Question

Suppose that the revenue (in dollars) from the sale of a product is given by

R = 60x + 0.8x2 − 0.001x3

where x is the number of units sold. How fast is the marginal revenue MR changing when x = 50?

Answer 1

To find how fast the marginal revenue is changing for the given revenue function when x=50, we calculate the second derivative of the revenue function and then substitute x=50 into it. The second derivative is 1.6 - 0.006x, which gives us a rate of change of 1.3 dollars per unit for the marginal revenue when 50 units are sold.

Step-by-step explanation:

The student asked about how fast the marginal revenue (MR) is changing when the number of units sold (x) is 50 for a revenue function given by R = 60x + 0.8x2 − 0.001x3. To find this rate of change, we will calculate the second derivative of the revenue function, as the second derivative represents the rate of change of marginal revenue.

First, we calculate the first derivative of the revenue function to find the marginal revenue:

MR = dR/dx = 60 + 1.6x - 0.003x2

Next, we find the second derivative to determine the rate of change of MR:

dMR/dx = d2R/dx2 = 1.6 - 0.006x

Substituting x = 50 into the second derivative:

dMR/dx at x = 50 = 1.6 - 0.006(50) = 1.6 - 0.3 = 1.3

Therefore, the marginal revenue is changing at a rate of 1.3 dollars per unit when 50 units are sold.