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? a) The derivative f'(x) represents the rate of change of cost with respect to the number of ounces of gold produced. Its units are:

b) The derivative f'(x) represents the instantaneous cost of producing x ounces of gold. Its units are:
c) The derivative f'(x) represents the total cost of producing x ounces of gold. Its units are:
d) The derivative f'(x) represents the average cost of producing x ounces of gold. Its units are:

Answer 1

The derivative f'(x) represents the marginal cost which is the additional cost of producing one more ounce of gold, with units in dollars per ounce. It demonstrates the rate at which production costs increase for each additional ounce produced.

Step-by-step explanation:

The derivative f'(x) of the cost function c = f(x), where x represents the number of ounces of gold produced, indicates the marginal cost of producing gold.

This derivative represents the rate of change of the cost, or the additional cost of producing one more ounce of gold.

The units of f'(x) are dollars per ounce because it calculates the cost for each additional ounce produced.

Considering an example, if the cost of producing gold increases from 40 ounces to 60 ounces, and total costs rise from $320 to $400, the marginal cost for each of the additional 20 ounces is $80 divided by 20, resulting in $4 per ounce.

Hence, the marginal cost curve is typically upward-sloping due to diminishing marginal returns, meaning that each additional ounce costs more to produce as production increases.