Calculo 2 De dos variables_9na Edición - Ron Larson
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SECCIÓN 12.1 12.1 Vector-Valued
Funciones vectoriales Functions
839
839
[Las graphs gráficas are labeled están marcadas (a), (b),(c), (c), a), and b), (d).] c) y d).]
11
1. r t
t 1 i t
1. r t
t 1 i t
22 jj 3tk a) a) z z
b) b)
z z
12.1 Exercises Ejercicios See See www.CalcChat.com for for worked-out solutions to to odd-numbered exercises.
En In In los Exercises ejercicios 1–8, 1 a find 8, the hallar the domain el dominio of of the the de vector-valued la función vectorial.
function. In En In Exercises los ejercicios 21–24, a match 24, asociar the the equation cada ecuación with its its con graph. su gráfica.
[The
2. 2. rt rrt t 44 4 t 2 t 2 i i t 2 t j j 6tk
44
44
3. 3. rt rrt t ln ln ti ti ee t jj tk tk
22
22
4. 4. rt rrt t sen
sin sin ti ti 44 cos cos tj tj tk tk
yy
5. 5. rt rrt t Ft Ft t Gt Gt t donde
where
−2 −2 2 2
−2 −2
yy
x
2
4
x 2 2
4
2
Ft Ft t cos cos ti ti sen
sin sin tj tj t t t k, k, Gt Gt t cos cos ti ti sen
sin sin tj tj
xx
6. 6. rt rrt t Ft Ft t Gt Gt t donde where
Ft Ft t ln ln ti ti 5tj 5tj 3t 3t 2 k, k, Gt Gt t i i 4tj 4tj 3t 3t 2 k
c) c) z z
d) d) z z
7. 7. rt rrt t Ft Ft t Gt Gt t donde
where
11
Ft Ft t sen
sin sin ti ti cos cos tj, tj, Gt Gt t sen
sin sin tj tj cos cos tk tk
8. 8. rt rrt t Ft Ft t Gt Gt t donde
where
44
22
Ft Ftt t3 3 i i tj tj tk, tk, Gt t t 3
1
G t 3
1
t t i i
t 2k
1
t 1 j t 2 1
t 1 j t 2 k
xx
11
yy
22
xx
44
yy
En
In In los Exercises ejercicios 9–12, 9 a 12, evaluate evaluar
(if (if (si es possible) posible)
the the la función vector-valued vectorial
en cada valor dado de t.
21.
21. rrt rtt ti ti ti 2tj 2tj t at of
t
function at each given value of
k, k, k, 2 22 ≤ ttt ≤ 2
2
t. t.
9. rt rt costi sintj k, 1 ≤ t ≤ 9.
22. 22. rrt t t ti i t j t t 1j
k, 1 t 1
1
r t 2 t sen
t j t 2 k, 1 t 1
1
9. r t 2 2 i i t t 11j
j
23.
a) r1 b) r0 c) rs 1
23. rrt rtt ti ti ti t t 2 j j e e 0.75t 0.75t k, k, k, 2 22 ≤ t t ≤ 2
2
(a) (a) rr11
(b) (b) rr00
(c) (c) rrs s 11
2t
d)
(d)
r2 t r2
24.
rt ti ln tj 2t 2t
(d) rr22 t t rr22
24. rrtt ti ti ln ln tj tj
≤ t ≤ 0.1 t 5
3 k, 0.1 t 5
3 k,
10.
10. 10. rt
rrt t cos
cos cos ti
ti ti 22 sen sin
sin sin tj
tj tj
25. Para pensar Las cuatro figuras siguientes son gráficas de la
a)
(a)
r0 b)
(b)
r4 c)
25. 25. Think About It It The four figures below are are
(c)
función vectorial rt 4 cos ti 4 sin tj t graphs
Asociar cada
d) r6 t r6
k.
of of the the
(a) rr00
(b) rr 44
(c) rr
vector-valued function rrt t 44 cos cos senti ti 44 sin sin tj tj t t44k.
k.
