Derivatives as Marginal Quantities

Rate of change made precise

A production example

From marginal products to marginal cost

Reading a tangent correctly

What exactly does the derivative $f'(x)$ measure about a function $f$ at the point $x$?

The workshop's output is $f(x) = 20x - x^2$. What is the marginal product of labor at $x = 8$?

Total output is $f(x) = 20x - x^2$. At what $x$ does output reach its maximum, and why?

Labor costs 2 dollars per unit and the marginal product of labor at the current input is 5. What is the marginal cost of output near that point?

For $f(x) = 20x - x^2$, compute the derivative $f'(5)$, and write the number.

For the same workshop function $f(x) = 20x - x^2$, compute total output $f(5)$, and write the number.

For the workshop output f of x equals 20x minus x squared, what is the marginal product at 8 workers?

At what labor level does the workshop's total output reach its maximum?

With labor at 2 dollars per unit, what is the workshop's marginal cost near 5 workers?

What does f prime of 12 equal to minus 4 tell the workshop?

The marginal product is 20 minus 2x. Find the worker level where it equals 12, and write the number.

Near 5 workers one more unit of output needs one tenth of a worker. Compute its cost at 2 dollars per worker and write it in decimals.