In business calculus, which definition best describes marginal cost?

• A. The rate of change of total cost with respect to output; if total cost is $C\left(q\right)$, then marginal cost is $C\prime \left(q\right)=\frac{dC}{dq}$.
• B. The total cost of producing the first unit of output; $C\left(1\right)$.
• C. The change in cost divided by the change in quantity over an interval; $\frac{C\left(q_2\right)-C\left(q_1\right)}{q_2-q_1}$.
• D. The average cost per unit produced; $AC\left(q\right)=\frac{C\left(q\right)}{q}$.

Answer: (A)

Verified step by step guidance

1. Understand that marginal cost represents how the total cost changes as the quantity of output changes. It is essentially the instantaneous rate of change of the total cost function with respect to output quantity.
2. Recall that if the total cost function is given by \(C(q)\), where \(q\) is the quantity produced, then the marginal cost is the derivative of \(C(q)\) with respect to \(q\).
3. Express the marginal cost mathematically as \(C\'(q) = \frac{dC}{dq}\), which means you take the derivative of the total cost function to find how cost changes for a very small change in output.
4. Differentiate this from average cost, which is the total cost divided by the quantity produced, given by \(AC(q) = \frac{C(q)}{q}\), and from the change in cost over an interval, which is a difference quotient rather than an instantaneous rate.
5. Conclude that marginal cost is best described as the rate of change of total cost with respect to output, represented by the derivative \(C\'(q) = \frac{dC}{dq}\).