Calculo 2 De dos variables_9na Edición - Ron Larson

SECCIÓN 12.4 Tangent Vectores Vectors tangentes and y vectores Normal normales Vectors 12.4 Tangent Vectors and Normal Vectors 867
867
12.4 Tangent Vectors and Normal Vectors 867
CAS CASEn In los Exercises ejercicios 63–66, y use 66, a usar computer un sistema algebra algebraico system to por graph computadora
y representar gráficamente la curva espacial. Entonces gura muestra una partícula que sigue la trayectoria dada por
the
72. Movimiento Motion 12.4 Along Tangent a lo largo Involute Vectors de una and of involuta a Normal Circlede Vectors un The círculo figure shows 867 La fi-
CAS In Exercises 63–66, use a computer algebra system to graph the 72. Motion Along an Involute of a Circle The figure shows
CAS
space In space Exercises curve.
curve. Then 63–66, Then find use T
Ta t computer t , N t , a
, N t , a T
hallar Tt, Nt, a T y a en el instante T , algebra , and a
and a N system at the
at the given to given graph time
time the
t.
t. 72. a Motion a particle
particle Along moving
moving an Involute along
along of a a Circle a path
path The modeled
modeled figure shows
by
dado N
by
t. Dibujar Tt y Nt
Sketch space Sketch curve. T t and
T t and Then N
N tfind t on
on N
the T the t space ,
space N t ,
curve. a curve. T ,
and a N
at the given time
t.
ra r t
t particle cos t
cos t moving t sen
t sen t, along t, sen t
sen a path t t cos
cos t modeled t . The figure
. The figure by
CAS en In Sketch Exercises la curva T t
en and 63–66, el N espacio. t
use on a the computer space curve.
algebra system to graph the 72. Motion
also rLa also t figura shows
shows cos Along muestra the
the t an vectors también Involute
vectors t sen v t
v t and t, los and sen of vectores a t
a t t Circle for t
for t vt t cos The 1y and
1 and at t figure
t.
tThe para 2. shows
2. tfigure
1 y
space Function
Función Function curve. Then find T t , N t , a T , and a at
Instante Time the Time
N given time t. a also t particle 2. shows the moving vectors y v t
along and a t
a for t path 1
and
modeled t 2.
by
y
Sketch Function
T t and N t on the space curve. Time
r t cos t t sen t, sen t t cos t . The figure
63. r t 4ti 3 cos tj 3 sen tk t
y
63. r t 4ti 3 cos tj 3 sen tk t 2
also shows the vectors v t and a t for t 1 and t 2.
63.
Function r t 4ti 3 cos tj 3 sen tk
Time t
2
t
y
64. r t 2 cos t i 1 sen t j t
64.
2 cos t i 1 sen t j
t = 1
3 k t
t = 1
63. r t 4ti 3 cos tj 3 sen tk 3 k 2
t
t
64.
r t 2 cos t i 1 sen t j
2
t x
65.
ti 3t 2 j
2 t x
65. r t ti 3t
64.
2 j
2 2
2 k
t
r t 2 cos t i 1 sen t j t
t = 1
3 k t 2
k
t
t = 1
3 k t 66.
t
66. r t t 2 t
i 2 i j 2tk
x
65. r t ti 3t 2 j
2 t 2
j 2tk
2 k
t 1
t t 1
x
65.
t ti 3t 2 j
2 t 2
t = 2
2 k
t = 2
66.
r t t 2 i j 2tk
t 1
WRITING
WRITING
ABOUT
ABOUT
CONCEPTS
CONCEPTS
t = 2
66. Desarrollo WRITING 67.
r t
Define
t 2 i
the ABOUT j de unit
2tkconceptos
tangent CONCEPTS
vector, the principal
t 1
unit normal
67. Define the unit tangent vector, the principal unit normal
(a) Find a
(a) Find a T and a
and a
t N = at
at
2 t 1 and t 2.
N t 1 and t 2.
67.
67. T
Definir vector, Define vector,
el and the and
vector
unit the
the unitario tangential
tangential tangente,
vector, and
and the normal
normal el vector
principal components
components unitario
unit normal
of
WRITING ABOUT CONCEPTS
normal of
(a) (b) Hallar Find Determine a a T y a N en t y t 2.
acceleration.
(b) Determine T
and whether a whether N
at t the 1
speed and
t of the 2.
particle is increasing or
the speed of the particle is increasing or
principal, acceleration. vector, and y las
the componentes
tangential tangencial
and normal y normal
components de la aceleración.
of
67. Define the unit tangent vector, the principal unit normal
Determinar decreasing
decreasing at si at each
each la rapidez of the indicated
of the indicated de la partícula values
values of aumenta of t. Give
t. Give reasons o reasons disminuye
for decreasing for your T
68. (b) Determine whether the speed of the particle is increasing or
68.
acceleration. How is the unit tangent vector related to the orientation of (a) Find a and a at t 1 and t 2.
vector, How is the and unit the tangent tangential vector and related normal to the components orientation of
your en cada answers.
answers. at N
uno each de of los the valores indicated indicados values of de t.
t. Give Dar reasons
razones
68.
