Calculo 2 De dos variables_9na Edición - Ron Larson

952 CAPÍTULO 13 Funciones de varias variables
En los ejercicios 51 a 56, encontrar el (los) punto(s) sobre la
superficie en la cual el plano tangente es horizontal.
En los ejercicios 57 y 58, demostrar que las superficies son tangentes
a cada una en el punto dado para demostrar que las
superficies tienen el mismo plano tangente en este punto.
En los ejercicios 59 y 60, a) demostrar que las superficies intersecan
en el punto dado y b) demostrar que las superficies tienen
planos tangentes perpendiculares en este punto.
61. Encontrar un punto sobre el elipsoide donde
el plano tangente es perpendicular a la recta con ecuaciones
paramétricas
62. Encontrar un punto sobre el hiperboloide
donde el plano tangente es paralelo al plano
67. Investigación Considerar la función
en los intervalos
a) Hallar un conjunto de ecuaciones paramétricas de la recta
normal y una ecuación del plano tangente a la superficie en
el punto (1, 1, 1).
b) Repetir el inciso a) con el punto
c) Utilizar un sistema algebraico por computadora y representar
gráficamente la superficie, las rectas normales y los planos
tangentes encontrados en los incisos a) y b).
68. Investigación Considerar la función
en los intervalos
a) Hallar un conjunto de ecuaciones paramétricas de la recta
normal y una ecuación del plano tangente a la superficie en
el punto
b) Repetir el inciso a) con el punto
c) Utilizar un sistema algebraico por computadora y representar
gráficamente la superficie, las rectas normales y los planos
tangentes calculados en los incisos a) y b).
69. Considerar las funciones
y
a) Hallar un conjunto de ecuaciones paramétricas de la recta tangente
a la curva de intersección de las superficies en el punto
(1, 2, 4), y hallar el ángulo entre los vectores gradientes.
b) Utilizar un sistema algebraico por computadora y representar
gráficamente las superficies. Representar gráficamente la
recta tangente obtenida en el inciso a).
70. Considerar las funciones
y
a) Utilizar un sistema algebraico por computadora y representar
gráficamente la porción del primer octante de las superficies
representadas por f y g.
b) Hallar un punto en el primer octante sobre la curva intersección
y mostrar que las superficies son ortogonales en este punto.
c) Estas superficies son ortogonales a lo largo de la curva intersección.
¿Demuestra este hecho el inciso b)? Explicar.
En los ejercicios 71 y 72, probar que el plano tangente a la superficie
cuádrica en el punto
puede expresarse en la forma
dada.
71. Elipsoide:
Plano: x 0x
a 2 y 0y
b 2 z 0z
c 2 1
x 2
a 2 y2
b 2 z2
c 2 1
x 0 , y 0 , z 0
gx, y 2
2 1 3x2 y 2 6x 4y.
f x, y 16 x 2 y 2 2x 4y
gx, y 2x y.
f x, y 6 x 2 y 2 4
2 3 , 3
2 , 3 2 .
2,
2 , 1 2 .
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
61. Find a point on the ellipsoid where the
tangent plane is perpendicular to the line with parametric
equations
y
62. Find a point on the hyperboloid where the
tangent plane is parallel to the plane
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y
(a) Find a set of parametric equations of the tangent line to the
curve of intersection of the surfaces at the point ,
and find the angle between the gradient vectors.
(b) Use a computer algebra system to graph the surfaces.
Graph the tangent line found in part (a).
70. Consider the functions
y
(a) Use a computer algebra system to graph the first-octant
portion of the surfaces represented by and
(b) Find one first-octant point on the curve of intersection and
show that the surfaces are orthogonal at this point.
(c) These surfaces are orthogonal along the curve of intersection.
Does part (b) prove this fact? Explain.
In Exercises 71 and 72, show that the tangent plane to the
quadric surface at the point
can be written in the
given form.
71. Ellipsoid:
Plane: x 0x
a 2
y 0 y
b 2
z 0 z
c 2 1
x 2
a 2 y 2
b 2 z 2
c 2 1
x 0 , y 0 , z 0 g.
f
gx, y
2
2
1 3x 2 y 2 6x 4y.
fx, y 16 x 2 y 2 2x 4y
1, 2, 4
gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
fx, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
fx, y
4xy
x 2 1 y 2 1
x 4y z 0.
x 2 4y 2 z 2 1
z 3 2t.
x 2 4t, y 1 8t
x 2 4y 2 z 2 9
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
63. Give the standard form of the equation of the tangent plane
to a surface given by
at
64. For some surfaces, the normal lines at any point pass
through the same geometric object. What is the common
geometric object for a sphere? What is the common
geometric object for a right circular cylinder? Explain.
