Calculo 2 De dos variables_9na Edición - Ron Larson

896 CAPÍTULO 13 Funciones de varias variables
896 Chapter 13
13 Functions of
of
896 Chapter 13 Functions of Several Several Variables
Variables
Redacción En los ejercicios 65 y 66, utilizar las gráficas de las
curvas Writing
de In
In
nivel Exercises
(valores 65
65
de and
and
c uniformemente 66,
66, use
use the
the graphs
espaciados) of
of the
the de level
level
Writing
la
función curves f ( c
In para -values Exercises
dar evenly 65
una descripción spaced) and 66, of
of use
de the
the the
una function
graphs posible f
of
gráfica to
to the write level
de f.
a
curves
¿Es description ( c
-values
única la of
of
gráfica a evenly possible spaced)
de f ? graph
Explicar of
of of la f.
f.
the Is
Is
respuesta.
the
the function
graph f
of
of to f
write unique?
a
description Explain.
of a possible graph of f.
Is the graph of f
unique?
Explain.
65. y
66.
y
65.
65. y
66.
66.
y
65. y
66.
y
x
x
x
x
x
x
67. Inversión En el 2009 se efectuó una inversión de $1 000 al
67.
67.
6% Investment
de interés compuesto In
In 2009, an
an
anual. investment
Suponemos of
of $1000
que el was
was
inversor made
paga
in
in a
67. Investment
una bond
tasa earning In
de impuesto 6%
6% 2009, compounded an investment
R y que la annually. of $1000
tasa de inflación Assume was anual that
that made the
the es buyer
in a
bond I. En
el pays
pays
earning
año tax
tax
2019, at
at rate
rate
6%
el valor R
compounded and
and V the
the
de annual
annually. la inversión rate
rate of
of en inflation
Assume
dólares is
is
that
constantes I.
I.
In
In
the the
the
buyer
year
year
pays de
2009 2019,
tax
es
the
the
at value
rate R
V
and of
of
the annual rate of inflation is In the year
2019, the value of the the
the investment investment in in
in constant constant 2009 2009
I.
dollars dollars
V
is is
is
V VI, 000 0.06 10
VI, R 1 000 1 0.06 1 R
10
VI, R 1 000 1 0.06 1 R
10
.
1 I
.
1 I
Utilizar Use Use
Use this this
this
esta function function
función of of
of two de two
two
dos variables variables
variables to to
to complete para complete
completar the the
the table.
table.
la tabla.
Tasa de
de
Tasa impuestos
de
impuestos
0
0
0.28 0.28
0.28
0.35 0.35
0.35
Tasa de
de inflación
Tasa de inflación
0
0 0.03 0.03
0.03 0.05 0.05
0.05
68.
68.
Inversión Investment
Se A
depositan principal
$5 of
of $5000
en una is
is cuenta
deposited in ahorro
in a savings
68. Investment a una
tasa account
de interés that
that A earns principal
compuesto interest of continuo at
at $5000 a rate
rate of r
of
is (expresado r
deposited (written as
as in
en a forma decimal),
savings
account decimal).
compounded that earns
La cantidad continuously. interest at
A(r, t) The
The a rate
después amount
of r
(written
de A
t r, años
r, t
after
after
as a decimal),
es A(r, t
years
t) is
is
compounded
5 Ar,
Ar,
000e t rt . Utilizar 5000e continuously. rt esta .
Use
Use
función this
this The
de function amount
A
dos variables of
of r, two
two t
after
para variables t
years is
completar to la
to
Ar,
tabla.
complete t 5000e the
the table.
rt .
Use this function of two variables to
complete the table.
Número de
de años
años
Número de años
Tasa Tasa 5 5 10
10 10 15
15 15 20
20
20
0.02
0.02
0.02
0.03
0.03
0.03
0.04
0.04
0.04
0.05
0.05
0.05
En In
In los Exercises
ejercicios 69–74, a 74, sketch
dibujar the
the
la graph
gráfica of
of
de the
the
la level
level
superficie surface
In de
nivel
f x,
x, Exercises y,
y,
f x, z
y, c
z at
at
69–74,
the
the
c para given
sketch
el value
the
valor of
of
de c.
c.
graph of the level surface
f x, y, z
c at the given value of
c. c que se especifica.
69.
69.
f x,
x, y,
y, z x y z,
z,
c 1
69. f x, y, z x y z, c 1
70.
70.
f x,
x, y,
y, z 4x
4x y 2z,
2z,
c 4
70. f x, y, z 4x y 2z, c 4
71.
71.
71.
f x,
x, y,
y, z x 2 y 2 z
f x, y, z x 2 y 2 z 2 2 ,
,
c c 9
9
72.
72.
f x,
x, y,
y, z x 2 1
72. f x, y, z x 2 1
4y 4y 2 z,
z,
c 1
4y 2 z, c 1
73. 73.
