If the cost of manufacturing x items is C(x)=x^3+20 x^2+90...

If the cost of manufacturing $x$ items is $C(x)=x^{3}+20 x^{2}+90 x+15,$ find the marginal cost function and compare the marginal cost at $x=50$ with the actual cost of manufacturing the 50th item.

Key Concepts

Cost Function

A cost function represents the total cost of producing a given number of items. It typically includes fixed costs and variable costs, and it expresses how the total cost changes as production volume changes. This function is fundamental in managerial economics and helps in planning production and budgeting.

Marginal Cost Function

The marginal cost function is the derivative of the cost function with respect to the number of items produced. It represents the rate at which the cost increases as one additional unit is produced, providing a close approximation to the cost of producing one more item under the assumption of infinitesimally small changes.

Differentiation

Differentiation is a key concept in calculus used to determine the rate at which a quantity changes. In the context of economics, it is applied to cost functions to derive marginal cost functions, thereby allowing for the assessment of incremental changes in production costs.

Comparison of Actual and Marginal Cost

Comparing the marginal cost, which is derived via differentiation, with the actual cost of producing an additional unit involves analyzing the difference between the derivative-based estimate and the finite difference obtained from the cost function. This comparison helps validate the use of the derivative as an approximation for small changes in the production quantity.

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