Calculo 2 De dos variables_9na Edición - Ron Larson

840 Chapter 12 Vector-Valued Functions
840 CAPÍTULO Chapter 12 12 12Vector-Valued Funciones Functions vectoriales
In 840 Exercises Chapter 27–42, 12 sketch Vector-Valued the curve represented Functions by the vectorvalued
En In In Exercises
function
los ejercicios 27–42,
and sketch
give the
a 42, dibujar the
orientation
the curve la represented
of the curve.
curva representada by by the the vector-
por la
función valued function vectorial t and y dar la orientación the de la of curva. the 27. r t
28. r t 5 t i tj
4 i and t give 1 j
the orientation of the curve.
In Exercises 27–42, sketch the curve represented by the vectorvalued
27. 27. rrt
function t t
29. t t 3 28. r t 5 t i tj
4 i t 2t 1 j 28. r t 5
30.
t 2 t i t 2 tj
4 i and t j
1 give j the orientation of the curve.
t j
31. 29. 29. rt t tcos 3 t
27. r t
i 3 i t 2 j 2 j3 sin j 32. 30. 28. 30.
rt r tt 2 ttcos 2 5t 2 ti ti tt i
i 2 t sin t 2 t 2 tj
tj
tjt tj
j
33.
31. r 3 4 i t 1 j
31. r cos cos sec ii i 332 sin sen
sin tan jj
jj
32. rt
32. r t 2 2 cos cos ti ti 2 2 sin sen
sin tj tj
29.
34.
34.
33.
r r t t 2 t
3 cos 3 i
sec 3 i
t
ti 2 j
i 2 sin tan 3 j
30.
tj rt r t
2
t
cos 2
j
3 ti
t i
2
t
33. r 3 sec i 2 tan j
sen sin 2 3 tj
t j
31.
35. 34.
rt r t
cos
t 2 t
i
cos 3 11i tii
3 sin
4t
j
sin 3 tj 22j j
32.
2t
r
t
33k
2 cos ti 2 sin tj
34. r t 2 cos 3 ti 2 sin 3 tj
k
33.
36. 35.
rt r ti
3
sec
2t
i
t
1 i5 5j
2 tan
j
4t 3tk
j
35. r t t 1 i 4t 22j j 2t 2t 33k
34.
37. 36.
rt r t 2
2
ti cos
cos ti 3 ti
2t 2
2
36. r t ti 2t 55 sin sen
sin
j tj 3
tj
j 3tk 3tk
tk
35. r t t 1 i 4t 2 j 2t 3 k
38. 37. 37.
r t ti 22 cos cos 3 ti cos ti tj 22 sin sin 3 tj tj sen tk tk
36. r t ti 2t 5 j 3tk
39.
39. 38. 38.
rt r t
2 ti ti sin
sin 3 ti
ti 3 cos cos 2 tj tj cos
cos 3 tj
tj 3 sen sen
e tk tk t t k
37. r t 2
sen
cos ti 2 40.
40. 39.
rt
t 3 sin tj tk
39. r t 22 isin sin ti 2tj
2tj ti 2 22tk
cos cos tj tj e
2tk e t t k
38. r t ti 3
41.
41. 40.
t 2 3
rt t,
t, ti 2 , 2 3 t cos tj
40. r t t 2 3
3
3 sen tk
i 2tj
2tk
39. r t 2 sin 2tk
42.
42. 41. 41.
rt r t cos
cos t, t, t 2 tt , 2 , 2 ti
2 3 t 3 sin
sin
t,
t,
sin
sin
t
t cos
cos
t,
t,
t
3 t 2 cos tj e t k
3
40. r t t 2 sen3
i 2tj
sen t
2tk
42. 42. r t cos cos tt t t sin sin t, t, sin sin tt t t cos cos t, t, t
CAS In Exercises 43–46, use a computer algebra t system to graph the
CAS En 41. los r tejercicios t, t 2 , 43 2 3 a 46, usar un sistema algebraico por computadora
In 42. In Exercises r ta fin cos 43–46, de representar t use t use sin a t, a computer sin gráficamente t t cos algebra t, t la system función to to vectorial graph the thee
identificar vector-valued la 1curva function común. and the 43. r t
vector-valued function t 3
and identify the common curve.
CAS CAS
2 t and identify 3
2 i tj
1
43. r t
2 t 3
rt 2 i tj
t 2 3 2 t the common curve.
