Calculo 2 De dos variables_9na Edición - Ron Larson

12.2 SECCIÓN Differentiation 12.2 Derivación and Integration e integración of Vector-Valued de funciones Functions vectoriales 849
12.2 Differentiation and Integration of Vector-Valued Functions 849
En In Exercises los ejercicios 49–52, a use 52, the usar definition la definición of the de derivative la derivada to para find In En Exercises los ejercicios 77–84, a prove 84, demostrar the property. la propiedad. In each case, En todos assume los
hallar rt. rt.
r, casos, u, and suponer v are differentiable que r, u y v son vector-valued funciones vectoriales functions derivables of t in space, de
In Exercises 49–52, use the definition of the derivative
t, que w es una función derivable de t, y que c es un escalar.
49. 50. rt t i 3 to find In w is Exercises a differentiable 77–84, real-valued prove the property. function of In t, each and ccase, is a assume
rt. 49. rt rt 3t 3t 2i 2i 1 1 t2 t 2 j j
t j 2tk r, u, and v are differentiable vector-valued functions of t in space,
w77.
is Da t differentiable crt crtreal-valued function of t, and c is a scalar.
51.
49. 50. rt rt t 3t t 2 , 0,
i 2t
2i 3 1 t 2 j
52. rt 0, sin sen t, 4t
77.
t crt crt
50.
t i 3 t j 2tk
78. D t rt ± ut rt ± ut
51. rt t
En los ejercicios 2 , 0, 2t
53 ta j 60, 2tk 52. rt 0, sin t, 4t
79. D
hallar la integral indefinida.
78.
t wtrt wtrt wtrt
t rt ± ut rt ± ut
51. rt t
53. 2 80. D
, 2t
52. rt 0, sin t, 4t
79.
t rt ut rt ut rt ut
In Exercises 53–60, find the indefinite integral.
54.
t wtrt wtrt wtrt
2ti j k dt
4t 3 i 6tj 4t k dt 81. D
80.
t rwt rwtwt
t rt ut rt ut rt ut
In Exercises 53–60, find the indefinite integral.
53.
55.
1
56. ln ti 1 j k
t i j t 54.
82. D
t dt
81.
t rt rt rt rt
2ti j k dt
32 k
4t t rwt rwtwt
dt
53. 54.
82.
t rt rt rt rt
3 i 6tj 4t k dt
83. D t rt ut vt rt ut vt
2ti j k dt
4t
57. 3 i 6tj 4t k dt
55. 56.
rt
83. D
2t 1i 4t 3 j 3t k
t rt ut vt rt
ut vt rt ut
1
dt
ln ti 1 j k vt
t ut vt
Si rt rt es una constante, entonces rt rt 0.
55. 56.
rt ut vt rt ut i j t 32 k dt
ln ti
85. Movimiento de una partícula Una partícula se mueve en el
58. 1 t dt
1
j k vt
t i j t t dt
84. If rt rt is a constant, then rt rt 0.
32 k dt
57.
e 2t 1i 4t t i sin tj cos tk dt
84. If rt rt is a constant, then rt rt 0.
57.
plano xy a lo largo de la curva representada por la función vec-
3 j 3t k
sen dt
85. Particle Motion A particle moves in the xy-plane along the
2t 1i 4t 3 j 3t k dt
curve represented by the vector-valued function
58. e torial rt t sin ti 1 cos tj.
59.
2
sec ti 1
60. e t sin ti e t cos tj dt
1 t j dt
85. Particle Motion A particle moves in the xy-plane along the
2 curve Usar represented una herramienta by de the graficación vector-valued para representar functionr.
58.
t i sin tj cos tk dt
rt t sin ti sen 1 cos tj.
sen
e t i sin tj cos tk dt
(a) Use a graphing utility to graph r. Describe the curve.
59. rtDescribir sin la ti curva. 1 cos tj.
En los ejercicios
sec2 ti 1 60. e
61 a 66, evaluar la integral definida.
(a) b) Hallar Use a graphing los valores utility mínimo to graph y máximo r. Describe de rthe y curve. .
59. 60.
t sin ti e t cos tj dt
1
1
2
sec ti
1
86. (b) Movimiento Find the minimum de una partícula and maximum Una values partícula of se mueve and en el
61.
1 t j dt 2 (b) Find the minimum and maximum values of r and .
e
8ti tj k dt 62.
t sin ti e t cos tj dt
1 t j dt 2 86. Particle Motion A particle moves in the yz-plane along the
In Exercises 61– 66, evaluate the definite integral.
r .
ti t 3 j t 3 k dt
curve represented by the vector-valued function
86. Particle plano yzMotion a lo largo A de particle la curva moves representada in the por -plane función along vectorial
rtrepresented 2 cos tj by 3 sin the tk. vector-valued function
the
0 1
1 1
rt 2 cos tj 3 sin tk.
yz
In Exercises 61– 66, evaluate the definite integral.
61. 2
curve
63.
8ti tj k dt 62. ti t
a cos ti a sen sin tj k dt 1
rt a) Describir 2 cos la tjcurva.
3 sin tk.
61. 0
62.
3 j t 3 k dt
sen
(a) Describe the curve.
01
1
2
8ti tj k dt
ti t 3 j t 3 k dt
(b) Find the minimum and maximum values of r and r.
63.
(a) b) Hallar Describe los the valores curve.
0
a cos ti a sin tj k dt
mínimo y máximo de r y r.
64.
4
1
02
87. Consider the vector-valued function
sec t tan ti tan tj 2 sen sin t cos tk dt
87. (b) Considerar Find the la minimum función vectorial and maximum values of r and r.
63.
4 a cos ti a sin tj k dt
rt e
0
87. Consider t sin ti e
the vector-valued t cos tj.