73.
f f x, x,
x, y, y,
y, z z 4x 4x
4x 2 2 4y 4y
4y 2 2 z z 2 2 ,
,
c c 0
0
74. 74.
74.
f f x, x,
x, y, y,
y, z z sen sen
sen x x z, z,
z,
c c 0
0
75. Explotación forestal La regla de los troncos de Doyle es uno
75.
75. de Forestry
varios métodos The
The Doyle
para Log
Log
determinar Rule is
is el one
one
rendimiento of
of several en methods
madera used
used
75. Forestry aserrada
to
to determine The
(en tablones-pie) the
the Doyle lumber Log
en yield Rule
términos of
of is a one
de log
log of
su (in
(in several
diámetro board-feet) methods
d (en pulgadas) in
in terms
used
to y of
of determine
su its
its
longitud diameter the lumber
L (en d
(in
(in
pies). inches) yield of
El número and
and a log its
its (in
de tablones-pie length board-feet) L
(in
(in
es
feet). in terms
The
The
of number its diameter of
of board-feet d
(in inches)
Nd, L
d 4
4 2 is
is
and its length L
(in feet). The
number of board-feet is
2L.
d 4
2 Nd,
Nd, L
2
d 4
L. L. Nd,
a) Hallar L
el número 4
L. 4 de tablones-pie de madera aserrada producida
por un tronco de 22 pulgadas de diámetro y 12 pies de
(a)
(a) Find
Find
longitud.
the
the number of
of board-feet of
of lumber in
in a log
log 22
22 inches
(a) Find in
in diameter the number and
and 12
12
of board-feet feet
feet in
in length.
of lumber in a log 22 inches
in diameter and 12 feet in length.
Evaluar N30, 12.
(b)
(b) Find
Find
N 30, 12
12 .
(b) Find
N 30, 12 .
Modelo de filas La cantidad de tiempo promedio que un
76.
76. Queuing
cliente espera Model
en una The
The
fila average
para recibir length
un of
of
servicio time
time that
that
es
a customer
76. Queuing waits in
in line
line Model for
for
Wx, y service The average is
is
length of time that a customer
waits in line for service is
x 1
y , x > y
Wx,
Wx, y
Wx, y
donde y es x
1
el ritmo y x
y , , x x o > > tasa y
y
media de llegadas, expresada como
número de clientes por unidad de tiempo, y x es el ritmo o tasa
where
y
is
is the
the average arrival rate,
rate, written as
as the
the number of
of
where media de servicio, expresada en las mismas unidades. Evaluar
customers y
is the per
per average unit
unit of
of time, arrival and
and
rate, x
is
is written the
the average as the service number rate,
rate,
of
customers cada una de las siguientes cantidades.
written in
in per the
the same
unit of units. time, Evaluate and
x
is each
each the of
of average the
the following.
service rate,
written in the same units. Evaluate each of the following.
a)
a) W 15,
15, 9
b)
b) W 15,
15, 13
13
c)
c) W 12,
12, 7
d)
d)
W 5,
5, 2
a) W 15, 9
b) W 15, 13
c) W 12, 7
d)
W 5, 2
Distribución de temperaturas La temperatura T (en grados
77.
77. Temperature Distribution The
The temperature T
(in
(in degrees
77. Temperature en cualquier punto (x, y) de una placa circular acero
Celsius) at
at any
any Distribution point
x,
x, y
The in
in a circular
temperature steel
steel T
de 10 metros de radio es T 600 0.75x 2 plate (in of
of degrees
0.75y 2
radius
Celsius) , donde x
10
10 meters at any is
is T point
600
600 x, y
0.75x in a y y se miden en metros. Dibujar 2 2 circular 0.75y steel
algunas de 2 2 ,
where plate x
of
and
and radius
y
are
are
10 las curvas isotermas.
measured meters in
in
is meters.
T 600 Sketch 0.75x some 2 of
of 0.75y the
the isothermal 2 ,
where x
curves.
and y
are
measured in meters. Sketch some of the isothermal curves.
Potencial eléctrico El potencial eléctrico V en cualquier punto
78.
78. Electric Potential The
The electric potential V
at
at any
any point x,
x, y
is
is
78. Electric (x, y) es
Potential The electric potential V
at any point x, y
is
5
Vx,
Vx,
Vx, y
y
5
Vx, y
25 25
x 2 2 y x 2 2 y . . 2 2 Dibujar the las curvas equipotenciales for de V 1
1 V 1 y
2 ,
1
3 ,
1
Sketch the equipotential curves for
V
1
2, 2,
V
1
3, 3,
and
and
V
1
44.
4.
Sketch the equipotential curves for
V
V
and
V
.
