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)? what are its units?
〇 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.
〇 f’(x) is the total production cost. its units are dollars.
〇 f’(x) is the total amount of gold produced. its units are ounces.
〇 f’(x) is the average production cost with respect to the number of ounces of gold produced. its units are dollars per ounce.
(b) what does the statement f’(700) = 12 mean?
〇 after 700 ounces of gold have been produced, the average production cost is $12/ounce. so the cost of producing 700 ounces is about $12/ounce.
〇 the total cost to produce 12 ounces of gold is approximately $700.
〇 after 700 ounces of gold have been produced, the rate at which the production cost is increasing is $12/ounce. so the cost of producing the 700th (or 701st) ounce is about $12.
〇 after 12 ounces of gold have been produced, the average production cost is $700/ounce. so the cost of producing 12 ounces is about $700/ounce.
(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 —select— because more efficient use is made of start - up costs as x increases. but eventually f’(x) might —select— due to large - scale operations.

Explanation:

Part (a)

The derivative \( f'(x) \) of a function \( y = f(x) \) represents the rate of change of \( y \) with respect to \( x \). Here, \( C = f(x) \) where \( C \) is the cost (in dollars) and \( x \) is the number of ounces of gold produced. So \( f'(x) \) is the rate of change of production cost with respect to ounces of gold produced. The units of a derivative \( \frac{dy}{dx} \) are units of \( y \) per unit of \( x \), so dollars per ounce.

• The second option is wrong because \( f(x) \) (not \( f'(x) \)) is total cost.
• The third option is wrong as \( f(x) \) is about cost, not gold amount.
• The fourth option is wrong because derivative is instantaneous rate, not average rate.

\( f'(x) \) is the instantaneous rate of change of cost with respect to ounces produced. So \( f'(700) = 12 \) means when \( x = 700 \) ounces (after 700 ounces are produced), the rate at which cost is increasing is 12 dollars per ounce. This approximates the cost of producing the next (700th or 701st) ounce.

• First option is wrong (it refers to average cost, but \( f'(x) \) is instantaneous rate).
• Second option is wrong (mixes up ounces and cost values).
• Fourth option is wrong (mixes up 12 and 700 in context).

In the short term, as production ( \( x \)) increases, there can be economies of scale or more efficient use of start - up costs (like machinery being used more optimally), so the marginal cost ( \( f'(x) \)) will decrease. In the long term, due to factors like depletion of easy - to - mine gold, increasing difficulty in extraction, or large - scale operation inefficiencies, the marginal cost \( f'(x) \) might increase.

Answer:

A. \( f'(x) \) 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.

Part (b)