Calculo 2 De dos variables_9na Edición - Ron Larson

SECCIÓN 15.5 Superficies paramétricas 1109
15.5 Ejercicios
En los ejercicios 1 a 6, relacionar la función vectorial con su gráfica.
[Las gráficas están marcadas a), b), c), d), e) y f).]
a) b)
c) d)
e) f)
En los ejercicios 7 a 10, hallar la ecuación rectangular de la
superficie por eliminación de los parámetros de la función vectorial.
Identificar la superficie y dibujar su gráfica.
En los ejercicios 11 a 16, utilizar un sistema algebraico por computadora
y representar gráficamente la superficie dada por la función
vectorial.
Para pensar
En los ejercicios 17 a 20, determinar cómo la gráfica
de la superficie
difiere de la gráfica de
(ver la figura) donde
y
(No es necesario representar s gráficamente.)
En los ejercicios 21 a 30, hallar una función vectorial cuya gráfica
sea la superficie indicada.
29. La parte del plano interior al cilindro
30. La parte del paraboloide interior al cilindro
x 2 y 2 9
z x 2 y 2 x 2 y 2 9
z 4
y
x
2
−2
−2
2
4
r(u, v)
z
0 v 2 .
0 u 2
u cos vi u sin vj u 2 k
ru, v
su, v
sen
x
y
2
2
2
−2
−2
1
z
In Exercises 1– 6, match the vector-valued function with its
graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
(a)
(b)
(c)
(d)
(e)
(f)
1.
2.
3.
4.
5.
6.
In Exercises 7– 10, find the rectangular equation for the surface
by eliminating the parameters from the vector-valued function.
Identify the surface and sketch its graph.
7.
8.
9.
10.
In Exercises 11–16, use a computer algebra system to graph the
surface represented by the vector-valued function.
11.
12.
13.
14.
15.
16.
Think About It
In Exercises 17–20, determine how the graph
of the surface
differs from the graph of
(see figure), where
and
(It is not necessary to graph s.)
17.
18.
19.
20.
In Exercises 21–30, find a vector-valued function whose graph
is the indicated surface.
21. El plano
22. El plano
23. El cono
24. El cono
25. El cilindro
26. El cilindro
27. El cilindro
28. El elipsoide
29. The part of the plane that lies inside the cylinder
30. The part of the paraboloid that lies inside the
cylinder x 2 y 2 9
z x 2 y 2
x 2 y 2 9
z 4
x 2
9
y 2
4
z 2
1
1
z x 2
4x 2 y 2 16
x 2 y 2 25
x 16y 2 z 2
y 4x 2 9z 2
x y z 6
z
y
0 v 2
0 u 2,
s u, v 4u cos vi 4u sen vj u 2 k
0 v 2
0 u 3,
s u, v u cos vi u sen vj u 2 k
0 v 2
0 u 2,
s u, v u cos vi u 2 j u sen vk
0 v 2
0 u 2,
s u, v u cos vi u sen vj u 2 k
y
x
2
−2
−2
2
4
r(u, v)
z
0 v 2 .
0 u 2
u cos vi u sen vj u 2 k
r u, v
s u, v
0 v 2
0 u
2 ,
r u, v cos 3 u cos vi sen 3 u sen vj uk
0 v 2
0 u ,
r u, v u sen u cos vi 1 cos u sen vj uk
0 v 3
0 u 1,
r u, v 2u cos vi 2u sen vj vk
0 v 2
0 u 2,
r u, v 2 senh u cos vi senh u sen vj cosh uk
0 v 2
0 u 2 ,
r u, v 2 cos v cos ui 4 cos v sen uj sen vk
0 v 2
0 u 1,
r u, v 2u cos vi 2u sen vj u 4 k
r u, v 3 cos v cos ui 3 cos v sen uj 5 sen vk
r u, v 2 cos ui vj 2 sen uk
r u, v 2u cos vi 2u sen vj
1
2 u 2 k
r u, v ui vj
v
2 k
r u, v 4 cos ui 4 sen uj vk
r u, v 2 cos v cos ui 2 cos v sen uj 2 sen vk
r u, v
ui
1
4v 3 j
vk
r u, v
ui
1
2 u v j vk
r u, v u cos vi u sen vj uk
r u, v ui vj uvk
y
2
2
2
z
x
y
4 4
4
−4
z
y
2
2
2
z
x
y
4 4
2
z
2
x
y
2
2
−2
−1
1
1
z
x
y
2
2
2
−2
−2
1
z
15.5 Parametric Surfaces 1109
15.5 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1– 6, match the vector-valued function with its
graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
(a)
(b)
(c)
(d)
(e)
(f)
1.
2.
3.
4.
5.
6.
In Exercises 7– 10, find the rectangular equation for the surface
by eliminating the parameters from the vector-valued function.
Identify the surface and sketch its graph.
7.
8.
9.
10.
In Exercises 11–16, use a computer algebra system to graph the
surface represented by the vector-valued function.
