Calculo 2 De dos variables_9na Edición - Ron Larson

SECCIÓN 13.9 13.9 Aplicaciones Applications de of Extrema los of of extremos Extrema of Functions of de of funciones Functions of de Two of of dos Two Variables variables Variables 969
969
13.9 Applications of Extrema of Functions of Two Variables 969
En
In Exercises In In Exercises
los ejercicios
39–42, 39–42,
39
use use
a 42,
the utilizar
result the result of
el
Exercise of of Exercise
resultado
37
del
to 37 37 find to to find
ejercicio
the least the least
37
45. 45.
45.
Modeling Modeling
Modelo matemático
Data Data A A meteorologist
Un meteorólogo
measures measures
mide la
the the
presión
atmospheric
atmospheric
atmosférica
P (en
para
squares squares
hallar
regression regression
el modelo
quadratic quadratic
cuadrático
for for
de
the the
regresión
given given
de
points. points.
mínimos
Use Use
cuadrados
the
the pressure pressure P (in P
kilogramos
kilograms (in kilograms
por
per per
metro
square square
cuadrado)
meter) meter)
a
at
una
altitude at at altitude
altitud
h
h
(in
h
(in
(en
In regression Exercises regression
para
capabilities 39–42, capabilities
los puntos
use
dados.
the of of result a of graphing a graphing
Usar
of
el
Exercise
programa
utility utility 37 to to to
de
confirm find to confirm
regresión
the least your
your
de
45. Modeling kilometers). kilometers).
kilómetros).
Data The The
Los
data
datos
A meteorologist data are se
shown are shown
muestran
below. measures below.
en la tabla.
the atmospheric
una
squares results. results.
herramienta
Use regression Use the graphing the graphing
de
quadratic
graficación
utility utility for to
para
plot to the to plot
confirmar
the given points the points. los
and and
resultados.
graph Use graph the
the pressure P (in kilograms per square meter) at altitude h (in
Utilizar
regression least least squares squares
la herramienta
capabilities regression regression of
de
quadratic. a quadratic.
graficación
graphing
para
utility
trazar
to confirm
los puntos
your
y
kilometers). Altitud, Altitud, hThe hdata 0are 0shown 5 below.
5 10 10 10 15 15 15 20 20 20
representar
results. Use the
la curva
graphing
de regresión
utility to
de
plot
mínimos
the points
cuadrados.
and graph the
39.
39. 2, 02, 2, , 0 ,, 1, 01, 1, , 0, ,, 10, 0, , 1, ,, 21, 1, , 2, ,, 52, 2, 5
10 least squares regression quadratic.
Altitud,
Presión, Presión,
h
P P10 0
332 10 3325 5
583 5 5832 10
376 2 3761 15
240 1 240517517
20
39. 40.
2, 40. 4, 0, 54, 4, , 1, 5 ,, 2, 0, 62, 2, , 0, 62, , 1, , 62, 2, , 1, 64, , 2, , 24, 4, 2, 2 5
a) Utilizar el programa de regresión de una herramienta de
40. 39. 4, 2, 5, 0 , 2, 1, 6, 0 2, 0, 6, 1 4, 1, 22 , 2, 5
(a) Presión, Use (a) Use the P regression the regression 10 332capabilities 5 capabilities 583 2 of 376 a of of graphing a a 1 graphing 240 utility 517
utility to find to to find
41. 41. 0, 00, 0, , 02, ,, 2 , 23, ,, 63, 3, , 64, ,, 12
12 12 42.
42. 0, 10 0, 0, 10 , 101, ,, 91, 1, , 92, ,, 62, 2, , 63, ,, 03, 3, 0 graficación para hallar una recta de regresión de mínimos
41. 40. 0, 4, 0, 5 2, , 2, 2, 3, 6 , 6, 2, 4, 6 , 12 4, 2
a least a a least squares squares regression regression line line for the for points the points h, ln h, h, P ln ln .
P ..
42. 0, 10, 1, 9, 2, 6, 3, 0
cuadrados para los puntos h, ln P.
a an 43. 41. 43. 43. Modeling
(a) Use the regression in capabilities is an of a graphing of utility to find
Modelo 0, 0 Modeling , 2, matemático 2 Data , 3, Data 6 , After 4, After
Después 12a new a new 42. turbocharger turbocharger
de que 0, 10 fue , desarrollado 1, for 9 , an for 2, automobile an 6 , automobile
un 3, nuevo
turbopropulsor para un motor de automóvil, se obtuvieron los
ah
0
(b) (b)
b) El
The The
resultado
result result in
del
part in part
inciso
(a)
a)
is (a)
es
an is
una
equation an equation
ecuación
of
de
the of the
la forma
form form ln
ln
Plnln P
engine engine was was developed, developed, the the following following experimental experimental data data were
were a ah least ah ah
b.
b. squares
Expresar
Write b. b. Write regression this this
esta
logarithmic logarithmic line for
forma logarítmica
form the form points
en
exponential exponential h, ln P
forma exponencial.
form. . form.
