Approximating incremental cost with derivative

TL;DR

The video explains how to calculate the marginal cost function and use it to approximate the cost of producing an additional unit of wood stain.

Transcript

The cost in dollars of producing x gallons of wood stain is given by C of x is equal to 3,200 plus 0.1x minus 0.001x squared plus 0.0004x to the third power. What is a formula for the marginal cost function C prime of x? So we really just have to take the derivative of C with respect to x, to think about how does C change as x changes. As our quant... Read More

Key Insights

• 🇨🇷 The marginal cost function represents the rate of change of the cost function with respect to quantity.
• 🇨🇷 Approximating the cost of producing an additional unit using the marginal cost function gives a close estimate.
• ☠️ The approximation assumes a constant rate of change, while the actual cost function may have varying slopes.

Questions & Answers

Q: What is the formula for the marginal cost function in wood stain production?

The formula for the marginal cost function can be derived by taking the derivative of the cost function with respect to the quantity produced. In this case, it is equal to 0.1 - 0.002x + 0.0012x².

Q: How does the marginal cost function help approximate the cost of producing an additional unit?

The marginal cost function represents the rate of change of the cost with respect to the quantity produced. By multiplying the marginal cost by the incremental change in quantity (in this case, one gallon), we can estimate the cost of producing the next unit.

Q: Why does the approximation using the marginal cost function differ from the actual cost of producing the 101st gallon?

The approximation using the marginal cost function assumes a constant rate of change, while the actual cost function may have a varying slope. This discrepancy arises because the cost of each additional gallon increases as the quantity produced increases.

Q: How can the exact cost of producing the 101st gallon be calculated?

To find the exact cost, subtract the cost of producing the 100th gallon from the cost of producing the 101st gallon. This gives the precise difference in cost.

Summary & Key Takeaways

• The video demonstrates how to find the marginal cost function by taking the derivative of the cost function for producing wood stain.

• The derivative is calculated step by step for each term in the cost function.

• The marginal cost is then used to approximate the cost of producing the 101st gallon of wood stain.