Answer:
(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
