Problem 65
Question
A company's marginal cost function is \(M C(x)=\frac{1}{2 x+1}\) and its fixed costs are \(50 .\) Find the cost function.
Step-by-Step Solution
Verified Answer
The cost function is \(C(x) = \frac{1}{2} \ln |2x + 1| + 50\).
1Step 1: Understand Marginal Cost and Cost Functions
The marginal cost function, denoted as \(MC(x)\), represents the derivative of the total cost function \(C(x)\). Therefore, to find \(C(x)\), we need to integrate the marginal cost function. The given marginal cost function is \(MC(x) = \frac{1}{2x + 1}\).
2Step 2: Integrate the Marginal Cost Function
To find the cost function \(C(x)\), we integrate the marginal cost function: \[C(x) = \int \frac{1}{2x + 1} \, dx\]To integrate \(\frac{1}{2x + 1}\), we use the formula for the integral of \(\frac{1}{ax + b}\) which is \(\frac{1}{a} \ln |ax + b| + C\) (where \(C\) is the constant of integration):\[C(x) = \frac{1}{2} \ln |2x + 1| + C_1\]
3Step 3: Consider Fixed Costs
Fixed costs are constant costs that do not change with the quantity of goods produced. It's given that the fixed cost is \(50\). To find the specific cost function \(C(x)\), we use the fixed cost to determine \(C_1\). At \(x = 0\), the cost is \(50\) (since fixed costs are incurred even when no goods are produced):\[C(0) = \frac{1}{2} \ln |2 \cdot 0 + 1| + C_1 = 50\]Since \(\ln |1| = 0\), we have:\[C_1 = 50\]
4Step 4: Write the Full Cost Function
Substitute \(C_1\) back into the expression for \(C(x)\):\[C(x) = \frac{1}{2} \ln |2x + 1| + 50\]This is the cost function incorporating both the marginal cost and the given fixed costs.
Key Concepts
Marginal CostIntegrationFixed CostsCalculus Applications
Marginal Cost
The concept of marginal cost is like examining the extra cost incurred for producing one more unit of a good. In simple terms, it's the cost required to increase production by just one unit. This can help businesses manage their production expenses wisely.
The marginal cost function, often expressed as \(MC(x)\), is integral to finding the total cost function of a product. The given exercise shows the marginal cost as \(MC(x) = \frac{1}{2x + 1}\). This means that the additional cost decreases as production increases, which can be a crucial insight for optimizing efficiency.
The marginal cost function, often expressed as \(MC(x)\), is integral to finding the total cost function of a product. The given exercise shows the marginal cost as \(MC(x) = \frac{1}{2x + 1}\). This means that the additional cost decreases as production increases, which can be a crucial insight for optimizing efficiency.
Integration
Integration is a calculus technique used to reverse the process of differentiation. In the context of our problem, we need to integrate the marginal cost function to find the total cost function \(C(x)\).
Integration helps us "sum up" the small costs (represented by the marginal cost function) to get the overall cost. So, stepping through, we integrate \(\frac{1}{2x + 1}\) using the formula for the integral of \(\frac{1}{ax + b}\), resulting in \( \frac{1}{a} \ln |ax + b| + C \). This gives our cost function as \(C(x) = \frac{1}{2} \ln |2x + 1| + C_1\).
This integration process is key to unlocking the broader cost structure from the marginal insights.
Integration helps us "sum up" the small costs (represented by the marginal cost function) to get the overall cost. So, stepping through, we integrate \(\frac{1}{2x + 1}\) using the formula for the integral of \(\frac{1}{ax + b}\), resulting in \( \frac{1}{a} \ln |ax + b| + C \). This gives our cost function as \(C(x) = \frac{1}{2} \ln |2x + 1| + C_1\).
This integration process is key to unlocking the broader cost structure from the marginal insights.
Fixed Costs
Fixed costs are those persistent expenses that do not change with the production volume. Whether a company produces a thousand units or none, these costs remain the same. This is important for understanding the base level of expenditure a business must cover.
In the exercise, it is given that the fixed costs amount to \(50\). When we calculated our integrated cost function, we accounted for fixed costs by setting \(x = 0\) and equating \(C(0)\) to \(50\). This helped us solve for \(C_1\), which we found to be \(50\), reflecting the consistency of fixed costs in the overall cost expression.
In the exercise, it is given that the fixed costs amount to \(50\). When we calculated our integrated cost function, we accounted for fixed costs by setting \(x = 0\) and equating \(C(0)\) to \(50\). This helped us solve for \(C_1\), which we found to be \(50\), reflecting the consistency of fixed costs in the overall cost expression.
Calculus Applications
Calculus finds extensive applications in economics and cost management. It allows us to model and analyze various economic phenomena quantitatively. In this scenario, using both integration and differentiation gives insight into cost structures. Such techniques are crucial for strategic decision-making.
The application of calculus to find the cost function involves integrating the marginal cost derivative to rebuild the total cost. It shows the power of calculus to address practical, real-world problems by helping businesses to strategize production for cost efficiencies effectively.
The application of calculus to find the cost function involves integrating the marginal cost derivative to rebuild the total cost. It shows the power of calculus to address practical, real-world problems by helping businesses to strategize production for cost efficiencies effectively.
- Allows detailed financial analysis
- Supports increasing outputs with decreased costs
- Aids in identifying cost-saving opportunities by understanding cost behavior
Other exercises in this chapter
Problem 64
Evaluate each definite integral. $$ \int_{0}^{\ln 5} e^{x} d x $$
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A country's imports are \(I(t)=30 e^{0.2 t}\) and its exports are \(E(t)=25 e^{0.1 t},\) both in billions of dollars per year, where \(t\) is measured in years
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Find a formula for \(\int e^{a x+b} d x\) where \(a\) and \(b\) are constants.
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Evaluate each definite integral. $$ \int_{1}^{2} \frac{(x+1)^{2}}{x} d x $$
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