given the function $g(x)=-x^{2}+8x + 22$, determine the average rate of change of the function over the…
given the function $g(x)=-x^{2}+8x + 22$, determine the average rate of change of the function over the interval $1leq xleq6$.
Answer
Explanation:
Step1: Recall average rate of change formula
The average rate of change of a function $y = g(x)$ over the interval $[a,b]$ is $\frac{g(b)-g(a)}{b - a}$. Here $a = 1$, $b=6$.
Step2: Calculate $g(1)$
Substitute $x = 1$ into $g(x)=-x^{2}+8x + 22$. $g(1)=-(1)^{2}+8\times1 + 22=-1 + 8+22=29$.
Step3: Calculate $g(6)$
Substitute $x = 6$ into $g(x)=-x^{2}+8x + 22$. $g(6)=-(6)^{2}+8\times6 + 22=-36+48 + 22=34$.
Step4: Calculate average rate of change
Use the formula $\frac{g(6)-g(1)}{6 - 1}$. $\frac{34 - 29}{6-1}=\frac{5}{5}=1$.
Answer:
$1$