65. Discuss the relationship between the tangent plane to a
surface and approximation by differentials.
x 0 , y 0 , z 0 .
F x, y, z 0
WRITING ABOUT CONCEPTS
66. Consider the elliptic cone given by
(a) Find an equation of the tangent plane at the point
(b) Find symmetric equations of the normal line at the
point 5, 13, 12 .
5, 13, 12 .
x 2 y 2 z 2 0.
CAPSTONE
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CAS
CAS
CAS
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
61. Find a point on the ellipsoid where the
tangent plane is perpendicular to the line with parametric
equations
y
62. Find a point on the hyperboloid where the
tangent plane is parallel to the plane
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y
(a) Find a set of parametric equations of the tangent line to the
curve of intersection of the surfaces at the point ,
and find the angle between the gradient vectors.
(b) Use a computer algebra system to graph the surfaces.
Graph the tangent line found in part (a).
70. Consider the functions
y
(a) Use a computer algebra system to graph the first-octant
portion of the surfaces represented by and
(b) Find one first-octant point on the curve of intersection and
show that the surfaces are orthogonal at this point.
(c) These surfaces are orthogonal along the curve of intersection.
Does part (b) prove this fact? Explain.
In Exercises 71 and 72, show that the tangent plane to the
quadric surface at the point
can be written in the
given form.
71. Ellipsoid:
Plane: x 0x
a 2
y 0 y
b 2
z 0 z
c 2 1
x 2
a 2 y 2
b 2 z 2
c 2 1
x 0 , y 0 , z 0 g.
f
gx, y
2
2
1 3x 2 y 2 6x 4y.
fx, y 16 x 2 y 2 2x 4y
1, 2, 4
gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
f x, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
f x, y
4xy
x 2 1 y 2 1
x 4y z 0.
x 2 4y 2 z 2 1
z 3 2t.
x 2 4t, y 1 8t
x 2 4y 2 z 2 9
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
63. Give the standard form of the equation of the tangent plane
to a surface given by
at
64. For some surfaces, the normal lines at any point pass
through the same geometric object. What is the common
geometric object for a sphere? What is the common
geometric object for a right circular cylinder? Explain.
65. Discuss the relationship between the tangent plane to a
surface and approximation by differentials.
x 0 , y 0 , z 0 .
F x, y, z 0
WRITING ABOUT CONCEPTS
66. Consider the elliptic cone given by
(a) Find an equation of the tangent plane at the point
(b) Find symmetric equations of the normal line at the
point 5, 13, 12 .
5, 13, 12 .
x 2 y 2 z 2 0.
CAPSTONE
CAS
CAS
CAS
CAS
1, 2, 4 5.
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
61. Find a point on the ellipsoid where the
tangent plane is perpendicular to the line with parametric
equations
y
62. Find a point on the hyperboloid where the
tangent plane is parallel to the plane
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y
(a) Find a set of parametric equations of the tangent line to the
curve of intersection of the surfaces at the point ,
and find the angle between the gradient vectors.
(b) Use a computer algebra system to graph the surfaces.
Graph the tangent line found in part (a).
70. Consider the functions
y
(a) Use a computer algebra system to graph the first-octant
portion of the surfaces represented by and
(b) Find one first-octant point on the curve of intersection and
show that the surfaces are orthogonal at this point.
(c) These surfaces are orthogonal along the curve of intersection.
Does part (b) prove this fact? Explain.
In Exercises 71 and 72, show that the tangent plane to the
quadric surface at the point
can be written in the
given form.
