Calculo 2 De dos variables_9na Edición - Ron Larson

SECCIÓN 12.1 12.1Vector-Valued Funciones Functions vectoriales 841
En In los Exercises ejercicios 75–80, a 80, determine determinar the el interval(s) (los) intervalo(s) on which en que the En In Exercises los ejercicios 89 and 89
12.1
y 90, dos two
Vector-Valued
partículas particles travel viajan
Functions
along a largo the space de
841
las
la vector-valued función vectorial function es continua. is continuous.
curvas curves de r tespacio and ur(t) t . y A u(t). collision Una colisión will occur ocurrirá at en the el point punto of de
intersección si ambas partículas están en P al mismo tiempo.
rt ti intersection P if both particles are at P at the same time. Do the
In Exercises 75–80, 1
rt t i t j ¿Colisionan las partículas? ¿Se intersecan sus trayectorias?
t j determine the interval(s) on which the In Exercises 89 and 90, two particles travel along the space
75. r t ti
76. r t t i t 1 j particles collide? Do their paths intersect?
vector-valued function t j is continuous.
curves r t and u t . A collision will occur at the point of
77. rt r t ti ti arcsen t 1k
intersection P if both particles are at P at the same time. Do the
1
arcsin tj t 1 k
89. r t) t 2 i 9t 20)j t 2 k
75. rt 2e t i e t 76.
particles collide? Do their paths intersect?
78.
r
r
t
t
ti
2e t r t t i t 1 j
ti j e j t j lnt t 1k 1 k
u t) 3t 4 i t 2 j 5t 4 k
77.
79. rrt rtt
ti ee t , t , arcsin tt 2 , 2 , tan tj tt
t 1 k
80. rt r t 8, 8, t, t, t 33 t
89. 90. rr(t t) t 2 tii t 2 9t j t20)j 3 k t 2 k
78. r t 2e t i e t ut) t) 3t 2t 4 i 3 i t 2 j 8tj 5t 12t 4 k 2 k
j ln t 1 k
WRITING ABOUT CONCEPTS
79. Desarrollo r t e t , t 2 , de tan tconceptos
80. r t 8, t, 3 t
90.
Para Think
r(t
pensar About
ti
It
t
En 2 j
los In
t 3 ejercicios Exercises
k
91 y and 92, dos 92, partículas two particles viajan travel
81. Consider the vector-valued function
a lo
81. Considerar la función vectorial
largo along u t) de the las space curvas 2t curves 3de i espacio r8tj t and r(t) 12t u t y . u(t). 2 k
WRITING r t tABOUT 2 i t CONCEPTS
3 j tk.
Si r(t) y u(t) se intersecan, ¿colisionarán las partículas?
rt t 2 i t 3j tk.
Think 91. If About r and Itu tIn intersect, Exercises will 91 the and particles 92, two collide? particles travel
81. Consider the vector-valued function
Write a vector-valued function s t that is the specified along 92. Si If the the las space partículas particles curves collide, colisionan, r t do and their ¿se u t intersecan paths . r t) and sus utrayectorias t intersect? r(t) y
rDar transformation t una t 2 i función t of 3vectorial
r. j tk. st que sea la transformación
u(t)?
especificada de r.
91. If r t) and u t intersect, will the particles collide?
(a) A vertical translation three units upward
True or False? In Exercises 93–96, determine whether the
Write a vector-valued function s t that is the specified ¿Verdadero o falso? En los ejercicios 93 a 96, determinar si la
a) Una traslación vertical tres unidades hacia arriba
92. statement If the particles is true collide, or false. do If their it is paths false, r explain t) and uwhy t intersect? or give an
transformation (b) A horizontal of r. translation two units in the direction of the declaración es verdadera o falsa. Si es falsa, explicar por qué o
example that shows it is false.
b) Una negative traslación x-axishorizontal dos unidades en dirección del
(a) A vertical translation three units upward
True dar un or ejemplo False? que In pruebe Exercises que 93–96, es falsa. determine whether the
(c) eje A xhorizontal negativo translation five units in the direction of the statement 93. If f, g, is and true hor are false. first-degree If it is false, polynomial explain functions, why or give then an the
(b) horizontal translation two units in the direction of the 93. Si ƒ, g y h son funciones polinomiales de primer grado, entonces
c) Una positive traslación y-axishorizontal cinco unidades en dirección del example curve that given shows by xit is ft, false. y gt, and z h t is a line.
negative x-axis
la curva dada por x f t, y g(t) y z ht es una recta.
