11. what is the average rate of change of f(x)=x² + 2x - 1 from x=-1 to x = 1.

Explanation:

Step1: Recall the average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is given by $\frac{f(b)-f(a)}{b - a}$. Here, $a=-1$, $b = 1$, and $f(x)=x^{2}+2x - 1$.

Step2: Calculate $f(a)$ and $f(b)$

First, find $f(-1)$: [ \begin{align*} f(-1)&=(-1)^{2}+2\times(-1)-1\ &=1-2 - 1\ &=-2 \end{align*} ] Then, find $f(1)$: [ \begin{align*} f(1)&=1^{2}+2\times1-1\ &=1 + 2-1\ &=2 \end{align*} ]

Step3: Calculate the average rate of change

Substitute $f(-1)=-2$, $f(1)=2$, $a=-1$, and $b = 1$ into the formula $\frac{f(b)-f(a)}{b - a}$: [ \begin{align*} \frac{f(1)-f(-1)}{1-(-1)}&=\frac{2-(-2)}{1 + 1}\ &=\frac{2 + 2}{2}\ &=2 \end{align*} ]

Answer:

$2$