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? f '(x) is the average production cost with respect to the number of ounces of gold produced. Its units are dollars per ounce. f '(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. f '(x) is the total amount of gold produced. Its units are ounces. f '(x) is the total production cost. Its units are dollars. (b) What does the statement f '(500) = 18 mean? The total cost to produce 18 ounces of gold is approximately $500. After 500 ounces of gold have been produced, the average production cost is $18/ounce. So the cost of producing 500 ounces is about $18/ounce. After 18 ounces of gold have been produced, the average production cost is $500/ounce. So the cost of producing 18 ounces is about $500/ounce. After 500 ounces of gold have been produced, the rate at which the production cost is increasing is $18/ounce. So the cost of producing the 500th (or 501st) ounce is about $18. (c) Do you think the values of f '(x) will increase or decrease in the short term? What about the long term? Explain. In the short term, the values of f '(x) will because more efficient use is made of start-up costs as x increases. But eventually f '(x) might due to large-scale operations.

Answer:a) The rate of change of the cost per ounce of gold produced. Dollars per ounce.b) At 500 ounces of gold, the cost per ounce is 18 dollars.c) Short term decrease; long term increase.Step-by-step explanation:a) The first derivative f'(x) is always a rate of change. You can think of it as indicator, how much changes the distance of an object when the time changes. (velocity) Or in this case how much changes the cost per ounce, when the production changes. (marginal cost). If C=f(x) registers the amount of dollars to make x ounces of gold, f'(x) will register how much it will cost to produce an extra ounce of gold. So the unit will be dollars per ounce.b) The difference with the answer given before is that without knowing f(x) is impossible to know if the cost per ounce is increasing or decreasing. What is certain is that the cost per ounce, when the production is 500 ounces, is 18 dollars per ounce. c) You can think of this situation as if when someone becomes a master of his/her craft. At the beginning it would be even painful to learn something knew, but one you know it, it becomes easier and easier. The same with this mining situation. At the beginning the cost would be huge, but then with the expertise and experience, the cost would decrease.However, in the long term we would expect inflation (prices changes over time). The cost to produce gold in the past century should have been low compared with the cost to produce gold currently. So we would expect the cost to increase in the long term.