Calculo 2 De dos variables_9na Edición - Ron Larson

13.1 Introduction to Functions of Several Variables 897
SECCIÓN 13.1 13.1 13.1 Introduction Introducción to to Functions a las funciones of of Several Several de varias Variables variables 897
897
84. Modeling Data The table shows the net sales x (in billions of
84. 84. Modeling Modelo matemático Data The table La tabla shows muestra the net las sales ventas (in netas billions x (en of
dollars), Data the total The assets table shows y (in billions the net sales of dollars), x (in billions and of the
dollars), miles dollars), millones the total de assets dólares), (in los billions activos of totales dollars), y (en and miles the
shareholder’s the total equity assets z (in ybillions (in billions of dollars) of dollars), for Wal-Mart and the for de
shareholder’s millones dólares) equity y los (in derechos billions of de dollars) los accionistas for z (en miles for
the years 2002 equity through z (in 2007. billions (Source: of dollars) 2007 for Annual Wal-Mart Report for
the de the millones years 2002 de dólares) through de 2007. Wal-Mart (Source: desde 2007 2002 Annual hasta el Report
for years Wal-Mart) 2002 through 2007. (Source: 2007 Annual Report 2007.
for (Fuente: for Wal-Mart)
2007 Annual Report for Wal-Mart)
Año 2002 2003 2004 2005 2006 2007
Año Año 2002 2002 2003 2003 2004 2004 2005 2005 2006 2006 2007 2007
x 201.2 226.5 252.8 281.5 208.9 345.0
x 201.2 201.2 226.5 226.5 252.8 252.8 281.5 281.5 208.9 208.9 345.0 345.0
y 79.3 90.2 102.5 117.1 135.6 151.2
y 79.3 79.3 90.2 90.2 102.5 102.5 117.1 117.1 135.6 135.6 151.2 151.2
z 35.2 39.5 43.6 49.4 53.2 61.6
z 35.2 35.2 39.5 39.5 43.6 43.6 49.4 49.4 53.2 53.2 61.6 61.6
Un A modelo for para these estos data datos is es
A model model for for these these data data is is
z f x, y 0.026x 0.316y 5.04.
z f x, x, y 0.026x 0.026x 0.316y 0.316y 5.04. 5.04.
(a) Utilizar Use a graphing una herramienta utility and de the graficación model to y approximate el modelo zpara
for
(a) (a)
aproximar
Use Use graphing utility and the model to for
the a given graphing
zvalues para
utility
los of valores x
and y. the
dados
model
de x
to
y
approximate
y.
z for
the given values of and y.
(b)
the
¿Cuál Which
given
de of
values
las the dos two
of x
variables variables
and y.
en in este this modelo model tiene has the mayor greater influencia
(b) (b) Which Which of the two variables in this has the greater
influence of
sobre
the on los shareholder’s two
derechos
variables
de equity? in
influence on equity? los
this
accionistas?
model has the greater
(c)
influence
(c) Simplificar Simplify
on
Simplify the la the
shareholder’s
expresión expression
equity?
de for f for (x, 95) fx, e 95 interpretar and interpret
and interpret su significado
its
(c) its
Simplify fx, 95
meaning
meaning en el in contexto
the in the expression context
the context del problema. of for the problem. fx, 95 and interpret its
of the problem.
85.
85. Meteorología Meteorology
meaning in the Los Meteorologists
context of the problem.
meteorólogos measure miden measure la presión the atmospheric
the atmosférica
85. Meteorology
en pressure
pressure milibares. in
in A millibars. Meteorologists
partir de From
From estas these
these observaciones
measure observations the elaboran
atmospheric they
they create mapas create
pressure
climáticos weather in maps
weather maps en
millibars.
los on
on que which
which se
From
muestran the these curves
the curves las
observations
curvas of equal
of equal de presión
they atmospheric create
atmosférica
pressure weather
pressure constante
maps (isobars) on
(isobars)(isobaras) are which drawn the
are drawn (ver (see curves
(see la figure).
figure). figura).
of On
On En
equal the
the el map,
map, mapa,
atmospheric the
the cuanto closer
pressure closer
más the isobars
the isobars juntas
(isobars)
the están the higher
higher las
are
isobaras
drawn the wind (see
the wind mayor speed. figure).
speed. es Match
Match la
On
velocidad
the points map, A,
points A, del
the B,
B, viento.
closer and C
the and
Asociar with isobars (a)
with (a) los highest the
highest puntos
higher pressure,
pressure, A, B
the
y
wind
C(b) con (b) speed. lowest
lowest a) la mayor
Match pressure,
pressure, presión,
points and A,
and (c) b) (c) B,
la highest and
highest menor
C
with
presión wind (a) velocity.
wind velocity. y
highest
c) la mayor
pressure,
velocidad
(b) lowest
del viento.
pressure, and (c) highest
wind velocity.
