Question

Suppose the dollar cost of producing x video cameras is C(x) = 500x − 0.003x² + 10−8x³. Estimate the marginal cost at production level x = 6500. (Round your answer to three decimal places.)

Answer 1

To estimate the marginal cost at production level x = 6500 in the given function C(x) = 500x − 0.003x² + 10−8x³, calculate the first derivative of the function and substitute x = 6500 into the derivative to find the estimated marginal cost of $1,267,500,461.

Step-by-step explanation:

To estimate the marginal cost at production level x = 6500 in the given function C(x) = 500x − 0.003x² + 10−8x³, we need to calculate the first derivative of the function, which will represent the marginal cost function. Let's find the derivative:

C'(x) = 500 - 0.006x + 30x²

Now, substitute x = 6500 into the derivative function to estimate the marginal cost:

C'(6500) = 500 - 0.006(6500) + 30(6500)²

C'(6500) ≈ 500 - 39 + 30(42250000)

C'(6500) ≈ 500 - 39 + 1267500000

C'(6500) ≈ 1267500461

Therefore, the estimated marginal cost at production level x = 6500 is approximately $1,267,500,461.