CAS In Exercises 43–46, use a computer 2 k algebra system to graph the
43. vector-valued 1
r t
2 t function 2 i tj and 3identify 3
44. r t ti
3
44.
ti
2 t 1
rt 3
t 1
2 2 t 2 k the common curve.
j
2 t 2 k
1
43. r t 3
44. r t ti
2 j
2 t 2 2 t 1
2 t 3
2 i 2 tj
j
3 2 45. r t sin ti
cos 2 t k
t 2 k
t 1
2 t j 1
3
45.
sin ti
2 cos t 1
2 t j 1
2 cos t 3
rt
3
2 cos t 3
k
3
2
44. r t ti 3
45. r t sin ti
3
46.
2 sin ti2 cos 2 cos t 1
tj2 t j 1
2 sin tk
2 cos t 3
sen 2 t 1
2 j
2 t 2 k
2 k
2
3
46.
CAS Think 45. 46. rt rr t
About t
2 sin It ti 22 In sin
sen sin Exercises ti ti 22 cos cos 47 tj tj and 2 2
48, 2 sin
use sin tk
a tk
2 cos t 1
2 t j 1
sen2 cos computer t 3
algebra k
2
system to graph the vector-valued function r t . For each u t ,
CAS CAS Think
Para pensar
About It It En
In
los In Exercises
ejercicios 47 47 and
y and 48,
48, 48, usar
use use un
a a computer
sistema algebraico
algebra
CAS 46. r t 2 sin ti 2 cos tj 2 sin tk
make
system
a conjecture about the transformation (if any) of the
por computadora
to to graph the the fin
vector-valued de representar
function
gráficamente
rrt t . . For For each
la función
u t t , ,
CAS graph
make Think of r t . Use a computer algebra system to verify your
vectorial a About a conjecture It In Exercises
rt. Para
about cada
the the 47
ut,
transformation and 48, use a computer
conjeturar sobre
(if (if any) algebra
la
of
transformación
of of to rrgraph (si
of the the
conjecture.
graph system t t . . Use
la Use the
hay)
a a computer vector-valued de la gráfica
algebra function
de
system
rt.
r t .
Usar
to to For verify each
un sistema
your
u t ,
1
47. conjecture. make
algebraico r t
a conjecture
2 por cos computadora ti 2
about
sin tj
the transformation (if any) of the
para 2tk verificar la conjetura.
graph of r t . Use a computer algebra 1 system to verify your
1
47.
1
conjecture. 47. (a) rrt t 22 cos cos 2 ti cos ti t 22 sin sin 1 itj
tj 2 sin
47. rt 2 cos ti 2 sin tj 1 2tk
2tk
(b)
1
(a)
t 2 cos tit 21 sin i tj tj 2tk
sen
1
(a) u t 2 cos t 1 i 2 2tk
1sin tj tj 2tk
47. a) r tut 2 2cos ti t 2 sin 1i tj 2 sin tj 1 2tk
(c)
2
(b)
cos tit i 2 2 sin sin
sen 2tk
tj t 2tk j
tk 1
(b) u t 2 cos ti 2 sin tj 2tk 2 t k
1
(d) b) (a) ut u t 12cos cos tit 21 sin tj 2tk 1
(c)
1
(c)
u t 2ti 2 cos cos2 sin t ti itj sen i 2 sin tj
22 sin 2 sin cos ttk
2tk
tjj
2 t (e) c) ut 21costi 1 2 sintj 1
(d)
1 2 t k
(b) u t 2 ti 2 sin
(d)
u t
62ti ti
2ti 22 sin sin 6 tj sin sentj 2tk
tj 22 cos cos 2tk
tk 2tk
48. rd)
1
(e) t ut ti 1 1
(c) u t 2 cos
2ti t6 1 t i 2 sin
(e) u t 6 cos j cos
2 ti ti 2t sin 3 ktj 66 sin
sin
2 tj tj
cos 1t j
sen tk 2 t k
1
2tk
(d) u t
2tk
48. (a) e) r tut u t ti 6 ti2ti cos t 2 1
j ti t
2t 3 1
48. r t ti t 2 12 sin tj 2
j 2t 3 k6 2sin j tj 1 2 t cos sen
3 2tk k
1
(e) u t 6 cos ti
48. (b) (a)
t t 2 1
rt ti t 2 ti 2 j tj
1 2t 3 16 sin tj
(a) u t ti t 2 k2t 22 3 1
jk
j 2 t 3 2 t 2tk
k 3 48. r t ti t
(c)
a) (b)
t 2 1
ut ti
ti 2 1
j
i t
t 2 2 (b) u t t 2 1
i tj j 2t 3 k
tj 2j 2t 2t 3 k 3 1 4
2t 3 k
(a) u t ti t (d)
b) (c)
ti t 2 1
ut t 2 i tj 2 1
j
1 1
(c) u t ti t 2 12 j
j 8t
2t 2t 3 3 k
2t 33
2
44 t 3 k
k
(b) u t t
(e)
c) (d)
2 1
i tj
tit i 2 1
ut ti t 2 j t
8t 3 2t 2 1
(d) u t ti t 2 12t j 8t 3 j
3 k
3 k 2 t
4k
3 k
(c) u t ti t 2 1
j 2t 3
(e)
d)
(e) u t t i i t 2 1
In Exercises ut 49–56, ti t 2 j represent 1 t j 2 t 3 8t 3 k
2 14 k
j 2 t
the plane 3 k
(d) u t ti t 2 1
j 8t 3 k
curve by a vectorvalued
In
function. (There are
In Exercises e)
ut ti 49–56, represent t 2 many
j 1 correct
2t the the 3 plane
answers.)