1032
1032
5.60
1032
B
5.60
5.60
1032
B
5.60
1036
1036
1036
1036
C
C
1024
1024
1024
1024
1036
1036
1036
1036 1032 1032
1032
1028
1028
1032
1028
1028
1012
1012
1016
1012
1016
1020
1016
1012 1020
1024
1020
1016 1024
1024
1020
1024
1028
1028
1028
1028
A
A
1024
1024
1024
1024
4.30
4.30
4.40
4.30
4.40
4.52
4.40
4.30
4.52
4.52
4.40
4.52
4.7 0
4.70
4.7 4.7 0
Figura Figure para for 85 85 Figura Figure para for 86 86
Figure Figure for for 85 85 Figure Figure for for 86 86
86. Lluvia Acid Rain ácida The La acidez acidity del of agua rainwater de lluvia is measured se mide in en units unidades
called Rain
86. 86. Acid Acid Rain The acidity of is in units
llamadas pH. A The pH pH. of acidity
Un 7 is pH neutral, of
de
rainwater
7 es smaller neutro,
is values valores
measured are menores increasingly
units
corresponden
acidic, pH. and A
called called pH. pH of is neutral, smaller values are a
pH larger acidez
of 7 values creciente,
is neutral, are y increasingly smaller
valores
values
mayores alkaline. are increasingly
a alcalinidad The map
acidic, acidic, and larger values are alkaline. The map
creciente. shows and curves El
larger
mapa of equal values
muestra pH are and las
increasingly
curvas gives evidence pH
alkaline.
constante that downwind The
y da
map
evidencia
heavily curves
of
shows shows curves of equal pH and gives evidence that of
de industrialized que
of
en
equal
la dirección
pH areas and gives
en the la acidity que
evidence
sopla has that
heavily areas the acidity has el been
been viento
downwind increasing.
de áreas
of
heavily
muy Using
Using industrializadas the industrialized level curves
the level curves la
areas
on acidez on the the
the map, ha map, acidity
ido determine aumentando.
has been the increasing. direction
the direction Utilizar of las of
Using
curvas the prevailing the
the de nivel
level winds
winds en
curves
el in mapa, in on the the
the para northeastern map,
determinar
determine United
United la
the
dirección States. direction
States. de
of
los
87.
the
vientos Atmosphere
prevailing
dominantes
winds
The en
in
contour el
the
noreste
northeastern
map de Estados shown
United
in Unidos.
States.
the figure was
87. 87. The contour map shown in the figure was
87. Atmósfera
Atmosphere computer generated El
The
contorno
contour using
using del data
data mapa
map collected mostrado
shown by in
by satellite en satellite the
la figura
figure instrumentation.
por Color generated
fue
was
generado
computer tation.
Color computadora is used using
is used to usando to data show collected
show una the
the colección “ozone by satellite
“ozone de hole”
hole” datos
instrumen-
in mediante Earth’s
Earth’s
instrumentación
tation. atmosphere. Color The is The del
used purple
purple satélite.
to show and
and El blue color blue the
areas se areas “ozone
usa para represent hole”
mostrar
in the Earth’s
the el lowest
lowest “agujero
atmosphere. levels
levels de of ozono” of ozone The
ozone en and
and la
purple
the atmósfera the green and blue areas
green areas de la
areas represent Tierra.
represent
Las the
the áreas highest the
highest púrpura
lowest levels.
levels. y
azul
levels (Source:
(Source: representan
of ozone National and
National los Aeronautics the más
green
bajos
areas and
and niveles
represent Space
Space de Administration)
ozono
the highest
y las
levels.
áreas
verdes
(Source:
representan
National Aeronautics
los niveles
and
más
Space
altos.
