Question

given the function $h(x) = x^2 + 3x - 1$, determine the average rate of change of the function over the interval $-7 \leq x \leq 5$.

Explanation:

Step1: Define average rate of change formula

The average rate of change of a function $h(x)$ over $[a,b]$ is $\frac{h(b)-h(a)}{b-a}$.
Here, $a=-7$, $b=5$, $h(x)=x^2+3x-1$.

Step2: Calculate $h(-7)$

Substitute $x=-7$ into $h(x)$:

$$\begin{align*} h(-7)&=(-7)^2+3(-7)-1\\ &=49-21-1\\ &=27 \end{align*}$$

Step3: Calculate $h(5)$

Substitute $x=5$ into $h(x)$:

$$\begin{align*} h(5)&=(5)^2+3(5)-1\\ &=25+15-1\\ &=39 \end{align*}$$

Step4: Compute average rate of change

Substitute values into the formula:

$$ \frac{h(5)-h(-7)}{5-(-7)}=\frac{39-27}{5+7}=\frac{12}{12}=1 $$

Answer:

$1$