Calculo 2 De dos variables_9na Edición - Ron Larson

En los ejercicios 1 a 8, dibujar la curva plana representada por
la función vectorial y dibujar los vectores y para el
valor dado de
Colocar los vectores de manera que el punto inicial
de esté en el origen y el punto inicial de esté en el
punto final de ¿Qué relación hay entre y la curva?
En los ejercicios 9 y 10, a) dibujar la curva en el espacio representada
por la función vectorial, y b) dibujar los vectores
y
para el valor dado de
9.
10.
En los ejercicios 11 a 22, hallar
En los ejercicios 23 a 30, hallar a) r(t), b) r(t) y c) r(t) r(t).
En los ejercicios 31 y 32 se dan una función vectorial y su
gráfica. La gráfica también muestra los vectores unitarios
y
Hallar estos dos vectores unitarios
e identificarlos en la gráfica.
Figura para 31 Figura para 32
En los ejercicios 33 a 42, hallar el (los) intervalo(s) abierto(s) en
que la curva dada por la función vectorial es suave.
33. 34.
35.
36.
37.
38.
39. 40.
41.
42.
En los ejercicios 43 y 44, usar las propiedades de la derivada
para encontrar lo siguiente.
a) b) c)
d) e) f)
43.
44.
En los ejercicios 45 y 46, hallar a) y b)
en dos diferentes formas.
i) Hallar primero el producto y luego derivar.
ii) Aplicar las propiedades del teorema 12.2.
45.
46.
En los ejercicios 47 y 48, hallar el ángulo entre y en función
de t. Usar una herramienta de graficación para representar
Usar la gráfica para hallar todos los extremos de la función.
Hallar todos los valores de t en que los vectores son ortogonales.
47. 48. rt t 2 i tj
rt 3 sin ti 4 cos tj
t.
rt
rt
ut j tk
rt cos ti sin tj tk,
ut t 4 k
rt ti 2t 2 j t 3 k,
D t [rt ut]
D t [rt ut]
ut 1 t i 2 sin tj 2 cos tk
rt ti 2 sin tj 2 cos tk,
rt ti 3tj t 2 k, ut 4ti t 2 j t 3 k
t > 0
D t [ rt) ],
D t [rt ut]
D t [3rt ut]
D t [r(t ut]
rt
rt
rt t i t 2 1j 1 4 tk
rt ti 3tj tan tk
rt e t i e t j 3tk
rt t 1i 1 t j t2 k
rt
2t
8 t 3 i 2t2
8 t 3 j
r 2 sin i 1 2 cos j
r sin i 1 cos j
r 2 cos 3 i 3 sin 3 j
rt 1
t 1 i 3tj
rt t2 i t 3 j
In Exercises 1–8, sketch the plane curve represented by the
vector-valued function, and sketch the vectors and for
the given value of
Position the vectors such that the initial
point of is at the origin and the initial point of is at the
terminal point of
What is the relationship between
and the curve?
1.
2.
3.
4.
5.
6.
7.
8.
In Exercises 9 and 10, (a) sketch the space curve represented by
the vector-valued function, and (b) sketch the vectors
and
for the given value of
9.
10.
In Exercises 11–22, find
11. 12.
13. 14.
15.
16.
17.
18.
19.
20.
21.
22.
In Exercises 23–30, find (a) (b) and (c)
23.
24.
25.
26.
27.
28.
29.
30.
In Exercises 31 and 32, a vector-valued function and its graph
are given. The graph also shows the unit vectors
and
Find these two unit vectors and identify them
on the graph.
31.
32.
Figure for 31 Figure for 32
In Exercises 33–42, find the open interval(s) on which the curve
given by the vector-valued function is smooth.
33. 34.
35.
36.
37.
38. 39.
40. 41.
42.
In Exercises 43 and 44, use the properties of the derivative to
find the following.
(a) (b) (c)
(d) (e) (f)
43.
44.
In Exercises 45 and 46, find (a) and
(b)
in two different ways.
(i) Find the product first, then differentiate.
(ii) Apply the properties of Theorem 12.2.
45.
46.
In Exercises 47 and 48, find the angle between and as
a function of Use a graphing utility to graph Use the
graph to find any extrema of the function. Find any values of
at which the vectors are orthogonal.
