If the cost of manufacturing x items is C(x)=x^3+11 x^2+40...

If the cost of manufacturing $x$ items is $C(x)=x^{3}+11 x^{2}+40 x+10,$ find the marginal cost function and compare the marginal cost at $x=100$ with the actual cost of manufacturing the 100th item.

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Key Concepts

Discrete versus Continuous Cost Analysis

While the marginal cost obtained by differentiation provides an approximation of the additional cost for a small increase in production, the actual cost of an additional unit is determined by the finite change, computed as the difference between the cost of producing x+1 items and x items. This concept highlights how continuous approximations may differ from discrete changes in practical scenarios.

Differentiation

Differentiation is the mathematical process of computing a derivative. In the context of cost functions, it is used to calculate the marginal cost, transforming the discrete problem of cost increments into a continuous framework that approximates the effect of producing an additional item.

Cost Function

A cost function represents the total expenditure incurred to produce a given number of items. It maps the quantity produced to its associated cost, incorporating fixed and variable costs, and is a foundational concept in both economics and optimization problems.

Marginal Cost

Marginal cost is defined as the derivative of the cost function with respect to the quantity produced. It indicates the instantaneous rate of change of cost as production increases, thereby providing an estimate of the additional cost incurred to produce one more unit when changes are very small.

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