Suppose that the cost, in dollars, for a company to produce x pairs of a new line of jeans is
C(x)=2200+4x+0.01x2+0.0002x3.
(a) Find the marginal cost function.
(b) Find the marginal cost at x=100.
(c) Find the cost at x=100.
(a) dC/dx
(b) and (c) are just plug and chug
(a) To find the marginal cost function, we need to take the derivative of the cost function C(x) with respect to x.
C(x) = 2200 + 4x + 0.01x^2 + 0.0002x^3
Taking the derivative of C(x) with respect to x:
C'(x) = d/dx (2200 + 4x + 0.01x^2 + 0.0002x^3)
= 0 + 4 + 0.02x + 0.0006x^2
Therefore, the marginal cost function is C'(x) = 4 + 0.02x + 0.0006x^2.
(b) To find the marginal cost at x = 100, we can simply substitute x = 100 into the marginal cost function C'(x):
C'(100) = 4 + 0.02(100) + 0.0006(100^2)
= 4 + 2 + 6
= 12.
Therefore, the marginal cost at x = 100 is 12 dollars.
(c) To find the cost at x = 100, we need to substitute x = 100 into the cost function C(x):
C(100) = 2200 + 4(100) + 0.01(100^2) + 0.0002(100^3)
= 2200 + 400 + 100 + 200
= 2900.
Therefore, the cost at x = 100 is 2900 dollars.
To find the marginal cost function, we need to differentiate the given cost function with respect to x. The marginal cost represents the rate at which the cost is changing with respect to the number of pairs produced.
(a) To find the marginal cost function, differentiate C(x) with respect to x:
C'(x) = d/dx (2200 + 4x + 0.01x^2 + 0.0002x^3)
Differentiating each term separately, we get:
C'(x) = 0 + 4 + 0.01(2x) + 0.0002(3x^2)
C'(x) = 4 + 0.02x + 0.0006x^2
Therefore, the marginal cost function is C'(x) = 4 + 0.02x + 0.0006x^2.
(b) To find the marginal cost at x = 100, substitute x = 100 into the marginal cost function:
C'(100) = 4 + 0.02(100) + 0.0006(100^2)
C'(100) = 4 + 2 + 0.06(10000)
C'(100) = 4 + 2 + 600
C'(100) = 606
Therefore, the marginal cost at x = 100 is 606 dollars.
(c) To find the cost at x = 100, substitute x = 100 into the given cost function:
C(100) = 2200 + 4(100) + 0.01(100^2) + 0.0002(100^3)
C(100) = 2200 + 400 + 0.01(10000) + 0.0002(1000000)
C(100) = 2200 + 400 + 100 + 200
C(100) = 2900 + 300
C(100) = 3200
Therefore, the cost at x = 100 is 3200 dollars.