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Lesson 10 Homework
Price Elasticity of Demand
Problem 1: In each of the problems compute the price elasticity of demand using the same approach as in
Example 171 on page 226.
a.
R(x) = 100 · x − 0.03 · x2 + 0.0001 · x3 ,
x = 1200,
p = 28
x = 2300,
p = 15
Show your work below:
b.
R(x) = 210 · x + 0.001 · x2 − 0.002 · x3 ,
Show your work below:
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c.
R(x) = 210 · x + 0.001 · x · ln(x) − 0.002 · x2 · e−x ,
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x = 2300,
p = 15
Elasticity for a Profit Model
Problem 2: In this problem, mathematical models of profit obtained from the production and sales of an item
are analyzed in terms of the point price elasticity of demand. Suppose that two mathematical models of a profit
function are given as follows.
P1 (x) = 85x − (0.003) · x2 · ln(x) − 1400
and
P2 (x) = 85x − (0.003) · x3 − 1400
Use the model that profit is equal to revenue minus cost with
C(x) = 15x + 1400
to write the revenue in terms of a price multiplied by the demand level, x, and then find the elasticity parameter
from this relationship (see Example 171). Compare the two models.
Show your work below:
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Differentials and Price Elasticity of Demand
Problem 3: Using the formula
η x · dp = p · dx
relating differentials in the elasticity application do the following problems.
a.
With η = 1/2, p = 20, x = 675, and the differential in demand dx = 0.003 compute the differential in
price, dp.
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b.
With η = 1/2, p = 20, x = 675, and the differential in demand dx = 0.0035 compute the differential in
price, dp.
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c.
With η = 1/2, p = 20, x = 675, and the differential in demand dp = 0.035 compute the differential in
price, dx.
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d.
Compare the results obtained in the first three parts of this problem.
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Productivity: Cobb-Douglas Models
Problem 4: Given the Cobb-Douglas model
P (x, y) = A · xa · y b
with A = 205, a = 0.42, and b = 0.58.
a. Compute P (50, 60).
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b. Suppose that P (x, y) = 500, compute y 0 (x).
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c. Suppose that P (x, y) = 500, compute x0 (y).
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d. Suppose that x = 55, compute P 0 (y).
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e. Suppose that y = 65, compute P 0 (x).
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Comparison of Cases: Cobb-Douglas Model
Problem 5: Using
P (x, y) = A · xa · y b
with A = 205, x = 50, and y = 60.
a. Given a = 0.27 and a + b = 1. compute P (x, y).
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b. Given a = 0.27 and a + b = 0.75. compute P (x, y).
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c. Given a = 0.27 and a + b = 1.05. compute P (x, y).
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d. Compare the results obtained in the other parts of the problem.
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Reciprocal Relation for Derivatives
Problem 6: For the Cobb-Douglas model
P (x, y) = 305 · x0.21 · y 0.79
with P (x, y) = 699 compute x0 (y) using implicit differentiation. Use the reciprocal relationship for derivatives
to compute y 0 (x).
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Interval Analysis of the Second Derivative
Problem 7: For each of the following functions, compute intervals on which the second derivative is positive or
negative, and points where the second derivative is zero. Determine which of these point(s) are inflection points.
This analysis is an interval analysis of the second derivative.
a.
f (x) = x · e−2x
2
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b.
f (x) = 3 · x2 − 2 · x3
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c.
g(x) =
e−2x
1 + e−2x
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Point of Diminishing Returns
Problem 8: Find the point of diminishing returns for each of the following functions.
a.
P (x) = 10 · x − 0.07 · x2 + 0.0001 · x3
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b.
g(x) =
e−x
1 + e−x
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Related Rates: Cobb-Douglas Revisited
Problem 9: In each of the following problem, compute the desired rate given the rest of the indicated information.
a.
Compute P 0 (t) given
0.27
P (t) = 145 · (x(t))
· (y(t))
0.73
and x0 (t) = 3.7 and y 0 (t) = −1.6.
Show your work below:
b.
Compute y 0 (t) given
0.27
P (t) = 145 · (x(t))
· (y(t))
0.73
and x0 (t) = 3.7 and P 0 (t) = −2.5.
Show your work below:
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