71. Ellipsoid:
Plane: x 0x
a 2
y 0 y
b 2
z 0 z
c 2 1
x 2
a 2 y 2
b 2 z 2
c 2 1
x 0 , y 0 , z 0 g.
f
gx, y
2
2
1 3x 2 y 2 6x 4y.
fx, y 16 x 2 y 2 2x 4y
1, 2, 4
gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
fx, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
fx, y
4xy
x 2 1 y 2 1
x 4y z 0.
x 2 4y 2 z 2 1
z 3 2t.
x 2 4t, y 1 8t
x 2 4y 2 z 2 9
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
63. Give the standard form of the equation of the tangent plane
to a surface given by
at
64. For some surfaces, the normal lines at any point pass
through the same geometric object. What is the common
geometric object for a sphere? What is the common
geometric object for a right circular cylinder? Explain.
65. Discuss the relationship between the tangent plane to a
surface and approximation by differentials.
x 0 , y 0 , z 0 .
F x, y, z 0
WRITING ABOUT CONCEPTS
66. Consider the elliptic cone given by
(a) Find an equation of the tangent plane at the point
(b) Find symmetric equations of the normal line at the
point 5, 13, 12 .
5, 13, 12 .
x 2 y 2 z 2 0.
CAPSTONE
CAS
CAS
CAS
CAS
f x, y
4xy
x 2 1y 2 1
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
61. Find a point on the ellipsoid where the
tangent plane is perpendicular to the line with parametric
equations
y
62. Find a point on the hyperboloid where the
tangent plane is parallel to the plane
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y
(a) Find a set of parametric equations of the tangent line to the
curve of intersection of the surfaces at the point ,
and find the angle between the gradient vectors.
(b) Use a computer algebra system to graph the surfaces.
Graph the tangent line found in part (a).
70. Consider the functions
y
(a) Use a computer algebra system to graph the first-octant
portion of the surfaces represented by and
(b) Find one first-octant point on the curve of intersection and
show that the surfaces are orthogonal at this point.
(c) These surfaces are orthogonal along the curve of intersection.
Does part (b) prove this fact? Explain.
In Exercises 71 and 72, show that the tangent plane to the
quadric surface at the point
can be written in the
given form.
71. Ellipsoid:
Plane: x 0x
a 2
y 0 y
b 2
z 0 z
c 2 1
x 2
a 2 y 2
b 2 z 2
c 2 1
x 0 , y 0 , z 0 g.
f
gx, y
2
2
1 3x 2 y 2 6x 4y.
fx, y 16 x 2 y 2 2x 4y
1, 2, 4
gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
fx, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
fx, y
4xy
x 2 1 y 2 1
x 4y z 0.
x 2 4y 2 z 2 1
z 3 2t.
x 2 4t, y 1 8t
x 2 4y 2 z 2 9
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
63. Give the standard form of the equation of the tangent plane
to a surface given by
at
64. For some surfaces, the normal lines at any point pass
through the same geometric object. What is the common
geometric object for a sphere? What is the common
geometric object for a right circular cylinder? Explain.
65. Discuss the relationship between the tangent plane to a
surface and approximation by differentials.
x 0 , y 0 , z 0 .
F x, y, z 0
WRITING ABOUT CONCEPTS
66. Consider the elliptic cone given by
(a) Find an equation of the tangent plane at the point
(b) Find symmetric equations of the normal line at the
point 5, 13, 12 .
5, 13, 12 .
x 2 y 2 z 2 0.
CAPSTONE
CAS
CAS
CAS
CAS
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
61. Find a point on the ellipsoid where the
tangent plane is perpendicular to the line with parametric
equations
y
62. Find a point on the hyperboloid where the
tangent plane is parallel to the plane
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y
(a) Find a set of parametric equations of the tangent line to the
curve of intersection of the surfaces at the point ,
and find the angle between the gradient vectors.
(b) Use a computer algebra system to graph the surfaces.
Graph the tangent line found in part (a).
70. Consider the functions
y
(a) Use a computer algebra system to graph the first-octant
portion of the surfaces represented by and
(b) Find one first-octant point on the curve of intersection and
show that the surfaces are orthogonal at this point.
(c) These surfaces are orthogonal along the curve of intersection.
Does part (b) prove this fact? Explain.
In Exercises 71 and 72, show that the tangent plane to the
quadric surface at the point
can be written in the
given form.
