Question

given the function f(x)=x^2 - 2x - 2, determine the average rate of change of the function over the interval -1 ≤ x ≤ 5.

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=-1$ and $b = 5$.

Step2: Calculate $f(a)$

Substitute $x=-1$ into $f(x)=x^{2}-2x - 2$.
$f(-1)=(-1)^{2}-2\times(-1)-2=1 + 2-2=1$.

Step3: Calculate $f(b)$

Substitute $x = 5$ into $f(x)=x^{2}-2x - 2$.
$f(5)=5^{2}-2\times5-2=25-10 - 2=13$.

Step4: Calculate the average rate of change

$\frac{f(5)-f(-1)}{5-(-1)}=\frac{13 - 1}{5+1}=\frac{12}{6}=2$.

Answer:

$2$