The cost of producing x ounces of gold from a new gold mine is C=f(x) dollars.
Do you think the values of f'(x) will increase or decrease in the short term? What about the long term? Explain.
To determine if the values of f'(x) will increase or decrease in the short term and long term, we need to assess the factors that influence the cost function and its derivatives.
In the short term, the cost of producing gold from a new mine is affected by various factors such as mining efficiency, labor costs, technology, equipment, and market conditions. Initially, the cost may be higher due to several reasons, such as the need to set up infrastructure, unoptimized production processes, or lack of economies of scale. However, as the operation becomes more established and efficient, the cost may decrease in the short term.
Considering the derivative f'(x), which represents the rate of change of the cost with respect to the amount of gold produced (x), if the cost decreases, it implies that f'(x) is decreasing in value. This indicates that producing additional ounces of gold is becoming relatively cheaper and more cost-effective.
However, in the long term, there are other factors to consider. As the mine continues to operate, it might encounter diminishing returns. This means that extracting more gold may become progressively more challenging or costly. Factors like increasing extraction depth, depletion of high-quality ore, or rising environmental and regulatory constraints can contribute to an increase in production costs in the long run.
Therefore, in the long term, it is more likely that the values of f'(x) will increase. As the mine exhausts its easily accessible and profitable resources, the incremental production cost may rise, leading to an upward trend in the derivative f'(x).
In summary, the values of f'(x) are expected to decrease in the short term due to operational improvements and economies of scale. However, in the long term, they are likely to increase due to diminishing returns and the realization of higher production costs.