Calculo 2 De dos variables_9na Edición - Ron Larson

858 CAPÍTULO 85812 Chapter Funciones 12vectoriales
Vector-Valued Functions
Vector-Valued 858 Functions Chapter 12 Vector-Valued Functions
50. a) Mostrar que 50. la rapidez (a) Show de la that partícula the speed es b. of the particle is b . 59. Investigación 59. Un Investigation objeto sigue una A particle trayectoria moves elíptica on an elliptical dada path given
ed of the particle 50. is (a) b) b Usar Show . una that herramienta the speed (b) Use of de the a 59. graficación particle graphing Investigation is utility b en . modo in A parametric paramétrico particle moves mode on 59. to an graph Investigation elliptical por la
the
función path given A vectorial particle the by vector-valued rt moves 6 on cos an function
elliptical 3 sen sin r tj. tpath 6 given cos ti by 3 sin tj.
utility in parametric (b) para mode Use representar a to graphing el the utility círculo circle in para for parametric b the vector-valued 6. 6. Try Probar mode different distintos to
function
graph values valores of the
r t.
Does 6 cos the the ti graphing a) vector-valued Hallar 3 sin tj. vt, vt, function (a) y Find at. r vt t , 6 vcos t , ti and a3 sin t . tj.
ry different values of de circle . Does . ¿Dibuja for the bgraphing
6. la Try herramienta utility different draw values (a) de the graficación Find circle of v.
Does t faster , más v the t for , rápido graphing and greater a los t values . of (a) ? b) Find Usar vuna t , herramienta v t ,(b) and Use ade t a . graficación graphing utility para completar to complete la the tabla. table.
rcle faster for greater values círculos utility of draw para ? 51. the los circle Find
valores faster the
mayores
acceleration for greater de ?
(b) Use values a vector graphing of and ? utility show to that complete its direction the (b) table. Use isa graphing utility to complete the table.
vector and 51. 51. show Find Hallar that the el its vector acceleration direction aceleración always is vector toward y mostrar and the show que center that su of dirección its the direction circle. es siempre
hacia el centro del círculo.
4 2 3
is
t 0
2
ter of the circle.
t 0
always toward the 52. center Show of that the the circle. magnitude of the acceleration vector 2 is b
t 0
2 .
2 4 2 3
52. Mostrar que la del vector aceleración es
t 0
ude of the acceleration 52. Show vector that the is bmagnitude 2 .
b2
of the acceleration vector is b4.
2 . 2 3
Rapidez 4 2 3
Circular Motion In Exercises 53 and 54, use the results of
Speed
ercises 53 and Circular Movimiento 54, use Motion the circular results In Exercises Exercises En oflos 49–52. ejercicios 53 and 5354, Speed y 54, use usar the los results resultados
de los 49–52. ejercicios 49 a 52.
tores velocidad (c) y aceleración Graph the para elliptical los valores path and de t the dados velocity en la and accelerat
of c) Representar Speed gráficamente la trayectoria elíptica y los vec-
Exercises
53. A stone weighing 1 pound is attached to a two-foot string tabla and del inciso b).
Una piedra que pesa 1 libra se ata
(c)
a
Graph
un cordel
the elliptical
de dos pies
path
de
and the velocity (c) Graph and the acceleration elliptical vectors path and at the velocity values of and t given acceleration in the table in part (b).
ound is attached 53. to A a stone two-foot weighing string 1 and is pound whirled is attached horizontally to a (see two-foot figure). string The and string will break under
d) Usar los resultados de los incisos b) y c) para describir la
largo y se hace girar horizontalmente (ver
vectors
la figura).
at the values
El cordel
of t
se
given in the table vectors in part at the (b).
