Final answer:
The marginal cost function MC1(Q) is 6 and MC2(Q) is computed using the chain rule resulting in 100(2Q3/2 − 5Q)(3−(2Q1/2) − 5). Marginal cost is crucial for economic decision-making in a business.
Step-by-step explanation:
To find the marginal cost functions for the two total cost functions provided, we need to take the first derivative of each total cost function. Marginal cost is the rate of change of the total cost with respect to the quantity (Q). For TC1(Q) = Q + 5Q, we calculate MC1(Q) by taking the derivative with respect to Q. The derivative of Q is 1, and the derivative of 5Q is 5. Therefore, MC1(Q) = 1 + 5 = 6.
For TC2(Q) = 2500 + 50(2Q3/2 − 5Q)2, we apply the chain rule to compute the derivative. The outer function is raised to the power of 2, and the inner function is 2Q3/2 − 5Q. The derivative of the inner function, with respect to Q, is 3−(2Q1/2) − 5. Applying the chain rule, MC2(Q) is 100(2Q3/2 − 5Q)(3−(2Q1/2) − 5). These calculations are essential for a business to understand how much additional cost is involved in producing one more unit of a product, which is a key factor in many economic decisions.