   Chapter 2.1, Problem 53E

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# 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?(b) What does the statement f ′ ( 800 ) = 17 mean?(c) Do you think the values of f ′ ( x ) will increase or decrease in the short term? What about the long term? Explain.

To determine

(a) To explain: The meaning of the derivative f(x) and  its units.

Explanation

The derivative f(x) is the instantaneous rate of change of C with respect to  ; that is rate of change of the production cost with respect to the number of ounces of gold.

f'x= limx  0C x

Since C  is measured in dollars and x  in ounce, it follows that the unit for f(x) is dollarsper ounce

Therefore, the units for f’(x) are dollars per ounce.

Given: The cost of producing x ounces of gold from a new gold mine is C = f(x) dollars.

Conclusion:  The f(x) is the instantaneous rate of change of C with respect to x; that is rate of change of the production cost with respect to the number of ounces of gold. Therefore, the unit of f(x) is dollars per ounce.

(b) To explain: The meaning of the statement of f (800) = 17

Solution: The statement f'(800) = 17 that means just after producing 800 ounces of gold, the rate at which the production cost is increasing is $17 per ounce. In other words the marginal cost once production has reached 800 ounces of gold is$17 per ounce.

The statement f'(800) = 17 that means after producing 800 ounces of gold, the rate at which the production cost is increasing is \$17 per ounce

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