Administration)
(Fuente: National
Aeronautics and Space Administration)
5.60
5.00
5.00
5.00
4.7 4.7 0
0 4.7 0
5.00
4.70
4.52
4.52
4.52
4.52
4.22 4.22
4.22 4.22
4.22 4.22
4.22 4.22
NASA
NASA
GeoQuest GeoQuest Systems, Systems, Inc.
Inc.
GeoQuest Systems, Inc.
Figure for 87
Figure Figure
Figura for for
para 87 87
87
(a) Do the level curves correspond to equally spaced ozone
(a) (a) Do Do ¿Corresponden the level curves las curvas de nivel to a equally los mismos spaced niveles ozone
levels? the level Explain. curves correspond to equally spaced ozone de
levels? ozono espaciados? Explain. Explicar.
(b)
levels?
Describe
Explain.
how to obtain a more detailed contour map.
(b) (b) Describe Describir how how cómo to to obtener obtain obtain a un more more contorno detailed detailed mapa contour contour más map. map. detallado.
88. Geology The contour map in the figure represents colorcoded
seismic The contour amplitudes map of in a fault the figure horizon represents and a projected color-
88. 88. Geology Geology Geología The El mapa contour de contorno map in the de la figure figura representa colorcodetudes
sísmicas seismic en código de of color fault de horizon una falla and horizontal y un
amplicoded
contour seismic map, amplitudes which is used of a in fault earthquake horizon studies. and a projected (Source:
contour contour mapa de map, contorno which proyectado is used in que se usa en studies. los estudios (Source: de terremotos.
from (Fuente: Adaptado de Shipman/Wilson/Todd, An An to In-
Adapted map, from which Shipman/Wilson/Todd, is used in earthquake An studies. Introduction (Source: to
Adapted Adapted
Physical from Science, Shipman/Wilson/Todd, Tenth Edition) An Introduction to
Physical Physical troduction
Science, Science, to Physical
Tenth Tenth Science,
Edition) Edition) 10a. ed.)
(a) Discuss the use of color to represent the level curves.
(a) (a) Analizar Discuss the el uso use de of colores to para representar the level las curvas curves. de nivel.
(b)
Discuss
Do the level
the use
curves
of correspond
to represent
to equally
the level
spaced
curves.
amplitudes?
(b) (b) Do ¿Corresponden Do the level curves las curvas de to nivel equally a amplitudes spaced Explain. the level curves correspond to equally spaced amplitudes? uniformemente
Explain. espaciadas? Explicar.
Explain.
True or False? In Exercises 89–92, determine whether the
True True ¿Verdadero or False? o falso? In En los ejercicios 89–92, 89 a 92, determinar whether the
statement or False? is true In or Exercises false. If it 89–92, is false, determine explain why whether or give the si
an
la
statement declaración is true es verdadera or false. o If falsa. it is false, Si es explain falsa, explicar why or por give qué an
example that is true shows or false. it is false. If it is false, explain why or give an o
example dar un ejemplo that that shows shows que it it demuestre is is false. false. que es falsa.
89. If fx 0 , y 0 fx 1 , y 1 , then x 0 x 1 and y 0 y 1 .
89. 89. If Si fx f x 0 , y 0 fx f x 1 , y 1 , then entonces 0 and x 1
y y 0 y 1 .
90.
If
If
fx
f is 0 ,
a
y 0
function, 1 ,
then
y 1 , then
f ax,
x 0
ay
x 1
a
and
90. If Si is f es una función, then entonces
ax, ay f ax, ay fx, 2 fx,
y 0
y .
y 1 .
90. a 2 f x, y.
91.
If
A
f is
vertical
a function,
line can
then f ax, ay a
intersect the graph 2 fx, y
of
.
z f x, y at most
91. 91. A Una vertical recta line vertical can intersect puede cortar the graph la gráfica of de z x, f x, at y most
once. vertical line can intersect the graph of z f x, y at most a lo
once. sumo una vez.
92.
once.
Two different level curves of the graph of z f x, y can
92. 92. Two Two Dos different diferentes level curvas curves de nivel of the de graph la gráfica of de
zx, f x, can
intersect. different level curves of the graph of z f y can y
intersect.
pueden intersecarse.