Chapter 2.6, Problem 47E

(a)

To determine

The meaning of derivative $f^{\prime }\left(x\right)$ and its units.

Explanation of Solution

The function $f\left(x\right)$ is the cost of production.

The cost of production is depends on number of ounces of gold produced.

That is, the cost of production is a function of ounces of gold produced.

Note that, the derivative $f^{\prime }\left(x\right)$ is the instantaneous rate of change of C with respect to x,

The derivative $f^{\prime }\left(x\right)$ is rate of change of the production cost with respect to the number of ounces of gold produced.

The instantaneous rate of change is equal to $f^{\prime }\left(x\right)=\underset{\Delta x\to 0}{\lim }\frac{\Delta y}{\Delta x}$.

Here $\Delta y$ is measured in dollars and $\Delta x$ is measured in ounces.

Thus, the units are dollars per ounces.

(b)

To determine

The meaning of $f^{\prime }\left(800\right)=17$.

Explanation of Solution

After 800 ounces of gold produced.

The rate at which the production cost is increasing is 17 dollar per ounce.

The cost of producing the 800^th ounce is about 17 dollar.

That is, when $x=800$, the production cost is increasing 17 times as fast as number of ounces of gold produced.

(c)

To determine

To check: Whether the derivative $f^{\prime }\left(x\right)$ will increase or decrease for short term and also for long term.

Explanation of Solution

In the short term,

The rate of production cost decreases as the production cost involves more the startup cost for the mass production. Hence, the values of $f^{\prime }\left(x\right)$ will decrease as x increases.

But, the derivative $f^{\prime }\left(x\right)$ might increase due to large-scale operations.