y (see figure). The is string whirled will horizontally break under a force (see figure). of 10 pounds. The string Find will the break maximum underspeed the stone can values (d) Use of tthe given results the of table parts in (b) part and (b). (c) to describe the geome
relación geométrica entre los vectores velocidad y aceleración
cuando la rapidez de la partícula aumenta y cuando
. Find the maximum a romperá force speed of con the 10 stone una pounds. fuerza can attain Find de without 10 the libras. maximum (d) breaking Use Hallar the speed la results the velocidad the string. of stone parts máxima
Use que Fwithout la ma, piedra breaking where puede m 32. alcanzar the string. sin que Use se Frompa ma, el where cordel.
relationship between the velocity and acceleration vectors
can Use (b) and F (c) ma, to (d) describe where Use the the results geometric of parts relationship (b) and (c) between to describe the velocity the geometric and acceleration vect
1
ing the string. attain
disminuye.
(Usar 1
m 32. donde m 1 .
relationship between the velocity and acceleration vectors when the speed of the particle is increasing, and when i
F ma, when the speed of the particle is increasing, when and the when speed it of isthe decreasing.
32 particle is increasing, and when it is
decreasing.
decreasing.
Para discusión CAPSTONE
1 lb
1 libra CAPSTONE
30 mphCAPSTONE
1 lb
60. Considerar una 60. partícula Consider que a se particle mueve moving sobre una on trayectoria an elliptical path describe
1 lb
2 ft
30 mph
60. Consider 30 a particle mph moving on an elliptical 60. Consider elíptica path a descrita described particle por by moving r t on a an cos elliptical t i b path sen described t j, where donde d dt is th
ft
2 pies ft
by r t a cos t i b sen t j, where by r td
dt a cos es is la the t velocidad iconstant b sen angular t j, where velocity. constante. d dt is the
constant angular velocity.
constant a) Encontrar angular el velocity. vector (a) Find velocidad. the velocity ¿Cuál vector. es la What rapidez is the de speed la of the particle
(a) Find the velocity vector. What is the speed (a) Find partícula? of the the particle? velocity (b) vector. Find What the acceleration is the speed of vector the particle? and show that its directio
300 pies
(b) Find the acceleration vector and show (b) b) that Find Encontrar its the direction acceleration vector is aceleración vector always and toward y show demostrar the that center its que direction of su the dirección
always está toward siempre the hacia center el centro of the de ellipse. la elipse.
ellipse.
300 pies
300 is pies always toward the center of the ellipse. is
Figure for 53 Figure for 54
Figure Figure for Figura 54 for
para
53
53 Figura
Figure for
para
54
54
54. A 3400-pound automobile is negotiating a circular interchange WRITING ABOUT CONCEPTS
obile is negotiating 54.
54.
A
Un a 3400-pound circular automóvil interchange de
automobile
3 400 of radius libras
is negotiating 300 está WRITING tomando feet 30 a circular
una miles ABOUT curva per interchange
circular hour CONCEPTS (see de figure). Desarrollo
WRITING Assuming de conceptos
ABOUT 61. CONCEPTS
In your own words, explain the difference between th
30 miles per hour of
300 (see radius
pies figure). de
300
radio Assuming feet
a
at the 30
30
millas roadway miles
por
per
hora
hour
(ver
(see
la
figure).
figura).
Assuming
Supuesto
the
que
roadway
la carretera
is level,
está nivelada,
find the
hallar
61.
is level,
In
force
la
between
fuerza
your
find
own
the
necesaria
words,
force between
the tires and
entre
explain
the
the
los
the
tires
difference 61. 61. and
In Con the
your las between own propias words, the palabras, velocity explain explicar of the an object la difference diferencia and its between entre speed. la velocidad
notde of un an objeto object and its speed.
the
find the force between the tires and road the such that the
road
neumáticos
such that
y el
the
pavimento
car stays
para
on the
que
velocity
car stays
circular
el automóvil
of an
on
object
the circular
path and
mantenga
and its
path
does not
la
speed.
and does
velocity
r stays on the circular path and does skid. not (Use F ma, where m 3400 32. ) Find the angle at
62. What y su rapidez. is known about the speed of an object if the angl
where m 3400 skid.
trayectoria 32. ) Find (Use the F
circular angle ma, which sin
where at derrapar. the
62.
m roadway 3400
(Usar What
32.
