The cost, in dollars, of producing x yards of a certain fabric is C (x) = 1500 + 12x - 0.1x2 + 0.0005x3. (a) Find the marginal cost function. C' (x) = (b) Find C' (500) and explain its meaning. What does it predict? C' (500) = and this is the rate at which costs are increasing with respect to the production level when x =. C' (500) predicts the cost of producing the yard. (c) Compare C' (500) with the cost of manufacturing the 501st yard of fabric. (Round your answers to four decimal places.) The cost of manufacturing the 501st yard of fabric is C (501) - C (500) = - 45,000 ≈, which is approximately C' (500).

(a) The marginal cost function is 12 - 0.2x + 0.0015x^2

(b) C' (500) = $287. C' (500) indicates the rate at which cost is rising with respect to production level of the fabric when x = 500.

C' (500) predicts the cost of producing the 500th yard of fabric.

(c) C' (500) = $287

The cost of manufacturing the 501st yard, C (501) = $287.6505.

C (501) is approximately the same as C' (500).

Step-by-step explanation:

(a) The marginal cost function is obtained by differentiating the cost function (C) with respect to the number of yards of fabric produced (x)

C (x) = 1500 + 12x - 0.1x^2 + 0.0005x^3

C' (x) = 12 - 0.2x + 0.0015x^2

(b) C' (x) = 12 - 0.2x + 0.0015x^2

C' (500) = 12 - 0.2 (500) + 0.0015 (500) ^2 = $287. This means that cost is rising at a rate with respect to the production level when x = 500.

C' (500) = $287 predicts that the cost of producing the 500th yard of fabric is $287.

(c) C (x) = 1500 + 12x - 0.1x^2 + 0.0005x^3

C (501) = 1500 + 12 (501) - 0.1 (501) ^2 + 0.0005 (501) ^3 = $287.6505 is approximately the same as C' (500) which is $287