Calculo 2 De dos variables_9na Edición - Ron Larson

968 968 CAPÍTULO Chapter 13 13 Functions Funciones of Several de varias Variables variables
968 Chapter 13 Functions of Several Variables
968 Chapter 13 Functions of Several Variables
22. Investigación Investigation Repetir Repeat Exercise el ejercicio 21 21 for con retail tiendas outlets de located ventas at al
22. menudeo Investigation localizadas Repeat en Exercise los puntos 21 4, for retail 0, 1, outlets 6, y 12, located 2. at
the points 4, 1, and 12, 22. the Investigation points 4, 0 ,
Repeat 1, 6 , and
Exercise 12, 2 21 .
for retail outlets located at
the points
4, 0 ,
1, 6 ,
and
12, 2 .
Desarrollo WRITING ABOUT de conceptos CONCEPTS
WRITING ABOUT CONCEPTS
23. Con
In your
las propias
own words,
palabras,
state
describir
the problem-solving
la estrategia para
strategy
la solución
for
23. WRITING In your own ABOUT words, state CONCEPTS the problem-solving strategy for
applied
de problemas
minimum
de
and
aplicación
maximum
de
problems.
23. applied In your minimum own words, and maximum state the problems.
problem-solving mínimos y máximos.
strategy for
24.
Con
In your las
own
propias
words,
palabras,
describe
describir
the method
el método
of least
de
squares
mínimos
for
24. In your
applied own
minimum words, describe
and maximum the method
problems.
of least squares for
cuadrados
finding mathematical
para elaborar
models.
24. finding In your mathematical own words, models.
describe modelos the matemáticos.
method of least squares for
finding mathematical models.
En In Exercises los ejercicios 25–28, a (a) 28, find a) the hallar least la squares recta de regression regresión line de and mínimos
In Exercises cuadrados 25–28, y (a) b) find calcular the least S, la squares suma regression de los errores line and
(b) calculate S, the sum of the squared errors. Use the regression al
cuadrado. (b) In calculate
Exercises Utilizar S, the 25–28, sum el programa (a) of the find squared the capabilities of graphing utility to para least errors. verify regresión squares Use regression
your results. de the una regression
herramienta
capabilities (b) calculate de graficación of a S, graphing the para sum utility of verificar the squared to verify los resultados. errors. your results.
Use the regression
line and
25. capabilities y
of a graphing utility 26.
to verify your y
results.
25. y
26.
y
(2, 3)
3 (2, 3)
4
25. y
26.
y
3
4
(2, 3)
3
2
3
3
4
(3, 2)
2
2 (3, 2)
(0, 1)
(−1, 1)
2
3(1, 1)
1
(0, 1) 1)
(−3, (−1,
0) 1) 1) 1
(1, 1)
2
(3, 2)
(−2, 0)
1
0)
1
(−3, 0)
2
(−2, 0) 0)
(0, 1)
(−1, 1)
(1, 1)
1
x
x
−3 −2
−1
1 2 3
−2
−1
(−3, 0) 1
(−2, 0) 1 2
−3
−2
−1
1
2
3
−2
−1 −1
1
2
−2
−1
−3 −2 −1 1 2 3
−2 −1 1 2
−2
−1
−2
27. y
28.
