Question

The cost, in dollars, of producing x yards of a certain fabric is C(x) = 1300 + 13x − 0.1x2 + 0.0005x3. (a) Find the marginal cost function.

Answer 1

The marginal cost function is found by differentiating the total cost function C(x) with respect to x, and the resulting marginal cost function is C'(x) = 13 - 0.2x + 0.0015x^2.

Step-by-step explanation:

Finding the Marginal Cost Function

To find the marginal cost function, we need to differentiate the total cost function, C(x), with respect to x. Given the total cost function C(x) = 1300 + 13x − 0.1x2 + 0.0005x3, the marginal cost function, C'(x), represents the rate of change of the total cost with respect to the number of yards produced, x.

Let's differentiate C(x):

1. The derivative of the constant 1300 is 0.
2. The derivative of 13x is 13.
3. The derivative of -0.1x2 is -0.2x (using the power rule).
4. The derivative of 0.0005x3 is 0.0015x2 (again using the power rule).

Combining these, the marginal cost function is C'(x) = 13 - 0.2x + 0.0015x2.

Thus, if you want to find the marginal cost at a specific level of production, simply substitute that quantity for x in the marginal cost function.