The cost function, in dollars, of a company that...

The cost function, in dollars, of a company that manufactures food processors is given by $C(x)=200+\frac{7}{x}+\frac{x^{2}}{7}, \quad$ where $x$ is the number of food processors manufactured. a. Find the marginal cost function. b. Find the marginal cost of manufacturing 12 food processors. c. Find the actual cost of manufacturing the thirteenth food processor.

The cost function, in dollars, of a company that manufactures food processors is given by $C(x)=200+\frac{7}{x}+\frac{x^{2}}{7}, \quad$ where $x$ is the number of food processors manufactured.
a. Find the marginal cost function.
b. Find the marginal cost of manufacturing 12 food processors.
c. Find the actual cost of manufacturing the thirteenth food processor.

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Key Concepts

Marginal Cost Function

The marginal cost function represents the instantaneous rate of change of the total cost with respect to the number of items produced. In other words, it gives an approximation of the additional cost incurred when one more unit is produced at a certain production level. This concept is especially useful in economics and business for decision-making and optimizing production levels.

Derivative

The derivative is a fundamental concept in calculus that measures the rate at which a function changes with respect to a variable. When applying differentiation to a cost function, the derivative yields the marginal cost function. This process transforms the discrete relationships in cost analysis into a continuous model, allowing for the use of calculus to approximate changes in costs.

Discrete Cost Difference

The discrete cost difference refers to the exact change in total cost when the production quantity is increased by one unit. This is calculated by taking the difference between the cost of production at two successive levels, for instance, the cost of producing 13 units minus the cost of producing 12 units. Although the marginal cost function provides an approximation, the discrete cost difference gives the actual change in cost for an incremental production increase.

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