y
27. y
28. y
(0, 4)
(5, 2)
4 (0, 4)
2
27. y
28. y (5, 2)
4
2
(4, 2)
(6, 2)
(0, (1, 4) 3)
(4, 2)
(5, (6, 2) 2)
3
4 (1, 3)
2
3
(3, 1) (4, 1)
(4, 2) (6, 2)
(1, 3)
1 (3, 1) (4, 1)
2
3
1
2
(3, 1) (4, 1)
(1, 1)
1
(1, 1
0)
(3, 0)
2 (1, 1)
1
(1, 0) (3, 0) x
(1, (2, 0)
(2,
1)
0) x
1
2 3
4
1
(1, 0) (3, 0) 5 6
1 2 1
2
3
4
(2, 3
0) 4 5 6
(2, 0)
1 2 3 4
1(2, 20)
3
4
5 6
1
2
3
4
(2, 0)
En In Exercises los ejercicios 29–32, a find 32, the hallar least la squares recta de regression regresión de line mínimos for the
cuadrados In Exercises
points. Use para 29–32,
the los regression puntos find the dados. least
capabilities Utilizar squares
of el programa regression graphing de line
utility regresión for the
to
de points. verify una
In Exercises
your herramienta Use the 29–32, regression results. de
find
Use graficación capabilities the least the graphing para
squares of
utility verificar a graphing regression
to plot los resultados. utility line for to
the
the points
Utilizar verify points. your and graph la herramienta
Use results. the regression Use
the regression de the graficación graphing capabilities
line. para utility trazar
of to a graphing plot los puntos the points
utility y representar
and verify graph la
to
your the recta regression results. de regresión.
Use line.
the graphing utility to plot the points
29.
and 0, graph 1, the 3, regression 4, line. 5, 29. 0, 0, 0 , 1, 1, 1 , 3, 4, 4 , 4, 2, 2 , 5, 5 5
30.
1, 3, 5, 30.
29.
1, 0, 0
0, , 3,
0 , 3, 3
1, , 5,
1 , 6 6
3, 4 , 4, 2 , 5, 5
31.
0, 4, 5, 8, 4, 10, 31.
30.
0, 6, 6
1, , 4,
0 , 3, 3
3, , 5,
3 , 0, 0
5, , 8,
6
4, 4, 10, 5 5
32.
6, 1, 3, 8, 11, 13, 32.
31.
6, 4, 4
0,
, 1,
6 ,
2, 2
4,
, 3,
3 ,
3, 3
5,
, 8,
0 ,
6, 6
8,
, 11, 4,
8, 8
10,
, 13, 8 8
5
32.
6, 4 , 1, 2 , 3, 3 , 8, 6 , 11, 8 , 13, 8
33. 33. Modelo Modeling matemático Data The En ages la tabla (in se years) muestran and las systolic edades blood
33. Modeling Data The ages x (in years) and systolic blood x (en
años) pressures y las presiones of seven men arteriales are shown sistólicas in the y de table.
33. pressures Modeling y of seven Data men The are ages shown x
(in the years) table. siete and hombres.
systolic blood
pressures y
of seven men are shown in the table.
Edad, 16 25 39 45 49 64 70
Edad, x 16 25 39 45 49 64 70
Presión Edad, x 16 25 39 45 49 64 70
Presión
arterial 109 122 150 165 159 183 199
arterial
Presión 109 122 150 165 159 183 199
sistólica, sistólica, arterial y
109 122 150 165 159 183 199
sistólica, y
a) Utilizar el programa de regresión de una herramienta de graficación
Use the para regression hallar la recta capabilities de regresión of a graphing de mínimos utility cuadrados to find
(a) Use the regression capabilities of graphing utility to find
(a)
the least squares regression line for the data.
para (a) the los Use least datos.
the squares regression regression capabilities line for of the a graphing data.
utility to find
(b) Use graphing utility to plot the data and graph the model.
(b) Utilizar Use
the
a graphing una least herramienta squares
utility
regression
to de plot graficación the
line
data
for para and
the
graph trazar data.
the los datos model. y
(c) representar Use the model modelo. to approximate the change in systolic blood
(c)
(b)
Use
Use
the
a
model
graphing
to approximate
utility to plot
the
the
change
data and
in systolic
graph the
blood
model.
pressure for each one-year increase in age.
c) Utilizar (c) pressure Use el the for modelo each para one-year to approximate aproximar increase la the in variación change age.
in en systolic la presión
blood
arterial pressure sistólica for por each cada one-year incremento increase de un in age. año en la edad.
