find the average rate of change for f(x) = x² + 7x + 10 from x = -5 to x = 2.\na 2\nb 4\nc 8\nd 14

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x = a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=-5$, $b = 2$, and $f(x)=x^{2}+7x + 10$.

Step2: Calculate $f(-5)$

Substitute $x=-5$ into $f(x)$: [ \begin{align*} f(-5)&=(-5)^{2}+7\times(-5)+10\ &=25-35 + 10\ &=0 \end{align*} ]

Step3: Calculate $f(2)$

Substitute $x = 2$ into $f(x)$: [ \begin{align*} f(2)&=2^{2}+7\times2+10\ &=4 + 14+10\ &=28 \end{align*} ]

Step4: Calculate the average rate of change

[ \begin{align*} \frac{f(2)-f(-5)}{2-(-5)}&=\frac{28 - 0}{2 + 5}\ &=\frac{28}{7}\ &=4 \end{align*} ]

Answer:

B. 4