Question

given the function $g(x) = x^2 + 6x + 1$, determine the average rate of change of the function over the interval $-4 \leq x \leq 0$.

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $g(x)$ over $[a,b]$ is $\frac{g(b)-g(a)}{b-a}$

Step2: Identify $a$, $b$ and compute $g(a)$

Here $a=-4$, $b=0$. Calculate $g(-4)$:
$g(-4)=(-4)^2 + 6(-4) + 1 = 16 - 24 + 1 = -7$

Step3: Compute $g(b)$

Calculate $g(0)$:
$g(0)=(0)^2 + 6(0) + 1 = 0 + 0 + 1 = 1$

Step4: Substitute into the formula

$\frac{g(0)-g(-4)}{0-(-4)}=\frac{1-(-7)}{0+4}=\frac{8}{4}$

Answer:

2