Calculo 2 De dos variables_9na Edición - Ron Larson

Review Exercises 759
En los ejercicios 33 a 36, hallar una representación paramétrica En los ejercicios 53 y 54, a) usar una
Review
herramienta
Exercises
de graficación
759
Review Exercises 759 Review Exer
de la recta o cónica.
para trazar la curva representada por las ecuaciones paramétricas,
b) usar una herramienta de graficación para hallar dx/dq,
33. In Exercises Recta: pasa 33–36, por 2, find 6 a parametric y 3, 2 representation of of the line
In Exercises 53 and 54, (a) use a graphing utility to to graph the
In Exercises 33–36, find parametric
In Exercises
representation
33–36, find
of
a
the
parametric
line
representation In dy/dq Exercises y dy/dx53 para of
and
the line
54, (a) use y In c) usar Exercises
graphing una herramienta 53
utility
and
to
54, graph
(a) de use graficación
represented
the
or In graphing util
34. or
Exercises conic.
conic.
33–36, find a parametric representation of the line curve In
Circunferencia: centro en (4, or 5); conic. radio 3
curve
Exercises represented 53 and by
para trazar to la by
54, the
recta the
(a) use parametric
tangente parametric
a graphing equations, curve a represented la curva equations,
utility to (b) cuando by
(b)
graph use
the use
the a
parametric equati
35. or conic.
Elipse: centro en 3, 4;
2,
longitud del
3, curve graphing
eje mayor horizontal 8 y
graphing
represented utility to find utility to find
by dx/d dx/d
the parametric , dy/d , and dy/dx
to dy/d
graphing and utility
dy/dx
equations, for
for
(b) /6,
to find to dx/d
/6,
use and
,
and
a
33. Line: passes through 2, 6 and 3, 2
33. Line: passes through 2, 53. 54.
longitud del eje menor 6
33.
and
Line:
3, dy/d , and dy/dx f
passes through 2, 6 and 3,
(c) graphing
2
use x a cot graphing x 2 sin at 4,
sen
(c) use graphing
utility to find utility utility
dx/d to graph
to graph
, dy/d the
(c) use
the
, and tangent
a
tangent
dy/dx for line to the
graphing
line
utility
to the
/6, curve
to
curve
and
34. 33. Circle: Line: passes center through at 4, 2, 5; 6radius and 3, 2
graph the tangent l
34. Circle: center at 4, 5;
36. Hipérbola: vértice en
34.
radius
foco
Circle: (c) when
en
center at 4, 5; radius 3 when
use a graphing /6.
y sen sin /6.
utility to graph the tangent line to the curve
when y 2/6.
cos 2
35. 34. Ellipse: Circle: center at at 4, 3, 3, 0, 4 ±4; 5;
;
horizontal radius 3
major 0, ±5
axis of of length 8 and
when
/6.
35. Ellipse: center of at 3, horizontal
35. Ellipse:
major
center
axis
at
of length
3, 4 ; horizontal
and 53. major x axis cot
of length 8 and 54.
x 2 sin
35. minor Ellipse: axis Longitud de arco En los ejercicios 53. 55 y 56, hallar la longitud de54.
37. Motor
minor axis
center of length
rotatorio
of length
at 3, 6
El 4 ; horizontal major axis of length 8 and 53. cot
54.
x
cot
sin
motor rotatorio minor fue axis inventado of length por 6 Felix 53. x 2 s
at 0, at 0, yx
sin cot 2
54. yx 2 cos sin
36. Hyperbola: minor axis of vertices length at 6 0, ±4 ; foci at 0, ±5
arco de la sin curva en el intervalo que se indica.
Wankel en década de los cincuenta. 36. Hyperbola: Contiene vertices un rotor at que es un
y sin 2
cos
36. Hyperbola: vertices at 0, ±4 foci at 0, ±5
y 2 co
0, ±4 ; foci at 0, y±5
sin 2
y 2 cos
36. Hyperbola: vertices at 0, ±4 ; foci at 0, ±5
triángulo equilátero modificado. El rotor se mueve en una cámara
Arc 55. x
Length
rcos
In
Exercises sen
55 and Arc
56.
