# The meaning of derivative f ′ ( x ) and its units. ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 2.6, Problem 47E

(a)

To determine

## The meaning of derivative f′(x) and its units.

Expert Solution

### Explanation of Solution

The function f(x) 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(x) is the instantaneous rate of change of C with respect to x,

The derivative f(x) 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(x)=limΔx0ΔyΔx.

Here Δy is measured in dollars and Δx is measured in ounces.

Thus, the units are dollars per ounces.

(b)

To determine

Expert Solution

### 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 800th 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

Expert Solution

### 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(x) will decrease as x increases.

But, the derivative f(x) might increase due to large-scale operations.

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