Suppose the marginal cost is $C ^ { \prime } ( x ) = e ^ { - 0.9 x }$ , where x is measured in units of 500 items and the cost is measured in units of $10,000. Find the cost corresponding to the production interval [1,000, 3,000]. Round to the nearest dollar.
A) $1,608
B) $1,447
C) $1,786
D) $1,476

Question

Quizplus · Accepted Answer

The cost is given by the integral of the marginal cost:$$C(x) = \int e^{-0.9x} dx = -\frac{1}{0.9}e^{-0.9x} + C$$where C is the constant of integration. To find the cost corresponding to the production interval [1,000, 3,000], we evaluate C(3,000) - C(1,000):$$C(3,000) - C(1,000) = \left(-\frac{1}{0.9}e^{-0.9(3,000)}\right) - \left(-\frac{1}{0.9}e^{-0.9(1,000)}\right) \approx 1,786$$Therefore, the cost corresponding to the production interval [1,000, 3,000] is $1,786.

Suppose the Marginal Cost Is C′(x)=e−0.9xC ^ { \prime } ( x ) = e ^ { - 0.9 x }C′(x)=e−0.9x

Suppose the Marginal Cost Is $C ^ { \prime } ( x ) = e ^ { - 0.9 x }$