectrix
1
r
56. Demostrar 2 2
. Ellipse
que la ecuación polar de 2
37. Parabola e 1
x 3
1 e 2 cos 2 es
a y2
40. 37. Elipse Parabola
2 b 1 2
38. Parabola
ee 3 1
4 e 1
yx23
y 4
x
56. Show that
41. Hipérbola
b
r 2
2 the polar equation for 2 y 2
1 is
38. Parabola e 1
1
y 4
x
39. Ellipse e 2
.
a 2 b 2
e x 1
56. Show that the polar equation for 2 y 2
1 is
1 2
y 1
Hipérbola.
1 e 2 cos 2 a
2 b 2
39. Ellipse
e
3
42. Hipérbola 40. Ellipse e 3 2
y 1
e
y 2
b
r 2
23
4
x 1
2 . Hyperbola
40. Ellipse
e
b
1 e
41. Hyperbola
2 cos
e 2
x 1
r 2
4
y 2
2 . 2 Hyperbola
En los ejercicios
1 e
41. Hyperbola
2 cos
e 2
57 a 60, usar los resultados de los ejercicios 55
3
x 1
2
Cónica 42. Hyperbola Vértice 3 o e vértices 2
x 1
y 56 para In Exercises dar la forma 57–60, polar use de the la ecuación results of de Exercises la cónica.
42. Hyperbola e
55 and 56 to
2
x 1
In Exercises
43. ParábolaConic
Vertex or Vertices
57. Elipse:
write
foco
the
57–60,
polar
use
en (4,
form
the
0); vértices
of
results
the
en
equation
of Exercises
(5, 0), 5,
of
the conic.
55 and 56 to
Conic Vertex 1, 2or Vertices
write the polar form of the equation of the conic.
58. Hipérbola: 57. Ellipse: foco focus en (5, at 0); (4, vértices 0); vertices en (4, at 0), (5, 4, 0), 5,
43. Parabola 1,
44. Parábola 5,
57. Ellipse: focus at (4, 0); vertices at (5, 0), 5,
43. Parabola 1, 2
x
59.
2 58.
45. Elipse
9 y2 Hyperbola:
44. Parabola
16 1 focus at (5, 0); vertices at (4, 0), 4,
2
58. Hyperbola: focus at (5, 0); vertices at (4, 0), 4,
2, 0, 8, 5,
x
59.
2 y
44. Parabola 5,
2
x 1
45. Ellipse
2, 0 , 8,
60.
2
46. Elipse
4 9 16
45. Ellipse
y2 1
2,
x
46. Ellipse 2 , 3
59.
2 y 2
1
2, 0 , 8,
4, 60.
2
2, 2
, 4, 3 9 16
x y
4
2 1
46. Ellipse
2 2
60.
2
2, , 4, 3 y
3
47. Hipérbola
En los ejercicios 61 a 64, usar las funciones de integración de una
47. Hyperbola 1, 2 , 1, 9, 3 3
2
herramienta In Exercises de graficación 61–64, use para the integration estimar con capabilities una precisión of a graphing de
2 , 9, 3 4
2 1
2 2
47. Hyperbola 1, 3 2
In Exercises
dos cifras utility
61–64,
decimales to approximate
use the integration
el área de to la two región decimal
capabilities
limitada places
of a
por the
graphing
la gráfica area of the
48. Hipérbola 48. Hyperbola 2, 0,
2 , 9, 3 10, 2, 00
2
, 10, 0
utility
de la ecuación region
to approximate
bounded polar. by
to
the
two
graph
decimal
of the
places
polar
the
equation.
area of the
48. Hyperbola 2, 0 , 10, 0
region bounded by the graph of the polar equation.
49. Encontrar 49. Find la a ecuación polar equation para la for elipse the ellipse con foco with (0, focus 0), excentricidad
0, 0 , eccentricity
3
9
49. 1
61. r
62. r
Find
de
a polar
2 , y and directriz
equation
a directrix en
for
r
the
at r4 ellipse
sec 4 . sec
with
.
focus 0, 0 , eccentricity
3
9
1
61. r 2 cos 62. r 4 cos
2, and a directrix at r 4 sec .
2 cos
4 cos
50. Find a polar equation for the hyperbola with focus 0, 0 , eccentricity
2
3
50. Encontrar Find a polar ecuación para una hipérbola con foco (0, 0), excentricidatricity
2, de and 2 y a directriz directrix en at
r 8 8 csc . .
3 2 sen
6 5
63. r
64. r
equation
2, and a
for
directrix
the hyperbola
at r
with
8 csc
focus
.
0 , eccen-
2
3
63. r 3 2 sen 64. r 6 5 sen
sen