71. Ellipsoid:
Plane: x 0x
a 2
y 0 y
b 2
z 0 z
c 2 1
x 2
a 2 y 2
b 2 z 2
c 2 1
x 0 , y 0 , z 0 g.
f
gx, y
2
2
1 3x 2 y 2 6x 4y.
fx, y 16 x 2 y 2 2x 4y
1, 2, 4
gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
fx, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
fx, y
4xy
x 2 1 y 2 1
x 4y z 0.
x 2 4y 2 z 2 1
z 3 2t.
x 2 4t, y 1 8t
x 2 4y 2 z 2 9
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
63. Give the standard form of the equation of the tangent plane
to a surface given by
at
64. For some surfaces, the normal lines at any point pass
through the same geometric object. What is the common
geometric object for a sphere? What is the common
geometric object for a right circular cylinder? Explain.
65. Discuss the relationship between the tangent plane to a
surface and approximation by differentials.
x 0 , y 0 , z 0 .
F x, y, z 0
WRITING ABOUT CONCEPTS
66. Consider the elliptic cone given by
(a) Find an equation of the tangent plane at the point
(b) Find symmetric equations of the normal line at the
point 5, 13, 12 .
5, 13, 12 .
x 2 y 2 z 2 0.
CAPSTONE
CAS
CAS
CAS
CAS
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
61. Find a point on the ellipsoid where the
tangent plane is perpendicular to the line with parametric
equations
y
62. Find a point on the hyperboloid where the
tangent plane is parallel to the plane
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y
(a) Find a set of parametric equations of the tangent line to the
curve of intersection of the surfaces at the point ,
and find the angle between the gradient vectors.
(b) Use a computer algebra system to graph the surfaces.
Graph the tangent line found in part (a).
70. Consider the functions
y
(a) Use a computer algebra system to graph the first-octant
portion of the surfaces represented by and
(b) Find one first-octant point on the curve of intersection and
show that the surfaces are orthogonal at this point.
(c) These surfaces are orthogonal along the curve of intersection.
Does part (b) prove this fact? Explain.
In Exercises 71 and 72, show that the tangent plane to the
quadric surface at the point
can be written in the
given form.
71. Ellipsoid:
Plane: x 0x
a 2
y 0 y
b 2
z 0 z
c 2 1
x 2
a 2 y 2
b 2 z 2
c 2 1
x 0 , y 0 , z 0 g.
f
gx, y
2
2
1 3x 2 y 2 6x 4y.
fx, y 16 x 2 y 2 2x 4y
1, 2, 4
gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
fx, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
fx, y
4xy
x 2 1 y 2 1
x 4y z 0.
x 2 4y 2 z 2 1
z 3 2t.
x 2 4t, y 1 8t
x 2 4y 2 z 2 9
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
63. Give the standard form of the equation of the tangent plane
to a surface given by
at
64. For some surfaces, the normal lines at any point pass
through the same geometric object. What is the common
geometric object for a sphere? What is the common
geometric object for a right circular cylinder? Explain.
65. Discuss the relationship between the tangent plane to a
surface and approximation by differentials.
x 0 , y 0 , z 0 .
F x, y, z 0
WRITING ABOUT CONCEPTS
66. Consider the elliptic cone given by
(a) Find an equation of the tangent plane at the point
(b) Find symmetric equations of the normal line at the
point 5, 13, 12 .
5, 13, 12 .
x 2 y 2 z 2 0.
CAPSTONE
CAS
CAS
CAS
CAS
Desarrollo de conceptos
63. Dar la forma estándar de la ecuación del plano tangente a
una superficie dada por
en
64. En algunas superficies, las rectas normales en cualquier punto
pasan por el mismo objeto geométrico. ¿Cuál es el objeto
geométrico común en una esfera? ¿Cuál es el objeto geométrico
común en un cilindro circular recto? Explicar.
65. Analizar la relación entre el plano tangente a una superficie
y la aproximación por diferenciales.
x 0 , y 0 , z 0 .
Fx, y, z 0
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
61. Find a point on the ellipsoid where the
tangent plane is perpendicular to the line with parametric
equations
y
62. Find a point on the hyperboloid where the
tangent plane is parallel to the plane
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y
(a) Find a set of parametric equations of the tangent line to the
curve of intersection of the surfaces at the point ,
and find the angle between the gradient vectors.
(b) Use a computer algebra system to graph the surfaces.
Graph the tangent line found in part (a).
70. Consider the functions
y
(a) Use a computer algebra system to graph the first-octant
portion of the surfaces represented by and
(b) Find one first-octant point on the curve of intersection and
show that the surfaces are orthogonal at this point.