43. in at datos
Modeling obtained obtained experimentales
Datafor speed After speed y in
siguientes
a yynew miles in miles turbocharger per per hour hour at
de velocidad
for two-second at
y
an two-second
en
automobile time time (b)
millas por (c) Utilizar The Use (c) Use a result graphing una a a graphing part
herramienta utility (a) utility to is
de plot to an to
graficación plot equation the original the original of
para data the trazar
data and form
and graph ln
los graph P
datos the
the
hora
engine intervals intervals
a intervalos
was x. developed, x. x.
x de dos
the
segundos.
following experimental data were
ah
originales exponential exponential b. Write
y representar model this model logarithmic in part in el
in modelo
part (b).
(b). form in exponential form.
exponencial del inciso b).
obtained for speed y in miles per hour at two-second time (c) (d) Si Use (d) If una your If a If graphing herramienta your graphing graphing utility utility de to utility graficación plot can can the fit logarithmic fit original fit puede logarithmic data ajustar models and models modelos graph to data, to the to lo-
data,
Tiempo, intervals
Tiempo, x x.
x 0 02 2 4 4 6 6 8 810
10 10
garítmicos exponential use it use to it it verify to to a model datos, verify the in result the utilizarla part result in (b). part in para in part (b). verificar (b). el resultado del
y 15 30 50 65 70
(d) inciso If your b). graphing utility can fit logarithmic of models to data,
Tiempo,
Velocidad, Velocidad,
x
y y 0 015 15
2
30 30
4
50 50
6
65 65
8 10
70 70
46. 46. Modeling Modeling Data Data The The endpoints endpoints of the of the interval interval over over which
which
46. distinct Modelo use distinct it vision matemático to vision verify possible is is the possible result Los are puntos in called are part called terminales (b). the near the near point del point and intervalo and far point far de point
Velocidad,
a)
(a)
Hallar Find (a) Find ya un
least a a 0 modelo least squares 15 squares cuadrático
regression 30 regression 50 quadratic
de
65 quadratic regresión
70for the for de data. the mínimos
data. Use Use a aa
46. Modeling of visión the of of eye. the se llaman Data eye. With With increasing punto The increasing endpoints próximo age, age, these y of punto these points lejano interval points normally del normally over ojo. change. which Con change. la
cuadrados
graphing graphing para
utility utility los
to
datos.
confirm to confirm Utilizar
your your results.
una results. herramienta de graficación
para confirmar los resultados.
distinct The edad, The table estos vision table shows puntos is shows possible the cambian. approximate the are approximate called La tabla near the muestra near points
points
ylos (in yand puntos y
inches) (in far inches) point próximos
the various y eye. (en ages With pulgadas) ages xincreasing (in xx
(in a years). varias years). age, these edades (Source: (Source: points x (en normally Ophthalmology años). change. (Fuente: &
&
for
for
(a) (b) Find (b) Use Use a graphing least a a graphing squares utility utility regression to plot to to plot the quadratic points the points and for and graph the graph data. the model. the Use model. a of various
b) Utilizar graphing una utility herramienta to confirm de graficación your results. para trazar los puntos The Ophtalmology Physiological table shows Optics) & the Physiological Optics) approximate Optics) near points y (in inches) for
44. 44. Modeling y Modeling
representar Data Data
el The modelo.