34. Modeling Modelo matemático Data store El manager gerente de wants tienda to know quiere the conocer demandla
34. Modeling demanda yData de una A barra store de manager energía wants en función to know del precio the demand x. Las
for energy bar as function of price x. The daily sales for
34. yventas for Modeling an diarias energy Data
a bar tres as precios
A a store function diferentes
manager of price
de
wants x. la The barra
to know daily de energía sales the demand for
three different prices of the energy bar are shown in the table. se
three muestran
y
for different an en
energy
la prices tabla.
bar of as the a function energy bar of are price shown x.
The in daily the table.
sales for
three different prices of the energy bar are shown in the table.
Precio, $1.29 $1.49 $1.69
Precio, x $1.29 $1.49 $1.69
Demanda, Precio, x 450 $1.29 375 $1.49 330
$1.69
Demanda, y 450 375 330
Demanda, y 450 375 330
(a) Utilizar Use the el regression programa capabilities de regresión of de una graphing herramienta utility de to graficación
the least para squares hallar regression la recta de line regresión for the de data.
find
(a) Use the regression capabilities of a graphing utility to find
(a)
mínimos cuadrados
Use para the los model datos.
estimate the demand when the price is
the Use least the squares regression regression capabilities line for of the a graphing data.
utility to find
(b)
(b)
b) Usar
Use
the el
the
least modelo
model
squares para
to estimate
regression estimar
the
line la demanda
demand
for the cuando
when
data.
the
el precio
price is
es
$1.59.
$1.59.
(b) $1.59.
Use the model to estimate the demand when the price is
35. Modeling Data An agronomist used four test plots to
35. Modeling
$1.59.
determine
Modelo matemático Data An
the relationship
Un agronomist agrónomo used
between
prueba four
the wheat
cuatro test
yield
fertilizantes plots to
(in
35. determine
bushels
en los Modeling campos the Data
per acre)
de relationship
and
cultivo An
the amount
para agronomist between determinar the used wheat
of fertilizer
la four relación yield
test
(in hundreds
entre y
plots (in
of
la
to
bushels
pounds
producción determine per
per acre).
de acre) the trigo and relationship
The
y the (en amount
results
bushels between of
are shown
por fertilizer acre) the
in the
y la x
wheat (in
table.
cantidad hundreds yield de fertilizante
bushels per x (en per acre). acre) cientos The and results de the libras amount are shown por of acre). fertilizer in the Los table.
xresultados (in hundreds se
of
of
y
(in
pounds
muestran Fertilizante,
pounds en per la acre). tabla.
1.0
The 1.5
results 2.0
are shown 2.5
in the table.
Fertilizante, x 1.0 1.5 2.0 2.5
Rendimiento, Fertilizante, x 32 1.0 41 1.5 48 2.0 53
2.5
Rendimiento, y 32 41 48 53
Rendimiento, y 32 41 48 53
Use the regression capabilities of graphing utility to find the
Use
least Utilizar
the regression
squares el programa
capabilities
regression de line regresión
of a
for the de
graphing
data, una and herramienta
utility to find
estimate the de yield graficación
the
least Use
for
squares the regression fertilizer para hallar capabilities
application la recta
line
of de
for
160 regresión
the of data, a graphing
pounds de
and
per mínimos
estimate utility
acre. cuadrados
the to yield
find the
for least
36. Modeling
para
a
los
fertilizer squares
datos,
application regression
Data
y estimar
The table
la
of line
shows
producción
160 for pounds the data,
the percents
para
per and acre.
estimate the yield
una and
aplicación
numbers
de
36. Modeling
for a fertilizer 160
(in
libras Data
millions)
de fertilizante The
application table
of women
por shows
of 160 in the
acre. the
pounds percents
per x
acre.
and numbers
work force for selected years.
