Question

The cost \(C\) (in dollars) of producing \(x\) units of a product is \(C=1.35 x+4570\) (a) Find the average cost function \(\bar{C}\). (b) Find \(\bar{C}\) when \(x=100\) and when \(x=1000\). (c) What is the limit of \(\bar{C}\) as \(x\) approaches infinity?

Short answer

\(a)\) The average cost function is \(\bar{C}(x) = \frac{1.35x + 4570}{x}\). \(b)\) The average cost when \(x=100\) is $46.20 and when \(x=1000\) is $4.905. \(c)\) The limit of the average cost function as \(x\) approaches infinity is $1.35.

Step-by-step solution

Formulate the Average Cost Function

The average cost function is given by the total cost divided by the number of units produced, \(\bar{C}(x) = \frac{C}{x}\). Substituting the given cost function \(C=1.35x+4570\) into this, the average cost function becomes \(\bar{C}(x) = \frac{1.35x + 4570}{x}\).

Compute the Average Cost for Specific Units Produced

To find the average cost for specific units of production, substitute the given values of x into the average cost function. \nFor \(x=100\), \(\bar{C}(100) = \frac{1.35*100 + 4570}{100} = 46.20\). \nFor \(x=1000\), \(\bar{C}(1000) = \frac{1.35*1000 + 4570}{1000} = 4.905\).

Find the Limit of the Average Cost Function as x Approaches Infinity

The limit as \(x\) approaches infinity is a measure of the behavior of the function for very large values of \(x\). The limit of the function \(\bar{C}(x)\) as \(x\) approaches infinity can be calculated using the properties of limits: \nWhen calculating the limit of a rational function where the degree of the numerator and the denominator are the same, the limit will be the ratio of the coefficients of the highest order terms. Here the highest order terms are \(1.35x\) in the numerator and \(x\) in the denominator: \nThus, \[\lim_{x \to \infty } \bar{C}(x) = \lim_{x \to \infty } \frac{1.35x + 4570}{x} = 1.35\].

Key concepts

Average Cost Function

In economics, understanding how costs behave as the production levels change is crucial. The average cost function represents the cost per unit of item produced, which helps businesses determine pricing and production strategies. It is obtained by dividing the total cost function by the quantity of items produced.

For a mathematical formulation, if the total cost function is given by a linear function like in our textbook example, where costs include both variable costs and fixed costs, the average cost function \( \bar{C}(x) \) is defined as: \[ \bar{C}(x) = \frac{C}{x} = \frac{1.35x + 4570}{x} \.\] It reveals how the cost per unit behaves with changes in production levels. This function will typically decrease as production increases, spreading the fixed costs over a larger number of units, until it reaches an optimal point. Beyond that, average costs might start to increase due to inefficiencies.

Limit of a Function

The concept of a limit in calculus is pivotal, especially when analyzing how functions behave as inputs approach a particular value or infinity. It provides insights into tendencies of function values under certain conditions without necessarily reaching those conditions. For instance, knowing the limit as production goes to infinity can be essential for long-run planning in economics.

Using limits with the average cost function allows businesses to predict their long-term average costs of production. To find the limit of \( \bar{C}(x) \) as \( x \) approaches infinity, you'd observe the highest power terms in the numerator and the denominator. If they are the same, like in our case with \( x \) in both, the limit as \( x \) approaches infinity will be the ratio of their coefficients: \[ \lim_{x \to \infty } \bar{C}(x) = \lim_{x \to \infty } \frac{1.35x + 4570}{x} = 1.35. \.\] This result implies that, as production grows indefinitely, the average cost will tend toward \(1.35\) dollars.

Rational Functions in Calculus

Rational functions are ratios of two polynomials and are a significant part of calculus. They show up frequently in problems relating to rates of change, optimization, and real-world applications like economic cost functions. The average cost function we have been discussing is an example of a rational function.

A rational function in the form of \( \frac{ax^n + (...)}{bx^n + (...)} \) tends to the ratio \(\frac{a}{b}\) as \( x \) increases without bound. This is because the other terms become insignificant compared to the dominant \(x^n\) terms. In our textbook example, the \(1.35x\) in the numerator and \(x\) in the denominator are the only terms that matter as \( x \) approaches infinity, hence the limit equals \(1.35\), the ratio of the coefficients of these terms.

Understanding rational functions in calculus is paramount for analyzing and interpreting the infinite behavior of functions, especially in diverse fields like economics, physics, and engineering.