47. 48. r t t 2 i tj
r t 3 sin ti 4 cos tj
t
t .
t.
r t
r t
u t j tk
r t cos ti sin tj tk,
u t
t 4 k
r t ti 2t 2 j t 3 k,
D t [r t u t ]
D t [r t u t ]
u t
1
t i 2 sin tj 2 cos tk
r t ti 2 sin tj 2 cos tk,
r t ti 3tj t 2 k, u t 4ti t 2 j t 3 k
t > 0
D t [ r t ],
D t [r t u t ]
D t [3r t u t ]
D t [r t u t ]
r t
r t
r t t i t 2 1 j
1
4tk
r t ti 3tj tan tk
r t e t i e t j 3tk
r t t 1 i
1
t j
t2 k
r t
2t
8 t 3 i 2t 2
8 t 3 j
r 2 sin i 1 2 cos j
r sin i 1 cos j
r 2 cos 3 i 3 sin 3 j
r t
1
t 1 i 3tj
r t t2 i t 3 j
y
x
z
2
2 1 1
2
y
z
1
1
1
t 0 1
4
r t
3
2 ti t 2 j e t k,
t 0 1
4
r t cos t i sen t j t 2 k,
r t 0 / r t 0 .
r t 0 / r t 0
r t
e t , t 2 , tan t
r t cos t t sen t, sen t t cos t, t
r t ti 2t 3 j 3t 5 k
r t
1
2 t 2 i tj
1
6t 3 k
r t 8 cos ti 3 sen tj
r t 4 cos ti 4 sen tj
r t t 2 t i t 2 t j
r t
t 3 i
1
2t 2 j
r t r t .
r t ,
r t ,
r t arcsen t, arccos t, 0
r t
t sen t, t cos t, t
r t
t 3 , cos 3t, sen 3t
r t e t i 4j 5te t k
r t 4 t i t 2 t j ln t 2 k
r t a cos 3 ti a sen 3 tj k
r t
1
t i 16tj t 2
2 k
r t 6ti 7t 2 j t 3 k
r t t cos t, 2 sen t
r t
2 cos t, 5 sen t
r t t i 1 t 3 j
r t t 3 i 3tj
r t .
t 0 2
r t ti t 2 j
3
2k,
t 0 3
2
r t 2 cos ti 2 sin tj tk,
t 0 .
r t 0 r t 0
t 0 0
r t e t , e t ,
t 0 0
r t e t , e 2t ,
t 0 2
r t) 3 sen ti 4 cos tj,
t 0 2
r t cos ti sen tj,
t 0 1
r t 1 t i t 3 j,
t 0 2
r t
t 2 i
1
t j, t 0 1
r t ti t 2 1 j,
t 0 2
r t t 2 i tj,
r t 0
r t 0 .
r t 0
r t 0 t 0 .
r t 0
r t 0
848 Chapter 12 Vector-Valued Functions
12.2 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
In Exercises 1–8, sketch the plane curve represented by the
vector-valued function, and sketch the vectors and for
the given value of
Position the vectors such that the initial
point of is at the origin and the initial point of is at the
terminal point of
What is the relationship between
and the curve?
1.
2.
3.
4.
5.
6.
7.
8.
In Exercises 9 and 10, (a) sketch the space curve represented by
the vector-valued function, and (b) sketch the vectors
and
for the given value of
9.
10.
In Exercises 11–22, find
11. 12.
13. 14.
15.
16.
17.
18.
19.
20.
21.
22.
In Exercises 23–30, find (a) (b) and (c)
23.
24.
25.
26.
27.
28.
29.
30.
In Exercises 31 and 32, a vector-valued function and its graph
are given. The graph also shows the unit vectors
and
Find these two unit vectors and identify them
on the graph.
31.
32.
Figure for 31 Figure for 32
In Exercises 33–42, find the open interval(s) on which the curve
given by the vector-valued function is smooth.
33. 34.
35.
36.
37.
38. 39.
40. 41.
42.
In Exercises 43 and 44, use the properties of the derivative to
find the following.
(a) (b) (c)
(d) (e) (f)
43.
44.
In Exercises 45 and 46, find (a) and
(b)
in two different ways.
(i) Find the product first, then differentiate.
(ii) Apply the properties of Theorem 12.2.
45.