56, Length
x
find 6
the cos
arc In Exercises
length of of the
37. Rotary Engine The rotary engine was developed by Felix
Arc Length
In Exercises 55 and 56, find the arc length
55
of the
37. and 56, find the a
que, Rotary en dos Engine dimensiones, The It rotary es un 37. epitrocoide. engine
Rotary was
Engine Usar developed una is The herramienta
by
rotary
Felix
engine
curve Arc
curve was
Length
developed
on the given
y on rsin the sengiven In
by
Exercises interval.
interval. Felix
55 and 56, find the arc length of the
37. Wankel Rotary curve on y the 6 sen sin given interval.
de Wankel
Engine the 1950s.
graficación in the para 1950s.
The It
trazar It
rotary features la features
engine a rotor,
cámara Wankel que rotor,
was which
in describen in which
developed is a modified
the 1950s. las is ecuaciones modified
by Felix
It in features a rotor, curve which
on the is
given
a modified
interval.
equilateral Wankel paramétricas.
0 ≤ ≤
0 ≤ ≤
equilateral
in triangle.
triangle.
the 1950s. The
is The
It rotor
rotor
features moves
moves
a rotor, in a chamber equilateral in chamber
which is that,
triangle. that,
a modified in two 55. x r cos sin 56. x 6 cos
The to in rotor
two
moves
55.
in a chamber cos
that, in
sin
two 55.
56.
x r cos cos
dimensions, equilateral sin 56. x 6 cos
x dimensions,
triangle. is epitrochoid.
cos 3 is an epitrochoid.
The rotor moves Use a
5 cos Use
dimensions, graphing
graphing
a chamber utility is
utility
that, to
an epitrochoid.
to
in graph
graph
two 55. xy r cos sin cos sin 56. xy
6 cos sin
the dimensions, chamber Use a graphing utility to graph
the chamber
is modeled
modeled
an epitrochoid. by the parametric
by the parametric
Use a graphing equations
Área de una superficie En los ejercicios y r sin 57 y 58, cos hallar el área y 6 sin
the chamber
equations
utility to graph
sin cos
sin
y
modeled by the parametric equations 0 r sin cos
0y
6 sin
the chamber modeled by the parametric equations
de la superficie generada por revolución 0
de la curva en torno 0
yx cos 3 5 cos
cos cos
0
0
x cos 3 5 cos
a) al eje x y b) al eje y.
of y x cos 3 5 cos
Surface Area In Exercises 57 and 58, find the area of the
and sen sin 3 5 sen sin .
Surface Area In Exercises 57
Surface and 58,
Area
find In
the Exercises
area of 57
the
and 58, find
and
surface Surface generated
and
surface 57. x generated
Area In by
t,
3t, by
Exercises revolving
0revolving 57 the
≤ t ≤ 2surface the
and curve
curve
58, find about
generated
about
the (a)
by
(a)
area the
revolving
the
of x-axis
-axis
the
and
the curve abou
38. Curva y sin serpentina 3 5 sin Considerar .
and surface (b) the
las ecuaciones paramétricas and (b) the
generated y-axis.
-axis.
by revolving the curve about (a) the x-axis
sin sin y sin 3 5 sin .
and (b) the y-axis.
x y 2 sin cot 3
y y 5 sin 4 sen sin .
cos , 0 < <
.
and
58.
(b)
the t, 2 cos
y
-axis.
,
y 2 sin sen
,
0 ≤ ≤
38. Serpentine Curve Consider the parametric equations
57. x t, y 3t, 0 t 2 2
38.
a) Serpentine
Usar una Curve
herramienta Consider graficación 38. Serpentine the parametric trazar Curve
equations
57.
t,
3t,
la curva. Consider the parametric equations
57. x t, y 3t, 0 t 2
38. Serpentine x 2 cot and
cot and
Curve y 4 Consider sin cos ,
sin cos Área
En los ejercicios 59 y 60, hallar el área de la región.
b) Eliminar el parámetro para to mostrar
x
the 0 < parametric < .
