Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 45

Suppose that the equation $c=f(x)$ represents the cost (in dollars) of producing $x$ ounces of gold from a new gold mine.



a.) State what is the meaning of the derivative $f'(x)$ and its corresponding units.



b.) What does the statement $f'(800) = 17$ mean.



c.) Do you think the values of $f'(x)$ wll increase or decrease in the short term? What about the long term? Explain.





$\quad$a.) The meaning of the derivative $f'(x)$ is the rate at which the cost is changing per ounce of gold produced.
Its unit is dollars per ounces.



$\quad$b.) $f'(800) = 17$ means that when 800 ounces of gold have been produced, the rate at which the
production cost is increasing at 17 $\displaystyle \frac{\text{dollars}}{\text{ounce}}$.



$\quad$c.) $f'(x)$ will decrease in short term suppose that the gold is abundant and the cost of production is cheap at that time.
However, $f'(x)$ will increase in the long term such that the amount of gold starts to deplete while the
demand increases.