A firm’s total cost (in dollars) to produce $q$ units is $C\left(q\right)=500+20q+0.5q^2$. What is the marginal cost function $MC\left(q\right)$?

• A. $MC\left(q\right)=20+q$
• B. $MC\left(q\right)=20q+q^2$
• C. $MC\left(q\right)=500+20q+0.5q^2$
• D. $MC\left(q\right)=20+0.5q$

Answer: (A)

Verified step by step guidance

1. Identify the total cost function given: \(C(q) = 500 + 20q + 0.5q^{2}\).
2. Recall that the marginal cost function \(MC(q)\) is the derivative of the total cost function with respect to quantity \(q\), so \(MC(q) = \frac{d}{dq} C(q)\).
3. Differentiate each term of \(C(q)\) separately: the derivative of the constant 500 is 0, the derivative of \$20q\( is 20, and the derivative of \)0.5q^{2}\( is \)0.5 \times 2q$.
4. Simplify the derivative expression to get \(MC(q) = 20 + q\).
5. Interpret the result: the marginal cost function \(MC(q)\) represents the additional cost of producing one more unit at quantity \(q\).