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Section 4.14 : Business Applications

4. The production costs, in dollars, per month of producing \(x\) widgets is given by,

\[C\left( x \right) = 200 + 0.5x + \frac{{10000}}{x}\]

What is the marginal cost when \(x = 200\) and \(x = 500\)? What do your answers tell you about the production costs?

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From the notes in this section we know that the marginal cost is simply the derivative of the cost function so let’s start with that.

\[C'\left( x \right) = 0.5 - \frac{{10000}}{{{x^2}}}\] Show Step 2

The marginal costs for each value of \(x\) is then,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{C'\left( {200} \right) = 0.25\hspace{0.75in}C'\left( {500} \right) = 0.46}}\] Show Step 3

From these computations we can see that is will cost approximately 25 cents to produce the 201st widget and approximately 46 cents to produce the 501st widget.