The cost of producing x ounces of gold from a new gold mine is C = f(x) dollars.

Question

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)? The derivative is the rate of change of the number of ounces of gold produced with respect to the production cost. The derivative is the cost of producing gold from a new mine. The derivative is the rate of change of the production cost with respect to the number of ounces of gold produced. The derivative is the number of ounces of gold produced from a new mine. What are its units? dollars ounces ounces per dollar dollars per ounce (b) What does the statement f'(800) = 15 mean? After $15 have been spent, the rate at which gold is being produced is increasing 800 ounces/dollar. After 800 ounces of gold have been produced, the rate at which the production cost is increasing is $15/ounce. The cost of producing 800 ounces of gold is $15. It costs $15/ounce to produce 800 ounces of gold. (c) Do you think the values of f'(x) will increase or decrease in the short term? Explain. In the short term, the values of f'(x) will decrease because more efficient use is made of start-up costs as x increases. In the short term, the values of f'(x) will increase because of high start-up costs. What about the long term? Explain. In the long term, the values of f'(x) might increase due to large-scale operations. In the long term, the values of f'(x) might decrease due to large-scale operations.

Explanation / Answer

(a) f '(x) means rate of change f(x) with respect to x that means in the above question f '(x) "means rate of change of production cost with respect to number of ounces of gold produced"

units of f '(x) is dollars per ounce

(b) f '(800)=15 means after 800 ounces of gold have been produced , the rate at which the production cost is increasing is $15/ounce

(c) in the short term the value of f '(x) will increase because of high startup costs

(d) in the long term, the values of f '(x) might decrease due to large scale operations

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