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).

Explanation:

The formula to find the average rate of change is

$\frac{f(b)-f(a)}{b-a}$

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).

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

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

$f(5)=-9$

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

$\frac{f(b)-f(a)}{b-a}=\frac{-9-3}{5-2}=\frac{-12}{3}=-4.$

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

Final answer:

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