Question

Suppose that the total profit in hundreds of dollars from selling x items is given by P(x) = 4x² - 6x +10. Complete parts a through d below. a. Find the average rate of change of profit as x changes from 2 to 4. $_____per item

Answer 1

The average rate of change of profit as x changes from 2 to 4 is $18 per item.

Explanation:

Let's call 2 x1 and 4 x2. The formula for the average rate of change from x1 to x2 is:

(f(x2) - f(x1)) / (x2 - x1).

Step 1: Find f(x1) and f(x2)

To fid f(x1) and f(x2), we plug in 2 and 4 for x in P(x):

Finding f(x1):

f(x1) = P(2) = 4(2)^2 - 6(2) + 10

f(x1) = P(2) = 4(4) - 12 + 10

f(x1) = P(2) = 16 - 12 + 10

f(x1) = P(2) = 14

Finding f(x2):

f(x2) = P(4) = 4(4)^2 - 6(4) + 10

f(x2) = P(4) = 4(16) - 24 + 10

f(x2) = P(4) = 64 - 24 + 10

f(x2) = P(4) = 50

Finding x2 - x1:

Since x2 = 4 and x1 = 2, the difference between these two numbers is 2 as 4 - 2 = 2.

Finding f(x2) - f(x1):

Since f(x2) = 50 and f(x1) = 14, the difference between these two numbers is 36 as 50 - 14 = 36.

Finding (f(x2) - f(x1)) / (x2 - x1):

Remember that f(x2) - f(x1) = 36 and x2 - x1 = 2. 36 / 2 = 18.

Thus, the average rate of change of profit as x changes from 2 to 4 is $18 per item.