Question

The revenue (in thousands of dollars) from producing x units of an item is modeled by R(x)=12x-0.01x^2. a. Find the average rate of change in revenue as x changes from 1002 to 1007. b. Find the marginal revenue at x=1000.

Answer 1

To find the average rate of change in revenue, calculate the change in revenue and divide it by the change in quantity. To find the marginal revenue, calculate the derivative of the revenue function and substitute the desired value of x.

Step-by-step explanation:

a. To find the average rate of change in revenue, we need to calculate the change in revenue and divide it by the change in quantity. So, the average rate of change in revenue as x changes from 1002 to 1007 is:

Average Rate of Change = (R(1007) - R(1002)) / (1007 - 1002)

b. To find the marginal revenue at x = 1000, we need to calculate the derivative of the revenue function with respect to x and then substitute x = 1000 into the derivative. So, the marginal revenue at x = 1000 is:

Marginal Revenue = R'(1000)

Answer 2

It is given in the question that

The revenue (in thousands of dollars) from producing x units of an item is modeled by

`R(x)=12x-0.01x^2`

a. Average rate of change is given by

`(R(1007)-R(1002))/(1007-1002)`

`= (1943.51-1983.96)/(5)`

`=(-40.45)/(5) = -8.09`

So the average rate of change of revenue is -8090dollars per unit .

b .To find the marginal revenue, we have to differentiate revenue function, that is

`R'(x)=12-0.02x`

And at x=1000, we will get

`R'(1000) = 12-0.02(1000)=-8`

S the marginal revenue is -8000 dollars.