79. 79. Función Cobb-Douglas de producción Production de Function
2,
Cobb-Douglas Use
Use the
the
3,
Utilizar Cobb-Douglas
4.
función
79. Cobb-Douglas de production producción function Production
de Cobb-Douglas (see
(see Example Function
(ver 5)
5) ejemplo to
to Use show the 5) that
that Cobb-Douglas
para if
if the
the mostrar number
que
production si of
of el units
units número of
of function labor de unidades and
and (see the
the Example number de trabajo of
of 5) units
units to show
y el número of
of capital that if the
de are
are number
unidades doubled,
de
of capital the
the units production of labor
se duplican, level
level and the
el is
is nivel also
also number de doubled.
of units of capital are doubled,
producción también se duplica.
the production level is also doubled.
80. 80. Función Cobb-Douglas de producción Production de Cobb-Douglas Function Show Mostrar that
that the
the que Cobb-
la función
Douglas de producción production Production de Cobb-Douglas
puede rees-
80. Cobb-Douglas cribirse como ln z function
Function z Cx
Cx
y ln C a ln x a a y z 1 1 Show
a a can Cx can
that a ybe be 1a rewritten
the Cobb-
as
as
Douglas production function
z Cx a y 1 a can be rewritten as
y . ln
ln ln z z ln
ln C a ln
ln 81. Costo* ln C a ln x x y
de construcción y y
y . . Una caja rectangular abierta por arriba
tiene x pies de longitud, y pies de ancho y z pies de alto.
81.
81. Construction Cost
Cost A rectangular box
box with
with an
an open
open top
top has
has a
81. Construction
Construir la base cuesta $1.20 por pie cuadrado y construir los
length of
of x
feet,
feet,
Cost a width
A rectangular of
of y
feet,
feet, and
and
box a
with height
an of
of
open z
feet.
feet.
top It
It
has costs
a
length
lados $0.75 por pie cuadrado. Expresar el costo C de construc-
$1.20
of per
per
x feet, square
a width foot
foot to
to
of build
y feet, the
the
and base
base
a height and
and
of $0.75
z feet. per
per
It square
costs
$1.20
ción de la caja en función de x, y y z.
foot
foot to
to
per build
square the
the sides.
foot to Write
build the
the
the cost
cost
base C
of
of
and constructing
$0.75 per the
the
square
box
box
foot
82. Volumen as
as a
to function
build the
Un of
of
sides.
tanque x,
x,
y,
y,
and
and
Write
de propano z.
z.
the cost C of constructing the box
as a function of x, y, and z. se construye soldando hemisferios
a los extremos de un cilindro circular recto. Expresar volu-
82.
82. Volume A propane tank
tank is
is
82. Volume A propane tank is constructed constructed by by
by welding
welding
men V del tanque en función de r y l, donde r es el radio del cilin-
hemispheres hemispheres to to
to the the
the ends ends
ends of of
of a right right
right circular circular cylinder. cylinder. Write Write the the
the
dro y de los hemisferios, y l es la longitud del cilindro.
volume volume V
of of
of
V
the the
the tank tank
tank as as
as a a function function of of
of r
r
and and
and l, l,
l,
where where r
r
is is
is
83. the the Ley
the radius radius
de los of of
of
gases the the
the cylinder cylinder
ideales and and
and
De hemispheres, hemispheres,
acuerdo con and and la
and
ley l
l
is is
is
de the the
the
los length
length
gases
of of ideales,
of the the
the cylinder.
cylinder.
PV kT, donde P es la presión, V es el volumen, T es
la temperatura (en kelvins) y k es una constante de propor-
83. 83.
83. Ideal Ideal Gas Gas
Gas Law Law
Law According According to to
to the the
the Ideal Ideal Gas Gas
Gas Law, Law,
Law,
PV PV
PV
kT, kT,
kT,
cionalidad. Un tanque contiene 2 000 pulgadas cúbicas de
where where P
P
is is
is pressure, pressure, V
V
is is
is volume, volume, T
T
is is
is temperature temperature (in (in
(in Kelvins),
Kelvins),
nitrógeno una presión de 26 libras por pulgada cuadrada y una
and and
and k
k
is is
is a a constant constant of of
of proportionality. proportionality. A A tank tank
tank contains contains 2000
2000
temperatura de 300 K.
cubic cubic inches inches of of
of nitrogen nitrogen at at
at a a pressure pressure of of
of 26 26
26 pounds pounds per per
per square
square
inch a) inch
inch
Determinar
and and
and a a temperature temperature
k.
of of
of 300 300
300 K. K.
K.
(a) (a) b)
(a) Expresar Determine
Determine
P k. como k.
k.
función de V y T y describir las curvas de
nivel.
(b) (b)
(b) Write Write P
P
as as
as a a function function of of
of V
V
and and
and T
T
and and
and describe describe the the
the level level
level
curves.
curves.
* En España se le denomina coste.