The table table shows shows the world the world populations populations y (in yy
(in
(b) Use a graphing utility to plot the points xand graph the model. various ages x (in years). (Source: Ophthalmology &
billions) billions) for five for five different different years. years. Let Let x x8
represent 8 represent the year the year
44. Modelo matemático La tabla muestra la población mundial y Edad, Physiological Edad, x x Optics) 16 16 1632 32 3244 44 44 50 50 50 60 60 60
44. Modeling 1998. 1998. (Source: Data (Source: U.S. The U.S. Census table Census shows Bureau, Bureau, the International world International populations Data Data Base) y (in
Base)
(en miles de millones) para cinco diferentes años. Considerar que
billions) for five different years. Let x 8 represent the year
y x = 8 representa el año 2008. (Fuente: U.S. Census Bureau, Edad,
Punto Punto
x
próximo, próximo, y y
16
3.0 3.0
32
4.7 4.7
44
9.8 9.819.719.7 50
39.4 39.4
60
1998. Año, Año, (Source: x x U.S. 1998 Census 19982000 Bureau, 20002002 International 200220042004 Data 2006
Base) 2006
International Data Base)
a) Hallar un modelo racional para los datos tomando el recíproco
o inverso de los puntos próximos para generar los pun-
Punto (a) próximo, Find (a) Find a rational a a yrational 3.0 model model 4.7 for the for 9.8 data the data by 19.7 taking by by taking 39.4 the reciprocals the reciprocals
y Año,
Población, Población,
x
y y
1998
5.9 5.9
2000
6.1 6.1
2002
6.2 6.2
2004
6.4 6.4
2006
6.5 6.5
of the of of near the near points points to generate to to generate the points the points x, 1x, x, y 1.
y Use y ..
Use the
the
tos x, 1y. Utilizar el programa para regresión de una herramienta
de graficación para hallar una recta de regresión de
(a) Find regression regression a rational capabilities model capabilities for of the a of of graphing data a a graphing by taking utility utility the to reciprocals find to to find a least a a least
(a) Población, Use (a) Use the regression the y regression 5.9 capabilities capabilities 6.1 of 6.2a of of graphing a a graphing 6.4 utility 6.5 utility to find to to find
of squares the squares near regression points regression to line generate line for the for the revised the points
revised data. x, data. 1The y . The resulting Use resulting the
the least the least squares squares regression regression line line for the for data. the data.
mínimos cuadrados para los datos revisados. La recta resultante
tiene la forma 1/y = ax + b. Despejar y.
a) Utilizar el programa de regresión de una herramienta de
regression line line has the has capabilities form the form 1 y 1of yyax a graphing ax axb.
Solve b. b.
utility Solve for
y. for to y. find y. a least
(a) (b) (b) the of of a to to graficación
Use Use the regression the regression
the para hallar
capabilities capabilities
la recta
of
de
a of graphing a graphing
regresión
utility utility
de mínimos
to find to find
line for the data.
(b) squares (b)
Utilizar Use Use a graphing regression
una
a a graphing line
herramienta utility utility to for
de plot to to the
graficación plot the revised data the data and data.
para and graph The
trazar graph the resulting
los model. the
datos
model.
cuadrados
the least the least squares squares
para los
regression regression
datos.
quadratic quadratic for the for data. the data.
line has the form 1 y ax b. Solve for y.
y representar el modelo.
be to b)
(b) (c) Utilizar Use the a graphing el a regression programa utility capabilities
de to plot regresión to the of data a
de
graphing and una graph herramienta utility the models. to find
(c) Do (c) Do you you think think the the model can can be used be used to predict to predict the the near near
(c) Use a graphing utility to plot the data and graph the models.
(b)
c) ¿Puede
Use a graphing
utilizarse a utility
el modelo to plot is para
the 70 data
predecir
and graph
el punto
the
próximo
model.
graficación
the least squares para hallar
regression
to el modelo
quadratic
cuadrático
for the de
data.
point point for a for person a person who who is 70 is years 70 years old? old? Explain. Explain.
(d) (d) Use Use both both models models to forecast to forecast the world the world population population regresión for the for de
the
(c) en Do una you persona think the de 70 model años? can Explicar.
(c)
to be used to predict the near
mínimos Use year a 2014. graphing cuadrados How utility do para do the to los plot two datos.
the models data and differ graph as you as the models.
47. 47. Use Use the Second the Second Partials Partials Test Test to verify to verify that that the formulas the formulas afor
a
year 2014. How do the two models differ as you extrapo-
extrapolate
into the future?
the rificar que las fórmulas para a y b proporcionadas en el teorema
47. and Usar point and el given criterio for in a person Theorem in de las who segundas 13.18 is 70 yield years derivadas a old? minimum. a Explain. parciales para ve-
c) (d) Utilizar Use late both into una the models herramienta future?
to forecast de graficación the world para population trazar los for datos
b b given in Theorem 13.18 yield a minimum.
y representar los modelos.