(d) (d) rr 66 t t rr 66
una de las gráficas con el punto en el espacio desde el cual se ve
11. rt ln ti Match each of of the the four graphs with the the point in in space from
11
la hélice. Los cuatro puntos son
11.
0, 0, 20, 20, 0, 0, (20, 0, 0)
the is are
20 ,
r t ln ti
t j
which the helix is viewed. The four points are
20 ,
11. r t ln ti
t j 3tk
y 20, 20,
10, 0, 020, 0 ,,
10. 20, 20, 0, 0, 00 ,,
and 10, 10, 20, 20, 10 10 . .
a)
(a) (a) r2
rr22
b)
(b) (b) r3
rr 33
c)
(c) (c) rt
r
r t t4
44
(a) (a)
z z
(b) (b)
z z
d)
a) z
b)
z
(d) (d) r1
rr1 t 1 t t r1
rr11
12.
12. 12. rt
rrtt t
t t i i t 32 t 32 jj e t4 e t k
a)
(a) (a) r0
rr00
b)
(b) (b) r4
rr44
c)
(c) (c) rc
rr
cc 2
22
d)
(d) (d) r9
rr9 t 9 t t r9
rr99
xx
x
En In In los Exercises ejercicios 13 13 13 and y 14, 14, find hallar rrt t rt . . .
y
yy
Generated by by Mathematica
y
14. 13. 13.
by rtr rtt t t t i i 3tj 3tj 4tk
Generated by Mathematica
Generada con Mathematica
Generada con Mathematica
13. 14. 14. rtr rt t sen
sin sin 3ti 3ti cos cos 3tj 3tj tk tk
(c) (c)
(d) (d) z z
c) d)
z
En In In los Exercises ejercicios 15–18, a represent 18, representar the the line el segmento from de recta Pto to Qdesde
by by aa
P vector-valued hasta Q mediante function una and función by by a a set set vectorial of of parametric y mediante equations. un conjunto
de ecuaciones paramétricas.
yy
15. 15. P0, 0, 0, 0, 00 , , Q3, 3, 1, 1, 22
16. 16.
P0, 0, 2, 2, 11 , , Q4, 4, 7, 7, 22
y
15. 17. 17.
P(0, 2, 0, 2, 5, 0), 5, Q 33 (3, , , Q( Q( 1, 2) 1, 1, 4, 4, 99
16. P (0, 2, 1), Q (4, 7, 2)
17. 18. 18.
P(2, 1, 1, 6, 5, 6, 8), 3), 8), QQ 3, (1, 3, 2, 4, 2, 559)
yy
18. P (1, 6, 8), Q (3, 2, 5)
xx
y
Generated by Think About It It In In Exercises 19 19 and 20, 20, find Is the
xby Mathematica
Generated by by Mathematica
rrt t ut t . . Is the
Generada con Mathematica
Generada con Mathematica
Para result pensar a a vector-valued En los ejercicios function? 19 Explain. y 20, hallar rt ut. ¿Es el 26. 26. Sketch the the three graphs of of the the vector-valued function
resultado una función vectorial? 11
Explicar.
19. 19. rrt t 3t 3t 11i
i 4t 4t 33 jj 4k, 4k,
utt t 2 t 2 i i 8j 8j t 3 t 3 k
26. rDibujar rtt ti ti
tres tj gráficas
tj 2k 2kas de
as la
viewed
función from
vectorial
each point.
rt ti tj 2k
19. 20. 20. rt rrt t 3t 33 cos 1i t, t, 22 sin sin 1 4tt, 3 t, jt t 4k 2, 2, ,
utut t 4 4 tsin 2 sin i t, t, 8j6 6 cos t 3 t, kt, t 2 t 2
(a) vistas (a) 0, 0, desde 0, 0, 20 20los (b) puntos. (b) 10, 10, 0, 0, 00
(c) (c) 5, 5, 5, 5, 55
20. rt 3 cos t, 2 sin sent, t 2,
ut 4 sen sin t, 6 cos t, t 2 a) 0, 0, 20 b) 10, 0, 0 c) 5, 5, 5
r