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Chapter 2: Problem 60

Marginal average cost. In Section \(1.6,\) we defined the average cost of producing \(x\) units of a product in terms of the total cost \(C(x)\) by \(A(x)=C(x) / x .\) Find a general expression for marginal average cost, \(A^{\prime}(x)\)

Short Answer

Expert verified
The marginal average cost is given by \[ A'(x) = \frac{xC'(x) - C(x)}{x^2} \].

Step by step solution

01

- Understand the Given Information

The total cost for producing x units is represented by the function C(x). The average cost A(x) is defined as the total cost divided by the number of units: \[ A(x) = \frac{C(x)}{x} \]
02

- Differentiate the Average Cost Function

To find the marginal average cost, differentiate A(x) with respect to x. Use the quotient rule for differentiation, which states: \[ \left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \] For \[ A(x)=\frac{C(x)}{x} \], assign \[ f(x) = C(x) \] and \[ g(x) = x \].
03

- Apply the Quotient Rule

Using the quotient rule, find \[ A'(x) = \frac{C'(x) \cdot x - C(x) \cdot 1}{x^2} \] Simplify the expression to get \[ A'(x) = \frac{C'(x) \cdot x - C(x)}{x^2} \]
04

- Write the Final Expression

The general expression for the marginal average cost is: \[ A'(x) = \frac{xC'(x) - C(x)}{x^2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

average cost function
The average cost function is a crucial concept in economics. It helps determine the cost per unit of production. If you have a total cost function, denoted as \(C(x)\), which represents the cost of producing \(x\) units, you can find the average cost function, denoted as \(A(x)\), by dividing the total cost by the number of units produced. Mathematically, it is expressed as:

\[A(x) = \frac{C(x)}{x}\]
  • The average cost function helps in understanding how costs behave as production changes.
  • It is useful for businesses to determine pricing strategies and optimize production costs.
Understanding and using the average cost function allows companies to improve profitability by managing their costs effectively.
total cost function
The total cost function, \(C(x)\), is fundamental for analyzing production economics. It represents the overall cost of producing \(x\) units of a product. Total costs typically include fixed costs (costs that don't change with the level of production) and variable costs (costs that change with the level of production). For example:

  • Fixed Costs: Rent, salaries of permanent staff, etc.
  • Variable Costs: Raw materials, hourly wages, etc.
When analyzing the cost behavior of a product, it's important to understand both the fixed and variable components of the total cost function. This understanding helps in determining the cost structure of production and aids in financial planning and decision-making.
quotient rule
In calculus, the quotient rule is essential for differentiating functions that are ratios of two differentiable functions. If you have a function \(\frac{f(x)}{g(x)}\), the quotient rule states:

\[\left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}\]
In the context of the given problem, we used the quotient rule to differentiate the average cost function. Given \(A(x) = \frac{C(x)}{x}\), let \(f(x) = C(x)\) and \(g(x) = x\). Applying the quotient rule, we get:

\[A'(x) = \frac{C'(x) \cdot x - C(x)}{x^2}\]
This formula helps find the marginal average cost, which tells us how the average cost per unit changes as the number of units produced changes. The quotient rule is a powerful tool in mathematical analysis and helps solve complex problems involving ratios.

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Most popular questions from this chapter

A university is trying to determine what price to charge for tickets to football games. At a price of 18 dollars per ticket, attendance averages 40,000 people per game. Every decrease of 3 dollars adds 10,000 people to the average number. Every person at the game spends an average of 4.50 dollars on concessions. What price per ticket should be charged in order to maximize revenue? How many people will attend at that price?

It is known that homing pigeons tend to avoid flying over water in the daytime, perhaps because the downdrafts of air over water make flying difficult. Suppose that a homing pigeon is released on an island at point \(C,\) which is \(3 \mathrm{mi}\) directly out in the water from a point \(B\) on shore. Point \(B\) is 8 mi downshore from the pigeon's home loft at point \(A\). Assume that a pigeon flying over water uses energy at a rate 1.28 times the rate over land. Toward what point \(S\) downshore from \(A\) should the pigeon fly in order to minimize the total energy required to get to the home loft at \(A\) ? Assume that Total energy \(=\) (Energy rate over water) \(\cdot\) (Distance over water) \(+\) (Energy rate over land) \(\cdot\) (Distance over land).

Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the \(x\) -values at which they occur. $$f(x)=x^{4}-8 x^{2}+3 ;[-3,3]$$

U.S. oil production. One model of oil production in the United States is given by $$ \begin{aligned} P(t)=& 0.0000000219 t^{4}-0.0000167 t^{3}+0.00155 t^{2} \\ &+0.002 t+0.22, \quad 0 \leq t \leq 110 \end{aligned} $$ where \(P(t)\) is the number of barrels of oil, in billions, produced in a year, \(t\) years after \(1910 .\) (Source: Beyond Oil, by Kenneth S. Deffeyes, p. \(41,\) Hill and Wang, New York, \(2005 .)\) a) According to this model, what is the absolute maximum amount of oil produced in the United States and in what year did that production occur? b) According to this model, at what rate was United States oil production declining in 2004 and in \(2010 ?\)

From a 50 -cm-by- 50 -cm sheet of aluminum, square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume?
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