64. 0
function
sec t tan ti tan tj 2 sin t cos tk dt
rt e t sen sin ti e t cos tj.
02
4
3
rt Show that e
65. 64. ti e 66.
t sin rtti and rt e t cos are tj. always perpendicular to each other.
2 sec t jt tan te ti t k dt
tan tj 2 sin 3 ti t cos tk t 2 j dt
Mostrar que rt y rt son siempre perpendiculares a cada uno.
0
0
65. ti e 66.
Show that rt and rt are always perpendicular to each other.
2
3
En 65. los t j te t k dt
ti t
ejercicios ti e 67 a 72, hallar rt 66. para 2 j dt
CAPSTONE
0
0
ti las condiciones t dadas. Para discusión
t j te t k dt
2 j dt
88. Investigation Consider the vector-valued function
CAPSTONE
In Exercises 0 67–72, find rt for the given 0 conditions.
rt ti 4 t Investigación Considerar 2 j.
la función vectorial r(t)
67. rt 4e 2t i 3e t 88. Investigation the vector-valued function
j, r0 2i
In 67. Exercises rt 4e67–72, 2t i 3e find
t j, rt r0for the 2i given conditions.
(a) (4 Sketch t 2 )j. the graph
rt ti 4 t 2 of rt. Use a graphing utility to verify
j.
rt 3t 6t k, r0 i 2j
your graph.
68. rt 3t Trazar la gráfica de r(t). Usar una herramienta de graficación
Sketch
67. rt 4e 2 2t j 6t
i 3e t k, r0 i 2j
j, r0 2i
(a) Sketch the graph of rt. Use a graphing utility to verify
(b) rt 32j, r0 6003i r0 your graph. para the vectors verificar r1, su gráfica. r1.25, and r1.25 r1 on
69. rt 32j,
68. rt 3t 2 r0 6003i 600j, r0 0
j 6t k, r0 i 2j
the graph in part (a).
Trazar los vectores r(1), r(1.25) y r(1.25) r(1) sobre la
70.
69. rt rt
rt4 4 32j,
cos tj tj
r0 3
sen
sin
6003i tk, r0 r0
600j, 3k,
r0 r0 r0
04j
4j
(b) Sketch the vectors r1, r1.25, and r1.25 r1 on
(c) gráfica Compare en the el inciso vector a). r(1 with the vector
71.
71.
70. rt rt
rt te
te
4 t2 icos tj t j 3 sin k, tk, r0 r0 1 2 i 3k, j the graph in part (a).
t2 i e t j k, r0 1 2i j k
r0 4j
(c) Comparar r1.25
Compare the el vector r(1) r(1 con with el the vector vector
71. 72.
rt te t2
72. rt 1 1 i
r1 2i
1 t i e t 1 j 2 t j k, 1
2 t k, r0 r1 1 2ij k
1 r1.25
72. rt 1 t i 1 2 t j 1 r1 .
2
r1 2i
1 t i 1 2 t j 1 t k, 1.25 1
r1 .
2 t k, 1.25 1
WRITING ABOUT CONCEPTS
True or False? In Exercises 89–92, determine whether the
73. State the definition of the derivative of a vector-valued
Desarrollo WRITING ABOUT de conceptos
CONCEPTS
statement ¿Verdadero is o true falso? or false. En los If ejercicios it is false, 89 explain a 92, determinar why or give si anla
function. Describe how to find the derivative of a vectorvalued
la function derivada and de give una its función geometric vectorial. interpretation. Describir cómo statement
True
declaración
or False?
es verdadera
In Exercises
o falsa.
89–92,
Si es
determine
falsa, explicar
whether
por qué
the
73. State the definition of the derivative of a vector-valued
example that shows it false.
o
73. Definir dar un ejemplo
is true
que
or
muestre
false. If
que
it is
es
false,
falsa.
explain why or give an
function. Describe how to find the derivative of a vectorvalued
function and give its geometric interpretation.
74. hallar How do la you derivada find the de integral una función of a vector-valued vectorial y dar function? su interpretación
geométrica.
89. its Si una derivative partícula vector se mueve is always a lo tangent largo de to una the esfera sphere. centrada en el
example 89. If a particle that shows moves it is along false. a sphere centered at the origin, then
75. The three components of the derivative of the vector-valued
74. How do you find the integral of a vector-valued function? 89. If origen, a particle entonces moves su along vector a sphere derivada centered es siempre at the tangente origin, then a la
74. ¿Cómo function se uencuentra are positive la integral at t t de una función vectorial? 90. The definite integral of a vector-valued function is a real number.
0 . Describe the behavior of u
its esfera. derivative vector is always tangent to the sphere.
75. The three components of the derivative of the vector-valued
75. Las at t tres t 0 . componentes de la derivada de la función vectorial
u son positivas en t t 0 . 0 .
91. rt rt
d
function u are positive at t t Describe the behavior of u 90. The La integral definite definida integral of de a una vector-valued función vectorial function es is un a real número number. real.
76. The z-component of the derivative Describir of el the comportamiento
vector-valued
dt
at t t d
de u en 0 .
t t 0 .
91.
function u is 0 for t in the domain of the function. What 92. If rt
dt
rt r and u are rt rt differentiable vector-valued functions of t, then
76. The z-component of the derivative of the vector-valued
76. La does componente this information z de la imply derivada about de the la función graph of vectorial u? u es
D t rt ut rt ut.
function u is 0 for t in the domain of the function. What
0 para t en el dominio de la función. ¿Qué implica esta información
acerca de la gráfica de u?
92. If Si rrand y u son are differentiable funciones vectoriales vector-valued derivables functions de t, of entonces t, then
does this information imply about the graph of u?
D t rt rt ut ut rt rt ut. ut.
r
r
r