68. ¿Cuál a How a curve?
acceleration. curve? is es
the Explain.
Explain.
unit relación
tangent entre
vector el vector
related unitario
to the orientation tangente y
of
la (b)
para
Determine for your las respuestas. answers.
whether the speed of the particle is increasing or
69.
69. orientación
a (a) curve? Describe Explain.
(a) Describe de the
the una motion
motion curva? of
of a Explicar. a particle if the normal component In Exercises
68. How is the unit tangent vector particle related if the to normal the orientation component of
In Exercises
decreasing 73–78,
73–78,
at each find
find
of the
the vectors
indicated vectors T
T
values and
and N,
of N, t. and
and
Give the
the
reasons unit
unit
69.
69. a)
(a) a curve? Describir of Describe of acceleration
acceleration Explain.
the movimiento
motion is 0.
is 0. of a particle de una
if partícula
the normal si
component
la componente
binormal En In binormal Exercises los ejercicios for vector
vector
your 73–78, answers.
B B
a 78, T find T
hallar
N,
N, the for
for vectors los the
the vectores
vector-valued
vector-valued T and T y N, y and function el
function
vector the unit
runi-
tario binormal at the
r t
t
(b) of Describe acceleration normal the de motion la
is 0.
aceleración of a particle es 0. if the tangential at the binormal
given
given vector
value
value BB
of
of T
t.
t. T N, N,
de for la función the vector-valued vectorial rt function
en el valor r t
69. (b) (a) Describe the the motion motion of a of particle a particle if the normal if the component tangential In Exercises 73–78, find the vectors T and N, and the unit
dado de t.
b) Describir of component component of
acceleration of movimiento acceleration acceleration is 0. de is is
una 0.
0.
at the given value of
t.
(b) Describe the motion of a particle
partícula
if the
si la
tangential
componente
tangencial de la aceleración es 0.
73. 74. r t ti t 2 j
3 r t vector
2 cos ti B 2 T sen N,
tj for tthe (b) Describe the motion of a particle if the tangential
2 k
vector-valued function r
binormal 73. t
at the r given t 2 value cos ti of t.
2 sen tj t
t
74. r t ti t 2 j
3 rt 3 t3k
t
component of acceleration is 0.
2 k
k
73. r t 2 cos ti 2 sen tj t
t
74. r t ti t 2 j
3
2 k
3 k
component of acceleration is 0.
CAPSTONE
CAPSTONE
73. r t
t 0 74. rt 0 t 1
t
ti t t
cos ti 2 sen tj t
2 2 j
3 0
2 k t 0 1
2
CAPSTONE
70. An object moves along the path given by
t 3 k
Para 70. An discusión
0 t object moves along the path given by
z
0 1
2 z
z
z
70. An object moves along the path given by
CAPSTONE r t 3ti 4tj.
t z
t z
70. Un r t objeto 3ti se 4tj. mueve a lo largo de la trayectoria dada por
0 0 1
4 2 4
70. An r t object 3ti moves 4tj.
1
along the path given by
1
Find v t , a t , T t , and N t (if it exists). What is the form
4
3z
Find r(t) v 3ti t , a t 4tj. , T t , and N t (if it exists). What is the form
3
z
1
rof Find of t the
the v path? 3ti path? t ,
a Is t 4tj. Is ,
the T the t speed
speed ,
and of N of
the t
the (if object
object it exists). constant
constant What or
or is changing?
changing? the form
3
4
1
y
Encontrar of the path? v(t), Is the a(t) speed T(t) y of N(t) the (si object existe). constant ¿Cuál or es changing?
la forma
1 1
2 y
2
Find 3
de la vtrayectoria? t , a t , T t ¿Es , and constante N t (if it o exists). variable What la velocidad is the form del
2 1
y
2
2
71. Cycloidal
of objeto? the path?
Motion
Is the speed
The
of
figure
the object
shows
constant
the path
or changing?
of a particle 3
y
71. Cycloidal Motion The figure shows the path of a particle x3
3 y
2
x
3
x
y
−1
x
1
2
modeled by the vector-valued function
−1
modeled by the vector-valued function
71. Cycloidal Motion The figure shows the path of a particle 3
x
3 y
x
−1
2
modeled by the vector-valued function
71. Movimiento rCycloidal r t
t tMotion t sen
cicloidal sen The
t, 1
t, 1La figure
cos
figura cos shows
t .
t muestra . the la path trayectoria of a particle
Figure
de una Figure 3 for 73
x for 73 3 y
Figure for 74
Figure x for 74
partícula modeled r t
representada by t the sen vector-valued t, por 1 la cos función function t .
vectorial
Figura Figure for para −1 73 73 Figura Figure for para 74
74
The figure also shows the vectors v t v t and a t a t at
The figure also shows the vectors v t v t and a t a t at 75. r t i sen tj cos tk, t
75. r t i sen tj cos tk, t 0
rthe 0 The t indicated
indicated figure t also values sen values shows of t, of 1 t.
t. the vectors cos t v . t
v t
and a t
a t
at
Figure for 73 Figure 4
75.
r t i sen tj cos tk,
t 4 for 74
y
76. r t 2e
La The y figura figure muestra also shows también the vectors los vectores v t vtvt v t and ay
t atat a t at 76. r t 2e t i t i e
e t t cos tj e
cos tj e t t 0
the indicated values of t.
sen tk, 4
t
sen tk, t 0 0
0 0
y
75. 76.
r tt i2e sen tj cos tk, t
en the los indicated valores values indicados of de t.
t i e t cos tj e t sen 0 tk,
t t.
