Calculo 2 De dos variables_9na Edición - Ron Larson

966 CAPÍTULO 13 Funciones de varias variables
966 Chapter 13 Functions of Several Variables
966 Chapter 13 Functions of Several Variables
EJEMPLO 3 Hallar la recta de regresión de mínimos cuadrados
EXAMPLE 3 Finding the Least Squares Regression Line
Hallar EXAMPLE la recta de 3regresión Finding de mínimos the Least cuadrados Squares para Regression los puntos Line (3, 0), (1, 1), (0, 2)
y Find (2, the 3). least squares regression line for the points 3, 0, 1, 1, 0, 2, and 2, 3.
Find the least squares regression line for the points 3, 0, 1, 1, 0, 2, and 2, 3.
Solution The table shows the calculations involved in finding the least squares
Solución SolutionLaThe tabla table muestra shows los the cálculos calculations necesarios involved para hallar in finding la recta the de least regresión squares de
regression line using n 4.
mínimos regression cuadrados line using usando n n 4. 4.
TECNOLOGÍA Muchas calculadoras
TECHNOLOGY
TECHNOLOGY Many calculators
have “built-in”
tienen
least
“incorporados”
squares
Many
regression
calculators programas
have “built-in”
programs. If de your regresión calculator
least squares de mínimos has
regression
such a
program, cuadrados. programs.
use it
If Se to
your puede duplicate
calculator utilizar the results
has unasuch a
of Example calculadora program,
3.
use con it to estos duplicate programas the results
of Example 3.
para reproducir los resultados del
ejemplo 3.
y (2, 3)
y
3
f(x) =
8 x +
47 (2, 3)
13 3 26 (2, 3)
8 47 2
f(x) = x + 3
13 8 26 47
f(x) = x +
(0, 2)
13 26 2
(0, 21
2)
(−3, 0) (−1, 1) (0, 2)
1
(−3, 0) (−1, 1) 1
(−3, −3 0) −2 (−1, 1) 1 2
x
−3 −2 −1 1 2
−3 −2 −1 1 2
Least Figura squares 13.77 regression line
Figure
Least
13.77
squares regression line
Figure 13.77
Recta de regresión de mínimos cuadrados
y
x
x
n
i1 n
i1
x
x
y
y
xy
xy
x 2
x 2
3
3
0
0
0
0
9
9
1
1
1
1
1
1
1
1
0
0
2
2
0
0
0
0
2
2
3
3
6
6
4
4
x i 2
x i 2
n
i1 n
i1
y i 6
y i 6
n
i1 n
i1
x i y i 5
x i y i 5
n
i1 n
i1
x i2 14
x i2 14
Aplicando
Applying Theorem
el teorema
13.18
13.18
produces
se obtiene
Applying
n n Theorem i i n 13.18 i n produces
n n
i1 x i
i1 i1 45 26
n n
i2
i1
n
i1
i 2 414 2 i y i n
x i n
y
i1
a
n n
x i
i1 i1 45 26
2 13
n n
x i2
i1
n
x
i1
i 2 414 2 8 i y i n
x i n
y i
i1 i1 i1 45 26
a
2 13
y
n n
x i2
i1
n
x
i1
i 2 414 2 8 2 13
and
and
b
n 1 n
y i a n
x
i1 i1
i 4 1 6 8
13 2 47
26 .
b 1 n n
y i a n
x
i1 i1
La recta de regresión de mínimos
i 1 4 6 8 13 2 47
b 1 26
n n
y .
i a n
x
cuadrados es f x 8 13 x 47
26 ,
i1 i1
i 1 4 6 8 13 2 47
26 . como se muestra en la
figura The least 13.77. squares regression line is f x 13x 8 47
26, as shown in Figure 13.77.
The least squares regression line is f x 13x 8 47
26, as shown in Figure 13.77.
■
■
13.9 13.9Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
Exercises Ejercicios
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
In Exercises En los ejercicios 1 and 2, 1 find y 2, the hallar minimum la distancia distance mínima from the del point punto al 9. Cost 9. Costos A home Un improvement contratista de contractor mejorías is caseras painting está the pintando walls las
to the plano In
plane
Exercises x x y 1
y and
1z z 2,
3.
find
3. (Sugerencia: (Hint:
the minimum
To simplify Para distance
the simplificar computations,
from los the cálculos,
to the minimizar plane
point
and
9. Cost paredes ceiling
A
of y home el a techo rectangular
improvement de una room. habitación contractor
The volume rectangular. is painting
of the El room volumen the walls
is de
minimize the square
x el ycuadrado of
1
the
z
distance.)