(c) These surfaces are orthogonal along the curve of intersection.
Does part (b) prove this fact? Explain.
In Exercises 71 and 72, show that the tangent plane to the
quadric surface at the point
can be written in the
given form.
71. Ellipsoid:
Plane: x 0x
a 2
y 0 y
b 2
z 0 z
c 2 1
x 2
a 2 y 2
b 2 z 2
c 2 1
x 0 , y 0 , z 0 g.
f
gx, y
2
2
1 3x 2 y 2 6x 4y.
fx, y 16 x 2 y 2 2x 4y
1, 2, 4
gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
fx, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
fx, y
4xy
x 2 1 y 2 1
x 4y z 0.
x 2 4y 2 z 2 1
z 3 2t.
x 2 4t, y 1 8t
x 2 4y 2 z 2 9
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
63. Give the standard form of the equation of the tangent plane
to a surface given by
at
64. For some surfaces, the normal lines at any point pass
through the same geometric object. What is the common
geometric object for a sphere? What is the common
geometric object for a right circular cylinder? Explain.
65. Discuss the relationship between the tangent plane to a
surface and approximation by differentials.
x 0 , y 0 , z 0 .
F x, y, z 0
WRITING ABOUT CONCEPTS
66. Consider the elliptic cone given by
(a) Find an equation of the tangent plane at the point
(b) Find symmetric equations of the normal line at the
point 5, 13, 12 .
5, 13, 12 .
x 2 y 2 z 2 0.
CAPSTONE
CAS
CAS
CAS
CAS
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
61. Find a point on the ellipsoid where the
tangent plane is perpendicular to the line with parametric
equations
y
62. Find a point on the hyperboloid where the
tangent plane is parallel to the plane
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y
(a) Find a set of parametric equations of the tangent line to the
curve of intersection of the surfaces at the point ,
and find the angle between the gradient vectors.
(b) Use a computer algebra system to graph the surfaces.
Graph the tangent line found in part (a).
70. Consider the functions
y
(a) Use a computer algebra system to graph the first-octant
portion of the surfaces represented by and
(b) Find one first-octant point on the curve of intersection and
show that the surfaces are orthogonal at this point.
(c) These surfaces are orthogonal along the curve of intersection.
Does part (b) prove this fact? Explain.
In Exercises 71 and 72, show that the tangent plane to the
quadric surface at the point
can be written in the
given form.
71. Ellipsoid:
Plane: x 0x
a 2
y 0 y
b 2
z 0 z
c 2 1
x 2
a 2 y 2
b 2 z 2
c 2 1
x 0 , y 0 , z 0 g.
f
gx, y
2
2
1 3x 2 y 2 6x 4y.
fx, y 16 x 2 y 2 2x 4y
1, 2, 4
gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
fx, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
f x, y
4xy
x 2 1 y 2 1
x 4y z 0.
x 2 4y 2 z 2 1
z 3 2t.
x 2 4t, y 1 8t
x 2 4y 2 z 2 9
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
63. Give the standard form of the equation of the tangent plane
to a surface given by
at
64. For some surfaces, the normal lines at any point pass
through the same geometric object. What is the common
geometric object for a sphere? What is the common
geometric object for a right circular cylinder? Explain.
65. Discuss the relationship between the tangent plane to a
surface and approximation by differentials.
x 0 , y 0 , z 0 .
F x, y, z 0
WRITING ABOUT CONCEPTS
66. Consider the elliptic cone given by
(a) Find an equation of the tangent plane at the point
(b) Find symmetric equations of the normal line at the
point 5, 13, 12 .
5, 13, 12 .
x 2 y 2 z 2 0.
CAPSTONE
CAS
CAS
CAS
CAS
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
61. Find a point on the ellipsoid where the
tangent plane is perpendicular to the line with parametric
equations
y
62. Find a point on the hyperboloid where the
tangent plane is parallel to the plane
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y
(a) Find a set of parametric equations of the tangent line to the
curve of intersection of the surfaces at the point ,
and find the angle between the gradient vectors.
(b) Use a computer algebra system to graph the surfaces.
Graph the tangent line found in part (a).
70. Consider the functions
y
(a) Use a computer algebra system to graph the first-octant
portion of the surfaces represented by and
(b) Find one first-octant point on the curve of intersection and
show that the surfaces are orthogonal at this point.
