Question

Average rate of change help!!

Answer 1

Given that the function is
`f(x)=x^(3)-6 x^(2)+4 x+7`

We need to determine the average rate of change over the interval
`0 \leq x \leq 5`

Value of f(x) when x = 0:

Substituting x = 0 in the function
`f(x)=x^(3)-6 x^(2)+4 x+7`, we have;

`f(0)=(0)^(3)-6 (0)^(2)+4 (0)+7`

`f(0)=0-0+0+7`

`f(0)=7`

Thus, the value of f(0) is 7.

Value of f(x) when x = 5:

Substituting x = 5 in the function
`f(x)=x^(3)-6 x^(2)+4 x+7`, we have;

`f(5)=(5)^(3)-6 (5)^(2)+4 (5)+7`

`f(5)=125-150+20+7`

`f(5)=2`

Thus, the value of f(5) is 2.

Average rate of change:

The average rate of change can be determined using the formula,

`Rate \ of \ change=(f(b)-f(a))/(b-a)`

where
`a=0` and
`b=5`

Thus, we have;

`Rate \ of \ change=(f(5)-f(0))/(5-0)`

`Rate \ of \ change=(2-7)/(5-0)`

`Rate \ of \ change=(-5)/(5)`

`Rate \ of \ change=-1`

Thus, the average rate of change over the interval
`0 \leq x \leq 5` is -1.