Problem 54 - Applied Calculus

Question

Cost A company's marginal cost function is \(M C(x)=x e^{-x / 2}\) and fixed costs are \(200 .\) Find the cost function. [Hint: Evaluate the constant \(C\) so that the cost is 200 at \(x=0 .\)

Step-by-Step Solution

Answer

The cost function is \( C(x) = -2xe^{-x/2} - 4e^{-x/2} + 204 \).

Step 1: Understanding the Marginal Cost Function

The marginal cost function given is \( MC(x) = x e^{-x/2} \). The marginal cost function represents the derivative of the total cost function \( C(x) \) with respect to quantity \( x \). Our aim is to find \( C(x) \).

Step 2: Integrating the Marginal Cost Function

To find the total cost function \( C(x) \), we need to integrate the marginal cost function. Compute the indefinite integral: \( \int MC(x) \, dx = \int x e^{-x/2} \, dx \).

Step 3: Solving the Integral

Utilize integration by parts, where \( u = x \) and \( dv = e^{-x/2} \, dx \). Thus, \( du = dx \) and \( v = -2e^{-x/2} \). The integration by parts formula is \( \int u \, dv = uv - \int v \, du \). Applying this, the integral becomes:\[ -2xe^{-x/2} + 2\int e^{-x/2} \, dx \].

Step 4: Evaluating the Second Integral

The second integral \( 2\int e^{-x/2} \, dx \) simplifies to \(-4e^{-x/2} \) (as \( \int e^{-x/2} \, dx = -2e^{-x/2} \)). So the integration becomes:\[ -2xe^{-x/2} - 4e^{-x/2} + C \].

Step 5: Setting the Fixed Cost Condition

To find the constant \( C \), use the condition that the fixed cost is 200 when \( x = 0 \). Substitute \( x = 0 \) into the cost function:\(-2(0)e^{-0/2} - 4e^{-0/2} + C = 200\), which simplifies to \(-4 + C = 200\).

Step 6: Solving for the Constant C

Solve \(-4 + C = 200\) to find \( C \). Thus, \( C = 204 \).

Step 7: Writing the Final Cost Function

The total cost function is therefore:\[ C(x) = -2xe^{-x/2} - 4e^{-x/2} + 204 \].

Key Concepts

Marginal Cost

The marginal cost is pivotal for understanding how the cost function changes with each additional unit produced. In the given problem, the marginal cost function is expressed as \( MC(x) = x e^{-x/2} \). This denotes the rate of change of the total cost relative to the change in quantity \( x \). It's like asking, "How much more will it cost to produce one additional item?" By knowing the marginal cost, businesses can make informed decisions on production to maximize profitability. In calculus, the marginal cost is the derivative of the total cost function \( C(x) \). Thus, integrating the marginal cost provides us the total cost function, which leads us to the next step.

Integration by Parts

The method of integration by parts is a calculus technique used to integrate products of functions. It stems from the product rule for differentiation and is useful when dealing with integrals of the form \( \int u \, dv \). In our exercise, we need to integrate \( \int x e^{-x/2} \, dx \). By choosing \( u = x \) and \( dv = e^{-x/2} \, dx \), we find that \( du = dx \) and \( v = -2e^{-x/2} \). Utilizing the integration by parts formula: \[ \int u \, dv = uv - \int v \, du, \] this breaks the problem down into manageable pieces. Calculating further gives \[ -2xe^{-x/2} + 2\int e^{-x/2} \, dx, \] simplifying to \[ -2xe^{-x/2} - 4e^{-x/2} + C, \] where \( C \) is the constant of integration. Each step returns us closer to determining the function \( C(x) \), our desired cost function.

Fixed Costs

Fixed costs are the static costs a company incurs that do not change with the level of output. In this example, the problem states that fixed costs are 200, regardless of production levels. To implement this information, we assume that at \( x = 0 \), our cost function \( C(x) \) must be 200. This condition helps solve for the constant \( C \) in our integrated total cost function. By substituting \( x = 0 \), the fixed cost equation \(-4 + C = 200\) arises from \(-2(0)e^{-0/2} - 4e^{-0/2} + C = 200\). Solving this gives us \( C = 204 \). It demonstrates how fixed costs play a critical role in shaping the total cost function, tying into non-variable production expenses. With this fixed cost insight, we complete the understanding by ensuring the total cost function aligns with real-world application at all production levels.