82. State eje ythe positivo definition of continuity of a vector-valued 94. If the curve given by x ft, y gt, and z h t is a line,
(c) A horizontal translation five units in the direction of the 93. 94. If Si f, la g, curva and hdada are por first-degree x f t, polynomial y g(t) y zfunctions, ht es then una recta, the
82. Dar function. la definición Give an de example continuidad of a vector-valued para una función function vectorial. that
then f, g, and h are first-degree polynomial functions of t.
positive y-axis
curve entonces given ƒ, gby y xh
son ft, funciones y gt, polinomiales and z hde t primer is a line. grado de t.
Dar is defined un ejemplo but not de continuous una función at tvectorial 2. que esté definida 95. Two particles travel along the space curves r t) and u t). The
82. State pero no the sea definition continua en of t continuity 2. of a vector-valued 94. 95. If Dos the partículas curve given viajan by xa través ft, de y las gt, curvas and de zespacio h t is r(t) a y line, u(t).
intersection of their paths depends only on the curves traced out
function. Give an example of a vector-valued function that
then La intersección f, g, and h are de first-degree sus trayectorias polynomial depende functions sólo de of las t. curvas
by r t and u t), while collision depends on the parameterizations.
is defined but not continuous at t 2.
95. Two trazadas particles por r(t) travel y u(t) along en tanto the space la colisión curves depende and de la parametrización.
The vector-valued function r t t
CAS 83. The outer edge of a playground slide is in the shape of a helix of
r t) u t). The
CAS 83. El borde exterior de una resbaladilla tiene forma de una hélice de
96.
intersection of their paths depends only on 2 i t sin t j t cos t k
radius 1.5 meters. The slide has a height of 2 meters and makes
the curves traced out
1.5 metros de radio. La resbaladilla tiene una altura de 2 metros
lies on the paraboloid x y
96. by La r tfunción and u t), vectorial while collision rt 2 depends t 2 z
i 2 .
one complete revolution from top to bottom. Find a vectorvalued
hace outer function una edge revolución of for a playground the helix. completa Use slide a is computer desde in the arriba shape algebra of hacia a system helix abajo. of to 96. The cuentra vector-valued en el paraboloide function x ry 2 t zt 22 . i t sin t j t cos t k
t on sen sin the t j parameterizations.
t cos t k se en-
CAS 83. The y
radius Encontrar graph 1.5 your meters. una function. función The (There slide vectorial has are a para many height la correct hélice. of 2 meters answers.) Usar and un sistema makes Slies E on C Tthe I Oparaboloid N P RxO JyE 2 C zT
2 .
one algebraico complete por revolution computadora from para top graficar to bottom. la función. Find a (Existen vectorvalued
muchas function respuestas for correctas.) the helix. Use a computer algebra system to PROYECTO DE TRABAJO
CAPSTONE
Witch of Agnesi
graph your function. (There are many correct answers.)
S E C T I O N P R O J E C T
84. Which of the following vector-valued functions represent
In
Para discusión
Bruja Section de 3.5, Agnesi you studied a famous curve called the Witch of
the same graph?
CAPSTONE
Witch Agnesi. In of this Agnesi project you will take a closer look at this function.
En la sección 3.5 se estudió una curva famosa llamada bruja de
84. Which ¿Cuál a) r tde of las the siguientes 3 cos t
following funciones 1)i 5 sen
vector-valued vectoriales t 2 j
functions representa 4k
la
Consider a circle of radius a centered on the y-axis at 0, a . Let
Agnesi. En este proyecto se profundiza sobre esta función.
the misma b) same r tgráfica?
In Section 3.5, you studied a famous curve called Witch of
graph? 4i 3 cos t 1)j 5 sen t 2)k
A be a point on the horizontal line y 2a, let O be the origin, and
Agnesi. Considérese In this project un círculo you will de radio take a closer centrado look en at el this punto function. (0, a) del
a) c) rrt t 33 cos cos t t 11)i i 55 sen sen t t 22j j 4k 4k
let B be the point where the segment OA intersects the circle. A
eje y. Consider Sea A un a circle punto of en radius la recta centered horizontal on ythe 2a, axis Oat
el origen Let y B
point P is on the Witch of Agnesi a if P lies on y- the horizontal 0, a . line
b) d) rrt t 4i 3 cos 32t cos t1 i 1)j 5 sen 5 sen 2t t 2 j2)k
4k
Ael be punto a point donde the el segmento horizontal OA corta y el 2a, círculo. let O be Un the punto origin, P está anden
through B and on the vertical line through A.
