Question

Find the average rate of change of f(x) = - x ^ 2 + 3x + 1 from x = 2 to x = 5 . Simplify your answer as much as possible.

Answer 1

Given:

`f(x)=-x^2+3x+1`
`The\text{ points are x=2 and x=5.}`

Aim:

We need to find the average rate of change of f(x).

Step-by-step explanation:

The formula to find the average rate of change is

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

Here b=5 and a=2.

Substitute x=2 in the function f(x) to find f(2).

`f(2)=-(2)^2+3(2)+1`

`f(2)=-4+6+1=3`

Substitute x=5 in the function f(x) to find f(5).

`f(5)=-(5)^2+3(5)+1`

`f(5)=-25+15+1`

`f(5)=-9`

`Substitue\text{ a=2, b=5, f\lparen2\rparen=3 and f\lparen5\rparen=-9 in }(f(b)-f(a))/(b-a).`

`(f(b)-f(a))/(b-a)=(-9-3)/(5-2)=(-12)/(3)=-4.`

The average rate of change of f(x) is -4.

Final answer:

The average rate of change of f(x) is -4.