Find the marginal and average function for each of the following functions. a). TC=3Q^2+5Q+10, b). TC=30+5Q-2Q^2+4Q^3
To find the marginal and average functions for each given cost function, we need to take the derivative of the cost function with respect to the quantity produced (Q).
a) TC = 3Q^2 + 5Q + 10
1. Marginal cost (MC): The derivative of the total cost function with respect to Q gives us the marginal cost.
MC = d(TC)/dQ = d(3Q^2 + 5Q + 10)/dQ
To find MC, we differentiate each term in the function with respect to Q, and the derivative of a constant is zero.
MC = 6Q + 5
2. Average cost (AC): The average cost can be calculated by dividing the total cost by the quantity.
AC = TC / Q = (3Q^2 + 5Q + 10) / Q
= 3Q + 5 + 10/Q
b) TC = 30 + 5Q - 2Q^2 + 4Q^3
1. Marginal cost (MC):
MC = d(TC)/dQ = d(30 + 5Q - 2Q^2 + 4Q^3) / dQ
Differentiating each term with respect to Q:
MC = 5 - 4Q + 12Q^2
2. Average cost (AC):
AC = TC / Q = (30 + 5Q - 2Q^2 + 4Q^3) / Q
= 30/Q + 5 - 2Q + 4Q^2