(c) These surfaces are orthogonal along the curve of intersection.
Does part (b) prove this fact? Explain.
In Exercises 71 and 72, show that the tangent plane to the
quadric surface at the point
can be written in the
given form.
71. Ellipsoid:
Plane: x 0x
a 2
y 0 y
b 2
z 0 z
c 2 1
x 2
a 2 y 2
b 2 z 2
c 2 1
x 0 , y 0 , z 0 g.
f
gx, y
2
2
1 3x 2 y 2 6x 4y.
fx, y 16 x 2 y 2 2x 4y
1, 2, 4
gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
fx, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
fx, y
4xy
x 2 1 y 2 1
x 4y z 0.
x 2 4y 2 z 2 1
z 3 2t.
x 2 4t, y 1 8t
x 2 4y 2 z 2 9
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
63. Give the standard form of the equation of the tangent plane
to a surface given by
at
64. For some surfaces, the normal lines at any point pass
through the same geometric object. What is the common
geometric object for a sphere? What is the common
geometric object for a right circular cylinder? Explain.
65. Discuss the relationship between the tangent plane to a
surface and approximation by differentials.
x 0 , y 0 , z 0 .
F x, y, z 0
WRITING ABOUT CONCEPTS
66. Consider the elliptic cone given by
(a) Find an equation of the tangent plane at the point
(b) Find symmetric equations of the normal line at the
point 5, 13, 12 .
5, 13, 12 .
x 2 y 2 z 2 0.
CAPSTONE
CAS
CAS
CAS
CAS
Para discusión
66. Considerar el cono elíptico dado por
a) Encontrar una ecuación del plano tangente en el punto
(5, 13, –12).
b) Encontrar ecuaciones simétricas de la superficie normal
en el punto (5, 13, –12).
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
61. Find a point on the ellipsoid where the
tangent plane is perpendicular to the line with parametric
equations
y
62. Find a point on the hyperboloid where the
tangent plane is parallel to the plane
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y
(a) Find a set of parametric equations of the tangent line to the
curve of intersection of the surfaces at the point ,
and find the angle between the gradient vectors.
(b) Use a computer algebra system to graph the surfaces.
Graph the tangent line found in part (a).
70. Consider the functions
y
(a) Use a computer algebra system to graph the first-octant
portion of the surfaces represented by and
(b) Find one first-octant point on the curve of intersection and
show that the surfaces are orthogonal at this point.
(c) These surfaces are orthogonal along the curve of intersection.
Does part (b) prove this fact? Explain.
In Exercises 71 and 72, show that the tangent plane to the
quadric surface at the point
can be written in the
given form.
71. Ellipsoid:
Plane: x 0x
a 2
y 0 y
b 2
z 0 z
c 2 1
x 2
a 2 y 2
b 2 z 2
c 2 1
x 0 , y 0 , z 0 g.
f
gx, y
2
2
1 3x 2 y 2 6x 4y.
fx, y 16 x 2 y 2 2x 4y
1, 2, 4
gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
fx, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
f x, y
4xy
x 2 1 y 2 1
x 4y z 0.
x 2 4y 2 z 2 1
z 3 2t.
x 2 4t, y 1 8t
x 2 4y 2 z 2 9
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
63. Give the standard form of the equation of the tangent plane
to a surface given by
at
64. For some surfaces, the normal lines at any point pass
through the same geometric object. What is the common
geometric object for a sphere? What is the common
geometric object for a right circular cylinder? Explain.
65. Discuss the relationship between the tangent plane to a
surface and approximation by differentials.
x 0 , y 0 , z 0 .
F x, y, z 0
WRITING ABOUT CONCEPTS
66. Consider the elliptic cone given by
(a) Find an equation of the tangent plane at the point
(b) Find symmetric equations of the normal line at the
point 5, 13, 12 .
5, 13, 12 .
x 2 y 2 z 2 0.
CAPSTONE
CAS
CAS
CAS
CAS
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
61. Find a point on the ellipsoid where the
tangent plane is perpendicular to the line with parametric
equations
y
62. Find a point on the hyperboloid where the
tangent plane is parallel to the plane
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y
(a) Find a set of parametric equations of the tangent line to the
curve of intersection of the surfaces at the point ,
and find the angle between the gradient vectors.
