Find the marginal and average function for each of the following function. TC=3Q^2 5Q 10

To find the marginal function and average function for a given function, we first need to determine the formula for the function itself. In the given example, the total cost (TC) function is provided as TC = 3Q^2 + 5Q + 10.

The marginal function represents the rate at which the total cost changes with respect to the quantity produced (Q). To find the marginal function, we need to take the derivative of the total cost function with respect to Q.

d(TC)/dQ = d(3Q^2 + 5Q + 10)/dQ

Taking the derivative of each term separately:

d(3Q^2)/dQ = 6Q
d(5Q)/dQ = 5
d(10)/dQ = 0 (since 10 is a constant)

Combining these derivatives, we get:

d(TC)/dQ = 6Q + 5

Hence, the marginal function for the given total cost function TC = 3Q^2 + 5Q + 10 is:

MC(Q) = 6Q + 5

Now, let's find the average function. The average function represents the average cost per unit of quantity produced. To obtain the average function, we divide the total cost function by the quantity produced (Q):

AC(Q) = TC(Q) / Q

Substituting the total cost function TC = 3Q^2 + 5Q + 10:

AC(Q) = (3Q^2 + 5Q + 10) / Q

Simplifying further:

AC(Q) = (3Q^2 / Q) + (5Q / Q) + (10 / Q)
AC(Q) = 3Q + 5 + (10 / Q)

Therefore, the average function for the given total cost function TC = 3Q^2 + 5Q + 10 is:

AC(Q) = 3Q + 5 + (10 / Q)