2 cot
que la
and
equations
57. x t, y 3t, 0 t 2
ecuación
y 4 sin
rectangular
de la curva serpentina es (a)
cos , 0 58. < x< 2 . cos , y 2 sin , 0
(a) x Use 2 cot a graphing and y utility 4 sin to cos graph , 0 the < curve. < .
58.
cos sin 58.
x 2 cos 2 , y 2 sin , 0
(a) Use graphing utility to graph the
4 Use
curve.
58. x 2 cos , y 2 sin , 0
x 2 a ygraphing 8x. utility to graph the 59. curve.
2
(b) (a) Use Eliminate a graphing the utility parameter to graph to to show the curve.
x 3 sen sin
60.
x 2
that the rectangular
cos of
(b) Eliminate of the parameter to
(b)
show
Eliminate is
that the
the rectangular Area In Exercises 59 and 60, find the area of the region.
(b) equation Eliminate of parameter to show ythat 2 the cos rectangular
Area yIn Exercises sen sin 59 and 60, find the area of t
En los ejercicios
equation
39
of
the serpentine
the
a 48,
serpentine
parameter curve
a) hallar
curve
to show is 4
Area In Exercises 59 and 60, find the area of the region.
dy/dxequation is
y that x
los puntos 2 of 2 the y rectangular 8x.
the de serpentine
8x.
tangen-curvcia horizontal, b) eliminar el parámetro cuando 2 y 8x.
59. x 3 sin
60. x 2 cos
Area
is 4 In
x 2 Exercises 59 and 60, find
y 8x.
the area of the region.
equation of the serpentine curve is 4 x
of sea 59. posible y
59. x 0 3 ≤
≤
60. x 2 cos
c) trazar la curva representada In por Exercises las ecuaciones 39–48, paramétricas.
(a) find dy/dx and all points 2 ≤ sin
60.
cos
In Exercises 39–48, (a) find dy/dx and all points of horizontal
≤
In Exercises 39–48, (a) find
dy/dx
and all points of horizontal 59. x
y 32 sin cos
of horizontal 2
60.
yx
sin 2 cos
tangency, In cos
y
2 cos
sin
y sin
tangency,
Exercises (b) (b)
39–48, eliminate eliminate
(a) find the the
dy/dx parameter
parameter
and all where
tangency, (b)
where
points possible,
eliminate
possible,
of horizontal and
the
and
parameter where
y 2 cos
possible, and
y sin
(c) tangency, sketch the y
y
(c) sketch the
(b) curve
curve
eliminate represented
represented
the parameter by the parametric
(c)
by
sketch
the parametric
where possible, equations.
the curve represented
equations.
and
by the parametric equations.
0
0
(c) sketch the curve represented 4t by the parametric 6, equations.
2 2
39. 40. x t 6, y t 4
2 32
39. 5t, 4t
39.
40.
x 2 5t,
6,
y 1 2
0
x 2 5t, y 1 4t
2
2 y 2
4t 40. x t 6, y t 3
2
y
39. x 2 y2
1
41. 2t 1
41. x
1
1
41.
42.
x
42. x
4
3
t , t2
y 2t 3 t , y
2t 42. x
4
3
t , y 5t, 2t y 1 3 4t 40. x t t , 6, y
y t 2
y
y
1
1
t2
t2
t2
41. x
42. x
34
23
t , 1
3
x
2
1
y 2t 3 t , y t2
2t −3 −2 −1 1 2 3
43. 2t 3
2
43. x
44. x 2t 1 1
2t 1 1
x
−1
43.
44.
x
2t 44. x 2t 1
43. x 2t 44. x 2t 2t 1
−3 −2 −1 1 1 2 3
x
−1
−3 −2 −1
1 −2
2t −1
1 1
1
−3 −2 −1 1 2 3
2t
1
−3 1 −2 −1−2
−3
−3 −2 −1
x
−1
y
1
x
−3 −2 −1 −1
x
y
y 2t
−1
t y
−1
−2
2
2 1 2t
t
2t
t 2
2
2 1 2t
−3 −2 −1 1 2 3
−3 −2 −1 1 2 3
2t
2t
t 2 −3 −2 −1 −1
x
2t −1
−3 −2 −1
−2 −1
y
y
−2 1 2 3
45.
x 5t 2
cos 2t
t 2 2t
−3 −2 −1−2
1 2 3
−1−3
−2
−2 −1
−3 −2
45.
cos
45. x 5 cos
−2
−3
45. x En los ejercicios 61 a 64, representar gráficamente −2 el punto
−3
y 53 cos 4 sen
−2
−3
sen
y 3 4 sen
en coordenadas polares y hallar in las coordenadas rectangula-
46. xy 10 3 cos 4 sen
In Exercises 61–64, plot the point in polar coordinates and find
46.