46.
In Exercises 47 and 48, find the angle between and as
a function of Use a graphing utility to graph Use the
graph to find any extrema of the function. Find any values of
at which the vectors are orthogonal.
47. 48. r t t 2 i tj
r t 3 sin ti 4 cos tj
t
t .
t.
r t
r t
u t j tk
r t cos ti sin tj tk,
u t
t 4 k
r t ti 2t 2 j t 3 k,
D t [r t u t ]
D t [r t u t ]
u t
1
t i 2 sin tj 2 cos tk
r t ti 2 sin tj 2 cos tk,
r t ti 3tj t 2 k, u t 4ti t 2 j t 3 k
t > 0
D t [ r t ],
D t [r t u t ]
D t [3r t u t ]
D t [r t u t ]
r t
r t
r t t i t 2 1 j
1
4tk
r t ti 3tj tan tk
r t e t i e t j 3tk
r t t 1 i
1
t j
t2 k
r t
2t
8 t 3 i 2t 2
8 t 3 j
r 2 sin i 1 2 cos j
r sin i 1 cos j
r 2 cos 3 i 3 sin 3 j
r t
1
t 1 i 3tj
r t t2 i t 3 j
y
z
2
2 1 1
2
x
y
z
1
1
1
t 0 1
4
r t
3
2 ti t 2 j e t k,
t 0 1
4
r t cos t i sen t j t 2 k,
r t 0 / r t 0 .
r t 0 / r t 0
r t
e t , t 2 , tan t
r t cos t t sen t, sen t t cos t, t
r t ti 2t 3 j 3t 5 k
r t
1
2 t 2 i tj
1
6t 3 k
r t 8 cos ti 3 sen tj
r t 4 cos ti 4 sen tj
r t t 2 t i t 2 t j
r t
t 3 i
1
2t 2 j
r t r t .
r t ,
r t ,
r t arcsen t, arccos t, 0
r t
t sen t, t cos t, t
r t
t 3 , cos 3t, sen 3t
r t e t i 4j 5te t k
r t 4 t i t 2 t j ln t 2 k
r t a cos 3 ti a sen 3 tj k
r t
1
t i 16tj t 2
2 k
r t 6ti 7t 2 j t 3 k
r t t cos t, 2 sen t
r t
2 cos t, 5 sen t
r t t i 1 t 3 j
r t t 3 i 3tj
r t .
t 0 2
r t ti t 2 j
3
2k,
t 0 3
2
r t 2 cos ti 2 sin tj tk,
t 0 .
r t 0 r t 0
t 0 0
r t e t , e t ,
t 0 0
r t e t , e 2t ,
t 0 2
r t) 3 sen ti 4 cos tj,
t 0 2
r t cos ti sen tj,
t 0 1
r t 1 t i t 3 j,
t 0 2
r t
t 2 i
1
t j, t 0 1
r t ti t 2 1 j,
t 0 2
r t t 2 i tj,
r t 0
r t 0 .
r t 0
r t 0 t 0 .
r t 0
r t 0
848 Chapter 12 Vector-Valued Functions
12.2 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
rt 0 / rt 0 .
rt 0 / rt 0
rt.
t 0 2
rt ti t 2 j 3 2k,
t 0 3
2
rt 2 cos ti 2 sin tj tk,
t 0 .
rt 0
rt 0
rt 0
rt 0 .
rt 0
rt 0
t 0 .
rt 0
rt 0
848 CAPÍTULO 12 Funciones vectoriales
sen
sen
sen
12.2 Ejercicios
In Exercises 1–8, sketch the plane curve represented by the
vector-valued function, and sketch the vectors and for
the given value of
Position the vectors such that the initial
point of is at the origin and the initial point of is at the
terminal point of
What is the relationship between
and the curve?
1.
2.
3.
4.
5.
6.
7.
8.
In Exercises 9 and 10, (a) sketch the space curve represented by
the vector-valued function, and (b) sketch the vectors
and
for the given value of
9.
10.
In Exercises 11–22, find
11. 12.
13. 14.
15.
16.
17.
18.
19.
20.
21.
22.
In Exercises 23–30, find (a) (b) and (c)
23.
24.
25.
26.
27.
28.
29.
30.
