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Problem 53 Medium Difficulty

The cost of producing $ x $ ounces of gold from a new gold mine is $ C = f(x) $ dollars.
(a) What is the meaning of the derivative $ f'(x) $? What are its units?
(b) What does the statement $ f'(800) = 17 $ mean?
(c) Do you think the values of $ f'(x) $ will increase or decrease in the short term? What about the long term? Explain.

Answer

a)(a) $f^{\prime}(x)$ is the rate of change of the production cost with respect to the number of ounces of gold produced. Its units are dollars per ounce.
(b) After 800 ounces of gold have been produced, the rate at which the production cost is increasing is $\$ 17 /$ ounce. So the cost
of producing the 800 th $(\text { or } 801 \text { st })$ ounce is about $\$ 17$
(c) In the short term, the values of $f^{\prime}(x)$ will decrease because more efficient use is made of start-up costs as $x$ increases. But
eventually $f^{\prime}(x)$ might increase due to large-scale operations.

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Video Transcript

Okay. First it's important to note that X. Is in ounces and F. Is in dollars. So if you're fine, if you're finding f prime years finding the derivative of F. With respect to X. That means what you're finding is the number of dollars per ounce, Okay, cost of mining gold per ounce. Okay. So not not the cost per hour or the cost per day. It's not a time thing but it cost per ounce. Okay so what does f. Prime 800 equals 17 mean well 800 oz. And so it would say uh to mine 800 ounces will cost yeah, $17 per ounce. Do you think the value of F. Prime of X will increase or decrease in the short term? What about the long term? Okay. So at first it's going to increase. Okay. The cost per ounce when you first start. Because you got to you know you got to get all the dirt out of the way and I don't know, I don't know what lines are like but at first it's going to increase and then it'll get to a place where you know you just go in and they're like oh there's a gold, I'll take it and then the the cost will decrease in the long term, explain because it's harder when you first start to get into the mind. All right. I hope that helps. If not just post your question again.

Linda H.

Oklahoma State University

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