Question

Suppose that the cost of drilling x feet for an oil well is C=f(x) dollars.
a) What are the units of f'(x)?
b) In practical terms, what does f'(x) mean in this case?
c) What can you say about the sign of f'(x)?
d)Estimate the cost of drilling an additional foot, starting at a depth of 300 ft, given that f'(300)=1000.

Answer 1

wat????

Answer 2

a) I would say the derivative results in units of dollars/foot

Answer 3

why though?

Answer 4

Because derivatives are instantaneous rates of change. A rate is one quantity divided by the other.

Answer 5

And so the output unit / input unit is the resultant unit of a derivative.

Answer 6

oh, ic thanks, mind helping me on b,c,and d pls?

Answer 7

For b) we like to call it the 'marginal cost' when talking about the derivative of cost.

Answer 8

oh, because its kind of related but not really? O_O

Answer 9

Marginal cost is important because profit is maximized when marginal cost equals marginal revenue.

Answer 10

oh ic ic. so would c) be that its the marginal revenue?

Answer 11

You typically expect marginal cost to go up due to the law of diminishing returns.

Answer 12

So it's sign is positive.

Answer 13

\[
f(301)-f(300)\approx f'(300)(301-300)
\]

Answer 14

So the linear approximation estimate would say drilling an additional foot would cost \[
1000(301-300) = 1000
\]

Answer 15

ah, lemme look at ur explanation carefully right now.

Answer 16

oh, i get it. thanks