3. de (Hint: la distancia.) To simplify the computations,
668.25
and la habitación cubic
ceiling
feet.
of es a
The
rectangular de 668.25 cost of pies wall
room. cúbicos. paint
The
is
volume El $0.06 costo of
per de the
square pintura room is de
minimize the square of the distance.)
foot
668.25 pared and the es cubic
cost de $0.06 of
feet.
ceiling por The pie paint
cost cuadrado of
is
wall
$0.11 y paint el per costo square
is de $0.06 pintura foot.
per
Find de square techo
1. 0, 1. 0, 0 0, 0
2. 1, 2. 2, 1, 3 2, 3
the
foot es room de and $0.11 dimensions
the cost por pie of
that
ceiling cuadrado. result
paint
in Encontrar is
a
$0.11
minimum
per las square dimensiones cost for
foot.
the
Find de la
1. 0, 0, 0
2. 1, 2, 3
habitación que den por resultado un mínimo costo para la pintura.
¿Cuál What es is el the mínimo minimum costo cost por for la pintura? the paint?
En los ejercicios 3 y 4, encontrar la distancia mínima desde el
paint.
the
What
room
is
dimensions
the minimum
that
cost for the
in
paint?
a minimum cost for the
In Exercises 3 and 4, find the minimum distance from the point
paint.
to the punto In Exercises
surface a la superficie 3 and 4, find z the 1 minimum
(Hint: 2x distance
In 2y. Exercise (Sugerencia: from the point
z 1 2x 2y.
4, use the En el 10. Maximum Volume The material for constructing the base
ejercicio to the surface 4, usar z la 1 operación 2x raíz 2y. de (Hint: una In herramienta Exercise 4, de use graficación.)
root feature of a graphing utility.)
material
the 10. Volumen máximo El material para construir la base de una
root feature of a graphing utility.)
of an
Maximum
open box
Volume
costs 1.5
The
times
material
as much
for
per
constructing
unit area the
the
base
of caja an abierta for
open
constructing
box cuesta costs 1.5 veces 1.5
the
times
sides. más as por For
much unidad a fixed
per de unit área amount
area que el as
ofmate-
the
3. 2, 2, 0
money
material C, para find construir for
the
constructing
dimensions los lados. of
the Dada the
sides.
box una of
For cantidad largest
a fixed
volume fija de amount dinero that
of C,
3. 2, 2, 0
hallar las dimensiones de la caja de mayor volumen que puede
4.
can
money
be made.
C, find the dimensions of the box of largest volume that
0, 0, 2
4. 0, 0, 2
can ser fabricada. be made.
11. Maximum Volume The volume of an ellipsoid
In En Exercises los ejercicios 5–8, find 5 a 8, three hallar positive tres números integers positivos x, y, and x, zythat
11.
y z que 11. Maximum Volumen máximo Volume El The volumen de of un an elipsoide ellipsoid
satisfy
In Exercises
satisfagan the given las conditions.
5–8, find three positive integers x, y, and z that x 2
condiciones dadas.
a y 2
2 2
b z2
satisfy the given conditions.
x 2
2
5. El producto es y la suma es mínima.
2
c 1 2
2 z2
5. The product is 27 and the is a minimum.
a y 2
2 b z2
2 2 c 1 2 2
5. The product is 27 and the sum is a minimum.
is 4abc3. For a fixed sum a b c, show that the ellipsoid
6. The 6. La sum suma is 32 es and y PP xyxy 2 z 2 is z a es maximum.
6. The sum is 32 and is máxima. a maximum.
of maximum
is es 4abc3. 4abc3. volume
For Dada a
is
fixed una a sphere. suma afija ab bc,
show c, mostrar that the que ellipsoid
P xy 2 z
elipsoide
maximum de volumen volume máximo is a sphere. es una esfera.
7. The 7. La sum suma is 30 es and y the la sum suma of de the los squares cuadrados is a es minimum.
of
mínima.
7. The sum is 30 and the sum of the squares is a minimum.
8. The 8. El product producto is 1 es and y the la suma of de the los squares cuadrados is a es minimum.
8. The product is 1 and the sum of the squares is a mínima. minimum.