(b) Use a computer algebra system to graph the surfaces.
Graph the tangent line found in part (a).
70. Consider the functions
y
(a) Use a computer algebra system to graph the first-octant
portion of the surfaces represented by and
(b) Find one first-octant point on the curve of intersection and
show that the surfaces are orthogonal at this point.
(c) These surfaces are orthogonal along the curve of intersection.
g.
f
gx, y
2
2
1 3x 2 y 2 6x 4y.
fx, y 16 x 2 y 2 2x 4y
1, 2, 4
gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
fx, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
fx, y
4xy
x 2 1 y 2 1
x 4y z 0.
x 2 4y 2 z 2 1
z 3 2t.
x 2 4t, y 1 8t
x 2 4y 2 z 2 9
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
63. Give the standard form of the equation of the tangent plane
to a surface given by
at
64. For some surfaces, the normal lines at any point pass
through the same geometric object. What is the common
geometric object for a sphere? What is the common
geometric object for a right circular cylinder? Explain.
65. Discuss the relationship between the tangent plane to a
surface and approximation by differentials.
x 0 , y 0 , z 0 .
F x, y, z 0
WRITING ABOUT CONCEPTS
CAS
CAS
CAS
CAS
1053714_1307.qxp 10/27/08 12:09 PM Page 952
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
61. Find a point on the ellipsoid where the
tangent plane is perpendicular to the line with parametric
equations
y
62. Find a point on the hyperboloid where the
tangent plane is parallel to the plane
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y
(a) Find a set of parametric equations of the tangent line to the
curve of intersection of the surfaces at the point ,
and find the angle between the gradient vectors.
(b) Use a computer algebra system to graph the surfaces.
Graph the tangent line found in part (a).
70. Consider the functions
y
(a) Use a computer algebra system to graph the first-octant
portion of the surfaces represented by and
(b) Find one first-octant point on the curve of intersection and
show that the surfaces are orthogonal at this point.
(c) These surfaces are orthogonal along the curve of intersection.
Does part (b) prove this fact? Explain.
In Exercises 71 and 72, show that the tangent plane to the
quadric surface at the point
can be written in the
given form.
71. Ellipsoid:
Plane: x 0x
a 2
y 0 y
b 2
z 0 z
c 2 1
x 2
a 2 y 2
b 2 z 2
c 2 1
x 0 , y 0 , z 0 g.
f
gx, y
2
2
1 3x 2 y 2 6x 4y.
fx, y 16 x 2 y 2 2x 4y
1, 2, 4
gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
fx, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
fx, y
4xy
x 2 1 y 2 1
x 4y z 0.
x 2 4y 2 z 2 1
z 3 2t.
x 2 4t, y 1 8t
x 2 4y 2 z 2 9
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
63. Give the standard form of the equation of the tangent plane
to a surface given by
at
64. For some surfaces, the normal lines at any point pass
through the same geometric object. What is the common
geometric object for a sphere? What is the common
geometric object for a right circular cylinder? Explain.
65. Discuss the relationship between the tangent plane to a
surface and approximation by differentials.
x 0 , y 0 , z 0 .
F x, y, z 0
WRITING ABOUT CONCEPTS
66. Consider the elliptic cone given by
(a) Find an equation of the tangent plane at the point
(b) Find symmetric equations of the normal line at the
point 5, 13, 12 .
5, 13, 12 .
x 2 y 2 z 2 0.
CAPSTONE
CAS
CAS
CAS
CAS
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
61. Find a point on the ellipsoid where the
tangent plane is perpendicular to the line with parametric
equations
y
62. Find a point on the hyperboloid where the
tangent plane is parallel to the plane
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y
(a) Find a set of parametric equations of the tangent line to the
curve of intersection of the surfaces at the point ,
and find the angle between the gradient vectors.
(b) Use a computer algebra system to graph the surfaces.
Graph the tangent line found in part (a).