10 cos
46. x 10 cos
In res Exercises correspondientes 61–64, plot al the punto. point
In Exercises polar coordinates of 61–64, plot
and
the point
find
in polar coord
46.
x
the corresponding rectangular coordinates of the point.
y 10 cos
In
the
Exercises
corresponding
61–64,
rectangular
plot the point
the
coordinates polar coordinates
corresponding
of the
rectangular
point.
and find
sen
10 sen
coordinates of th
y 10 sen
the corresponding
47. x cos 5, 47.
cos 3
3
47. x cos 3
61. 5, 3 rectangular coordinates of the point.
y 10 sen
61. 5, 47. x y 4 cos sen 3
2
61. 5, 3 sen 3
3
61. 5, 3 y 4 sen 3
2
48.
xy e4 6, 48.
t sen 3
2
t
62. 6, 7 48. x e t
62. 6, 7 62. 6, 48. x
6
y e t
t
t
62. 6, 7 6
y e t
63. 3, 3, 1.56 6
y e 63. 3, 1.56
t
of 63. 3, 1.56
En In los Exercises ejercicios 49–52, a find 52, hallar points todos (if los any) puntos of horizontal (si los hay) and de
2, In Exercises 49–52, to find points In Exercises (if any)
49–52,
of horizontal
find to all
and 64. 63. 2, 3, 1.56 2.45
points (if 64. any) of 2, horizontal 2.45 and
tangencia In vertical tangency horizontal to the y vertical curve. Use a la a curva. graphing Usar utility una herramienta
vertical your
to confirm
64. 2, 2.45
vertical
Exercises tangency
49–52,
to the
find
curve.
all points
vertical
Use graphing
(if any) of
tangency
utility
horizontal
to the
to
curve.
confirm
and 64. 2, 2.45
Use a graphing utility to confirm
of your de graficación results.
results.
tangency para to the confirmar curve. Use los a resultados. graphing utility to confirm In Exercises 65–68, the rectangular coordinates of a point are
your results.
In En Exercises los ejercicios 65–68, a the 68, se rectangular dan In las Exercises coordenadas of coordinates 65–68, rectangulares of
the point
rectangular of are de coordinate
your
49.
results.
t,
2t given. In Plot the point and find two sets of polar coordinates of the
x 5 t, y 2t
49.
given.
Exercises
un punto. Plot Representar the
65–68,
point and
the rectangular gráficamente find two
given.
sets
coordinates
Plot
of el punto polar
the point
coordinates
of a point
y hallar and dos find
of
are
pares two
the
t,
2t 2
2
sets of polar co
2,
2t 49. x 5 t, y 2t 2
point given. for 0 < 2 .
point de coordenadas for
Plot the point
polares and find two sets of polar coordinates of the
50. 49. x 5t 2, t, y 2t t del punto point para for 0 < 2 .
50.
2,
3 2
3 2t
2t
50. x t 2, y t 3 2t
point 4, for
0 < 2 .
0, 51. 50.
x 65. 4, 4
66. 0, 7
t2 2, 2 sen y , t 3 y 2t 1 cos
65. 4, 51.
65.
66.
4,
0,
4
sen 66. 0, 7
51.
cos
x 2 2 sen , y 1 cos
1, 3, 52. 51.
67. 65. 4, 1, 34
68. 66. 0, 3, 7
x
3
2 2 sen cos ,
y 12 sen cos 2
67. 1, 52.
67.
68.
1, 3
3, cos sen 68. 3,
52.
52.
x 2 2 cos , y 2 sen 2 67. 1, 3
68. 3, 3
x 2 2 cos , y 2 sen 2
/6,
sin
cos
Ejercicios de repaso 759
/6.