(e) u t t i t 2 1
j 2 tk
3 k curve by by a a vector-
49. valued y function. x 5 (There are are many 50. correct 2x answers.) 3y 5 0
En In los Exercises ejercicios 49–56, a 56, representar the plane curva curve plana by por a vectorvalued
49. una yy función function. xx 52 5
2 52.
medio
51. 49. de vectorial. (There are (Hay many muchas 50. 50. correct
y2x 2x 4
respuestas 3y answers.) 3yx 2 55correctas.)
00
51. 51. y x
x 2
5
2 22 52.
52. y
2x
y 4
4 x
3y
x 2
49. y x 5
50. 2x 3y 2
5 0
51. y xx 2 2 22 52.
y 4 x 22
53. x 2 y 2 25
54.
x 2
y 2
x 2 2 y 2 4
53. 53. x
55. x 2 2 yy 2 2 25
25
54. x
1
54. x x
56.
2
x 22 2 2 2 y 1y 2 2 4
4
16
x 55. 4
y 2
9
x 56.
2 16
55. 2 2
y 2
1
16 y y2
2
x
56.
2 y 2
53. x 1
16
2 y
4
2 125
54. x 2
99
16
2 y 2 4
In Exercises 57 and 58, find vector-valued functions 16 forming the
x
boundaries 55. 2 y 2
x
of the region in the figure. 56.
2 y
1
2
State the interval 1
In
En In Exercises
for the
los 16 ejercicios 457 57 and and 57
58, 58, y
find
58, hallar
vector-valued funciones 9 functions 16vectoriales forming que
the
describan
los límites
the
parameter
boundaries
of
of
each
of the the region
function.
de la región
in in the the figure.
en la figura.
State the
Dar the interval el intervalo
for for the the correspondiente
Exercises y of of 57 each and
57. parameter In
al parámetro
function.
58, find vector-valued
de cada 58. función. y functions forming the
boundaries of y = the x 2 region in the figure. State the interval for the
57. 57. 5 y
y
58. 12 y
parameter of each function.
58.
y
4 y y = = x x 2 2 10 x 2 + y 2 = 100
5
5
12
8
12
57. 3 y
58. y
4
10
4
6
10 x x 2 2 + + y y 2 2 = = 100
100
2 y = x 2 8
3
4
8
35
12
1
2
6
6
24
10
2
45° x 2 + y 2 = 100
x
4
84
x
13
1 1 2 3 4 5
2
62
2 45° 4
45° 6 8 10 12
2
x
x
4
x
x
2
4
6
8
10
12
In Exercises 1 11 22
33
44
55
2 4 6 8 10 12
59–66, sketch the space curve 2 represented 45° by the
x
x
intersection
In In Exercises
of the surfaces. Then represent the curve by a
1 259–66, 3 4 sketch 5 the the space curve 2 represented 4 6 8 10 12by by the the
vector-valued
intersection En los ejercicios of
function
of the the 59 surfaces. a using 66, dibujar the
Then
given la represent curva parameter. en the el the espacio curve by representada
In Exercises
by a a
vector-valued Surfaces
por la 59–66, function intersección sketch using de the the las space given superficies. curve parameter.
Parameter
represented Después representar
intersection la curva of por the medio surfaces. de una Then función represent vectorial the curve usando by a el
by the
59.
parámetro vector-valued
zSurfaces
x 2 dado.
y 2 function
, x y
using
0
the given parameter.
xParameter
t
60. 59. 59.
zz x x 2
Superficies
2 yy 2 , 2 , zxx 4yy 00
xx
2 ttcos t
Surfaces
Parámetro
60.