47. Use the Second Partials Test to
13.18 llevan a un mínimo.
d) Utilizar ambos modelos para estimar la población mundial
n verify that n the 2 formulas for a
year 2014. How do the two models differ as you extrapolate
into the future?
i
x2
and Hint: b given Use in the Theorem fact that 13.18 n n
n 2
Hint: Use the fact that n n
n 2
xyield 2
i a minimum. x i .
x2
i x i . x i .
i 1i i 11
i 1i i
en el año 2014. ¿Cómo difieren los dos modelos cuando se
Sugerencia: Considerar el hecho que
extrapola hacia el futuro?
n n 11
x 2 n
i ≥
i1
x
i1
i 2 Hint: Use the fact that n n
n 2
x2
.
I J i x i .
S ESC ET CI TO IN O N P RPO RJ OE JC ET
C T
PROYECTO Building a DE a Pipeline TRABAJO
S E C T I O N P R O J E C T
An An oil oil company oil company wishes wishes to construct to to construct a pipeline a a pipeline from from its its offshore its offshore
Construcción Building a Pipeline
facility facility A to Aits to to its refinery its refinery
de un
B. The B. B.
oleoducto
The offshore offshore facility facility is 2 is is miles 2 miles from
from
Una An shore, oil shore, empresa and company and the petrolera refinery the wishes refinery is to desea 1 is is construct mile 1 construir mile inland. inland. a pipeline Furthermore, un Furthermore, oleoducto from its A and desde offshore A
and B are
su B
are
plataforma facility 5 miles 5 miles Aapart, to A apart, hasta its refinery shown as as su shown refinería B. the The B. figure. the La offshore figure. plataforma facility está is a 2 miles millas from de la
costa, shore, y and la refinería the refinery está is 1 milla 1 mile tierra inland. adentro. Furthermore, Además, A y and B están B are a
A
A
5 millas miles apart, de distancia as shown una in de the otra, figure. como se muestra en la figura.
A
2 mi A2 2 mi mi
5 mi 55 mi mi
2 millas P
P
2 5 millas
x xx
5 mi
1 mi 11 mi mi
P P
x
B
B
x
1 mi 1 milla
B
B
i 1
The The cost cost of building of of building the pipeline the pipeline is $3 is is $3 million $3 million per mile per mile in the in in the
water water and and $4 $4 million $4 million per mile per mile on on land. on land. So, the So, cost the cost of the of of pipeline the pipeline
depends depends The El costo on on the on de of location the building construcción location of the point of of pipeline del point P, oleoducto where P, P, is where $3 it million es meets it $3 it meets millones the per shore. the mile por shore. in What milla the
What
water would en el would and mar, be the $4 be be y $4 million most the millones most economical per economical mile por milla route land. route en of tierra. So, the of of pipeline? the Por cost pipeline? tanto, of the el pipeline costo del
depends oleoducto Imagine on Imagine depende the that location that you de you are la of localización to are point write to to P, write a where report a del a report punto it meets the to to P oil the en the oil company oil la shore. orilla. company What about ¿Cuál
about
would this sería this problem. la be ruta problem. the más most Let económica Let xeconomical be xxthe be be distance the para route distance el shown oleoducto? of the shown in pipeline? the in in figure. the figure. Determine
Determine
the cost the Imagine Imaginar cost of building of of that building que you the hay are pipeline the que to pipeline write redactar from a report from Aun to Ato informe P, to to the and P, P,
oil and the para company cost the la cost from empresa
about from P to
Ptoto
this B. petrolera Analyze B. problem. B.
Analyze acerca some Let
some de xsample be este the sample problema. distance pipeline pipeline Sea shown routes xroutes la in and distancia the and their figure. their mostrada Determine
corresponding en la
the costs. figura. costs. For Determinar of building For instance, instance, the el what costo pipeline what is the de is is construir from cost the cost of A to the of of el P,
most oleoducto the and most the direct cost direct de route? from A route? a P, Then Py to
Then el
B. use costo Analyze use calculus de calculus P a some B. to Analizar determine to to sample determine alguna pipeline the route the trayectoria routes of the of of and muestra pipeline the their pipeline para that corresponding
that el minimizes oleoduc-
minimizes
costs. the to y the cost. sus For cost. costos Explain instance, Explain correspondientes. all what all steps all is steps the of cost your of of Por your of development ejemplo, the development most ¿cuál direct and es route? and el include costo include Then
anyla
any
use relevant ruta calculus relevant más graphs. directa? to graphs. determine Utilizar the después route el of cálculo the pipeline para determinar that minimizes la ruta
the del cost. oleoducto Explain que all minimiza steps of el your costo. development Explicar todos and include los pasos any del
relevant desarrollo graphs. e incluir una gráfica pertinente.
i 1