In Exercises 31 and 32, a vector-valued function and its graph
are given. The graph also shows the unit vectors
and
Find these two unit vectors and identify them
on the graph.
31.
32.
Figure for 31 Figure for 32
In Exercises 33–42, find the open interval(s) on which the curve
given by the vector-valued function is smooth.
33. 34.
35.
36.
37.
38. 39.
40. 41.
42.
In Exercises 43 and 44, use the properties of the derivative to
find the following.
(a) (b) (c)
(d) (e) (f)
43.
44.
In Exercises 45 and 46, find (a) and
(b)
in two different ways.
(i) Find the product first, then differentiate.
(ii) Apply the properties of Theorem 12.2.
45.
46.
In Exercises 47 and 48, find the angle between and as
a function of Use a graphing utility to graph Use the
graph to find any extrema of the function. Find any values of
at which the vectors are orthogonal.
47. 48. r t t 2 i tj
r t 3 sin ti 4 cos tj
t
t .
t.
r t
r t
u t j tk
r t cos ti sin tj tk,
u t
t 4 k
r t ti 2t 2 j t 3 k,
D t [r t u t ]
D t [r t u t ]
u t
1
t i 2 sin tj 2 cos tk
r t ti 2 sin tj 2 cos tk,
r t ti 3tj t 2 k, u t 4ti t 2 j t 3 k
t > 0
D t [ r t ],
D t [r t u t ]
D t [3r t u t ]
D t [r t u t ]
r t
r t
r t t i t 2 1 j
1
4tk
r t ti 3tj tan tk
r t e t i e t j 3tk
r t t 1 i
1
t j
t2 k
r t
2t
8 t 3 i 2t 2
8 t 3 j
r 2 sin i 1 2 cos j
r sin i 1 cos j
r 2 cos 3 i 3 sin 3 j
r t
1
t 1 i 3tj
r t t2 i t 3 j
y
z
2
2 1 1
2
y
z
1
1
1
t 0 1
4
r t
3
2 ti t 2 j e t k,
t 0 1
4
r t cos t i sen t j t 2 k,
r t 0 / r t 0 .
r t 0 / r t 0
r t
e t , t 2 , tan t
r t cos t t sen t, sen t t cos t, t
r t ti 2t 3 j 3t 5 k
r t
1
2 t 2 i tj
1
6t 3 k
r t 8 cos ti 3 sen tj
r t 4 cos ti 4 sen tj
r t t 2 t i t 2 t j
r t
t 3 i
1
2t 2 j
r t r t .
r t ,
r t ,
r t arcsen t, arccos t, 0
r t
t sen t, t cos t, t
r t
t 3 , cos 3t, sen 3t
r t e t i 4j 5te t k
r t 4 t i t 2 t j ln t 2 k
r t a cos 3 ti a sen 3 tj k
r t
1
t i 16tj t 2
2 k
r t 6ti 7t 2 j t 3 k
r t t cos t, 2 sen t
r t
2 cos t, 5 sen t
r t t i 1 t 3 j
r t t 3 i 3tj
r t .
t 0 2
r t ti t 2 j
3
2k,
t 0 3
2
r t 2 cos ti 2 sin tj tk,
t 0 .
r t 0 r t 0
t 0 0
r t e t , e t ,
t 0 0
r t e t , e 2t ,
t 0 2
r t) 3 sen ti 4 cos tj,
t 0 2
r t cos ti sen tj,
t 0 1
r t 1 t i t 3 j,
t 0 2
r t
t 2 i
1
t j, t 0 1
r t ti t 2 1 j,
t 0 2
r t t 2 i tj,
r t 0
r t 0 .
r t 0
r t 0 t 0 .
r t 0
r t 0
848 Chapter 12 Vector-Valued Functions
12.2 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
In Exercises 1–8, sketch the plane curve represented by the
vector-valued function, and sketch the vectors and for
the given value of
Position the vectors such that the initial
point of is at the origin and the initial point of is at the
terminal point of
What is the relationship between
and the curve?
1.
2.
3.
4.
5.
6.
7.
8.
In Exercises 9 and 10, (a) sketch the space curve represented by
the vector-valued function, and (b) sketch the vectors
and
for the given value of
9.
10.
In Exercises 11–22, find
11. 12.
13. 14.
15.
16.
17.
18.
19.
20.