11.
12.
13.
14.
15.
16.
Think About It
In Exercises 17–20, determine how the graph
of the surface
differs from the graph of
(see figure), where
and
(It is not necessary to graph s.)
17.
18.
19.
20.
In Exercises 21–30, find a vector-valued function whose graph
is the indicated surface.
21. El plano
22. El plano
23. El cono
24. El cono
25. El cilindro
26. El cilindro
27. El cilindro
28. El elipsoide
29. The part of the plane that lies inside the cylinder
30. The part of the paraboloid that lies inside the
cylinder x 2 y 2 9
z x 2 y 2
x 2 y 2 9
z 4
x 2
9
y 2
4
z 2
1
1
z x 2
4x 2 y 2 16
x 2 y 2 25
x 16y 2 z 2
y 4x 2 9z 2
x y z 6
z
y
0 v 2
0 u 2,
s u, v 4u cos vi 4u sen vj u 2 k
0 v 2
0 u 3,
s u, v u cos vi u sen vj u 2 k
0 v 2
0 u 2,
s u, v u cos vi u 2 j u sen vk
0 v 2
0 u 2,
s u, v u cos vi u sen vj u 2 k
y
x
2
−2
−2
2
4
r(u, v)
z
0 v 2 .
0 u 2
u cos vi u sen vj u 2 k
r u, v
s u, v
0 v 2
0 u
2 ,
r u, v cos 3 u cos vi sen 3 u sen vj uk
0 v 2
0 u ,
r u, v u sen u cos vi 1 cos u sen vj uk
0 v 3
0 u 1,
r u, v 2u cos vi 2u sen vj vk
0 v 2
0 u 2,
r u, v 2 senh u cos vi senh u sen vj cosh uk
0 v 2
0 u 2 ,
r u, v 2 cos v cos ui 4 cos v sen uj sen vk
0 v 2
0 u 1,
r u, v 2u cos vi 2u sen vj u 4 k
r u, v 3 cos v cos ui 3 cos v sen uj 5 sen vk
r u, v 2 cos ui vj 2 sen uk
r u, v 2u cos vi 2u sen vj
1
2 u 2 k
r u, v ui vj
v
2 k
r u, v 4 cos ui 4 sen uj vk
r u, v 2 cos v cos ui 2 cos v sen uj 2 sen vk
r u, v
ui
1
4v 3 j
vk
r u, v
ui
1
2 u v j vk
r u, v u cos vi u sen vj uk
r u, v ui vj uvk
y
2
2
2
z
x
y
4 4
4
−4
z
y
2
2
2
z
x
y
4 4
2
z
2
x
y
2
2
−2
−1
1
1
z
x
y
2
2
2
−2
−2
1
z
15.5 Parametric Surfaces 1109
15.5 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1– 6, match the vector-valued function with its
graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
(a)
(b)
(c)
(d)
(e)
(f)
1.
2.
3.
4.
5.
6.
In Exercises 7– 10, find the rectangular equation for the surface
by eliminating the parameters from the vector-valued function.
Identify the surface and sketch its graph.
7.
8.
9.
10.
In Exercises 11–16, use a computer algebra system to graph the
surface represented by the vector-valued function.
11.
12.
13.
14.
15.
16.
Think About It
In Exercises 17–20, determine how the graph
of the surface
differs from the graph of
(see figure), where
and
(It is not necessary to graph s.)
17.
18.
19.
20.
In Exercises 21–30, find a vector-valued function whose graph
is the indicated surface.
21. El plano
22. El plano
23. El cono
24. El cono
25. El cilindro
26. El cilindro
27. El cilindro
28. El elipsoide
29. The part of the plane that lies inside the cylinder
30. The part of the paraboloid that lies inside the
cylinder x 2 y 2 9
z x 2 y 2
x 2 y 2 9
z 4
x 2
9
y 2
4
z 2
1
1
z x 2
4x 2 y 2 16
x 2 y 2 25
x 16y 2 z 2
y 4x 2 9z 2
x y z 6
z
y
0 v 2
0 u 2,
s u, v 4u cos vi 4u sen vj u 2 k
0 v 2
0 u 3,
s u, v u cos vi u sen vj u 2 k
0 v 2
0 u 2,
s u, v u cos vi u 2 j u sen vk
0 v 2
0 u 2,
s u, v u cos vi u sen vj u 2 k
y
x
2
−2
−2
2
4
r(u, v)
z
0 v 2 .