Fshould is
)
known Find
ma,
the be donde about
banked angle
mthe at
speed so that
of 62. 62.
no
an What object
lateral
¿Qué is se if known conoce the angle about acerca between the de speed la the rapidez velocity of an de object un and objeto acceleration if the si angle ángulo
entre is the los (a) velocity vectores acute and and velocidad (b) acceleration obtuse? y aceleración vectors es is (a) agudo acute y
vectors is (a) acut
should be banked which
3 400/32.) so that the
Hallar
roadway no lateral el frictional ángulo
should
de force be
peralte
banked is
between
exerted necesario the
so on that
para the
velocity
no tires que
lateral
ninguna
the fuerza automobile. force
of
and
the
acceleration
automobile.
vectors between
rted on the tires of frictional
de fricción
is exerted
lateral
on the
sea
tires
ejercida and (b)
of the
sobre obtuse?
automobile.
los neumáticos and b) obtuso? (b) obtuse? 63. Consider a particle that is moving on the pat
del automóvil. 55. Shot-Put Throw 63. Consider The path a of particle a shot thrown that at is an moving angle 63. 63. Consider Redacción is
the a Considerar particle path r 1 t that una x tpartícula is i moving ytj que se ztk. on mueve the sobre pathla
e path of a shot 55. thrown Shot-Put Lanzamiento at Throw angle de peso The is path La trayectoria of a shot r 1 thrown tde un xobjeto at t ian angle ytj lanzado isztk.
con r 1
trayectoria t x t ir 1
1 t ytj (a) xti Discuss ztk. ytjany changes ztk. in the position, velocity, or accele
un ángulo es r t v
1
0 cos (a) t i Discuss h any v 0 sin changes t in
1
2 gt2 the position, j
(a) velocity, a) Discuss Analizar or acceleration
any todo changes cambio ation in en the of la position, the posición, particle velocity, velocidad if its position or acceleratioración
by the of de the vector-
la particle partícula if valued its si su position function posición is given está r by the vector-
o acele-
is given by the vecto
h v 0 sin t r t v 0 cos t i h v
rt 2 gt2 j
v 0 cos t i
0 sin t
h v 0 sin t 2 1 gt2 of jthe particle if its position is given
sen
2
2 t dada r 1
por 2t . la función
vectorial function r 2 (b) t r 2 t Generalize r 1 2t. r 1 2t . the results for the position functio
where v gt2 j
0 is the initial valued speed, function h is the initial r 2 t height, r 1 2t . t is the time valued
speed, h is the initial where height, v is the initial speed, h is the initial height, t is the time
donde 0
t is the time
seconds, and g
v 0 es la rapidez inicial, h es la (b)
is
altura Generalize
the acceleration
inicial, t the
due
es el results
to gravity.
tiempo
en segundos y g es la aceleración r
for the
Verify
(b) position that
Generalize Generalizar function los the resultados results r 3 t for a rla 1
función the t . position posición function
he acceleration due in to seconds, gravity. and Verify g is that the the shot acceleration will remain due to gravity. Verify that
in
3 debida t
the air
r
for
1 a la t
a
.
total of
gravedad.
r 3 t t r 1 t. t .
the air for a total the of shot will remain in the air for a total 2 of
Verificar que el objeto permanecerá v en el aire
in 2 2gh v t 0 sin v 0 sin 2 2gh
seconds
2 v t 0 sin v 0 sin 2 2gh
64. When t 0, an object is at the point 0, 1 and has a veloc
seconds sen sen
g
seconds
g
64. When segundos t 0, an object is at the point 64. 0,
64. When 1 and
Cuando t has
t 0,
a
0, an velocity
un object vector
objeto is está at vthe 0
en point i.
el punto 0, 1It (0, and moves
1) y has tiene a with velocity an acceleration
and will travel
un vector
vector a horizontal v 0 distance i. of It moves with vector an acceleration
velocidad v 0v(0) i. aof
t
i. It Se moves sin ti
mueve con with cos t j.
aceleración an Show acceleration that the path
a(t) sen ti of the object is a cir
ontal distance of and y recorrerá will travel una a distancia horizontal horizontal distance a of tde
v
2 sin ti cos t j. Show that the path aof cos t the
t j. sin object
Mostrar ti is a cos circle.