la bruja de Agnesi si P se encuentra en la recta horizontal a través de
c) r t 3 cos t 1 i 5 sen t 2 j 4k
let B be the point where the segment OA intersects the circle. A
point B(a) y Show en P la is recta that the
on the vertical point
Witch a A
of través is traced
Agnesi de A. out by the vector-valued function
if P lies on the horizontal line
d) r t 3 cos 2t 1 i 5 sen 2t 2 j 4k
85. Let r t and u t be vector-valued functions whose limits exist through a) Mostrar r A
B and que 2a on cot el the punto vertical i A2aj,
está line descrito 0 through < < por A.
la función vectorial
as t → c. Prove that
(a) Show
rwhere A
that
2a is the the cot
point angle i
A is
2aj, that traced OA0 makes out
<
by
<
the with vector-valued the positive x- function axis.
85. Sean Let limr rt rt
tand y ut u t funciones be vector-valued lim rvectoriales t limfunctions ucuyos t . límites whose existen limits exist cuando
Demostrar que
t→c t→c t→c
(b) rShow 0 < <
donde A that 2a the cot point i B2aj,
is traced out by the vector-valued function
es el ángulo formado por OA con el eje x positivo.
as t → c. Prove that
86. Let r t and u t be vector-valued functions whose limits exist b) Mostrar where r B que is a the sin el angle 2punto i that Ba está 1OAdescrito cos makes 2 j, with por 0 la the < función positive < vectorial . x-axis.
lim
lím as t rt ut lím rt lím
ut.
t→c r→ t c. Prove u t that lim
t→c r t lim
t→c u t .
(b) (c) Show Combine that the point B is traced out by the vector-valued function
r B a the sen sin 2i results a1 of parts cos (a) 2j, and 0 (b) <
find the vector-
B
86. Sean Let limr rt rt
tand y ut u t tfunciones be vector-valued lim r vectoriales t lim ufunctions t cuyos . límites whose existen limits exist cuando
as t → c.
Prove Demostrar that que
c) Combinar utility to graph los resultados this curve de for los aincisos 1. a) y b) para hallar la fun-
t→c t→c t→c
rvalued function a sin 2 ir
afor 1 the cos Witch 2 j, of 0 Agnesi. < < Use . a graphing
87. Prove that if r is a vector-valued function that is continuous at (c) Combine ción vectorial the r() results para of la parts bruja (a) de Agnesi. and (b) Usar to find una the herramienta vectorvalued
de graficación function para r representar lim r and lim
lim
lím c, then rt r t r
ut is t continuous lim
lím rt r t at c.
lím lim
ut.
t→c t→c t→c u t .
(d) Describe the limits
for the Witch esta curva of Agnesi. r para
.
→0
→ Use a a 1. graphing
utility to graph this curve for
87. 88.
Prove Demostrar Verify that
that if que the
r is si converse
a rvector-valued es una of función Exercise
function vectorial 87 is not
that continua true by
is continuous en finding c, entonces
vector-valued
a (e) Eliminate the parameter
at d) Describir los límites lim lím r a and 1. determine
y lím lim r. the rectangular
→0 →
c, then r es is continuous continua function en r such
at c. that r is continuous at c but r equation of the Witch
c.
(d) Describe the limits
and lim
e) Eliminar el parámetro lim of Agnesi. Use a
r
y determinar
→
la r graphing
ecuación . utility to graph
→0
is not continuous at c.
this function for a 1 and compare your graph rectangular with that de
88. Verify Verificar that que the el converse recíproco of de Exercise lo que se 87 afirma is not true en el by ejercicio finding 87 a (e) Eliminate the parameter and determine the rectangular
la obtained bruja de in Agnesi. part (c). Usar una herramienta de graficación para
vector-valued no es verdad encontrando function r such una función that r vectorial is continuous r tal que at c r but sea r equation of the Witch of Agnesi. Use a graphing utility to graph
representar esta función para a 1 y comparar la gráfica con la
is continua not continuous en c pero at rc.
no sea continua en c.
this function for a 1 and compare your graph with that
obtenida en el inciso c).
obtained in part (c).
< .