cost, revenue, and profit are in dollars and x s the number of units

cost: if the cost function for a particular good is C(x)= 3x^2 +6x + 600, what is the (a) marginal cost function (b) marginal cost if 30 units are produced? (c) interpretation of your answer in part (b)?

To find the marginal cost function, we need to take the derivative of the cost function with respect to x. Let's go step by step:

(a) Marginal cost function (MC):

The marginal cost (MC) represents the additional cost incurred from producing one more unit of a good. In this case, we will need to take the derivative of the cost function C(x) = 3x^2 + 6x + 600 with respect to x.

To do this, let's differentiate C(x) term by term:

dC(x)/dx = d(3x^2)/dx + d(6x)/dx + d(600)/dx

The derivative of x^2 with respect to x is 2x, and the derivative of a constant term (like 600) is zero. So, we have:

dC(x)/dx = 6x + 6

Hence, the marginal cost function is:

MC(x) = 6x + 6

(b) Marginal cost if 30 units are produced:

To find the marginal cost when x = 30 (30 units are produced), we substitute the value of x into the marginal cost function:

MC(30) = 6(30) + 6
MC(30) = 180 + 6
MC(30) = 186

Therefore, the marginal cost when 30 units are produced is 186 dollars.

(c) Interpretation of the answer in part (b):

The interpretation of the marginal cost (MC) is that it represents the additional cost of producing one more unit of the good. In this case, when 30 units are produced, the marginal cost is 186 dollars, which means that the cost of producing the 31st unit would be an additional 186 dollars.

To find the marginal cost function, we need to find the derivative of the cost function, C(x), with respect to x.

(a) Marginal cost function:
To find the derivative of C(x), we differentiate each term of the function separately, using the power rule:

C(x) = 3x^2 + 6x + 600

Differentiating each term:
dC/dx = d/dx(3x^2) + d/dx(6x) + d/dx(600)
= 6x + 6 + 0

Therefore, the marginal cost function is:
MC(x) = 6x + 6

(b) Marginal cost if 30 units are produced:
To find the marginal cost at a specific value of x, we substitute x = 30 into the marginal cost function:

MC(30) = 6(30) + 6
= 180 + 6
= 186

Therefore, the marginal cost when 30 units are produced is $186.

(c) Interpretation of the answer in part (b):
The interpretation of the marginal cost is the additional cost incurred by producing one extra unit of the good. In this case, if 30 units are produced, the cost of producing the 31st unit would be $186.