0 u 2
u cos vi u sen vj u 2 k
r u, v
s u, v
0 v 2
0 u
2 ,
r u, v cos 3 u cos vi sen 3 u sen vj uk
0 v 2
0 u ,
r u, v u sen u cos vi 1 cos u sen vj uk
0 v 3
0 u 1,
r u, v 2u cos vi 2u sen vj vk
0 v 2
0 u 2,
r u, v 2 senh u cos vi senh u sen vj cosh uk
0 v 2
0 u 2 ,
r u, v 2 cos v cos ui 4 cos v sen uj sen vk
0 v 2
0 u 1,
r u, v 2u cos vi 2u sen vj u 4 k
r u, v 3 cos v cos ui 3 cos v sen uj 5 sen vk
r u, v 2 cos ui vj 2 sen uk
r u, v 2u cos vi 2u sen vj
1
2 u 2 k
r u, v ui vj
v
2 k
r u, v 4 cos ui 4 sen uj vk
r u, v 2 cos v cos ui 2 cos v sen uj 2 sen vk
r u, v
ui
1
4v 3 j
vk
r u, v
ui
1
2 u v j vk
r u, v u cos vi u sen vj uk
r u, v ui vj uvk
y
2
2
2
z
x
y
4 4
4
−4
z
x
y
2
2
2
z
x
y
4 4
2
z
2
x
y
2
2
−2
−1
1
1
z
x
y
2
2
2
−2
−2
1
z
15.5 Parametric Surfaces 1109
15.5 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1– 6, match the vector-valued function with its
graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
(a)
(b)
(c)
(d)
(e)
(f)
1.
2.
3.
4.
5.
6.
In Exercises 7– 10, find the rectangular equation for the surface
by eliminating the parameters from the vector-valued function.
Identify the surface and sketch its graph.
7.
8.
9.
10.
In Exercises 11–16, use a computer algebra system to graph the
surface represented by the vector-valued function.
11.
12.
13.
14.
15.
16.
Think About It
In Exercises 17–20, determine how the graph
of the surface
differs from the graph of
(see figure), where
and
(It is not necessary to graph s.)
17.
18.
19.
20.
In Exercises 21–30, find a vector-valued function whose graph
is the indicated surface.
21. El plano
22. El plano
23. El cono
24. El cono
25. El cilindro
26. El cilindro
27. El cilindro
28. El elipsoide
29. The part of the plane that lies inside the cylinder
30. The part of the paraboloid that lies inside the
cylinder x 2 y 2 9
z x 2 y 2
x 2 y 2 9
z 4
x 2
9
y 2
4
z 2
1
1
z x 2
4x 2 y 2 16
x 2 y 2 25
x 16y 2 z 2
y 4x 2 9z 2
x y z 6
z
y
0 v 2
0 u 2,
s u, v 4u cos vi 4u sen vj u 2 k
0 v 2
0 u 3,
s u, v u cos vi u sen vj u 2 k
0 v 2
0 u 2,
s u, v u cos vi u 2 j u sen vk
0 v 2
0 u 2,
s u, v u cos vi u sen vj u 2 k
y
x
2
−2
−2
2
4
r(u, v)
z
0 v 2 .
0 u 2
u cos vi u sen vj u 2 k
r u, v
s u, v
0 v 2
0 u
2 ,
r u, v cos 3 u cos vi sen 3 u sen vj uk
0 v 2
0 u ,
r u, v u sen u cos vi 1 cos u sen vj uk
0 v 3
0 u 1,
r u, v 2u cos vi 2u sen vj vk
0 v 2
0 u 2,
r u, v 2 senh u cos vi senh u sen vj cosh uk
0 v 2
0 u 2 ,
r u, v 2 cos v cos ui 4 cos v sen uj sen vk
0 v 2
0 u 1,
r u, v 2u cos vi 2u sen vj u 4 k
r u, v 3 cos v cos ui 3 cos v sen uj 5 sen vk
r u, v 2 cos ui vj 2 sen uk
r u, v 2u cos vi 2u sen vj
1
2 u 2 k
r u, v ui vj
v
2 k
r u, v 4 cos ui 4 sen uj vk
r u, v 2 cos v cos ui 2 cos v sen uj 2 sen vk
r u, v
ui
1
4v 3 j
vk
r u, v
ui
1
2 u v j vk
r u, v u cos vi u sen vj uk
r u, v ui vj uvk
y
2
2
2
z
x
y
4 4
4
−4
z
y
2
2
2
z
x
y
4 4
2
z
2
x
y
2
2
−2
−1
1
1
z
x
y
2
2
2
−2
−2
1
z
15.5 Parametric Surfaces 1109
15.5 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1– 6, match the vector-valued function with its
graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
(a)
(b)
(c)
(d)
(e)
(f)
1.
2.
3.
4.
5.
6.
In Exercises 7– 10, find the rectangular equation for the surface
by eliminating the parameters from the vector-valued function.
Identify the surface and sketch its graph.
7.
8.
9.
10.
In Exercises 11–16, use a computer algebra system to graph the
surface represented by the vector-valued function.
11.
12.
13.
14.
15.
16.
Think About It
In Exercises 17–20, determine how the graph
of the surface
differs from the graph of
(see figure), where
and
(It is not necessary to graph s.)
17.
18.
19.
20.
In Exercises 21–30, find a vector-valued function whose graph
is the indicated surface.
