Question

what is the average rate of change for each function on the given intervals? (answer using a simplified fraction)

1. y=-x + 10 on - 3≤x≤-1
2. y=x² + 4x - 1 on 2≤x≤5

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$.

Step2: Solve for question 8

For $y=-x + 10$ on $-3\leq x\leq - 1$, let $a=-3$ and $b = - 1$. First find $y(-3)$ and $y(-1)$.
$y(-3)=-(-3)+10=3 + 10=13$.
$y(-1)=-(-1)+10=1 + 10=11$.
Then the average rate of change is $\frac{y(-1)-y(-3)}{-1-(-3)}=\frac{11 - 13}{-1 + 3}=\frac{-2}{2}=-1$.

Step3: Solve for question 9

For $y=x^{2}+4x - 1$ on $2\leq x\leq5$, let $a = 2$ and $b = 5$. First find $y(2)$ and $y(5)$.
$y(2)=2^{2}+4\times2-1=4 + 8-1=11$.
$y(5)=5^{2}+4\times5-1=25+20 - 1=44$.
Then the average rate of change is $\frac{y(5)-y(2)}{5 - 2}=\frac{44-11}{3}=\frac{33}{3}=11$.

Answer:

1. -1
2. 11