21.
22.
In Exercises 23–30, find (a) (b) and (c)
23.
24.
25.
26.
27.
28.
29.
30.
In Exercises 31 and 32, a vector-valued function and its graph
are given. The graph also shows the unit vectors
and
Find these two unit vectors and identify them
on the graph.
31.
32.
Figure for 31 Figure for 32
In Exercises 33–42, find the open interval(s) on which the curve
given by the vector-valued function is smooth.
33. 34.
35.
36.
37.
38. 39.
40. 41.
42.
In Exercises 43 and 44, use the properties of the derivative to
find the following.
(a) (b) (c)
(d) (e) (f)
43.
44.
In Exercises 45 and 46, find (a) and
(b)
in two different ways.
(i) Find the product first, then differentiate.
(ii) Apply the properties of Theorem 12.2.
45.
46.
In Exercises 47 and 48, find the angle between and as
a function of Use a graphing utility to graph Use the
graph to find any extrema of the function. Find any values of
at which the vectors are orthogonal.
47. 48. r t t 2 i tj
r t 3 sin ti 4 cos tj
t
t .
t.
r t
r t
u t j tk
r t cos ti sin tj tk,
u t
t 4 k
r t ti 2t 2 j t 3 k,
D t [r t u t ]
D t [r t u t ]
u t
1
t i 2 sin tj 2 cos tk
r t ti 2 sin tj 2 cos tk,
r t ti 3tj t 2 k, u t 4ti t 2 j t 3 k
t > 0
D t [ r t ],
D t [r t u t ]
D t [3r t u t ]
D t [r t u t ]
r t
r t
r t t i t 2 1 j
1
4tk
r t ti 3tj tan tk
r t e t i e t j 3tk
r t t 1 i
1
t j
t2 k
r t
2t
8 t 3 i 2t 2
8 t 3 j
r 2 sin i 1 2 cos j
r sin i 1 cos j
r 2 cos 3 i 3 sin 3 j
r t
1
t 1 i 3tj
r t t2 i t 3 j
y
z
2
2 1 1
2
y
z
1
1
1
t 0 1
4
r t
3
2 ti t 2 j e t k,
t 0 1
4
r t cos t i sen t j t 2 k,
r t 0 / r t 0 .
r t 0 / r t 0
r t
e t , t 2 , tan t
r t cos t t sen t, sen t t cos t, t
r t ti 2t 3 j 3t 5 k
r t
1
2 t 2 i tj
1
6t 3 k
r t 8 cos ti 3 sen tj
r t 4 cos ti 4 sen tj
r t t 2 t i t 2 t j
r t
t 3 i
1
2t 2 j
r t r t .
r t ,
r t ,
r t arcsen t, arccos t, 0
r t
t sen t, t cos t, t
r t
t 3 , cos 3t, sen 3t
r t e t i 4j 5te t k
r t 4 t i t 2 t j ln t 2 k
r t a cos 3 ti a sen 3 tj k
r t
1
t i 16tj t 2
2 k
r t 6ti 7t 2 j t 3 k
r t t cos t, 2 sen t
r t
2 cos t, 5 sen t
r t t i 1 t 3 j
r t t 3 i 3tj
r t .
t 0 2
r t ti t 2 j
3
2k,
t 0 3
2
r t 2 cos ti 2 sin tj tk,
t 0 .
r t 0 r t 0
t 0 0
r t e t , e t ,
t 0 0
r t e t , e 2t ,
t 0 2
r t) 3 sen ti 4 cos tj,
t 0 2
r t cos ti sen tj,
t 0 1
r t 1 t i t 3 j,
t 0 2
r t
t 2 i
1
t j, t 0 1
r t ti t 2 1 j,
t 0 2
r t t 2 i tj,
r t 0
r t 0 .
r t 0
r t 0 t 0 .
r t 0
r t 0
848 Chapter 12 Vector-Valued Functions
12.2 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
In Exercises 1–8, sketch the plane curve represented by the
vector-valued function, and sketch the vectors and for
the given value of
Position the vectors such that the initial
point of is at the origin and the initial point of is at the
terminal point of
What is the relationship between
and the curve?
1.
2.
3.
4.
5.
6.
7.
8.