21. El plano
22. El plano
23. El cono
24. El cono
25. El cilindro
26. El cilindro
27. El cilindro
28. El elipsoide
29. The part of the plane that lies inside the cylinder
30. The part of the paraboloid that lies inside the
cylinder x 2 y 2 9
z x 2 y 2
x 2 y 2 9
z 4
x 2
9
y 2
4
z 2
1
1
z x 2
4x 2 y 2 16
x 2 y 2 25
x 16y 2 z 2
y 4x 2 9z 2
x y z 6
z
y
0 v 2
0 u 2,
s u, v 4u cos vi 4u sen vj u 2 k
0 v 2
0 u 3,
s u, v u cos vi u sen vj u 2 k
0 v 2
0 u 2,
s u, v u cos vi u 2 j u sen vk
0 v 2
0 u 2,
s u, v u cos vi u sen vj u 2 k
y
x
2
−2
−2
2
4
r(u, v)
z
0 v 2 .
0 u 2
u cos vi u sen vj u 2 k
r u, v
s u, v
0 v 2
0 u
2 ,
r u, v cos 3 u cos vi sen 3 u sen vj uk
0 v 2
0 u ,
r u, v u sen u cos vi 1 cos u sen vj uk
0 v 3
0 u 1,
r u, v 2u cos vi 2u sen vj vk
0 v 2
0 u 2,
r u, v 2 senh u cos vi senh u sen vj cosh uk
0 v 2
0 u 2 ,
r u, v 2 cos v cos ui 4 cos v sen uj sen vk
0 v 2
0 u 1,
r u, v 2u cos vi 2u sen vj u 4 k
r u, v 3 cos v cos ui 3 cos v sen uj 5 sen vk
r u, v 2 cos ui vj 2 sen uk
r u, v 2u cos vi 2u sen vj
1
2 u 2 k
r u, v ui vj
v
2 k
r u, v 4 cos ui 4 sen uj vk
r u, v 2 cos v cos ui 2 cos v sen uj 2 sen vk
r u, v
ui
1
4v 3 j
vk
r u, v
ui
1
2 u v j vk
r u, v u cos vi u sen vj uk
r u, v ui vj uvk
x
y
2
2
2
z
x
y
4 4
4
−4
z
y
2
2
2
z
x
y
4 4
2
z
2
x
y
2
2
−2
−1
1
1
z
x
y
2
2
2
−2
−2
1
z
15.5 Parametric Surfaces 1109
15.5 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1– 6, match the vector-valued function with its
graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
(a)
(b)
(c)
(d)
(e)
(f)
1.
2.
3.
4.
5.
6.
In Exercises 7– 10, find the rectangular equation for the surface
by eliminating the parameters from the vector-valued function.
Identify the surface and sketch its graph.
7.
8.
9.
10.
In Exercises 11–16, use a computer algebra system to graph the
surface represented by the vector-valued function.
11.
12.
13.
14.
15.
16.
Think About It
In Exercises 17–20, determine how the graph
of the surface
differs from the graph of
(see figure), where
and
(It is not necessary to graph s.)
17.
18.
19.
20.
In Exercises 21–30, find a vector-valued function whose graph
is the indicated surface.
21. El plano
22. El plano
23. El cono
24. El cono
25. El cilindro
26. El cilindro
27. El cilindro
28. El elipsoide
29. The part of the plane that lies inside the cylinder
30. The part of the paraboloid that lies inside the
cylinder x 2 y 2 9
z x 2 y 2
x 2 y 2 9
z 4
x 2
9
y 2
4
z 2
1
1
z x 2
4x 2 y 2 16
x 2 y 2 25
x 16y 2 z 2
y 4x 2 9z 2
x y z 6
z
y
0 v 2
0 u 2,
s u, v 4u cos vi 4u sen vj u 2 k
0 v 2
0 u 3,
s u, v u cos vi u sen vj u 2 k
0 v 2
0 u 2,
s u, v u cos vi u 2 j u sen vk
0 v 2
0 u 2,
s u, v u cos vi u sen vj u 2 k
y
x
2
−2
−2
2
4
r(u, v)
z
0 v 2 .
