If the cost of manufacturing x items is C(x)=x^3+21 x^2+110...

If the cost of manufacturing $x$ items is $C(x)=x^{3}+21 x^{2}+110 x+20,$ find the marginal cost function and compare the marginal cost at $x=100$ with the actual cost of manufacturing the 100th item.

Key Concepts

Approximation vs. Actual Cost

The marginal cost calculated by the derivative provides an approximation of the cost incurred by producing one more unit. Comparing this approximation with the actual cost of producing the unit can reveal discrepancies attributable to non-linearities or discrete production effects, illustrating the limitations of using derivatives for precise predictions in certain contexts.

Differentiation

Differentiation is a mathematical process used to compute the derivative of a function. In the context of cost functions, it helps determine the marginal cost by measuring how the cost changes as the production quantity changes.

Cost Function

A cost function is a mathematical expression that represents the total cost of production based on the number of items produced. It includes fixed and variable components, and it provides a basis for predicting how changes in output affect the overall cost.

Marginal Cost Function

The marginal cost function indicates the change in total cost resulting from producing one additional unit of output. It is obtained by differentiating the cost function with respect to the number of items. This concept is essential for making decisions about increasing or decreasing production.

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