In Exercises 9 and 10, (a) sketch the space curve represented by
the vector-valued function, and (b) sketch the vectors
and
for the given value of
9.
10.
In Exercises 11–22, find
11. 12.
13. 14.
15.
16.
17.
18.
19.
20.
21.
22.
In Exercises 23–30, find (a) (b) and (c)
23.
24.
25.
26.
27.
28.
29.
30.
In Exercises 31 and 32, a vector-valued function and its graph
are given. The graph also shows the unit vectors
and
Find these two unit vectors and identify them
on the graph.
31.
32.
Figure for 31 Figure for 32
In Exercises 33–42, find the open interval(s) on which the curve
given by the vector-valued function is smooth.
33. 34.
35.
36.
37.
38. 39.
40. 41.
42.
In Exercises 43 and 44, use the properties of the derivative to
find the following.
(a) (b) (c)
(d) (e) (f)
43.
44.
In Exercises 45 and 46, find (a) and
(b)
in two different ways.
(i) Find the product first, then differentiate.
(ii) Apply the properties of Theorem 12.2.
45.
46.
In Exercises 47 and 48, find the angle between and as
a function of Use a graphing utility to graph Use the
graph to find any extrema of the function. Find any values of
at which the vectors are orthogonal.
47. 48. r t t 2 i tj
r t 3 sin ti 4 cos tj
t
t .
t.
r t
r t
u t j tk
r t cos ti sin tj tk,
u t
t 4 k
r t ti 2t 2 j t 3 k,
D t [r t u t ]
D t [r t u t ]
u t
1
t i 2 sin tj 2 cos tk
r t ti 2 sin tj 2 cos tk,
r t ti 3tj t 2 k, u t 4ti t 2 j t 3 k
t > 0
D t [ r t ],
D t [r t u t ]
D t [3r t u t ]
D t [r t u t ]
r t
r t
r t t i t 2 1 j
1
4tk
r t ti 3tj tan tk
r t e t i e t j 3tk
r t t 1 i
1
t j
t2 k
r t
2t
8 t 3 i 2t 2
8 t 3 j
r 2 sin i 1 2 cos j
r sin i 1 cos j
r 2 cos 3 i 3 sin 3 j
r t
1
t 1 i 3tj
r t t2 i t 3 j
y
z
2
2 1 1
2
y
z
1
1
1
t 0 1
4
r t
3
2 ti t 2 j e t k,
t 0 1
4
r t cos t i sen t j t 2 k,
r t 0 / r t 0 .
r t 0 / r t 0
r t
e t , t 2 , tan t
r t cos t t sen t, sen t t cos t, t
r t ti 2t 3 j 3t 5 k
r t
1
2 t2 i tj
1
6 t3 k
r t 8 cos ti 3 sen tj
r t 4 cos ti 4 sen tj
r t t 2 t i t 2 t j
r t
t 3 i
1
2t 2 j
r t r t .
r t ,
r t ,
r t arcsen t, arccos t, 0
r t
t sen t, t cos t, t
r t
t 3 , cos 3t, sen 3t
r t e t i 4j 5te t k
r t 4 t i t 2 t j ln t 2 k
r t a cos 3 ti a sen 3 tj k
r t
1
t i 16tj t 2
2 k
r t 6ti 7t 2 j t 3 k
r t t cos t, 2 sen t
r t
2 cos t, 5 sen t
r t t i 1 t 3 j
r t t 3 i 3tj
r t .
t 0 2
r t ti t 2 j
3
2k,
t 0 3
2
r t 2 cos ti 2 sin tj tk,
t 0 .
r t 0 r t 0
t 0 0
r t e t , e t ,
t 0 0
r t e t , e 2t ,
t 0 2
r t) 3 sen ti 4 cos tj,
t 0 2
r t cos ti sen tj,
t 0 1
r t 1 t i t 3 j,
t 0 2
r t
t 2 i
1
t j, t 0 1
r t ti t 2 1 j,
t 0 2
r t t 2 i tj,
r t 0
r t 0 .
r t 0
r t 0 t 0 .
r t 0
r t 0
848 Chapter 12 Vector-Valued Functions
12.2 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
In Exercises 1–8, sketch the plane curve represented by the
vector-valued function, and sketch the vectors and for
the given value of
Position the vectors such that the initial
point of is at the origin and the initial point of is at the
terminal point of
What is the relationship between
and the curve?