0 cos
2gh
que t j. la Show trayectoria that the del path objeto of the es object un círculo. is a circle.
sin
2gh v 2
pies.
sin feet.
True or False? In Exercises 65–68, determine whether
0 cos
g
sin feet.
2gh
sin sin
2
2
feet.
True or False? In Exercises 65–68, determine whether the
g
2 True or False? In Exercises
v
2
sen sen
0 65–68, determine whether statement the is true or false. If it is false, explain why or give
v
2
v
2
0 statement 0 is true or false. If it is false, statement
¿Verdadero explain is why o
true
falso? or or give false.
En an los
If it
ejercicios
is false, explain
65 a 68,
why
determinar
or give an
si la
56. Shot-Put Throw A shot is thrown from a height of h 6 feet example that shows it is false.
56. Lanzamiento de peso Un peso es lanzado desde una altura de declaración es verdadera o falsa. Si es falsa, explicar por qué o
shot is thrown 56. from Shot-Put a height Throw of h 6A feet with shot an is initial thrown
example speed from
that of a height vshows of
it feet is false.
0 45 h per 6 feet second and example at an angle that shows it is false.
h 6 pies con rapidez inicial v pies por segundo y con dar un ejemplo que 65. pruebe The acceleration que es falsa. of an object is the derivative of the speed.
f v feet per with second an initial and at speed an angle of v feet 0
per
45
0 45
of 0
42.5 45 second and at angle
of
un ángulo de
with the horizontal.
con la
65.
with
horizontal.
The acceleration
the horizontal.
Find the
Hallar
total time
el
of
tiempo
an
Find
object
the total
of travel
total
is the
time
65. derivative
of travel
The acceleration of the speed. of an object is the derivative of the speed.
horizontal. Find the total 42.5 time of travel and the total distance traveled.
66. The velocity of an object is the derivative of the position.
de recorrido y la distancia horizontal recorrida.
65. La aceleración de un objeto es la derivada de la rapidez.
al distance traveled.
66. The velocity of an object is the derivative
and the total horizontal distance traveled.
66. The of velocity the position. of an object is the derivative of the position.
57. Prove that if an object is traveling at a constant speed, its
67. The velocity vector points in the direction of motion.
57. Demostrar que si un objeto se mueve con rapidez constante, sus
66. La velocidad de un objeto es la derivada de la posición.
ect is traveling 57. at Prove a constant that if speed, an object velocity its
67. The velocity vector points in the direction
is traveling and acceleration at a constant vectors speed, are orthogonal. its
67. The of velocity motion. vector 68. If points a particle in the moves direction along of motion. a straight line, then the velocity
tion vectors are orthogonal. vectores velocidad y aceleración son ortogonales.
67. El vector velocidad apunta en la dirección de movimiento.
velocity and acceleration 58. Prove vectors that
68.
an are If
object
a orthogonal. particle
moving
moves
in a
along
straight
a straight
line at 68. line,
a If constant
then a particle the velocity moves and along acceleration a straight vectors line, are then orthogonal. the velocity and
moving in a 58. Demostrar que un objeto que se mueve en línea recta a velocidad
constante tiene aceleración nula.
los vectores velocidad y aceleración son ortogonales.
68. Si una partícula se mueve a lo largo de una línea recta, entonces
58. straight Prove line that at an a constant
acceleration vectors are orthogonal.
object speed moving has an in acceleration a straight line of 0. at a constant
acceleration vectors are orthogonal.
tion of 0. speed has an acceleration of 0.
42.5
2