Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.5.60

Chapter 4, Problem 4.5.60

Business and Economics
60. Production level Prove that the production level (if any) at which average cost is smallest is a level at which the average cost equals marginal cost.

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Define the average cost (AC) as the total cost (C) divided by the quantity produced (Q), i.e., AC = C/Q.

Define the marginal cost (MC) as the derivative of the total cost with respect to quantity, i.e., MC = dC/dQ.

To find the production level where the average cost is smallest, take the derivative of the average cost with respect to quantity, d(AC)/dQ, and set it equal to zero to find critical points.

Use the quotient rule to differentiate AC = C/Q, which gives d(AC)/dQ = (Q * dC/dQ - C) / Q^2.

Set the derivative d(AC)/dQ equal to zero and solve for Q, which simplifies to Q * MC = C, or AC = MC, proving that the production level where average cost is smallest is where average cost equals marginal cost.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Average Cost

Average cost is the total cost of production divided by the number of units produced. It is a measure of the cost per unit of output and is crucial for determining the efficiency of production. Understanding average cost helps in analyzing how costs behave as production levels change.

Marginal Cost

Marginal cost is the additional cost incurred by producing one more unit of a good or service. It is derived from the derivative of the total cost function with respect to quantity. Marginal cost is essential for decision-making, as it helps determine the optimal level of production where profits are maximized.

Optimization in Calculus

Optimization involves finding the maximum or minimum value of a function. In the context of production, it is used to find the level of output that minimizes average cost. This often involves setting the derivative of the average cost function equal to zero and solving for the production level, which is where average cost equals marginal cost.

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