36. 36. y
(Source: Modelo (in Modeling millions) matemático Data of women The U.S. Bureau of La in table
Labor tabla the shows
Statistics) muestra work force the los percents for porcentajes selected x
and xyears.
numbers y los
(Source: números y
(in millions) U.S. y (en Bureau millones) of women of Labor de in mujeres Statistics)
the work en force la fuerza for selected laboral years. en
Año determinados (Source: U.S. años. 1970
Bureau (Fuente: 1975
of Labor U.S. Bureau 1980
Statistics) of 1985 Labor Statistics)
Año 1970 1975 1980 1985
Porcentaje, Año 43.3 1970 46.3 1975 51.5 1980 54.5
1985
Porcentaje, x 43.3 46.3 51.5 54.5
Número, Porcentaje, x 31.5 43.3 37.5 46.3 45.5 51.5 51.1
54.5
Número, y 31.5 37.5 45.5 51.1
Número, y 31.5 37.5 45.5 51.1
Año
1990 1995 2000 2005
Año
1990 1995 2000 2005
Porcentaje, Año
57.5 1990 58.9 1995 59.9 2000 59.3
2005
Porcentaje, x 57.5 58.9 59.9 59.3
Número, Porcentaje, x 56.8 57.5 60.9 58.9 66.3 59.9 69.3
59.3
Número, y 56.8 60.9 66.3 69.3
Número, y
56.8 60.9 66.3 69.3
(a) Use the regression capabilities of graphing utility to find
(a) Use the regression capabilities of a graphing utility to find
a) Utilizar the least el squares programa regression para regresión line for the de data. una herramienta de
(a) the Use least the squares regression regression capabilities line for of the a graphing data.
utility to find
(b) graficación According to para this hallar model, la approximately recta de regresión how many de mínimos women
(b) cuadrados According
the least enter the para to
squares this
labor los model,
regression force datos. approximately
line for the how
data.
many women
for each one-point increase in the
b) Según
(b) enter According the
percent este labor of women modelo,
to force this model,
in the ¿aproximadamente for each approximately one-point
labor force? cuántas increase how many
mujeres in women the
ingresan percent enter of a
the
la women fuerza
labor in laboral
force the labor for por
each
cada force?
incremento
one-point increase
de un punto
in the
37. Find system equations whose solution yields the coefficients
a, b, and for the least squares regression quadratic
37. Find en a el
percent
system porcentaje of
of women
equations de mujeres
in the
whose en
labor
la solution fuerza
force?
laboral? yields the coefficients
Hallar Find a, un a sistema
37. 37.
for
de
the
ecuaciones
points
cuya solución proporcione
ax 2 b,
system and c
of for equations the least whose squares solution regression yields quadratic
the coefficients
ax for the points x n
bx los coeficientes a, b y c para el modelo 1 1 cuadrático 2 2 de regresión n y by minimizing 2 a, bx b,
and c
c
for the least
the sum
1 , squares y 1 , x 2 , yregression 2 , . . . , x quadratic n , y n
by de minimizing ymínimos ax 2 cuadrados
bx the sum
c
for y the points ax 2 bx x 1 , y 1 c, xpara 2 , y 2 , los . . puntos . , x n , y n
n
x 1 , by y 1 , minimizing x 2 , y 2 n, . . the . , x sum n , y n minimizando la suma
Sa, b, i
Sa, b, c n i ax2 i bx i 2 Sa, b, c
1
y i ax2 n i bx i c 2 .
i 1 y i ax i2 bx i c 2 Sa, b, c y .
i ax2 i bx i c 2 .
i1
i 1
CAPSTONE
CAPSTONE
38.
Para The
discusión
sum of the length and the girth (perimeter of cross
38. CAPSTONE
The sum of the length and the girth (perimeter of a cross
38. section) of package carried by delivery service cannot
38. La
section)
suma The sum de
of
las
a of package
longitudes the length carried
y and el tamaño
by the a delivery girth (perímetro (perimeter service
de una cannot of a sec-
cross
ción exceed 108 inches. Find the dimensions of the rectangular
exceed section) transversal)
108 inches. of a package de
Find
un paquete
the carried dimensions
llevado by a delivery por
of the
un
rectangular
service servicio cannot de
entrega package of largest volume that may be sent.
package exceed a
of 108 domicilio
largest inches. volume
no Find that
puede the may dimensions exceder
be sent.
108 of the pulgadas.
rectangular
Encontrar package las of dimensiones largest volume del that paquete may be rectangular sent. de más
grande volumen que puede ser enviado.