70. Consider the functions
fx, y 16 x 2 y 2 2x 4y
1, 2, 4
gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
fx, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
fx, y
4xy
x 2 1 y 2 1
x 4y z 0.
x 2 4y 2 z 2 1
z 3 2t.
x 2 4t, y 1 8t
x 2 4y 2 z 2 9
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
63. Give the standard form of the equation of the tangent plane
WRITING ABOUT CONCEPTS
CAS
CAS
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1053714_1307.qxp 10/27/08 12:09 PM Page 952
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
fx, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
fx, y
4xy
x 2 1 y 2 1
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
CAS
CAS
1053714_1307.qxp 10/27/08 12:09 PM Page 952
In Exercises 51–56, find the point(s) on the surface at which the
tangent plane is horizontal.
51.
52.
53.
54.
55.
56.
In Exercises 57 and 58, show that the surfaces are tangent to
each other at the given point by showing that the surfaces have
the same tangent plane at this point.
57.
58.
In Exercises 59 and 60, (a) show that the surfaces intersect at
the given point, and (b) show that the surfaces have perpendicular
tangent planes at this point.
59.
60.
61. Find a point on the ellipsoid where the
tangent plane is perpendicular to the line with parametric
equations
y
62. Find a point on the hyperboloid where the
tangent plane is parallel to the plane
67. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
68. Investigation Consider the function
on the intervals
y
(a) Find a set of parametric equations of the normal line and an
equation of the tangent plane to the surface at the point
(b) Repeat part (a) for the point
(c) Use a computer algebra system to graph the surface, the normal
lines, and the tangent planes found in parts (a) and (b).
69. Consider the functions
y
(a) Find a set of parametric equations of the tangent line to the
curve of intersection of the surfaces at the point ,
and find the angle between the gradient vectors.
(b) Use a computer algebra system to graph the surfaces.
Graph the tangent line found in part (a).
70. Consider the functions
y
(a) Use a computer algebra system to graph the first-octant
portion of the surfaces represented by and
(b) Find one first-octant point on the curve of intersection and
show that the surfaces are orthogonal at this point.
(c) These surfaces are orthogonal along the curve of intersection.
Does part (b) prove this fact? Explain.
In Exercises 71 and 72, show that the tangent plane to the
quadric surface at the point
can be written in the
given form.
71. Ellipsoid:
Plane: x 0x
a 2
y 0 y
b 2
z 0 z
c 2 1
x 2
a 2 y 2
b 2 z 2
c 2 1
x 0 , y 0 , z 0 g.
f
gx, y
2
2
1 3x 2 y 2 6x 4y.
fx, y 16 x 2 y 2 2x 4y
1, 2, 4
gx, y 2x y.
f x, y 6 x 2 y 2 4
2
3 , 3 2 , 3 2 .
2, 2 , 1 2 . 0 y 2 .
3 x 3
fx, y
sen y
x
1, 2,
4
5 .
1, 1, 1 .
0 y 3.
2 x 2
fx, y
4xy
x 2 1 y 2 1
x 4y z 0.
x 2 4y 2 z 2 1
z 3 2t.
x 2 4t, y 1 8t
x 2 4y 2 z 2 9
4x 2 y 2 16z 2 24, 1, 2, 1
x 2 y 2 z 2 2x 4y 4z 12 0,
z 2xy 2 , 8x 2 5y 2 8z 13, 1, 1, 2
2, 3, 3
x 2 y 2 2z 7,
x 2 y 2 z 2 8x 12y 4z 42 0,
1, 1, 0
x 2 y 2 z 2 6x 10y 14 0,
x 2 2y 2 3z 2 3,
z
xy
1
x
1
y
z
5xy
z 4x 2 4xy 2y 2 8x 5y 4
z x 2 xy y 2 2x 2y
z 3x 2 2y 2 3x 4y 5
z 3 x 2 y 2 6y
952 Chapter 13 Functions of Several Variables
63. Give the standard form of the equation of the tangent plane
to a surface given by
at
64. For some surfaces, the normal lines at any point pass
through the same geometric object. What is the common
geometric object for a sphere? What is the common
geometric object for a right circular cylinder? Explain.
65. Discuss the relationship between the tangent plane to a
surface and approximation by differentials.
x 0 , y 0 , z 0 .
F x, y, z 0
WRITING ABOUT CONCEPTS
66. Consider the elliptic cone given by
(a) Find an equation of the tangent plane at the point
(b) Find symmetric equations of the normal line at the
point 5, 13, 12 .
5, 13, 12 .
x 2 y 2 z 2 0.
CAPSTONE
CAS
CAS
CAS
CAS