Further Reading on the Quotient Rule

A firm's total cost of production is given by the function $C(Q) = 20 + 8Q + 2Q^2$, where Q is the quantity of output. By defining average cost as $AC(Q) = C(Q)/Q$ and applying the appropriate rule for differentiation, which of the following expressions correctly represents the slope of the firm's average cost curve?

Calculating the Slope of the Average Cost Curve

Analyzing the Slope of an Average Cost Curve

A firm's average cost (AC) is defined as its total cost function, C(Q), divided by the quantity of output, Q. To find the slope of the average cost curve, one must differentiate the AC function, $AC(Q) = \frac{C(Q)}{Q}$, with respect to Q. Arrange the following mathematical operations in the correct sequence as dictated by the rule for differentiating a quotient.

If a firm's total cost function is represented by C(Q), the slope of its average cost curve is correctly calculated by finding the marginal cost, C'(Q), and then dividing that result by the quantity of output, Q.

A firm's average cost (AC) is defined as total cost, C(Q), divided by output, Q. The slope of the AC curve is found by differentiating the function $AC(Q) = \frac{C(Q)}{Q}$ with respect to Q. Match each mathematical term from this calculation to its correct economic or mathematical description.

Interpreting the Slope of the Average Cost Curve

A firm's total cost is given by the function $C(Q) = 50 + 10Q + 2Q^2$. The slope of its average cost curve, derived by differentiating the average cost function $AC(Q) = C(Q)/Q$ with respect to Q, can be written in the form $\frac{A \cdot Q^2 - 50}{Q^2}$. The value of the coefficient A is ____.

A company's total cost to produce a good is described by the function $C(Q) = 4Q^2 + 10Q + 100$, where Q is the quantity of output. The slope of the average cost curve is found by differentiating the average cost function, $AC(Q) = C(Q)/Q$, with respect to Q. At what quantity of output (Q) is the slope of the average cost curve equal to zero, indicating that average cost is at its minimum?

Evaluating Cost Analysis Methodologies