Question 3.9.1: Analyzing the Marginal Cost of Producing a Commercial Produc...

Calculus: Early Transcendental Functions

Analyzing the Marginal Cost of Producing a Commercial Product

Suppose that

C(x)=0.02 x^2+2 x+4000

is the total cost (in dollars) for a company to produce x units of a certain product. Compute the marginal cost at x = 100 and compare this to the actual cost of producing the 100th unit.

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The marginal cost function is the derivative of the cost function:

C^{\prime}(x)=0.04 x+2

and so, the marginal cost at $x=100 \text { is } C^{\prime}(100)=4+2=6$ dollars per unit. On the other hand, the actual cost of producing item number 100 would be C(100) − C(99). (Why?) We have

C(100)-C(99)=200+200+4000-(196.02+198+4000)

$=4400-4394.02=5.98 \text { dollars }$ .

Note that this is very close to the marginal cost of $6. Also notice that the marginal cost is easier to compute.

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