Question

hw 24 - marginals section 3.10: problem 3
(1 point)
the cost function of a certain commodity is ( c(x)=83 +.16x-.0006x^{2}+.000003x^{3} ).
(a) find ( c^{prime}(100) ). (what does this mean?)
(b) approximate the cost of producing only the ( 101^{st} ) item.
(a) ( c^{prime}(100)=square ) dollars/item
(b) approx. cost of producing the ( 101^{st} ) item ( =square ) dollars
note: you can earn partial credit on this problem.
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Explanation:

Step1: Differentiate the cost function

Using the power rule \((x^n)^\prime = nx^{n - 1}\), if \(C(x)=83 + 0.16x-0.0006x^{2}+0.000003x^{3}\), then \(C^\prime(x)=0.16-2\times0.0006x + 3\times0.000003x^{2}\), so \(C^\prime(x)=0.16-0.0012x+0.000009x^{2}\).

Step2: Evaluate \(C^\prime(100)\)

Substitute \(x = 100\) into \(C^\prime(x)\):

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\(C^\prime(100)\) represents the marginal - cost when \(x = 100\) items are produced. It is the approximate cost of producing the \(101^{st}\) item.

Step3: Approximate the cost of the \(101^{st}\) item

By the definition of marginal cost, the approximate cost of producing the \(101^{st}\) item is \(C^\prime(100)\)

Answer:

(a) \(C^\prime(100)=0.13\) dollars/item
(b) Approx. cost of producing the \(101^{st}\) item \(=0.13\) dollars