2 2 cos t
z 4
Parameter
61. 60. xz 2 xy 22 y4, 2 , z 4x 2 x 2 sin cos tt
59. z x 62. 61. 4xx 2 2 y 4y
y 2 2 2 , x y 0
x t
61. x 2 y 2 4,
4, z 2 z 16, xx 2 sin 2 x z 2 zx t2 sin t
60. z x 63. 62. x 2 4x 2 y 2 y 2
4y
2 z 2 , z 4
x 2 cos t
62. 4x 2 4y 2 2 zz 2 2 4, 16, 16, x xz x z z 22 xzz
1tt
sin t
61. x 64. 63.
2 y
x 2 y 2 4, z x
2 z
2 10, 2 x 2 sin sen t
63. x 2 y 2 z 2 4, 4, xx zyz 2
4
x
x 21 1 sin sin t t
62. 4x 65. 64.
2 4y 2 z 2 z
2 4, 2 y 2 16, x z 2
10, z 2 x 4
2 z t
64. x 2 y 2 z 2 10, x yy 44
x t 22first sin sin octant t
63. x 66. 65.
2 y 2 yz 2 z
2 z4, 2 2 4, x z 2 x 1 sin sen t
65. x 2 z 2 y
y16, 2 t first 2 zxy z 2 2 4
4
x
t first octant
64. x 2 y 2 z 2 10, x y 4
x 2 sen sin t
66.
67.
66. x Show 2 2 yy that 2 2 zz the 2 2 16,
vector-valued 16, xy xy 4 x
t first first octant
65. x 2 z 2 4, y 2 z 2 4
function x
t (primer first octant octante)
67. 66. 67. rShow x t 2 that y ti that 2 the the 2t z 2 vector-valued cos 16, tj xy 2t sin function 4
tk x
t (primer first octant octante)
67. Mostrar que la función vectorial
67. lies r
Show rtton ti the that ti cone the
2t 2t cos
vector-valued cos 4xtj 2 tj y2t 2 2t sin sin zfunction
2 tk.
tkSketch the curve.
68. Show lies
rt ti 2t cos tj 2t sin tk
lies on on that the the the cone
vector-valued 4x 4x 22 yy 2 sen
r t ti 2t cos tj 2t 2 sin
function zz 2 tk . 2 . Sketch the the curve.
68. 68. rShow se t encuentra ethat t cos the the en ti vector-valued el cono e t sin 4xtj 2 function ey 2 t k
z 2 . Dibujar la curva.
lies on the cone 4x 2 y 2 z 2 . Sketch the curve.
68. Mostrar que la función vectorial
68. lies r
Show rtton ethe that e t t cos cone the cos ti
vector-valued ti z 2 eex t 2t sin sin tj ytj 2 function . Sketch ee t k t the curve.
lies
rt e t cos ti e t sin tj e t k
In Exercises lies on on the the cone
69–74, z
find z 22 x
the x 2 sen
r t e t cos ti e t 2 sin y
limit tjy 2 . 2 . Sketch
(if e the
it t kexists).
the curve.
se encuentra en el cono z 2 x 2 y 2 . Dibujar la curva.
69. In In Exercises lím lies ti on the 69–74, cos cone tj find z 2 sen tk xthe 2 limit y 2 . (if Sketch (if it it exists).
the curve.
t→
69. En In 69. Exercises lím los límejercicios ti ti cos cos 2tj tj a 74, sen evaluar tk el límite.
70. lím t→ 3ti
t→2 t 2 1 j sen tk 1
t→
69–74, find the
t k limit (if exists).
2
70. 69.
2
70. lím lím
lím 3ti
3ti
cos tj
t→2 t 2 j 1
1 cos
71. lím t t t→2
t
2 t
i 3tj
2 1 j sen
1
tk
t k
k
t→0 2
1
t
70. lím 3ti
71.
1 cos cos t t
71. lím lím t 2 t→2 t 2 i
t→0
i t3tj
2 k
72. lím t→0
ln t1 j 1
t k
t i
t
t→1 t 2 1 j t 1
1 cos t t1 k
71. lím t ln
72.
ln t t
72. lím
2 i 3tj
lím t t i i
t→1 t 2 1 j 1
sen t
73. lím e t 1 t→1 t t
i
2 1 j 1 k
t→0 t
j e
t→0 t
t t 1 k
k
ln t
72. lím sen
73.
sen t t
73. lím lím ee t t i
i t i jj
ee t→0
1t
t→0
74. lím e t i
t→ t j t
t t→1 t 2 1 j 1
kt
1 k
sen t t
1
2 1 k
73. lím e 1
74. 74. lím lím
t i
e t t i i
t→ t j t
t 2 1 t→
j j et
t→0 t
t k
t 2 1 k
1
74. lím e t i
t→ t j t
t 2 1 k
y 2