Problem 61 - Calculus and its applications

Question

Belvedere, Inc., determines that the marginal \(\operatorname{cost}, C^{\prime},\) of producing the \(x\) th thermos is given by $$C^{\prime}(x)=x^{3}-2 x$$ Find the total-cost function, \(C,\) assuming that \(C(x)\) is in dollars and that fixed costs are \(\$ 7000\).

Step-by-Step Solution

Answer

The total cost function is \( C(x) = \frac{x^4}{4} - \frac{x^2}{2} + 7000 \).

Step 1: Understand Marginal Cost

The marginal cost function, \( C'(x) = x^3 - 2x \), represents the rate of change of the cost with respect to the number of thermoses produced. To find the total cost function \( C(x) \), we need to integrate the marginal cost function.

Step 2: Integrate the Marginal Cost Function

To determine the total cost function \( C(x) \), we perform indefinite integration of the marginal cost \( C'(x) \).\[C(x) = \int C'(x) \, dx = \int (x^3 - 2x) \, dx.\]Using the power rule to integrate, this becomes:\[C(x) = \frac{x^4}{4} - \frac{x^2}{2} + C.\]Here, \( C \) is the constant of integration.

Step 3: Apply the Fixed Costs

We are given that the fixed costs are \$7000. To find the constant \( C \) in our integrated function, we apply this fixed cost. Since \( C(0) \) represents the total cost when 0 thermoses are produced (only fixed costs), we set:\[C(0) = \frac{0^4}{4} - \frac{0^2}{2} + C = 7000.\]Thus, \( C = 7000 \).

Step 4: Write the Total Cost Function

Now that we know the constant of integration, we substitute it back into the expression for \( C(x) \):\[C(x) = \frac{x^4}{4} - \frac{x^2}{2} + 7000.\]This is the total cost function that includes both the variable and fixed costs.

Key Concepts

Understanding Marginal Cost

Marginal Cost is a concept that tells you how much the cost changes when we produce one additional unit. Think of it as the cost for the next item, in this case, a thermos. In mathematical terms, the marginal cost function, denoted as \( C'(x) \), is the derivative of the total cost function \( C(x) \). It gives the rate at which the total cost changes with each additional unit produced.

For Belvedere, Inc., the marginal cost is given by \( C'(x) = x^3 - 2x \). This function is interpreted as:

• \( x^3 \) indicates that the cost increases quite rapidly as production increases, due to the cubic term, which grows very fast.
• \( -2x \) reduces the cost a bit, indicating a linear offset to the fast-growing cubic cost.

Overall, understanding this function helps us predict how much additional production will affect the costs.

Integration to Find Total Cost

Integration allows us to determine the Total Cost from the Marginal Cost. Essentially, integration can be viewed as the opposite operation of differentiation. When you have a marginal cost function \( C'(x) \), integrating this will give you the total cost function \( C(x) \).

For our example, we integrate \( C'(x) = x^3 - 2x \) to form \( C(x) \). The process involves applying the power rule for integration:

• Integrating \( x^3 \) becomes \( \frac{x^4}{4} \).
• Integrating \( -2x \) becomes \( -\frac{x^2}{2} \).

This results in the total cost function \[C(x) = \frac{x^4}{4} - \frac{x^2}{2} + C\]. The constant \( C \) here is crucial as it incorporates fixed costs, which leads us to understand it better in the next section.

Applying Fixed Costs

Fixed Costs are those expenses that do not change with the level of production. Whether you make one thermos or thousands, the fixed costs remain constant. In this scenario, Belvedere, Inc. has fixed costs of \(7000.

When integrating the marginal cost function, a constant of integration, often denoted as \( C \), appears. This constant is essential for adjusting the total cost function to include the fixed costs. To determine this constant:

• We understand that \( C(0) \), the cost for producing 0 thermoses, should equal the fixed costs, which are \)7000.
• Thus, substituting into the equation, we set \( C(0) = 7000 \), giving us \( \frac{0^4}{4} - \frac{0^2}{2} + C = 7000 \).

Solving this, we find \( C = 7000 \), solidifying our total cost function:
\[C(x) = \frac{x^4}{4} - \frac{x^2}{2} + 7000\]Now, this function incorporates both variable costs, depending on production, and the fixed costs.