Question

f(x)=x^{2}-x - 1
what is the average rate of change of f over the interval -1≤x≤1?

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, $a=-1$, $b = 1$.

Step2: Calculate $f(-1)$

Substitute $x=-1$ into $f(x)=x^{2}-x - 1$. Then $f(-1)=(-1)^{2}-(-1)-1=1 + 1-1=1$.

Step3: Calculate $f(1)$

Substitute $x = 1$ into $f(x)=x^{2}-x - 1$. Then $f(1)=1^{2}-1-1=-1$.

Step4: Calculate the average rate of change

Using the formula $\frac{f(b)-f(a)}{b - a}$, we have $\frac{f(1)-f(-1)}{1-(-1)}=\frac{-1 - 1}{1+1}=\frac{-2}{2}=-1$.

Answer:

$-1$