If the cost of manufacturing x items is C(x)=x^4+14 x^2+60...

If the cost of manufacturing $x$ items is $C(x)=x^{4}+14 x^{2}+60 x+35,$ find the marginal cost function and compare the marginal cost at $x=50$ with the actual cost of manufacturing the 50th item.

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Key Concepts

Marginal Cost Function

The marginal cost function represents the instantaneous rate at which production costs change with respect to output quantity. It is obtained by differentiating the total cost function, and it approximates the additional cost of manufacturing one more unit when production increases by a small amount.

Differentiation

Differentiation is a fundamental concept in calculus used to calculate the derivative of a function. In the context of cost functions, it is applied to compute the marginal cost, which indicates how the cost changes as the number of produced items increases.

Discrete vs Continuous Analysis

Discrete vs Continuous Analysis refers to the comparison between the marginal cost calculated using the derivative (a continuous approximation) and the actual cost computed by finding the difference between the costs of producing consecutive quantities. This comparison highlights that while marginal cost can provide an estimate for the cost of an additional unit, it may differ from the actual change when the cost function is nonlinear or when production increments are finite.

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