77. r t 4 sen ti 4 cos tj 2tk, 4 0 0
t
77.
4 sen ti 4 cos tj 2tk, t 0
0 3
y y
76. 77.
r tt 2e 4 t sen i ti e t cos 4 tj cos tj e t sen 2tk,
t t 0 3 0
1
0 t = 1
3
t = 2 t = 1
2 t = 1
78. r t 3 cos 2ti 3 sen 2tj tk, t 1
t =
78. 77. r t 43 sen cos 2ti 43 cos sen tj2tj 2tk, tk, t 0
t 0
2
t = 1
3
0 4
t = 3
4
t = 2
78. r t 3 cos 2ti 3 sen 2tj tk, t 0 3
1 1
2
t = t =
3
4
t =
2 t 2 = 1t = 1 2
79. Projectile Motion Find the tangential and normal components
of acceleration for a projectile fired 4 at an angle with the
x
79. 78.
3
rProjectile Movimiento t 3 cos Motion 2ti de un 3 proyectil
Find sen 2tj the tangential tk, Hallar t las
and
componentes
normal components
Projectile cial y
x
0 tangen-
3t =
t =
79.
2
of
normal
acceleration Motion de la aceleración
for Find a projectile the tangential de un
fired
proyectil
at and an angle normal disparado
with compo-
x
con
the
2
horizontal at an initial speed of v What are the components un
horizontal nents of acceleration ángulo con
at an
la horizontal
initial for speed a projectile y con
of v
rapidez 0 . 0 .
What fired at inicial
are an the angle v
components with the
0 . ¿Cuáles son
x
79. Projectile Motion Find the tangential and normal components
of acceleration for a projectile fired at en an su angle altura máxima? with the
x
when the projectile is at its maximum height?
when horizontal las componentes
the projectile at an initial cuando
is at speed its
el
maximum of
v 0 .
What proyectil está
height? are the components
1
3
(a) Find a and a at t 1 t 1, and t
80. when Projectile the projectile Motion is Use at its your maximum results from height?
Exercise 79 to find
3
(a) Find a T
and a N
at t 2,
t 1, and t 2.
80. Projectile
Movimiento
Motion
de un proyectil
Use your results
Utilizar
from
los resultados
Exercise 79
del
to
ejercicio
find
T N 2,
2.
horizontal
the tangential
at an
and
initial
normal
speed
components
of v 0 . What
of
are
acceleration
the components
1
3
for a
(a) (b) Find Determine a whether the speed of the particle is increasing or when the (b) Determine whether the speed of the particle is increasing or
tangential 79 the para projectile hallar
and is normal las at componentes its maximum components tangencial height? of acceleration y normal
for and at and
80. Projectile Motion Use your results from Exercise 79 to find
T a N t 2, t 1, t 2.
de
a
projectile fired at an angle of 45 with the horizontal at an la
decreasing Hallar decreasing y a at each
en t of 1 t 1 y t 3 aceleración de un proyectil disparado con un ángulo de 45° con
N 2 , the
2 .
projectile the tangential fired and at an normal angle components of 45 with of the acceleration horizontal at for ana
(b) Determine
(a) Find and at whether each at of the the 1 speed indicated values
indicated of the and values particle 3 of t. Give reasons
a a t t 1, t of t. is Give increasing reasons or
80. Projectile initial speed Motion of 150 Use feet your per second. results from What Exercise are the components 79 to find
for your T answers. N 2,
2.
initial projectile la horizontal
speed fired of
con
150 at an rapidez
feet angle per second. of
45
inicial de
What with the 150
are
pies
the horizontal por
components at an
segundo.
En for decreasing cada your uno answers. at each of the indicated values of t.
Give reasons
the tangential and normal components of acceleration for a
(b) Determine whether de los the valores speed indicados of the particle de t, determinar is increasing si or la when the projectile is at its maximum height?
when initial ¿Cuáles
the speed son
projectile of 150 las componentes
is feet at its per maximum second. What cuando
height? are the components
for your answers.
projectile fired at an angle of 45 with el the proyectil horizontal está at en ansu
rapidez decreasing la at each partícula of the aumenta indicated o values disminuye. of t. Give Dar reasons razones when the projectile is at its maximum height?
initial altura speed máxima? of 150 feet per second. What are the components
para for your las respuestas. answers.
when the projectile is at its maximum height?