0 u 2
u cos vi u sen vj u 2 k
r u, v
s u, v
0 v 2
0 u
2 ,
r u, v cos 3 u cos vi sen 3 u sen vj uk
0 v 2
0 u ,
r u, v u sen u cos vi 1 cos u sen vj uk
0 v 3
0 u 1,
r u, v 2u cos vi 2u sen vj vk
0 v 2
0 u 2,
r u, v 2 senh u cos vi senh u sen vj cosh uk
0 v 2
0 u 2 ,
r u, v 2 cos v cos ui 4 cos v sen uj sen vk
0 v 2
0 u 1,
r u, v 2u cos vi 2u sen vj u 4 k
r u, v 3 cos v cos ui 3 cos v sen uj 5 sen vk
r u, v 2 cos ui vj 2 sen uk
r u, v 2u cos vi 2u sen vj
1
2 u 2 k
r u, v ui vj
v
2 k
r u, v 4 cos ui 4 sen uj vk
r u, v 2 cos v cos ui 2 cos v sen uj 2 sen vk
r u, v
ui
1
4v 3 j
vk
r u, v
ui
1
2 u v j vk
r u, v u cos vi u sen vj uk
r u, v ui vj uvk
y
2
2
2
z
x
y
4 4
4
−4
z
y
2
2
2
z
x
y
4 4
2
z
2
x
y
2
2
−2
−1
1
1
z
x
y
2
2
2
−2
−2
1
z
15.5 Parametric Surfaces 1109
15.5 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1– 6, match the vector-valued function with its
graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
(a)
(b)
(c)
(d)
(e)
(f)
1.
2.
3.
4.
5.
6.
In Exercises 7– 10, find the rectangular equation for the surface
by eliminating the parameters from the vector-valued function.
Identify the surface and sketch its graph.
7.
8.
9.
10.
In Exercises 11–16, use a computer algebra system to graph the
surface represented by the vector-valued function.
11.
12.
13.
14.
15.
16.
Think About It
In Exercises 17–20, determine how the graph
of the surface
differs from the graph of
(see figure), where
and
(It is not necessary to graph s.)
17.
18.
19.
20.
In Exercises 21–30, find a vector-valued function whose graph
is the indicated surface.
21. El plano
22. El plano
23. El cono
24. El cono
25. El cilindro
26. El cilindro
27. El cilindro
28. El elipsoide
29. The part of the plane that lies inside the cylinder
30. The part of the paraboloid that lies inside the
cylinder x 2 y 2 9
z x 2 y 2
x 2 y 2 9
z 4
x 2
9
y 2
4
z 2
1
1
z x 2
4x 2 y 2 16
x 2 y 2 25
x 16y 2 z 2
y 4x 2 9z 2
x y z 6
z
y
0 v 2
0 u 2,
s u, v 4u cos vi 4u sen vj u 2 k
0 v 2
0 u 3,
s u, v u cos vi u sen vj u 2 k
0 v 2
0 u 2,
s u, v u cos vi u 2 j u sen vk
0 v 2
0 u 2,
s u, v u cos vi u sen vj u 2 k
y
x
2
−2
−2
2
4
r(u, v)
z
0 v 2 .
0 u 2
u cos vi u sen vj u 2 k
r u, v
s u, v
0 v 2
0 u
2 ,
r u, v cos 3 u cos vi sen 3 u sen vj uk
0 v 2
0 u ,
r u, v u sen u cos vi 1 cos u sen vj uk
0 v 3
0 u 1,
r u, v 2u cos vi 2u sen vj vk
0 v 2
0 u 2,
r u, v 2 senh u cos vi senh u sen vj cosh uk
0 v 2
0 u 2 ,
r u, v 2 cos v cos ui 4 cos v sen uj sen vk
0 v 2
0 u 1,
r u, v 2u cos vi 2u sen vj u 4 k
r u, v 3 cos v cos ui 3 cos v sen uj 5 sen vk
r u, v 2 cos ui vj 2 sen uk
r u, v 2u cos vi 2u sen vj
1
2 u2 k
r u, v ui vj
v
2 k
r u, v 4 cos ui 4 sen uj vk
r u, v 2 cos v cos ui 2 cos v sen uj 2 sen vk
r u, v
ui
1
4v 3 j
vk
r u, v
ui
1
2 u v j vk
r u, v u cos vi u sen vj uk
r u, v ui vj uvk
y
2
2
2
z
x
y
4 4
4
−4
z
y
2
2
2
z
x
y
4 4
2
z
2
x
y
2
2
−2
−1
1
1
z
x
y
2
2
2
−2
−2
1
z
15.5 Parametric Surfaces 1109
15.5 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1– 6, match the vector-valued function with its
graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
(a)
(b)
(c)
(d)
(e)
(f)
1.
2.
3.
4.
5.
6.
In Exercises 7– 10, find the rectangular equation for the surface
by eliminating the parameters from the vector-valued function.
Identify the surface and sketch its graph.
7.
8.
9.
10.
In Exercises 11–16, use a computer algebra system to graph the
surface represented by the vector-valued function.
11.
12.
13.
14.
15.
16.
Think About It
In Exercises 17–20, determine how the graph
of the surface
differs from the graph of
(see figure), where
and
(It is not necessary to graph s.)
17.
18.
19.
20.
In Exercises 21–30, find a vector-valued function whose graph
is the indicated surface.
