Problem 67 - Applied Calculus

Question

BUSINESS: Cost The marginal cost function for a computer chip manufacturer is \(M C(x)=1 / \sqrt{x^{2}+1},\) and fixed costs are \(\$ 2000 .\) Find the cost function.

Step-by-Step Solution

Answer

The cost function is \( C(x) = \sinh^{-1}(x) + 2000 \).

Step 1: Understand the Problem

We are given the marginal cost function \(MC(x) = \frac{1}{\sqrt{x^2 + 1}}\) and fixed costs of $2000. We need to find the total cost function \(C(x)\).

Step 2: Integrate the Marginal Cost Function

The marginal cost \(MC(x)\) is the derivative of the cost function \(C(x)\). To find \(C(x)\), integrate the marginal cost function.\[ \int MC(x) \, dx = \int \frac{1}{\sqrt{x^2 + 1}} \, dx \]

Step 3: Calculate the Integral

Calculate the integral to find that:\[ \int \frac{1}{\sqrt{x^2 + 1}} \, dx = \sinh^{-1}(x) + C \]Here, \(C\) is the integration constant.

Step 4: Determine the Total Cost Function

The cost function \(C(x)\) is: \[ C(x) = \sinh^{-1}(x) + K \]where \(K\) is the constant of integration.

Step 5: Adjust for Fixed Costs

Since the fixed costs are \$2000, we set \(K = 2000\). Therefore, the cost function becomes:\[ C(x) = \sinh^{-1}(x) + 2000 \]

Key Concepts

Marginal Cost

Marginal cost is a crucial concept in economics, particularly in production and cost analysis. It represents the cost of producing one additional unit of a good. In mathematical terms, it is the derivative of the total cost function. For a computer chip manufacturer, knowing the marginal cost helps in understanding how cost changes with production volume. If the marginal cost is low, producing more units might be profitable. In the given exercise, the marginal cost function is provided as:\( MC(x) = \frac{1}{\sqrt{x^2 + 1}} \)This function describes how the cost of manufacturing each additional chip varies with the number of chips produced.

Integration

Integration is a mathematical operation used to find the total size, length, area, or volume of objects. In economics, integration is often used to deduce the total cost function from the marginal cost function. When you integrate a marginal cost function, you essentially sum up all the little pieces of costs across different units, thus giving us the total cost function. In the exercise, we integrate the marginal cost:\[\int \frac{1}{\sqrt{x^2 + 1}} \, dx = \sinh^{-1}(x) + C\]Here, \(\sinh^{-1}(x)\) is the result of the integration, with \(C\) being the integration constant. This shows how the total cost function is constructed by comprehensively accounting for each additional unit's production cost.

Fixed Costs

Fixed costs refer to expenses that do not change with the amount of goods produced. They are incurred even if the production level is zero, such as rent or salaries. Understanding fixed costs is essential because they must be covered regardless of production volume. In business, knowing fixed costs helps determine pricing strategies and breakeven points. In this exercise, we're given fixed costs of $2000. To incorporate fixed costs into the total cost function, we add them to the integral of the marginal cost function. Therefore, the total cost function from our exercise is: \[C(x) = \sinh^{-1}(x) + 2000\]This equation clearly separates the changeable operational costs from the constant overhead costs, providing a full picture of the production expenses.