what is the average rate of change of the function f(x) = 2x² + x + 5 over the interval -2 ≤ x ≤ 2? -4 1 4 -1

Answer

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, $a=-2$, $b = 2$, and $f(x)=2x^{2}+x + 5$.

Step2: Calculate $f(2)$

Substitute $x = 2$ into $f(x)$: $f(2)=2\times(2)^{2}+2 + 5=2\times4+2 + 5=8 + 2+5=15$.

Step3: Calculate $f(-2)$

Substitute $x=-2$ into $f(x)$: $f(-2)=2\times(-2)^{2}+(-2)+5=2\times4-2 + 5=8-2 + 5=11$.

Step4: Calculate the average rate of change

$\frac{f(2)-f(-2)}{2-(-2)}=\frac{15 - 11}{2+2}=\frac{4}{4}=1$.