Question

Find the average rate of change of the function f ( x ) = 1 x 2 + 2 x + 2 , on the interval x ∈ [0,2].

Answer 1

Given :$f ( x ) = x^2 + 2 x + 2$

Formula for average rate of change is

$Average =\frac{f(x_2)-f(x_1)}{x_2-x_1}$

x1 and x2 is the given interval.

Given interval is [0,2].

x1 = 0, x2=2

$f ( x ) = x^2 + 2 x + 2$

$f(0) = 0^2 + 2(0) + 2=2$

$f(2) = 2^2 + 2(2) + 2=10$

$Average =\frac{10-2}{2-0}$ = 4

Average rate of change is 4

Answer 2

The given function is

$f(x) = x^2 +2x+2$

First we have to find f(0) and f(2).

And for that, we put 0 and 2 for x. That is,

$f(0)=0^2+2(0)+2, f(2) =2^2 +2(2) +2  \\ f(0) = 2, f(2) = 10$

Now we use the formula for average rate of change, which is

$\frac{f(2)-f(0)}(2-0} = \frac{10-2}{2-0} = \frac{8}{2} = 4$