1.
2.
3.
4.
5.
6.
7.
8.
In Exercises 9 and 10, (a) sketch the space curve represented by
the vector-valued function, and (b) sketch the vectors
and
for the given value of
9.
10.
In Exercises 11–22, find
11. 12.
13. 14.
15.
16.
17.
18.
19.
20.
21.
22.
In Exercises 23–30, find (a) (b) and (c)
23.
24.
25.
26.
27.
28.
29.
30.
In Exercises 31 and 32, a vector-valued function and its graph
are given. The graph also shows the unit vectors
and
Find these two unit vectors and identify them
on the graph.
31.
32.
Figure for 31 Figure for 32
In Exercises 33–42, find the open interval(s) on which the curve
given by the vector-valued function is smooth.
33. 34.
35.
36.
37.
38. 39.
40. 41.
42.
In Exercises 43 and 44, use the properties of the derivative to
find the following.
(a) (b) (c)
(d) (e) (f)
43.
44.
In Exercises 45 and 46, find (a) and
(b)
in two different ways.
(i) Find the product first, then differentiate.
(ii) Apply the properties of Theorem 12.2.
45.
46.
In Exercises 47 and 48, find the angle between and as
a function of Use a graphing utility to graph Use the
graph to find any extrema of the function. Find any values of
at which the vectors are orthogonal.
47. 48. r t t 2 i tj
r t 3 sin ti 4 cos tj
t
t .
t.
r t
r t
u t j tk
r t cos ti sin tj tk,
u t
t 4 k
r t ti 2t 2 j t 3 k,
D t [r t u t ]
D t [r t u t ]
u t
1
t i 2 sin tj 2 cos tk
r t ti 2 sin tj 2 cos tk,
r t ti 3tj t 2 k, u t 4ti t 2 j t 3 k
t > 0
D t [ r t ],
D t [r t u t ]
D t [3r t u t ]
D t [r t u t ]
r t
r t
r t t i t 2 1 j
1
4tk
r t ti 3tj tan tk
r t e t i e t j 3tk
r t t 1 i
1
t j
t2 k
r t
2t
8 t 3 i 2t 2
8 t 3 j
r 2 sin i 1 2 cos j
r sin i 1 cos j
r 2 cos 3 i 3 sin 3 j
r t
1
t 1 i 3tj
r t t2 i t 3 j
y
z
2
2 1 1
2
y
z
1
1
1
t 0 1
4
r t
3
2 ti t 2 j e t k,
t 0 1
4
r t cos t i sen t j t 2 k,
r t 0 / r t 0 .
r t 0 / r t 0
r t
e t , t 2 , tan t
r t cos t t sen t, sen t t cos t, t
r t ti 2t 3 j 3t 5 k
r t
1
2 t 2 i tj
1
6t 3 k
r t 8 cos ti 3 sen tj
r t 4 cos ti 4 sen tj
r t t 2 t i t 2 t j
r t
t 3 i
1
2t 2 j
r t r t .
r t ,
r t ,
r t arcsen t, arccos t, 0
r t
t sen t, t cos t, t
r t
t 3 , cos 3t, sen 3t
r t e t i 4j 5te t k
r t 4 t i t 2 t j ln t 2 k
r t a cos 3 ti a sen 3 tj k
r t
1
t i 16tj t 2
2 k
r t 6ti 7t 2 j t 3 k
r t t cos t, 2 sen t
r t
2 cos t, 5 sen t
r t t i 1 t 3 j
r t t 3 i 3tj
r t .
t 0 2
r t ti t 2 j
3
2k,
t 0 3
2
r t 2 cos ti 2 sin tj tk,
t 0 .
r t 0 r t 0
t 0 0
r t e t , e t ,
t 0 0
r t e t , e 2t ,
t 0 2
r t) 3 sen ti 4 cos tj,
t 0 2
r t cos ti sen tj,
t 0 1
r t 1 t i t 3 j,
t 0 2
r t
t 2 i
1
t j, t 0 1
r t ti t 2 1 j,
t 0 2
r t t 2 i tj,
r t 0
r t 0 .
r t 0
r t 0 t 0 .
r t 0
r t 0
848 Chapter 12 Vector-Valued Functions
12.2 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
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