21. El plano
22. El plano
23. El cono
24. El cono
25. El cilindro
26. El cilindro
27. El cilindro
28. El elipsoide
29. The part of the plane that lies inside the cylinder
30. The part of the paraboloid that lies inside the
cylinder x 2 y 2 9
z x 2 y 2
x 2 y 2 9
z 4
x 2
9
y 2
4
z 2
1
1
z x 2
4x 2 y 2 16
x 2 y 2 25
x 16y 2 z 2
y 4x 2 9z 2
x y z 6
z
y
0 v 2
0 u 2,
s u, v 4u cos vi 4u sen vj u 2 k
0 v 2
0 u 3,
s u, v u cos vi u sen vj u 2 k
0 v 2
0 u 2,
s u, v u cos vi u 2 j u sen vk
0 v 2
0 u 2,
s u, v u cos vi u sen vj u 2 k
y
x
2
−2
−2
2
4
r(u, v)
z
0 v 2 .
0 u 2
u cos vi u sen vj u 2 k
r u, v
s u, v
0 v 2
0 u
2 ,
r u, v cos 3 u cos vi sen 3 u sen vj uk
0 v 2
0 u ,
r u, v u sen u cos vi 1 cos u sen vj uk
0 v 3
0 u 1,
r u, v 2u cos vi 2u sen vj vk
0 v 2
0 u 2,
r u, v 2 senh u cos vi senh u sen vj cosh uk
0 v 2
0 u 2 ,
r u, v 2 cos v cos ui 4 cos v sen uj sen vk
0 v 2
0 u 1,
r u, v 2u cos vi 2u sen vj u 4 k
r u, v 3 cos v cos ui 3 cos v sen uj 5 sen vk
r u, v 2 cos ui vj 2 sen uk
r u, v 2u cos vi 2u sen vj
1
2 u 2 k
r u, v ui vj
v
2 k
r u, v 4 cos ui 4 sen uj vk
r u, v 2 cos v cos ui 2 cos v sen uj 2 sen vk
r u, v
ui
1
4v 3 j
vk
r u, v
ui
1
2 u v j vk
r u, v u cos vi u sen vj uk
r u, v ui vj uvk
y
2
2
2
z
x
y
4 4
4
−4
z
y
2
2
2
z
x
y
4 4
2
z
2
x
y
2
2
−2
−1
1
1
z
x
y
2
2
2
−2
−2
1
z
15.5 Parametric Surfaces 1109
15.5 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1– 6, match the vector-valued function with its
graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
(a)
(b)
(c)
(d)
(e)
(f)
1.
2.
3.
4.
5.
6.
In Exercises 7– 10, find the rectangular equation for the surface
by eliminating the parameters from the vector-valued function.
Identify the surface and sketch its graph.
7.
8.
9.
10.
In Exercises 11–16, use a computer algebra system to graph the
surface represented by the vector-valued function.
11.
12.
13.
14.
15.
16.
Think About It
In Exercises 17–20, determine how the graph
of the surface
differs from the graph of
(see figure), where
and
(It is not necessary to graph s.)
17.
18.
19.
20.
In Exercises 21–30, find a vector-valued function whose graph
is the indicated surface.
21. El plano
22. El plano
23. El cono
24. El cono
25. El cilindro
26. El cilindro
27. El cilindro
28. El elipsoide
29. The part of the plane that lies inside the cylinder
30. The part of the paraboloid that lies inside the
cylinder x 2 y 2 9
z x 2 y 2
x 2 y 2 9
z 4
x 2
9
y 2
4
z 2
1
1
z x 2
4x 2 y 2 16
x 2 y 2 25
x 16y 2 z 2
y 4x 2 9z 2
x y z 6
z
y
0 v 2
0 u 2,
s u, v 4u cos vi 4u sen vj u 2 k
0 v 2
0 u 3,
s u, v u cos vi u sen vj u 2 k
0 v 2
0 u 2,
s u, v u cos vi u 2 j u sen vk
0 v 2
0 u 2,
s u, v u cos vi u sen vj u 2 k
y
x
2
−2
−2
2
4
r(u, v)
z
0 v 2 .
0 u 2
u cos vi u sen vj u 2 k
r u, v
s u, v
0 v 2
0 u
2 ,
r u, v cos 3 u cos vi sen 3 u sen vj uk
0 v 2
0 u ,
r u, v u sen u cos vi 1 cos u sen vj uk
0 v 3
0 u 1,
r u, v 2u cos vi 2u sen vj vk
0 v 2
0 u 2,
r u, v 2 senh u cos vi senh u sen vj cosh uk
0 v 2
0 u 2 ,
r u, v 2 cos v cos ui 4 cos v sen uj sen vk
0 v 2
0 u 1,
r u, v 2u cos vi 2u sen vj u 4 k
r u, v 3 cos v cos ui 3 cos v sen uj 5 sen vk
r u, v 2 cos ui vj 2 sen uk
r u, v 2u cos vi 2u sen vj
1
2 u 2 k
r u, v ui vj
v
2 k
r u, v 4 cos ui 4 sen uj vk
r u, v 2 cos v cos ui 2 cos v sen uj 2 sen vk
r u, v
ui
1
4v 3 j
vk
r u, v
ui
1
2 u v j vk
r u, v u cos vi u sen vj uk
r u, v ui vj uvk
y
2
2
2
z
x
y
4 4
4
−4
z
y
2
2
2
z
x
y
4 4
2
z
2
x
y
2
2
−2
−1
1
1
z
x
y
2
2
2
−2
−2
1
z
15.5 Parametric Surfaces 1109
15.5 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1– 6, match the vector-valued function with its
graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
(a)
(b)
(c)
(d)
(e)
(f)
1.
