given the function $g(x)=-x^{2}+5x + 14$, determine the average rate of change of the function over the…
given the function $g(x)=-x^{2}+5x + 14$, determine the average rate of change of the function over the interval $1leq xleq9$.
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 $\frac{g(b)-g(a)}{b - a}$, where $a = 1$ and $b = 9$.
Step2: Calculate $g(1)$
Substitute $x = 1$ into $g(x)=-x^{2}+5x + 14$. $g(1)=-(1)^{2}+5\times1 + 14=-1 + 5+14=18$.
Step3: Calculate $g(9)$
Substitute $x = 9$ into $g(x)=-x^{2}+5x + 14$. $g(9)=-(9)^{2}+5\times9 + 14=-81+45 + 14=-22$.
Step4: Calculate the average rate of change
$\frac{g(9)-g(1)}{9 - 1}=\frac{-22-18}{8}=\frac{-40}{8}=-5$.
Answer:
$-5$