James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

926 chapter 14 Partial Derivatives
80. If u − e a 1x 1 1a 2 x 2 1∙∙∙1a n x n
, where a 2 1 1 a 2 2 1 ∙ ∙ ∙ 1 a 2 n − 1,
show that
− 2 u
1 −2 u
−x1
2 −x2
2
81. The diffusion equation
1 ∙ ∙ ∙ 1 −2 u
−x 2 n
−c
−t − D −2 c
−x 2
− u
where D is a positive constant, describes the diffusion of heat
through a solid, or the concentration of a pollutant at time t at
a distance x from the source of the pollution, or the invasion of
alien species into a new habitat. Verify that the function
csx, td −
1
s4Dt
e 2x 2 ys4Dtd
is a solution of the diffusion equation.
82. The temperature at a point sx, yd on a flat metal plate is given
by Tsx, yd − 60ys1 1 x 2 1 y 2 d, where T is measured in 8C
and x, y in meters. Find the rate of change of temper ature with
respect to distance at the point s2, 1d in (a) the x-direction and
(b) the y-direction.
83. The total resistance R produced by three conductors with resistances
R 1, R 2, R 3 connected in a parallel electrical circuit is
given by the formula
Find −Ry−R 1.
1
R − 1 R 1
1 1 R 2
1 1 R 3
84. Show that the Cobb-Douglas production function P − bL K
satisfies the equation
L −P
−L 1 K −P − s 1 dP
−K
85. Show that the Cobb-Douglas production function satisfies
PsL, K 0d − C 1sK 0dL by solving the differential equation
(See Equation 6.)
dP
dL − P L
86. Cobb and Douglas used the equation PsL, Kd − 1.01L 0.75 K 0.25
to model the American economy from 1899 to 1922, where L
is the amount of labor and K is the amount of capital. (See
Example 14.1.3.)
(a) Calculate P L and P K.
(b) Find the marginal productivity of labor and the marginal
productivity of capital in the year 1920, when L − 194 and
K − 407 (compared with the assigned values L − 100 and
K − 100 in 1899). Interpret the results.
(c) In the year 1920, which would have benefited production
more, an increase in capital investment or an increase in
spending on labor?
87. The van der Waals equation for n moles of a gas is
SP 1
n 2 a
V 2 DsV 2 nbd − nRT
where P is the pressure, V is the volume, and T is the temperature
of the gas. The constant R is the universal gas constant
and a and b are positive constants that are characteristic of a
particular gas. Calculate −Ty−P and −Py−V.
88. The gas law for a fixed mass m of an ideal gas at absolute
temperature T, pressure P, and volume V is PV − mRT, where
R is the gas constant. Show that
−P
−V
−V
−T
−T
−P − 21
89. For the ideal gas of Exercise 88, show that
T −P
−T
−V
−T − mR
90. The wind-chill index is modeled by the function
W − 13.12 1 0.6215T 2 11.37v 0.16 1 0.3965Tv 0.16
where T is the temperature s°Cd and v is the wind speed
skmyhd. When T − 215°C and v − 30 kmyh, by how much
would you expect the apparent temperature W to drop if the
actual temperature decreases by 1°C? What if the wind speed
increases by 1 kmyh?
91. A model for the surface area of a human body is given by the
function
S − f sw, hd − 0.1091w 0.425 h 0.725
where w is the weight (in pounds), h is the height (in inches),
and S is measured in square feet. Calculate and interpret the
partial derivatives.
−S
−S
(a) s160, 70d
(b) s160, 70d
−w −h
92. One of Poiseuille’s laws states that the resistance of blood flowing
through an artery is
R − C L r 4
where L and r are the length and radius of the artery and C is
a positive constant determined by the viscosity of the blood.
Calculate −Ry−L and −Ry−r and interpret them.
93. In the project on page 344 we expressed the power needed by a
bird during its flapping mode as
Psv, x, md − Av 3 1 Bsmtyxd2
v
where A and B are constants specific to a species of bird, v is
the velocity of the bird, m is the mass of the bird, and x is the
fraction of the flying time spent in flapping mode. Calculate
−Py−v, −Py−x, and −Py−m and interpret them.
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