2.
3.
4.
5.
6.
In Exercises 7– 10, find the rectangular equation for the surface
by eliminating the parameters from the vector-valued function.
Identify the surface and sketch its graph.
7.
8.
9.
10.
In Exercises 11–16, use a computer algebra system to graph the
surface represented by the vector-valued function.
11.
12.
13.
14.
15.
16.
Think About It
In Exercises 17–20, determine how the graph
of the surface
differs from the graph of
(see figure), where
and
(It is not necessary to graph s.)
17.
18.
19.
20.
In Exercises 21–30, find a vector-valued function whose graph
is the indicated surface.
21. El plano
22. El plano
23. El cono
24. El cono
25. El cilindro
26. El cilindro
27. El cilindro
28. El elipsoide
29. The part of the plane that lies inside the cylinder
30. The part of the paraboloid that lies inside the
cylinder x 2 y 2 9
z x 2 y 2
x 2 y 2 9
z 4
x 2
9
y 2
4
z 2
1
1
z x 2
4x 2 y 2 16
x 2 y 2 25
x 16y 2 z 2
y 4x 2 9z 2
x y z 6
z
y
0 v 2
0 u 2,
s u, v 4u cos vi 4u sen vj u 2 k
0 v 2
0 u 3,
s u, v u cos vi u sen vj u 2 k
0 v 2
0 u 2,
s u, v u cos vi u 2 j u sen vk
0 v 2
0 u 2,
s u, v u cos vi u sen vj u 2 k
y
x
2
−2
−2
2
4
r(u, v)
z
0 v 2 .
0 u 2
u cos vi u sen vj u 2 k
r u, v
s u, v
0 v 2
0 u
2 ,
r u, v cos 3 u cos vi sen 3 u sen vj uk
0 v 2
0 u ,
r u, v u sen u cos vi 1 cos u sen vj uk
0 v 3
0 u 1,
r u, v 2u cos vi 2u sen vj vk
0 v 2
0 u 2,
r u, v 2 senh u cos vi senh u sen vj cosh uk
0 v 2
0 u 2 ,
r u, v 2 cos v cos ui 4 cos v sen uj sen vk
0 v 2
0 u 1,
r u, v 2u cos vi 2u sen vj u 4 k
r u, v 3 cos v cos ui 3 cos v sen uj 5 sen vk
r u, v 2 cos ui vj 2 sen uk
r u, v 2u cos vi 2u sen vj
1
2 u 2 k
r u, v ui vj
v
2 k
r u, v 4 cos ui 4 sen uj vk
r u, v 2 cos v cos ui 2 cos v sen uj 2 sen vk
r u, v
ui
1
4v 3 j
vk
r u, v
ui
1
2 u v j vk
r u, v u cos vi u sen vj uk
r u, v ui vj uvk
y
2
2
2
z
x
y
4 4
4
−4
z
y
2
2
2
z
x
y
4 4
2
z
2
x
y
2
2
−2
−1
1
1
z
x
y
2
2
2
−2
−2
1
z
15.5 Parametric Surfaces 1109
15.5 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1– 6, match the vector-valued function with its
graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
(a)
(b)
(c)
(d)
(e)
(f)
1.
2.
3.
4.
5.
6.
In Exercises 7– 10, find the rectangular equation for the surface
by eliminating the parameters from the vector-valued function.
Identify the surface and sketch its graph.
7.
8.
9.
10.
In Exercises 11–16, use a computer algebra system to graph the
surface represented by the vector-valued function.
11.
12.
13.
14.
15.
16.
Think About It
In Exercises 17–20, determine how the graph
of the surface
differs from the graph of
(see figure), where
and
(It is not necessary to graph s.)
17.
18.
19.
20.
In Exercises 21–30, find a vector-valued function whose graph
is the indicated surface.
21. El plano
22. El plano
23. El cono
24. El cono
25. El cilindro
26. El cilindro
27. El cilindro
28. El elipsoide
29. The part of the plane that lies inside the cylinder
30. The part of the paraboloid that lies inside the
cylinder x 2 y 2 9
z x 2 y 2
x 2 y 2 9
z 4
x 2
9
y 2
4
z 2
1
1
z x 2
4x 2 y 2 16
x 2 y 2 25
x 16y 2 z 2
y 4x 2 9z 2
x y z 6
z
y
0 v 2
0 u 2,
s u, v 4u cos vi 4u sen vj u 2 k
0 v 2
0 u 3,
s u, v u cos vi u sen vj u 2 k
0 v 2
0 u 2,
s u, v u cos vi u 2 j u sen vk
0 v 2
0 u 2,
s u, v u cos vi u sen vj u 2 k
y
x
2
−2
−2
2
4
r(u, v)
z
0 v 2 .
