The revenue (in dollars) from the sale of x car seats for infants is given by the following function.

R (x)= 64x - 0.020 x^2 0 <= x <= 3200

a. Find the average change in revenue if production is changed from 1,000 car seats to 1,050 car seats.
b. Use the four-step process to find R (x).
c. Find the revenue and the instantaneous rate of change of revenue at a production level of 1,000 car seats, and interpret the results.

a) Average rate of change: 23 $/seat

b)

c) R(1000) = 44,000 $, R'(1000) = 23 $seat

Explanation:

a)

The average rate of change of a function f(x) in a certain interval is given by

The function that gives the revenue in dollars from the sale of x car seats is

where

0 <= x <= 3200

We want to calculate the average rate of change in the interval

Calculating the value of the revenue function for these two values,

Therefore, the average rate of change is

b)

Here we want to find R'(x), the derivative of R(x).

Step 1:

Evaluating :

Step 2:

Evaluating :

Step 3:

Evaluating

Step 4:

Find the limit for :

c)

When the production level is 1000 car seats,

x = 1000

Therefore the revenue is:

The instantaneous rate of change of revenue instead is given by the derivative of R(x), found in part b):

Therefore, substituting x = 1000,

The two data can be interpreted as follows:

- When the production level is 1000 car seats, the revenue is 44,000$

- When the production level is 1000 car seats, the revenue is changing by 23$/new seat produced