Question

Find the average rate of change

Answer 1

(a) The average rate of change from 0 to 2 is 6

(b) The average rate of change from 3 to 5 is 24

(c) The average rate of change from -3 to 0 is -9

Explanation:

The formula for the average rate of change (AROC) is given by:

`AROC=(f(b)-f(a))/(b-a)`

(a):

Finding the AROC from 0 to 2

To find the average rate of change from 0 to 2, we use f(2) and f(0), while we plug in 2 for b and 0 for a:

`(f(2)-f(0))/((2-0))\\ \\(((3(2)^2+4)-(3(0)^2+4)))/((2-0)) \\\\((16-4))/((2))\\ \\(12)/(2)\\ \\6`

Thus, the average rate of change from 0 to 2 is 6.

(b):

Finding the AROC from 3 to 5:

To find the average rate of change from 3 to 5, we use f(5) and f(3), while we plug in 5 for b and 3 for a:

`(f(5)-f(3))/((5-3))\\ \\(((3(5)^2+4)-(3(3)^2+4)))/((2-0)) \\\\((79-31))/((2))\\ \\(48)/(2)\\ \\24`

Thus, the average rate of change from 5 to 3 is 24.

(c):

To find the average rate of change from -3 to 0, we use f(0) and f(-3), while we plug in 0 for b and -3 for a:

`(f(0)-f(-3))/((0-(-3)))\\ \\(((3(0)^2+4)-(3(-3)^2+4)))/((0+3)) \\\\((4-31))/((3))\\ \\(-27)/(3)\\ \\-9`

Thus, the average rate of change from -3 to 0 is -9.