0 u 2
u cos vi u sen vj u 2 k
r u, v
s u, v
0 v 2
0 u
2 ,
r u, v cos 3 u cos vi sen 3 u sen vj uk
0 v 2
0 u ,
r u, v u sen u cos vi 1 cos u sen vj uk
0 v 3
0 u 1,
r u, v 2u cos vi 2u sen vj vk
0 v 2
0 u 2,
r u, v 2 senh u cos vi senh u sen vj cosh uk
0 v 2
0 u 2 ,
r u, v 2 cos v cos ui 4 cos v sen uj sen vk
0 v 2
0 u 1,
r u, v 2u cos vi 2u sen vj u 4 k
r u, v 3 cos v cos ui 3 cos v sen uj 5 sen vk
r u, v 2 cos ui vj 2 sen uk
r u, v 2u cos vi 2u sen vj
1
2 u 2 k
r u, v ui vj
v
2 k
r u, v 4 cos ui 4 sen uj vk
r u, v 2 cos v cos ui 2 cos v sen uj 2 sen vk
r u, v
ui
1
4v 3 j
vk
r u, v
ui
1
2 u v j vk
r u, v u cos vi u sen vj uk
r u, v ui vj uvk
y
2
2
2
z
x
y
4 4
4
−4
z
y
2
2
2
z
x
y
4 4
2
z
2
x
y
2
2
−2
−1
1
1
z
x
y
2
2
2
−2
−2
1
z
15.5 Parametric Surfaces 1109
15.5 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1– 6, match the vector-valued function with its
graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
(a)
(b)
(c)
(d)
(e)
(f)
1.
2.
3.
4.
5.
6.
In Exercises 7– 10, find the rectangular equation for the surface
by eliminating the parameters from the vector-valued function.
Identify the surface and sketch its graph.
7.
8.
9.
10.
In Exercises 11–16, use a computer algebra system to graph the
surface represented by the vector-valued function.
11.
12.
13.
14.
15.
16.
Think About It
In Exercises 17–20, determine how the graph
of the surface
differs from the graph of
(see figure), where
and
(It is not necessary to graph s.)
17.
18.
19.
20.
In Exercises 21–30, find a vector-valued function whose graph
is the indicated surface.
21. El plano
22. El plano
23. El cono
24. El cono
25. El cilindro
26. El cilindro
27. El cilindro
28. El elipsoide
29. The part of the plane that lies inside the cylinder
30. The part of the paraboloid that lies inside the
cylinder x 2 y 2 9
z x 2 y 2
x 2 y 2 9
z 4
x 2
9
y 2
4
z 2
1
1
z x 2
4x 2 y 2 16
x 2 y 2 25
x 16y 2 z 2
y 4x 2 9z 2
x y z 6
z
y
0 v 2
0 u 2,
s u, v 4u cos vi 4u sen vj u 2 k
0 v 2
0 u 3,
s u, v u cos vi u sen vj u 2 k
0 v 2
0 u 2,
s u, v u cos vi u 2 j u sen vk
0 v 2
0 u 2,
s u, v u cos vi u sen vj u 2 k
y
x
2
−2
−2
2
4
r(u, v)
z
0 v 2 .
0 u 2
u cos vi u sen vj u 2 k
r u, v
s u, v
0 v 2
0 u
2 ,
r u, v cos 3 u cos vi sen 3 u sen vj uk
0 v 2
0 u ,
r u, v u sen u cos vi 1 cos u sen vj uk
0 v 3
0 u 1,
r u, v 2u cos vi 2u sen vj vk
0 v 2
0 u 2,
r u, v 2 senh u cos vi senh u sen vj cosh uk
0 v 2
0 u 2 ,
r u, v 2 cos v cos ui 4 cos v sen uj sen vk
0 v 2
0 u 1,
r u, v 2u cos vi 2u sen vj u 4 k
r u, v 3 cos v cos ui 3 cos v sen uj 5 sen vk
r u, v 2 cos ui vj 2 sen uk
r u, v 2u cos vi 2u sen vj
1
2 u 2 k
r u, v ui vj
v
2 k
r u, v 4 cos ui 4 sen uj vk
r u, v 2 cos v cos ui 2 cos v sen uj 2 sen vk
r u, v
ui
1
4v 3 j
vk
r u, v
ui
1
2 u v j vk
r u, v u cos vi u sen vj uk
r u, v ui vj uvk
y
2
2
2
z
x
y
4 4
4
−4
z
y
2
2
2
z
x
y
4 4
2
z
2
x
y
2
2
−2
−1
1
1
z
x
y
2
2
2
−2
−2
1
z
15.5 Parametric Surfaces 1109
15.5 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS