Marginal cost The cost C (in dollars) of producing x units...

Marginal cost The cost $C$ (in dollars) of producing $x$ units of a product is given by $C=3.6 \sqrt{x}+500 .$
(a) Find the additional cost when the production increases from 9 to 10 units.
(b) Find the marginal cost when $x=9 .$
(c) Compare the results of parts (a) and (b).

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Key Concepts

Cost Function

A cost function expresses the total cost incurred by a firm in relation to the quantity of goods produced. It typically includes both fixed costs and variable costs, and enables businesses to analyze how production volume impacts overall expenditures.

Marginal Cost

Marginal cost is the additional cost incurred by producing one more unit of output. It is a fundamental concept in economics and helps firms decide how much to produce by linking production levels to incremental costs.

Derivative

The derivative is a mathematical tool used to measure the instantaneous rate of change of a function with respect to one of its variables. In the context of cost functions, taking the derivative allows us to calculate the marginal cost as a function of the production level.

Approximation Using Marginal Cost

When the production change is small, the derivative (or marginal cost) provides a close approximation to the actual additional cost calculated by the difference in cost function values. This approximation is particularly useful in decision-making and economic analysis.

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